MANAGEMENT OF FIXED
INCOME SECURITIES
Prof.Jitendra Mahakud
Humanities and Social Sciences
IIT Kharagpur
INDEX
S.NO
TOPICS
PAGE.NO
Week 1
1
Lecture 01: Introduction to Fixed Income Securities
4
2
Lecture 02: Risks in Fixed Income Securities Investments
24
3
Lecture 03: Bond Valuation - I
40
4
Lecture 04: Bond Valuation - II
58
5
Lecture 05: Bond Price Relations
71
Week 2
6
Lecture 06: Bond Returns-I
84
7
Lecture 07: Bond Returns-II
101
8
Lecture 08: Bond Returns -III
118
9
Lecture 09: Bond Returns - IV
134
10
Lecture 10: Level of Interest Rate Determination-I
146
Week 3
11
Lecture 11: Level of Interest Rate Determination-II
160
12
Lecture 12: Level of Interest Rate Determination-III
176
13
Lecture 13: The Term Structure of Interest Rates-I
191
14
Lecture 14: The Term Structure of Interest Rates-II
205
15
Lecture 15: The Term Structure of Interest Rates-III
219
Week 4
16
Lecture 16: The Term Structure of Interest Rates - IV
233
17
Lecture 17: The Term Structure of Interest Rates-V
245
18
Lecture 18: The Term Structure of Interest Rates-VI
257
19
Lecture 19: The Term Structure of Interest Rates-VII
274
20
Lecture 20: The Term Structure of Interest Rates-VIII
290
Week 5
21
Lecture 21: Determination of Theoretical Spot Rate and Forward
1
306
Yield Curves
22
Lecture 22: Interest Rate Risk
321
23
Lecture 23: Bond Duration
337
24
Lecture 24: Bond Convexity
358
25
Lecture 25: Default Risk
374
Week 6
26
Lecture 26: Short-Term Government Bond Markets
392
27
Lecture 27: Long-Term Government Bond Markets
411
28
Lecture 28: Issuance and Trading Mechanism of Government
Securities
428
29
Lecture 29: Short-Term Corporate Bond Markets
445
30
Lecture 30: Medium and Long-term Corporate Bond Markets
465
Week 7
31
Lecture 31: Issuance and Trading Mechanism of Corporate Securities
485
32
Lecture 32: Municipal Bonds-I
505
33
Lecture 33: Municipal Bonds-II
523
34
Lecture 34: Debt Securities of Commercial Banks
539
35
Lecture 35: International Bonds
558
Week 8
36
Lecture 36: Bonds with Embedded Options
575
37
Lecture 37: Binomial Valuation of Callable Bonds-I
591
38
Lecture 38: Binomial Valuation of Callable Bonds-II
608
39
Lecture 39: Binomial Valuation of Puttable and Sinking-Fund Bonds
626
40
Lecture 40: Binomial Valuation of Convertible Bonds
648
Week 9
41
Lecture 41: Estimation of Binomial Trees-I
669
42
Lecture 42: "Estimation of Binomial Trees-II
687
43
Lecture 43: Estimation of Binomial Trees-III
702
44
Lecture 44: Estimation of Binomial Trees-IV
719
2
45
Lecture 45: Estimation of Binomial Trees-V
734
Week 10
46
Lecture 46: Bond Investment Strategies-I
746
47
Lecture 47: Bond Investment Strategies-II
764
48
Lecture 48: Bond Investment Strategies-III
781
49
Lecture 49: Bond Investment Strategies-IV
802
50
Lecture 50: Bond Investment Strategies-V
820
Week 11
51
Lecture 51: Mortgage Backed Securities-I
841
52
Lecture 52: Mortgage Backed Securities-II
858
53
Lecture 53: Mortgage Backed Securities-III
880
54
Lecture 54: Mortgage Backed Securities-IV
899
55
Lecture 55: Mortgage Backed Securities-V
919
Week 12
56
Lecture 56: Fixed Income Securities Derivatives-I
939
57
Lecture 57:Fixed Income Securities Derivatives-II
966
58
Lecture 58: Fixed Income Securities Derivatives-III
988
59
Lecture 59: Fixed Income Securities Derivatives-IV
1011
60
Lecture 60: Fixed Income Securities Derivatives-V
1038
3
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 01
Lecture No # 01
Introduction to Fixed Income Securities
Good morning. So, we can discuss a particular subject that is called the management of fixed
income securities. Because, all of you know that whenever, we make investments in the market,
we have different kinds of assets, we always look at. We have the stocks, we have the bonds, we
have the derivatives instrument, and all these things. And the choosing of those securities
depends upon the objectives of that particular investor.
So in that particular context, the investors who are very much inclined towards income
maximization always concentrate on a particular type of security which is called fixed income
securities. Mostly, it gives a constant flow of income over, relatively it is less risky than the other
type of instruments that are available in the market particularly, the stocks and the derivatives
instrument. (Refer Slide Time: 01:16)
4
After going through this particular session, we will be covering the different types of
fixed-income securities that are traded in the market. And mostly, the most important fixed
income security is the bond.
So what are the characteristics of the bond and what are the different types of bonds which are
traded in the market? Then another type of security, we have is a part of the debt and part of the
equity which is called the preferred stock.
Although, in many of the cases, we have seen that is not considered a debt instrument but we will
just briefly discuss what that preferred stock is and what are the different types of preferred
stocks which are traded in the market. Then, we have certain kinds of products that we call,
structured products. These are the mortgage-backed securities that comes from the process of
securitization where the different pool of funds are considered and accordingly, the different type
of assets are issued against that particular pool of funds or pool of the assets whatever we have.
We have mortgage-backed securities, we have asset-backed securities and all these things come
under the structured products then we have certain products which are fixed income securities,
derivatives particularly.
Derivatives instruments or the price of those particular derivatives are based upon the different
types of fixed income securities which are available in the cash market. So these are called fixed
income derivatives mostly the derivatives are like interest derivatives, credit default stocks, and
all these things that come under the part of the fixed income derivatives. So here is what
basically we are going to do. We will be discussing the different types of derivatives which are
designed on the basis of fixed income securities.
And as well as how the pricing of the securities is done and how the securities are traded in the
market if you want to minimize the risk in this particular market then what are the different kinds
of strategies or what are the different kinds of assets you have to consider? so that is the fixed
income derivatives part.
(Refer Slide Time: 03:37)
5
So, after this, certain things you will remember like what is coupon, what is debenture, what is
mortgage-back securities, what is asset-backed securities, and what is interest derivative? These
are the major things that basically we will be covering up some of the concepts that basically will
be learning from this particular session. Okay.
(Refer Slide Time: 03:59)
So, let us come to our discussion on fixed income securities. Already, what I told you is that by
the name itself if you see that what do you mean by the fixed income securities? The fixed
income means it gives a kind of fixed income or regular income over a period of time. So that is
6
why we call it, it is a kind of financial claim with regular cash flow of a fixed amount at fixed
intervals or dates.
For example, if you take a bond, the bond has a coupon feature and the coupons are paid against
the principal or the par value of the bond. Let the par value of the bond is 1000 rupees, if the
coupon is 10% and the coupon is paid annually then every year on a particular date the investor
will be receiving 100 rupees from the bond issuer. So periodically there is a fixed income that
basically you would be receiving in a regular interval or the fixed interval at a particular date.
We call these fixed income securities which give a regular cash flow and the cash flow amount is
also fixed and is paid in a particular interval or on a particular date. So already what I told you
there are different types of fixed income securities what always we see and the whole particular
subject will be dealing with mostly all types of fixed income securities that are traded.
And we will look at the different nitty-gritty, different investment aspects, the pricing aspects,
and the strategic aspects whenever we will discuss all type of instruments which comes under
fixed income securities. So the first instrument which comes under the fixed income security is
the bond and already I told you about the preferred stock which is a hybrid instrument it has an
equity feature as well as the bond feature.
Then we have the structured products then we have the fixed income derivatives. So one by one
we will see and over the particular course, we will be discussing sequentially the different
aspects of this particular instrument. Like the pricing aspect or the evaluation aspect, we will
discuss the strategy aspect and we will discuss the trading aspects. And all these things which are
basically a part of the investment process will be discussed in that particular case. And mostly
how the risk basically the investor is going to minimize whenever they are using this instruments
in their investment portfolio.
(Refer Slide Time: 07:00)
7
Then what is a bond? Already all of you know what is bond; you have a fair idea about the
bond. The bonds are basically a kind of instrument which imposes fixed financial obligations on
the bond issuers. If any company or a government issues a bond then on a periodical basis they
have to pay a certain amount to the bond investor or the bondholder. That is why the owner of
the bonds generally is called the debt holders or the creditors.
So, on a periodical basis, a fixed amount of money or a fixed amount of return you will be
receiving from the bond issuer. So this is a kind of financial obligation the bond issuer has and
what are those fixed obligations: the fixed obligations are the fixed amount of interest which is
nothing but the coupon and another one is the repayment of the fixed amount of the principal at
the end of the maturity.
If you have invested in a bond and the bond is going to be matured after 5 years or after 10 years
then after this maturity period generally your principal will be returned back to you. So that is
why they have two fixed obligations: one is your payment of the fixed amount of interest which
the bond issuer has to pay them periodically then the repayment of the principal at the end of the
maturity.
8
So these are the two fixed obligations always the bond issuer has. And the bondholder will be
receiving these two different financial claims whenever they are holding that bond with them.
(Refer Slide Time: 08:45)
Then we have to look at certain characteristics of the bond whenever you define a bond, a bond
has certain kinds of characteristics, there are some basic features always we look at whenever
typically we define a bond. Already I told you the bond has a coupon, the coupon will be fixed in
the beginning and how much, coupon the bond issuer will be giving. Then it will have a term to
maturity up to what period the bond you can hold.
And or you can say that you maximum period, the bond has for redemption that is called the
term to maturity then you have the principal or the par value of the bond. So these are the three
different characteristics you will look at for the bond typical bond features basically we can say.
But there are different types of bonds there are let zero coupon bonds so there is no such kind of
coupons involved in that. But, the bond is always redeemed at the time of maturity and it is
issued at a discount and redeemed at par. So that is a typical type of different type of bonds we
can also see. But in general, if you see that whenever we picture or we look at a bond thing or we
discuss about a bond then always we see that the bond should have a coupon, the bond should
have a term to maturity and the bond should have a principal. So, these are the three things
always involved with respect to the bond.
9
Then the bond can be a secured bond. A secured bond in the sense it should be backed by certain
kinds of assets or certain kinds of mortgages. That means it is if there is any default then those
assets can be liquidated and the money can be repaid back to that particular investor. So that is
called the secured bond or we can also call it the senior bond. Then we have the unsecured
bonds. Mostly, the corporate sectors generally whenever they issue these long term bonds which
in our language we call debentures, are mostly unsecured bonds.
And the unsecured bond means it is not backed by any kind of asset or any kind of collateral
there is no such collateral involved against this unsecured bond. So, if there is any default then
there is a, or the default risk of that particular type of bond is relatively higher in comparison to
the secured bonds. Then the bonds can be also divided into two types, on the basis of the
ownership, we can call a bond a bearer bond and we can also call a bond a registered bond.
So what do you mean by this bearer bond? The bearer bond means here there is no such record
with the bond issuer that who is holding that bond. And always there is some kind of coupons are
associated with that bond and on every periodical basis the bond investor has to send that to the
bond issuer and against that the bond issuer will pay the coupons. So the bond issuer does not
have to keep the record of the ownership of that particular bond. So these are called the bearer
bond.
But whenever we are talking about the registered bond mostly most of the bonds which are
traded in the market are registered bonds. So registered bonds mean here the issuer basically
maintains the record of the ownership and the coupon or the interest which are directly paid to
the owner's account. So there is no need to always send this claim to the bond issuer to get back
your interest or to get back your money. So that is called the registered bond. So, most of the
bonds in the market are called registered bonds.
(Refer Slide Time: 12:37)
10
Then if you define the bond market where the bonds are traded, the bonds can be defined on the
basis of their maturity. That means there are some bonds which are short-term bonds and there
are some bonds that are long-term bonds. And short-term bonds are mostly the money market
instruments generally we refer like your treasury bills, commercial papers, certificates of
deposits, and all these things.
The features of this kind of instruments and the details about their process or concepts related to
this instrument will be discussed gradually. But these are the different short-term bonds which
are available as a part of the money market instruments and they are basically traded in the
market to fulfill certain requirements for the different types of investors. Then we have the
long-term bonds these are the capital market instruments mostly if you see that the government
dated security long-term government bonds. Then the corporate bonds, most of the corporate
bonds are basically long-term bonds so these are basically a part of the long-term bonds. So you
can also classify them in various ways on the basis of the issuer, the type of the issuer. So in that
case the long-term bonds are the capital market instruments. And short-term bonds are money
market instruments.
(Refer Slide Time: 14:09)
11
Then if you are talking about the issuer's point of view, who is basically issuing the bond if you
want to classify that that will have more popularity and more meaning in terms of defining the
bonds across the globe. So in this case, if you talk about India, we have government securities,
which is a highly developed market and a quite old market. And the reserve bank of India is
responsible for issuing that bond on behalf of the government of India. So these are called
government securities.
We have the corporate bonds then we have the public sector unit bonds. We have classified this is
also a part is a part of the corporate bond but the PSU bonds have certain characteristics so that is
why we have separated them from the normal corporate bonds. So we have on the basis of the
issuer in India we have the three types of bonds which are available. One is your government
securities, one is corporate bonds and the other one is the public sector unit bonds.
But, if you look at the US, it is more or less the same but they have a very popular bond we have
called the municipal bonds. They have also government securities or government bonds which
basically they call the federal government and its agencies who are eligible to issue these bonds
on behalf of the government. Then we have the companies which are situated in the US are
domestic corporations that operate in the US market and they are also eligible to issue this bond
then we have the municipal bonds.
12
So these are the three different broad categories, there are various types of bonds but these are
the major types of bonds which are available in the US market. And that is the way the
classification of the bonds can be made with reference to the issuer's point of view.
(Refer Slide Time: 16:13)
Then we already told you about preferred stock, I will just give you a brief idea that preferred
stock means it is a kind of instrument where the investor gets a fixed dividend on a periodical
basis. So that is why it is a hybrid instrument, it is not a pure fixed income securities instrument.
But still, we should have some basic idea about what exactly the preferred stock is! Preferred
stock means that like other equity holders or shareholders we do not have any kind of ownership
of the company.
But we will have this kind of right to get a certain amount of fixed dividend on a periodical
basis. And that is why they do not have any voting rights. But if there is a chance of liquidation
of the company then the company is going to be liquidated or the company has liquidated then in
that case what happens is that after the bondholders the next financial claims always prefer
stockholders have.
First, the money will be paid to the preferred shareholders or preferred stockholders then it will
go to the common shareholders or the equity holders. So that is why it has a kind of fixed income
13
security characteristics but it is not purely fixed income security that basically we can say. So
there are different types of preferred stocks which are available in the market.
(Refer Slide Time: 17:45)
If you see what are those different types of preferred stock we have the cumulative and
non-cumulative preferred stocks. What do you mean by this cumulative preferred stock or
preferences if it is a cumulative preference share if a dividend is skipped in any period then it has
to be paid subsequently? For any reason, if the company was not able to pay that particular
dividend then the dividend basically will be paid in the subsequent period.
But if it is non-cumulative then if for some reason some kind of unforeseen reason the company
was not able to pay that particular kind of dividend then again the dividend will not be repaid.
Then we have the convertible and non-convertible preferences convertible preference share
means that it can be converted into an ordinary share or equity common equity with certain kind
of terms and conditions which are fixed at the time of issue of such shares.
First, with some terms and conditions, after a point of time, this particular preferred shares can
be converted into ordinary shares which are called convertible preferences. Then you have
redeemable and non-redeemable, redeemable preferences basically mature in a fixed period of
time and for all practical purposes, it is regarded as debt security like a debenture. So if you are
holding it after a certain period of time this particular instrument will be matured.
14
So in that particular case further you will not be receiving any dividend from the company
against that particular preference share because the maturity period is already over for that. So
that case it is called the redeemable preferences and it is more or less a kind of debt equity or
debt equity like the debentures. Then you have the participating and non-participating
preferences, the participating preferences basically can always have a higher dividend than the
fixed one if the company makes a good profit.
There is a provision, at the time of issuance, that it is defined whether it is a participating
preference here or a non-particular participating preference share. If it is a participating
preference share then what will happen is that, if the company for any reason makes a good
profit then the participating preference shareholders can get a higher dividend. So the dividend
payments are not fixed in this particular context.
So that is why we can say that there is a chance of getting high dividends in this particular
context. That is why it may not be a fixed kind of security or fixed income security that basically
we can conclude if it is a participating preference share. So these are the different types of
preferences what we see.
(Refer Slide Time: 20:52)
Then we will come to the different kinds of structured products which are a part of the
fixed-income securities market. Whenever you talk about structured products, there are certain
15
products which are part of the residential mortgage-backed securities and some of the
instruments which are basically these non-residential mortgage-backed securities.
So whenever we come to discuss residential mortgage-backed securities, generally, in this case,
the cash flow depends on the cash flow of an underlying pool of mortgages. For example;
whenever the bank and other entities give a loan against any kind of residential purpose. In that
case, many banks or many agencies can always come together and make a pool of mortgages.
And against this pull of mortgages whatever cash flow basically, they will be receiving from that
cash flow they can basically issue certain kinds of financial assets. In the form of shares or in the
form of, some of the participating certificates. So there are different types of mortgage-backed
securities we can in detail about the pricing and all these things will be discussed further in
future sessions. But I will just give you a brief idea of what exactly these mortgage back
securities are.
So there are three types of mortgage-backed securities; we have collateralized mortgage
obligation, we have mortgage-pass-through securities and we have stripped-mortgage-back
securities. So broadly whenever we come to these mortgage-backed securities these are the three
different types of mortgage-backed securities always we can find we can observe in the market.
(Refer Slide Time: 22:56)
16
So if you see that there is a cash flow already I told you there is a cash flow involved in the
mortgages? So what are those cash flows? So what do you mean by a mortgage? All of you
know that the mortgage is basically a pledge of real estate to secure the loan originated for the
purchase of that particular real asset. That means if you are buying a house then the house is your
mortgage.
So if at any point in time you are not able to repay the loan either in terms of interest or in terms
of the principal. Then the particular loan giver or the lender has the authority to liquidate that
particular asset and recover this particular amount whatever you have not repaid. So that is why
we call, it a pledge. So the mortgage basically provides already I told you the mortgage basically
provides you, the lender what we can call the mortgagee, the right to foreclose on the loan and
seize the property in order to ensure that the loan is paid up, if the borrower fails to make the
payments of any point of time if the borrowers failed to make the payments. Then they can seize
the property and get the money back by selling those kinds of assets or by liquidating those kinds
of assets which we call, mortgages.
If you look at the mortgages already told you there are two types of mortgages, one is residential
mortgages and non-residential mortgages. And nonresidential mortgages I am talking about
commercial or industrial mortgages. And what is that cash flow involved against this? One is
your interest which basically you should pay on a periodical basis to the lender. Then the
scheduled repayment of the principal because you have to pay; the principal amount also
periodically. Then there is a possibility of the prepayment of the loan. If you have taken a loan, at
any point in time, you are also eligible to prepay that particular loan. And there are various
reasons either you are getting a better house so that is why you want to sell your house and buy
the new one. Or you can get a better interest rate from another lender so there is a refinancing
you want to make which is why you want to make prepay the loan. Or all of sudden you get
back certain money and because of this, you want to always get rid of all those kinds of loans or
the things whatever you have done. So that is why there is that possibility so that is why these are
the different reasons but because of this the prepayment of the loan also can be possible. So these
are the different cash flows.
17
(Refer Slide Time: 25:49)
So what are mortgage-pass-through securities? It is basically created when one or more holders
of mortgages form a collection of the mortgage and sell the shares in the pool. And the cash flow
from the pool already told you, is interest, principal, and prepayments. And payments are made
to the security holders each month. An agency pass-through securities are the securities with
various specific agencies like the government national mortgage association, federal home loan
Mortgage Corporation, and federal national mortgage association. All these are agencies
particularly based in the US because; these markets are more developed in US markets. So they
are the agencies who are involved in this particular process. But there are some non-agency
pass-through securities these are issued by commercial banks, there are some private channels, or
thrift institutions which are existing in the market, they can always go for or issue this kind of
securities also.
(Refer Slide Time: 26:54)
18
Then, we have the collateralized mortgage obligations (the CMO). So these instruments are
basically structured so that there are several classes of bondholders with varying maturities. And
the different bond classes generally are called the tranches. So different classes of bonds are
created which have different degrees of repayment and interest rate risk. And the bonds include
sequential pay bonds, floating rate bonds, accrual bonds, etc.
So this is the way different classifications are made and those on that basis basically the bonds
are issued or the securities are issued and accordingly the investments are made in the market.
(Refer Slide Time: 27:46)
19
Then we have the stripped mortgage-backed securities, so these are created by altering the
distribution of the principal and interest from a pro-rata distribution to an unequal distribution.
So we will explain the nitty-gritty or complexity of these instruments gradually. But if you see
there are two types of stripped mortgage-backed securities we observe. One is synthetic coupon
pass-through and interest-only or principal-only securities.
Whenever you call the synthetic coupon pass through kind of stripped mortgage back security,
this is nothing but the unequal distribution of coupon and principal, resulting in a synthetic
coupon rate that was different from the underlying collateral. And this principal-only class
security receives only the principle from the underlying mortgage. And the interest-only class of
security receives only the interest from that particular underlying mortgage. So this is the way
the stripped mortgage-backed securities are classified.
(Refer Slide Time: 29:01)
Then we have the commercial mortgage-backed securities. Here, it is backed by the pool of
mortgage commercial mortgage loans on income-producing properties like shopping centers,
hotels, warehouses, health care facilities, etc. So these loans are unsecured, the lender can only
look to the income-producing property which is backing the loan and the principal payments. So
we have two types of this commercial mortgage-backed securities, one is multi-property single
borrowers and multi-property conduits. And the conduits are the commercial lending entities that
20
are established for the sole purpose of generating collateral to securitize; in the securitization
process basically, they play the role.
(Refer Slide Time: 29:51)
Then we have asset-backed securities. So here, these are collateralized by the assets not the
mortgage loans. So they have been structured as pass-through and as a structure with multiple
bond classes called the pay-through which are similar to the CMOs. And the common types of
asset-backed securities are credit card receivables, home loans, equity loans, and automobile
loans, these are basically other assets also you can consider student loans, leasing, and equipment
leasing. So, all these things can be considered only as asset-backed securities. And the
collateralized debt obligation is basically an asset-backed security, it is backed by the diversified
pool of one or more of the following types of debt obligations like high yield corporate bonds,
emerging market bonds, residential mortgage-backed securities, asset-backed securities,
commercial mortgage-backed securities, anything can be considered to construct this
collateralized debt obligations which popularly we call it the CDO's.
21
Then we have the fixed income securities. Mostly, we look at the different types of securities
which come under the fixed income derivatives. We have the interest rate futures, we have
interest rate options, we have interest rate swaps, we have credit derivatives, and all. We will
discuss the pricing, the valuation, the investment process of all those kinds of securities, the
advantages, disadvantages of these instruments, and all these things over these particulars in
different sessions that we will discuss in detail.
(Refer Slide Time: 31:34)
So that is why basically we have seen that the fixed income securities provide the fixed cash
flow in regular intervals. These are more appropriate for income maximization and the broad
22
types of fixed income securities include bonds, structured products, and interest rate derivatives.
And this is the way basically the overall fixed income securities market can be classified or can
be defined.
(Refer Slide Time: 32:01)
So these are the references that basically you have to go through to know the details about this
particular thing. And we will be using this particular book also for discussion of the different
sessions of this particular course. Thank you.
23
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 01
Lecture No # 02
Risks in Fixed Income Securities Investments
Welcome!
Again, Good morning to all of you,
So, in the previous class, we have just discussed about the overview of this particular subject
on fixed income security. We have seen that different broad instruments like bonds, we have
the structured products, 0 and we have the fixed income securities derivatives. So these are
the major instruments which are available in the fixed income securities market.
But, as all of you know that whenever, we always go and invest in the fixed income securities
market, we are also exposed to certain amount of risk or certain types of risk always we look
at or we face. So today, we will be discussing about it: what are the different types of risks
generally always associated with the fixed income securities and how these risk are basically
defined. And gradually, we will be discussing that how those risks can be managed or this
can be mitigated by taking the different portfolio positions or by using the different strategy
by that the income maximization will be possible.
(Refer Slide Time: 01:26)
So today mostly, I will be discussing with you regarding the different type of risk which are
associated with the fixed income securities investments. Whenever, you are making the
24
investment in the fixed income securities market then what are the different probable risk we
are exposed to or we may face. And always we should be careful about those type of risk by
that this will not largely affect our expected return what we are trying to derive from this
particular market.
So these are basically the basic objective or basic thing what always we keep in mind. So that
is basically what we are going to discuss today.
(Refer Slide Time: 02:08)
So from this, we will discuss certain concepts, certain keywords basically you can keep in the
mind like market risk, Bid-ask spread, currency risk, credit risk, inflation risk. There are
many types of concepts basically we will be discussing in this particular session and which
are nothing but different type of risk what the investor always may face whenever they are
trying to invest in the fixed income securities market.
(Refer Slide Time: 02:42)
25
So, let us see, that what are the broad types of risk we face are and how basically we are
going to define them. You see, whenever we invest in a fixed income security market, the
basic objective of the investor is to get some return.
So what is the return? What are the different components of that particular return what they
are going to expect from that particular investment? Obviously, whenever you have
purchased a particular bond then obviously you are going to sell that bond. The first cash
flow, first component or first income component will be receiving whenever you will sell the
bind that is the market value of that particular bond. If you are selling that bond in the market
then what is the price you are getting? What is the market price of the bond at that particular
point of time? So that is the first component.
Second, obviously, the cash flow which is nothing but the coupon payment whatever you
have received whenever you are basically with or you are holding that particular bond up to
the maturity or up to the selling point, you are getting certain kind of periodical cash flow
from this. So that is the cash flow received from the security over the time that is basically
nothing but the part of the coupons. Then whenever you have received the coupon, the
coupons also can be reinvested in the market at a market interest rate.
So whenever you are making the reinvestment in the market then obviously you can earn
some additional income. So what is the additional income from the reinvestment you are
basically getting? So that also you have to keep in the mind. The additional income from the
reinvestment of the periodic cash flows which is nothing but the coupons always we look at.
26
So here, it is income there is a space so you keep that thing in the mind. It is basically
additional income. So, additional income from the reinvestment of the periodic cash flows is
nothing but the interest on interest. The coupon you got as a part of interest and utilizing that
particular interest again in the market, you are getting some interest against that. So over and
above the interest whatever you are getting that is called the interest on interest. So that is
basically the additional income what we get.
(Refer Slide Time: 05:19)
So whenever we are investing in the fixed income securities, what are the different types of
risk probable risk which are associated with that? We have many risks. we are exposed to
market risk or other name of the market risk also can be called the interest rate risk. So the
interest rate risk is related to the price risk and the reinvestment risk. Then we have the
default risk or the credit risk that bond issuer at any point of time can go for default. So that is
why we are exposed to credit risk. We have a maturity risk because we have the certain
bonds of the long term bonds. So the yield variation may happen, the return variation may
happen across this type of the bonds. There is a difference which may arise between short
term and long term bonds that is why there is a maturity risk involved.
Inflation risk, at any point of time, the inflation can increase can decrease. So, accordingly,
your real value of that particular return what you are getting from the bond that may change.
That is why we are also exposed to the inflation risk.
27
Liquidity risk, whether really the bond is really marketable or not, at lower transaction cost
we are able to sell the particular bond or particular fixed income securities in the market. So
that is the part of the liquidity risk.
Then if you are holding the foreign bonds, there is some foreign exposure then we are also
exposed to exchange rate risk. So if there is a fluctuation in terms of exchange rate then the
values of the cash flow are also going to be changed. So that is why we are exposed to the
exchange rate risk.
Then if the bond has any kind of features or the fixed income securities has any features call
features or put features. So then those kinds of bonds are also relatively riskier from the
investment point of view. So in that case, we can expose to a call risk.
So at any point of time, the bond can be called back by the issuer and that will be more
problematic for the bond investor if at that particular point of time, the interest rate in the
market is very low. They could have sold the bond in the market at a higher price. And
whatever money they will get back from the issuer whenever they are going to invest in that
particular money in the market, it will not be profitable for them. So that is why there is a
huge risk we are also exposed to, that is called the call risk.
Then you have the volatility risk. The volatility risk means, due to the fluctuations of the
volatility of the interest rate and all these things, our value of the bond also gets affected and
that is more appealing or more suitable of the bonds which have embedded options.
Then we have the political risk or the legal risk what we can call. Then we have event risk. A
typical event can happen to a particular company which may affect the other type of aspects
of the bond investment. That is called event risk.
Then finally, the sector risk means that a particular sector whenever you are investing for
some reason if the sector has not performed well then in that particular context, we are also
exposed to certain type of risk of that particular investment. So that is called the sector risk.
So these are the broad lists, we have classified certain type of risk which are associated with
the fixed income security investment. So we will see how those particular risks are defined.
(Refer Slide Time: 09:00)
28
Already, I told you that the market or the interest at risk the market risk is basically has two
components. There is a combination of the price risk and the reinvestment risk. Because,
whenever, the interest rate increases the price of the bond declines that means the investor is
exposed to more price risk. That is why there is an inverse relationship between the bond
price and interest rate.
So in that case, what will happen that the once the interest rate is increasing if you want to
sell the bond at any point of time whenever the interest rate is increasing then price of the
bond in the market goes down. So that is why the value of that particular investment goes
down. But, at the same time, whenever you are getting the coupons these coupons whatever
you are getting, you see, coupon rate are fixed, you will be getting a fixed amount of coupon
in very month or every quarter or it can be every 6 months or every year and that money can
be reinvested in the market; you can use it for some other investments. So, whenever you are
using it for the other investment if really that interest rate has increased then getting some
better return from the reinvestment will be always higher. So the reinvestment risk goes
down, that is why what basically we can say that if the rate is increasing we are exposed to
more price risk but at the same time we are less exposed to the reinvestment risk.
Reinvestment risk is basically going down in that particular context so the price risk and
reinvestment risk is basically that is why opposing each other. It will always basically
compensate the reinvestment return whatever we get due to the high interest rate that will
compensate the loss in the value of that particular bond in the market. So if you look at the
reinvestment risk, it is generally higher for the long maturity and high coupon bonds.
29
So in that case if you are talking about the high coupon, let the coupon rate is 15% at the
same time the interest rate has gone down. Then you are getting a high cash flow in the
periodical basis. But, you do not enough revenue, you can generate by the reinvestment
process. So in that case that is why the reinvestment risk is higher for the high coupon
bearing bonds. So in the long maturity bonds also because the interest rate cannot be
predicted accurately at any point of time the interest rate can change or interest rate can move
in a particular direction what you have not expected. In that case, what will happen, that we
can also face certain problem with respect to the value of the bond because of the interested
fluctuations. So that is why the reinvestment risk is generally higher for the long maturity and
the high coupon bonds. That actually you can always keep in the mind.
(Refer Slide Time: 12:15)
Then we have what basically you call it, the credit risk, it is a major risk. Whenever we invest
in the fixed income securities, we are very much exposed to the credit risk. So credit risk
means it is the mostly the default risk which is basically the risk of default on issuers
obligations. Let company X has issued the bond because of the certain problem, the company
X was not able to repay the interest or the principal to the bond holder. Then we are highly
exposed to the risk of default or the credit risk exposed is quite high. So that arises due to the
risk of default of the issuers obligations they are obliged to pay you interest in the form of
coupon and they should also pay you the principal amount at the end. But because of some
reason if they are not able to repay either coupon or the principle or both then we are always
exposed to the default risk which is basically major part of the credit risk.
30
And another component, another thing also which basically arises credit raise that is called
the credit spread risk what exactly it means? It means the risk aspect or the risk which is
basically arises due to an increase in the spread between the particular bonds what basically
you have invested are the comparative bonds which are basically traded in the market.
So that is why the risk attributable to an increase in the spread due to the risk that the bond
value will decline or the bond performance will be worse than the other bonds at that
particular point of time which are traded in the market. So that basically is a created a kind of
spread between this returns what you are expected from that particular bond wherever you
have invested and the bonds which are available in the market; the competitory bonds or the
bonds which are comparable to the bonds where you have been invested earlier. So that
basically happens when? That happens basically whenever the credit rating of the bond
declines or some kind of forecasting made by the company or made by the some kind of the
analyst about the company which is not a positive kind of news or positive kind of impression
creates in the mind of the investors.
So the risks which are attributed to the lowering of the credit rating are called the downgrade
risk. So there is a possibility because of the certain financial parameters, because of certain
kind of exposure towards the market, there is a possibility that the rating of the bond can go
down. So if the rating of the bond can go down then that also creates a kind of impression in
the mind of the investor. The return or the bond may not do well in the market or the return
will be lower. There is high chance that the bond, this particular issuer is going to be
defaulted. So in that case the credit risk is basically going to be increased.
(Refer Slide Time: 15:39)
31
So, here, if you see, how do basically judge or how to gauge the credit risk? You see that you
can use your bond rating or credit watches or credit outlooks; you can use all to understand
that what kind of issuer it is or whether there is any possibility of default. So for long term
bonds credit rating, it is basically a forward looking assessment for the probability of default
and the relative magnitude of loss which basically a default can create. If there is a default
then what kind of loss basically can be incurred? But for short term credit rating, it is
basically assessment of the probability of default only. So the relative magnitude of loss is
not there because it is generally the shorter bonds are maturity period is less than 1 year. So at
any point of time, before real thing can happen, the particular bond can be redeemed and
accordingly, you can get back your proceedings what you are expecting from that particular
bond. So that is why risk of default is the only concept in that particular context. But,
whenever the long term bond, because you have to hold it and in that case, the loss basically
comes into the picture. The magnitude of loss basically comes into the picture with this
probability of default. So the credit rating agencies generally provide the information about
the default rates and the default loss rate.
They provide the information about the default rate and the default loss rate. What is the
default? The default rate is nothing but is the percentage of the bonds of a given rating at the
beginning of the period that have defaulted at the end of the period. Whenever they have
basically issued the bond whatever rating was there and in the end of the period basically how
many companies or how many percentages of the bonds which are basically available then
which are going to be defaulted. So percentage of bonds of a given rating at the beginning of
the period which are defaulted at the end of the period, that is basically called the default rate.
32
And default loss rate is what? It is basically a measure of the magnitude of the potential of the
loss that a default can occur. So the credit rating agencies can provide this kind of
information to you and this information can be assessed or can be utilized by the investor to
understand that what kind of credit risk is exposure this particular company has? Or what
kind of exposure a credit risk exposure the bond has? Accordingly, you can decide that
whether it is a good candidate for investment or not?
(Refer Slide Time: 18:52)
The next is your basically maturity risk and inflation risk. What is maturity risk? So I already
told you interest rate very across the different maturity of the bonds. The yields basically vary
across the different maturity of the bonds. Long term maturity bonds and short term maturity
bonds you will find that there is a return differentials. Why there is a return differential in all
these things? That basically we will be discussing further which; is part of the term structure
interest rate theories.
Then they the long term bonds have a different interest rate than the certain bonds, why also
we can say that the bonds have the same maturity also have the different rates or different
interest rates, why that thing happened. That basically will discuss. But if you observe, longer
term bonds are riskier than the shorter term bonds. The long term bonds are riskier than the
short term bonds. The reason is that the future cannot be predicted perfectly or interest rate
prediction cannot be done accurately. In that case, there is a high chance that anything can
happen in the long run and the bond market is going to be affected by that. So because of this
we say that investment in a long term bond is always riskier than investment in the short term
bonds. So that is why we can bring this liquidity premium and the risk premium because of
33
high risk the long term bond should give more return. Because we are forgoing our liquidity
in today’s context to invest in the long term bond, we should get some premium. So those
nitty-gritty, the mathematics of that, the conceptual understanding of that, will be discussing
in future sessions. That why basically this thing is added, how this is added, how to measure
that and all type of concepts we will be discussing.
And the next thing is that the inflation risk. What is inflation risk? The inflation risk basically
arises due to the variations in the value of cash flow because of change in the inflation. For
example your bond is giving 10% coupon and the interest rate was fixed from the beginning
so that is why you know that every year or every 6 months you will be getting 10% coupon
from that particular bond.
But let, the inflation rate while investing in the bond the inflation rate was let 5% but now
because of some reason in the market the inflation rate has increased from 5% from 11%.
You are getting a return of 10% and in the market the inflation rate is 11%. What does it
mean? It means that your real rate of return what you are getting from that particular
investment is negative.
So that is why that will affect the value of that particular cash flow. So that is why there is a
variation in the value of the cash flow due the change in the inflation rate, due to change in
the purchasing power in the particular market. And if you observe, that is why there are
different type of bonds: let the coupon is fixed in some cases and the coupons may not be
fixed the coupons may be floating. There are some inflation linked bonds also that will be
discussing further. But whenever you will find that a fixed coupon bonds, the fixed are more
exposed to the inflation risk. In the long, run the inflation can go or the direction of the
inflation can go to any extent or it can move any direction that cannot be predicted before.
But if you are going for a floating rate bonds then there is a chance of changing the interest
rate due to the change in the inflation and other kind of macroeconomic parameters.
So that is why the floating rate bonds have the lower level of inflation risk in comparison to
bonds having the fixed coupon rates. That is another risk always we are exposed to whenever
we are investing in the fixed income securities market.
(Refer Slide Time: 23:19)
34
Then we have the liquidity risk that already all of you might have known. Liquidity risk is the
risk where the bond holder basically always exposed to or will have whenever they the
problem in terms of selling the bond or they will have to sell a bond below its true value.
What should be the true value of the bond? Either they are selling it below the true value
because there is no such market for that or they are not able to sell it for some reason. Even if
they want to sell they cannot sell it. There is no proper market available for that or this
particular bonds cannot be easily converted into a liquid asset like cash. So in that case, we
are facing the issue of liquidity risk and whenever we in the overall market, whenever we
look at the liquidity mostly always we refer to the transaction cost. So if the transaction cost
is low then the marketability of that particular asset will be high, the market is more
transparent. So in that case, the trading in all these things will be easier. We have the proper
buyers, we have the proper sellers and always we can sell that particular asset or we can buy
that particular asset with the minimum cost. So in that case, one of the best measures of the
liquidity is the bid-ask spread. The bid-ask spread can be used as the measure of the liquidity.
So wider will be bid ask spread, wider will be the spread, greater is the liquidity risk because
spread higher means there is a high transaction cost in the market. So that is why it is difficult
to make that particular asset marketable. So in that particular context, the liquidity risk will
be higher. From the overall market perspective if you look at because the many traders or
many kind of investors who are there or many kind of brokers who are there they basically
quote the different prices: different bid price, different ask price. So from the overall market
prospective, the market bid-ask spread is generally defined as the difference between the best
bid price- the best bid price in the sense it is the higher price at which the dealer is willing to
buy the security and the lowest ask price which is nothing but the lowest offer price at which
35
one of the dealer is willing to sell the security. So that is called the market bid ask spread
which is the difference between the best bid price and the best ask price. That is the way
overall market liquidity can be judged or can be examined. So this is basically about the
liquidity risk.
(Refer Slide Time: 26:19)
Then already I told you that exchange rate risk or other name of that we can call the currency
risk also. Generally the non-home currency or the foreign currency dominated bonds are
exposed to the currency risk. Because if Rupee versus Dollar let your particular investment is
dominated with dollar, let the value of the Rupee has gone down. Then accordingly what will
happen that if you are paying in terms of the dollar, your cash flow in terms of the dollar then
you have to pay. Accordingly, your exposure basically is going to be affected by that or your
cash flow is going to be affected by that. Let 1 Dollar was Rs.80 and now let Rupee become
very strong then it became 1 Dollar = Rs.70. So in that case your cash flow will also get
affected. It can go also opposite directions.
So in that case, if any time the exchange rate fluctuates, if your investments are dominated by
the foreign nominated bonds then what will happen that always your cash flow gets affected
and as well as there is a proper link between interest rate and the exchange rate. If there is any
kind of fluctuations in the market then that also creates some kind of supply and demand
imbalances with respect to the investments, foreign investments and other things. And
because of this, the exchange rate also gets affected. So that is why the currency risk or the
exchange rate risk has lot of implications from the bond investments or the fixed income
securities investment point of view.
36
(Refer Slide Time: 28:13)
Then we have the volatility risk and the political risk. Already I told you the bonds which are
having the embedded options: they have the call feature and they have the food feature and
all these things. Or they have the options; they are more exposed to volatility risk. Because
the expected interest rate volatility increases the value of the options and generally the
adverse price impact of the change in volatility on the price of the security is called the
volatility risk.
So, that we will discuss whenever you go for the valuation of embedded options. We will see
that how the cash flows are getting affected due to the volatility of the interest rate and the
other factor which are the major factors which drive the price of the underlying assets. Then
we have the political risk. Political risk means any kind of we can say that tax structure, any
kind of regulation, any kind of stability with the government. All kinds of things basically
come into the picture. Political risk includes the tax risk and regulation risk.
So, the bonds which are having the embedded options are more exposed to the volatility risk.
Because the price of that particular asset; or value of the particular asset depends upon the
cash flow of the underlying assets and all these things.
And the expected interest rate; if the interest rate is going to be volatile which is a major
factor that affects the price increases the value of the options. And the adverse impact of the
change in volatility on the price of security generally called the volatility risk. So that has lot
37
of implications on the investments of the fixed income securities. Then we have the political
risk which includes the tax risk and regulation risk.
Any kinds of regulatory change, any kind of changes in terms of tax rates, will also have
implications on the investments of the fixed income securities or the return of the fixed
income securities.
(Refer Slide Time: 31:27)
Then event risks already all of you have idea that any kind of natural disaster, any kind of
takeover activities, merger activities or any kind of restructuring. Or in the restructuring
process we can take example of leverage buyout. For example, any company wants to acquire
another company, to acquire that they have gone for very high amount of loans that basically
have the implications on their rating and all these things.
So that will have the impact on the value of this particular bonds whatever they have issued
on the return of the bonds also whatever they have issued and as well as the credit risk of the
particular bond.
And there is a sector risk which is basically the possibility of adverse differential movement
of particular sectors generally known as the sector risk. And that may arises because of many
external factors like any kind of uncertainty, any kind of fluctuations in the macroeconomic
fundamentals.
That will have the impact on a particular specific structure. That is why we should not
concentrate on a particular sector, we should concentrate on the diversified sectors, by that,
the risks can be diversified and the return can be maximized.
38
(Refer Slide Time: 32:41)
So what basically we have seen here, there are various type of risk which affect the returns
from the investments in the fixed income securities. And all type of risks should be carefully
measured and considered while making the investment in the market. And basically we will
be discussing gradually in the forthcoming sessions.
(Refer Slide Time: 33:03)
So this is the reference basically what you have to see for the detailed discussion on this
issue.
Thank you.
39
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 01
Lecture No # 03
Bond Valuation - 1
Good morning, welcome back to the next session of the particular subject on the management of
fixed income securities.
(Refer Slide Time: 00:26)
So in the previous sessions, we discussed about the overview of the fixed income securities
market and as well as the different type of risk what generally we always face whenever we
invest in this particular market. So today, we can start the discussion with respect to the different
types of the fixed income securities and how the pricing of those securities are done and how the
return of those securities, are basically calculated.
So among all those markets whatever we have discussed, the most important market is the bond
market. Then gradually we can move towards the mortgage backed securities and the fixed
income derivatives and all. But, today's discussion will mostly focus on the valuation of the
bonds.
(Refer Slide Time: 01:14)
40
So, here, if you see that whenever we go for the evaluation of the bonds, generally today's
discussion, largely cover up the different aspects of the valuation of the bonds. And after the
discussion, you will come to understand about the different kind of features of the bond like the
coupons, like the term to maturity, how the effective rate is calculated, then, concept of
continuous compounding and all these things. Basically, the major of this particular discussion
today!
(Refer Slide Time: 01:47)
41
Bond Valuation
The value of a bond is the present value of its future cash flow (CF):
M
CFt
CF1
CF2
CFM
B
V0 = ∑
=
+
+ ⋯+
t
1
2
(1 + R)
(1 + R)
(1 + R)
(1 + R)M
t=1
where :
CF1 = cash flow at t; principal and / or coupon
R = required return
M = term to maturity
So let us go to the bond valuation. So all of you know, that whenever we go for the valuation of
the bond. Like other assets whether it is a stock or whether it is a bond or any kind of asset, the
value of a particular asset is nothing but the present value of its future cash flows. So whenever
we talk about the bond part, what is the cash flow involved whenever you talk about the bond.
So now we talk about the bond, largely the cash flow is basically the coupon and as well as the
principal or the par value of that particular bond what we are going to expect or we are going to
get in the end of the maturity. So, In the generic way, if you want to find out a formula for the
evaluation of the bond that is basically nothing but the cash flow what basically we are getting
that is the coupon and as well as the principal and that has to be discounted with respect to the
required rate of return or the market interest rate, what generally in our term we call it the yield.
So if you discount this particular cash flow with respect to the discount rate which is nothing but
the market interest rate. Then what basically we can get we can get the present value of that
particular cash flow and this present value of cash flow is nothing but the price of the bond. Then
another important feature here is the term to maturity, up to what period the particular cash flow
has to be discounted. So that cash flow will be discounted up to the term to maturity of that
particular bond.
So therefore if you this look at this particular formula that value of the bond will be:
Value of the Bond =
𝐶𝐹𝑡
𝑡
(1+𝑅)
42
Where, 𝐶𝐹𝑡 = cash flows at the time period
R= required rate of return
t= time period
So now already what basically we have discussed, we have two types of cash flow: one is your
coupon, other one is the principal or the par value or the price value those words are used
interchangeably whenever we go for the valuation of the bonds.
(Refer Slide Time: 04:04)
Bond Valuation Cont...
So then if you look at this and you assume that the coupon basically is fixed and it is paid let
annually. And the principal what basically you, are expecting that principle you will be getting at
the time of the maturity. Then we have two components in terms of the present value calculation
of the cash flow: one is your C which is basically the annual coupon what you are receiving. And
another one is the F represents the face value of the bond or the par value of the bond.
So now what basically we can do the face value of the bond or par value of the bond you will be
realizing once the bond will be matured. So at the end of the maturity, you will be getting the
face value of the bond or the par value of the bond. So now first of all, the C has to be discounted
periodically over the period of time up to what time you are holding that bond.
43
And in the end of the maturity you will be getting your face value and that has to be discounted
with respect to that particular maturity period and that mature maturity period is represented as
m.
𝑀
𝐶
𝐹
𝑉0𝐵 = ∑
+
(1 + 𝑅)𝑇 (1 + 𝑅)𝑀
𝑖=1
So that is what basically and here you remember this bond basically is making the fixed coupon
payments each year and you are realizing or is you are receiving the maturity at the time of
maturity you are receiving your principle. So that is the two things basically you have to keep in
the mind.
(Refer Slide Time: 05:45)
Then if you see if you have to expand it mathematically then how this particular expansion can
be possible such as:
𝑀
𝐶
𝐹
𝑉0𝐵 = ∑
+
(1 + 𝑅)𝑇 (1 + 𝑅)𝑀
𝑖=1
𝑀
𝑉0𝐵 = 𝐶 ∑
𝑖=1
1
𝐹
+
𝑡
(1 + 𝑅)
(1 + 𝑅)𝑀
𝐹
(1 + 𝑅)𝑀
1 − 1/(1 + 𝑅)𝑀
𝐹
]+
𝑉0𝐵 = 𝐶 [
𝑅
(1 + 𝑅)𝑀
𝑉0𝐵 = 𝐶[𝑃𝑉𝐼𝐹𝑎𝑎 (𝑅, 𝑀)] +
44
So now if you get the bonds par value if you get the coupon or coupon rate or if you get the yield
to maturity or the discount rate and if you get the term to maturity then you can calculate the
value of that particular bond at that particular point of time. That particular period, the price of
that particular bond can be calculated by using this four kind of factors or four kinds of
components.
What basically we require for the evaluation of the bond: one is your coupon, second one is the
par value of the bond, third one is the term to maturity, and fourth one is the discount rate. So
these are the four things what basically we need.
(Refer Slide Time: 07:43)
Then let us see one example that how this particular value of the bond can be calculated so if you
see that how this particular bond value is calculated. So in this case let us we take one example
where,
Par value of the bond= 1000(Face value of the bond)
Term to maturity= 10 years
Coupon rate= 9%, Coupon payment= 90
Interest rate= 10%
45
So now what else we need we have the par value, we have the term to maturity, we have the
coupon rate and we have the interest rate at which the particular bond will be discounted right.
So now, what is the formula already we just now we have seen the value of the bond will be
Value of the bond =
𝑉0𝐵 =
𝑀
∑
𝑖=1
10
∑
𝑡=1
𝐶
𝐹
+
(1 + 𝑅)𝑇 (1 + 𝑅)𝑀
90
1000
+
𝑡
(1 + 0.10)
(1 + 0.10)10
= 90 [
1−1/(1⋅1)10
0.1
]+
1000
(1+0.10)10
= 553.011 + 385.5433
= 938.5543
Hence, the value of the bond is 938.5543.
So the value of the bond this is basically this is rupees 938.5543 so if the bond will be traded in
the market at that particular point of time then the value will be this is the price of the bond. So
here one thing you remember that the coupon is basically paid annually and the par value of
bond are 1000. So in that case the market price of the bond or the intrinsic value of that particular
bond will be 938.5543.
And here we have basically gone for the two components one is discounting the coupons and
another one the discounting the par value of that particular bond.
(Refer Slide Time: 13:26)
46
But let us see that if the coupons are let paid semi-annually. Many a times, we have observed that
generally coupons are paid semi-annually, every 6 months basis, the bond issuer may pay the
coupon. So then whenever the coupons are paid semi-annually, we have to make certain
adjustments with respect to that formula. Whatever formula we have derived when the bonds
basically make the annual coupons in that case we have made the certain adjustments.
What are those adjustments, we can make? One is your number of period maturity period is
basically is mentioned on the basis of the years but now, it is basically 6 months basis the
coupons are paid. So that the number of periods should be doubled that is number one. Number
two, the coupons basically what you are getting in 6 months that should be half because in the
annual you are getting 90 rupees at a 9% coupon rate. But whenever you the coupon is paid
semi-annually then the annual coupon rate also should be half then the discount rate also should
be half. Because if you annually you are getting 10% then semiannual you should get 5%. So
there are 3 things basically what we have to make the adjustment whenever we are talking about
the semi-annual coupon payments.
So now if you see your formula will be automatically changed.
VB0 =
2𝑀
∑
𝑖=1
𝐶 𝐴 /2
𝐹
+
𝐴
𝑡
(1 + (𝑅 /2))
(1 + (𝑅 𝐴 /2))2𝑀
47

Where, CA = Annual coupon rate, RA = Annual discount rate, M = Term to maturity in years
and F = Face value
So in the same formula, if you observe here, what basically we have done, your m was the term
to maturity in years so automatically your maturity period has become 2 M you observe. Your
coupon was annual coupon rate you have taken let CA, A represents another coupon that would
be divided by 2 the annual coupon that whatever we have that is the 2.
So here your CA represents your annual coupon rate, your RA represents annual discount rate and
M is the term to maturity in years. Then automatically in this case the period will be 2 M then F
is the face value or the par value of that particular bond. Now let us with the same example if
you see that how that particular thing worked out.
(Refer Slide Time: 16:07)
For example, F= 1000, M= 10 years, CA= 9%, Ra= 10% and Coupons are paid semi annually.
VB
20
∑
𝑖=1
=
90/2
1000
+
𝑡
(1 + 0.1/2)
(1 + 0.1/2)20
48
=
20
∑
𝑡=1
45
1000
+
𝑡
(1.05)
(1.05)20
1
1000
= 45[[1 − 1.0520] +(1.05)20
.05
= 560.7995 + 376.8895
=937.6889 or approx.(937.69)
Hence, the value of the bond is 937.69.
This is basically in terms of rupee that is the value of bond if you are paying the coupon semiannually. If you are paying the coupon semi-annually then that is basically your value of the
bond, little bit different from the valuation whatever we have done whenever we have used the
annual coupon payments. But we will not find much differences but little bit differences will be
there because the cash flow what basically you are getting that frequency of the payment of the
cash flow gets changed. So because of that some deviation you can find in terms of the value of
that particular bond.
(Refer Slide Time: 20:35)
So now we can generalize that formula we have taken the number of frequency of the coupon
payment is 2 per year. Now we can extend it even more frequently for example to determine the
value of the bond paying interest 4 times a year, we can increase your periods 4 times. And also
49
the coupon payments we can divided by 4, discounted also will be divided by 4 like that. So now
if you see that then in general if N is equal to the number of payments per year that means the
compounding per year.
Then M is equal to the maturity in years then age N is equal to your number of periods to
maturity let that is N M and your R A be the discount rate quoted on an annual basis. So if you
express the general formula for valuation of a bond. Then it will:
V03 = ∑nM
t=1
1 − 1(1 + (𝑅 𝐴 /n))
𝑛𝑚 𝑉 0 = 𝐶 /n [
t+
𝑅 𝐴 /n
(1 + (R 𝐴 /n)) (I + (R 𝐴 /n))
C𝐴 /n
F
𝐵
𝐴
𝑛𝑀
]+
𝐹
(1 + (𝑅 𝐴 /n))
𝑛𝑀
Note : M = maturity in years
n = number of payments per year
Then here already I told you that m is the maturity in years and n is equal to the number of
payments per year. How many times the coupon is paid accordingly the particular formula can be
generalized if the coupons are paid n times in a particular year.
So that is what basically we can generalize it whether it is 4 times or it is 2 times or would be 6
times, depending upon that this kind of adjustments you have to make and accordingly the value
of that particular bond can be calculated.
(Refer Slide Time: 22:49)
50
Then we are talking about a concept of the compounding frequency. so what do we mean by this
compounding frequency? So whenever you talk about the 10% annual rate, it is basically the rate
at which one annualized compounding; it is the rate with one annualized compounding. So if you
are using one annualized compounding then what basically you are earning, you are earning 10%
every year.
Then if you have invested 100 rupees, then the 100 rupees will grow to 110 rupees, after 1 year
this particular value will become 110. So that will be how you get this that is 100 into 1.1 that is
110. So if you are going by the simple annual rate you are expressing with the semi-annual
compounding then what will happen? You will get 5% every 6 months and if you assume that
the interest is being reinvested again because you get some money after 6 months and that money
can be reinvested in the market.
So what will happen in this case, in this case your 100 rupees will grow to 110.25 after one year.
In the case of 1 year annualized compounding or one annualized compounding what we got we
got 110 but whenever you are using semi-annual compounding, you are basically getting 110.25
rupees
In case of monthly compounding, the future value of the bond will be:
Future Value of the bond=100(1+R)2
0.10
= 100(1+ 2 )2
= 100(1.05)2
= 110.25
Hence, in case semiannually compounding, the future value of bond will be little bit higher than
in case of an annually compounding.
Then the value has become little bit more value you will be getting because you are using that
particular money whatever you got after 6 months that has been reinvested in the market then
your value has become 110.25 rupees.
(Refer Slide Time: 25:12)
51
So like that if you increase your compounding let you are going for monthly compounding. In
case of monthly compounding, the future value of the bond will be:
100(1+R)n
0.10
= 100(1+ 12 )12
= 100(1.0083)12
= 110.47
In case of daily compounding, the future value of the bond will be
100(1+R)n
0.10
= 100(1+ 365 )365
= 100(1.00027)365
= 110.52
If your compounding frequency will increase then automatically your value this ending value is
going to be increasing. Because we are talking about the concept of the reinvestment so that also
you have to keep in the mind.
So compounding frequency basically will affect the investment value, the ending investment
value of the investor.
(Refer Slide Time: 26:40)
52
So here another concept you have to keep in the mind that is called the effective rate. So what is
effective rate? The effective rate basically is the rate which includes the reinvestment of interest
or compounding. So here if you want to calculate this effective rate, it is:
Effective rate= [1+(RA/n))n -1].
Your RA means it is the simple annual rate and from there you can calculate your effective rate.
(Refer Slide Time: 27:22)
So, let you assume that our cash flow is continuously compounded. When the compounding
becomes very large then generally we approach towards the continuous compounding. So
53
wherever there is continuous compounding if you want to calculate the future value for any
investment of let 8 dollars for M years from now the future value will become:
F V = A eRM .
= 100 eRM ,
= 100 e0.01*1,
= 100*2.71825.01 ,
= 110.52
(Refer Slide Time: 28:26)
So like that if you go for the calculating the present value from that if you know the future value
whatever you are going to get then the present value will be:
The present value of a future receipt (FV) with continuous compounding is
A = PV =
FV
= FV e−RM
eRM
So given the continuous compounding, let R = 10% and your bond is going to pay 100 rupees 2
years from now then what is the value of the bond?
What is the current worth of 100 particular bonds? The current worth of the particular bond will
be:
Given continuous compounding, if 𝑅 = 0.10, a bond paying ₹100 two years from now would currently
be worth:
𝑃𝑉 = ₹100e−(0.10)(2) = ₹81.87
54
So if a bond which is paying 100 rupees each year for 2 years then what is the value of that
particular bond now? Then it will be:
Similarly, a bond paying ₹100 each year for two years would be currently worth:
2
𝑃𝑉 = ∑ 𝑅𝑠 ⋅ 100𝑒 −(10𝑥(1) = 𝑅𝑠 ⋅ 100𝑒 −(10×1) + 𝑅𝑠 ⋅ 100𝑒 −(10×2) = 𝑅𝑠. 172.36
𝑖=1
That means it will give you the current worth of that particular bond will be 172.36 and you will
be getting the ending value will be 200 rupees at the end of the maturity. So that is what basically
what way we use this continuous compounding.
(Refer Slide Time: 30:12)
Let us one small example we can we can take with respect to this.
Let, the term to maturity= 10years
Discount rate = 10%
Coupon rate is 9%
Future value = 1000
Then, V=
𝑚
∑ 𝐶 𝐴 𝑒 −𝑟𝑡 + 𝐹𝑒 −𝑅𝑚
𝑡=1
=
𝑚
∑ 90𝑒 −0.1𝑡 + 1000𝑒 −0.10.10
𝑡=1
= 908.82
Hence, the present value of the bond will be 908.82.
55
So that will basically become your 908.82. It is the value of the bond. So in the same example
whenever we have taken this concept of continuous compounding then what basically we got
that the value of the bond will become 908.82 with the term to maturity of the 10 years and the
face value is 1000 and coupon is 9% and the discount rate is 10%.
(Refer Slide Time: 32:16)
So what basically we have seen here, the bonds are priced according to the mathematics of
present value and the future value. And the effective price and yield of the bond generally
depends on the frequency of the compounding. Greater the compounding frequency greater
would be the amount of interest and accordingly the value of the bond also will to be changed.
(Refer Slide Time: 32:16)
56
These are the references what you can you can go through then the other discussion with respect
to bond valuation, we will discuss in the next class. Thank you.
57
Management of fixed Income Securities
Prof: Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 01
Lecture No # 04
Bond Valuation-II
Welcome back. In the previous class, we discussed about the valuation of the bond and there we
have seen that the value of the bond gets changed whenever the frequencies of the coupon
payments get changed. And as well as basically the compounding frequency gets changed so
accordingly the value of the bond always will vary.
(Refer Slide Time: 00:42)
So today we will be discussing about the certain things that how to do the valuation of a zero
coupon bond if the bond does not pay any coupon and also the valuation of the bond as non
coupon dates. That means sometimes also we go and invest in a bond where already this
particular coupon debt is over or the coupon date and the next coupon date is basically always
you are buying the bond between these 2 dates. And how the bond prices are quoted and the
concept of the discount yield?
(Refer Slide Time: 01:23)
58
So these are the things what basically will be discussing, that is why after that you will get to
know what is zero coupon bond? That is the concept of the day count convention what is day
count convention the concept of a clean price concept of the accrued interest and the full
price. So these are the certain concepts or certain keywords what basically we will come to
know after the discussion of this particular class.
(Refer Slide Time: 01.48)
So let us see that what do we mean by zero coupon bond so this type of bonds do not make
any periodic coupon payments, no coupons are paid against this zero coupon bond. The bond
is basically issued at a particular price and after end of the maturity you will get your face
value of the bond or the par value of the bond. So that means many a times or mostly the
bond is issued at discount and redeemed at par.
59
So the investor basically will realize the interest as the difference between the maturity value
and the purchase price so there is a purchase price and there is a maturity value. So the
investor will get the difference only so these bonds are also called the zero discount bonds.
And the zero coupon bonds also called the pure discount bonds the other name but popularly
it is known as the zero coupon bond.
So if you go for a zero coupon bond how the value of a zero coupon bond is calculated the
value of a zero coupon bond is basically F by 1 + R to the power M it is very simple. In
comparison to the coupon basic bonds, the value of a zero coupon bond is relatively simpler
so here F is the maturity value and your R is the required rate or the discount rate and M is
the maturity period.
So, if you only get this face value of the bond then you get this required rate of return or this
discount rate or you get your maturity period. Then this particular value of the bond zero
coupon bond can be calculated. So comparatively this is easier to calculate.
(Refer Slide Time: 03:44)
So let us take 1 small example let I will give you that let M = 10 years and your discount rate
discount rate let 10% then what is the value of the bond? Then value of the bond and also the
face value has to be given to you that is 1000. If maturity period is 10 years and it is a zero
coupon bond then what is the value? That is 1000 divided by 1 + R to the power M so this is
basically your R that means your 1000 divided by 1.1 to the power 10 that will give you let
385.54.
60
So the value of the zero coupon bond will be Rupees 385.54 in comparison to the coupon
bearing bonds, and valuation of a zero coupon bond is much simpler. So that actually you
have to keep in the mind.
(Refer Slide Time: 05.12)
Then already I told you that if any bond zero coupon bond which is having maturity less than
1 year then what basically we will do? So in that case we use a concept called the day count
convention. Assume on march 1 a zero coupon bond is promising to pay 1000 Rupees on
September 1 and trading at an annual rate of 8% then what is the value of that particular
bond? So in that case you can go for then what is the maturity period?
From March 1 to the September 1 right so then that will give you that 184 days so then the
value of the bond is F by 1 + R to the power m and what is the m here the m is basically 184
days then that is why we have second 184 divided by 365. Then it is 1.08 to the power 184
divided by 365 then whatever value we will get that will be the value of the bond. So the day
count convention is defined as the way in which the ratio of the number of days to maturity or
days between the dates to the number of days in the reference period that is basically year is
calculated.
That concept is called the day count convention so day count convention is actual days to the
maturity to the actual days in a year that is, actual by actual that is what we have done here
that is 184 divided by 365. And day count convention also can be considered assuming 30
days in a month that also can be considered in this particular case. So that means for year it is
360 days so here we have to consider a convention of 30 by 360.
61
So there are certain bonds where basically this 30 days month concept the 30 by 360 days
concept is used like corporate bonds in US. They use this particular kind of day count
convention. We will discuss all these things in detail further. But you have to keep in the
mind that the day count convention all will also have the impact on the value of that
particular bond. In this case if you going for a recount convention it may not be coming 184
days it may be coming some different dates.
So because we are assuming every month is 30 days so March 1 to March 31st instead of then
April we can consider 30 days may instead of 31 days we consider 30 days like that. So in
this case we might have taken 180 days instead of taking 184 days. So the day count
convention will have the impact on the value of that particular bond that actually you have to
keep in the mind.
(Refer Slide Time: 09:14)
So that’s why this has lot of implication in that particular case and this part is more important
you have to keep in the mind. Because the investor who basically buys the bond between the
coupon payments. So in that case he should compensate the seller of the bond for the coupon
interest and from the time of the last coupon to the settlement date of the bond. Because let
every 6 months basis coupon is paid and coupon debts are fixed so if you are buying the bond
for example March 1 is the coupon date and September 1 is the coupon date.
Let you have bond in bought the bond in July then from March 1 he has received the coupon
but from March 1 to July let 15 in that particular period the coupon he has not received the
coupon basically received in the September 1. But in that period his or she should be
62
compensated against that because the coupon he has not realized. So that particular thing is
called the accrual interest which is basically nothing but the days from the last coupon date is
divided by the total days between the coupon dates * the coupon.
So whenever we are basically or the investor is buying the coupon between these 2 coupon
dates or in trading of the bonds on the non-coupon date the amount the buyer basically pays
to the seller that is the basically has 2 components. One is your agreed upon price plus the
accrued interest and if you take the total that is called the full price or the other name of that
is the dirty price.
So the full price has 2 components 1 is clean price plus the accrued interest right very
minutely you observe that that full price has 2 components 1 is clean price other 1 is the
accrued interest.
(Refer Slide Time: 11:45)
So here how basically we calculate these valuations of the bonds at the non coupon dates how
the price is calculated? At the coupon date price is easier because coupon is realized you have
to discount it with respect to discount rate. And the maturity is basically known the period is
known. But now what is happening here our coupon is not yet realized next coupon. And we
have basically trying to sell the bond between that coupon dates.
So then what we should do? There are 2 methods for calculations of that particular price of
the bond what is those 2 methods? First of all, determine the number of days between the
settlement and the next coupon date. Let July was the coupon date and next coupon date is
September. And in between how many dates how many days are there? Determine the
63
number of days in the coupon period let coupon is paid annually or semi-annually whatever
then you compute that V which is nothing but the number of days between the settlement and
the next coupon divided by the number of days in the coupon period.
For example if a bond is paying the semi-annual coupons then what is the value of that
particular bond in this case? It will be summation t = 1 to N C by 1 + r to the power t - 1 + v
+ F by 1 + R to the power N - 1 + v and if you expand it then you will get c by 1 + R to the
power V + C by 11 + R to the power 1 + V + C by 1 + R to the power 2 + V and so on up to
your C by 1 + R to the power N - 1 + p + f y 1 + R to the power N -1 + v. So here in this case
how basically we can we can calculate this value of that particular bond.
(Refer Slide Time: 14:24)
Let us take 1 example I will take certain things here let the last coupon date was first march
right next coupon date is first September. I already told you let you assume that this is a
corporate bond that means your day count convention is 30 by 360. It will be the day count
convention is 30 by 360 ratio let the settlement date. I will give you let the settlement debt is
seventeenth July 2020 maturity is let 2026 6 years period basically. Already you have to
consider 30 by 360 ratio that means 30 days in a month.
That basically we are considering right so that means you have how many days? In the July
how many month days are remaining your 30 - 17. 13 days august we have 30 days then we
have September 1 that is 1 day. So effectively we have 44 days from settlement date to next
coupon date is 44 days. Then what is your V? Your V will be 44 by 180 coupon payment
64
period is 180 that that will be 0.2444. Let I will give you the par value of the bond, par value
is let 100 Rupees. Right? let yield is let 10%.
Then how you can calculate the value of the bond then what is the value of the bond. So how
the value of the bond will be calculated so what you need? You have a period which is t we
need t - 1 then cash flow then we have the present value of the cash flow. So, period 1 t - 1 0
then this will be your period will be 0.24444, cash flow will be your cash flow will be 4 then
present value of the cash flow. You can calculate 4 by this that will be 3.952578 for period 2,
t - 1 will be 0 then this will be 1.24444, cash flow will be 4 then this will be your 3.764360.
Like that, you can go up to your period 12. This will be your 11.24444 your last cash flow
will be Rs.104. 100 Rs is your face value. And 4 Rs is your coupon. That will give you that
will give you your 60.085757. If you add up this it will give you ninety 4.559097. So, that
will basically will give you the value of the bond on that particular day if the settlement is
basically done between the 2 different coupon dates.
(Refer Slide Time: 20:14)
And we have also another method what is called the method 2. Here what we do we go to the
move to the next coupon date and determine the value of the bond at that date based on the
future coupons, add the coupon at the next coupon rate to the value of the bond and discount
the bond, loss the coupon back to the current rate backward. You can also do that here what
basically you can do.
(Refer Slide Time: 20:53)
65
Let us go by the same example we have the certain things, we have to see then we have to
find out the value of the bond at next coupon date in 44 days. Then what is the value? That is
your t = 1 to 11 4 by 1.05 to the power t+ your 100 divided by 1.05 to the power 11 that will
give you 91.693586. Then what will be the value of the bond? Value of the bond at next
coupon date plus your coupon paid at that date you can add your coupon here that is your
91.693586 plus your 4 that will give you 95.693586.
Now what basically you have to do you have to find out your current value of the bond
current value of the bond, which is nothing but the full price.That full price will become your
95.693586 divided by 1.05 to the power what 0.24444. That will give you same value your
94.559 right, so that is basically your current value of that particular bond so either of these 2
methods whatever method you can use you will get the same value then, you can also
calculate your accrued interest from this. Your accrued interest also can be calculated from
here.
That is your 4 into what? your number of days from the last coupon number of days from the
last coupon means it is 180 - 44 that is 136 that means 4 into 136 divided by 180 that is
3.0222 so, that is basically your accrued interest value what you get and your this is your full
price or dirty price then your clean price will become 94.559 - 3.0222 it will be 91.5368. But
in US, the bond prices generally quoted with the clean price but the accrued interest, the
buyer has to pay to the seller that actually you have to keep in the mind.
So that is the way basically this valuation of this bonds between the 2 coupon dates are
calculated.
66
(Refer Slide Time: 24:56)
Then the question is how the bond price are quoted many traders if you see they quote the
bond prices as a percentage of their par value. For example, if a bond is selling at par it would
be quoted at 100 that means 100% of the par. If a bond with a face value of 1000, which if it
is quoted like 81 by 8 it is not 80 - 1 by 8 it is 81 by 8, that means it will be selling at
0.80125. So, that means the selling price will be 0.80125 into 1000 that is 801.25.
That means at that time the bond is traded in the market at a price of 801.25 that is the way
basically you have to interpret this.
(Refer Slide Time: 25:58)
Generally they are quoted in the basis point all of you know what is basis point? Basis point
is nothing but it is 1 by 100 of a percentage point so, 6.5% means it is 6% + 50 basis point or
67
650 basis point so, increase in the yield from 6.5% to 6.55% means it is an increase of the 5
basis point so many a times, the basis point concept is used instead of using the percentage
things that actually also you have to keep in the mind.
(Refer Slide Time: 26:37)
Then there is a whenever it is quoted, there is a bid price that is ask price so the bid price is
the whatever price the dealer is willing to pay for the bond. And the ask price is the price that
the dealer is willing to sell the bond and the bid yield is the return which is expressed as a
percentage of the par value. That the dealer wants if she buys the bond and many a case the
yield is generally annualized.
And the ask yield basically the rate at that the dealer is offering to sell the bond. They want to
sell the bond so then for treasury bills and some other securities bid and ask yields are quoted
as the discount yield. We will discuss more on this whenever you discuss about the markets
so the annual discount yield is basically your F - P 0 divided by F into 360 by days to
maturity. We are using the 30 by 360 ratio in this case.
F is the final value and P is the purchase price and days to maturity if you know then the
annual discount yield basically is calculated from that. So this is the way the bond prices are
quoted and this is the way basically we can assume or we can define the bid price, ask price,
bid yield, ask yield discount yield all these things.
(Refer Slide Time: 28:12)
68
So if you know the discount yield then price also can be calculated from there. You can solve
that equation for the bond price P 0 = F into 1 – R D into days to maturity divided by 360.
Just now whatever formula I have shown you the same formula can be used also if you know
the yield then from there, the price can be calculated. If you know the face value, if you know
the yield, then the price of the bond can be calculated accordingly. So that is basically the
relationship between the price and yield.
(Refer Slide Time: 28:53)
So whatever we have discussed that the present value of a zero coupon bond is the present
value of the face value of the bond only. So, in the context of the valuation of the bonds at
non coupon dates, the price the buyer pays the seller is called the full price which is a
comprised of 2 :- 1 is your accrued interest. And we have the clean price and the buyer must
compensate the seller for the portion of the next interest payment the seller has earned but
69
will not receive because the issuer will send the next coupon payment to the buyer. So that is
why the amount is called the accrued interest that particular amount is defined as the accrued
interest. And there are 2 formulas or 2 methods whatever we have discussed through which
the price of the bond between the 2 coupon dates can be calculated.
(Refer Slide Time: 29:51)
So this is the reference what basically you can follow for the detailed discussion on this.
Thank you.
70
Management of Fixed Income Securities
Prof: Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 01
Lecture No # 05
Bond Price Relations
Good morning, so, we have discussed about the bond valuations or how the valuation of the
bond is carried out on the basis of the different characteristics of the bond and as well as on
the basis of the coupon payments. But one thing to remember is that the price of the bond will
get change on the basis of the change in interest rate, change in the coupon, and as well as
change in the maturity.
And you will always find the different kinds of relationships whenever you find that keeping
other things remain constant if you change other variables, then how the price of the bond
basically behaves differently which we basically call, the bond price relationships.
(Refer Slide Time: 01:13)
So, in today’s discussion we will cover of that relationship among the bond price, interest rate
coupon, maturity and yield. So, there are different kinds of relationship we can observe or the
characteristics wise the price of the bond or the price change of the bond which basically will
be different.
(Refer Slide Time: 01:38)
71
So in this context, you will come to know about certain keywords like discount bond,
premium bond, price sealed curve so those things basically will be discussed in today's
session.
(Refer Slide Time: 01:53)
So, let us see that the first relationship you know that in the examples whatever we have
taken the previous class if you recall. We have taken your coupon was 9% discount rate was
10% and the par value of the bond was Rs.1000 and what we got? Whenever we are paying
your coupon annually, we got the price somewhere around 938.55 or something and
whenever we are paying the coupon semi-annually we got 937.6889.
So, that means the value of the bond is less than the par value of the bond.The value of the
bond what we got that is basically lesser than the par value of the bond, why we get that?
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Because, your coupon was less than the discount rate, your coupon was 9% and your interest
rate or the discount rate (R) was 10%. Because the coupon was less than the discount rate that
means your V is less than F, we got the value is of the bond is less than the par value.
So in this case, we call it the discount bond. But, if your coupon is equal to the par value, let
coupon is let 9% let the discount rate is 9% in that case, the market price of the bond will be
equal to the par value of the bond that is called the par bond. If you are writing the bond is
issued at par that means by default your coupon is equal to your discount rate then if your
coupon rate is greater than the discount rate, then you will find the market value of the bond
will be more than the par value of the bond which is called the premium bond. So, C R is the
coupon rate, R is the discount rate, V is the value of the bond and F is the par value of the
bond. So, accordingly on the basis of the relationship between the coupon and the discount
rate whether coupon is more than or less than the discount rate.
Accordingly, the particular bonds are defined as whether it is a discount bond or whether it is
a par bond or whether it is a premium bond. Already we have seen that the bond we have
taken in the example is basically a discount bond. Because the coupon is less than the
discount rate that is why; the market value of the bond is less than the par value of the bond.
(Refer Slide Time: 04:52)
So, therefore, whatever thing we have discussed so when the yield in the marketplace rises
above the coupon rate the price of the bond basically adjusts in such a way that investors are
trying to realize that purchase with some additional return and when the yield in the market
73
rate is below the coupon the bond basically always sell above its par value. So, if the required
rate does not change then what will happen to the value of the bond from time to time?
From the time the bond is purchased to the maturity for a bond which is selling at par, as the
bond will move closer to the maturity, the bond will continue to sell at par. That means the
price of the bond will remain constant as the bond move towards the maturity. So, whatever
interest rate will be there if you are holding the bond up to the maturity then what will happen
that the bond will always continue to sell at the par closer to the maturity period.
And the bond price will not remain constant if it is selling at a premium or the discount. So, a
discount bond increases its price as it moves toward the maturity assuming that the required
rate does not change and a premium bond decreases its price as it moves towards the maturity
assuming that the required does not change. So, these are the different characteristics that
basically you have to keep in the mind.
So, that means the bond which is selling at par if; you are going towards the maturity always
the bond will sell at the par that means the price of the bond will remain constant as the bond
moves towards the maturity. But a discount bond, it will increase the price as it moves
towards the maturity assuming the required rate does not change and the premium bond
decreases the price as it moves towards the maturity (assuming the required return does not
change). That 3 things basically you have to keep in the mind. So, that is basically the first
bond price relationship what we can get.
(Refer Slide Time: 07:48)
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The second one is already we have observed that. There is an inverse relationship between
the bond price and the rate of return. If your discount rate will increase, then, the value of the
bond will go down keeping your coupon constant. Obviously, coupon is constant which is
mentioned at the beginning and if your discount rate will decrease, then the value of the bond
will increase.
That means change in the value of the bond with respect to change in the discount rate is less
than 0. Delta V by Delta R is less than 0.
𝛥𝑉
𝛥𝑅
< 0,
Where, 𝛥𝑉 = change in the value of the bond
𝛥𝑅 = change in the discount rate
So, if R is increasing, the value is declining. If R is declining, the value is increasing. So, in
that case, we can have an inverse relationship between the price of the bond and the rate of
return.
(Refer Slide Time: 08:46)
I will show you that in our same example, if you take that how this thing is basically is going
to be changed. For example, we have taken the par value that is let 1000 then your coupon is
9% and maturity is 10 years. Right? Let with the different yield we will calculate the
particular bonds price. Let I will take year. Let it the price of bond where the yield is 11%.
Here your price of the bond when the yield is 9% and then your price of the bond when your
yield is let 7 %.
75
If you see let years to maturity is 10 then if you go by the same way, whatever way we have
calculated the value of the bond, here only R I am changing that is 11% you will find if your
coupon is 9% your yield is 9 % then the value of the bond will be 1000. So, if you calculate
your value of the bond whenever yield is 11% you will get 882.22. But, whenever it was 7%
you will get 1140.47.
That means if your yield has become 7 which is lesser than the coupon then the value of the
bond has gone up above the par value which is 1000 Rupees. Let if its maturity is 7 years, it
will become 905.76 this will not change; It will be 1107.49. If it is let 3 years you will get
951.13. this will remain the same but this will become 1052.49. let it is 1 year then it is
981.98 but it will become same 1000 and it is 1018.69.
If it is a 0 coupon bond let 0 maturity period is not 0 coupon bond let the maturity period is 0
and that time everywhere the price of the bond will become 1000. So what we have seen? If
your yield is increasing, then, the value of the bond is going down. But whenever your yield
is decreasing, the value of the bond is going up. So, that is why there is an inverse
relationship between the yield and the value of the bond.
So, if you basically see this thing that is called the price yield relationship or that is called the
price yield curve. So the yield and value of the bond has a kind of inverse relationship always
we can we can observe. OK? So, here what basically we have seen whenever your interest
rate is changing let us see that how this price yield curve basically looks like.
See price yield curve basically looks like this way it will basically looks like this.OK? So
here, wherever your let this is your R and this is your value of the bond it is basically the
convex to the origin. So, let if it is 1000 then it will be definitely 9% which is equal to the
coupon rate. Let it has become 11% then, you will find that this particular value will become
how much? 882.22.
So, now, how basically we can draw this? We can draw this in this way what basically you
can do you can take the interest rate, you can calculate the bond value, you can take the
interest rate and the bond value. And here, we have taken your face value is equal to 1000
coupon is equal to 9% let maturity is 10 years and coupon basically paid annually. So, then at
7%, it will be already you have calculated 1140.47. If you take 8%, it will be 1067.10, 9%
1000, 10%, it will be 938.55.
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If you recall we have already calculated this 11%, we have 882.22, 12%, 830.49. So,
accordingly, if you draw this, then, you will find a curve like this. Let this is at 9% let this is
8% then this value will basically at rate of 8%, it will be 1067.10 like that. So, that is
basically called the price yield curve. So, the price yield curve basically shows that there is an
inverse relationship. It shows basically the relationship between the yield and the value of the
bond and that relationship is basically inverse the convex curve basically proves that.OK.
(Refer Slide Time: 15:57)
So, then, another relation we can also see the greater the bonds maturity if the bonds maturity
is increasing, then, greater its price sensitivity to the interest rate change. Greater will be the
aim. greater the epsilon. Epsilon is the percentage change in the value of the bond divided by
percentage change in the interest rate. So, here, the investor will realize the greater capital
gains and capital losses on the long-term securities than the short-term securities when the
interest rate changes by the same amount. OK?
So, that means the sensitivity of the longer-term maturity bonds with respect to change in the
interest rate is always more than the shorter modes.
(Refer Slide Time: 17:00)
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So, let us take one example in this case also we will continue with that. Our par value is let
1000 or the face value is 1000, coupon is let 9 %, maturity is 10 years and discount rate is
10%. So, in this case, what is the value of the bond? Already we have calculated this value of
the bond we have calculated that is rupees 938.55 coupons is paid annually annual coupon
payment. Annual coupon payment.
Let your discount rate has become 9% from 10% it has come down to 9% then what will be
the value of the bond? For simplicity, I am taking this. The value of the bond become 1000
because now the coupon is equal to the discount rate then, what is the percentage change in
the value of the bond?
Percentage change in the value of the bond=
1000−938.55
938.55
= 6.85%
Let another bond you take another bond you take for simplicity, let let the maturity period is
1 year. If the maturity period will be one year, then what is the value of the bond? It will be
990.91 at the rate of 10% discount rate. Now, the discount rate has changed to 9% then the
value of the bond will become 1000? value of the bond become 1000? Then, what is the
percentage change?
Percentage change in the value of the bond =
1000−990.91
990.91
= 0.9%
So, whenever the bonds maturity was 10 years due to one percent interest rate change, the
value of the bond has changed by 6.35%. But, whenever the bonds maturity period was 1
year, the percentage change in the value of the bond has become 0.9%. So, that means greater
the maturity, greater the sensitivity of the change in the value of the bond or the price of the
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bond basically will be more sensitive. With the same amount of interest rate change, the
sensitivity is more for the longer-term maturity bond than the short-term maturity bond. OK.
(Refer Slide Time: 21:08)
Then, another thing is the smaller the bonds coupon rate, greater its price sensitivity to
interest rate changes. Bonds having low coupons are more sensitive to change in the interest
rate in comparison to the bonds having high coupon rates. Lower the coupon rate, greater will
be the sensitivity. Lower the coupon rate, greater will be the sensitivity. How basically it
happens?
(Refer Slide Time: 21:46)
Let us take one example you consider 2 bonds. Right? Then, here, the maturity period is let
10 years, discount rate is 10% & par value let Rupees 1000. Let bond A has a coupon rate
10% then automatically the price of the bond will be same with the par value that is
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1000.OK? Fine? Let the bond B the coupon is coupon rate is 2%. OK? Then, what is the
price of the bond? Will be automatically you can if you calculate then it will be, let you will
get 508.43.
How you got this?
10
∑
𝑡=1
20
1000
+
= 508.43
𝑡
(1.1)
(1.1)10
Let interest rate has come down to 9% then, what is the value of the bond-A?
10
∑
𝑡=1
100
1000
+
= 1046.18
𝑡
(1.09)
(1.09)10
Then what is the proportional change?
The proportional change =
1046.18−1000
1000
= 0.064 𝑜𝑟 6.4
Now, you come to bond B. It is for the bond A. Now if you go to bond B, what is the value
will be at a interest rate change to 9%?
10
∑
𝑡=1
Then the proportional change =
20
1000
+
= 550.76
(1.09)𝑡 (1.09)10
550.76−508.43
508.43
= 8.3%
So, that basically will give you 8.3%. That means the lower coupon bonds price is more
responsive to a given interest rate change than the price of the higher coupon bonds. So,
responsiveness is more for the lower coupon bonds than the higher coupon bonds.
(Refer Slide Time: 26:40)
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Then, we have the bond price and the interest rate relationship whatever we have seen for a
specific absolute change in interest rates, the proportionate increase in the bond prices when
rates fall exceeds the proportionate decrease in the bond prices when; in interest rise. So, the
proportionate difference increases with maturity and is larger the lower bonds periodic
interest payments. So, for the identical absolute change ininterest rates, a bond holder will
realize greater capital gain when rates decline and capital loss when rates increase.
(Refer Slide Time: 27:28)
So, if you see that I will just give you one small example here. Let there is bond the maturity
is 20 years, coupon is 4%, the par value is 1000.Right? Let I will give you the YTM( the
yield to maturity), the value of the bond & percentage change in total value. So, let it was 3%
it has changed to 4%. Let YTM was 6% it has changed to 8% ,let YTM was 9 % it has
changed to 12% or let it was again 9% to 10%. 9% to 10%.
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If you see here, in this particular bond at a rate of 3%, coupon is 4% that is fixed, we get
1164, at 4 % this is automatically 1000. At 6%, it is 769.At 8%, it will be 604. At 9%, it is
545. at 12%, it is 398 at 9% it is your 545 and at 10% it is 485. Then, if you change in the
total value if you see, it is -14.1%, this is -21.5%, this is -27% this is -11 %. what basically it
says? Its if you observe here, in the first 3 cases, this one the YTM percentage change is same
33.33%.
3 to 4% 4 - 3 by 3, 6 8 - 6 by 2, 8 - 6 divided by 6 so, if you see, the percentage change is
basically 33.3%. Here, it is 100 basis point, it is 200 basis point, it is a 300 basis point. But,
the percentage change is same. So, now, when the rates change by constant percentage, the
change in the bond price is larger when the rates are at higher level. You see whenever it was
9% to 12%, the change was -27 but, here, it is -21.5.
But, here, it is -14.1%. But, percentage change in the interest rate is same. Right? The YTM
basically has changed by constant 33.3% but now let you compare between; 3% to 4% & 9%
to 10%. If you assume a constant basis point change in yield, you get the opposite results.
Right? From 9 % to 10%, the change in the value of the percentage change in the total value
of the bond is -11% but from 3% to 4% it is -14.1%. It is reverse.OK?
So, that is basically another interesting relationship what we can observe whenever we go for
the one price relationship with respect to the change in the interest rate that is another thing
basically what you can keep in the mind.
(Refer Slide Time: 32:05)
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So, what basically we have discussed today? If you go for a conclusion, then, when the
interest rate changes, price moves inversely and the proportionate price move is relatively
greater when the rate fall compared to rates rise and the proportionate magnitude of the price
move increases with maturity and decreases with the size of the coupon payments that we
have observed and the proportionate difference increases with maturity and is larger the
lower a bonds periodic interest payment that is nothing but the coupon.
So, that is what basically what we have observed that on the basis of the characteristics of the
bond in terms of the coupon into maturity, if the interest rate will change, then, the bond
percentage in the change in the bond prices will be different that is what basically you have to
keep in the mind.
(Refer Slide Time: 33:10)
So, for the references, you can follow this book you can get the broader idea about this
particular concepts. OK? And the next class will be talking about the yield concepts or the
return concepts of the bond and the other results related to that . Thank you.
83
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 02
Lecture No # 06
Bond Returns-I
Welcome back, in the previous class we discussed about the different approaches or different
methods which are used for the bond evaluation. And today, we will be discussing about
certain concepts related to the bond returns. So, how the different types of yields are
associated with the bond that is basically the basic objective of the today's discussion.
(Refer Slide Time: 00:42)
So here if you see that there are different concepts what we are going to discuss today one is
your current yield. Then you have yield to maturity then you have yield to call then you have
yield to put. Here the yield to call and yield to put means if any bond has any kind of call
features already you must have the idea that there are certain bonds which are the call
features. After certain point of time the bond can be called back by the issuer and generally
the issuer always can go back for calling the bond whenever there is a low interest rate
scenario in the market.
So that is why there is some risk involved in that particular context. And there is a call price
already we know that is related to that particular bond and accordingly the yield or the return
from that particular bond can be calculated. Then same thing also the put feature: it gives the
kind of right to the bond investor to sell back that particular bond to the issuer. And already
84
there is a price involved in that in the beginning in the indenture provision that is mentioned
and accordingly the yield of that particular bond also can be calculated. So these are the
different concepts what we will be going to discuss today.
(Refer Slide Time: 02:11)
So then if you see that after this there are certain keywords or certain concepts you will learn
in this discussion process. That is your average days to maturity which is used to calculate the
yield to maturity of the bond that is the approximation formula that we will try to derive then
the concept of the bond equivalent yield, then yield to worst. So these are the some of the
things what you will earn after the discussion of this today's session.
(Refer Slide Time: 02:44)
CR =
Annual Coupon
F
85
So let us see that what is current yield? The other name of that thing is also nominal yield. So
you know whenever you talk about the bond the basic features of the bond if you look at. The
bond has a par value, it has a coupon rate, it has a term to maturity and it has the discount rate
and the discount rate basically depends upon the market interest rate. The price of the bond
largely depends upon the fluctuations of the interest rates in the market. So here the question
is that what is the basic difference between the coupon rate and the yield what we are
receiving from this particular coupon? Are they different?
One thing you remember, although, we are little bit using this word in a similar way but the
coupon rate whenever you talk about the coupon rate is calculated on the basis of the face
value of the bond or the par value of the bond. For example, if you are getting 90 Rupees per
annum with a par value of 1000 Rupees then we can say that the coupon rate is 9%. 90
divided by 1000.
So coupon rate is basically calculated with reference to the face value or the par value of the
particular bond. But whenever, we talk about the current yield or yield from the coupon that
is always calculated with respect to the market price of the bond. The coupon rate which is
fixed in the beginning whenever the bond is issued that time, that thing is basically linked to
the face value of the bond that is the basic difference between these 2 that actually you have
to keep in the mind. So therefore, the current yield is the ratio of its annual coupon to its
closing price. And whenever you talk about the coupon rate the coupon rate is the proportion
of annual coupon payment to the bond face value already what example I have given to you.
If there is a bond with no maturity or the repayment of the principal then that is basically
called the perpetuity or the consol. So in that case how basically we calculate that the value of
that particular bond is nothing but it is C by R, R is the discount rate, C is the cash flow. So if
the bond which is priced in the market equal to this value of that particular bond, the rate on
that particular bond will be it is just R = C by V.
So in general if you look at or what is the basic objective of differencing between the yield
and the coupon rate? One is with respect to the market value, other one is the with respect to
the face value that is the basic difference what basically you can keep in mind.
(Refer Slide Time: 06:03)
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10
Pb = ∑
t=1
90
1000
+
= 938.55
(1 + 0.1)t
(1 + 0.1)10
Coupon Yield =
Rs. 90
= 0.0958926
Rs. 938.55
Let us take one example, how basically it is different. For example if you if you see that with
a particular bond we can say that let the coupon is 9% then maturity is 10%, the par value is
1000 Rupees. And we have the discount rate that is basically your 10%. So in that case
already we have taken this example in the previous sessions also what we have seen that if
you find the price of the bond that you have to discount your T = 1 to 10. Your coupon which
is 90 divided by 1+R to the power T that means 0.1 to the power T + your 1000 divided by 1
+ 0.1 to the power 10. So in this case what we have seen that already we have calculated this
that will give you 938.55 that means the price of the bond in the market at that particular
point of time is 938.55. So now here the coupon rate is 9% but what is the coupon yield or
current yield from the bond?
The yield is basically what your 90 divided by 938.55 because 90 Rupees is the annual
coupon what you are getting so 90 divided by 938.55. Let they will be getting. 0.095898926.
So basically you will get 9.58926%, that is basically your current yield but coupon already I
told you that is 90 divided by your 1000 that is your 9%.
87
So that is the basic difference between current yield and the coupon rate that is what basically
always keep in the mind because one is related to the face value other one is related to the
market price.
(Refer Slide Time: 09:05)
M
P0B = ∑
t=1
C
F
+
t
(1 + YTM)
(1 + YTM)M
So then the next is basically the most important concept in the bond basically we always use
that is called the yield to maturity. Always we use this word, yield to maturity. What is
basically yield to maturity? Yield to maturity or YTM in short what we call, it is a particular
rate which basically equates the price of the bond to the present value of the bonds cash flow.
The bonds cash flows are what?
The coupon payments what; you are getting and the first value of the bond in the end at the
time of maturity. So which particular interest rate which make that particular price of the
bond equal to the present value of the cash flow. That is basically called theYTM generally, it
is similar to the internal rate of return what already you might have heard or you might have
studied in the financial management or the corporate finance kind of papers.
So therefore, the yield on any investment basically is the interest rate which will make the
present value of the cash flow from the investment equal to the price of the investment or the
cost of the investment. So that is basically what we call it the yield and that is called the yield
to maturity. So if you recall whenever we are going for this kind of for discussion.
88
We go back to our previous example or the equation what we have used for evaluation of the
price of the bond that is your summation T = 1 to M C by 1 + YTM to the power T+F by
1+YTM to the power M. M is the maturity period, F is the first value, C is the cash flow or
the couponand YTM is the yield to maturity. So now, if you know the price data, you know
the coupon, you know the face value, you know the maturity period then you can calculate
YTM.
From this particular formula, it will be easy for you to calculate this YTM if the market price
is known to you, at what price the bond is traded in the market that is known to you and
coupon rate is known to you and the face value is known to you then automatically your
YTM can be calculated. So let us see that how this YPM basically it will be calculated and
what is the concept what we are trying to derive through this YTM.
(Refer Slide Time: 11:50)
So YTM is basically a kind of effective rate of return, then what this YTM includes? Yield to
maturity basically includes the return what we get it from the coupon and the capital gains or
losses from that particular investment what we are making in that particular bond and the
reinvestment of the coupons at the calculated YTM. So the particular kind of interest rate will
combine all those aspects or the return or the output what we are going to receive from this
particular investment.
To return from the coupon capital gain or losses and the reinvestment of the coupons are the
calculated YTM. So these are the 3 things basically which is included for the measurement of
the YTM that actually you should always keep in the mind.
89
(Refer Slide Time: 12:49)
10
938.55 = ∑
t=1
90
1000
+
t
(1 + r)
(1 + r)10
45
Semi-annual Coupon payments
1000
937.69 = ∑20
t=1 (1+r)t + (1+r)20
Let us take one example we go ahead with the same data
what basically we have used. Let the par value of the bond or the face value of the bond that
is Rupees 1000 right then you have the maturity that is let 10 years then coupon is 10%.Then
I have already told you the price of the bond that means the bond is traded in the market at a
price of 938.55. Now let us put that thing in our bond price equation that 938.55 that means
here the YTM is not given to you. But the bond price data is available to you, at what price
the bond is traded in the market? So in this case if you see that then automatically if you go
by our equation T = 1 to 10, 90 divided by 1+R to the power T+1000 divided by 1+R to the
power 10. So then if you solve this equation whatever value of the R we calculate that is
basically your YTM and that R value will you will be getting 10%.
So at the 10% rate whenever you are going for the valuation of the bond you got the price of
938.55. Now what has happened you get the 10% is the YTM which is basically giving you
this particular present value of that particular bond using the other kind of data or the inputs.
Same way, you can go for also the semi annual. This is basically the annual coupons the
coupon is paid basically annually, annual coupon payments if semi-annual coupon payments
let the price of the bond we have 937.69. Then that will be your equal to your other things
you mentioned then automatically. It will become 20 it will be 45 divided by 1+R to the
90
power T+1000 divided by 1+R to the power 20. So if you solve it then you can get your semi
annual price basis you can find out your yield you will get 0.05.
Then if you make it annualized then it will be annualized YTM that will be 2 into 0.05 that is
your 0.1 that means 10%. But if you go for the effective annual rate if you recall what is the
effective annual rate? The effective annual rate will be 1.05 to the power 2 minus 1 that will
be basically giving you little bit higher rate that is 0.1025, 10.25%. So that effective rate
concept already we have discussed in the previous session.
So if you go by this way, whatever way basically we calculate this, what we can call it the
rate at which the price of the bond is equal to the present value of the cash flows that is
basically called the yield to maturity.
(Refer Slide Time: 17:12)
ARTM =
C + [(F − P0B )/M]
(F − P0b )/2
Then how basically this is a trial error basis we calculate so to avoid that trial error
calculations some approximation way basically you can calculate your average raise to
maturity. Or we can say that the approximate yield how much you can receive it. Unless the
cash flows are constant, there is no algebraic solution to finding the YTM. So the YTM is
found basically through one iterative process like trial and error method.
But the ARTM, the average rate to maturity formula can give you some approximate yield to
maturity of that particular bond. And what is this particular formula, ARTM formula? We
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generally take the coupons the annual coupons what we got that is your what we call it the 90
Rupees in our case. Face value of the bond that is the price of the bond that is the term to
maturity this is again same face value the priceof the bond divided by 2.
(Refer Slide Time: 18:28)
90 + [(1000 − 938.55)/10]
(1000 + 938.55)/2
45+[(1000−937.69)/20]
(1000+937.69)/2
Annualized ARTM = 0.049663 ∗ 2 = 0.099325
So now in our example if you see let us check it using this ARTM what is the coupon
payment annual you are getting the annual coupon payments is basically you are getting 90. F
= 1000 your price of the bond is equal to your 938.55 and your M or term to maturity is 10
years. So if you put that particular formula here that is 90 which is basically our coupon then
we have + your 1000 - your 938.55. So that is basically your difference between the face
value and this market price divided by M. M means it is the 10. Then that will be divided by
your 1000 + your 938.55 by 2. Approximately if you calculate this you will be getting 0.0992
that means 9.92 %. In our case generally we got how much 10%. So 9.92 % is more or less
close to that. So use your ARTM formula, find out the approximate yield and after that you
can use some trial error basis exact YTM you can calculate from that particular figure.
92
So if you go by your semi annual payment basis also you can calculate semi-annual coupon
payment is 45 +1000-937.69 by 20. If you recall then divide by your 1000 + 937.69 divided
by 2 that will give you a value of 0.049663. So if you go for annualisation of that, the
annualized ARTM will give you 0.049663 into 2 that will give you 0.099325 which is
9.9325% which is more or less close to this 10% what basically we are calculating from our
formula. So this is the way the ARTM formula is used to find out the approximate yield to
maturity of that particular bond. So this is basically one of the way we can calculate the YT.
(Refer Slide Time: 21:44)
Then we come to the concept of bond equivalent yield. So, generally, the rates on the bond
are often quoted as a simple annual rate with no compounding. For bonds with semi-annual
coupon payments, the rate can be found by solving for the YTM and a bond using the 6
months cash flowsthen you can multiply that particular rate by 2 just now whatever we have
done so that is called generally the bond equivalent yield.
So just now, whenever, we multiply that 2, because the coupons are paired semi-annually. So
that concept basically is called the bond equivalent yield. If the 10 year 9% bond with semi
annual coupon payments and trading at 937.69, it would have a YTM for a 6month period of
5% and bond equivalent yield of 10% but the effective rate is 10.25%.That already we have
calculated that is 1.05 square -1 that already we have seen that is 10.25%.
So the bonds with different payment frequencies often have their rates expressed in terms of
the bond equivalent yield, by that their rates can be compared with each other on a common
basis that is the use of that bond equivalent yield. So first of all, we calculate this bond
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equivalent yield even if the coupons are paid in the different frequency to compare it and on a
common basis. By that we can choose that how much yield we are generating from the
different type of the bonds so that is called the cons bond equivalent yield
(Refer Slide Time: 23:38)
Then we have yield to call already I told you what do you mean by this callable bonds, many
bonds generally have the call features .They have a call option if there is a call option
available then it will allow the issuer to buy back that particular bond at a specific price
which is known as the call price. And that call price is mentioned in the beginning whenever
the bond was issued. That is why, when the bond may be called and at what price it will be
called that will be specified inthe indenture or the agreement from the beginning part.
So for some issues, the call price is the same but generally the call price may also vary. For
other callable bonds, the call price may depend on the time when the bond is called. Because
you see the maturity period is 10 years the bond is called back after 5 years, minimum 5 years
is mentioned, after 5 years at any time the bond can be called back or the issuer can buy back
that particular bond. So in that case they will mention a price so if the bond will be called
back after 6 years there is a different price, if the bond will be called back after 8 years there
is a different price some cases. But some cases at any time the bond will be called back they
will mention the same price whether it will be called back after 5 years or after 7 years, the
price will be same; the call price will be same but there may be a schedule on the basis of the
maturity period, on the basis of the time, may be the price will may also vary.
(Refer Slide Time: 25:26)
94
N
P0B = ∑
t=1
CFt
CP
+
(1 + YTC)t
(1 + YTC)N
So in this case how basically we calculate the yield to call? And the any yield which is related
to that particular bond that is basically called the yield to call. So this is the rate where we can
get by assuming that the bond basically is called back on an identified date, which is called
the call date. So then we go for our original formula for the price of the bond that is P B =
your t = 1 to N CF t divided by 1+YTC to the power t here we are using YTC.
And instead of face value we are using here basically the call price and what is the N year?
The N is not the maturity period, the N is the number of periods to the identified call date so
that is the difference basically what you can get it from here. So let us see that how this thing
is basically varying.
(Refer Slide Time: 26:34)
95
10
937.69 = ∑
t=1
45
1100
+
(1 + YTC)t
(1 + YTC)10
Simple annualized YTC = 0.0610575 ∗ 2 = 0.122115 = 12.2115%
So in the same example, we have maturity period is 10 years, coupon is 9% then let the call
price is mentioned. The call price is let mentioned 1100 and the bond can be fast callable in 5
years and let the interest is paid semi-annually. So in this case if you want to use this formula
that 937.69 is equal to t = 1 to 1 How many periods are remaining? That is the 5 years that
means 6 months that is you have the 10.
Then 45 divided by 1 YTC to the power t + here in the place of 1000 we will be using 1100
which is the call price 1+YTC to the power 10. If you solve it you will get 0.610575. Now if
you make it simple analyzed YTC, then how much you will get? 2 into 0.0610575 that is
0.122115 that means 12.2115% so if the bond will be callable in 5 years then the actual yield
from that particular bond will be 12.2115%. But the yield to maturity of that particular bond
was 10% that we have seen because there is a call feature generally yield is relatively high.
(Refer Slide Time: 28:50)
96
Same thing can be applicable for the yield to put also so any bond can be putable. If there is a
put feature that means, it allows the bond holder the right to sell the bond back to the issuer at
a specified price that is called the put price. Again like callable bonds, the putable bonds can
have a constant put price or a put schedule depending upon the time period and whenever the
bond is putable the convention is to always calculate the yield to put.
(Refer Slide Time: 29:31)
N
P0B = ∑
t=1
CFt
PP
+
(1 + YTP)t
(1 + YTP)N
So in this case, we can use this same formula but one difference is basically in the place of
the call price we are using the put price in the price of YTC we are using the YTP. And the
97
number of period to the identifier put date that is basically your N. So we equate the present
value of the bonds cash flow to the assumed put date plus the present value of the put price at
the put date which should be equal to the price of the bond. So if you solve this equation then
you can find out your yield to put.
(Refer Slide Time: 30:12)
10
937.69 = ∑
t=1
45
950
+
(1 + YTP)t
(1 + YTP)10
2 ∗ 0.0491387 = 0.09807741
Let us see that how this basically works, let the put price is 950. So now it is semi annual
coupon example we have taken 937.69 and let the put it is putable in 5 years that is the same
thing we are taking then 5 into 2 that is 10. This is your 45 divided by 1+YTP to the power t
+ your 950 divided by 1+YT p to the power 10. So if you solve it you will find 0.0490387
make it if you make it analyze then 2 into this 0.0490387 that will be giving you 0.09807741.
So that is basically giving you this yield to put. So, now, the yield to put is lesser than the
yield to maturity .That what we have taken because into maturity was 10 % now your yield to
put is 9.8077%. So, that is the concept what we use for yield to put.
(Refer Slide Time: 31:16)
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So what is yield to worst? The convention is for an investor to calculate the YTM, the yields
to all possible call dates and the yields to all possible put dates. The minimum of all the
yields is generally called the yield to worst. The minimum of all the yields, what basically we
get it from the different possible call dates? If the prices are different then we call it the yield
to worst. So that basically is a concept youhave to keep in the mind.
(Refer Slide Time: 32:09)
So what basically we have discussed that the current yield of a bond is nothing but the annual
coupon to closing price. YTM is basically always equates the price of the bond to the present
value of the cash flow. Yield to call is the rate which is obtained by assuming that the bond is
called on an identified call date. And yield to put also the particular rate which equates the
present value of the bonds cash flow to the assumed put date plus the present value of the put
price at the put date equal to the bond price.
99
So these are the things what basically we have discussed and we may have a put schedule we
might have a call schedule also. Accordingly also the each, we can calculate our yield with
respect to that particular put option or yield with respect to that call option.
(Refer Slide Time: 33:04)
So, this is the reference what basically you can follow for the detailed analysis on this.
Thank you.
100
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 02
Lecture No # 07
Bond Returns - II
Welcome back, we have discussed certain return concepts in the previous class particularly
the bond return concepts. We discussed about the current yield, we discussed about the yield
to maturity then we have the yield to call and we have discussed also yield to put. We will
continue with our discussion.
(Refer Slide Time: 00:40)
There are some other concepts what always we use in the bond analysis one is called the cash
flow yield. And we have yield to maturity on the zero coupon bonds, we have the bond
portfolio yield if you are holding a portfolio of the bonds then how the yield of that particular
portfolio can be calculated. Then we have a concept of the total return. So these are the other
concepts what we will be discussing today.
(Refer Slide Time: 01:08)
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So there are certain keywords you will be learning in this particular session: we have the
amortized securities, the concept of the spot rate and the concept of the realized return. So
these are the different keywords what basically we will be learning in this particular
discussion.
(Refer Slide Time: 01:31)
So let us see, what is cash flow yield? You know that there are different kind of fixed income
securities always we see. So there are certain fixed income securities whose cash flow
generally includes the scheduled principle payments prior to the maturity. Those kinds of
securities are called the amortized securities. Generally, those cash flows are always include
the principal and the interest payments. And which are those amortized securities? The
amortized securities include generally the securitized securities like your mortgage backed
securities or other assets backed securities. In the beginning of the session, I have discussed
102
with you, there are certain kind of fixed income securities which are recently quite emerging
these are called the mortgage backed securities and the asset backed securities. So, whenever,
we calculate the yield of that particular kind of securities, generally, these yields are basically
calculated with respect to the cash flows which are involved with respect to that kind of
instruments.
And why generally we do that because many of the securities of the prepayment options.
Because, let, one security is based upon the housing loans and may be this particular person
who has taken the loan, they can go for the prepayments. So, whenever this kind of thing
arises this basically makes the cash flow which consists of the interest schedule interest
principal then we have some prepaid principal.
So therefore, the cash flows of this kind of securities are not fixed over the life of the
securities. It is always constantly changing on the basis of nature of cash flow what we are
receiving from the original instrument on which these particular securities are basically
designed or constructed. So therefore, that particular concept is quite important with respect
to the yield with respect to this mortgage backed securities cash flow. So that is called the
cash flow yield.
(Refer Slide Time: 04:04)
So let us see, what basically how it basically this works? If for example we assume that the
cash flows are constant then the yield can be determined by solving for the yield that equates
the present value of the cash flow to the current price that already you know. Your price of a
particular security is nothing but it has to be discounted with respect to this yield to maturity.
103
Then if there is a series of cash flow then it is summation t = 1 to m CF by 1 + YTM to the
power t.
PB0 =
𝑁
∑
𝑡=1
𝐶𝐹
(1 + 𝑌𝑇𝑀)𝑡
So the cash flow may be your coupon as well as the principal in the generic bond but here in
this case we have the different type of cash flow. The cash flow with respect to the interest
payment, cash flow with respect to the principle repayments, then with prepayments, then
you might have a cash flow with respect to the payment of the principal. So, all kinds of
things are involved with respect to that.
So in this case, how basically the cash flow yield can be calculated? I will give a simplistic
case then further we can go into the complexity of that. Because, the process of securitization,
itself is a complex process. So we will discuss how basically this process works but today I
will just certain brief idea about that.
(Refer Slide Time: 05:42)
Let there is a fully amortized bond which is available then here let the maturity is 10 years.
And interest against this particular kind of cash flow you are getting 8% and we are doing it
on the semi-annual basis. Let, the principle is let 100 Rupees. So my question is that find out
the cash flow. So if you get your cash flow from there you can calculate your yield. So now
104
100 Rupees which is basically your principle so if you go by our formula then this is nothing
but the coupon into your 1 – generic bond wise but in case of we are using the cash flow this
is 1 – 1 by 1 + R to the power N divided by R.
1−
100 = CF[
CF =
1
(1+𝑅)𝑁
𝑅
100
]
100
=
1
1−
(1+𝑅)𝑁
[
]
𝑅
[
1
1−
(1+0.08/2)20
0.08/2
= 7.358175
]
Hence, cash flow is 7.388175.
So here that means if you take cash flow this side then your cash flow will be 100 divided by
this 1 – 1 by 1 + R to the power N divided by R. Now we have 100 which is available then 1
– 1 by 1 + R means it is 0.08 coupon is paid basically semi annual bases the interest is paid.
When 0.08 by 2 to the power 20 maturity period is 10 means 6 months rate is number of
period will be 20 divided by 0.08 by 2.
So if you calculate you will get this 7.358175 which are basically you cash flow because
there is no coupon concept. So your cash flow is coming 7.358175. Let, you assume, this
particular bond is priced in the market with a price of 95 in that particular point of time. Then
now your job is to find out the yield from that particular bond. Particularly, we are talking
about the semi-annual because the interest is paid semi-annually.
So in this case what basically we have to do we have to equate it with the price of the bond
that is 95 your t = 1 to 20 then 7.358175 divided by your 1 + YTM to the power t. So if you
solve it then you will find let 0.04586656.
Bond is priced in the market at Rs. 95.
𝐶𝐹
Price = ∑𝑁
𝑡=1 (1+𝑌𝑇𝑀)𝑡
7.358175
 95 = ∑20
𝑡=1 (1+𝑌𝑇𝑀)𝑡
 YTM = 0.04586656
So, semi-annual yield of the bond is 0.0458665.
105
Hence, that is your semi-annual yield what you get it from that particular bond. Because, in
the beginning case, for the generic bond, we know the coupon and accordingly, that coupon is
directly used for calculation of the particular bond price. And then we try to equate it to the
price of the bond at what interest rate which will equate the price of the bond with the present
value of the cash flow. On that basis, we can call it YTM. But, in this case, what you are
doing? First, we have to find out the cash flow involved with respect to that particular bond
then after that you have to find out the yield. So if you make it annualized then you just
multiply here 0.04586656 that will give you 0.091733, 9.1733%.
Annualised or Cash flow yield= 2 × 0.04586656 = 0.091733 or 9.1733%.
So, this is basically called the cash flow yield and that is generally related to the mortgage
back securities. But, the complexity of the mortgage securities or asset backed securities and
the pricing of that all these things will be discussing in the future sessions.
(Refer Slide Time: 10:54)
So, then we also discussed about the zero coupon bond; the valuation of the zero coupon
bond that already we have discussed. And, one thing you remember, the rate on zero coupon
bond generally called is spot rate. Then, the solution if you want to find out the YTM of the
zero coupon bond; And zero coupon bonds also called pure discount bond that already we
have discussed because there is no coupon involved in that. The bond is issued at a particular
price at a discount and it will be redeemed at par so that is why it is called the pure discount
bond. So, if you find out this particular yield of a zero coupon bond how basically we can go
ahead? Already, you know this part, your price of a zero coupon bond is equal to your face
106
value of the bond or the par value of the zero coupon bond which is the maturity value
divided by 1 + YTM to the power M.
Then your 1 + YTM to the power M is basically F by P 0 B, I have just taken to that side and
here it is coming this side this is F by P 0 B means is the price of the bond. Then you YTM
will be automatically F by P 0 to the power 1 by M -1. So automatically F by P0 to the power
1 by M – 1 that will give you the yield to maturity of that particular zero coupon bond, it is
the zero coupon bond shield. Right?
Algebraic solution to the YTM on ZCB will be:
𝐹
P0B = (1+𝑌𝑇𝑀)𝑀
 (1 + 𝑌𝑇𝑀)𝑀 =
𝑚
𝐹
𝑃𝐵
0
𝐹
 𝑌𝑇𝑀 = √𝑃𝐵 – 1
0
𝐹 1/𝑀
 YTM = [𝑃𝐵]
–1
0
Whatever you will get, that is known as zero coupon bond shield.
So, in this case,
Let, how basically; because there is no such coupon involved with respect to this in that case.
So if you want to calculate the YTM on a zero coupon bond then how basically we can
calculate this?
(Refer Slide Time: 13:00)
107
Let us take the maturity value of the face value of a zero coupon bond is 1000. Let the
maturity period is 3 years, let this bond is trading the zero coupon bond is trading at a price of
Rupees 800. That means that is the initial price at which the bond is basically purchased, then
what is your YTM? Already, as per the formula, this is very simple 1000 divided by 800 to
the power your 1 by 3 – 1 that will be 0.0772.
𝐹 1/𝑀
YTM = [𝑃𝐵]
–1
0
100 1/3
 YTM = [800]
– 1 = 0.0772
If you assuming semi-annual compound then what basically will happen? Then the simple
annual or what we call it a bond equivalent rate, what would be? That will be what? That will
be automatically your 1000 divided by 800 to the power 1 by 6 – 1 that will 0.03789 then
your bond equivalent yield will be how much? 2 into this that is 0.075782.
If you assuming semi-annual compound then bound equivalent rate can be calculated as:
𝐹 1/𝑀
YTM = [𝑃𝐵]
–1
0
1000 1/6
 YTM = [ 800 ]
– 1 = 0.03789
Then annual bond equivalent rate will be 2×0.03789 = 0.075782.
But, there is also possibility the maturity period of that particular bond is less than 1 year. We
have discussed that part before we have used this concept of day count conventions. Let, I
will another examples. Let the par value of a zero coupon bond is 100 Rupees then at the end
of 182 days the bond is trading at 96 Rupees. And use this 1 year = 365 days then what will
be the YTM of that bond? Obviously it will be 100 divided by 96 to the power 365 by 182 –
1 that will give you 0.0853.
1000 365/182
YTM of the bond will be = [
800
]
-- 1 = 0.0853.
So, depending upon the takeout convention if the particular bond maturity period is less than
1 year on that basis we can calculate that what will be the yield of that particular bond.
(Refer Slide Time: 16:34)
108
So, then let us see that sometime also the zero coupon bonds are continuously compounded.
So, what is the solution? Algebraic solution to the YTM on a zero coupon bond with
continuous compounding, this part already you know.
Your F is equal to this whenever we talk about the final value of that particular bond that
generally we call it that then P 0 B e to the power Rt = F. Then e to the power Rt because our
job is to find out R then e to the power Rt = F by P 0 then you take the log in both sides you
take ln both sides.
Then ln e to the power Rt = ln F by P 0 now ln e to the power Rt = Rt = ln F by P 0. Then Rt
if you find then R is equal to what? R = your ln F by P 0 b divided by t. Directly, you can
calculate this yield to maturity of that particular 0 coupon bond.
Algebraic solution to the YTM on a zero coupon bond with continuous compounding:
𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡, 𝑃0𝑏 𝑒 𝑅𝑡 = F
𝐹
 𝑒 𝑅𝑡 = 𝑃𝑏
0
𝐹
 𝑅𝑡 = ln[ 𝑏]
𝑃0
𝐹
 𝑅 = ln[𝑃𝑏] ÷ t
0
109
(Refer Slide Time: 17:56)
So, then in our example whatever we have taken let the bond is trading at the price of 96.
You par value is already we have taken that is 100 Rupees, then let maturity we have taken
that is 182 days. Then what is your R? R = your ln 100 by 96 divided by 182 by 365 because
we are assuming 1 year = 365 days.
R=
100
]
96
182
365
𝑙𝑛[
On that basis you can calculate your R you can calculate the R from this so this is what
basically you can use or you can calculate whenever we are assuming that this particular zero
coupon bond is continuously compounded.
So, depending upon this what we call it the compounding frequency, the yield also can be
changed. So the concept of yield can be calculated in this way so that is about the yield from
the zero coupon bond.
(Refer Slide Time: 19:24)
110
Then let us see about the portfolio yield. Let, somebody is holding a portfolio bond instead of
holding a single bond somebody is holding a portfolio of the bonds. Then how the portfolio
of the bonds yield can be calculated? Remember, the bond portfolio yield is not the weighted
average of the yield to the maturity of the bonds comprising this portfolio what generally
people do? People calculate the yield of each bond then accordingly they calculate the
weights that how much fund they have invested against that particular bond. And giving the
weights, they can calculate the weighted average but that is not basically the yield of we can
say that bond portfolio yield. What is the bond portfolio yield? Then bond portfolio yield
basically is a how basically we can solve it?
You can solve it by making the present value of the portfolio cash flow equal to the market
value of the portfolio. So you have to find out what is the market value of the particular
portfolio. And after that you can see that what is the cash flow you are going to receive
against that particular portfolio whatever we have? Then accordingly the bond portfolio yield
can be calculated.
(Refer Slide Time: 20:55)
111
So let us see what basically the concept is let there is a portfolio which has two bonds in the
portfolio. So what basically I told you how generally we do this? So let bond 1, I will give
you certain features let the maturity is 2 years, coupon is 5% and this bond is issued at par.
What does it mean? If the bond is issued at par means the coupon is equal to with the
discount rate. So which means the yield is also 5% or the discount rate is 5%. Let, the par
value is 100 Rupees.
Let, there is a bond 2, here the maturity is 3 years then your coupon is let 10%, let it is a
premium bond, the price is 107.87 then your par value = 100. And obviously your yield is
taken as 7% so what is the cash flow will be generated in 3 years from the bond. Cash flow
generated in 3 years, tell me? Obviously, in the first bond, how much we are getting? 5% that
means 5 first year from the bond 1. From the second bond, it is how much? It is, the coupon
is 10%, then it is 10. That is 15 Rupees in the end of the first year you are getting this. In the
second year how much you are getting first bond is matured then you will get 105 and from
here you getting 10 that is 115. And in the third year, this bond 1 is matured; this bond is
already matured then from this bond 2, how much you got 10 Rupees the coupon, 100 of the
par value that is the 110.
So, the cash flow from the two bonds in the first year is: 15, second your: 115, third year:
110. Then, what is the market value of the portfolio in that maturity? That is 100 + your
107.87 that will be your 207.87. So this 207.87 = 15 divided by 1 + y to the power 1 + you
115 1 + y to the power 2 + 110 divided by 1 + y to the power 3. If you solve this y, then you
will get your yield from that particular folio.
112
Portfolio yield will be calculated as:
15
115
110
207.87 = (1+𝑦)1 +(1+𝑦)2 + (1+𝑦)3
After solving this above equation, we get y= 0.062
So, that is basically called the portfolio. But, If you would have gone for weight then what
basically you would have got? The W1 into YTM 1 + W2 into YTM 2 that means your 100
divided by 207.87 into 0.05 + your 107.87 divided by 207.87 into 0.07 it would have been
given different result 0604 which is not the correct approach.
The portfolio yield can also be calculated frlm the weighted average approach:
Y = W1.YTM1 + W2.YTM2
100
107.87
= 0.05(207.87 ) + 0.07(207.87)
=0.064
So this approach is basically not the correct approach, so, we have to basically go by that
way. So, this is the way basically the bond portfolio yield can be calculated.
(Refer Slide Time: 25:47)
So, then the next one we have a concept of the total return. let me tell you, what the total
return is?. In the simplistic way, let you are holding a bond, bond’s maturity period is let a
particular year. But, you want to sell your bond before that. If you want to sell your bond
before that, how much extra return you can get it from there. So that is why it is called the
realized yield, it is actual yield what you are getting.
113
It is basically a measure of the yield which is obtained by assuming the cash flow to be
reinvested that means the coupon whatever you are getting receiving in between, that will be
reinvested to the investor’s horizon, that means up to what period you are holding the bond.
And, at the horizon of the bond, you are selling that bond and you are not holding the bond
up to that maturity.
So, that is why, the total return or realized return what you are getting from that particular
bond. First of all, we have to calculate or estimate the horizon value; total monetary return
and the bond price at the horizon. So, given the current price of value and the horizon value,
solving for the rate, we can calculate the total rate. Let us, it will be clearer for you whenever
I will take example. I will explain it through the example.
(Refer Slide Time: 27:17)
Let, as in investor, you buy by a bond: the maturity period is 7 years, coupon is let 11%.
Coupon payment is annual, right! and the discount rate or market interest rate is let 10%, then
the par value of the bond is 1000 Rupees. Assume that, immediately after you purchase the
bond, the market interest rate has gone down; it has decreased to 9.5%. You have already
bought the bond. The discount rate has gone down to 9.5% and it remains constant
throughout the maturity until the investors basically sell the bond.
Let the investors need the money or is going to sell the bond, after 5 years you will sell the
bond. And he is basically going to reinvest that coupon payment up to that particular 5 years
on the basis of the 9.5%. So what is the price of the bond? At what price he has bought the
114
bond whenever he has started investing? In that case, your coupon was 11% and market was
discount rate was 10%.
So if you go by our formula, that t = 1 to 7 the coupon is how much 110 each year, 1.1
discount rate was 10% to the power t + your 1000 divided by 1.1 to the power how much? 7
years. So if you calculate, it will give a value of 1048.864.
Price of bond = ∑7𝑡=1
100
(1.1)1
+
1000
(1.1)7
= 1048.684
Hence, the price of the bond is 1048.684.
This is your purchase price of the bond. This is the market price of the bond at which you
have basically bought the bond that time the interest rate was 10%.
So now you have sold the bond for 5 years. So, what is the selling price of the bond? How
will you calculate the selling price of the bond, still another 2 years are remaining, coupon is
110 you are getting? So, the remaining years, in the first year, you getting a coupon of 110
and you have to discount it with respect to 9.5% and this discounted at 9.5%. Second year
you will be getting you are getting back your 1000 + 110 that means 1110 you have also
discount it at a rate of 9.5%.
So now if you discount it if you calculate the present value of cash flow of 110 and 1110 then
the price at the end of the 5 year will be you get it is 1026.209. So, this is the price what you
got then after that you got your coupons, the coupon you got after end of the first year, how
much you got- 110 Rupees. That you have reinvested at a rate of 9.5% in the market then that
is 110 into for how many years for 4 years?
For 4 years you have interested it, 1.095 to the power 4, you got let 158.143. Next year, you
got coupon in the end of the second year you got a coupon, that you have interested it for 3
years to the power 3 that you got 144.423. Next year, you got another coupon that you have
reinvested it for 2 years that you got 131.893. And at the end of 2 years you got another one
095 that will give your 120.45 and in the fifth year coupon you got 110 Rupees.
So if you add it up you get your coupon as well as you have the coupon is reinvested at the
rate of 9.5% which is prevailed in the market. So in total you got your how much it is
basically your 664.908.
115
The value of the coupon reinvested can be calculated as:
=110(1.095)4 +110(1.095)3 +110(1.095)2+110(1.095)1+110
= 158.14 + 144.423 + 131.89 + 120.45 + 110
= 664.908
So, now you have sold the bond at a price of this from the coupon and the reinvestment of the
coupon you got this much. Then the total money what you got that is 1026.209 plus your
664.908.
Total money received (Horizon value) = 664.908 + 1026.209 = 1691.114.
So that will give you 1691.117, the total money what you got from this particular bond
investment what you have sold after 5 years. And what price you have purchased? You have
purchased it as 1048.684. So, now, if you calculate the total return, your total return will be
your horizon value. How much you got? That is 1691.117 divided by your 1048.684 it is the
ending value, this is the beginning value to the par your 1 by horizon period that is basically 5
years 5 – 1.
1691.117 1/5
Total return = [1048.684 ]
–1
=10.029%
That basically will give you 10.029% so that means if even if you are holding the bond of the
maturity you would have got the yield what basically is mentioned there and the par will be
that 1000 Rupees would have got. But if you have sold the bond at the end of the 5 years then
how much total return you have generated that is basically called the realized return, exactly
how much return you have realized from that particular investment. So that is called the
realized return.
So that thing is basically happening and that return will vary on the basis of the change in the
interest rate in the market and also your horizon period when you have sold that particular
bond.
(Refer Slide Time: 35:07)
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So, what we have discussed today? The cash flow for amortized securities are not fixed over
the life of the securities. The rate on a zero coupon bond generally called this spot rate and
the portfolio return or the portfolio yield generally is calculated by solving the rate that will
make the present value of portfolio cash flow equal to the market value of the portfolio. And
the total return is always equal to the calculated YTM if the cash flows can be reinvested or
the calculated YTM and bond can be sold at the calculated YTM. The total return always
calculated on the basis of 3 components: one is your selling price at the horizon rate and the
coupons whatever you have received and the reinvestment if the coupons or the return from
the reinvestments of the coupons. So these are the three things, three components have to be
added whenever the realised return is calculated.
(Refer Slide Time: 36:05)
This is the reference what basically you can follow. Thank you.
117
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 02
Lecture No # 08
Bond Returns-III
Good morning. Welcome back, in the previous class, we discussed about certain concepts
which are related to the calculation of the bond returns or the bond yield. There we have
discussed about particularly the current yield, then we have yield to maturity then as well as
we also discussed about the yield of the zero coupon bond. Then we discussed about the
concept of the total return.
But, here what basically we have taken this example in the context of the yield to maturity is
constant across the particular period. But, in the true sense, if you observe that the interest
rates always varies in the market and whenever the interest rate will change automatically the
yield of that particular bond also will change. So let us see that if these particular interest
rates in the market will change then how the total return of the particular bond is calculated.
(Refer Slide Time: 01:23)
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So, therefore, in today's context, we will discuss about the market risk and total return. So
here market risk means fluctuations of the interest rate. So if there is a change in the interest
rate how the total returns of the particular bond is going to be changed? And as well as
another concept we will discuss about the horizon analysis. So here we will discuss about the
different type of bonds which are available in the market and the investor will choose what
kind of bond to minimize this particular interest rate risk what basically they are always
exposed to.
(Refer Slide Time: 02:04)
And after this, you will be knowing about the 2 types of risk and how this particular risk is
playing a major role in terms of the bond return? That is basically your price risk and the
reinvestment risk which are basically the components of the interest risk. If there is a change
in interest rate then the bond price will be changed and as well as the reinvestment of the
coupon that value is also going to be changed. So we will see that how exactly the price risk
is and if there is a fluctuation of the interest rate then what kind of bond is suitable for
investor for the investment.
119
(Refer Slide Time: 02:45)
So here already what in the beginning I told you, we will be discussing about the total return
with different reinvestment rate. Because, whenever we calculate the total return what exactly
the total return is? Total return is generally always calculated on the basis of the yield to
maturity and here we have assumed the cash flows are always reinvested at the calculated
YTM and the bond also is sold at the calculated YTM.
So whatever yield to maturity is there if in between before the maturity the investor is going
to sell the particular bond, they can sell the bond on the basis of the yield to maturity already
what yield to maturity was there whenever they have bought the bond. So that is the basically
way we calculate the total return. So, the total return basically what it basically then it
explains? It explains that the YTM the yield to majority basically captures the return from the
coupons, the capital gains and the reinvestment of the cash flows at the calculated YTM.
Because there are 3 components we have seen whenever we calculate the bond return, total
return of the bond, we have seen there are 3 components. One is your coupons whatever you
are getting then you have the capital gain. The capital gain is basically will be calculated on
the basis of what price you have bought the bond and what price you have sold the bond.
Accordingly, the capital gain can be calculated.
Then whatever coupons you are receiving over the period of time or in the periodical basis,
whatever amount of interest on interest or how you are going to reinvest that particular
coupon in the market that will give you the reinvestment of the cash flows. And all these
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things are calculated on the basis of the calculated YTM. But the interest rates do change
over the time. So, in that context, the total return is not exactly equal to the calculated YTM.
Whatever yield to maturity you are receiving or you have with the bond that is not exactly
equal to yield to maturity of the particular bond whenever you have started investing in that
particular bond. So, if the coupons are expected to be reinvested at the different rates of the
bond or the bond sold at a different YTM: whenever. Let your bond maturity period is 10
years and you have sold the bond at the end of 7 years. Then in that particular year, the yield
to maturity of the bond if you observe that basically will depend on the market interest rate.
So, therefore, we cannot always use that YTM in the beginning whatever we have considered
or what price we have basically bought the bond, that concept cannot be used for calculation
of the total return. So, in that case if you see the total return may not be equal to the initial
yield to maturity of that particular bond whenever you have bought it.
So there may be a variation. The return may be more or return may be less but that basically
will depend upon that whether the 2 types of risk whatever we are talking about one is your
price risk and the reinvestment, which particular effect: the price effect is dominating the
reinvestment effect or the reinvestment effect is dominating price effect. That will decide that
the whether we should get the more return or we will get the less return.
So that is basically the general concept always we should keep in the mind. So, let us take
one example, it will be more clearer for you that what basically we are trying to explain in the
context of the total return with the different reinvestment rate.
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(Refer Slide Time: 07:04)
So, let us take this particular thing with a simple case then we can move towards the other
thing. So let there is a bond, let this maturity period of the bond is 4 years, and let the coupon
rate is 10% and coupon is paid annually. For simplicity, let we assume, bond is basically
selling at par. What does it mean? It implies that your yield to maturity is also 10%. If the
bond is selling at par that means the yield to maturity is also 10% because for if the particular
bond which is traded at par that means the coupon is equal to the into maturity.
Let the par value of the bond is 1000 rupees, right! so let the horizon period of the investor
horizon period is 3 years that means the investor is going to sell the bond after 3 years but
actual bonds maturity period is 4 years. So here what does it mean? So if the horizon period
is 3 years and the coupon is basically paid annually that means and the interest rate does not
change. The interest rate will not change. For simplicity, we are taking that assumption then
what this investor will do? They will reinvest the coupon at 10% because yield to maturity is
not changed. So, they will reinvest the coupon at the rate of 10% and also selling the bond at
the end of the 3 years at the rate of 10%. So here everything is basically at the 10%. So here
the question is that how basically if you are selling the bond at the end of the 3 years then
what is the selling price of the bond?
How much cash flow is remaining? You have 100 rupees coupon, only 1 year is remaining
and plus 1000 what basically will be getting in the on the basis of the par value divided by
your only 1 year is left then 1.1 to the power 1.
Selling price=
100+1000
1.11
=1000
Hence, the selling price of bond is 1000.
122
So you will be getting exactly 1000 rupees. Why you are getting? Because all the rates we
have kept it same.
So now what is the coupon value at the horizon period? The coupon value at horizon means
after 3 years you get the coupon how many coupons you will be receiving 100 rupees at end
of the 1 year, another 100 rupees end of the 2 years, another 100 rupees for the end of the 3
years. And that 100 rupees what you get at end of the 1 year that will be again reinvested for
2 years and for the second year it will be reinvested for one year and third year coupon
basically is not going to be reinvested, immediately, you have just received the coupon.
So if directly you want to calculate, what is the value you will be getting? That is nothing but
the coupon times the future value of the annuity, the coupon times the future value of the
annuity that formula basically we can use. So, here 100 rupees is the coupon then what is the
future value of annuity? That is your 1.1 to the power 3-1 divided by 0.1 that means at a time
I am calculating the total amount of coupon and as well as the reinvestment income whatever
you are getting from that particular coupon.
(1.1)3
1
Coupon value at time horizon: 100[ 0.1 − 0.1]= 331
So that will be: you are getting 331.Where basically you get this formula? You see, end of the
first year, you will get 100 rupees and that 100 rupees will be reinvested. Then 1.1 to the
power 2 that will be 121 and in the second year basically you get 100 rupees then that will be
1.1 again you have reinvested for 1 year. That you will be getting 110 rupees and in the third
year you will be getting 100 rupees so that if you add all these three that will give you 331.
The value of the coupon if reinvested at ytm can be calculated as:
First year, the coupon is invested for two years. So the value of reinvested coupon at the end
of three years will be: 100(1.1)2 =121
Second year, the coupon is invested for two years. So the value of reinvested coupon at the
end of three years will be: 100(1.1)1 =110
Third, the coupon received is not invested because it is received at the end of third year. So
the value of this coupon is 100.
When we add the three values, we get 331.
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That means your coupon: total coupon you have received that is 300 and your reinvestment
income reinvestment income that you are getting 31 rupees. And you have sold the bond the
selling price of the bond is 1000 rupees. And now the total value of the coupon in the horizon
period that is 331 rupees your reinvestment income is 31 and coupon is 300. So now your
total return will be what will be your total return?
Already you know that then your 1000+331 divided by 1000, this 1000 rupees basically your
buying price of the bond. Whenever you have started investing because par value of the bond
is 1000 and the bond is selling at par. So you have bought the bond at a price of 1000 to the
power 1 by 3-1.
1000+331 1/3
Total return = [
1000
]
- 1 = 10%
That will be giving you 10 %. Why we got 10% because we have kept the YTM unchanged.
We have also kept that the coupon is again reinvested at that particular rate that means almost
all the rates are same. So, because of this, your total return will be equal to the YTM the
initial YTM at which the bond was basically issued. Right! So this is what basically for
simplicity we have taken these assumptions. Now, let us see, is there any change if the
coupon is paid semi-annually? Let the coupon is paid semi-annually, if the coupon is paid
semi-annually then is there any change in terms of the formula?
(Refer Slide Time: 14:55)
So let us see, for example, the coupon is basically paid semi-annually then what will be the
initial price of the bond? Now with the same example, we have taken coupon is paid semi-
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annually right. If coupon is paid semi-annually then what is the initial price of the bond? How
will you calculate this? Already you know that that is your t = 1 to 6 50 divided by your 1.05,
10% of the coupon now divided by 2 to the power t + your 1000 divided by 1.05 to the power
6.
50
1000
Initial Price= ∑6𝑡=1 (1.05)1 + (1.05)6 = 1000.
So this will also give you exactly your 1000 rupees. Now, what is the coupon value? I mean
this price you are basically selling the bond then what is the coupon value? The same formula
you can use 50 into 1.05 to the power 6 - 1 divided by 0.05, that will give you 340.1.
(1.05)6−1
Coupon value = 50 [
0.05
] = 340.1
It will be little bit higher; the reason is basically the coupon is reinvested semi-annually. So
now your investment income is how much?
Your reinvestment income will be 40.1 and your coupon is 300 that is fixed; coupon is 300
right. So now what is your HD price that basically will get 50 divided by your 1.05 + your in
the end how much you are going to get? 100 rupees plus your coupon 1050 divided by 1.05
to the power 2 times it will be reinvested because 6 month species that will give you 1000
rupees.
50
1050
HD price= 1.05 + 1.05 =1000.
So now the total value what basically you are getting? 1000+340.1, that will be 1340.1. Then
your total return will 1340.1 divided by 1000 to the power 1 by 6 - 1, it will be also giving
you 0.05.
1340.1 1/6
Total return = [ 1000 ]
- 1 = 0.05.
Then simple annual rate = 2×0.05=0.10 or 10%.
Effective annual rate = (1.05)2 − 1= 0.1025 or 10.25%
So your simple annual rate will be 10% simple annual rate will be 0.05 into 2 that is 0.1 that
means 10%. And if you go for the effective annual rate will be how much that will basically
give you 1.05 power 2 - 1 that is 0.1025 that is 10.25%. But effective annual rate will be
10.25% but exactly you will be getting your10% only in this case. So therefore what basically
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we have seen that whenever we are calculating this particular total return assuming that YTM
remains constant across the period then your total return is equal to your YTM.
(Refer Slide Time: 19:27)
But in the true sense, there is uncertainty that the actual return of a bond which may be
deviated from the expected return because of change in the interest rate which is basically
known as the market risk. So the change in interest rates has 2 effects that already told you
one is your price risk and the reinvestment risk. Let us see in the same example what will
happen if the interest rate will change?
126
(Refer Slide Time: 20:04)
So if the interest rate will change, let the interest rate has gone up you have bought the bond.
But after that the interest rate has increased to rate 12%, if the interest rate has increased to
12%. After you purchase the bond and let the coupon is paid semi-annually. Whenever you
have bought the bond that time the yield was 10% but now after you bought the bond the
price or the interest rate has increased to 12%, then what will be the coupon value at the
horizon period is 3 years?
That we have taken then it will be 50 into 1.05 to the power 6 right it will be 1.05 or it will be
1.06 because now the interest rate has increased to 12% divided by 2 that is 6% then 1.06 to
the power 6-1 divided by 0.06 so that will give you 348.77.
(1.06)6−1
Coupon value at HD = 50 [
0.06
] = 348.77.
So the coupon value is 348.77.
Reinvestment income = 348.77 – 300 = 48.77
50
1050
HD price = 1.06 + 1.06 = 981.67.
The reinvestment income and HD price are 48.77 and 981.67 respectively.
So now your reinvestment income has become 348.77-300 that is 48.77. Now your what is
your HD price? At the end of the 3 hours or what price you are going to sell the bond?
That is 50 divided by 1.06+1050 divided by 1.06 to the power 2 square of this now this will
give you 981.67 or automatically the price will decline because the interest rate has gone up.
So the price basically will be getting whenever you sell the bond that is 981.67.
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Then total value how much is 981.67 + 348.77 that will give you 1330.44. So, now what is
your total return.
Total return will be 1330.44 divided by 1000 at what price you have bought the bond to the
power 1 by 6-1 that will give you 0.048735.
1330.44 1/6
Total return= [ 1000 ]
− 1 =0.048735.
Simple annualised return = 2×0.048735 = 0.097471 or 9.74%
So if you make it analyzed return that; will be 2 into 0.048735 that will give you
0.097471.So that means 9 up 9.74% approximately. So now what you have observed here the
yield to maturity in the beginning was 10% but whenever the interest rate has gone up from
10 to 12%, the total return if you are selling the bond at the end of the 3 years the total return
has become 9.74%. What does it mean? It means that you get some better return in terms of
reinvestment of the coupons that value has increased; you got 48.77 which was more than the
previous value. But at the same time the price of the bond has come down whenever price of
the bond has gone down. Now if what you can judge? You can judge that price effect is more
than the reinvestment effect. The return what you got in terms of the investment of the
coupons that is not sufficient enough to compensate the loss whatever you have made in
terms of the change in the prices due to increase interest rate. So therefore, we have to
understand that what exactly the price effect and the reinvestment effect are.
(Refer Slide Time: 25:14)
128
So whenever you talk about this price risk already what I told you, the interest rate changes
basically will affect the price of a bond and generally this is called as the price risk. So here,
the investor’s horizon period which was 3years is different from the bonds maturity which
was 4 years. So, then what will happen that the investor will uncertain about the price what
he or she will receive from the selling the bond because the horizon period is less than the
maturity period.
Or what price he or she has to pay for a new bond, if their horizon period is greater than the
maturity period because they need some money maybe after 5 years. So they can go for
another bond, so in that case, they are very much uncertain that what kind of yield or what
kind of return they are going to get if the horizon period and the maturity period is not
matching. So already you know that the price of a bond is inversely related to the interest rate
and is more prices responsive to a change in interest rate if it has a longer term to maturity
and its coupon rates are less. Already all we have discussed that part that the price
responsiveness of the less coupon bond and long maturity bonds are more than the high
coupon and the shorter maturity bonds. So the interest rate change then the price effect on the
total return will be negative. That means the lower rates increase the horizon price and
therefore the total return and with the effect being greater for bonds with greater terms of
maturity and the lower coupon rate that actually you have to keep in mind.
(Refer Slide Time: 27:16)
129
Then we have the reinvestment risk. So, in the reinvestment risk, the interest rate change
basically affect the return; the investor expects from the reinvesting the coupon that is called
the reinvestment risk. If the investor buys a coupon bond, he or she is automatically subject to
the market risk. If the interest rate changes then the interest effect on the total return will be
direct. That means greater rates increase the reinvestment in return and the total return may
be increased because of that.
With the effect being greater for the bonds with greater coupon rates, the higher the coupons
then automatically their investment return will be more that is called the reinvestment risk.
(Refer Slide Time: 28:03)
So one way to evaluate the market risk for a bond is to estimate the bonds total return in a
given interest rate scenario that is called the horizon analysis. So in this case, what these
investors do? Investors basically go for analysing one or more bonds or bond portfolios and
they try to see that or try to compare the different bonds based on their planning or planned
horizon period and the expectations what they have concerning about the market. Then
accordingly they decide that which bond is better for them.
(Refer Slide Time: 28:53)
130
Let me give you one small example, in this case, let there is a bond, the par value is 100
rupees, horizon period is let 2 years and semi-annual coupon payments and yield are same in
all maturities. And coupons are different and maturities are different. Let we have considered
the yield is 7.5% and it has declined to 5% and it has increased to 10%. There are 3 scenarios
that are called the scenario analysis so I will give you some figure then you can find out.
Scenario analysis of different bonds:
Bond
A
B
C
D
Annual
coupon
rate
10%
7.5%
10%
5%
maturity
3years
5years
10years
15years
Bond
price at
7.5%
106.61
100
117.37
77.71
Total returns
At 5%
At 7.5%
At 10%
7.23
8.59
10.85
13.8
7.5
7.5
7.5
7.5
7.7
6.48
4.47
2.01
Let bond your annual coupon rate then you have the maturity then you have the bond price at
7.5% then you have the total return. Already we have calculated this. Let at 5% at 757.5%,
then you have the R 10%. Let bond A, annual coupon rate is 10%, maturity period is 3 years
then the bond price at 7.5% is 106.61, total return whenever we calculate at 5% we got 7.23
and at 7.5% automatic it will be same at 10% it is 7.77.
131
For bond B, 7.5% that maturity period is 5 years we got the price is 100 it is 8.59% 8.59%,
7.5% then you have 6.46,6.48. Then bond C, it is 10% that let it is 10 years price become
117.37 it is 10.85 this is 7.5, 4.47. Bond D: 5%. Let 15 years is the 10 maturity price becomes
77.71 it is 13.8 it is 7.5 and it is 2.01. So in this is a scenario of the different bonds with the
different coupon and different maturity but the yield we have kept is at 7 point bond price is
calculated on the basis of 7.5%.
So whenever the interest rate will change at 5% and it has gone down to 5%, 10%, this is the
total return what already we have calculated. So in these case what basically observations we
got?
(Refer Slide Time: 32:35)
We got the bond A has the smallest deviation in total returns with the lowest rate being
7.23% and highest being 7.77%. So bond A’s total return also decreases when rates decrease
and increases when rates increase. What it gives? It basically suggests that the bonds interest
on interest effect dominates the price effect. But in contrast, if you see the longer term bond
with the last bond D, has the greatest market risk with the range of the total return varies from
2.01% to 13.8%.
And with its total return increasing when rates decrease and it is decreasing when rates
increase which implies that the price effect dominates his interest on the interest effect. So
here we have to understand, what this investor or one manager do? They go for a scenario
building analysis, they try to find out that which one is less exposed to market risk or for
which bond is beneficial for them for the investment. That is basically called the horizon
analysis.
132
(Refer Slide Time: 34:03)
So what we have discussed today? If the coupons are expected to be invested at different
rates, then the total return may not be equal to the YTM and the interest rate change has 2
effects one is price risk another investment risk. And the price risk is nothing but due to the
interest rate how the price of the bonds is going to be changed. And the reinvestment effect is
nothing but the due to the interest rate how the reinvestment return from the coupon is going
to be change.
(Refer Slide Time: 34:36)
So, this is the reference what you can go through for the detailed discussion on this.
Thank you.
133
Management of fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 02
Lecture No # 09
Bond Returns- IV
Welcome back. So we discussed about the horizon analysis as well as how the total return is
different from the YTM whenever the interest rate will change and the horizon period of
the investor is different from the maturity period that we have discussed in the previous
class.
(Refer Slide Time:00:37)
So, today in this particular session, we will be discussing about the concept of the spot
rate and how the price of the bond is calculated with respect to the spot rate. So if there is a
deviation from the price that we have calculated using the spot rate from the market price
then there is a chance of arbitrage opportunity. Then what exactly that arbitrage opportunity
and how that arbitrage opportunity is basically can be utilized by the investor. So those
concepts are important from the investment point of view.
(Refer Slide Time: 01:11)
134
So let us see, that is why we will use these two keywords, you will understand that is your
arbitrage and the second one is the bootstrapping. Bootstrapping is basically a method
through which we calculate this spot rate. This spot rate is calculated using the method of the
bootstrapping.
(Refer Slide Time: 01:31)
Let us see, what exactly the spot rate is? So three things you will understand for this
discussion: one is your spot rate, equilibrium price of the bond and the concept of arbitrage.
So, what is the spot rate? Generally, we consider the spot rate is nothing but the rate on a
zero-coupon bond. And otherwise, it can be defined as the discount rate for a cash flow at the
specific maturity. That basically, we can call the spot rate but it is basically a rate on a zerocoupon bond. So whenever we calculate the price of a bond; the intrinsic price or the
equilibrium price of a bond is basically obtained by discounting the bond’s cash flow by the
135
spot rates. So we have to understand, we have to measure the spot rate already what we have
taken: the spot rate is nothing but the rate on the zero-coupon bond.
And the equilibrium price of the bond is calculated on the basis of the spot rate which is
prevailed in the market with respect to that particular cash flow. So if the price does not hold
that means the bond is traded in the market at a different price and you have calculated the
price of the bond on the basis of the spot rate which is different from that particular market or
the market price. Then there is a chance of arbitrage opportunity and how that advantage in
terms of that particular arbitrage opportunity can be created or how this arbitrage opportunity
exists? That can be always carried out by buying the bond and stripping that particular bond
into a series of zero-coupon bonds and selling them. Depending upon the particular price
difference which one is more and which one is less you can adopt either of these two
methods. One is buying the bond and strip that particular bond into a series of the zero
coupon bond and selling them. Or buying the zero coupon bonds bundling them and then
selling these bundles bonds. Either of these 2 ways it depends that which one is more and
which one is less that basically, you have to keep in the mind. So therefore, using the spot
rate first, you calculate this equilibrium price of the bond and then compare this particular
price of the bond with the market price then see how that particular arbitrage opportunity can
be created. Right!
(Refer Slide Time: 04:40)
So let us see, how basically this works? Let me give you an example in this case, how
basically you can do this? Let, there are 3 zero coupon bonds: each bond principal value each
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with a principle of rupees 100 right and they are trading at annualized spot rates. Let, the spot
rate for one year that S1 is 7%, S2 is let 8% and S3 is let 9%. Right. And there is a couponbearing bond and this bond’s maturity period is 3 years and the coupon is 8%.
Then what is the price of the bond? You can calculate that is your 8 divided by your 1.07 to
the power 1 + 8 divided by 1.08 that is 7% that is 8% that is 9%1.08 to the power 2 + the last
1 you will get 108 divided by 1.09 to the power 3. We are discounting it with respect to the
spot rates, then you get let 97.73 that is this one is basically your equilibrium price, right.
Price of the Bond( P0)=
8
(1.07)1
+
8
(1.08)2
+
108
(1.08)3
= 97.73
Hence, equilibrium price of the bond is 97.73.
Suppose a bond of 3 years maturity with an 8% coupon is traded in the market at a price of
95 Rupees. Let, 3 year bond with 8% coupon is traded in the market at a price of 95 rupees
but your equilibrium price is 97.73 what you have calculated using the spot rates. So then
what is the situation here? The market price is different from the equilibrium price so there is
a chance of arbitrage? Then, how the arbitrage can be created or how you can create the
arbitrage? So what basically you can do, first of all, number 1, buy the bond from the market
buy the bond for rupees 95 which is available. Right! Then, what basically you can do? You
form 3 stripped zero coupon bonds 0 coupons and sell them. How we can make it? Number 1:
one year zero coupon bond with your face value of Rupees 8 then, your selling price will be 8
by 1.07 that is 7.4766. Second: 2 year zero, I am using the short form zero, with your face
value 8 again then what is your selling price? Selling price has become 8 by 1.08 to the
power 2 that you get it 6.8587.
Number 3: your 3 years zero and here your face value will be 108 then your selling price will
be 108 divided by 1.09 to the power 3 that will give you 83.3958. So now what will happen?
This is basically second step so for simplicity let us write a this is a, b this is c then in the
third step what basically you will do you sell these trip bonds sell of the strip bonds the total
value if you add it then it will give you 97.73.
Then how much profit that is basically arbitrage means? It is basically risk free profit you
have generated 97.73 - 95 that is 2.73.
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The arbitrage profit can be calculated as:
Step1: Buy the bond for Rs 95
Step 2: Form three stripped zero coupon bonds and sell them
a) Case1- One year zero coupon bond with face value of Rs8, then here selling price
8
will be = (1.07)1 = 7.4766
b) Case2- two years zero coupon bond with face value of Rs8, then here selling price
8
will be = (1.08)2 = 6.8587
c) Case3- three years zero coupon bond with face value of Rs108, then here selling
108
price will be = (1.09)3 = 83.3958
Step 3: Sale of stripped bond at Rs (7.4766+6.8587+83.3958) = 97.73app.
Step 4: Due to arbitrage, the risk free profit = 97.73 – 95 = 2.73
You have bought the bond at a price of 95 strips them into 30 coupon bonds sell them then
you get 97.73 then straight forward you get a profit of 2.73 rupees. This is the way basically
you can create this opportunity and get some kind of riskless profit from this.
(Refer Slide Time:12:48)
Suppose the market price of the 3 years 8% coupon bond is traded in the market at a price of
100. Suppose we will continue with this discussion. Suppose the market price of the same
bond is 100 rupees then how the arbitrage opportunity can be created? This is the question,
how it can be created? We will just do the reverse one, buy three zero coupon bonds, your F
= 100 + value is equal to 100. So number one: 8% of one year zero coupon bond will be what
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is the cost? Cost will be 100 divided by 1.07 first yours 1 = 1.07 into your 0.00 8 that will
give you 7.4766.
Then your 8% of 2 year zero coupon bond that cost will be 100 divided by 1.08 square into
0.08 that will give you exactly 6.8587 itself. Third it will be obviously 100 8% of 3 year zero
coupon bond and what is the cost? That will be 100 divided by 1.09 to the power 3 into your
1.08 that will give you 83.3958. The total cost if you add it will be 97.73 then second what
basically you can do bundle these bonds and sell them as a 3 year 8% coupon bond for face
value of 100.
For 100 Rupees which is the market price then your profit will be your riskless profit you
have generated that is your 100 rupees minus your 97.73 that is 2.27.
In this case, the market price of the same bond is Rs100. The arbitrage opportunity can be
created in the following steps:
Step1: Buy three zero coupon bonds whose face values are Rs 100.
100
a) Case1- 8% of 1 year zero coupon bond and the cost will be = (1.08)1 × 0.08 =
7.4766
100
b) Case2- 8% of 2 year zero coupon bond and the cost will be = (1.08)2 × 0.08 =
6.8587
100
c) Case3- 8% of 3 year zero coupon bond and the cost will be = (1.08)3 × 0.08 =
83.3958
Step 2: Bundle these three bonds and sell them as a 3 year, 8%coupon bond for Rs100.
Step 3: Total purchase cost of three zero coupon bonds (7.4766+6.8587+83.3958) =
97.73app.
Step 4: Due to arbitrage, the risk less profit = 97.73 – 95 = 2.73
This is the profit you have created. The first case, we have stripped this particular coupon
bond into 3 zero coupon bonds. So here in this case, you are buying three zero coupon bonds
,bundling that 3 zero coupon bonds to a coupon bond then sell it in the market. So, why
basically you are doing it?
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Here, in this case, the market price is different than the equilibrium; price previous case, the
market price was less than the equilibrium price. And equilibrium price means whatever price
basically we have charged on the basis of these spot rates. And the equilibrium price is
always calculated using the spot rates which are prevailed in the market. So this is the way
the arbitrage opportunity can be created and the investor can always earn some riskless profit
through that particular process.
(Refer Slide Time:17:41)
So here the next question is that what basically we have seen here, in the first case what we
have seen? Given this risk-free opportunity, arbitragers should implement this strategy of
buying and stripping the bond until the price of the coupon bond was bid up to equal its
equilibrium price that is 97.73. And once it will reach 97.73, the arbitrage would disappear.
So automatically, the price will change looking at the equilibrium price and the market price.
Because, if many people will go ahead with this strategy automatically it will increase the
price from 95 to 97.73 then in the end the arbitrage basically will disappear.
And in the second case, what we have seen? Arbitrageurs basically will implement this
strategy of buying the zero coupon bonds or zeros, bundling these bonds and selling the 3
years coupon bond until the price of the 3 years coupon bond was bid down to equal its
equilibrium price that is 97.73. Then the arbitrage would disappear. End of the day, the
arbitraries will disappear but within that particular period, the profit riskless profit can be
generated by the investors by taking this kind of position. That actually you can keep in the
mind. In the end automatically this will create a certain kind of demand in the market and
once the demand-supply, things will change the price will automatically come back to
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equilibrium. But it will take some time then within that particular time this investor can get
the advantage from this particular arbitrage opportunity.
(Refer Slide Time:19:41)
So now the question is that how to estimate the spot rates? Why basically, the problem arise?
Because you see whenever we are going for evaluation with the spot rates and creating the
different types of the stripped securities. But in the true sense in a practical sense, there are
not enough; longer-term zero coupon bonds available to determine the spot rates on higher
maturities. Zero coupon bond may be available for one year period, 2 year period but for
longer-term maturity zero-coupon bonds here we have taken for first year zero coupon bond
rate 7% second year 8% third year 9%. But that is not always possible for the investor to find
out a zero coupon bond whose maturity period is 15 years. And that can be utilized as a spot
rate for the calculation of the equilibrium price. So because of this the long-term spot rates
have to be estimated; we have to estimate the long-term spot rates and one of the estimation
approaches that can be used, which is a basically sequential process is called boot strapping.
So, given the bond’s rate, a coupon bond with the next highest maturity is used to the implied
spot rate. That means we are taking the help of the rates which is prevailed from the coupon
rate to calculate the spot rates of the longer-term maturity zero coupon bonds. Then another
coupon bond with the next highest maturity is then used to find the next spot rate like that it
will go on.
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So that is the process basically what we follow whenever we use this boot strapping method
for calculation of the spot rate for the longer maturity zero coupon bonds and that thing again
can be utilized to calculate the equilibrium price of the bond.
(Refer Slide Time:21:51)
So let us see that how this particular bootstrapping method is utilized or is used to calculate
the longer-term spot rate. Let I will give you certain things here let suppose bond 1: you
assume suppose there is a bond one that is the one year zero coupon bond selling price, let
Rs.100 and the maturity value is 107. Let there is a bond 2: there is a 2 year zero coupon
bond your selling price selling at par that is rupees 100 and coupon rate is let 8% it is a
coupon bond, it is a 2 year coupon bond, coupon rate is and it is selling price means Rs.100
means it is selling at par.
Then there is a bond 3: that is the 3 years coupon bond it is also selling at par that Rs.100 and
the coupon rate is at 9%. These are 2 coupon bonds and the first one is basically zero coupon
bond, it was bought at a price of 100 and the maturity value will be 107. Then what basically
what is your S1 then? S1 will automatically your 107 divided by 100 - 1 right that is your 7%.
First rate it is S1 is 7% this is your S1.
107
The spot rate (S1) can be calculated when we are going for 1 year zero coupon bond= 100 – 1
= 7% 0r 0.07.
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Right, now what is S2? S2 is equal to your principal value is 100 selling price is 100 then that
means it is 100 is equal to year 8 by coupon is 8%, 8 by 1.07 + 108 divided by 1 + S2 to the
power 2 is it clear 1 + S2 to the power 2 but S2 we do not know then we have to find out the
S2 then that will be is equal to 100 = your 7.48 by 1.07 that is 7.48 + 108 divided by 1 + 1 +
S2 square. Then that is equal to that means it will be 92.52 = 108 divided by 1 + S2 square
then yours 2 = 108 divided by your 92.52 to the power 1 by 2 - 1 that will be 0.08428042.
Similarly, the spot rate (S2) can be calculated when we are going for 2 year at 8% coupon
bond as: 100 =
8
(1.07)1
 100 = 7.48 +
 92.52 =
+
108
(1+𝑆2)2
108
(1+𝑆2)2
108
(1+𝑆2)2
 S2 = 0.08042 or 8.042%
So this is the spot rate if you are going for a 2 year zero coupon bond then your spot rate will
be 8.04% 0.08042. Now we go for the third one what is your S3? S3 will be then again you
solve this equation 100 = 9 by 1.07 + your 9 by one point now we have to use this 108042
square plus your 109 divided by 1 + S3 to the power 3. So if you solve it then automatically
this part will come 83.88 = your 109, 1 + S3 to the power 3 then your S3 = 109 divided by
your 83.88 to the power 1 by 3 - 1 that will give you 0.0912. So this is your S3.
Similarly, the spot rate (S3) can be calculated when we are going for 3 year 9% coupon bond
as: 100 =
9
(1.07)1
 100= 83.88 +
+
9
(1.08042)2
+
109
(1+𝑆3)3
109
(1+𝑆3)3
1/3
109
]
 S3 = [83.88
-1
 S3= 0.0912 or 9.12%
The equilibrium price of other 2 and 3 years bond can be obtained by the 3 year spot rates. So
in this case if you want to calculate the equilibrium price how will you calculate the
equilibrium price? Now you can calculate let you have a bond whose yield to maturity is 10%
YTM. Then now you have got the S1, S2, S3 you can find out your equilibrium price that
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means your 10 by 1.07 that the par value is 100, then it will be 10 by 1.077,10 by 1.08042 to
the power 2 + 110 divided by 1.0912 to the power 3 if you solve it you will get 102.57.
The equilibrium price of three year coupon bond can be calculated by using spot rates as
discount rates. And here, S1, S2 and S3 are 0.07,0.08042 and 0.0912 respectively.
𝐶𝑜𝑢𝑝𝑜𝑛
𝐶𝑜𝑢𝑝𝑜𝑛
The equilibrium price (Po) = (1+𝑆1)1 + (1+𝑆2)2 +
10
10
𝐶𝑜𝑢𝑝𝑜𝑛+𝑚𝑣
(1+𝑆3)3
110
= (1.07)1 + (1.08042)2 + (1.0912)3
= 102.57
Hence, the equilibrium price is 102.57.
So this is your equilibrium price. We are using this spot rate for calculation of an equilibrium
price of a bond whose par value is 100, your coupon is 10% and you are using this spot rate
as the discount rate. So then you got your equilibrium price 102.57. So then you compare at
that time what is the market price of this 10% coupon and 3 years maturity bonds, then
accordingly your arbitrage can be created.
So this is what basically the bootstrapping method, like that for bond 4 you can take a 4
years coupon bond, you know this coupon rate, you know this par value or the selling price
from that you can use this S1, S2, S3 to calculate your S4 then like that is 5. So that is the
way basically this bootstrapping method works.
(Refer Slide Time: 29:47)
So what we have basically discussed here? The equilibrium price of a bond is nothing but, we
can get this equilibrium price by discounting bonds cash flow with the spot rates. And the
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arbitrage opportunities created when the equilibrium price is different from the price which is
prevailed in the market. And bootstrapping approach is generally used to estimate the spot
rates, there we use the different maturity periods coupon bonds for calculation of this spot
rates for the particular kind of analysis or the zero coupon rates.
(Refer Slide Time: 30:24)
So this is the reference what you can follow for this. Thank you.
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Management of fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 02
Lecture No # 10
Level of Interest Rate Determination-I
Welcome back. So in the previous class, we discussed about the calculation of the spot rate
which is generally used to calculate the equilibrium price of the bond. Then accordingly, we
can understand that whether the arbitrage opportunity can be created or not. So what do we
have seen here? The interest rate is the major factor which is very much important for the
bond analysis or the price of the bond or for yield calculation of the bond.
So in that particular context, we have to understand, what is the behaviour of that particular
interest rate? So whenever we talk about the interest rate behaviour, there are 2 things
basically comes to our mind. One is your structure of your interest rate and another one is the
level of the interest rate. We will discuss the structure of the interest rate little bit later
because that basically decides that on the basics of the characteristics of the bond, the rate of
interest or the yield or the return changes.
But whenever, we talk about the structure of the interest that is basically the overall market
interest rate. How the overall level interest rate level can be determined by the market rate?
So we will go from a macro to micro perspective from the micro point of view first of all we
have to understand that what exactly the level of interest rate is? Then we can move towards
this structure point.
(Refer Slide Time: 01:55)
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So here, in this case if you see that today’s discussion will be based upon the two things:
there are basically 3 different theories. But, today’s discussion will basically cover up the two
different theories and then will move towards the other kind of aspects that. One is your
classical theory of interest determination and another one is your loanable bond theory.
So these are the two theories what we will be discussing today and then we will move
towards the other type of theories which are prevailed in the market.
(Refer Slide Time: 02:37)
So after this you will be understood about the concept of savings, concept of investments,
concept of the bond demand curve and bond supply curve. So these are the things basically
you will come to know after the discussion of today’s topic. So you know already, I told you,
there are broadly three theories we always use to understand this level of interest rate how the
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overall market interest is determined. In that particular context, we generally always look at
these three different theories: one is your classical theory, second one is the loanable font
theory and third one is the Keynesian theory.
(Refer Slide Time: 03:29)
So let us see that what this classical theory talks about? According to classical theory, the
interest rate is real phenomena. What does it mean? That means it is determined by the real
factors. Real factors in the sense we talk about savings, we talk about the investments, all
kind of macro economy concepts you might have already aware about. Those things basically
determine the interest rates and interest rate is determined with the equilibrium of the market.
Then when the market is in the equilibrium? The market is in the equilibrium basically
whenever the supply is equal to demand. Then, what is the supply here and what is the
demand here? Here, it is the supply of the savings and the demand for the investments
whatever money is saved that money will be invested in the market. So that means the
interest rate will be determined in that phase whenever there is a savings is equal to
investment.
So it is the supply of the savings and demand for investment that will determine the
equilibrium rate of interest in the market.
(Refer Slide Time: 04:59)
148
So here what basically overall we are talking about what do you mean by this aggregate
savings? If you go by normal economic identity, you know y is equal to we consider c + s.
You total income is equal to consumption plus savings then how you get the savings, it is y –
c, y = income and c = consumption.
According to classical economic theory, Y = C+S
Where, Y = Income
C= consumption
S= savings
So whatever money is not consumed that is saved, that is the logic basically we adopt. The
classical approach basically tells whatever money is not basically used for consumption that
will be saved. So that is why, the aggregate savings is the difference between the total
national income and the total consumption expenditure. And Savings may be affected by
whom? Who basically go for savings? It is the individual’s households, business sector or
corporate sector, government any entity in the economy can go for the savings.
So why somebody will save if somebody have some surplus money? Why they will go for the
savings? Because, they are expecting some reward from this and they are not basically
utilizing that money now. They are keeping that money with somebody with the bank or with
other financial institutions expecting that some kind of reward they will get. So that is why
according to classical theory, interest rate is a reward for sacrifice or waiting involved in act
of supplying savings.
149
That means whenever I am not utilizing that money now, I am foregoing my consumption.
Why I am foregoing my consumption? Because, I want some returns. Because that money
will be kept somewhere and I will get some return out of this. So that is why bank gives you
interest because you are not utilizing that particular money and that is why some reward is
given to you.
And who basically and where the bank gets the money to give you the interest because there
are some other entities mostly the corporate sector if you look at, they need money for their
investment or to produce. In the production process, they take the money from the banks and
bank basically takes the money from the savings whatever we are making and that money is
utilized in the market or utilize in the production process. And against that because they have
taken the loan from the bank they will pay certain thing to that bank and from that, bank will
pay some interest to us. So, money or the particular kind of savings whatever we are making
that savings are basically demanded by the investor or the investment. So therefore, why they
invest? Because, they want to produce; once they produce they will sell and they will get the
price, also they will get some profit. And from the profit, they can generate some surplus
again which can be saved. So in that process, the business sector also can save something
there is a corporate saving. So the supply basically is from saving side and demand basically
comes from the investment side. So wherever the savings is equal to investment there the
equilibrium interest rate basically is created.
(Refer Slide Time: 08:58)
150
And if you see that, the saving schedule is upward sloping and investment schedule is
downward sloping. Why this saving schedule is upward sloping? Because if the interest rate
will be more then savings will be more; if bank will give more interest then you will be
inclined to save more. So in that process, the supply is basically directly proportional to the
interest rate. But if you look at the demand side: if the interest rate will be more then what
will happen! then the demand will go down. Because, loan will be expensive; the corporate
sector may not go for higher demand. That is why the investment schedule is downward
sloping. The saving schedule is upward sloping and investment schedule is downward
sloping and wherever the interaction between the savings and investment that means saving is
equal to investment. There the equilibrium interest rate is determined in which particular
interest rate makes this supplies side or supplies the savings is equal to demand for
investments. That is basically equilibrium interest rate which can be prevailed in the market.
So that is basically the basic essence of the classical theory. So if the savings is equal to
investment the particular interest rate wherever the savings is equal to interest rate that is the
equilibrium interest rate which can be prevailed in the market at a particular point of time.
(Refer Slide Time: 10:54)
But the question here is that, all the money which are saved cannot be invested? Out of them,
some money can be loanable and some money is not loanable or cannot be loanable. And that
is why according to loanable fund theory; they said that it is not the total savings and
investment which determine the interest rate. It is the interest rate is generally determined by
the demand and supply of the loanable funds.
151
So mostly, it is basically related to this supply and demand in the bond market. So the interest
rate is largely determined by this supply and demand in the bond market. In this regard, this
theory basically is more realistic or may be more practical than the classical theory of
interest. And also the loanable fund theory discards the independence of the interest rate from
the behaviour of the money and banks.
They said there is some dependence. So according to this, the real supply and demand curves
which basically determine the interest rates should have added to them the component of the
supply of the savings which is associated with the creation of the new money or the credit in
the system. We will explain this- what basically here we mean.
(Refer Slide Time: 12:51)
So in this context, they are talking about the demand and supply in the bond market. So what
is the demand and supply curve in the bond markets? Let graphically if you want to show,
this, the demand curve or the bond demand curve what it shows? It shows the relationship
between the quantity demanded and the price when all economic variables are held constants.
Other factors which are affecting the price of the bond that basically remains constant. In that
particular point of time, what is the price and demand for the bond relationship that is
basically called the bond demand curve? And usually already you know that the bond
demand curve has the usual downward sloping. What is indicates? That the lower price of the
prices of the bond everything else being equal the quantity demanded is higher. Many people
will be ready to invest in that particular bond if the price is low. Expecting that they will get
more return in the future.
152
And what is bond supply curve? It basically shows the relationship between the quantity
supplied and the price when all other economic variables are held constant. And it is usually
upward sloping, what does it means: If the price increases then the quantity supplied also will
increase. Right! Because price and supply side are directly related; the price is more
everybody wants to supply that and if the price is higher the demand will be less that is the
usual economic principle already all of us know. So if the price is higher that means the
interest rate is lower in case of the bonds. If the price is higher the interest rate is low if the
price is low then interest rate is high. So, both basically move in the opposite directions. So in
this case, what basically we have seen that the demand and supply of the bond market will
decide that which; is the equilibrium interest rate.
So let us see that how basically this thing works that is called the market equilibrium that
means when the market equilibrium occurs. The market equilibrium will occur when the
amount that people are willing to buy which is basically the demand side equals the amount
that people are willing to sell that is basically the supply side at a given price. At a given
price, the amount that what the people are willing to buy or the amount that what people are
willing to sell?
So that basically gives you the market equilibrium. So, in the bond market, when does the
market equilibrium occur? The market equilibrium occurs when the quantity of the bonds
demanded equals the quantity of the bond supplied. And the interest rate of that corresponds
to that particular price is called the equilibrium or the market clearing interest rate. The
interest rate of that corresponds to that particular price is called the equilibrium or the market
clearing interest rate. What basically it means?
(Refer Slide Time: 17:15)
153
Let me explain it in a diagrammatic way what basically here what I was trying to explain you.
Let us draw a diagram, let I write this is your quantity of the bonds and here we will let the
price of the bond here let us take the interest rate. So here what happen if I supply demand
curve, we have drawn that is basically here let for example this is 100 this is 200 let this is
300. So here we have taken the price so what we have seen that?
The price is more than the demand will be less. And which is the demand curve? This is your
demand curve which is downward sloping and this is your demand curve and this is your
supply curve. So here what basically we are trying to say that this is your equilibrium point
and this is basically price which we are getting that is your equilibrium price and this is your
equilibrium interest rate.
So here let 100 so in this case this is the price. So in this price you are demand is if you are
trying to see if let the price is let here I am writing 700. So here if you see those 100
quantities are supplied at the price of 700 but the demand is more maybe 500. But let the
prices increase to 800 so here your supply has increased let that this has become 200. But if
you look at the demand, the demand that has come down to 400 from 500 to 400 so this will
go on and in that time the interest rate is basically we are taking in this side.
It is increasing but in the lower side so this is the price going in this side so let in that time,
the interest rate was let 9%. So here the interest rate was let 80% so let it is 7%. It is
increasing but in the lower what in this vertical axis if you go down it is increasing. So here
the price is increasing in the upward side here the interest rate is increasing in the downward
154
side. Is it clear? It was 9% ss price was 700 it was 8% price was 800 at 7% maybe this price
will be 900 like that.
So the interest rate is less, price is more if the interest rate is more the price is less. So that
means here in this case your equilibrium interest rate is 7% and equilibrium price is 900.
Further if you go up may be you will go up here let the price this will be late 180 let the
prices become 950. So this is your demand part but now supply is very high let 430. So here
the equilibrium interest rate is determined that is 7% is corresponds to this particular price
900. Is it clear?
So that is your equilibrium interest rate, that is your demand side and this is your this is
basically your supply setting this is your demand setting which is downward sloping and this
is upward sloping. Now for example if you little bit for your clarity now you have mixed of
both price and interest rate.
(Refer Slide Time: 23:27)
In the loanable theory context if I draw it only in one part, 1 vertical axis, 1 horizontal axis.
Let this is your quantity and this is your interest rate it is going up that means so this is your
equilibrium quantity this is your equilibrium interest rate. So here the interest rate is basically
increasing or decreasing let it was 5% then it becomes 6% then you become let 7% like that.
Accordingly your demand supply so this is what, this is basically your supply of the loanable
funds.
But this is basically talks about the demand for the bonds and this is your demand for
loanable fund and this is basically your supply of the bonds. In bracket, we are writing to
155
make a synonymous with respect to the previous diagram. So loanable bond theory is linked
to supply and demand for the bonds that is what basically we are trying to do. So that is
equilibrium point at which the interest rate is basically determined.
So here what basically we are talking about that loanable fund theory talks between the
equality between the supply side and the demand side of the bonds.
(Refer Slide Time: 25:43)
Then there is a possibility when the quantity demanded or supplied changes as a result of a
change in price of a bond. Then we have a movement along this demand or the supply curve.
Just now we have seen but the demand and supply curve can shift when the quantity
demanded or supplied changes at each given price or interest rate of a bond in the response to
change in some other factors which are affecting the demand and supply of the bonds.
When any of the factors will change, there is shift in the demand curve or there maybe shift
in the supply curve and a new equilibrium value of the interest rate can be determined or can
be decided. So that is why, moving along this demand curve or moving along the supply
curve is different than shifting the demand curve and shifting the supply curve.
(Refer Slide Time: 27:09)
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For example, if you see let me tell you that what basically I mean here? If I draw a demand
curve, this is your demand curve. So if your price will change let this was the price of 500
and this is quantity let 1000 but when the price has become 600, the quantity has become let
850. This is your p, this is your demand this is your quantity and in that time your supply is
also changing. So for example if we draw a supply curve like this so this is your equilibrium
point before at this price at this quantity.
Now whenever the price will change the supply is also changing at this price this much was
the supply but at this price the supply is this much. So what is happening? So we are moving
either through this demand curve or through this particular supply curve. And this is
happening because of the change in the price but we are not basically shifting the supply
curve from this to this demand curve from this to this or supply curve from this to this or
from this side we are not changing that.
We are only moving across the demand curve or we are moving across the supply curve and
that thing happens whenever the price changes. But there are other factors which also affect
the demand and supply of the bonds. So if those things will change then what will happen?
That this will shift the demand curve and the supply curve .So that is what basically always
we have to understand that when this or what factors basically change the or shift the demand
curve of the bond or the supply curve of the bond. This is what basically we have to see.
(Refer Slide Time: 29:41)
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So what basically we have discussed? That, according to classical theory, supply of the
savings and the demand for investment which will determine the interest rate but loanable
fund theory basically tells that the interest rate determined by the demand and supply of the
loanable funds. Both bond demand curve basically shows the relationship between the
quantity demanded and the price when all other economic variables had constant.
And supply curve also show the relationship between quantity supplied and the price when all
other economic variables held constant. So further we will be discussing about what factor
basically shift is one supply curve and the demand curve by that a new equilibrium point can
be established.
(Refer Slide Time: 30:27)
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These are the references what basically you have to go through for the comprehensive or the
detailed discussion on this. Thank you.
159
Management of Fixed Income Securities
Prof. Jitendra Makud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 03
Lecture No # 11
Level of Interest Rate Determination – II
Welcome back. So, in the previous class, we started the discussion on the level of interest
determination particularly how the market interest rate is determined. So, in that part, we have
discussed about the three theories which generally tries to explain interest rate. One is your
classical theory second one is your loanable fund theory. We have discussed about the classical
theory and also we started the discussion on the loanable fund theory, and in the loanable fund
theory, what we have discussed that the demand and supply side of the bonds particularly the
supply of the bonds and demand for the bonds that will decide that how much interest rate will
be prevailed in the market and accordingly, the yield of the particular bond can be determined.
(Refer Slide Time: 01:12)
So, in today’s discussion, we will be discussing about that what are those factors which are
affecting the demand for bonds and what are those factors which are affecting the supply of the
bonds and in what context the demand curve and the supply curve of the bonds are going to shift.
(Refer Slide Time: 01:31)
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So, in that particular context, if you see there are certain keywords you will come to know one is
your wealth or the income what generally we always consider. Which general the major factor
which is which basically affects the demand and supply of the bonds? Then the business cyclic
conditions, inflation, investment opportunity, liquidity all those factors which are quite important
from the shifting of the demand and supply of the bonds in the bond market.
So, you will have a kind of comprehensive idea about all those factors and how these particular
factors are going to play the role in terms of the determination of interest rate in terms of the
supplying supply side of the bonds and the demand for the bonds.
(Refer Slide Time: 02:20)
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So, let us see that what are those factors basically affect we can say that the demand for the
bonds. You see that in the previous class, we have discussed that there is a supply curve there is
a demand curve and there are 2 types of movements can happen in that case. One is the
movement across this particular curve or on that particular curve and another one is total shifting
of this particular supply and demand curve these are the 2 different things.
So, whenever your price will be changed, so, the change in the price is not going to shift the
demand and supply of the bond or the curve what basically we derive for the supply side and
demand side that is not going to be changed. But, apart from the factors like price, the other
factors how what are those other factors we generally affect the supply of the demand for bonds?
If those factors will change then what will happen? Automatically, the demand for the supply of
the curve is going to be changed. The demand curve and the supply curve are going to be
changed. So, if you see, there are major 4 factors we can consider although, there are other
factors also but there are major broad 4 factors which can affect the demand curve for the bonds
or which can shift the demand curve for the bonds.
One is your income or the wealth and another one is the expected return on the bonds related to
the other alternative assets like stocks and other kind of financial assets which are traded in the
market. Then, we have the riskiness of the bond how much risk is involved with respect to the
bond with in comparison to the other assets which are traded. Then, the last but not the least we
have the liquidity so liquidity of the bonds related to the other alternative assets.
So, these are the major factors which may have the impact which generally affect the demand for
the bonds and that factors are responsible for shifting the demand curve for the bonds. So, one by
one if you see that how these factors are basically going to affect?
(Refer Slide Time: 04:46)
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So, let us because the income or the wealth and the business cycle these are also linked. So; all of
you know that although there are different 4 phases of the business cycle all of you might have
the idea about that. We have an expansionary phase and or we can if you start it like this we have
a kind of phase which from the beginning, if you start it, there is a kind of boom phase or the we
can say that the recovery phase.
Then, it will reach the top or the peak then it will start declining then finally, it will reach the top.
So, then again the same cycle will go on for the next period. But, broadly if you see that, we have
categorized this business cyclic condition in 2 phases; one is your expansionary phase and
another one is your contractionary phase or the recessionary phase. So, these are in general terms
we define always whenever we talk about the categorization of the business cyclic conditions.
So, whenever there is a expansionary phase, so obviously there are enough opportunity in the
market and the income level or the wealth level of the different stakeholders in the market
generally go up and if the income level will go up. So, there is a possibility that the demand for
bonds may rise because people have enough money to invest and on the basis of their objective,
they may be inclined to invest more in the bond market.
So, in that case, we can assume that the demand for the bonds may rise so if the demand for want
will raise then, the demand curve for the bonds basically shift to the right one. So but if you look
at the recessionary phase so in that case we are expecting that lot of investment opportunities not
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there. And because of this the wealth or income level of the individuals and as well as other
market participants may go down.
So, in that case, the because there is not enough money available for the investment then the
demand for bonds may fall. So, if it will fall, then the demand curve will shift to left. So, in that
case, what basically we have seen that the higher the income higher the wealth, the demand
curve will shift rightward and lower the wealth or lower the income, the demand curve will shift
to left one.
So, then another one is that the expected return there is another factor if your expected return will
be higher in the future you are expecting that your return will be more in the future. Then what
will happen that? That will also have the impact on the demand curve for the bonds. So, higher
expected interest rate in the future basically lower the expected return for the long term bonds
because the price of the particular bond is going to be affected.
So, all of you know that that is there is a relationship between the interest rate and the price of
the bond and accordingly the demand for bonds also gets affected. So, higher expected interest
rate in the future will lower the expected return for the long term bonds so, therefore, there is a
decrease in the demand and in that case, the demand curve will shift to leftward but the lower
expected interest rate in the future will increase the demand for the long term bond and that will
shift the demand curve to the right one.
So, that is just a reverse relationship what we can expect in this particular context. So, here
income and the expected interest rate or the income or the return what we are going to expect
from the different long term bonds these are the 2 major factors what just now we have discussed
which can have the impact on the shifting of the demand for the bonds or demand curve of the
bonds.
(Refer Slide Time: 08:47)
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Then the other factor if you see that we have the change in the expected return on the other
assets. Already I told you the other assets means let we are talking about the stocks we have we
can also take the other financial assets like derivatives and other instruments which are also
traded in the market. So, changes in the expected return on other assets also can shift the demand
curve for the bonds.
So, in this case, that if you are expecting that your return from the other assets will be more, then
what will happen? People will be interested more to invest in those assets. Then obviously the
demand for the bonds will go down but if you are expecting that the return from the bonds
expected return from the bonds will be higher then automatically people will be inclined to
invest in the bonds more.
So, that is why there is a relationship we can establish between the exchange in the expected
return on the other assets and the demand curve for the bonds. Then we have the inflation which
is a major factor which can affect almost all the financial assets return. So, if you are expecting
or somebody is expecting that there is a increase in the expected rate of inflation, then what will
happen? That will basically decline or lower the expected return from the bonds.
So, whenever the expected return from the bonds will go down, then that will cause the demand
to decline. So, in that case, the demand curve will shift to the left if it will decline, then it will
shift to the left if the demand is increasing then it will shift to the right. So, extracted inflation is
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another factor which is affecting the demand curve or which is responsible for shifting the
demand curve for the bonds.
Then we have the riskiness of the bonds. So, how much risk is involved? Any investment
whenever we are making in the market, that is basically is always prone to some amount of risk
there is a market rate there are credit risk there are different type of risk already we have
discussed. So, if your riskiness of the bond is higher, than obviously the demand for bonds will
fall then again automatically the demand curve will shift to the left.
But if you assume that the riskiness of the alternate alternative assets which; are available in the
market which are traded in the market. So, if you compare it with the riskiness of the bonds, so
then also you can decide that whether the demand for bonds willing will increase or it will
decline. So, if you assume that there is a increase in the riskiness of the alternative assets then
what will happen? That in comparison to the riskiness of the bonds, if the riskiness of the order
or other alternative assets are more then in that case, people will be more inclined to invest in the
bond market. So, that is why the demand for bonds will rise and in that case, the demand curve
will shift to the right.OK? So, what we have discussed? We have discussed there is a wealth
effect or the income effect. So, there is a effect which is coming from the expected return what
we are getting from the bonds in comparison to the other assets.
We have discussed about the expected inflation or the impact of expected inflation on the
demand for the bonds. Then we have also discussed about the riskiness of the bond in
comparison to the other assets which are available in the market for the investment. So, these are
the some of the factors which are responsible for shifting.
(Refer Slide Time: 12:41)
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Then the other factor already what i told you that is basically what we call the liquidity. So, what
do you mean by liquidity? Liquidity means how fast this particular asset can be converted into
cash. So, that means there should be enough market or the marketability of the particular assets
should be higher. So, if there is a requirement that particular asset can be easily marketed or
easily can be sold or it can be bought so by that the liquidity of the particular asset will be more.
So, if you fill that or if you find that there is increased liquidity of the bonds, then obviously the
demand for bonds will increase. So, the liquidity of the bonds will increase the demand for the
bonds. So, definitely in that time, the demand curve will shift to the right. Similarly, if you if you
see that the increase liquidity of the alternative assets in comparison to the liquidity of the bonds.
If you find that the there is a high liquidity of other assets which are available in the market, then
that will lower the demand for the bonds and shift the demand curve to the left.
So, the liquidity is another factor which is also responsible for shifting the demand curve for the
bonds. So, overall, what basically we can say? Income then we have inflation we have expected
return we have the riskiness of the bond then we have the liquidity. These are the major factor
factors which are responsible for shifting the demand curve for the bonds. Then accordingly, if
the demand curve will change then that will have the impact on the equilibrium price then with
respect to that your interest rate also is going to be changed.
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So, that we have discussed that how the price and interest rate are linked so if your interest rate is
increasing, their price is declining. If the price is higher, then the interest rate will decline. So,
there is an inverse relationship always we will get between the interest rate and the prices.
(Refer Slide Time: 14:41)
So, let us summarize it that in the diagrammatic form that how this thing basically works in that
particular context. So, already what I told you that one factor is basically your income or the
wealth. So, let us take a certain variable let I will make a table like this let this is variable then
we have the change in the variable. OK? Then we have change in quantity demanded then we
have shift in the demand curve. OK?
So, let the first variable whatever we have just now discussed let us take income or the wealth.
Let the wealth has increased then what we have seen? The change in the quantity demanded of
the bond will also will increase. So, then if you show it in the diagrammatic form, then how
basically how the demand what is the shape of the demand curve? The shape of the demand
curve is downward sloping. OK?
So, this is your bond which is demanded and this is your basically price “p”. So, in that case, if
your price is increasing, then accordingly, your demand for the bond is going to be changed. So,
here so let this is your B D 1 which was the demand curve before now what has happened? Let
the income level has gone up. So, in that case, the demand curve will shift it out rightward. So,
this will be B D 2 that means there is a shift like this. OK?
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So, then we have another factor we have discussed that is your expected interest rate. OK? So, let
if it has gone up, then bond demand will basically go down then in that case, what will happen?
That the same curve if you draw, then if this is your b d 1, then this will shift it shift to left ward.
That will be your b d 2. So, this is your bond this is your price. So, like that you have expected
inflation. OK?
So, in this case also, if it will go up, then the change in quantity demanded will go down. So, the
same type of curve you will find. OK? This is your b d 1. Then this will shift the leftward. Then
you will find this b d 2 so there is a leftward moment so that is your bond that is your p price.
Right? So, like that you can draw this demand curve that how this particular movement can take
place.
(Refer Slide Time: 18:22)
Then we have the other factors also that we have riskiness of the bonds relative to other assets
relative to other assets. Let riskiness of the bond has gone up then the quantity demanded will go
down then what will happen that the demand curve will shift leftward. So, this is your b d 1 this
is your b d 2 so this is your bond this is your p then we have the liquidity. So, the liquidity of the
bond is more, demand will be more and the curve will shift rightward. So, it will be a rightward
movement. So, this is your bond this is your p this is your b d 1 this is your bond demand 2.
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So, like that the, if any of the factors will change, then accordingly, the movement of this
particular different type of bonds will happen. So, depending upon the change in the quantity
demanded, your demand curve will shift if it will increase, the quantity demanded will increase.
Then automatically it will have a rightward shift. So, if the demand quantity demanded of the
bond will decline, then it will have a leftward shift. OK? So, that is the way basically the demand
for the bond which is going to be changed or the bond demand curve will shit.
(Refer Slide Time: 20:01)
Then what are those factors which shift the supply curve of the bonds? There are many factors
but we have highlighted some of the major factors here. One is your expected profitability of
investment opportunity and we have expected inflation and the government activities means
government policies. You see who supplies the bonds it is the corporate sector mostly or the
bigger entities who basically supplies the bonds. Investor can be anybody investor can be another
financial institutions or it can be an individual. But individual never supplies the bond supply
side of the bond generally comes from the business sector or the government. So, in that case, the
profitability or the investment opportunity in the market that is a major concern for them whether
they want to go for supplying the bonds or not.
(Refer Slide Time: 21:08)
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So, what it exactly means? If you see let you take the same thing like business cycle. If there is a
business cycle expansion, then what will happen? The supply of the bonds will increase why?
Why it will increase? Because, this business sector will have the enough opportunity to invest
their money and by that process the profitability can be enhanced. So, to increase the profitability
and to utilize the investment opportunities which are available in the market, they need more
financing.
So, because of this they want to issue more bonds. So, in the business cyclic expansion, the
supply of the bonds may increase and in that case, the supply curve will shift towards the right.
But in recession when there is fewer expected profitable investment opportunity, the supply of
the bond will fall then automatic the supply curve shifts to the left. That is just reverse the
argument or logic is same.
So, in that particular regard, we have also seen the expected inflation has the impact on the
demand for the bonds then the expected inflation also has the impact on the supply of the bonds.
So, an increase in the expected inflation causes the supply of the bonds to increase and the
supply curve to the shift to the right it is just reverse whatever just now we have discussed in the
context of the demand for the bonds. Right?
And higher government deficits higher government deficits increase the supply of the bonds and
shift the supply curve to the right any government policies always have the impact because that
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will have the impact on the price of the bonds. So, if the price is more, then supply will be more.
OK? So, in that case, what will happen? That whenever we look at the different type of policies
what the government has it will have the implications on the prices of the bonds that will
automatically have the impact on the demand and supply of the bonds and here also the supply
side is more affected by that. So, higher government deficit increase the supply of the bonds that
is why the there is a shift from towards the right. But if there is a surplus, then why the
government will issue the bonds? They do not need finance they do not need money. So, you
might have linked it to the for example open market operations which is instrument of the
monetary policy.
What basically government does? Government needs the kind of selling and buying the bonds to
regulate the money supply. And sometimes also government if there is a deficit government
wants to issue the bonds to finance the deficits. So, in that case, we have taken this example in
the case of government deficit or the surplus. So, if already there is a surplus, then there is no
need to supply the bonds. So, that will decline the supply of the bonds and automatically the
supply curve will shift towards the left.
Then what basically we have seen here? We have seen that the again the business cyclic
conditions because that has linked to the investment opportunity and the profitability of the
companies. So, because of this there is a chance that the supply curve gets affected. We have
seen that inflation which has also the impact on the supply side of the bonds. Then most
important is the government policies or government activities.
So, if there is a deficit, then obviously to finance the deficit, they will issue more bonds but if
there is a surplus then they do not want to finance this there is no requirement of the financing
that particular thing. So, in that particular context, there is a shift towards the left or the supply of
the bond will go down.
(Refer Slide Time: 25:32)
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The same thing you can you can also draw whatever way we have drawn. You can make a table
like that let we have the variable then your change in variable then we have change in quantity
supplied. OK? Then your shift in the supply curve. So, now you tell me that first factor what we
have taken- The profitability. OK? (Profitability of the investment) So, obviously if it will
increase your supply also will increase.
So, what is the shape of the supply curve? The shape of the supply curve should be upward
sloping. Right? So, this is your bond supply this is generally your pi price. So, if it is your let bs
1 then it will shift rightward then it will move here. So, this is your b s 2. Right? So, then we
have we have discussed about the expected inflation. If this will increase, then automatically here
also the supply will increase because it will have the impact on the price.
Then the same thing will happen here. Right? So, this is your b s 1 this is your b s 2 the move is
like this. Right? Then we have we have taken the let you have taken government deficit. If it is
more, then what will happen? In the case of government deficit, they need more financing
because of that they will issue or they will supply more bonds. So, the same thing will happen.
This is your, this is your b s 1 this is your b s 2 the move is like this.
But for example, you have taken government surplus. So, if this will increase, then obviously the
change in the quantity supplied will decline then the supply curve will move to leftward. So, this
is the way the supply curve will move apart from the price what are the other factors which are
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affecting the demand and supply side of the bonds or demand for the bonds or supply of the
bonds? If those factor will change, then there will be a shift but only prices you will move only
on that particular curve.
So, this curve is not going to be shifted. So, that thing you have to keep in the mind. So,
automatically if the supply will change and demand will be changed then the interest rate the
equilibrium interest rate also is going to be changed. A new equilibrium interest will be
determined in the market and this is a basically part of the loanable fund theory. In the
beginning, we have started the discussion that the loanable fund theory is basically linked to the
bond market equilibrium.
The supply of the bonds and the demand for the bonds and these are the factors which are going
to affect the supply side and already we have discussed the factors which are responsible for the
demand side.OK?
(Refer Slide Time: 29:13)
So, what basically we have discussed? We have basically discussed that the which are the major
factors which basically change the equilibrium interest rate. That basically include your income
or the wealth, expected return on the bonds relative to alternative assets. Risk of the bonds
related to alternative assets, liquidity of the bonds related to alternative assets, expected
profitability of investment opportunity and the government activities.
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So, all these factors are responsible for change in the interest rate in the market. Why they are
responsible for changing interest in the market? Because these; factors are responsible for the
demand for and supply of the bonds in the corporate bond market.OK? So, this is basically the
essence of the loanable fund theory.
(Refer Slide Time: 30:06)
OK these are the references you can go through for the comprehensive idea about this particular
topic or about particular issue OK?
Thank you.
175
Management of Fixed Income Securities
Prof: Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 03
Lecture No # 12
Level of Interest Rate Determination – III
Welcome back. So, in the previous class, we discussed about the different factor which are
affecting the demand for and supply of the bonds. Accordingly, your equilibrium interest rate
in the market gets changed.
(Refer Slide Time: 00:37)
So, in today’s discussion, we will be discussing about the another theory which is also is used
for determination of the interest rate in the market that is called the Keynesian theory and also
the concept of liquidity trap that you might have the idea but again I will just discuss this
things again that what exactly the liquidity trap is? And how; it is basically relevant in terms
of the interest rate determination
(Refer Slide Time: 01:07)
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and after this discussion, we will come through basically the different concepts like
transactions demand for money, precautionary demand for money, speculative demand for
money, supply of money. These are the keywords what basically will be using for this
particular discussion.
(Refer Slide Time: 01:31)
Let me tell you what basically the Keynesian theory talks about regarding the interest rate.
You just recall that whenever we discussed about the classical theory the classical theory tells
the interest rate is a complete real phenomena. The real factor in the economy is going to
determine the level form of interest rate. The level of interest rate is determined by the real
factors. So, in that case, we have basically taken the variables like total savings and the total
investments and this savings is basically is the supply side and investment is the demand side.
Whenever there is a equality between the saving and investment, our equilibrium interest rate
177
in the market is determined. OK? But whenever we talk about the Keynesian theory, Keynes
theory basically talks about or totally, he is not in favour of the concept that interest rate is a
real phenomenon or interest rate is determined using the real factors.
That is why according to Keynes, interest rate is purely a monetary phenomenon. OK? So,
the Keynesian economics mostly talks about the short-run dynamics. According to the
Keynes or according to the Keynesian theory, interest rate in the short run is determined by
the all those monetary factors which are available. Mostly, it is the interaction between the
supply of the money and the demand for money in the economic system.
But Keynes Keynesian theory mostly concentrated on the demand for money. Why? Because
according to Keynes, the supply of money is exogenously determined which is driven by or
which is determined by the regulatory authority the monetary authority. So, that is why the
interest rate is completely depends on the actions of the monetary authority and the attitude of
the economic units or the stakeholders or the market participants likes individuals, corporate
sector and other bodies who are interested to hold the money or demand the money for some
specific reasons. The demand basically comes from different economic entities and supply
basically comes from the monetary authority. So, that is why the demand for money and
supply of money that is going to determine the interest rate. The real factors are not
significant in terms of the determination of interest rate in the market. That is the basically
the basic essence of the Keynesian theory.
(Refer Slide Time: 04:43)
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So, in that context, what is the assumptions what basically Keynes has taken? That you just
see that what is the assumption that Keynes has taken? One basic assumption Keynes has
taken that only there are 2 main categories of assets are available in the market. One is money
another one is the bond and the total wealth of the economy must be equal to the total
quantity of the bonds plus the money in the economy because only there are 2 assets are
available money and bond.
So, if your only 2 types of assets are available then total wealth in the economy must equal
the total quantity of the bonds plus the money in the economy which basically equals the
quantity of the bond supplied plus the quantity of the money supplied. Right? Total bond
means it is basically the quantity of the bond supplied plus the quantity of the money supplied
then the quantity of the bonds and money that people want to hold. That means it is basically
demand side. That must also equal the total amount of the wealth. So, that means the quantity
of the bonds and money supplied must be equal the total quantity of the bonds and money
demanded. Whichever if there is an equilibrium, then there are 2 assets then the total money
supplied and total bond supplied if you take a summation of that, that is equal to your total
demand for money and as well as the total demand for the bonds.
(Refer Slide Time: 06:34)
So, if you look at that identity, what basically it says that BS represents your bonds supplied
and MS represents your money supplied is equal to your bond demanded that is BD + MD.
MD represents the money demanded, BD represents basically the bond demanded.

BS + MS = BD + MD
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
BS - BD = MD - MS
Supply is equal to demand. If there is a equilibrium, then only interest rate can be determined.
So, let I have taken this BD into the left hand side then what will happen that BS – BD is equal
to your MD – MS.
For example, if the market for money is in the equilibrium, let MS = MD.
Then, MD – MS = 0
If MD – MS = 0 then what will happen?
Then BS – BD also is equal to 0.
So, if your BS – BD = 0, that means BS = BD.
It means that the bond market is also in the equilibrium. If MS = MD, then BS also should be
equal to BD.
One restrictive assumption here the other alternative assets which are nowadays we are
observing in a market Keynes basically has not considered those assets. If there are 2 assets
only one is money and other one is bond so in that case what will happen? That always you
will observe that there is equilibrium in the money market.
If the money is in the equilibrium, then the bond market also will be in the equilibrium. So,
the demand for money equal to the supply of money then demand for bond is equal to the
supply of bonds. That is the argument what Keynes has given and already what I told you that
the money supplied is determined by the monetary authority. It is exogenously there is a
debate but according to Keynes, the supply of money is determined by the monetary
authority.
So, we are much more concerned about the demand for money. So, we have to look at what
are those factors which drive the demand for money in the market.
(Refer Slide Time: 09:07)
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So, then in that context, Keynes has given lot of importance to the demand for money and the
demand to hold the money is generally called the liquidity preference the other name of the
Keynesian theory is also called the liquidity preference theory. The demand to hold money is
called the liquidity preference. And why we need money? Why we hold the money? What are
the reasons for holding the money?
We hold the money for what reasons? First reason is the transactions day to day transactions
and as well as we also keep some money for our precautionary reasons for some unforeseen
reasons and as well as we also need money for speculative reasons because we want to
enhance our wealth by making the investments. So, money is demanded for 3 reasons: for
transactions, for precaution and for the speculation. That is the 3 motives what Keynes has
highlighted whenever he talked about the concept of the demand for money.
(Refer Slide Time: 10:35)
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So, then let us discuss what is do mean by transactions demand for money? What do you
mean by transaction for money? Transactions demand for money basically what? Here the
consumer basically needs for transactions purpose mostly for buying and selling of the goods
and services and what are those major factors which affect your transactions demand for
money? Obviously income if your income is more then you will consume more spending
habits and alternative mode of the opportunities through which the spending can be made.
In today’s contest, if you see because of the online transactions and all people are basically
spending more. If their income is more, it takes fraction of second to buy the commodities
and people some people are inclined to buy many things even if they have the things. So,
spending habits is important factor for consumption. Then the time interval after which the
income is received how frequently you are receiving your income.
In short period of time you receive some whether your income you are receiving periodically
in terms of months or in terms of week or in terms of day that will also have the impact on
your transactions. Banking developments why banking developments were talking about?
Sometimes, if the banks are highly developed or the financial sector is developed, people
have the alternative kind of use of the money so, sometimes, the consumption may go down
they may go for the savings if there is an availability and the industrial stock remains how the
industrial development or corporate sector is developed. So, you can take all those factors
which are affecting consumption. Income is the major factor, spending is another factor. So,
these are basically the transactions demand for money factors why people basically or what
factors basically drive the transactions demand for money.
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And Keynes has highlighted certain factors but you can add also more number of factors into
this which are responsible for the consumption behaviour of individuals. Right!
(Refer Slide Time: 13:15)
Then second thing is precautionary demand for money and why the precautionary demand for
money is required? Precautionary demand whatever money we are keeping in bank that may
not be all money for precautionary reasons that may be for the speculative reasons also to
some extent. But the precautionary reason means we are talking about the particular kind of
unforeseen contingencies.
We don’t know what will happen in the future so, because of that some kind of money has to
be kept with us to fulfil that requirement that is basically called the precautionary motives.
Then what are those factors which may affect the precautionary demand for money? First of
all, the nature of business what kind of business you are doing. If it is highly risky business
let a corporate sector which is doing a business which is highly we can say that highly
volatile in nature.
The nature of the product what they are producing they are highly seasonal or there may be
prone to high fluctuations because of some kind of factors any small changes in the economy
is going to affect the business largely. So, they will keep more money for precautionary
reason. Access to the money market you see which is the immediate market the financial
entities always rely upon to fulfil the short term requirement mostly the money markets.
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But there are many countries where access to money market is only restricted to some entities
you don’t have that much access or the sufficient amount of resources you may not get from
that particular market whenever we need it. There is a restrictive market the market
participates in the money markets are quire limited. So, because of the limited access maybe
they are always inclined to have certain kind of what we can say that money what they need
to fulfil their precautionary motives.
Degree of conservatism:-How conservative you are in terms of your business in terms of
taking the risk in the market. So, if you are highly conservative that means your riskiness or
exposure towards the risk is relatively less, then in that case, you may not need more money
for the precautionary motives. But sometimes, if you are not conservative, you are very much
aggressive in terms of investments and other kind of activities in the market. So, in that
particular point of time, some money has to be kept to fulfil or to overcome the risk if there is
any kind of risk you may face in the market that can be overcome by the money which is
available for the precautionary reason. Then degree of liquidity of the assets: - how liquid the
asset is whatever you are holding. So, if your assets are not that much liquid, they are not that
way marketable in this particular market or at any point of time, you are not able to get back
your cash by liquidating that asset then also you need some money to be kept with you to
fulfil that particular immediate requirement.
Then obviously the income is another factor which always have the impact on transaction
motive and as well as the impact on the precautionary motive. More the income you have
then obviously some amount of money always you will be interested to keep for fulfilling the
precautionary demand for the individuals or the market participants. So, these are the factors
which are affecting the precautionary demand for money.
(Refer Slide Time: 17:10)
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Then the other factor basically talks about that is the speculative demand for money. So, what
is the speculative demand for money? Here basically the money which is people kept for
making speculation in the financial market particularly you are talking about the investments
and there is inverse because already in the beginning, we have taken only one asset is
available in the financial market for investment that is bond because Keynes has taken that
the only 2 types of assets are available one is your bond another one is money and there is a
inverse relationship between the interest rate and the speculative demand for money and the
Keynesian theory is largely concentrated on the transactions and speculative demand. The
explanations on precautionary demand for money although Keynes has highlighted that
because Keynes has not discussed much about this. So, the relationship between interest rate
and speculative demand for money if you see that is generally called the liquidity preference
curve. OK?
So, the interest elasticity of speculative demand for money increases when the interest rate
declines. The interest elasticity of speculative demand for money increases or the interest rate
declines why? If the interest rate will increase or decrease if interest rate will decrease, then
the price of the bond will be high. OK? So, in that case, you will have a impact on the
demand for bonds and as well as the demand for cash or the money whatever people are
trying to hold.
So, that’s why there is a kind of relationship you can establish between the change in the
interest rate and as well as the demand for money in the market at that particular point of
time. So, in this particular context, what basically Keynes has done.
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(Refer Slide Time: 19:23)
Keynes has derived a concept called the liquidity trap. What this liquidity trap is? The interest
rate at which the speculative demand for money becomes perfectly elastic that is called
liquidity trap. So, if you are reaching in a particular interest rate level that means that is called
the liquidity trap interest rate, then the wealth holders hold their entire wealth in terms of
money instead of holding the interest bearing bonds.
So, at the liquidity trap rate of interest, money becomes perfect substitute for the bonds. You
see that whenever we talk about this if interest rate is higher then what will happen? The
price of the bond will be lower. In that case, people will be inclined to invest in the bond
market because they will it will be cheaper for them to invest but gradually, if the interest rate
will decline and reach in a certain stage, in that case, if you will observe, that the demand for
money is perfectly elastic.
It will not basically affect this kind of we can say that whether they will go for investing in
the bond market. That means it will just work like a substitute. The money basically it will be
substituted by the bonds. So, there is a perfect substitution happens in that particular zone that
is basically called the liquidity term. So, that concept all of you might have the idea but we
just because this particular topic is linked to this that’s why we are discussing this.
So, in that context, the relevance of the liquidity trap is quite relevant whenever we talk about
the interest rate determination using the Keynesian theory. But now the question is that what
is the question here?
(Refer Slide Time: 21:39)
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What factors basically change the equilibrium interest rate as per the Keynesian theory? So,
Keynesian theory talks about the exogeneity of the money supply and only thing is we can
play with the demand for money and there are certain motives for the demand for money. So,
that’s why, Keynes if you summarize Keynes theory, Keynes largely talks about the concept
called the income effect. What is the income effect here?
The higher level of income causes the demand for money to increase and that’s why the
demand curve will shift to the right and another thing also price effect the rise in the price
level causes the demand for money to increase and again the demand curve will shift to the
right. Price level means we are talking about the price of the goods and services. Right? So,
an increase in the money supply which is generally triggered by the central bank or monetary
authority will shift the supply curve up for the money to the right.
If money supply will increase and that is in the hand of the regulatory bodies or the monetary
authority or the central bank, then what will happen? That this will be a vertical axis this will
be a vertical line which is basically touching your horizontal axis of that particular kind of
curve. So, in that case, if there is a change in money supply, then that particular vertical line
may shift. So, when money supply increases, everything else remains equal then interest rate
will decline.
If money supply increases, everything else remains same the interest rate will decline that
means if interest rate will decline the money supply will increase that also you can say. If you
are using money supply as an exogenous variable then it will go in this way. If money supply
increases then interest rate will decline. Right? But if you assume that the money supply is
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not exogenous money supply is indigenous in that particular case maybe the reverse thing
may happen and if the other way you can discuss it that the interest rate will decline then the
money supply will increase not a reverse thing but other way around we can discuss that. Can
start from the interest rate and go to the money supply. But in this case, we are coming from
one supply to the interest rate. OK?
(Refer Slide Time: 24:45)
So, let us again look at this things from the diagrammatic point of view that already I have
told you there are income effect and the price effect and we have the money supply there are
3 things. So, let the same way we have taken your variables then we have the change in the
variable OK? Then we have change in money demand and money supply. OK? Then we have
change in interest rate then we have the shift in the curve. So, first variable we have taken
major variable that is income if it is increasing, then your demand for money will increase
and your interest rate will increase then now if you see that how basically it will so, here your
money supply is constant. It will not have an impact on the money supply. So, this is your
money supply and money demand, this is your interest rate. So, now previously let the this
was your demand curve 1 so, whenever income will change then what will happen that it will
basically the demand for money will shift towards the right. Right?
So, previously this was the equilibrium interest rate this is let i1 so, this become now this is i2
the new equilibrium has been established the interest rate has been changed interest rate has
gone up. So, then we have the price level. So, let it has gone off then obviously your demand
for money will increase. Here also, if you find, the interest rate will increase because the
same thing will happen supply is constant. OK?
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This is your Md 1, this is your Md 2 so, obviously, this was your i1 this will become i2 so,
the interest rate has changed. Let we have taken the money supply because it is in the control
of the authority. So, if this will increase, then money supply will increase then in that case,
what will happen? The interest rate will decline. If you observe here, let this was your money
supply before now the money supply this was your demand, this is your Ms, and this was
your equilibrium interest rate. Now due to increase in the money supply, it will shift like this.
This is Ms 1, this is Ms 2, Right? Demand is fixed. So, in that case, this was your i1 now the
interest rate has declined to i2. So if you increase your money supply keeping your demand
for money it remains same then what will happen? That I will it will basically decline the
interest rate because we are assuming that the money supply is the total control of the
monetary authority.
Whenever they want, they can change it. OK? So, this is the way basically the demand for
money and supply of money curve can shift. Mostly, the income, price, money supply these
are the major factors which are responsible for shifting the demand for money and the supply
of money according to the Keynesian framework. OK?
(Refer Slide Time: 28:52)
So, what basically we have discussed today? As per Keynesian theory, the interest rate is
determined by the interaction between the supply of money and the demand for it in the
economic system and there are 3 motives behind the demand for money. There is a
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transactions motive, precautionary motive and speculative motive and money supply is in the
total control of the monetary authority which is completely exogenous to the system.
Then at the liquidity trap rate of interest, money becomes a perfect substitute for the bonds
and the income, price of commodities and the government policies are the major determinants
of interest rate and already I have told you according to Keynes also, the interest rate
determination is a completely a monetary phenomenon and it has a lot of significant in terms
of the equality for the demand of money and the supply of the money. OK?
(Refer Slide Time: 29:53)
So, these are the references what you can go through for the detail discussion on this.
Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 03
Lecture No # 13
The Term Structure of Interest Rates- 1
Welcome back. So, in the previous class, we discussed about the Keynesian theories of
interest determination which is popularly known as the liquidity preference theory and also
we discussed about the classical theory and the loanable fund theory which basically tries to
explain that how the interest rate in the market is determined. So, broadly, we discussed about
the 3 theories: one is your classical theory then we discussed about the loanable fund theory,
then we discussed about the Keynesian theory.
Overall, if you see, today, we will start a discussion on a different note what we call the term
structure interesting theory. But before that, I just wanted to share something with you.
(Refer Slide Time: 01:10)
If you want to make a empirical framework of the determination of the interest rate the level
form of interest rate, then how this particular model will look like?- number 1, number 2 we
will start the discussion on the term structure of interest rate which is more relevant from the
fixed income securities point of view.Right? Because if you see that whenever we discussed
the bond yield concept the total return concept, then the concepts related to the different type
of what we can the spot rate calculations, bootstrapping method and all these things.
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So, those things are linked to the term structure of interest rate. Further, you will come across
that how these particular things are basically linked.
(Refer Slide Time: 02:07)
So, after this, you will understand certain things about the risk premium, yield curve and the
concept of the maturity spread. So, those things basically will come to know after this
discussion of the today’s sessions.
(Refer Slide Time: 02:28)
So, already I told you that you see that we have discussed about the three different theories of
interest rate determination in the market. But if I want to formulate a model on the
determinants of interest rate, how the model looks like and what are those fundamental
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determinants of interest rates? Fundamental factors which; drive this interest rate or which
affect the interest rates.
Here, I have just simply write a simple equation. Here, your i represent your interest rate and
RFR basically the risk-free rate of interest, EI represent the expected rate of inflation then
your risk premium you know what does it mean? It means these 2 the risk-free rate of return
and this expected inflection that what I am talking about these two basically comes from the
economic forces, economic factors.Right?
But whenever you talk about the risk premium, this basically based upon the bond
characteristics, the types of the bond, the basic nature of the bond, how basically what type of
bonds we are dealing with? Whether it is a corporate bond or it is a government bond or it is a
long-term bond or it is a Short-term bond. So, those things basically responsible for
determination of this particular interest rate, which basically a major part linked to the termstructure theory.
So, we have some economic forces some forces which come from the bond prospect typical
type of bond prospect. So, this is the way we can decompose the factors.
(Refer Slide Time: 04:40)
So, now what are those economic forces which are basically responsible? If you want to
formulate a model empirical model for the determination of interest rate in a simple model,
so, in that case, you will find that the real growth rate of the economy, which is again a proxy
for the business cycle and other things then that we consider then the tightness or ace of the
capital market, what is the condition of the capital market?
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The capital market is quite developed, we can say that we can have a highly liquid capital
market we have transaction cost are less. In that case, that will have the impact, that how the
investors are inclined to invest in different type of securities, how they are inclined to invest
in stocks? How they are inclined to invest in bonds? What is the return what they are
expecting from the different assets? How they are going to basically change?
So, in that particular case, the tightness or the age of the capital market is a very important
factor which can have the impact on the interest rate. Obviously, expected inflation how the
inflation is going to affect the interest rate that we have discussed a lot. Then we have the
supply and the demand for the loanable funds that we have discussed extensively. There are
many factors which are affecting the supply and the demand for the loanable funds. So, you
see this is a simple model what I am trying to discuss with you.
But there are other factors also which may have the implications and may have the impact on
the interest rate determination and they can be considered as a economic factors. You see if
you talk about India, the foreign capital inflow FII. So, these are also the responsible factor
which can affect the interest rate. Another factor the government policies you see monetary
policy authority can change any policy rate.Any policy rate change have the implications on
the market rates.
For example, you talk about India if the repo rate will be changed, then it will have the
impact on the call money rate because call money rate is an intermediate target. If the call
money rate will be changed, then it will have the impact on the lending rate and the deposit
rate. Then in that process, the whole market interest rates overall market interest rate is going
to be changed.
So, the government policies monetary policy or the fiscal policy capital expenditure and other
things also will have the implications on the interest rate. FII (Foreign Institutional Inflows),
which is coming to India that basically also have the implications on the interest rate. So,
there are many factors and those factors may vary from country to country on the basis of the
types of the country on the basis of the regulatory norm and as well as the conditions the
market conditions in that particular context.
So, that actually you can formulate a model you can also see the bond characteristics what
type of bond it is credit quality? What kind of rating the bond has it is a triple A rated bond or
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triple B rated bond or double B rated bond or only A rated bond what kind of bond it is
accordingly, your interest rate will change.
Term to maturity: All of you know that we have discussed a lot that the long-term bonds are
more risky than the short-term bonds because interest rate is not predictable accurately . If the
interest rate cannot be predicted accurately then what will happen? Then it will have the
implications on the the interest rate of the bond or the return from the bond. So, in that case,
if somebody is interested to invest in the long-term bond market or the issuer wants to attract
this investor to invest in the long-term bond market when they have to provide a certain
premium.
The risk premium has to be given, so the higher term to maturity linked to the higher risk
premium. Then we are talking about the indenture provisions.The indenture provision means
this means, what type of agreement or contract the bond has whether the bond has a call
feature, or it has a put feature, or is there any kind of options which are linked to that
particular bond? That will have also the implications on the interest rate involved with respect
to that bond or the risk premium what you are getting if you are going to invest in that bond.
So,that is that is another thing you have to keep in the mind that is called the indenture
provisions of the bond. Then, if you are investing in a foreign bond, then you are also
exposed to the exchange rate risk and the country risk. Country risk means you are talking
about the political risk another type of risk which can prevail which is specific to that
particular country. So, you can formulate your model based on aggregate macro-economic
factors and as well as some of the factors which are responsible for the risk premium
component.
In the beginning,I started the discussion with that that i is equal to your risk spread rate of
return which is the minimum return what you are going to get plus your we can say that the
inflation because expected inflation what we talk about then we have some risk premium.
There are many factors which can affect the risk premium and some of the factors are
responsible from the bond prospective or individual bond perspective.
So, these are the different characteristics which can affect the risk premium of the bond. So,
overall, you can formulate your model in that particular context.OK?
(Refer Slide Time: 11:17)
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So, let us come back to our discussion, what basically we are trying to discuss in today's
session that is basically your term structure interest rate theory or term structure interest rates.
We will come back to the theory later. So, what basically exactly the term structure interest
rate theory means or what do you mean by the term structure of interest rate? The term
structure of interest rate basically gives your idea that why there is a return difference of the
bonds with the different maturity keeping other things remains constant.
So, to understand that, we have to first understand, what does it mean? What do you mean by
the term to maturity, all of you know that but what is the term to maturity? The term to
maturity means it is the basically time which is remaining on a bonds’ life.OK? And in
general in the market if you look a,t there are three types of classifications you will find in
terms of the maturity one is your short-term bonds, intermediate or long term bonds.
You see periods are varying some people say that the short-term bonds the maturity period is
1 to 3 years.In some countries, it takes 1 to 5 years.In U.S., it goes from 1 to 5 years. The
intermediate term, somebody say that it is 3 to 5 years or 3 to 7 years but in U.S., it goes from
5 to 12 years and longer term bond, it goes beyond 12 years.
So, there is no proper definition that what up to what maturity we consider this is a short-term
bond this is a long term bond this is intermediate term bond but these are the classifications
you can make. Intuitively you can consider. Ok. If the bond’s maturity is up to 3 hours, we
can consider it is a short term; from 3 to 5 years, we can consider it is a intermediate term or
5 more than 5 years, it is a long term that you can decide there is no specific rule of thumb
which can give you the idea that which one is long term and which one is short term.
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So, there is a spread between the 2 maturity segment bonds which are traded in the market.
Because the long term maturity bonds gives more return than the short term maturity bonds
or intermediate term maturity bonds give more return than short term. Or if you compare
between the long term and intermediate term, you will get more return in the long term that is
the general notion and that spread is called the maturity spread.
You see I am talking from the general perspective but that may not happen also there are
certain kind of situations where this kind of rule may not prevail, that will discuss that what
context this may not prevail. But there is a spread which exists between the two maturity
segments that is called the maturity spread. Right?
(Refer Slide Time: 14:25)
So, in that particular case, if you formally define what the term structure of interest rate
means? How the interest rate differences can be observed between the similar bonds with
different maturity? Remember similar bonds this is quite important. Second line, if you see
the term structure of interest rate is a static function that relates the term to maturity to the
yield to maturity for a sample of bonds at a given point of time. Time is important at a given
point of time, that is important.
Similar type of bonds that is important, the term structure interest rate basically gives you the
idea why there is a differences in terms of the yield of the same type of the bonds with the
different maturity only maturity is different. But the characteristics of the bond are similar
because if you change your characteristics, automatically return will be changed.Right?
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So, because of this, you have to focus on these 2 and this term structure of interest rate what
basically we are discussing here if you plot this YTM, again the term to maturity of the
similar bonds that is called the yield curves.Why is it quite popular? We are very much aware
about the concept of yield curve. Yield curve is nothing but shows the relationship between
the yield to maturity and term to maturity of the similar bonds.
So, yield curve is a graphical representation of the term structure of interest rate the term
structure of interest rate is if you want to graphically represent it, then yield curve is basically
trying to explain that.Right?
(Refer Slide Time: 16:34)
Now the question here is how the yield curve can be constructed? OK? The yield curve can
be constructed from the current observations which are available in the market. For example,
one could take all outstanding corporate bonds from a group in which the bonds are almost
identical in all respects expect their maturity. Already I told you in all other aspects, all other
kind of characteristics are more or less similar, only the maturities are different then you can
generate a yield curve.
Here, all the bonds are corporate bonds maybe the industry is also more or less same from the
similar industry.OK? But only difference is that the maturities are different. So, if you draw a
curve between the yield to maturity and term to maturity, then you can get your current yield
curve of particular bond, the corporate bonds. The investors who are basically interested in
the long run average yields instead of the current ones.
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Deal curve could be generated by taking the average yields over a sample period.That sample
period, here, I have taken seven years you can take 5 years you can take 10 years there is no
fixed time.OK? You take a average yield of a particular period then plot these averages
against the maturity if you want to have a long run picture. These are practically happens in
the market to understand that how this long run yield is behaving in the market.
How the long run yield curve looks like? You see yield curve is a very powerful technique
for prediction of the interest rate, prediction of the business cycle etc…We call yield curve is
a public good because everybody can use the yield curve to understand the behaviour of the
market in a
particular way. They can predict the interest rate. It is also considered as a
strategy in the market for the investment for prediction of the interest rate and another
approach is to generate a spot yield curve from the spot rates. That I will discuss with you
later. Whenever we already we have derived the spot rates and also will derive the implied
forward rate from the spot rate and after that, again we will go for plotting the yield curve by
using the spot rates. If you recall, how the spot rates are determined? Spot rates are decided
on the basis of the bootstrapping methods and in this case, we can also have a spot yield
curve which can be drawn from the spot rates.Right? So, these are some of the examples
whatever I have given by which the yield curve can be formulated or can be constructed.
(Refer Slide Time: 19:57)
What is generally the shape of the yield curve? The yield curve can be positively sloped. That
means the long term rates are higher than the short term rates and such kind of yield curves
are called generally the normal yield curve which are upward sloping yield curves. These are
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upward sloping and these are called the normal curves. They are usually convex from below
and the YTM is going to be flattering out in the higher maturity.
The curve will be relatively more flatter whenever the maturity period will be more that is
called the normal yield curve where the long-term bonds will give more return than the shortterm bonds. We can have a negatively slopped yield curve also with what we call the inverted
yield curve. Here, the short term rates give the greater returns than the long-term ones but
these are very unlikely but that happens in some countries some period those kinds of curves
we have observed we have seen.
So, these curves are basically known as the downward sloping yield curve. These curves also
tend to be convex where the yields are flattering out at the higher maturity, but the shape of
the yield curve is downward sloping that is another safe the yield curve can take. There are
various reasons one of the reasons is high inflation rate I can share with you, when the
purchasing power of the people is very low, in that time, that country can always experience
or there is a possibility that it can experience a negatively sloped yield curve or a downward
sloping yield curve but in
regular context, in the normal conditions, you will not observe
this inverted yield curve. Always, we find a upward sloping yield curve where the long term
bonds will give you more return than the short term bonds and also the yield curve can flat
irrespective of the maturity. Deal to maturity is not going to be changed completely
horizontal to the x axis. So, that is also another step which is again a hypothetical case.
Occasionally, the yield curve also can take a more complicated shape. Here, it can have both
positively sloped and negatively sloped versions. In some parts, it is positively sloped and in
some parts, it is negatively sloped in a particular period. So, this is basically called the
humped yield curve. That also you can experience where the some period, the long term rates
are more than the short term rates and some period, the short term rates are more than the
long term rates that possibility is also there.
You see these are very interesting kind of concept which can give a better idea about the
economic system.So, that is why the shape of the yield curve speaks many things and we can
judge about the economic conditions about the interest rate scenario and other things using
this particular kind of yield curve concept or this by reading the yield curve or by analyzing
the silicon. So, these are the different shapes what the yield curve can take.
(Refer Slide Time: 24:05)
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Let us see that how it looks like:- In the x-axis, you have term to maturity M.M represents the
term to maturity here, it is your YTM it is your upward sloping yield curve and this is your
downward sloping yield curve which is the inverse yield curve. This is a normal yield curve
and it can be flat and that is a humped yield curve this can go up to this may be, it can go like
this. In the beginning, it will be maybe It will go little bit in the upward sloping then started
go like this so, this is called basically the humped one. This is called the humped yield curve
that possibility is also there but this is very rare. In a particular period, either you can get a
normal yield curve or you can get inverse yield curve.The flat yield curve is also again a kind
of hypothetical conditions.
(Refer Slide Time: 25:16)
So, which are the factors which can affect the shape of yield curve? The types of the bonds
what you are considering whether it is triple a-rated bond or whether it is a triple b-rated
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bond or only b-rated bond like that. That is a factor because the triple-rated bond always
gives lesser return than the triple b-rated bond because the riskiness of that kind of bond is
relatively low. Economic conditions whether economic growth or recession is going on in the
economy or there is a tight monetary conditions the monetary policy is very much restrictive.
There is a contractionary monetary policy, is operating in the system that also will have a
impact on this. Most important is that we will discuss more about these two theories: the
maturity preferences of the investors and the borrowers or these source maturity preferences.
Whether they are more inclined more interested to invest in the short-term bonds or they are
more inclined to invest in the long term bonds the horizon period, particularly the investment
horizon period that is also quite important that also you have to keep in the mind.
Because of the demand in one segment is higher then obviously there is a return difference
you will find may be short-term bonds will highly demanded that is why the short-term
bonds’ return will be more than the long-term bonds because the investment horizon period
for the investor will be shorter than the shutter. So,that is why, there is a demand for the
short-term bonds and long-term bonds demand is not there so, in that case, maybe the return
will be more in that case and also the investors or the borrower’s expectations about the
future rates, inflations and the state of the economy. So, these are some of the broader factors
what basically will give you the idea that how this particular shape of the yield curve looks
like, but you keep in the mind the shape of the yield curve is explained through the term
structure interest rate theories and we have 3, 4 theories which basically try to explain
that.We have a market segmentation theory we have a preferred habitat theory, we have a
pure expectations theory, we have a liquidity preference theory, or liquidity premium theory
what we call it.
So, these are the different theories which basically try to explain the shape of the yield curve.
So, we will discuss extensively about this, what these specifically theories talk about and how
this long-term interest rate is determined through these different theories and what are the
basic essences of this particular theory that basically over the sessions we will discuss. But
these are the common factors you can keep in the mind which can have the influence on the
shape of the yield curve.OK?
(Refer Slide Time: 28:34)
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So, what we have discussed today? We have discussed the factors which are affecting the
interest rate determination that mostly, there are some factors are economic factors and some
factors are specific to the bond characteristics and the yield curve basically shows the
relationship between the yield to maturity and term to maturity of the similar bonds at a given
point of time. Again, I am emphasizing on that and the shape of the yield curve generally
depends on the types of the bond, economic factors and the expectations about the future and
further will be discussing how this shape of the curve is explained through different types of
theories like your market segmentation theory or preferred habitat theory or the pure
expectation theory or the liquidity premium theory. OK?
(Refer Slide Time: 29:22)
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So, these are the references you can go through for a detailed discussion on this and further
will be discussing about the how the different theories are able to explain this yield curve or
the shape of the yield curve we can close today.
Thank you.
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Managements of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology Kharagpur
Module No # 03
Lecture No # 14
The Term Structure of Interest Rates – II
Welcome back. In the previous class, we started the discussion on the term structure interest rate
theories. There basically, 2 questions we always try to answer one is that why the differences
always be observed between the long term and short term interest rates and on the basis of that,
there are different type of yield curves we have observed or the shape of the yield curve basically
varies. So, what is the basic reason for that? Why basically the shape of the yield curve is
different?
So, to answer that particular questions, there are different theories basically were developed and
all those theories basically include the market segmentation theory. Then you have the preferred
habitat theory then you have the pure expectations theory then you have the liquidity premium
theory. So, over the sessions, we will be discussing about all those theories and how those
theories basically is trying to explain the shape of the yield curve or the relationship between the
long term and the short term yield of the different type of bonds.
(Refer Slide Time: 01:29)
So, in today’s discussion, we will start with the discussion on the market segmentation theory
and in this particular session, you will come across different keywords like maturity segment,
substitution effect, what is the relationship between economic expansions with the term to or the
yield to maturity or what is the role of the economic recession on the determination of the
interest rate or the yield to maturity? So, these are the things what basically will be covering of in
today’s discussion.
(Refer Slide Time: 01:36)
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(Refer Slide Time: 02:07)
So, let us see that what this market segmentation theory basically talks about. So, the market
segmentation theory what basically I told you that this is the one of the theories which try to
explain the difference between the long term and short- term interest rates or long term or shortterm yield in the bond market. But one thing you keep in the mind that this theory basically
assumes that the markets are segmented by the maturity. That means whatever short-term, long term, medium term bonds are available in the market accordingly, the markets are classified.
Another thing is that whenever you talk about this, the yield curve is determined by the supply
and demand conditions which are specific or unique to each maturity segment. That means the
supply and demand factors to determine the yield curve of a short-term bond market is different
from the long term bond market. So, therefore, there is no such dependence or no substitution
happens between the short term and long term bonds or the medium term and the long term
bonds like that.
But one thing it assumes there may be interdependence between the markets in different sectors
within that particular maturity segment. So, here for example if you see that if you are talking
about a short-term bond market, there may be a substitution between the short-term treasury
bonds and the commercial papers both are short term. The investor can decide between these two
depending upon the rates or the return what they are going to receive from these 2.
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So, which one gives the better return they can go and invest in that particular type of security. In
the same way, in the long term market, the substitution may take place between the long term
corporate bonds or the long term government bonds. So, that particular thing is basically
decided by on the basis of the returns what they are going to derive. So, one thing you always
overall, you should keep in the mind that the supply and demand factors which determine the
interest rate in the short term bond market is different from the supply and demand forces which
are operating in the long term bond market.
So, the yield curves are completely independent across the different 2 segments and the segments
are classified on the basis of the term to maturity of the bonds which are traded in that particular
market.
(Refer Slide Time: 04:58)
So, here, if you see that, whenever we talk about this concept let us give one example. You
assume that the yield curve for the corporate bonds having the similar quality which is very
important what do you mean by the similar quality? Similar quality in this sense let the bonds
rating are more or less similar. Because if you are comparing between a high rated bond with the
lower rated bond, you will not find the actual differences in terms of yield because that also is a
factor which decide which one give you the more returns.
So, if the bonds having the similar quality if you are dividing the market into 2 parts let there is a
short term bond market and there is a long term bond market. So let us see what happens within
that particular segment. Let you have a short-term market then in the short-term market, which
are the supply side and which are the demand side or the particular kind of factors which are
affecting the supply of the bonds and which are the factors which are affecting the demand for
the bonds?
So, whenever you talk about the supply side, the supply of the certain corporate bonds for
example that depends upon the business demand for the short-term assets. Why the business
sectors demand the certain assets? Because they have to basically fulfil the requirements like
inventories account receivables etc which are the short term requirements of the different
business units and for to finance that particular activities, they want to always issue these shortterm bonds and which one, basically; coming; from the demand side? The demand side basically;
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coming from the or the demand for the short term corporate bonds that basically comes from the
investors who basically is looking to invest their excess cash for a short period of time. Their
investment horizon period is relatively short. So, the corporate sector issues the bond to fulfil
their short-term requirements and the investor whose horizon period is relatively less relatively
short, they basically demand for that particular type of bonds and they were interested to invest
in that type of bonds and generally this demand for the short term bonds by the investor and
supply of the bonds by the corporations ultimately determine the interest rate on that particular
segment on that particular market. So, here what you have seen? You have seen some of the
requirements from the corporate side which basically compel them to supply the bonds and we
have seen certain kind of demand side factors where the investments is for short term.
So, those basically factors are determining the demand side. So, wherever there is a interaction
happens between these 2, the supply of such bonds by the companies and the demand for the
short-term bonds by the investor finally determine the interest rate
(Refer Slide Time: 08:13)
The same thing also if you see that how basically this works for example if you see that whatever
thing we have explained if I write that let the short term bonds. So, if you have a short-term bond
market. So, you can make it in 2 ways both from the supply side and demand side. So, how the
supply basically comes? The supply comes from the corporate sector and why they basically
supply? They basically financing of the short term requirements. OK?
And short-term requirements already we have discussed that is inventories then particularly this
working capital needs or the accounts receivables etc. So, in the demand side what we have
discussed the demand part why basically investors? Having short term horizon investment
horizon period or short term horizon. So, they need the money in the short-term basis. So, that is
why those kind of investors will go for demanding for this kind of bonds and the corporate sector
basically supply for this rejects.
Then wherever there is a interaction happens, so, in that case, the interest rate in that particular
market will be determined. So, that is the way basically the short-term interest rate is determined
in the short term segment.
(Refer Slide Time: 10:21)
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The same thing can happen in the long term segment also and why the corporate sector supplies
the long term bonds? The short term bonds are supplied by the corporate sector because of the
certain requirements like inventories account receivables etc. But in the long term bond case, the
corporate sector basically supplies the bond for the capital expenditure mostly for the plant
expansion, equipment purchase or acquisition of other companies. So, for these reasons, they
need some kind of financing and to provide the finance for this kind of activities, they can go for
supplying the long term bonds. So, the demand for such bonds generally which comes from the
investor side that basically either come directly or indirectly through this different institutions
like pension funds because the pension funds basically invest the money for a long term purpose
mutual funds, insurance companies all these kind of entities they note they always need the
money for long term reasons their investment horizon period is relatively longer because they
have the long term liabilities and also the investment horizon period is very long and again the
supply and demand of this type of bonds will decide the interest rate in that particular segment.
OK? So, that is the way basically the interest rate in the short-term market determined and this is
the way the long term interest rate is also determined. So, that is why there is no such
interdependence happens in these 2 different segments that is the basic essence of the market
segmentation theory.
(Refer Slide Time: 12:21)
In the same way, you can also write down that thing that here you have the long term bonds. So,
in the long term bond again you have a supply part and you have a demand part and already what
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basically we have discussed? The supply basically comes for the financing of capital
expenditure. Right? And demand basically comes from the investor side that has the long term
liabilities and long term investment horizon period. OK?
So, wherever the supply is equal to demand in that particular segment, the long term interest rate
will be determined accordingly. So, that is the basic essence of the market segmentation theory.
(Refer Slide Time: 13:25)
So, the short-term bond market is unaffected by the rates determined in the intermediate or the
long term market and vice versa. So, the independent assumption whatever the market
segmentation theory takes that is basically based on the premise that the investors and the
borrowers particularly the issuers have a strong need to match the maturities of their assets and
liabilities because their assets and liabilities period has to be matched. So, that is why this
particular independent assumption is based on that particular kind of logic.
So, the positions and the shape of the yield curves generally depend on the factors which
determine the supply and demand for the short-term, intermediate and the long-term bonds. So,
they are independent. So, the shape of the long term yield curve and shape of the short-term yield
curve generally is or we cannot say that short term and long term yield curve the yield curve
basically shows the relationship between the yield and term to maturity but the interest rate in the
long term market and the interest rate in the short-term market that basically is decided on the
basis of the supply and demand forces within that particular segment. That is the basic essence of
the market segmentation theory. Keep in the mind there is no such concept called short term
yield curve and long term yield curve but yield curve basically talks about the relationship
between yield and term to maturity and here the interest rate in the short-term market and longterm market is decided on the basis of the supply and demand forces within that particular
segment that actually you have to keep in the mind.
(Refer Slide Time: 15:23)
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Then if you see, what are those factors? Those factors can be the business cycle those factors can
be the state of the economy it can be the inflation or the expected inflation. The credit risk of that
particular bond which basically talk about the quality of the bond, liquidity of the particular
bond, how the bond is marketable in this particular market and also the sales and purchase by the
central bank because once the sales and purchase means that is a part of the monetary policy if it
is short term, it is monetary policy and for long term regions also sometimes the central bank
wants to use this bond instruments to control the inflation and other things. So, once you change
these factors or these factors will be changed, then that will cause the change in the structure of
the interest rate and which will be reflected by the shift and twist of the yield curve. And once
these factors will change the shape of the yield curve will change.
In a particular initial condition may be the shape of the yield curve looks different but once these
factors will change then the yield curve also gets changed the shape of the yield curve may
change or the yield curve may also shift. So, that is the basic factor which are affecting the shape
of the yield curve.
(Refer Slide Time: 17:03)
So, here if you see for example let the yield curve was initially positively sloped then any
economic or financial change which increase the short term let treasury bill rate then what will
happen? That will cause the yield curve to become flatter Right? But if the yield curve is initially
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negatively sloped, then what will happen? That the rate change; would cause the curve to
become even more negatively sloped. Right?
So, it depends upon the initial shape of the yield curve and accordingly any kind of factor which
will going to be change in the market that will have a larger impact on the shape of the yield
curve. But one thing you keep in the mind a change in economic or financial factors has not only
a direct impact on one sector. It also has an indirect impact on another sector. OK? So, that
means if the Treasury bill rates will be changed, that will also have the impact on the demand for
the corporate bonds market particularly the short term bond market. That is also going to be
changed.
(Refer Slide Time: 18:33)
So, that particular thing what we call it that is basically in our language we call it the substitution
fact. We will see that how basically that effect works. So, these are the factors which are
affecting the yield curve shift. Already I told you the broad factors we have economic expansion
we have economic recession, credit risk, credit tightening, deficit financing with short-term
securities, deficit financing with long-term securities deficit financing means I refer to if there is
any deficit with the government and government also can go for purchasing the long term
securities if there is a surplus and obviously we have the typical instruments of the monetary
policy like open market operation. So, depending upon the liquidity condition in the market, the
central bank can go and purchase the short-term treasury bills or they can also go and sell this
long-term treasury bills. Sale of the short-term treasury bills and purchase of the short term
treasury bills not the long treasury bills.
So, these are the generally the short term treasury bills are used in the short term market which is
a part of the monetary policy instrument.
(Refer Slide Time: 19:49)
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So, let us see that how the economic expansion is going to affect the yield to maturity. So,
economic expansion what will basically happen if there is an expansion in the economy? This
will increase the demand in the corporate sector for both short term and long term assets because
there are many investment opportunities. So, in that case, many companies will go for issuing
more short-term bonds to finance their larger requirements in terms of inventories account
receivables and all and they also issue more long-term bonds to finance their increase in the
investment in plants, equipment and other long term assets. So, in the bond market, what will
happen? All these actions will cause the increase in the supply of the both short-term and longterm bonds because they need money. So, because of that they issue this kind of bonds so that is
why the supply of these bonds will increase.
So, at the initial interest rate, the increase in the bonds outstanding creates an excess supply.
Demand may not be there may not be there much in that particular time so what will happen? If
there is excess supply, then the price will go down because demand is more than the supply. So,
the price will go down then the yield to maturity will go up. There is an inverse relationship
between yield to maturity and price:- if the price will be lower, than yield to maturity will be
more.
And the excess cause the corporate bond prices to fall and rates to rise until a new equilibrium is
reached.OK?
(Refer Slide Time: 21:41)
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So, in this case, how the substitution effects work in this case? What here we have seen? They
have let the whenever you observe that the long term and the short term corporate bonds
increase, then the short term and long term government securities become relatively less
attractive. Right? So, in that case, what will happen? The demand for the short term and long
term government bond decrease creating excess supply in both the markets at the initial rates.
Like that, the corporate bond markets the excess supply in the government security market will
cause their prices to decrease and the rates to rise until a new equilibrium is obtained. That
means any change in the government securities market will have the impact on the corporate
sector also. So, here what will happen that how basically that mechanism will work?
(Refer Slide Time: 22:56)
The economic expansion has a tendency to increase both short term and long term rates for the
corporate bonds and by a substitution effect; it increases the short-term and long-term treasury
rates. OK? So, the economic expansion will cause the yield curves for both the sectors to shift
up. But here one thing you keep in the mind we are only talking about the supply side but let you
bring the demand factor into the consideration.
If you assume that the economic condition also will affect the demand for bonds, then what we
can conclude? The expansion would cause the interest rate to increase provided that the supply
impact on the interest rate dominates the demand impact. So, if the demand impact dominates,
then an expansion could push the rates down. So, which one is dominating that we have to see
whether the supply is dominating and the demand is dominating accordingly the shifting of this
particular yield curve will take place.
(Refer Slide Time: 24:07)
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So, let us see what is the mechanism works here? Let in the corporate bond market there is a
increase in capital formation more corporate bonds sold which will lead to excess supply for the
corporate bonds. Then the corporate bond prices will decrease and yield will increase so that will
be upward shift in the corporate yield curve. In the same time ,in the government bond market,
what will happen? There is a substitution effect which will work as the corporate yield increase
the demand for treasury securities decrease and the treasury bond prices decrease then finally
your yield will also increase so that will be upwards stripped in the treasury yield card. So that is
the way basically the both the corporate bond market and the government bond market is getting
affected due to the change in the economic conditions.
(Refer Slide Time: 25:06)
So, we can also see how this particular shape basically looks like? So let this is your maturity
and this is your interest rate.Right? So, let this is your initial yield curve let this is for the shortterm bond this is for the long term bond so what will happen? Due to the economic expansion, so
I am talking about here it is corporate bond market. So, due to this economic expansion, what
will happen? So, there will be the price will basically decline because there is excess supply.
So, there will be a movement upward movement of this particular yield curve both in terms of
the short term segment and the long term segment. So, interest rate basically will automatically
decline in that particular increase if the price is declining, then the intertest rate is going to be
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increasing. The same thing will also prevail in the government bond market in the bomb and
bond market also you will observe.
So, this is your let short term bond this is your long term bond. So, again the movement will be
like this. So, there will be upward shift of the curve this is your short term this is your long term
this is your maturity this is your interest rate. Right? So, that is the way basically this will work.
(Refer Slide Time: 27:00)
The same thing you can also see with respect to economic recession. Obviously the capital
formation will decline then fewer corporate bonds will be sold excess demand for the corporate
bonds then corporate bond prices will increase then yield will decline. So, there will be a
downward shift in the corporate yield curve then the same thing will happen in the government
bond market there will be a substitution effect. When the corporate yields decrease, the demand
for government securities increase then government bond prices increase then yields become
declining.
So, there will be downward shift in the government yield curve. So, again if you are bringing this
demand factor into the consideration, then we have to see that whether the supply effect is
dominating or the demand effect is dominating. So, finally accordingly the yield curve will
move.
(Refer Slide Time: 27:55)
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So, you can also show it through the diagram it is just reverse of that let here it is on is initially
yield curve then there will be a downward shift. This is for the short term then this is for the long
term. So, this is your maturity this is your interest rate. So, this happens in the corporate bond
market and the same kind of observation you will find in the government bond market. OK?
Again the shifting will be in the downward side. This is your short term bond and this is your
long term bond. OK? The movement will be like this. This is your interest rate this is again your
maturity.
So, that is the way the movement basically will happen both in the corporate bond market and
the government bond market.
(Refer Slide Time: 29:05)
So, what basically we discussed today? So, the market segmentation theory says that the yield
curve is determined by the supply and demand conditions which are unique to each majority
segment and this theory is based on the idea of unique or independent markets and the factors
affecting the yield basically vary across the maturity segments and an economic condition causes
the yield curves for the corporate and government sectors to sit depending upon the expansion
and recession. There will be a downward or the upward moments. OK?
(Refer Slide Time: 29:44)
This is the reference what you can go through for the detailed discussion.
Thank you.
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218
Management of Fixed Income Securities
Prof: Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 03
Lecture No # 15
The Term Structure of Interest Rates – III
Welcome back. So, in the previous class, we discussed about the market segmentation theory
and how the market segmentation theory is trying to explain the shape of the yield curve or it
is a part of the term structure interest rate theories and in that part, we have discussed about
the how the economic expansion or economic recession is going to affect or going to shift the
yield curve and as well as we also try to explain certain other factors which also play the role
in terms of the shifting of the yield curve. Or there are certain other factors which also are
responsible for shifting of the yield curve.
(Refer Slide Time: 01:14)
So, in today’s discussion, we will be covering of or will continue with the same discussion. I
try to bring the other factors into the consideration that how the other factors also is affecting
the yield curve or how the other factors basically play the role in terms of the shifting of the
yield curve. So, in that case, one of the most important factors is deficit financing of the
government.
Whenever there is a deficit, government tries to overcome that particular deficit using certain
kind of instruments in the market and whenever they try to use those instruments, how those
particular activities is going to change shape of the yield curve.
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(Refer Slide Time: 02:05)
So, you will come across certain other things into the consideration like other keywords what
you will come across like credit risk, credit tightening, deficit financing all kinds of terms
basically will be using in today’s session. So, let us see that how the credit risks or the credit
tightening are going to affect the yield curve or how the yield curve is going to be shifted due
to the credit risk and the credit tightening.
(Refer Slide Time: 02:45)
So, if to analyse that particular thing in continuation with our previous discussion, we have
taken a 2 sector economy let there are 2 sectors which are operating in the particular system
one is government other one is corporate and there are 2 segments one is short term segment
and the long term segment. One thing you keep in the mind that we are discussing about the
independence of demand and supply forces across this particular segments that means the
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demand and supply forces of the short term segment determine the interest rate in that
segment and the demand and supply forces of the long term segment will determine the long
term interest rate. So, these are the 2 things basically you have to keep in the mind. So,, in
that context, we will covering of 3 things here within this particular framework in a 2 sector
economy government and corporate and there are 2 segments one is short term segment
another one long term segment. For the simplicity, we have basically considered we have
assumed in this particular way one factor is the credit risk and the credit tightening.
Second is that how the government can finance the deficit using the short term securities and
how this is going to affect the shifting of the yield curve and as well as the deficit financing
with the long term securities. So, these are the 3 factors what we are going to discuss today
within this 2 sector framework. So, all of you know that or in the beginning of this particular
subject, we discussed about the concept of the credit risk.
(Refer Slide Time: 04:50)
What exactly the credit risk is all about? The credit risk means basically the probability of
default that means if the issuer for some reason is not able to pay the interest or the principal
as per the agreement or as per the indentured provision of the bond, then we can say that that
particular issuer is exposed to the credit risk. So, generally if you see that if there is a
probability of credit risk which is going to increase in the future, then what will happen that
will have the impact on the demand and supply of the bonds?
Obviously this particular thing may prevail in the market whenever there is economic
recession so, whenever there is a recession; the investment opportunity in the market is
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relatively less. So, because of the less investment opportunity maybe investors are less
inclined to invest in the financial sector or financial market and then what will happen that
will have the impact on the yield or interest rate of that particular security.
So, if you assume that there is a credit risk which is involved in the market and because of the
economic recession there is a kind of tendency towards the increasing the credit tightening.
Then what will happen? That will lead to the increase in the demand for the short term and
long term bonds why? Because all of you know that whenever there is a credit tightening or
there is some kind of uncertainty or uncertain condition prevails in the market, the credit
exposure of the corporate bonds increases but because government is the guarantor or
government is in the backup of the government securities then what will happen? That the
people have more faith or investors have more faith on investing in the government securities
both in the short term and as well as in the long term. So, what will happen? The demand for
both short term and long term government bonds are going to increase.
But the demand for the corporate bonds will decline again keep in mind we have assumed a 2
sector economy government and the corporate. So, because of high credit risk and credit
tightening the demand for government bonds have increased and the demand for the
corporate bonds are declined. So, that thing will happen then what is going to happen in that
particular case.
So, whenever the demand for the government bonds (both short term and long term) will
increase that will basically lead to the rightward shift in the short term and long term
government bond curve. That already we have discussed whenever the demand for the bonds
will increase, that there will be a rightward shift of the demand curve. So, the rightward shift
in the short term and long term government bond curve and if there will be a rightward shift
if the demand of the government bonds will increase then what will happen? It will basically
lower or it will try to reduce the government bond rates. The yield of the government bond
will decline. So, then what will happen? It will reduce or it will shift the government bond
curve down and a leftward shift basically will be observed in the short term and long term
corporate bonds.
That will also further reduce the corporate yield. So, in that case, any kind of situation in
terms of the credit risk or the credit tightening will have the impact on the yield of the
government bond and as well as the corporate bond.
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(Refer Slide Time: 10:04)
So, then what basically will happen in that case? Or what we have observed in this case? The
combination of recession accompanied by an increase in the credit risk would lead to a lower
government bond yield and also the corporate yield and it will widen this spread between the
corporate bond and the government bond. OK? And if you can utilize that particular same
theoretical consideration assuming a particular economy which is expanding or economies
moving from recession to economic expansion, then what will happen?
It will reduce the credit risk and the credit tightening. Accordingly, the demand for the
corporate bonds also will increase. So, the reverse result you can find the or the opposite
result you can find whenever there is a expansionary economy or there is a economic
expansion which is prevailing in the market. So, what basically we have observed here? And
how this particular mechanism basically works here?
(Refer Slide Time: 11:24)
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That credit tightening or the credit risk exposure has the impact on the shape of the yield
curve. Or there is a yield curve shift can prevail because of this. Now, we can come back to
our discussion on the deficit financing. You know the government needs certain money for
the public expenditure and whenever there is a certain kind of deficit arises, government
needs certain resources to fulfil that deficit or finance the deficit and what is the convenient
way of financing the deficit? The most convenient way of financing the deficit is selling the
government securities and if you talk about the government securities, the most used
government securities let you take the example of India generally if you consider about the
deficit financing, mostly we can use this short term treasury bills.
So, if the government wanted to finance the deficit by selling this short term treasury bills,
then what will happen? It will increase the supply of the treasury bills. The short term
treasury bills supply is going to increase because government is ready to finance the deficit
by selling that. So, if the supply of the treasury bills will increase, then what will happen?
The price of that particular security will go down because supply is more than the demand
now the demand may not be that much in the market. So, if the demand is not much in the
market but supply is more, then what will happen? That will have the impact on the price that
logic already you know. So, the price of the short term government securities will go down.
So, if the price of short term securities or short-term treasury bill will go down, then what
will happen? It will increase the yield of the Treasury bill higher will be the interest rate on
the Treasury bill. So, it will basically increase the treasury bill rate.
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In the same time whenever this particular thing prevails in the short term government bond
market, what will happen in the corporate bond market? So, in the same time, the higher rate
on the short term government securities will lead to a crowding out effect. So, if there is a
crowding out effect, what will happen? It will decrease the demand for the short term
corporate securities and automatically it will decrease the demand for this short term
corporate securities.
And finally what will happen? In turn that would lead to an excess supply in that particular
market particularly the short term corporate bond market. Because the short term corporate
bond holders try to sell their corporate bonds in order to buy the higher yielding government
securities. That means government will crowd basically will go for crowding out the
investments or the kind of bond investments which is happening in the corporate bond
market.
Because there is a high interest rate which is prevails or high yield is prevailing in the
government securities market. So, when the bond holders will try to sell their short term
corporate bonds then the prices of such bonds will decrease and what will happen? It will
cause the rates on the short term corporate bonds to rise until a new equilibrium is
established.OK?
So, in that case, what basically we can find there will be new equilibrium which will establish
whenever the deficit financing will be done using the short term government securities like
the treasury bills.
(Refer Slide Time: 16:17)
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So, let us see what particular mechanism works here so what we have observed here? This
sale of the short term government securities increases both short term government and the
short term corporate yield rates. That is the thing what basically we have observed. So, now
the short term interest rate has gone up. So, already we have assumed as per the market
segmentation theory, the 2 markets are completely independent the short term market and the
long term market both are independent.
So, if both are independent, then what will happen? The adjustment the total adjustment
which is going to happen to the government sale of short term securities that will occur only
through the increase in the corporate and treasury bill rates. The long term corporate bond
market or the interest rate in the corporate bond market or interest rate in the long term
government securities market are not going to be affected.
So, in general, what we can observe if you assume that initially the corporate and the
Treasury bill yield curves were initially led flat. Then the government’s action will cause the
yield curve to become negatively sloped because the interest rate in two markets are going to
increase.
(Refer Slide Time: 18:04)
So, what is the summary you can get from here? Whatever thing just now we have discussed.
The government basically sells the short term treasuries or T-bills what we can say then the
T-bill prices will decline because the supply is more than the demand that is why the T-bill
prices will decline. If the T-bill prices will decline, the yield will increase. So, if the yield will
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increase, then there will be tendency for the Treasury bill yield curve to become negatively
sloped.
And substitution effect which is going to play the role in the corporate bond market how? As
the short term treasury bills yield increase, the demand for the short-term corporate securities
decline. Then obviously if the demand in the short-term corporate securities will decline, then
the short term corporate bond prices will decrease and obviously their interest or interest rate
also is going to increase.
Then there will be a tendency for the corporate bond yield curve also to become negatively
sloped. So, what we have observed through the substitution effect? Any kind of activities that
is happening in the government securities market will have the implication on the corporate
bond market. So, now, if you assume that there is a flat yield curve, the yield curve will
become negatively sloped in this particular condition.
So, now we can see that this shape of this particular yield curve how basically it will move.
So, let this is a Treasury bill yield curve and other one is the corporate bond yield curve and
we have taken a assumption that there was a flat yield which was prevailed in this 2 markets
initially. That means your y LT = y ST then what has happen? Because of the selling of the
short-term treasury bills now what will happen that the price has gone down. If the price of
the bond has gone down, then the interest rate has gone up.
So, this is the movement what basically you can observe in this case. So that is what basically
what we can see that this is the movement basically is going to happen both in the treasury
bill market and the corporate bond market and this thing will basically will happen because of
the substitution effect because of the substitution effect and that is basically happen because
of the crowding out effect.
Because when you find that the government bond market looks very attractive, people will
move from basically corporate bond market to the government bond market. So, that is the
way basically this yield curve shifting will happen or the slope of the yield curve is going to
be changed. OK? So, then let us see that this thing we have discussed with respect to deficit
financing through this short term securities and here our assumption is the long term
securities market is not going to be affected. The reason is as per the market segmentation
theory, both the markets are independent to each other. So, any kind of adjustment
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mechanisms which happens in the short term market that is not going to make any kind of
change in the long term bond market. That basically we have discussed in this particular case.
(Refer Slide Time: 22:20)
Let us see if you are going to finance their deficit in the long term securities you want to
finance the deficit using the long term securities and how this is going to affect the yield
curve? So, now to finance the deficit government has to sell the securities that are clear
because they need the money because of that they will sell the bond and against that they will
get the finance.
So, now the government basically sells the long term bonds. So, whenever government will
sell the long term bonds, long term bond prices will decline and obviously the yield will
increase. Just now, we have discussed that thing in the short term segment also. So, then what
will happen? There will be a tendency for yield curve to become positively sloped. OK? So,
in this case, what basically here we have seen the same thing prevail in the corporate bond
market through the substitution effect through the substitution effect.
When the long term government bond yield will increase, the demand for the long term
corporate securities will decline and the long term corporate bond prices will decrease and
their yields will increase and then you will find that there is a tendency for the corporate yield
curve to become positively sloped because you will find that if you understand that thing in a
different way, the reverse thing will happen whatever thing has happened in the short term
bond market.
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In the short term bond market, the yield curve become negatively sloped but whenever you
come to the selling the long term securities, you will find that the yield curve basically is
positively sloped.
(Refer Slide Time: 24:30)
And how the shape of the yield curve will look like just now, we have seen these things from
the short term investment point of view in short term securities point of view. Now let you
look at the long term securities point of view. In the long term securities context, what we
have seen if this is your let the maturity this is the short term one and this is a long term one
and this is your initial yield curve which is basically a flat yield curve. So, whatever
reasoning or understanding we have just discussed.
In that case, what we have seen that the long term yield has increased. Here also the long
term because of the substitution effect. So, this thing will prevail because of the substitution
effect. Now here also the long term yield has increased. So, whenever the long term yield has
increased in this case, here also we have seen the long term yield has increase then what will
happen? The long term bond interest rate which was here now it has increased to this level.
It has increased to this level. So, there is a upward movement in both the cases. So, because
of any kind of activities with respect to the selling of the long term bonds in the government
securities market that will have the impact on the interest rate prevails with respect to the
corporate bond market and that is basically what we call it the substitution effect. So, in both
the cases, what we have seen? Whenever you are selling the short term securities for the
financing, we find that the yield curve is negatively sloped because the short term interest rate
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has gone up and whenever we are going for this same activities in the long term securities
market or selling the long term securities in the government bond market, we can find that
there is a positively sloped yield curve because the long term interest rate has increased. So,
this is the thing basically which prevails in terms of the deficit financing through the selling
of the short term securities and the long term securities. OK?
(Refer Slide Time: 27:15)
So, what basically we have discussed here? The sale of the long term treasury securities
increases both the long term government and long term corporate rates that just now we have
seen and as usual because all kind of activities which is happening in the long term market
that is not going to affect the activities in the short term market because both the markets are
independent. OK?
So, the total adjustment to the government sale of long term securities will occur through the
increase in the long term corporate and government bond rates. This short term government
bond rate or the short term corporate bond rate are not going to be affected due to the
activities in the long term market. That is why the short-term rate remains constant that is
why the shape of the slope of the yield curve basically moves in this way.
So, if you are given a corporate and government bond yield curve that is initially flat, the
government’s action basically causes the yield curves to become positively sloped. OK?
(Refer Slide Time: 28:35)
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So, what we have discussed here? The combination of recession accompanied by an increase
in the credit risk will lead to the lower government bond yield and possibly the corporate
yield also and it will widen the spread between the corporate bond and the government bond.
And the sale of the short term government securities increases both short term government
and the short term corporate rates and the sale of the long term government securities
increases both the long term government and the long term corporate rates.
And accordingly in the short term market, your yield curve will be negatively sloped and in
the short-term market, it will be negatively sloped and in the long term market, it will be
positively sloped. Again I repeat in the short term market, the yield curve will be negatively
sloped and in the long term market,it will be positively sloped. So, this is what basically we
have discussed that means the deficit financing has a significant role and whenever we are
making the deficit financing by selling the short term or the long term securities, it will
basically change the slope of the yield curves both in the government segment and as well as
the corporate segment. OK?
(Refer Slide Time: 29:59)
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So, these are the reference what you can go through for the detailed analysis on this.
Thank you.
232
Managements of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology - Kharagpur
Module No # 04
Lecture No# 16
The Term Structure of Interest Rates - IV
Welcome back! in the previous discussion, we have discussed about if there is government
deficit then the deficit financing can be done through either selling the certain government
securities or selling the long-term government securities and whenever the government basically
sells the short-term securities or the long-term securities accordingly the slope of the yield curve
gets changed.
So, in case of the selling the short term securities the yield curve become negatively sloped and
in the context of selling the long term securities yield curve becomes positively sloped that is
what basically we have discussed in the previous class.
(Refer Slide Time: 01:06)
And in today’s discussion, let us discuss that if there is a surplus. then how the government’s
activities going to affect the yield and how the slope of the yield curve basically is going to be
changed that is basically our discussion. In the previous class, we discussed about the concept of
the deficit and today we will be discussing about the surplus.
(Refer Slide Time: 01:35)
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Let us see that, what are those keywords will come across in this particular discussion.
Already,we have discussed this crowding out effect the same thing will again come also here, the
concept of open market operation, the expansionary monetary policy, and contractionary
monetary policy. All those kinds of terms basically will be covering up in today’s session.
(Refer Slide Time: 02:02)
And again, we will continue with our discussion with the assumptions that there is a 2 sector
economy, government and corporate and there are 2 segments one is short-term other one is the
long term. Only thing is here, we are bringing 3 things here, one is if there is a surplus then
obviously government can try to utilize their surplus by purchasing the long term securities Or
they can also use the short-term securities which are in the context of the monetary policy.
Because, in our term we always assume that the instrument which is used to control the money
supply and economy are mostly done through the use of the short term securities. So, depending
upon the availability of the money supply or whether the government wants to increase the
money supply or they want to decrease the money supply, accordingly they use that particular
instrument one is your contractionary another one is the expansionary.
So, one of the instruments basically is the open market operation. So, that’s why there are three
things the last one the contractionary open market operation,with a sale of short-term treasury
bills that is completely similar with the deficit financing part which we have discussed in the
previous session. But these 2 are basically different or how the yield curve shape or the slope of
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the yield curve is going to be changed because of this kind of actions that the government is
trying to make.
That is basically is different from the analysis whatever we have done in terms of the deficit
financing part. So, we have 2 sectors 2 segments. each sector has these 2 segments, one is short
term other one is long term and we have 2 sectors, one is government other one is corporate.
(Refer Slide Time: 04:34)
Let us see, there is a surplus if government has certain surplus money or government has any
kind of surplus resources then how they can use it? They can always use it to buy the long term
government securities. So, if they will go and buy more long term government securities then
what will happen? Obviously, there is a decrease in the supply of the long term government
securities because now the government is demanding.
So, there is a decrease in the supply as the government is going to reduce the more number of the
long term bonds or supply of more number of long term bonds. The supply of the long term
securities will go down. So, whenever the supply will go down what will happen? Now the
supply is less than the demand even if it is initial demand if the supply is reducing then,
obviously demand will be more than supply. So, then what will happen?
It will basically push the prices of the long term government securities and the long term
securities prices will be up. So, whenever the prices of the long term securities will be up, then
obviously what will happen? The long-term government bond yield will go down. There will be
reduction in the yield of the long government bonds because now the supply is less demand is
more.
Now let us see what is going to happen in the corporate bond market. Recall the substitution
effect. So, whenever the long-term government securities yield will go down or it is lower then
what will happen? It will lead to an increase in the demand for the long term corporate securities
because the yield in the government securities market is low. So, now maybe investors are more
inclined to go and invest in the corporate securities.
So, that is why there is a possibility that there is a increase in the demand for the long term
corporate securities. Then what will happen,if they will be more interested to invest in the
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corporate long term corporate securities? Then, there will be excess demand in that market. The
demand of the long term corporate securities will increase. So, then what will happen?
That will basically lead to whenever many bond holder or bond investor try to buy the long term
corporate bonds then,what will happen in the end? The prices of such bonds will increase
because the demand is now more than the supply. Demand is there but the corporate sector may
not supply that much. So, in that case what will happen? The prices of such bonds are going to
increase. if the prices of such bonds will increase then, obviously what will happen? The yield on
the long-term corporate bonds will fall in the end until a new equilibrium is reached
So, initiation was started by buying the long-term government bonds through the substitution
effect. it has the implication on the return or the yield of the corporate bonds. So, in both the
segments now the yield has come down.
(Refer Slide Time: 08:50)
So then, if you summarize it then what basically we have observed here? The government buys
the long-term bonds and that basically will happen in the government securities market. So,
whenever government is buying the long-term bonds, then the long-term bond prices will
increase because the supply is less than the demand. Because of the high demand, the long-term
bond prices will increase and whenever the long-term bond prices will increase, the yield will
decline.
And obviously your government bond yield curve to become negatively sloped because the short
term bond market is not going to be affected; they are independent. Through the substitution
effect what we have seen? when the long-term government bond yields decrease, there will be
huge demand for the long-term corporate securities. That will lead to the increase of the long
term government bond prices and declining their yields.
So, in that case again there is a tendency of the corporate yield curve to become negatively
sloped. You recall! whenever we were selling, the long-term bonds yield curve was positively
sloped. in the previous class we discussed that. we have discussed, whenever the government is
selling more long term bonds the yield curve becomes positively sloped. But now, whenever
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government is buying more long term bonds yield curve becomes negatively sloped because the
price will increase and the yield will decline.
(Refer Slide Time: 11:02)
So then, how that yield curve shape will look like ? because you know this is the initial yield
curve, which was flat. This is the assumption whatever we have taken. So, this is your Short-term
bond and this is your long-term bond. so this was the initial interest rate which was there, little
bit it has moved. So, this is the initial interest rate. Then what has happened? Because of this
demand-supply adjustment which has happened in the long term market, what we have seen?
The short-term interest rate was there wherever it was there, but the long term interest rate has
come down. So now what has happened ? the shape of the yield curve gets changed. Now this is
a negatively sloped yield curve what is really basically we have observed and this is basically
will link to this. Because, here there is a substitution effect, which will work. Because of the
substitution effect, whatever thing has happened in the long-term government securities market,
the same thing will prevail in the corporate securities market.
So, here again the same thing. Here you have the short term, you have the long term, you have
the short term interest rate and here you have the long term interest rate. So, there is a downward
moment. This is what basically we have seen whenever there is a surplus and government wants
to utilize that particular surplus by buying the more long-term securities.
(Refer Slide Time: 13:26)
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So, what we have seen here? the purchase of the long-term government securities decreases both
the long term government and the long term corporate rates. It will decrease both the government
and the long term government and long term the corporate rates and the short term rates are not
going to be affected because they are independent as per the assumptions of the market
segmentation theory and the total adjustment to the government’s forces of the long-term
securities will occur through the decrease in the long term corporate and the long term compound
trades.
The same thing what we are repeating again and again because, that is the assumption we have
taken from the beginning. So, if the corporate and the government bond yield curves were
initially flat then the governments this kind of action will cause the yield curves to become
negatively sloped. just now what basically we have discussed.
(Refer Slide Time: 14:44)
So then, let us see what is going to happen in terms of the monetary policy? All of you are very
much through about the monetary policy what is the monetary policy? Monetary policy means
the basic objective of the monetary policies to control the money supply. why the government
wants to control the money supply? because they want to control the inflation and also the
increase the output. So, the objective of the monetary policy is multi-dimensional. but in today’s
context, if you look at the Indian context also we are largely relying upon the infliction targeting.
But the overall objective of the monetary policy based upon the increase in the output and as
well as the controlling the inflation. So, there are many instruments what we use, the instruments
are like Repo rate reverse Repo rate. All of you might have heard, these are the part of the
marginal liquidity adjustment mechanism. Even there is another rate called the marginal standing
facility.
Nowadays, central bank is also using the other type of instruments instead of using the reverse
Repo rate other thing. They are using another instrument that is called the deposit standing
facility. all kinds of instruments and what is the basic aim of this thing? That if the Repo rate will
change the Repo rate will increase then, what will happen? then it will have the impact on the
intermediary rate like call money rate.
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If the call money rate gets affected, then it will have the impact on the lending rates of the
commercial bank .. If the lending rates of the commercial banks gets; affected then it will have
the impact on the demand for loans and other things. Then accordingly, the total money supply
of the economy gets affected and another instrument one of the instruments what basically, we
can say that is called the open market operation.
In the open market operation, you see the liquidity adjustment mechanism is completely a shortterm phenomenon. The repo market which is used by the monetary authority that is the overnight
operations mostly under this liquidity adjustment facility. But whenever we are using another
instrument of the monetary policy let that is open market operations the open market operation
can be a certain concept can be a long-term concept.
So, either government can use the short-term government securities buying or selling or they can
use the long-term government securities also to control the money supply. Either they can reduce
the money supply or they can increase the money supply by utilizing these rates. So, if the
central bank is engaged in let you assume, that expansionary means they want to increase the
money supply. So, increasing the money supply means if you using the liquidity adjustment
facility mechanism.
Then, obviously they have to reduce the different rates like Repo rate and other rates or even
another instrument they can also use CRR. They have to reduce those rates and accordingly the
demand for loans and other will increase because of the lower interest rates. And if they are
going for a contractionary part that means they have to increase those rates but if they are using
the open market operation then here what they have to do?
If they are going for expansionary open market operation they have to buy the short term
government securities. that means more money they have to inject to the system and how they
can do this? they can do this only buying the short term government securities. So, whenever
they will buy more short-term government securities, then you will find that the slope of the
yield curve will change and there will be a tendency of the yield curve to become positively
sloped.
So, if the central bank is buying the long term securities as a part of the open market operation
then the yield curve may be tends to become negatively sloped. But if you go by the
contractionary open market operation, where they are going to sell this short term and long term
securities then the reverse thing will happen. So, in the contractionary open market operation
there would be a tendency for the government bond yield curve to become negatively sloped.
If the central bank is ready to sell some of its holding in the short term bills and it will be
positively sloped. if it were to sell some of the long-term security holdings. And the open market
operations what the monetary authority does or the behalf of the government the central bank
does that will also change the yield curve for the corporate securities through a soft register
effect. The same thing also will work here. you see all these things are quite similar quite linked.
So, due to the substitution effect, the yield of the corporate securities also will get changed.
239
(Refer Slide Time: 20:53)
So let us see what will happen in the expansionary open market operation and how it will affect
the yield curve? Already you know that, whenever you are going for the expansionary open
market operation, the central bank will buy the short-term treasury bills. So, if they will buy the
short-term treasury bills then what will happen? The table prices will increase and demand for
treasury bills will increase. So, in that case the treasury bill prices will increase and yields
basically will decline whenever the price is increasing the yield will be declining.
So, then what will happen? your yield curve to become positively sloped whenever the central
bank was selling the short-term treasury bills. In the deficit financing case we have seen yield
curve was negatively sloped. Now come to the corporate bond market what is going to happen in
that case ? again the same substitution effect will work here when the yield and the short-term
treasury bills will decline. The demand for the short-term corporate securities will increase
because investor may be more interested to invest in the corporate short-term securities
Then, when they will invest more on the short-term corporate securities, then the price of the
certain corporate securities will increase because the demand is more. Then obviously yield will
decline. Then obviously your yield curve will become positively sloped. So, either you can go
for the selling or buying, the substitution effect always works and there is a linkage can be
created between the government bond market and the corporate securities market.
(Refer Slide Time: 23:16)
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So, now you have seen. let us see through the diagrammatic way that, how this thing prevails.
Again, this is the initial yield curve. So now what has happened? this mechanism or adjustment
is happening in the short-term government securities market. the government is this
expansionary part, the government is buying this short-term securities. So, because of this the
demand is more. That is why the price has increased. So, if the price has increased the yield has
come down. So, this was the initial short-term interest rate.
Now it has come down to this, because the long term market is completely independent that will
remain there. So, then the curve basically will move the slope of the curve will get changed and
again this will basically will have the link with the corporate bond yield curve. Here, again the
your substitution effect will work. If the substitution effect will work, then obviously the same
thing will prevail in the market and the adjustment basically will happen in the short-term
corporate bond market by that the short term corporate yield will come down.
And this is the way the slope of the yield curve will get changed. So, this will happen in terms of
the expansionary monetary policy.
(Refer Slide Time: 25:20)
So, the long-term market is not affected that already we have discussed. So, if the both the
treasury and the corporate yield curve are initially flat, then the expansionary open market
operation will result in new positively sloped yield curve.
(Refer Slide Time: 25:41)
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Contractionary open market operation. Already we have discussed that. but it is same with the
deficit financing part. The central bank has to sell the short-term treasury bills. The treasury bill
prices will decline because why? Because, that why the treasury bill prices will decline? because
the bank is selling more treasury bills, the supply is more than the demand. So, that is why the
price will decline and the yield will increase.
So, that is why the treasury yield curve to become negatively sloped. So, in the corporate bond
market the substitution effect will work. So, because the yields on the short-term treasury
securities have increased, the demand for the short-term corporate securities will decline. So,
then the price of the short-term corporate security bonds will decrease and their yield will
increase. This is the prices of short term we can write bond one decrease and the yields will
increase. Then what will happen that the corporate will curve also to become negatively sloped ?
the same thing will prevail in both the markets.
(Refer Slide Time: 27:12)
So, you can also show these things to the diagram. in the same diagram whatever we have used
for the deficit financing part using the short-term securities. So here, you have seen that this was
your initial yield curve, this is your initial short-term interest rate, this is the long-term interest
rate. So, because of this adjustment, this short term interest rate has gone up. Long-term interest
rate remains there because it is independent. Then the slope become basically this gets changed
or the slope of the yield curve gets changed.
So, this thing will have the impact on this. so this happening, because of the again I am writing
here, this is because of the substitution effect. In both the markets, you will find the same kind of
adjustment mechanism will work.
(Refer Slide Time: 28:05)
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So, long term market is not going to be affected. So, because of that the initial rate will remain
intact there. So, if both treasury and
corporate yield costs were initially flat, then the
contractionary open markets operations will result in a new negatively sloped yield curve.
Negatively sloped curve will prevail in that particular market that actually you have to keep in
the mind. So, we have discussed about the if there is a surplus by the government, then how the
yield curve is going to be changed due to the government actions whenever they are utilizing that
surplus?
And whenever also they are operating or using the instruments like open market operation as a
part of the monetary policy instrument. Then how the slope of the yield curve also gets changed
both in the case of expansionary and as well as the contractionary?
(Refer Slide Time: 29:07)
So, in the conclusion what basically we discussed here, the purchase of the long term treasury
securities or the government securities decreases both long-term government and the long-term
corporate rates. Open market operation changes the yield curve of the government as well as the
corporate bond markets. And if both treasury and corporate yield curves are initially flat, then
expansionary open market operation will result in a new positively slopped yield curve.
And in the case of the contractionary case,t hey will result a new negatively sloped yield curve.
So, these are the different factors which basically will shift the yield curve and as well as the
slope of the yield curve gets changed. One is your selling the certain securities or buying the
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certain securities or selling the long term securities or buying the long term securities. So, that is
why any monetary policy instrument like open market operation has larger implications on the
set of the yield curve. In the bond markets, both long term and short-term and as well as in the
case of corporate and as well as the government.
(Refer Slide Time: 30:27)
So, these are the references what you can go through for idea about this. thank you.
244
Managements of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology - Kharagpur
Module No # 05
Lecture No # 17
The Term Structure of Interest Rates V
Welcome back! So, in the previous discussion we have discussed about the first part of the term
structure theory which basically talks about the market segmentation theory. There we have seen
that, how the markets are segmented and as well as the demand-supply forces of the short-term
market is not going to affect the demand-supply forces of the long-term market. So, the interest
rates which are determined, these are determined with respect to that particular segment.
And other things we have discussed that, within that segment there may be a kind of substitution
take place. That means if you are operating in a short-term market, you can substitute your kind
of securities with the treasury bills with the commercial papers. If you are operating in a longterm market, you can substitute your bond with long-term government bonds with the long term
corporate bonds.
(Refer Slide Time: 01:14)
So, then the other part of the term structure theory we have that is called the preferred habitat
theory, which is basically a extension of the market segmentation theory. in the true sense that is
linked to the market segmentation theory. Little bit we can relax certain assumption whatever we
have taken in the context of the market segmentation theory.
(Refer Slide Time: 01:44)
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So, in the preferred habitat theory if you see, that mostly will be dealing with the short-term yield
and long- term yield concept and how the long-term yield is related to the short term yield. That
particular concept would be used in this particular session.
(Refer Slide Time: 01: 57)
So, already what basically I told you? that what is the basic essence of the preferred habitat
theory? Here, there is a typical assumption that the investors and the issuers or the borrowers
whatever what you can use for that. They are willing to give up their desired maturity segment
and assume the market risk if the rates are attractive. In the previous case, what we have seen?
that everybody has their investment horizon period and they want to stick to that particular
segment and the demand-supply forces of that segment will determine the interest rate in that
segment.
But, here what basically the preferred habitat theory is trying to explain, that the investor can
also move from one segment to another segment or the issuer also can move from one segment
to another segment if the risk is compensated. So, they are going to take certain kind of risk and
already we know that, risk is nothing but the market risk, which means the interest rate risk. So,
if they are going to take more risk and the risk is compensated with the extra return what they are
going to get? so that, means they can move from one segment to another segment.
That means the yield curve basically you are going to determine through this preferred Habitat
theory, that basically gives you the idea that how the long-term interest rate is linked to the short
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term interest rate. So, in that particular context that is the basic assumption what basically we
have taken. So, therefore the preferred habitat theory basically always tells that the investors and
the issuers will be always ready to forgo their perfect hedges and shift out their preferred
maturity segment when supply and demand conditions in different maturity segments do not
match.
So what does it mean? For example, there is a huge demand in the short-term market where
supply is not there. But there is excess supply in the long-term market but the demand is not
there. In that particular context, there may be a shift can happen between these 2 segments. And
the investments horizon period which is basically matching the liability and the assets time
period or the maturity that also can be changed.
So, in that particular context, a new equilibrium point can be established with the interaction of
both the short-term yield curve or we can say that short-term market and also the long-term
market. so that is the basic essence of the preferred habitat theory.
(Refer Slide Time: 05:00)
So, let us see that what basically happens in that particular segment with certain kind of
examples. Already I told you, that it is an extension of the market segmentation theory. So, if
you assume that the economy is poorly hedged. poorly hedged in the sense what? Let the more
investors want short-term investments and more borrowers or the issuers want to borrow the long
term. Then, obviously, neither of these markets are in the equilibrium. Whether the short-term
market or the long-term market is in the equilibrium.
In that case, what basically will happen? both the short-term and long-term interest rate will
change and the market will clear or the equilibrium position will arrive when the investors and
borrowers give up their particular hedge. That means, I want the money for one period of time,
may be for one year. my maturity period or my horizon period is one year. But I am ready to give
up that particular segment whenever I am compensated by that.
Because, whenever I forgo that particular horizon period or my desired period that means I am
going to take more risk. So, in that particular context, that risk should be compensated with the
more return. So, that is why in the true sense the short-term and long-term interest rate is going
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to be changed because there is a kind of disequilibrium happens in both the markets. So, that is
the basic thing what basically the preferred habitat theory tells.
(Refer Slide Time: 06:55)
So, let us see with an illustration that how basically it happens? we have let consider a situation
in which, on the demand side, the investors in the corporate securities, let somebody interested to
invest in the corporate securities and they prefer the short-term to long-term instruments. The
investors are more interested to invest in the short-term instruments than the long-term
instruments that mean their horizon period is relatively short.
But on the supply side,Let you assume the companies have a greater need to finance their long
term assets than the short-term that means, that company wants to go for the capital expenditure
and all these things and because of that they want to issue more long-term bonds. So, to finance
their long-term activities they are going to issue more long-term bonds. So, that means here what
basically we have seen in the shorter market the investors are more inclined to go and invest in
the short-term bonds but the supply is not there.
Because in the supply side who are going to supply the bonds they are more inclined to supply
the long-term bonds. So, then what basically you can observe here? If you combine these 2 then
the relative performance between these 2 markets will cause an excess demand for the short-term
bonds and an excess supply of the long-term bonds. And obviously, somewhere the market is
basically is disequilibrium. Both the markets are in the disequilibrium condition.
So, therefore some adjustment has to take place and that adjustment will change this interest rate.
So, there will be a shift about the interest rate and that adjustment would make both the markets
in the equilibrium and then accordingly the yield curve can be derived.
(Refer Slide Time: 09:20)
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So, let us see what basically happens in this case? if this kind of condition you will assume, Then
in the long-term market what will happen? that the excess supply for the bonds would force the
issuers to lower the bond prices because supply is more than the demand. So, in that case what
will happen? the price of the bond will decline. if the price of the bond will decline then it will
increase the yield because there is an inverse relationship between the prices and yield.
If the price is declining, then obviously your yield will increase then whenever the yield will
increase then what will happen? It will induce some investors to change their short-term
investment demands. whenever they will observe that they are getting more yields in the longterm market then what will happen? They can shift from the short-term segment to the long-term
segment because it looks very attractive for them . that attraction basically makes them to change
their preferred zone preferred term to maturity will be changed.
Now come to the short-term market. Short-term market there is excess demand but supply is not
there. So, if the supply will not be there then what will happen ? that excess demand will
basically cause the bond prices to increase because supply is more than the demand in that
particular segment. Then, whenever the bond price will increase then obviously the interest rate
will fall. So, whenever the interest will fall then what will happen? that will induce the
companies to finance their long-term assets by selling the short-term claims or certain bonds.
In first case, the investors will be interested to invest in the long-term segment because the.longterm segment is giving more yields to them. The second case, whenever the bond price will
increase because of the excess demand in the short-term market and the interest rate basically is
going to decline that will basically induce the issuers to finance their long-term assets using the
short-term bonds.
A series of the short-term bonds they can sell and they want to finance those particular activities
through that. So, in both the markets this kind of adjustment is going to take place. So, whenever
this kind of adjustment will take place then what will happen at the end? Ultimately the
equilibriums in both the markets will be reached with the long-term rates higher than the shortterm rates and a premium basically is necessary. why basically this thing will observe?
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Because, a premium is necessary to compensate the investors and the issuers for the market risk
they have assumed. Whenever, because the long-term investment is riskier than the short-term
investment, whenever they are taking more risk then what will happen? That the yield from the
long-term market is going to be more than the yield of the short-term market. this is what
basically we can observe whenever this kind of scenario will arise.
(Refer Slide Time: 13:04)
So let us see, how basically exactly this demand and supply curve in both the market is going to
be changed ? If you look at these 2 figures, what these 2 figures basically tell? Here, we have a
Short-term market and we have a long-term market then we have taken the instruments like
corporate bonds. And here, if you see the bond demand in each segment is assumed to be
depending on the own segment’s interest rate and as well as the rate which is prevailed in the
other segment.
Because, there is a kind of interaction which is going to happen between these 2 markets. And
bond supply also in each segment is assumed to depend inversely on its own rate and On the rate
in the other segment. That is the common relationship between the supply and the interest rate
and the demand and the interest rate. So let us if you see that let there is an initial rate which is
available in both the markets.
So, the initial rate let we have represented is the y0 ST represents the initial yield in the shorter
market, and y0 LT represents the initial rate in the long-term market. That is the basically
assumption whatever we have taken. So, now if you see that what basically happens here? That
initial rate which basically here that is your y0 ST and here it is your y0 LT. That is the initial
rate which is available.
Now, already we have seen there is excess demand in the short-term market and there is an
excess supply in the long-term market. So, now if you see in this particular rate, So that is your
supply curve this is your demand curve so that means this much is demanded up to this to this
and this much is basically supplied. So, that means there is an excess supply which is happening
and there is no such equilibrium is prevailing in this particular rate in this particular market.
Then what will happen?
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And in the same thing, you can observe also here in this case, your demand is less and the supply
is more. So, this is your demand part and this total is the supply part. so here, there is excess
supply and here, there is excess demand. So, now what will happen? because of the excess
demand the price will go up and because of the excess supply the price will go down. So, then
what will happen? whenever the price will change then obviously your interest rate will move.
So, now what has happened? with respect to that particular price, you will prevail a new interest
rate. In these particular 2 segments a new interest rate basically will be arrived. So, let us see
how that particular interest rate in both the markets are going to be changed and what context
this particular movement will take place in which direction this particular movement will take
place?
(Refer Slide Time: 17:02)
Let us come this explanations of this particular diagram. so, already what we have discussed that,
the excess supply in the long-term market will force the issuers to lower their bond prices. Then,
it will increase the long-term bond yield and the excess demand in the shorter market will cause
the short-term bonds price to rise and yields to fall. So, in that case the long-term bond yields
will increase and short-term bond yields will decline.
So, then some investor will change their short-term investment demands or the preference which
will increase their demand for higher yielding the long-term bonds. So that is why, there is a
movement down the long-term bond demand curve and decreasing their demand for the lower
yielding short-term bonds basically what basically will happen with respect to the short-term
demand curve? The short-term demand curve will have the movement up the short term demand
curve. The position will be changed that way. So, now what basically we have seen? because of
this kind of interactions the interest rate in 2 different segments have been changed.
(Refer Slide Time: 18:39)
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So, in that case if you see on the supply side. first part, we have discussed about the demand side.
in the supply side what will happen? The decrease in the short-term rate and the increase in the
long-term rate will induce the companies to finance their long-term assets, by selling the shortterm assets. that means this would basically lead to a substitution in which the companies will
increase the sale of lower-yielding short-term bonds.
That means, there is a movement of in terms of the short-term supply curve of the bond and
decrease their sale on the higher-yielding long-term bonds. So that is why the long-term supply
curve movement will be down. Then ultimately what will happen? the equilibrium in both the
markets will be reached with the long-term rates higher than the short-term rates. The
equilibrium in both the markets will be reached with the long-term rates higher than the shortterm rates.
So, that is what basically we have observed whenever we go for this kind of interaction between
the short-term and the long-term segments.
(Refer Slide Time: 20:11)
So, then in that case what basically if you summarize these things then what basically we have
observed here? Already, we assume that is a poorly hedged situation there is disequilibrium
happening in both the segments. That means, the investors before the short-term investments and
the issuers basically prefer to issue the long-term securities that means they want to sell more the
long-term securities.
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So here, if you see the investors prefer the short-term part that means, the demand is more in the
short-term segment and the borrowers or issuers prefer the long-term segment so which will
basically create this kind of situation. So that means, there is excess demand in the short-term
part. So, if there is a excess demand in the short-term segment then the price of the bond in the
short-term segment will be up and the yield will go down.
So, whenever the yield will go down, then what will happen? This will attract more long-term
borrowers because there is a demand which is prevailed in that particular segment. The
companies who are issuing this long-term bonds, they will be inclined to issue more long-term
bonds. In the same case, if you see the other part the excess supply in the long-term segment
which is coming from this part, that will basically reduce the price in the long-term segment.
In the long-term segment, price will decline. Then, the yield will increase. The yield is going to
be increasing. then what will happen? That will attract more short-term investors to invest in that
type of bonds because the yield is more price is less. So, that is why they will be more inclined to
invest in these kinds of bonds and in the other part in the demand part, If you see this will attract
the more companies to issue the short-term bonds to finance their long-term activities. So that
means there is some dynamics which will happen in these 2 different segments.
(Refer Slide Time: 23:08)
Then, what basically we can observe your yield curve safe is going to be changed. Initially, we
have assumed a flat yield curve, where your y0 ST = y 0 LT; that is the initial yield curve. then
what will happen? Because of the demand-supply dynamics which has happened in that
particular segment the short-term yield will go down and long-term yield will go up. So now if
you observe, so this particular y0 ST will go down to y 1 ST and here this y0 LT will go up to
the y 1 LT.
So, now you will have if you draw this with respect to the interest rate and term to maturity
which is our yield curve then what basically we will observe here? we will have an upward
sloping yield curve. Initially, there was a flat yield curve that is the assumption whatever we
have taken but whenever this kind of situation will arise, then you will find that the movement of
the yield curve basically will go in this direction.
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So, then what we have got from here? if there is some disequilibrium happens in both short-term
and long-term market due to the change in the demand and change in the supply or there is
disequilibrium in terms of the supply and demand dynamics, then what will happen? That
because of the interdependence between the 2 different segments and we can say that, this will
have some kind of for the investors or the issuers will change their behavior.
And because of their change in the behavior, then what will happen? that they will forgo their
preferred zone and they will move to the other zone which will be attractive to them. So, here in
this case what will happen? that because the corporate borrowers are ready to provide more
short-term bonds to finance their long-term activities. And because of the high yield the shortterm investors will be interested to invest in the long-term bond. then what will happen? you will
observe this kind of yield curve in this particular segment. This is the way the yield curve can be
derived through this preferred habitat theory.
(Refer Slide Time: 26:11)
Then, you see that this is a real practical situation. this preferred habitat theory, basically always
whatever suggestion they have given that basically we can see in the practical situation also. The
preferred habitat theory if you see that the yield curve is positively sloped if investors prefer
short-term to long-term investments and issuers prefer to issue the long-term bonds. And that
generally happens .why it generally happens? because the investors are risk averse in nature.
And as already we know also, we have discussed in the previous sessions that the long-term
maturity bonds tends to be more sensitive to interest rate changes. And in general, the companies
have more long-term financing needs because their short-term financing needs like working
capital financing that basically they can always fulfill using the different other sources, also like
bank financing and other sources.
But, whenever they want to finance their long-term activities, mostly one of the reliable source is
the debt financing and in that case that basically in that particular context they will go for the
long-term debts. And, if you already know from your financial management and other subjects
that, we assume that debt is a cheaper source than the equity. So, because of less risk in nature so
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because of that they can always go for the debt financing what your pecking order theory and
other theory basically talks about, you must have the idea about that.
So, that is the practical situation that always will find more demand in the short-term market and
more supply in the long-term market. But assume that, there is a reverse situation there is a more
demand in the long-term market and more supply in the short-term market for some reason then,
in that case, your yield curve will be negatively sloped. In that case, your yield curve will
negatively sloped but that is very unlikely in the sense. in a general sense, this thing happens in
the market.
And we can say that because of these differential preferences always the market is poorly
hedged. Always whatever horizon period the investor fixes that horizon period may not be
prevailed in the market always. So always there is disequilibrium in these 2 markets and we can
observe that there is some dynamics always happens between these 2 segments and accordingly,
your shape of the yield curve gets changed.
And if you look at that this kind of dynamics regularly happens in the aggregate sense and
accordingly the shape of the yield curve is always determined. But in general, we can say that the
long-term bonds always give more returns or more yield than the short-term bonds because of
this reason, according to the preferred habitat theory.
(Refer Slide Time: 30:00)
So, what basically we have discussed here? That according to the preferred habitat theory,
investors and borrowers or the issuers will be induced to forgo their perfect hedges and shift out
their preferred maturity segments when supply and demand conditions in different maturity
markets do not match. And the long-term rates are always higher than the short-term rates as a
premium is necessary to compensate the investors and the issuers for the market risk what
basically they have assumed.
Whatever market is they consider because of this always the long-term bonds give more yield or
more returns than the short-term ones.
(Refer Slide Time: 30:45)
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So, this is the reference what basically you can go through for the detailed discussion and we will
continue with the discussion with the other theories of term structure interest rate in the
forthcoming sessions thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 04
Lecture No # 18
The Term Structure of Interest Rates-VI
Welcome back! So, in the previous class we have discussed about the preferred habitat theory,
which is necessarily an extension of the market segmentation theory.
(Refer Slide Time: 00:35)
In today's lesson, we will be discussing another popular theory what we call the pure
expectations theory. That means, how the expectations of the investors and the issuers about the
future interest are going to change the yield curve or decide the shape of the yield curve or the
slope of the yield curve. So, that is the basic essence of the pure expectations theory. If you are
expecting the interest rate is going to be up, what the shape of the yield curve will look like?
If you are expecting that the interest rate is going to down, then what should be the shape of the
yield curve? If you assume that the interest rate is not or you are expecting that the interest rate is
not going to change, then what will be the shape of the yield curve? So, basically, the basic
objective of the particular session or what in a general sense we call the pure expectations theory.
In this case, there are certain things we will discuss; the concept of the implied forward rate,
concept of locking strategy and the bond investors expectations.
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In the beginning, I am just telling you that in this particular session we will discuss about, how
the expectations of the investor are going to affect the yield curve? Or what is the response of the
investor expectations about the interest rate changes are going to affect the yield curve? That is
the basic essence. we will not consider the expectation of the issuers. That we will discuss in the
future sessions. But in today's class, we will be discussing only the expectation of the investors
that actually you can keep in the mind.
(Refer Slide Time: 02:48)
So, what is pure expectations theory basically talking about? The pure expectations theory
basically explains the impact of the investor and borrower expectations on the term structure of
interest rate or the shape of the yield curve. How the yield curve shape is going to be changed if
the expectations of the investors or the issuers are going to be changed? That is the basic essence
of the pure expectations theory and as per the definition; the pure expectations theory basically
tells that the yield curve is governed by the condition that the implied forward rate is equal to the
expected spot rate.
There are 2 things you keep in the mind. one is implied forward rates and another one is the
expected spot rate. So, whatever spot rate is going to happen after 2 years that basically, we have
to decide today and for today that particular rate is basically called the forward rate, and that is
implied upon something, and what is that something? That something is the spot rate of the
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today. The forward rate is what basically we are going to calculate on the basis of the spot rate
that is called the implied forward rate.
So, on the basis of the pure expectations theory, we assume that the implied forward rate
whatever, we are calculating today or we are deciding today should be equal to the expected spot
rate.
(Refer Slide Time: 04:48)
So let us see, what is the implied forward rate? So, whenever you talk about the implied forward
rate, already what I told you? Implied forward rate is the future rate which is implied by today's
interest rate structure. That is why I told you that its implied upon today's spot rate, today’s
interest rate. So, that is why we use this word implied. It is conditioned upon today's spot rate.
Let us see, how it is basically attained? We can explain it using a concept called the locking-in
strategy.
What is this locking-in strategy? You assume the current yield to maturity on a 2 year zero
coupon bond is 9% and the current yield to maturity of a 1 year zero coupon bond is 10%.
Because already we have discussed that this spot rate is nothing but zero coupon rates. That was
already in the previous sessions we have discussed. So, that is why the zero coupon rates are
considered as these spot rates. So, here a 2-year zero coupon bond the maturity period is 2 years,
that yield is 9% and the 1-year zero coupon rate let you assume that, is 10%.
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So, if you go for an annual compounding and use this geometric mean concept, all of you know
what is geometric mean and arithmetic mean. So, using the geometric mean; the implied forward
rate on a 1 year bond. One year from now will be how much? So, your spot rate for the 1 year
zero coupon bond and the spot rate of the 2 year zero coupon bond that data is available. So, you
are calculating the one year from now, what will be the implied forward rate?
So, what is the formula we can use that your YTM 2 is equal to your 1 + YTM 1into 1 + the
forward rate which is 1 year from now, you are calculating the power 1 by 2-2? So, that means
your f1 is equal to your 1 + YTM 2 square divided by 1 + YTM 1 - 1. Then what is f 11 + f 11 =
1.09 square divided by 1.10, because 9% is the 2-year spot rate and 10% is the 1 your spot rate 1, that will give you 8%, right? But, what does it basically imply? That basically explains this
locking-in strategy.
Implied forward rate=
(1+𝑌𝑇𝑀2 )2
=
(1+𝑌𝑇𝑀1 )
(1.09)2
(1.10)
−1
− 1 = 0.08
(Refer Slide Time: 08:16)
You use these rates and try to explain this. Let, what the investor can do go for a short sell by
borrowing 1-year bond and selling it at a market price of 909.09, how we have arrived at this
909.09? That is your 1000 divided by 1.10 or you can also borrow 909.09 rupees at the rate of
10%, which is basically your one-year rate.
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The market price of the one-year bond= 1000/1.10= 909.09
The 2 year bond will be trading at that particular time with a market price of 841.68, how we got
this go for the valuation of zero coupon bond that is your let you assume that face value is 1000.
1000 divided by 1.09 square that will give you 841.68.
The market price of the two-year bond= 1000/(1.09)2= 841.68
So, what you can do with 1.08 issues of the 2 year bond? That is your 909 whatever you have
borrowed 909.09 divided by 841.68, which means 1.08 issues of the 2 year bond. You can get it
from now.
No.of issues of the two year bond= 909.09/841.68= 1.08 issues
1 year later, what you can do? Cover the short cell or pay the loan by paying the holder of the 1year bond, which is your 1000 and 2 years later, because you are holding 1.08 issues. 2 years
later receive the principle on the maturing this 2-year bond that is 1.08 into 1000 that is 1080.
Basically you can do you can get some profit out of this.
Principle on the maturing two-year bond issues of (1.08) (1000)= Rs 1080.
(Refer Slide Time: 10:34)
Now, with this locking-in strategy, you see the investors or investor does not make any kind of
cash transactions until the end of the first year because he has gone for short selling. When he
covers the short cell, then only he has to go for these cash transactions. So, when he covers the
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short cell, he needs the money to repay that particular money. So, in the present, the investor
simply initiates the strategy in the beginning.
So, the investment of 1000 is made at the end of the first year, and in turn, the return on
investment is the principal payment of 1080 which is 1.08 holdings of the 2 year bonds, which
comes one year after the investment, that means your short sale coverage is made. So, the return
on the investment or the return on rate of return on this 1 year investment is what your 1080 1000 divided by 1000 that will give you 8%.
Return on the investment= (1080-1000)/1000=8%
So, if the particular individual is going to use this locking in strategy, then the 8% rate of return
on 1-year investment to be made 1 year in the future can be attained. That is the way he got the
rate of return of 8%. That concept is called the locking in strategy; through that basically we
have observed that your implied forward rate was 8%. Now, he is getting a return of 8%
whenever we are going to use this kind of strategy in the market.
(Refer Slide Time: 13:03)
Now, we will come back to our pure expectations theory which basically we have said, that the
implied forward rate should be equal to the expected spot rate. So, to explain that there are
certain assumptions have been taken. Let you consider a market consisting of only 2 bonds. A
risk-free 1-year zero coupon bond and a risk-free 2-year zero coupon bond .let the principal of
the 100 rupees.
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And supply and demand conditions are such that, both the 1 year and 2-year bonds are trading at
8% yield to maturity. Suppose, the market is expecting the yield curve is going to shift to 10% in
the next year. But, this thing is not factored into the current investment decisions now and the
market is risk neutral. That means the investors do not require a risk premium in investing in
risky securities.
That is the assumption of whatever we are taking? That means, they will accept an expected rate
on a risky investment that is equal to the risk rate. That is why you call it that a market is risk
neutral and investors with 1 year and 2 year horizon periods are willing to consider alternative
segments, that means the bond demand is a function of the expected rates. And another major
assumption is issuers have a strong maturity segment segmentation preference and are not
willing to assume the interest rate risk.
That means, the issuers are not going to change their segment, the expectations have no role from
the issuer perspective. Only from the investor perspective do the expectations basically place the
role. Right! So, these are the basic assumptions that basically we are taking.
(Refer Slide Time: 15:19)
With these assumptions let us see, how the impact of the market expectations will be there on the
interest rate structure. There are 2 segments we have seen. we have assumed one is the investor
with a horizon period of 2 years. So, he has 2 alternatives one alternative is to buy a 2-year bond
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at 8% or buy a series of 1-year bonds today at 8%, and 1 year bond 1 year later, let at the rate of
10%.
Your horizon period is 2 years at a time you can invest in a bond that will give you an 8% return
or you can invest in a bond that is giving you 8% for 1 year and after one year again you invest
another 1-year bond that will give you 10% return. Then the overall yield to maturity if you see
in the series of the bonds in the second alternative, you will get an average return of 9% that is
1.08 into 1.01 to the power one by 2-1, we are using the geometric mean concept, then that is
9%.
YTM2: series = [(1.08)(1.10)1/2 ] − 1 = 0.09
So, in a risk-neutral world, investors with a horizon period of 2 years would prefer this series of
1 year over the 2 years bond. Because it is getting more return, instead of investing in a 2 years
bond at a rate of 8%, you will get 9%. If, you will go for a series of bonds.
(Refer To Slide Time: 17:06)
Then another alternative we have, let us have another investor which has a horizon period is 1
year. Then, he has also 2 alternatives. You can buy a 1-year bond at the rate of 8% or buy a 2year bond at a rate of 8%. Let the price is 100 divided by 1.08 to power 2, that is 85.734. then
sell it 1 year later at an expected price of 100 divided by 1.1. The interest rate which is going to
prevail for the 2 years bond is 10%. Then you will get 90.91.
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Price of two-year bond at 8%= 100/(1.08)2= 85.734
Expected price = 100/(1.10)= 90.91
Then, what is the return expected return he will he or she will get that is 90.91- your 85.734
divided by 85.734 which is 6%?
The expected rate of return=
90.91−85.734
85.734
= 0.06
So, then what the investor will do is the investor with a horizon period of 1 year would also
prefer a 1-year bond over the 2 year spot. Because he is getting only 6% in the second case first
case is getting 8%.
(Refer Slide Time: 18:27)
Then, what basically we have seen? in a risk-neutral market with an expectation of higher rates
next year, both investors with 1-year horizon dates and investors with 2-year horizon dates
would purchase the one-year instead of the 2 years bonds. So, then what will happen? if many
investors will do this then, there will be an increase in the demand for the 1-year bonds and a
decrease in the demand for the 2-year bond. And that will continue until the average annual rate
on the 2 years bond is equal to the equivalent annual rate from the series of the 1-year
investments.
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To the rate expected on the 2-year bond held for 1 year where generally the 1 year bond rate is
equal to the rate expected on the 2 year bond held for one. That particular process will continue
unless this particular equality will not arrive.
(Refer Slide Time: 19:42)
So, if you see then what will happen in this particular case? The investors with a horizon period
of 2 years and those with a horizon period of 1 year would prefer 1-year bonds over the 2-year
bonds that we have seen. The price of the 2-year bond will go down because the demand for the
price of the bond will go down. So, that means your yield of the 2-year bond will go up and the
demand for 1-year bond will be up and the price also will be up then the yield will go down. So,
in that case, if you see the yield curve that will become positively sloped.
The first-year rate was 8%, the second-year rate was 10%, and if you observe that there will be a
shift in the yield curve, the yield curve becomes positively sloped. That is basically what we can
observe in this particular case.
(Refer Slide Time: 20:54)
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There are possible equilibrium scenarios that can happen in this case. One scenario is the
adjustment may take place solely in the 2-year market or adjustment also can take place in the 1year market. In the example whatever we have considered if the price on a 2-year bond will fall
down in such a way that it traded on a yield to maturity of 9% and the rate on a 1-year bond
stayed at 8%, then investors with the 2-year horizon dates would be indifferent between the 2
year bond yielding a certain 9% and a series of 1 year bonds yielding 10% and 8% for an
expected rate of 9%.
And investors with 1 year horizon dates would also be indifferent between the 1 year bond which
is yielding 8% and 2 year bond purchased at 9% and sold 1 year later at 10% for an expected
rate of 8%. Is it clear?
(Refer Slide Time: 22:12)
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So, in that case, what is going to happen? If the market responds to the expectations only in
terms of a change in the 2-year bond, then the equilibrium yield on the 2-year bond will be 9%.
Because, the demand for the bond will go down, the price will go down, YTM will go up and the
YTM 2 will go up until, the YTM 2 is equal to your YTM of the 2 series of the bonds, which has
a maturity period of 1 year. That will be equal to 9%.
Then when YTM 2 = 9% and YTM 1 = 8%, then your f 11 will be 10% so, this is the
equilibrium mechanism what you can observe in that particular case which will happen in the 2year bond case.
(Refer Slide Time: 23:16)
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Now, the other scenario is that can happen also in the 1-year market. Let the adjustment will take
place in the 1-year market. Then what will happen? If the price on the 1-year bond increased
such that it traded at a YTM of 6%, what we have calculated before and the rate on a 2-year bond
stayed at 8%, Then investors with 1-year horizon dates would be indifferent between a 1-year
bond which is yielding a certain 6% and 2-year bond which is priced at let 1000 divided by 1.08
square that is it is 857.34.
Then, they would sell 1 year later at an expected 1-year yield of 10% to earn an expected return
of 6%. So, at 1-year and 2-year yields of 6% and 8% respectively, the investors with 2-year
horizon dates would be indifferent between a 2-year bond yielding a certain 8% and a series of 1
year bonds yielding 6% and 10% for an expected rate of 8%. So, in this adjustment scenario, the
impact of market expectations of higher rates would be to push the short term 1 year rates down
to 6% which was before the 8%.
(Refer Slide Time: 24:51)
So, you can also explain these things in this way. The demand for the 1-year bond has increased.
So, that is why the price has increased, the YTM has gone down and the YTM or the YTM will
go down until the YTM 1 is equal to your YTM 1 holding for the 2 years and 1 year that will be
equal to 6%, then when YTM 1 = 6% and your yield to maturity of the 2 years bond is 8% then
your implied forward rate will be 10%. So, our 2-year spot rate is what we assumed was 10% in
the beginning.
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Now, we have seen that if you are trying to find the equilibrium in the 2 markets, then that spot
rate what we call the expected spot rate should be equal to the forward rate. If that will is equal
to the forward rate, then this kind of situation will prevail in the market. That is what basically
we are trying to explain. So, in that particular context what basically we have seen? if the
expectations in the market are going to be changed and this expectation is coming from the
investor side, then this kind of situation will prevail in the market and this is the way basically
your equilibrium dynamics will work.
(Refer Slide Time: 26:27)
In this example, what we have seen? if the equilibrium yield to maturity on the 2-year bond is
9% and the equilibrium YTM on 1 year bond is 8%, then the implied forward rate is 10%. in
both the cases we have seen, Which is the same as the expected rate on a 1-year bond and 2 years
and 1 year from now. Similarly, if the equilibrium YTM on the 2-year bond is 8% and 1 year
bond is 6%, then the implied forward rate is also 10%.
So, the equilibrium yield curve will be governed by the condition that the implied forward rate is
equal to the expected spot rate, which is basically 10%; which is the basic essence of the pure
expectations theory. The equilibrium yield curve basically will be attained or will be governed
by the condition that the implied forward rate is equal to the expected spot rate which is 10% in
this particular case.
(Refer Slide Time: 27:40)
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We can derive it with our formula whatever we have seen or we have discussed in the beginning
your YTM 2 = 1 + YTM 1 into 1 + f 11 to the power 1 by 2-1 then your f 11 = 1 + YTM whole
square divided by t + 1+ y 10 1 - 1. Now, your f 11 in the first case that is 1.09 in the 2 years
market, if you are talking about the equilibrium or the adjustment is taking place, then it is 1.09
square divided by 1.08 - 1 that is 10% that means 0.10 second case it is 1.08 square divided by
1.06 – 1 that is the 10%.
YTM2 = [(1 + 𝑌𝑇𝑀1 )(1 + 𝑓11 )]1/2- 1
F11 =
F11=
(1+𝑌𝑇𝑀2 )2
(1+𝑌𝑇𝑀1 )
(1.09)2
(1.08)
−1
− 1 = 0.10
Or
F11=
(1.08)2
(1.06)
− 1 = 0.10
So, in both the cases whether the adjustment is taking place in the 1 year bond market or
adjustment is taking place in the 2 year bond market that does not make any difference. In both
the cases the implied forward rate is equal to the expected spot rate that is the 10%.
(Refer Slide Time: 28:53)
271
So, what basically conclusions we can derive in this case. The implied forward rate is the future
rate that is implied by 2 days interested structure that we have discussed. And the yield curve is
basically governed by the condition that implied forward rate is equal to the expected spot rate.
Whether the adjustment is taking place in the 1-year bond market or the adjustment is taking
place in the 2-year one market, that does not make any difference in both the markets you will
find that the implant forward rate is always equal to the expected spot rate.
That is what basically what we can see from the concept of the pure expectations theory. Further,
we will continue with this discussion whenever we assume that both the investors and this
issuer’s expectation are also going to be changed. Here, we have assumed that the resource
expectation is not playing any role. Only the investors, expectations we are considering but if
you are going to accommodate that thing also.
Then how it is going to change the concept of the yield curve and how the yield curve is going to
be shifted? and what is the adjustment mechanism will work in that particular context that
basically will be discussing in the future?
(Refer Slide Time: 30:24)
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So, these are the references what you can go through for detailed discussion on this, Thank you.
273
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 04
Lecture No # 19
The Term Structure of Interest Rates – VII
Welcome back! So, in the previous class, we started the discussion on the pure expectations theory
and we have discussed that how the expectations about the future by the bond investors is going
to change or going to affect the yield curve. And what we have seen that the implied forward rate
should be always equal to the expected spot rate whenever there is any equilibrium. So, that is the
way basically we have always explained this concept of the pure expectations theory.
(Refer Slide Time: 00:54)
In today’s class will be continuing with this particular discussion.
(Refer Slide Time: 01:00)
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And through this particular class you will understand how the bond issuer’s response is going to
affect the yield curve, whenever they are also expecting that the interest rate is going to change in
the future. In the previous case, we have assumed that only the expectations of the bond investors
has some kind of impact or some kind of response towards the concept of the income. But, now
we are incorporating both; bond investor and as well as the bond issuer. In that case also what is
going to happen, that is the basic objective of the today's discussion.
(Refer Slide Time: 01:52)
We will continue with the same example whatever we have we have taken. Let bond issuers also
consider the expectation, and like the bond investors they are also risk neutral. The concept of risk
neutral already we have discussed. Let we have taken again this hypothetical situation, the issuer
275
consider the expectations in their current financing decisions and they have 2 alternatives. One is
they can issue a 2-year bond which is giving 8% return or a series of 1-year bond.
1 year today at 8 % and 1 year bond one year later at 10% for an average of the 9%. That already
we have calculated. Then, what will happen from the issuer perspective? This issuer obviously
will prefer to issue the 2-year bond because that is lower. The investor can go for the other
alternative but the issuer perspective they will go for a bond which is basically giving the 8%
because in the second case they have to give 9%. So, the 2 years bond issuer, prefer to issue the 2
year bond at a yield of 8%. is it clear?
(Refer Slide Time: 03:42)
Second case, let there is a 1-year borrower. first case we have considered the horizon period of the
bond issuer is 2 years. Second case, let we have assumed that the particular borrower horizon
period is 1 year. then again 2 alternatives we have; 1 year bond at a rate of 8% Or a 2-year bond
at 8%. For example, borrowing 857.33= 1000/(1.08)2 and then buy that particular bond back in the
market one year later at a rate of 10% and the price of the bond is 909.09=(1000/1.10), then paying
the one-year borrowing rate of 6%(=(909.09/857.3388)-1).
So, in that case a 1-year borrower would prefer to issue 2-year bonds instead of 1 year because the
yield is only 6% in that case, effectively he has to pay 6% only.
(Refer Slide Time: 05:07)
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So, then what we have observed in this case? There are 2 markets in this case; one is a 2-year
market, second one is a one-year market. So, what will happen in the 2 year market? The
expectation of higher rates will cause the supply of the 2 years bond to increase. Then, there will
be a rightward shift in the bond supply curve in that particular segment that will ultimately lower
the price in increasing the 2-year yield.
But, at the same time that will reinforce the demand impact where the demand and price of 2 years
bonds are decreasing. That we are linking it in the previous class, which basically causing the 2year yields to increase because that time basically we have seen that the preference of the investor
perspective is different. If you come back to the 1-year market then what will happen? the
expectation of higher rates would cause the supply of 1-year bond to decline.
Because you are expecting that the rate will be more than the leftward shift in the 1-year bond
supply curve which will increase their price and reduce the yield, the 1-year yield will be reduced.
And again, if you look at the demand side the demand impact. The demand and price of the 1-year
bonds are also increasing causing this 1-year yields to decline. So, in the 2 year market and 1 year
market this kind of scenario is going to be prevailed.
(Refer Slide Time: 07:29)
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So, let us see that how that particular mechanism works? Now, if you look at this what will be
happening in the equilibrium? Already we have seen there is an increase in the supply of the 2year bond, that means the bond curve will shift rightward, the 2-year bond supply curve will shift
to the right ward. So, here if you see that there is a 2-year bond market. if you look at it, then you
will find that there is a shift in the demand curve and a decrease in the supply of the 1-year bond.
So, that’s why there is a leftward shift in the 1-year bond supply curve. 1-year market the bond
supply curve will shift the leftward right. So, if you combine these 2, the demand adjustment of 1year bond is increasing, that means the 1-year bond demand curve is shifting right. So, that is the
way basically it will shift right. And 2-year bond demand curve is decreasing so that is why this is
basically shifting left. So, that is a moment which is going to happen.
And this same process will continue until the annual rate of series of the one-year bonds or oneyear loans. if you see, or the one-year bonds rate is equal to the rate extracted on the 2-year bond
and 1 year bond. This particular process basically will go on. So, because of the change in the
demand-supply conditions in one-year bond market, in 2 year bond market this kind of scenario is
going to be prevailed and this is the way the adjustment mechanism basically will work.
(Refer Slide Time: 10:16)
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Now, if you see what basically exactly happens here? Here what is the assumption we have taken?
There is a increasing expectations, that means the yield was 8% & yield is going to increase to
10%. That is why the expectations of rates is increasing from 8% to 10% that is the assumption
whatever we have taken, that is the thing you keep in the mind. Then investors with horizon period
of 2 years and also with the horizon period of 1 year would prefer one-year bonds over the 2-year
bonds.
That is from the investor perspective that we have seen in the previous class. Now, the 2-year
borrower would prefer to finance the 2-year bonds and 1-year borrower would prefer also to
finance with the 2-year bonds. Right? So, from the demand side it is the preference is coming for
the one-year bond and from the supply side it is coming with the 2 years. is it clear? Now, how the
market basically will be responding towards that, what is the markets response?
Now, what is happening? The demand for the 2-year bond is low because the demand for the 1year bond is higher from the demand side, from the investor side. So, the demand for the 2-year
bond is low then the price of the bond will go down and the YTM will go up. And the demand for
the one-year bond will be up and the price of one year bond will be up then the YTM will be down.
here the YTM will be up and here the YTM will be down.
Come to the supply side, which is coming from the issuer perspective. The demand for 2-year bond
supply is up so that is why the price of the 2 years bond will be down. Then, your YTM will be
279
down. sorry! the YTM will be up. Sorry! the supply of the 2 years bond will be up, so that is why
the price will be down because the demand is not there so then what will happen? that the YTM
will be up.
If the YTM will be up, then what will happen? in the one-year market if you observe the supply is
less price will be higher, then the YTM will be down. So, then what will happen in that case? if
you summarize this, the yield curve becomes also positively sloped. Initially, it was 8% now it has
gone up to 10%. So, now what has happened in the 2 markets? if you look at the one-year market
the yield has come down and the 2-year market the yield has gone up.
So, in that case what basically here we are trying to say? If you are expecting both the issuer and
as well as the investor are expecting that the interest rate will be up. Then, that will have the impact
on the short-term yield and as well as the long-term yield. The short-term yield is going down and
the long-term is going to be up. So, like in the previous class in the demand case or the investor
perspective whenever we have discussed.
The response of the issuers or the borrowers to the expectation of higher rates contributes to the
steepening of the income. If you are expecting the issuer is also expecting the interest rate will be
up and the investor is also expecting that interest it will be up. Then it will further make this yield
curve steeper that is the thing basically you have to keep in the mind.
(Refer Slide Time: 14:56)
280
Let us see, if you are expecting that the interest rate is going to be down. Here, we have seen that
the interest rate is going to be up. Let you assume there is a flat yield curve and that is basically
your 10%. let the market expectation is like that it will shift down to 8% in the next year. Then,
what will happen? then the investor with a 2-year horizon date would prefer the 2-year bond at a
10% to a series of one-year bond which can yield an expected rate of 9%.
(E(R)= [(1.10)(1.08)]1/2-1=0.09)
So if you look at the investor with a one-year horizon they will prefer buying a 2-year bond that
has an expected rate of 12%. How you got this 12%? Let we assume that principal is 100 then 100
divided by 1.1 square that will the price will be 82.6446. Then your expected price will be 100
divided by 1.08 that is 92.5926. Then expected return will be your 92.5926 – 82.6446 divided by
82.6446 that will give you 1 % to the one year bond which is yielding only the 10% now.
P2= 100/(1.10)2=82.6446
E(P11)= 100/1.08=92.5926
Expected return will be= [(92.5926-82.6446)/82.6446=0.12]
Here, P2=Price and E(P11)= expected price
But, if you are going for your horizon period is one year, they should prefer a 2 years bond that
has an expected return of the 12 % because it is giving you a yield of the 12%. So, both one-year
and 2-year investors prefer to buy 2-year bonds instead of one year.
In the previous case whatever we have seen both the one-year and 2-year investors would prefer
to buy one year bond instead of the 2-year bond,whenever there is expectations about the interest
rate is higher right. So, now what is the mechanism which basically works here? how this particular
concept or understanding basically works here?
(Refer Slide Time: 17:48)
281
So, now what will happen from the issuer perspective? let us see the borrower perspective. Let this
borrower's horizon period is 2 years, so a 2-year borrower would favor the financing with a series
of one-year bond at a rate of 10% and 8% for an average of 9%([(1.10)(1.08)]1/2-1=0.09) to a 2year bond of the 10%. But, if the issuer’s horizon period is one year, they would prefer the current
financing with one-year bonds at 10% to issuing a 2-year bond at 10%.
How they can do this? for example, they can borrow 826.4463=1000/(1.10)2. Then buying the
bond back in the market one year later when the rates are at 8% and the price of the bond will
become 1000 divided by 1.08 that will give you 925.9259(=1000/1.08). And effectively paying a
one year borrowing rate of 12%(925.9259/826.4436)-1).
So, both one-year and 2-year borrowers will prefer to issue the one-year bonds instead of the 2
years. Here, in the first case the interest rate is 9%, second case it is 10% or second case it is if you
are going for the 2 years bond they have to pay 12%. So, from the investor side, what we have
observed? we have observed that, they will prefer the 2-year bond and from the issuer perspective
we have seen that, they will prefer the 1 year bond. Both type of investors having the one-year
horizon period and 2-year horizon period, they will prefer to issue the 1-year bond instead of the
2 years bond. Right?
(Refer Slide Time: 20:25)
282
Then what will happen in that case? So, whenever you are expecting there is a lower rate that will
basically cause the supply of the 2 years bond to decrease. That means that, there was a leftward
shift in the bond supply curve and an increase. Obviously, whenever the supply is going to decline
the price will be up. Because their preference is basically the one-year bond then if the price will
be up then it will increase this 2-year yield.
And the supply of one-year bond to increase because their preference is 1-year bond. So, that is
why they are ready to supply more 1 year bond. that is why there is a rightward shift in the oneyear bond supply curve, that will decline the price and increase the 1-year yield. So, these
adjustments basically along with the demand adjustments of 1-year bond demand is increasing and
2-year bond demand is increasing would continue until the rate on the 2-year bond equal the
average rate from the series of the 1-year investments.
Or until the rate on the one-year bond, we called the expected rate from the holding a 2-year bond,
one year this same equilibrium condition as the bond investors. That means, when the implied
forward rate is equal to the expected spot rate. So that, particular condition will prevail up to that
time this adjustment will go up and the price movement will go up and corresponds to that
particular price whatever interest rate will be prevailed in the market, that interest rate will be
called the equilibrium interest rate.
(Refer Slide Time: 22:50)
283
So, let us see that how basically it works? so whatever basically we have explained that is explained
through this particular 2 pictures or 2 diagrams. You come back to this part. we are expecting that
the interest rate is declining from 10 to 8%. Then, the investors with a horizon period of 2 years
and those with a horizon period of one year would prefer the 2-year bond over the one-year bond,
that is from the investment side.
And this issuer perspective, they would prefer to finance with one-year bond. investor need 2-year
bonds, but issuer wants to supply one year bond. If you are expecting that the interest rate is going
to be down or so then, what will happen both in the supply side and demand side? The demand for
one-year bond will be down, then the price will be down, then YTM will be up; yield to maturity
will be up.
And same thing what will happen in the supply side of the one year bond? investors are ready to
supply more one year bond, then it will be up, price will be down, then your YTM will be up. So,
what basically here it will happen, that it will have the YTM of the one year bond is going to be
up. Come back to the 2-year market the demand is more that's why price is more, then YTM will
be down. Supply will be less, then price will be up, then YTM will be down. both the things both
in the same direction. is it clear?
So, then what will happen that the yield curve becomes negatively sloped? so, whenever the
expected interest rate or the expectations about the market is up, then the yield curve is positively
284
sloped, but whenever the market expectation is there that, interest rate is going to be down then,
your yield curve becomes negatively sloped.
So, what we have observed here? if the current yield curve is flat at 10% and there was a market
expectation that it would shift down to the 8% next year then, the expectations of the lower rates
would cause the yield curve to become negatively sloped.
(Refer Slide Time: 26:13)
So, now here we have given certain situations. if the market responds to the expectations only in
terms of change in the 2 years bond, then what will happen? The demand will be up, price will be
up, then YTM will be down and this YTM 2 in the 2-year market will be down until your YTM 2
is equal to your YTM of the 2 series bonds that will be equal to 9%. So, then YTM 2 is a 9% YTM
1 was 10%, which was before, then your f 11 will be 8%.
That is the way basically we have here, it is calculated f 11 = 8%. If the market response to the
expectations is only in terms of change in the 1-year bond, then the equilibrium yield on the oneyear bond will be 12%. How? The demand is less, price will be less, then it will be up, then if this
thing will go up until this f 11 will reach at 8%. So, in that case what basically here you can see?
Ok! this is basically your price of one; this is basically your YTM1, because we are talking about
one year bond so that will be equal to your 8%.
285
So, now whether the one-year, 2-year yields are 8%, 9%, 12% and 10% or some are in between,
the resulting yield curve will satisfy the condition that, the implied forward rate is equal to the
expected spot rate of the 8%. in any of the cases. The first case, we have seen that it is if 9% we
got 8% is the forward rate implied forward rate second case also if it is one year rate will be 12%,
then also we are getting it is it is 8%. So, that is what basically what we have observed in this
particular case.
(Refer Slide Time: 28:49)
If you assume that there is no change in the interest rate and you assume that currently the yield
curve is positively slopped, Then with the rate on the 1-year bond at 8% and rate on the 2-year
bond is at 9%. This time, suppose that the market expects no change in the yield, then that means
the expectation has not yet been reflected in the yield curve. Then, in that scenario the investors
with horizon dates of 2 years would prefer the 2-year bond yielding 9% to a series of 1 year bond
yielding 8% now and 8% expected at the end of the end 1 year, which is 2 year equivalent of the
8%.
Similarly, investors with 1-year horizon dates would prefer the 2-year bond held for 1 year, which
is expected 10% compared to only 8% from the 1-year bond, if you are assuming that no interest
rate happens in that particular case.
(Refer Slide Time: 29:51)
286
So, given the preference of the 2-year bond, the market response would occur in which the demand
for 2-year bond would rise, which in turn increasing the price and lowering the yield. And the
demand and price of the 1-year bond would decrease, which is basically increase the yield. So, the
investor would flatten this positively sloped yield curve, given the expectation of no change in the
yield curve.
Bond suppliers also would also contribute to the flattening this particular positively sloped yield
curve given the expectations of no change in the yield curve. So, both demand side, supply side,
they will contribute for a flat yield curve in that particular context, if you are assuming that there
was a positively sloped yield curve before.
(Refer Slide Time: 30:40)
287
So, what we have seen? the response of the issuer and borrowers to the expectations of higher rates
contribute to the steeping of the yield curves .and the expectation of lower rates will cause the
yield curve to become negatively sloped. And investor would flatten this positively yield curve
given the expectation of no change in the yield curve. So, that is what basically we have observed
whenever the expectations from the both demand side and supply side comes here and as well as
from the investor side and also the issuer side will be taken into the considerations.
(Refer Slide Time: 31:19)
So, this is the reference what you can go through. Thank you.
288
289
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 04
Lecture No # 20
The Term Structure of Interest Rates – VIII
Welcome back! So, in the previous class we have discussed about the pure expectations theory or
we started the discussion on the pure expectations theory. Then, we will continue with that
particular discussion and we will learn something about the features of this pure expectations
theory. Then, we move forward the other theories of term structure interested like liquidity
premium theory and other things.
(Refer Slide Time: 00:45)
So, in this particular session we will cover up the concepts like the features of the pure
expectations theory. Then, how we can forecast the future spot yield curve and as well as the
forward yield curve? Then we will discuss about the liquidity premium theory. So, all these
concepts basically will be covering up in this particular session.
(Refer Slide Time: 01:12)
290
So, we will come across certain keywords like your spot yield curve, expected rate of return, the
concept called Hedgeable rate, risk premium. All these things basically we will come across.
Whenever; we go ahead with this particular discussion with respect to this particular subject or
particular session.
(Refer Slide Time: 01:35)
So, let us come to the discussion on the features of the pure expectations theory. So, in the
previous class, we discussed about the basic essence of the pure expectations theory. Whenever,
the investors and the bond issuers expect something about the movement of the interest rate in
the market, then how the yield curve is going to move or how the particular yield curve generally
shifts? That part we have discussed.
291
So, here what basically we have seen from the previous class discussion? That, in equilibrium
the yield curve basically reflects the current expectations about the future rates. So, we have
introduced the concept called implied forward rate and the expected spot rate. And what we have
seen? That, in the equilibrium the implied forward rate should be equal to the expected spot rate.
So, what is the implication we have drawn from there when the market is in the equilibrium ,the
yield curve is positively sloped?
Or we can say that if in the equilibrium the yield curve is positively sloped, then the market
basically expected the higher rates in the future. So, if you observe that, there is a positively yield
curve prevailed in the market, then we can say that the expected rates are going to be higher than
the current rate. So, when it is negatively sloped, then obviously we can expect that the market
basically expected the lower rate.
So, that is the observations what basically we have drawn, whenever we are trying to discuss
about the pure expectations theory. So, if the pure expectation theory basically strictly holds,
then the expected future rates would be equal to the implied forward rates. That basically just
now whatever I told you. So, what basically here we are trying to imply from this particular kind
of concept that if you are given a spot yield curve?
Then, investor could use that spot yield curve or they can use this pure expectations theory to
estimate the next periods spot yield curve by determining the implied forward rate. So, if they
can determine the implied forward rate, then the next period spot yield curve can be determined
in the market. So, that is the basic essence of the pure expectations theory or basic feature of the
pure expectations theory.
(Refer Slide Time: 04:54)
292
So, here let us see how this pure expectations theory is helping us to forecast the future spot yield
curve? let us take one example. So, here if you observe what basically is given to you one is that
there are, let maturity period is given and against this maturity the spot rates are given. And what
is our job? our job is to find out the expected spot rate 1year from the present and the expected
spot rate 2 years from the present. So, these are the 2 things what basically we have to determine.
So, this table shows the spot, rates on the bonds whose maturity is basically is varying from 1
year to 5 years which is given in the column 2. And from these rates the expected spot rates can
be generated for the bonds 1 year from the present which should be we have to fill the table
column 3. And the 2 years from the present which has to be reported in the column 4. And the
expected spot rates whatever are shown are equal to their current spending implied forward rates.
So, that is the essence of the pure expectations theory whatever we have discussed. Let us see
how the expected spot rates one year from the present and the expected spot rates the two year
from the present can be estimated by using this particular data on the basis of the pure
expectations theory?
(Refer Slide Time: 06:55)
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So, if you see the rates are given to you and already, we have discussed this particular formula,
whenever we started the discussion on the pure expectations theory. So, here if you see that for
example, we are calculating f12. So, here the f12 if you are calculating. So, in the concept wise
as per the pure expectations theory whatever we have seen year s3 which is the spot rate for the
period 3.
Should be 1 + s1 into 1 + f 11 + into 1 + f 12 to the power 1 by 3 – 1. So then year s 3 will be 1 +
S2 whole square into 1 + f 12 to the power 1 by 3 - 1 so this part. So now from here our job is to
find out your f 1 2 then your f 12 will become 1 + s 3 to the power 3 divided by 1 + S2 to the
power 2 we are deriving this from this particular equation that is – 1. 1 + S3 to the; power q
divided by 1 + S2 to the power 2 or the square – 1. Then, in the previous table, it has been given
year s3 is equal to 11% right this spot rate or the maturity period 3 that is 11%, then it is 1.11 to
the power 3 divided by and year S2 is 10.5% then 1.105 to the power 2 - 1 then we got the 0.12
that means 12%.
Calculation of f12: S3=[(1+S1) (1+f11) (1+f12)]1/3-1
S3=[(1+S2)2 (1+f12)]1/3-1
f12 =
(1+𝑠3 )3
(1+𝑠2 )2
−1
(1.11)3
f12 =(1.105)2 − 1 = 0.12
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Like that, if you are trying to calculate the f 32 then it will be 1 + S1 into 1 + f 11 into 1 + f 12
into 1 + f 13 into 1 + f 14 to the power 1 by 5 – 1.So, if you are going to solve this then
ultimately you will find your f 32 is equal to your 1 + S5 to the power 5 divided by 1 + S2 to the
power 2 whole to the power 1 by 3 - 1. So, your S5 is 12 % which was given in the previous
table and year S2 = 10.5%. then using putting those particular values, you got your f 32 = 0.13.
Calculation of f32: S5=[(1+S1) (1+f11) (1+f12) (1+f13) (1+f14)]1/5-1
S5=[(1+S2)2 (1+f32)]1/5-1
(1+𝑠 )5
F32 =(1+𝑠5 )2 − 1
2
F32 =
(1.12)5
(1.105)2
− 1 = 0.13
So, from this we can derive a general formula then your f M t is equal to your 1 + S M + t to the
power M + t divided by 1 + S t to the power t whole to the power 1 by M - 1.
General Formula:
FMt =[
(1+𝑠𝑀+𝑡 )𝑀+𝑡 1/M
1+𝑠𝑡 𝑡
]
-1
So, this formula has been derived by using this concept of the pure expectations theory. So now
you can calculate the all those kinds of rates whatever was blank in the previous table. So, here
we have given the 2 cases. you can also use the other you can use this formula to calculate the
other rates which has to be filled out in that particular table.
(Refer Slide Time: 10:43)
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So, now your spot rate was given so now whatever we have calculated your f 11 we got 11.1
11% and f 32 we got 13%. So, like that you can calculate f 21, you can calculate f 31, your f 41, f
12,
f 22 like that. So, like that you can use that particular formula and derive all those expected spot
rates 1 year from present and the expected spot rate 2 year from the present that actually you can
estimate using that particular formula.
(Refer Slide Time: 11:29)
Now, we see that how to determine this expected rates of return using this forward rates? Right?
So, according to this pure expectations theory, if you assume that the market is risk neutral we
know that, the implied forward rate is equal to the expected spot rate and the expected rate of
return for holding any bonds for one year is equal to the current spot rate on one year bonds. So,
let us take one example. what is the expected rate of return from purchasing let a 2-year zero
coupon bond at the spot rate of 10.5% which was given to you year s 2 is 10.5%?
And selling it one year later at an expected one-year spot rate equal to the implied forward rate
that is f 11, that is your 11% and you take the phase value of 100 rupees right. So, your expected
price P 11 will be how much? your forward rate is 11 %. so then if you are using that one for
discounting then the value will be 90.09 right. Then, what will be the price on that particular
day? you have to use the discount rate for the spot rate for the period 2 so that means your price
will be 100 divided by 1.105 to the power 2.
E(P11)= (100/1.11)=90.09
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P20= 100/(1.105)2= 81.8984
Because basically it is a 2 year bond zero coupon bond your face value divided by 1 + r to the
power 2 two year zero coupon bond then and the beginning the price is p 20 the price is basically
81.8984. Then how much expected return you got your 90.90 – 81.8984 divide by 81.8984 that
will be 10%. So this is the same rate which basically is obtained from investing in a 1 year bond
right.
Expected rate of return will be= (90.09-81.8984)/81.8984= 0.10
So, that is what basically what we are trying to justify here, that how the expected spot rate
basically what we are trying to measure it is equal to the forward rate as per the pure
expectations theory. And you will get the same return what was given for holding a one-year
bond, if you are using this particular implied forward rate for your calculation.
(Refer Slide Time: 14:41)
Now, if same way you want to calculate rate, what is the expected return rate of return from
holding a 3 year bond from one year? Then selling it at the implied forward rate of f 21 right of
11.5% what already we have calculated. Then, you take the phase value of 100, then same logic
you can apply your 100 divided by 1.115 square that will give you the expected price on that
particular day. Then, initial price that is p 30 that is 100 divided by 1.11, which is the S3.Right?
3 years zero coupon bond then that will give you 73.1191. then what is the return you got that
80.43596 - 73.1191 divided by 73.1191 = 10%.
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E(P21)= (100/1.115)2=80.43596
P30= 100/(1.11)3= 73.1191
Expected rate of return will be= (80.43596-73.1191)/73.1191= 0.10
That means, any of the bonds with spot rate shown in this particular exhibit or the table would
have expected rates for one year of 10%. If the implied forward rate is used as the estimated
expected rate, then you will get the same return expected your one-year return will be same.
Similarly for other things any bond held for 2 years and sold at its forward rate will earn the 2year spot rate; that is 10.5%. If you see the example, let a 4 year bond persist at the spot rate of
11.5% and expected to be sold 2 years later at f 22, what we have calculated that is 12.5%, then
obviously it would trade at an expected rate of 10.5% which is the same as the current two year
spot rate. So, the forward rates can be used to determine the expected rates of this particular
bonds so that is the way basically it is estimated.
(Refer Slide Time: 17:05)
So, this particular rate whatever basically we have just now used, the analyst many a time refer
this forward rate, as the hedgable rates. these particular rates are called as the hedgable rates. The
most practical use of the forward rates are the expected spot rate yield curves, basically
generated from the forward rate is that they provide the cut off rates. And these rates are
generally very much important for the investment decisions. How for example, if an investor has
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a one year horizon, then they should only consider investing in the 2 year bond is horizon period
is 1 year.
But, they can consider also a 2 year bond if the investor expects 1 year rates one year later to be
less than your forward rate which is 11%. Even if his horizon period is 1 year, they can go for the
2-year bond if this condition will prevail. So that means, this forward rate has been used as a cut
off rate so accordingly he or she can take the decisions. So, therefore the forward rates basically
always serve as a good cut off rate for evaluation of the investments in the market that actually
you have to keep in the mind.
(Refer Slide Time: 18:47)
So, in this context the basic essence of the pure expectations theory, if you see we can calculate
the long term yield to maturity by using the current spot rate and the implied forward rates. So,
the long term YTM generally is nothing but a geometric mean of today's rate and the implied
forward rates. So, therefore you can write a formula that YTM = 1 + YTM 1 which is the current
spot rate into 1 + f 11 into 1 + f 12 into 1 + f 13 so on up to 1 + f 1 m - 1 to the power whole to
the power 1 1 by m – 1.
The long-term YTM expressed as a geometric average of today’s rate and implied forward rates,
fMT:
YTMM= [(1+YTM1) (1+f11) (1+f12) (1+f13) …… (1+f1, M-1)]1/M - 1
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Take an example let a 3-year zero coupon bond which is trading at 800 rupees, which will give
you 1000 rupees which is the principal at the maturity and it is annually compounded. So, I will
look at the YTM straight forward, you can say that the particular rate which can give you 1000
rupees if you are investing 800 rupees today. it is simply one 1000 divided by 800 to the power 1
by 3 - 1 that is 7.72 %.
YTM= [1000/800]1/3 – 1= 7.72%
There are other ways also these 800 rupees can grow to 1000 rupees at the end of the three
years.how? How basically it can happen?If the rates on current one year bond rate is 10 % rates
on one year bond 1 year from now are expected to be let 8% and rates on one year bond 2 years
from now that is r 12 that is expected to be 5.219%. Then your 800 rupees investment also can
grow to 1000 in the 3 years time because the 800 into 1.1 into 1.08 into 1.05219 that will give
you 1000 rupees so that if this 8 % is f 1 1 this 5.219% is f 1 2 and so on.
800(1.10) (1.08) (1.05219) = 1000
(Refer Slide Time: 21:24)
So here this YTM of 7.72% can be considered or can be viewed as a geometric average of the
10%, 5% and 5.219% is it. So you can derive this in this way your price into 1 + YTM m to the
power m is equal to your phase future value of this particular bond which is 1000 rupees. That is
nothing but your price of the bond into 1 + YTM 1 into 1 + r 1 1 + into 1 + 1 12 + 1 + r 13 up to
your 1 + r 1 m – 1.
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Then if you take this one into this side then it will be f by p 0 that will be equal to this then now
your 1 + YTM 3 here in our example to the power 3 is equal to your 1.0772 your 100 by 800
right is equal to your 1.10 into 1.08 into 1.05219. So finally what; basically we have seen your
YTM 3 is equal to your 1.10 into 1.08 into 1.05219 to the power 1 by 3 - 1 that is (()) (22:56). So
that means the long term interest rate is a geometric mean of the current spot rate and the implied
forward rates so that is the basically thing what basically you have to keep in the mind.
YTM of 7.72% can therefore be viewed as the geometric average of 10%,8% and 5.219%:
P0B(1+YTMM)M= F= P0B[[(1+YTM1) (1+R11) (1+R12) (1+R13) …… (1+R1, M-1)]
𝐹
(1+YTMM)M = 𝑃 𝐵 = [(1+YTM1) (1+R11) (1+R12) (1+R13) …… (1+R1, M-1)]
0
1000
(1+YTM3)3 = (1.0772)3 = 800 = [(1.10) (1.08) (1.05219)]
YTMM = [(1+YTM1) (1+R11) (1+R12) (1+R13) …… (1+R1, M-1)]1/M – 1
YTM3= [(1.10) (1.08) (1.05219)]1/3 -1 =0.0772
(Refer Slide Time: 23:15)
Then, the other theory is basically your liquidity preference theory and why basically the
liquidity preference theory was emerged? Because, whenever we have explained this pure
expectations theory it always assumes that, there is a risk neutral world. So, that is why we can
consider that is the limitation of the pure expectations theory because it does not consider the
risk.
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And where the risk comes? because all of you know that investing in a long-term bond is riskier
than the investment in a short-term bond. that all of us know, because the interest rate is very
much uncertain. And also, we have explained in the previous sessions that the long-term bonds
are more price sensitive to interest rate changes than the certain bonds. So, if anybody is holding
a long-term bond he is always exposed to more risk in comparison to the person or investor who
is holding a short-term bond.
That is why the price of the long-term securities or long-term bonds always tends to be more
volatile and therefore more risky, the price volatility is more for the long-term bonds. So, that is
why there is an emergence of the liquidity preference theory which is also popularly called as the
risk premium theory. So, according to this liquidity premium theory always that should be
premium involved in the long-term bonds.
That means, definitely if anybody is investing in the long-term bond that long term bonds should
give more return than the short-term bonds because there is a risk premium, there is a liquidity
premium that is basically involved in that particular case. But, the investor also invest in the long
term bond, investor also invest in the short term bond. then when? in what condition the investor
will go and invest in the long term bond?
The investor will go for a long-term bond obviously if the long-term bonds will give higher rates
than the expected future rate. So, what we have seen? that the expected future rate should be
implied forward rate as per the pure expectations theory. But as for the liquidity premium theory,
they said that if anybody is investing in the long term bonds then, whatever calculations we are
doing on the basis of the current spot rate, there we have to add the premium.
So, the risk premium concept if you are bringing into the considerations then this particular
premium should be increasing with the term to maturity. So, higher the bonds term to maturity
more the return that basically we should expect. So, what basically then this theory suggests? the
yield curve is governed by the condition that the expected spot rate is equal to the forward rate
plus a liquidity premium.
Previous case what we have seen? that expected spot rate is equal to the forward rate but as per
the liquidity premium theory the expected spot rate should be equal to the forward rate plus the
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liquidity premium which is basically nothing. But the maturity spread what we call it that is your
return from the long-term bond, the return from the short-term bond. And if you assume that
investment in the longer period is riskier than the investment in the short period or the
investment in the short-term bond then the premium should be increasing with the maturity.
So, therefore as per the liquidity preference theory or liquidity premium theory what basically we
can call it always the long-term bonds should give more return than the short-term bonds because
there is a risk involved. So, the limitations what basically we have seen whenever we have
discussed about the pure expectations theory. there we have seen there is a risk neutral world.
But you assume that the risk is also playing the role and the investor who is investing in the longterm assets.
This will be compensated against the extra risk what they are taking whenever they are holding
the long-term bonds or they are trying to invest in the long-term bond. So, that particular extra
return what they are expecting that basically we call it the liquidity premium and that premium is
basically increasing with the maturity. So, that is the thing basically what basically you have to
keep in the mind.
(Refer Slide Time: 29:14)
Then what basically we have discussed in today's class? that when the equilibrium yield curve
was positively sloped the market always, expect higher rates in the future. Well then, it is
negatively sloped then the market, basically expect a lower rates in the future. Then, if the pure
303
expectations theory holds, then the expected future rate should be equal to the implied forward
rates.
And the as for the pure expectations theory, the yield to maturity can be expressed as a geometric
mean of or the average of the today’s rate and the implied forward rate. And the implied forward
rate is equal to the expected spot rate, as per the pure expectations theory. But whenever, we
have brought the liquidity premium theory that basically tells that there is a liquidity premium
for the long-term bonds over the short bonds.
So, in view of this long term bonds always give more return than the short term bonds to
compensate that particular extra risk what the investor takes. So, we can also say that the
liquidity premium theory is an extension of the pure expectations theory. So, you can assume
your expected spot rate you can determine your expected spot rate or you can calculate your
implied forward rate using the pure expectations theory.
But, we have to add that risk premium into that to find out the long term interest rate. that is what
basically always you have to keep in the mind. So, these are the different theories which,
basically tries to explain the term structure of the interest rates. Why there is a difference
between the long-term rate and this short term rate and as well as the shift of the yield curve in
this particular way can be determined.
(Refer Slide Time: 31:20)
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So, these are the references you can go through for the detailed discussion on this. Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 05
Lecture No # 21
Determination of Theoretical Spot Rate and Forward Yield Curves
Welcome back! So, in the previous class we discussed about the features of the pure expectations
theory then as well as the liquidity preference theory or the liquidity premium theory. So, in this
particular session, we will be discussing about the how to determine this theoretical spot rate and
the forward yield curve?
(Refer Slide Time: 00:46)
So, here mostly the 2 concepts will be covered up. one is your derivation of the theoretical spot
rate curve and the forward yield curve.
(Refer Slide Time: 00:59)
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And you will come across certain keywords like your benchmark yield curve, bootstrapping,
implied forward rate. again, so all these things basically you will come across in this particular
discussion.
(Refer Slide Time: 01:13)
Let me tell you what do you mean by the theoretical spot rate and why it is called a theoretical
spot rate? The concept of the theoretical spot rate, basically comes to understand or to derive a
base rate. Because, whenever the market participants invest in the market, they have some idea
about the basic characteristics or basic behaviour of the yield curve from where they will start.
So, to understand the basic behaviour of the yield curve, always we need a kind of base yield
curve and that base yield curve is derived from these, portraits which are prevailed in the market.
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So, if there is a current spot rate, may be 2-year spot rate or one-year spot rate which is prevailed
in the market. And using that particular spot rate we try to determine the shape of the curve in the
beginning. And, after that the role of expectations and other things will come into the picture; the
liquidity premium and other concepts will come into the picture. But, in the beginning, we have
to understand that, how the shape of the yield curve will look like using the spot rate which is
prevailed in the market.
So, if you recall to some extent, we discussed about this part. whenever I was discussing with
you about the methods of bootstrapping, which basically used to calculate this kind of rates
which is going to prevail in the future. The same thing I will just repeat. but why basically we are
discussing this? because that concept is basically used to also derive the forward yield curve. So,
using a bootstrapping method, a spot rate yield curve can be derived. So, that is basically what in
our language, we call it the theoretical spot rate curve.
And already I told you. this is used by the practitioners to price these financial instruments and
also used by the dealers to identify the arbitrage opportunity. That is why it is called a bench
mark yield curve, using the bootstrapping method, whatever spot rate yield curve we are
deriving, that is used as a benchmark yield curve. So, once the theoretical spot rate curve is
derived or determined then, this can be used as a base rate yield curve or base income and other
yield curves are generated by adding the estimated risk and the liquidity premiums.
Very minutely you observe that, unless that base yield curve is not available, then where are
these premiums and to which rate this premiums will be added? So, because of that, the base rate
yield curve is very much important. And also, the particular spot rate curve is used to estimate
the implied forward rates and also it will generate the forward yield curve. So, it has lot of use.
So, the benchmark yield curve or the spot rate or the theoretical spot rate yield curve
determination is very much important.
Whenever we are trying to determine the long-term rates in the market or the investors are trying
to price the securities which are available in the market, which are basically depending upon the
interest rates which are prevailing in the market.
(Refer Slide Time: 06:07)
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So, in that context let us take one example and we will see that, how this particular thing is
working? here we have taken 6 assets. The first 2 assets are the zero-coupon bonds and next four
assets are the coupon bearing bonds. And the maturity period is given. let the first zero coupon
bond the maturity period is 6 months, then next one is 1 year, then 1.5 years, then 2 years, 2.5
years and 3 years only 6, 6 months maturity is increasing.
First 2 bonds are the zero-coupon bonds so they do not pay any coupons in between, that is why
there is no coupon involved in that and another four bonds they are giving certain coupons. So,
because these are all 6 months basis given, that is why here it is the semiannual coupon payments
have been given to you 2.75, 2.875, 3, 3.125. so, these are the semiannual coupons.
Now, the YTM; the yield to maturity of all those bonds are given. For 6-month zero coupon
bond yield to measure it is 5%, for one year it is 5.25%, for 1.5 coupon bearing bonds, it is 5.5,
5.75, 6% 6.25%. only 0.25% are increasing on the basis of the maturity. The face value of the
zero coupon bonds are 100 rupees and we are assuming that coupon bearing bonds are issued at
par. That is why, the phase value of the bond is basically is given 100 and the current price is
also 100.
So, these are the current prices. So, these are also 100 for the coupon bearing bonds but because
we are assuming these bonds are issued at par. And already, you know that these are the price of
the zero coupon bonds because the face value is given, this annualized YTM is given. Then, you
309
can find out your price on this face value divided by 1 + r to the power t your r is given here.
Right?
So, in that case you can find out the price of the zero-coupon bond and here, we have calculated
that the price of zero these 2 zero coupon bonds with the different maturity is 97.561, one is
94.947. Now our job is or our basic objective is to calculate this, these rates how these rates are
calculated? Already, the rates we have mentioned here but our basic discussion is how to
calculate this spot rates of the different zero-coupon bonds or the coupon-bearing bonds which
are traded in the market?
So already, below it is already highlighted there are 2 zero coupon with maturity 6 months. zero
coupon bonds basically with a maturity of 6 months and one year which are trading at yields of
5%, 5.25%. you can mention, it is zero coupon bonds. And these rates can be used as a spot rate
because generally the zero-coupon bonds rates are used as a spot rate that already we have
discussed in the previous sessions.
So that is why the, for maturities of 0.5 here we are representing at s 0.5 and for 1 year maturity
we are representing at s1. Right? And the other bonds which are reflected here these are the
coupon bonds that trading at par. So, that is why they therefore their yield is equal to their
coupon rates. And also, the price is also equal to the face value of the bond. is it clear? So, then
our objective is to how to calculate or how to find out these spot rates which are already given to
you?
(Refer Slide Time: 10:58)
310
So how we can calculate? let find the spot rate for 1.5 years. let the question is find the spot rate
for the 1.5 years? So, if you are going to calculate the spot rate for 1.5 years, how we calculate?
Taking the zero-coupon taking the coupon bond with a maturity of 1.5 years and your annual
coupon rate of 5.5 % that means semi-annual coupon is 2.75 .right? 5.5 divided by 2 that is 2.75.
Then, you set the par value of this 1.5 year bond equal to the present value of its cash flows
which are discounted at known spot rates.
There are 2 spots are available one is with a maturity period 0.5 and another one is 1 year S 0.5
and s 1. And one annual spot rate for 1.5 years. is it clear? that is s 1.5. So, now what basically
you can you can say? that p 1.5 years the price of the bond of the maturity 1.5 years should be
equal to your cash flow at 6 months. That is 0.5 divided by your 1 + s 0.5 divided by 2 to the
power 1 + C F 1 divided by 1 + S 1 divided by 2 to the power 2 + c f 1.5 divided by 1 + S 1.5
divided by 2 to the power 3. Because it is semiannual coupons what we are considering.
Now we say we know the price of the bond which is issued at par that is 100, price of the bond is
known. And the cash flow at 6 months for a first zero coupon bond which is S. what basically we
are getting? that is 2.75 second also we are getting 2.75 are getting my point. The zero-coupon
bonds rates will be considered below but above the cash flows what you are getting from the
coupon bearing bonds that is your 5.5% annually that means that is 2.75 semiannually.
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The first 6 months you will get 2.75, second 6 months you got 2.75 and obviously at the maturity
you will be getting your face value that is 100 + 2.75; that is 102.75. Which are the rates at
which it will be discounted the spot rate for S 0.5 that is given here. That is 0.05 divided by 2,
that is the spot rate for the 0.5 years bond, that is the spot rate for the one year bond zero coupon
bond. And this rate we have to determine that s 1.5 that basically we have to determine.
So, now what basically we have seen whenever these 2 things we have calculated we got
94.705956, that basically we have taken from the other side. That is 100 - this will we are getting
this much 94.705956 that will be equal to 102.75 divided by 1 + S 1.5 divided by 2 to the power
3. Then from there we can calculate that your s 1.5 that will be calculated on the 6 months basis
you multiplied by 2. That will give you 0.0551, which is basically spot rate for the coupon
bearing bonds, which maturity is 1.5 years. is it clear?
Estimation of spot rate for 1.5 years bond:
𝐶𝐹
𝐶𝐹
𝐶𝐹
P1.5 = (1+(𝑠 0.5/2))1 + (1+(𝑠 1.0/2))2 + (1+(𝑠 1.5/2))3
0.5
1.0
1.5
2.75
2.75
102.75
100 =(1+(0.05/2))1 + (1+(0.0525/2))2 + (1+(𝑠
1.5 /2))
3
102.75
94.705956 = (1+(𝑠
1.5/2))
102.75
3
1
S1.5 = 2[[94.705956 ]3 − 1 ] = 0.0551
(Refer Slide Time: 15:28)
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Like that, you can calculate also for the other maturity period bonds the spot rates can be
calculated. The spot rate for the 2-year bond, we repeat the process using the 2 year bond paying
semi-annual coupons, that is 2.875 selling at par. Then, now you got your spot rate for 1.5 years.
that spot rate now can be used for calculations of the spot rate for the 2-year maturity period.
And whatever to your maturity spot rate you will find that is your 5.577%, then that can be used
to calculate your 2.5 years maturity bonds.
Then again, whatever rate you will get it for the 2.5 years that we got it S 2.5, 6.03% that 6.03%
can be used again to calculate for the 3 years spot rate that is 6.3% like that. You can see all
these things are already mentioned in the last column of this particular table which has given in
the previous slide.
So, this is the process whatever we have followed to calculate the spot rates of the different
maturity and only we have 2 zero coupon bonds which are available, which we consider at S 0.5
and S 1 and from there the other rates are calculated.
(Refer Slide Time: 16:59)
Once this is calculated what basically we can do? we can calculate the forward yield curves? So,
I am here basically, we are giving you the general formula for deriving this forward yield. So, in
general, if you are using implied semiannual forward rates on m-year bond purchased t years
from the present. Let then this can be always found by using this formula your f m t = 1 + S M +
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t divided by 1 + S t 1 + S M t to the power n divided by 1 + S t to the power i whole to the power
1 by N - i – 1.
(1+𝑠𝑀+𝑡 )𝑁
Forward yield = [
(1+𝑠𝑡 )𝑡
]1/(𝑁−𝑖) - 1
And here, your S represents your semiannual spot rate, m represents time to maturity in years, t
represents the time period from the present to the forward rate in years. N is equal to your
number of semi-annual periods to m + t then i is equal to your number of semiannual periods to
the t. So, this is the general formula. So, let us apply this and find out the forward rates and from
this the forward yield curves can be determined.
(Refer Slide Time: 18:38)
We continue with our same example. Already, you have seen that we have 6 bonds, 2 zero
coupon bonds and another four coupon bonds. These are the maturity which has given to you
these are the spot rates whatever we have calculated. So, these are the spot rates and these are the
semiannual rates from here this divided by 2 we have basically calculate the semiannual rates.
6 months from now, the maturity is 6 months and 6 months from now that is basically t so let we
are representing it f 0.5 0.5. Forward implied forward rate on a 6 months bond, 6 months from
now then, how basically we calculate it? This can be obtained by defining the one-year spot rate
as the geometric average of the current 6 month spot rate. Let the first question is using the semiannual spot rate what will be the implied forward rate on the 6 months bond?
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And the implied forward rate that already we know this 6 month rate implied forward rate on a 6
month rate that is called the 6 months forward. Then your S 1 is nothing but your 1 + S 0.5 into 1
+ f 0.5, 0.5 to the power 1 by 2 - 1 that already you know. We have already discussed that then
from there if you want to calculate your f 0.5 0.5 that means your implied forward rate on the 6
months bond the 6 months from now. That will be 1 + S 1 whole square divided by 1 + S 0.5 - 1
and your S 1 = 0.2625 0.02625 the semiannual basis this one.
Then it is 1.02625 to the power 2 divided by 1.025 which is basically your spot rate for a 6
months bond that is 0.025 in the semiannual basis. Then if you solve this then you will find this
is 0.27502 right. So if you make it annualized then you will get your 2 into 0.027502 that is
0.55003 and approximately this values are calculated because this is a very the fractions are quite
long. So, like that you can calculate the other rates the other implied forward rates can be
calculated.
S1 = [(1+S0.5) (1+f0.5 0.5)]1/2 – 1
f0.5 0.5 =
f0.5 0.5 =
(1+𝑠1)2
(1+𝑠0.5 )
−1
(1.02625)2
(1.025)
− 1 = 0.027502
Annualized f0.5 0.5 = (2) (0.027502) = 0.055003
(Refer Slide Time: 21:52)
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For example, if you see what is the implied forward rate on a one-year bond purchase 6 months
from now f M t = f 1, 0.5. Then, how you can solve it? You can solve it by 1.5 were geometric
mean for f 1.5 you now. we can use this formula here m = 1 here t = 0.5 n = 3. That is basically
the number of semiannual periods to M + t and i = 1. So then your f M t become your m = 1 year
t = 0.5 years so that is why f 1, 0.5 right. So that means it is M + t your t = 0.5 years then it is 1 +
S 1.5 to the power n.
N = 3 to the power 3 divided by 1 + S 0.5 to the power 1 right. Your t is equal to t = 1 sorry t =
0.5 i = 1 that is why it is 1 + S 0.5 to the power 1 divided by to the power 1 by n – i. N = 3 i = 1 1 then if you solve this then S 1.5 already we have that is 0.027550 then 1.027550 divided by
1.025 and here you take the cube of this that 1.027550 to the power t 3 divided by 1.025 to the
power 1 to the power 1 by 2 - 1 then your f 1, 0.5 = 0.028827.
So if you make it annualized then it will be multiplied by 2 then you will get 0.057655. So this
5.6755% is the implied forward rate on a one year bond purchased 6 months from now is it.
fM = [
(1+𝑠𝑀+𝑡 )𝑁
(1+𝑠𝑡 )𝑡
(1+𝑠
]1/(𝑁−𝑖) - 1
)3
f1.0.5 = [ (1+𝑠1.5 )1 ]1/(3−1) - 1
0.5
(1.027550)3
f1.0.5 = [
(1.025)1
]1/2 - 1
f1.0.5 = 0.028827
(Refer Slide Time: 24:40)
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Now for example I will tell you to calculate implied forward rate on a 6 months maturity bond
0.5-year bond purchased one year from now. Then, here your m = 0.5, then your t = 1, n = 3
because the number of semiannual periods to m + t and i = 2. Because the number of semiannual periods to t because it is a purchased one year from now. Now your S 1.5 that already
given that is 0.027550 so 1.027550 to the power 3 divided by your 1 + S 1 to the power 2
because here your t = 1 then it is 1.02625 to the power 2 to the power 1 by n – i. N = 3 and i = 2
that is why it is to the power 1 only - 1 that will give you 0. 030155.So, if you make it annualized
then it will become 2 into 0.030155 into 2 that will give you 0.06031. So, like that the other rates
can be calculated from this, particular spot rates whatever we have derived.
fM = [
(1+𝑠𝑀+𝑡 )𝑁
(1+𝑠𝑡 )𝑡
(1+𝑠
]1/(𝑁−𝑖) - 1
)3
1.5
f0.5.1 = [ (1+𝑠
]1/(3−2) - 1
)2
1
(1.027550)3
f0.5.1 = [ (1.02625)2 ]1- 1
f0.5.1 = 0.030155
Annualized f0.5.1 = (2) (0.030155) = 0.060310
(Refer Slide Time: 26:28)
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Let I will give a little bit more, longer period or the other kind of forward rates let you are
calculating the implied forward rate on 1 year bond percentage 1 year from now. Your M = 1 and
also t = 1 so then here what is happening your M = 1 t = 1 n = 4 because the number of
semiannual periods to M + t. M = 1 and the t = 1 that means total how much 4 periods 6 months
basis we are paying right then i = 2 because the number of semiannual periods to t
So now what will happen your f 1 1 whenever you are calculating you have to consider S m + t,
m is equal to 1 t = 1 then we have to calculate s 2 to the power n. N is equal to your periods that
is the 4 periods that is why we are taking 4 divide by 1 + S 1 to the power 2 because this is s t. T
= 1 then 1 + S 1 to the power 2 whole to the power 1 by 4 - 2 – 1.
Then your f 1 1 will become we have the S 2 rate already available the spot rate we have derived
it then 1 + 0.027885 to the power 4 divide by 1.2625 to the power 2 whole to the power 1 by 2 1 and that will give you your 029523. Then your if you make it analyze then you can multiply 2
here then you will get this particular rate like that the other rates basically you can calculate from
here.
fM = [
(1+𝑠𝑀+𝑡 )𝑁
(1+𝑠𝑡 )𝑡
]1/(𝑁−𝑖) - 1
(1+𝑠 )4
f1.1 = [ (1+𝑠2 )2 ]1/(4−2) - 1
1
(1.027885)4
f1.1 = [ (1.02625)2 ]1/2 - 1
f1.1 = 0.029523
Annualized f1.1 = (2) (0.029523) = 0.059045
So, we have given the 3 cases the 1-year bond purchased from 1 year bond purchased from 6
months from now 6 months bond purchases 6 months from now. So, like that there are different
alternatives what basically, we have taken here and accordingly the implied forward rates are
calculated using the spot rates the benchmark rates whatever we have derived in the beginning.
(Refer Slide Time: 29:11)
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So, how it can be used so if the implied forward rate 6 months from now are realized then an
investor with 6 months horizon would earn the same semiannual rate. That is 2.5% obtained on
the 6 months bill or the coupon whatever we are taking coupon bond by buying any maturity
bond and selling it 6 months later at its implied forward rate. Similarly, if the implied forward
rates one year from now are realized then the investor with a one-year horizon would earn the
same semi-annual rate of 2.655% obtained on 1 year by buying any maturity bond and selling it
one year later I did simply forward it.
So, one of the practical uses of this forward curve is that it can be used to identify the hedgable
rates which are useful in determining the alternative investments. So that is the practical use of
this particular forward yield curve.
(Refer Slide Time: 30:18)
319
So, what basically we discussed, we derived a spot rate yield curve where the in which this spot
rates are estimated using the bootstrapping, which generally called the theoretical spot red curve
and also it is used as a benchmark yield curve. And the spot rate curve is generally used to
estimate the implied forward rates and also generate the forward yield curves. And already, just
now we see that the forward curve can be used to identify the hedgable rates, which is very much
important in terms of the determination of the alternative investments.
(Refer Slide Time: 30:59)
So, these are the references you can go through for today's session. thank you.
320
Managements of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology Kharagpur
Module No # 05
Lecture No # 22
Interest Rates Risk
Good morning! So, in the previous class, we discussed about the how to derive this spot rate
yield curve? And from there also, we discussed about how to derive forward yield curve?
Today, we will start the discussion on the particular important risk, what the fixed income
securities market investor always face, that is called the interest rate risk. But, the other name
of that thing is also the market risk.
Largely, whenever we are investing in the fixed income securities we are more exposed to
interest rate risk because the interest rate fluctuations with the impact of the pricing and other
things.
(Refer Slide Time: 01:05)
So, in this particular session, we will discuss largely about the concept of the market risk or
the interest rate risk. And you will be coming across these particular words like price effect or
price risk, interest on interest effect and the reinvestment risk etc. So, these are the different
keywords for basically we will come across, whenever this particular session will be going
on.
(Refer Slide Time: 01:35)
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You see? what is the market risk? The market risk is nothing but it is the uncertainty that the
realized return will deviate from the expected return because of the interest rate changes.
Your total return concept already we have discussed. the total return whatever you are
getting, if you are expecting this much return to get and end of the day your total return is not
exactly equal to your expected return.
That means, we are exposed to the market risk. So, that is basically formally we defined the
market risk. So, the return on a bond basically comes from 3 sources mostly; coupons,
interest what basically we earned, whenever you are reinvesting the coupons, that is called
the interest on interest and the capital gain or the losses at what price you have bought the
bond and what price you have sold the bond.
So, these are the 3 major components of the return always we see. so whenever, there is a
interest rate change that will affect basically the interest on interest, the capital gain or the
loss part. Because coupons are not going to be affected as coupon is fixed. So, only these 2
components are going to be affected. So, because of this the return can be derivate. So, that is
basically what we call it the market risk.
(Refer Slide Time: 03:20)
322
So now, if you see whenever there is a interest rate change, it has 2 effects on the bonds
return one is price effect, another one is interest on interest effect. And what is price effect?
because the interest rate change affect the price of the bond. if interest rate will go up the
price of the bond will go down or the interest rate will go down the price of the bond will go
up. So, there is an inverse relationship between the interest rate and the price of the bond.
So if the price of the bond is getting affected we call it the price risk. if you are holding up to
the maturity then, your return is not going to be affected. But, if you are going to sell the
bond in between; before the maturity, then what basically will happen? you are exposed to
some amount of price risk or some amount of interest on interest risk or the investment what
we call it.
So that is why, if the investors horizon period is different from the bond maturity date, then
the investor generally is uncertain about the price what he or she is going to receive from
selling the one. If your horizon date or horizon period is less than the maturity period also the
price he will have to pay if somebody is going to buy a bond. If the horizon period is greater
than the maturity period, both ways the price effect is basically working there.
Then, the other one is interest on interest effect. So, the interest rate change will affect the
return, the investor basically expects from the investing the coupon that is called the
reinvestment risk. So, if the interest rate will go up then what will happen? The return from
the reinvestment of the coupon will be increasing but the interest rate will go down, then if
whenever you are receiving the periodic coupons that coupons will be basically reinvested in
the market at the lower rate.
323
That is why your reinvestment return will go down but your price of the bond will go up. So,
the investment effect and the price effect works in the opposite direction.
(Refer Slide Time: 06:25)
So let us see, how basically this works? I have just summarized here in this particular case.
So here, what basically we have seen? let your delta r represent the change in interest rate.
So, if your there are 2 components it will affect the interest on interest and also affect the
price change in the price. The change in the interest rate will have the impact on the change
in price and change in the interest on the interest rate. In this, interest on interest means the
reinvestment on the coupons.
So, if the r is increasing, then your interest on interest rate is increasing or price is declining
but if r is declining, then your interest on interest rate is declining and price is increasing. So,
now the price effect and interest on interest effect, they are moving in the opposite direction
whenever there is a change in the interest rate.
(Refer Slide Time: 07:37)
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So, let us what happens in that particular context why basically this thing happens in the
market? You consider an investor whose horizon period is 3.5 years who buys a 10 years
maturity bonds and coupon is 10%. And the par value of the bond is 1000 Rupees which is
going to yield 10%. Your expected return from this is the 10%. And you assume that initially
the yield curve is flat which is giving 10% return and there are no changes in the yield curve
in the forthcoming years.
Then, obviously what will happen? That 10% will be automatically realized because the
interest rate is not going to be changed. So, if no change is happening then what will happen?
That the investor will be getting the coupons for the 3.5 years at a rate of 10% and the total
value of this particular coupon will be 347.16 at the year 3.5 and how we got it? They will
got 300 rupees coupon and your 47.16 is the interest which is earned from the reinvesting the
coupon.
Annual coupons that 100 rupees each for 3 years he got, and this 300 rupees has been
reinvested in the market. Then, he got a interest of 47.16 then the interest on interest basically
the reinvestment return amount what they got from the coupon that is 47.16 and the coupon is
300 then total 347.16 they got. Now the investor have to sell the bond because the horizon
period is 3.5 years and after 3.5 years he has to sell the bond, then we have to find out the
price in which he has sold the bond.
So, the investor will be able to sell the original 10 years bond now the maturity left out that is
6.5 years. Then at what price you will sell the bond? There are 2 ways you can calculate it but
you can also calculate it since the bond is sold in a non-coupon rate. Because coupon rate is
325
annually then what this investor can do? You can determine the price by discounting. Then
we can say that the value of the bond at the next coupon date is 4 years.
So, therefore the P4 you can calculate then this P4 will be your summation t = 1 to 6 because
the 6 years are remaining that is 100 divided by 1.1 to the power t + 1000 divided by 1.1 to
the power 10 is it clear. So that will give you the 1000 rupees generally it is the price of the
bond. But P 3.5 will be how much? That will be your only the 6 months are left out to reach
this particular next coupon date.
That will be 1000 + 100 divided by 1.1 to the power 0.25 so then you can find out the price at
the end of the 3.5 years or you can also discount it because it is not in the coupon date then
this is the way you can find out the price of the bond. The price of the bond has come
1048.81. so, at this price the investor has sold the bond. he has bought the bond at a price of
1000 and he has sold the bond at the price of 1048.81. And he got the coupon plus
reinvestment of the coupon that is 347.16.
Calculation of selling price of 10-year bond at the end of 3.5 years:
P4 =∑6𝑡=1
100
(1.10)𝑡
+
1000
(1.10)10
= 1000
1000+100
P3.5= (1.10)0.5 = 1048.81
(Refer Slide Time: 12:16)
Now what happen you can find out your total return. from this total return is you are this to
1048.81 + 347.16 divided by 1000 to the power 1 by 3.5 -1 that is yours 10%. And why you
got 10% total return? because your yield curve has not shifted. the yield curve remains there
at 10%. that is why your total return is realized.
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Total return = [
1048.81+347.16 1/3.5
1000
]
– 1 =0.10
(Refer Slide Time: 12:48)
HD value = 100 (1.10)2.5+ 100 (1.10)1.5 + 100 (1.10)0.5+ P3.5B = 1395.97
1000+100
P3.5B = (1.10)0.5 = 1048.81
P4B =∑6𝑡=1
TR3.5 = [
100
(1.10)𝑡
+
1395.97 1/3.5
1000
]
1000
(1.10)6
= 1000
– 1 =0.10
So this is the summary basically has been given so this is the way the horizon value has been
calculated and this is your total return basically you are getting.
(Refer Slide Time: 13:02)
327
Suppose shortly after investor purchase the bond the flat 10% yield curve shifted to 12% and
remained there for the 3.5 years. your horizon period is 3.5 years whenever you have bought
the bond the interest rate was 10% but all of sudden it has shifted up to 12%. And remain
there are the 3.5 years then what will happen? First of all you find out your horizon value
right so 100% coupon you are getting 100 into 1.12 to the power 2.5 because 12% has
become the interest rate and you have reinvested it for 2.5 years whatever coupon you have
got in the end of the first year.
Then plus 100 into 1.12 to the power 1.5 because that thing you have invested second year
coupon you have invested reinvested it 1.4 for 1.5 years that is 100 into 1.12 to the power 1.5
+ your 100 into 1.12 to the power 0.5 plus your price of the bond at the period 3.5 years. So
what is the price of the bond at the end of the 3.5 years that will be your 961.70 that we have
calculated here that is 961.70.
So now your and how about this because price of the bond at the end of the 4 year that is 4
year price that will be 6 years is remaining that is t = 1 to 6 that is 100 divided by 1.12 to the
power t + 1000 divided by 1.12 to the power 6 because 12% is the now the discount rate that
is got 917.77. So this price will be 917.77 + 100 divided by 1.12 to the power 0.5 that you got
961.7 is it clear?
So now the total return will become so if you add this 961.70 here your P 3.5 B price of the
bond at which the investor has sold the bond. Then he got total value has become 1318.81
and you have bought the bond at the price of 1000 which was given in the beginning because
the bond basically was that time the yield was 10% and it was issued at par. Then your total
return has become 13.181 divided by 1000 to the power 1 by 3.5 that will become 8.23%.
If 10% yield curve shifted upto 12%, then total return at the end of 3.5 years will be:
HD value = 100 (1.12)2.5+ 100 (1.12)1.5 + 100 (1.12)0.5+ P3.5B = 1318.81
P3.5B =
917.77+100
= 961.70
(1.12)0.5
P4B =∑6𝑡=1
TR3.5 = [
100
+
𝑡
(1.12)
1318.81 1/3.5
1000
]
1000
(1.12)6
= 917.77
– 1 =0.0823
So, in this context what we have seen, there is a capital loss of 38.30 and partially the loss can
be offset by the gains what basically you are getting by reinvesting your coupon at the rate of
12%. So combined the investor horizon value has become 1318 which is 77.16 less than the
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horizon value, what basically we got in the previous case, whenever the yield curve was flat
that is 1395.97.
So, whenever the rate was remained constant at 10% your total value was 1395.97. now, the
value has become total value has become 1318.81. So, there is a loss in that particular case
that is why the total return as come down to 8.23%. Let us take a reverse case here the
interest rate has gone off now let the interest rate has gone down to 8%.
(Refer Slide Time: 17:50)
Let the yield curve has shifted down from 10% to 8% and remain there for 3.5 years then
what will happen? So, obviously your horizon value will be changed that is 100 into 1.08 to
the power 2.5 because 8% has become the interest rate and 100 into 1.08 to the power 1.5 +
100 and 1 positive per 0.5. Then, your price of the bond at 3.5 years that will become 1484.82
because the price of the bond at the end of the 3.5% you got 1147.44.
Same way we have calculated first we calculate the P 4 price of the bond at the end of the 4
that is your t = 1 to 6 100 divided by 1.08 to the power t + 1000 divided by 1.08 to the power
6 that will give you 1092.46. Then here 1092.46 + 100 which is the coupon what you are
going to get divide by 1.08 to the power 0.5 that will give 1147.44 the price of the bond bas
gone up.
Because the interest rate has gone down so in this case your horizon value has increased from
1484.82 which was 1395 point something in the whenever the yield curve was flat. So that is
why the total return has increased from 10% to 11.96%
If 10% yield curve shifted down to 8%, then total return at the end of 3.5 years will be:
329
HD value = 100 (1.08)2.5+ 100 (1.08)1.5 + 100 (1.08)0.5+ P3.5B = 1484.82
P3.5B =
1092.46+100
(1.08)0.5
P4B =∑6𝑡=1
TR3.5 = [
100
(1.08)𝑡
= 1147.44
+
1484.82 1/3.5
1000
]
1000
(1.08)6
= 1092.46
– 1 =0.1196
So, what you have observed here? What is the implication or the findings you got in this
particular 2 examples.
(Refer Slide Time: 19:51)
In this example, what basically we have received? that the interest rate change has 2 opposite
effects. one is inverse price effect and one is direct interest on interest effect. And the total
return varies directly or inversely to the change in the interest rate that will depend on which
effect is dominating whether the price effect is dominating or the interest on interest rate
effect is dominating. that will decide so, that your total return will be more or total return will
be less whatever basically you have expected from the beginning.
So, in this example whatever we have taken the inverse price effect is dominating that is why
whenever the interest rate is going up the total return is going down. Whenever the interest
rate is going down the total return is going up, so, this basically causes the total return to vary
inversely with the change in the interest rate. That is the findings what we got from the
example, whatever we have taken. So now, let us see is there any possibility that the interest
on interest effect also can dominate the price effect or not. And that is also possible.
(Refer Slide Time: 21:33)
330
Let us take another example where the interest on interest effect which is dominating the
price effect. Let we have taken a bond whose maturity period is 4 years and coupon is 20%
and the yield curve was flat at 10%. Let you assume the same thing that the yield curve shift
to 12% then you have to find out the price 20% of the coupon the periodic cash flow is 200
per annum. 200 into 1.12 to the power 2.5 + 200 into 1.12 to the power 1.5 + 200 into 1.12 to
the power 0.5 plus your price of the bond at the end of the 3.5 years.
That will give you horizon value now what is the price of the bond your price of the bond
basically has become this at the end of the 3.5 years that is basically your; what we call it the
1133.89. So your total value of the bond was basically we got it that is your 1848.12 and the
price of the bond is basically in the beginning you have bought it that is 1317. So now your
total rate has become 1848.12 divided by 1317 to the power 1 by 3.5 -1 that is 10.16%.
If 10% yield curve shifted up to 12%, then total return at the end of 3.5 years will be:
HD value = 200 (1.12)2.5+ 200 (1.12)1.5 + 200 (1.12)0.5+ P3.5B = 1848.12
1000+200
P3.5B = (1.12)0.5 = 1133.89
P0B =∑4𝑡=1
TR3.5 = [
200
(1.10)𝑡
+
1848.12 1/3.5
1317
]
1000
(1.10)4
= 1317
– 1 =0.1016
What does it mean? It means the interest rate has gone up but still your return has gone up
from 10% to 10.16%. So what it implies? It implies that the interest on interest rate effect is
dominating the price effect.
let the interest rate has gone down to 8%. Then the price of the bond at the end of the 3.8
years has become this 1154.70 and the total value at the horizon has become 1829.45 little bit
331
it has gone down because the reinvestments return what you are getting from the coupon
investment that is little bit has gone down but the price has increased.
So now the total value got 1828.45 and your price of the bond is 1317 then at this price you
have bought the bond then obviously your total return has become 9.845%.
If 10% yield curve shifted down to 8%, then total return at the end of 3.5 years will be:
HD value = 200 (1.08)2.5+ 200 (1.08)1.5 + 200 (1.08)0.5+ P3.5B = 1829.45
1000+200
P3.5B = (1.08)0.5 = 1154.70
P0B =∑4𝑡=1
TR3.5 = [
200
(1.10)𝑡
+
1829.45 1/3.5
1317
]
1000
(1.10)4
= 1317
– 1 =0.09845
What you have realized here? If your interest rate will go down in the previous case the
return has gone up because the price effect has dominating the reinvestment effect or the
interest on interest rate effect.
And in this case what we have observed? Even if the interest rate has come down the return
what you got that is lesser than the 10% return. what was there in the beginning? whenever
the yield curve was flat at 10%. So clearly in this example we have seen, that the interest on
interest rate effect which is dominating the price effect. that is why it will result in the direct
relationship between the total return and the interest rate changes.
So that means here you see that is your high coupon rate and term to maturity is also low that
is 4 years. So, in that case your interest on interest rate effect generally dominates the price
effect. Previous case, we have seen the coupon was 10% term to maturity also 10% so in that
case the price effect was dominating the interest on interest effect.
(Refer Slide Time: 26:44)
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Then let us see, if there is any possibility that interest on interest effect which can equals the
price effect. It is also possible to select a bond in which the interest on interest rate price
effect exactly offset to each other. We will discuss extensively more about this whenever we
go for the bond portfolio strategy that is generally called the immunization strategy. But, I
will just give an example here, if the price effect will exactly offset this interest on interest
rate effect then the total return will not change.
When there is a yield curve shift just after then purchase even if the interest rate will change
the total return will remain same. Suppose, the investor basically purchased a 4 years
maturity bond of 9% annual coupon yield is 10% and the price is let 968.30 to assume. So, if
the flat yield curve immediately shifted to 12% or 8% or any other rate then the total return
will remain at 10%. how it will work?
(Refer Slide Time: 28:18)
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You just see let the interest rate has shifted to 12%. 3.5 years is our horizon period that we
are keeping it. coupon is 9% then you got 90 into 1.12 to the power 2.5 + 90 into 1.12 to the
power 1.5 + 19 to 1.12 to the power 0.5 plus your price of the bond at the end of the 3.5 years
that has become 1351.35. Because the price of the bond at the end of the 3.5 years is 1029.95
that is the price of the bond that basically is here we have added.Price of the bond already we
know that is 968.30 is it clear? So now the total return has become this 1351.35 divided
968.30 to the power 1 by 3-5 – 1 that is 10%.
If 10% yield curve shifted up to 12%, then total return at the end of 3.5 years will be:
HD value = 90 (1.12)2.5+ 90 (1.12)1.5 + 90 (1.12)0.5+ P3.5B = 1351.35
1000+90
P3.5B = (1.12)0.5 = 1029.95
P0B =∑4𝑡=1
90
(1.10)𝑡
+
1000
(1.10)4
= 968.30
1351.35
TR3.5 = [ 968.30 ]1/3.5 – 1 =0.10
Let the yield curve as shifted to 8% then if you go by your understanding or our analysis 90
into 1.80 to the power 2.5 + 90 into 1.08 to the power 1.5 + 90 into 1.08 to the power 0.5 plus
the price of the bond which is basically the price that is 1352.49.Then your total return has
become 1.352.49 divided by 968.30 to the power 1 by 3.5 that is 10%.
If 10% yield curve shifted down to 8%, then total return at the end of 3.5 years will be:
HD value = 90 (1.08)2.5+ 90 (1.08)1.5 + 90 (1.08)0.5+ P3.5B = 1352.49
P3.5B =
1000+90
(1.08)0.5
P0B =∑4𝑡=1
= 1048.85
90
(1.10)𝑡
+
1000
(1.10)4
= 968.30
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1352.49
TR3.5 = [ 968.30 ]1/3.5 – 1 =0.10
Because price of the bond is 968.30. how it is possible? The investor can do this by buying a
bond that pays a coupon and as a maturity different than the investors horizon period and up
to what particular what should be horizon period the investor should always keep and that
should always keep which particular period that also has to be understood.
So, that is basically we call it the bond immunization strategy, where the concept of duration
and other things comes into the picture. that we will discuss in the forth coming sessions but
that is possible. Price effect can exactly offset this interest on interest effect that possibilities
are there.
(Refer Slide Time: 31:01)
So, what basically we have discussed here? the interest on whenever there is a change in
interest rate it has 2 opposite effects on total return, one is inverse price effect, other one is
the direct interest on interest effect. And the total return will vary directly or inversely to
interest rate changes that depends on that which effect is dominating. The price effect is
dominating then the interest rate will increase then the return will go down if the interest on
interest rate is dominating then the opposite result will prevail in the market.
So, that is the way basically the concept on interest rate risk can be explained. So, these are
the 2 effects always we observe and that is very much important from the investment point of
view.
(Refer Slide Time: 31:58)
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So, these are the reference what you can go through for the detailed discussion on this and in
the future classes, we will be discussing about the other aspects which are related to the bond
risk. Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 05
Lecture No # 23
Bond Duration
Welcome back! and good morning. So, in the previous class, we discussed about the concept of
the market risk which is nothing but, the due to the interest rate change how the price of the bond
basically changes or the total return of the bond changes? So, there we have seen there are 2 effects
work, one is your price effect other one is the reinvestment effect or interest on interest effect.
So, depending upon the interest rate moments and as well as the 2 effects which are working are
the depending upon the dominance of those particular effects. Whether price effect dominance, or
the interest on interest rate effective dominance, the total return of the bond basically will change.
So, today we will be discussing about the other concept which is related to the bond risk, that is
basically we call it the bond duration.
(Refer Slide Time: 01:15)
So, today we will be covering of the concept of the bond duration. So, in this particular context,
we will discuss about what exactly the bond duration is. And how the bond duration is measured
and what is the use of the bond duration? And also, how we can calculate the duration of a portfolio
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of the bonds or we can say that portfolio duration or duration of the bond portfolio. So, these are
the different concepts, what basically will be discussing in today's session.
(Refer Slide Time: 01:52)
So, you will come across certain concepts or certain keywords with respect to this particular
session that is basically your Macaulay’s duration, modified duration, and the portfolio duration.
So, these are the different keywords you will come across throughout this particular session.
(Refer Slide Time: 02:14)
So, let us see that what exactly the bond duration is? So, all of you know that we have a concept
called the term to maturity. term to maturity is nothing but the period at which the bond is going
to be mature. So, like that the duration is also a time measure, bonds duration is also a time measure
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but it is basically a weighted time measure. So, in this case what basically we can say that the
bonds duration is basically a weighted average of the bonds time period.
Bonds duration is defined as the weighted average of the bonds time period. So how we can give
the weights? The weights are basically given on the basis of the time periods relative present value
of its cash flow. So, in general, if you see it is nothing but a weighted average on a present value
basis of the time to full recovery of the principal and the interest payments on a bond. So, therefore
it generally measures the weighted average maturity of a bonds cash flow on a present value basis.
If you look at this particular formula, this formula is used to calculate the concept of the duration.
So, here if you see it is written summation t = 1 to n t into the present value of the cash flow at the
time period t divided by the price of the bond. So, what basically we do? we calculate the present
value of the cash flow in each period. Then, we try to find out this weight by dividing that particular
each period time period with respect to the price of the bond then we multiply that with the time.
D= ∑𝑁
𝑡=1 𝑡
𝑃𝑉(𝐶𝐹𝑡 )
𝑃0 𝑏
So, then we take a summation of all then we can calculate the duration of this particular bond. So
that is why it is called a weighted average of the bonds time period and the weights are basically
given on the basis of the present value basis.
(Refer Slide Time: 04:47)
So, let us see that what exactly it measures? If you see that the bonds duration is basically a measure
of the effective maturity that incorporates the timing and the size of the bonds cash flow. The size
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of the bonds cash flow and also the time both the factors are taken into consideration whenever we
calculate the duration of the particular bond. So, it captures the combined impact of the interest
rate size of the interim payments which is coming out of the coupons and maturity on the bonds
price volatility.
Conceptually if you see duration is nothing but it is a measure of the interest elasticity in
determining the bonds market value. it is basically measure of the interest elasticity in determining
a bonds market value. So, if the securities or the bonds duration is known, then the investor can
estimate the size of change in the value or the price of the particular bond for different interest rate
changes.
So, if there is a interest rate change, how the bond price is going to be changed, that you can judge
if you have the duration data available with you. So that is the basic use of the particular duration.
So, that is why the concept of duration is quite important from the risk management perspective.
In the previous class if you remember, we have discussed that a particular bond where the price
effect is equal to the interest on interest effect or the reinvestment effect.
So, how basically you can choose that bond and what type bond basically will give this kind of
scenario, in that case your duration generally helps. So that is why that concept or that particular
strategy is called the immunization strategy, that will discuss further. But you just keep in the mind
duration is very much relevant. duration is very much important from that perspective.
(Refer Slide Time: 07:38)
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So let us see that how exactly the duration is measured? because I have given you this particular
formula. But here if you see that whenever we calculate this duration let we have taken a bond
where the term to maturity is 4 years, coupon is 9%, par value of the bond is 1000 rupees and the
yield to maturity is 10%. So, in that case if you if you observe let the coupons are paid annually,
it is an annual coupon payment bond.
So, in the first year you will get a coupon of 90 rupees. in the end of the first year, second year you
get 90, in the end of the third year you get 90 and in the end of the fourth year you get 1090,
because 1000 is the par value of the bond. So now what you can do your yield is 10%? so you find
out the present value of that particular cash flow, what you got in the first year. So, then it is 90
divided by 1.1 yes then it will be 81.818. then second year end of the second year it will be 90
divided by 1.1 square, then you will get 74.380.
Third year 90 divided by 1.1 to the power 3 you get 67.618. And in the fourth year you get your
coupon and as well as the principal. then if you are using your yield for discounting it then you
can get your 744.485 right. So, then if you add up all these then you get this price of the bond. So
that is basically your price of the bond. so price of the bond has come 968.30. So now what you
can do as per the formula what basically we discussed in the previous slide, you find out the
proportion of each cash flow with respect to that particular price.
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Then this one will be 81.818 divided by 968.30, that will give you this one this will give you
74.380 divided by 968.30, you will get this. Then 67.618 divided by 968.30 will give you this then
finally 744.485 divided by 968.30, that will give you this. Now, what you can do you multiply this
value with respect to the time so 0.84496 by 1 you get this 0.76815 multiplied by 2 you get this
then 0.69832 multiplied by 3 you get this then 0.768857 into 4 you get this.
Then approximately your duration has come 3.52 if you add up all these, then you get a summation
of all. You get a summation of all then that basically will be 3.52. then this 3.52 is basically the
bonds duration, it is a bonds duration .so that is the way the duration of the bond is calculated.
(Refer Slide Time: 11:27)
So, now you see how we can calculate the duration of a bond portfolio? here what calculation we
have shown? that is basically the duration of a single bond. So, whenever you are calculating the
duration of a bond portfolio that is also very simple. it is nothing but the weighted average of each
of the bonds division. And how the weights are given? the weights are given on the basis of the
proportion of bonds, you have invested in each bond.
Dp= ∑𝑖=1 𝑤𝑖 𝐷𝑖
Let you have 3 bonds available. one bond duration is 2, another bond is 3, another bond is four.
then let total 1000 rupees you are investing. Then you are allocating let 300 rupees for bond 1, 400
rupees for bond 2, and 300 rupees for bond 3, then in that case what basically you can do? you
find out the each proportion 30% 40% and 30%. Then each bond has the duration available, then
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0.3 multiplied by 2 plus 0.4 multiplied by 3 plus 0.3 multiplied by 4, that are the duration which
has been given to you for each type of bonds. then your portfolio duration can be calculated.
So, instead of selecting a specific bond with a desired duration, one investor also could determine
the allocations for each bond in his or her portfolio, that would yield the desired portfolio duration.
That also somebody can do instead of investing in a single one, you can choose the multiple bonds
and where you can get your desired portfolio duration from that particular context.
(Refer Slide Time: 13:27)
So otherwise already i told you, because the duration has a different use, also it talks about how
the movement of the price of the bond is going to be changed. That you can judge if you have the
duration data available with you. So, that is what duration is also considered as a price sensitivity
measure. So, already we have defined that duration is nothing but a weighted average of a bonds
time period. but it is also important measure of the volatility; price volatility of the bond.
So, whenever we are thinking from that perspective, the duration as a volatility measure, in that
case, what basically we do or how we can define the duration? In that case, duration is basically
defined as the percentage change in the bond price given a small change in yield. That percentage
change in the bond price is percentage of delta p. delta p is the change in the bond price. p 0 is the
original price of the bond. then your d y which is represented as the small change in the yield or
small change in the interest rate.
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So, in that case if you want to calculate the duration of the bond mathematically, then what you
have to do? you have to take the derivative of the price equation? You take the first derivative of
the price equation of a bond with respect to the yield. So, that means with respect to the yield how
the price, if there is a change in the yield how the price of the bond is going to be changed?
You take the first derivative of that particular bond price formula and divide by the bond price and
express that particular value in absolute form, so that will also give you the value of the duration.
So, it is nothing but the first order derivative of the bond price equation with respect to the change
in the yield divided by the price of the bond. That basically will give you also the concept of the
duration or the measure of the duration.
(Refer Slide Time: 15:58)
So, in this case just now what basically we discussed then how we can duration can be defined.
Duration is equal to your d p by p divided by d y. So that is basically nothing but your 1 by 1 + y
the summation of t = 1 to n t the present value of the cash flow at the period t divided by the price
of the bond. So, this particular term already we have seen that is basically your duration in the
formula whatever we have seen.
Duration =
𝑑𝑝/𝑃0
𝑑𝑦
=
1
(1+𝑦)
[ ∑𝑁
𝑡=1 𝑡
𝑃𝑉(𝐶𝐹𝑡)
𝑃0 𝐵
]
So, here you have, you are getting extra term, that is 1 by 1 + y you are getting a 1 extra term that
is 1 by 1 + y. So, that is why that is little bit different. what formula we have just discussed in the
previous slide and now what formula we are deriving here. So, here already I told you d p by p is
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a percentage change in the price of the bond. So you can write your 0 I think I have so d p by p 0
is the percentage change in the price of the bond.
And d y is equal to your small change in the yield and your n is equal to the number of periods to
the maturity that is basically we can also say m. So, with bracketed expression that just now
whatever I have written, that is basically is the weighted average of the time period which is
defined in the last section as the duration just now what I told you. So, here why there is a change
in the formula because the 2 duration concepts are different.
In the first case, whenever we have calculated the duration on the basis of this formula, that is
called basically your Macaulay’s duration. And the equation which basically defines the
percentage change in the bond price due to a small change in the yield in the absolute value that is
called basically the modified duration. So, this part is called the Macaulay’s duration and this
divided by 1 + y or multiplied by 1 by 1 + y that are called the modified duration.
So, whenever we are calculating the duration as a price sensitivity measure using the concept of
the first order derivative, generally we are calculating the modified duration. And using this
formula whenever we are calculating the duration that is basically called the, what we call it that
is basically called this Macaulay’s duration. These are the 2 different concepts what basically you
have to keep in the mind.
(Refer Slide Time: 19:48)
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So, what we have seen here? the modified duration is equal to the Macaulay’s duration divided by
1 + y. where y basically is nothing but the yield to maturity. So, the proportional change in the
bond price to the change in yield that is your d p by p by d y is negative, that means there is a
inverse relationship between the price and yield. So that is why there is a convention to express
the modified duration in the absolute form right.
Modified Duration =
1
(1+𝑦)
[ 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 ′ 𝑠 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛]
Macaulay’s Duration = ∑𝑁
𝑡=1 𝑡
𝑃𝑉(𝐶𝐹𝑡)
𝑃0 𝐵
So already, I told you that modified duration is nothing but the Macaulay’s duration divided by 1
+ y and the Macaulay’s duration is t = 1 to n t into p v or the present value of the cash flow or the
period t divided by the price of the bond that part already we have discussed.
(Refer Slide Time: 20:48)
So, let in the same example, whatever we have taken if your Macaulay’s duration, we have
calculated that is your 3.52 and your yield is 10% then your modifier duration will be 3.52 divided
by 1.1 that will give you 3.20. 3.20 is the modified duration and 3.52 was the Macaulay’s duration.
So, whenever we are using duration as a price sensitivity measure mostly, we are calculating this
modified duration not the Macaulay’s duration.
Modified Duration =
Modified Duration =
1
(1+𝑦)
1
(1.10)
[ 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 ′ 𝑠 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛]
[ 3.52] = 3.20
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(Refer Slide Time: 21:32)
So, that is why if you show it in a diagrammatic way, the modified duration is nothing but is the
slope of the price yield curve divided by the p. Because, the slope of the price yield curve is nothing
but d P by d y divided by p, that is basically called the modified duration. so the percentage change
in the price that is percentage change in percentage of delta p divided by d y. So, this is basically
wherever this particular this is the slope of this particular curve price yield curve that is basically
your d P by d y.
Duration =
𝑑𝑝/𝑃
𝑑𝑦
=
%∆𝑃
𝑑𝑦
Or you can use d P 0 by d y, whatever depends on the notations, whatever you are using then divide
by your p that will give you the modified duration of this particular form. So, that is why, if you
are measuring the slope of the yield curve and divide it with respect to the price then you can
calculate the duration. So already, you know that what is your yield price yield curve; one is your
price one axis other axis is the yield to maturity. So, that is basically you call it the price rate curve.
that is the way basically the modified duration can be formulated or can be calculated.
(Refer Slide Time: 23:12)
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So, if you see that the price of a bond which basically pays the coupon in each period and its
principle at the maturity. Already, we know the formula how the price is calculated? that is your p
0 b is equal to c into 1 - 1 by 1 + y to the power n divided by y + f by 1 + y to the power n. N is
the term to maturity, that already we have discussed.
1
P0B = C [
1−(1+𝑦 )
𝑦
𝑁
]+
𝐹
(1+𝑦)𝑁
So, if you take the first derivative of this equation and divide that particular thing through p then
express that particular resulting equation in the absolute value that will basically provide you the
measure of the duration for a bond paying the principal at the maturity.
So, that basically will get your modified duration is equal to your c by y square into 1 – 1 by 1 +
y to the power n + your n into f - c by y divided by 1 + y to the power n + 1. So, that is basically
the derivatives first order derivative what you are calculating from this equation. then you divide
it with respect to the price that will give you the modified duration of that particular coupon paying
bond and the principal is basically paid at the maturity. So, directly you can calculate your modified
duration using this particular formula. this particular equation that basically you can do whenever
you are using duration as a price sensitivity measure.
Modified Duration =
𝐶
𝑁[𝐹−( )]
𝑐
1
𝑦
[1−
]+
𝑦2
(1+𝑦)𝑁 (1+𝑦)𝑁+1
𝑃0 𝐵
348
(Refer Slide Time: 25:02)
Now, if you see what kind of properties we can get it in terms of the duration? So, the measure of
the duration basically is defined in terms of the length of the period between the payments, it is a
time measure. If the cash flow is distributed annually, duration is reflected in years. if cash flow is
semiannually, then duration basically reflects the half years. But general convention is always
duration is expressed as an annual measure.
So, how you can calculate the annualized duration? The annualized duration is nothing but
whatever duration you are calculating on the basis of the frequency of the coupon payments that
you first calculate and divided by the number of payments per year, that will give you the annual
duration of that particular bond.
(Refer Slide Time: 26:07)
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So, let us come to an example. so if you question is let find the modified duration measured in half
years for a 10 year, 9% coupon bond selling at par. let f is equal to 100 with coupon payments
made semiannually. Then 9% of the coupon so that is why half yearly coupon is c that is 5.5 then
your divided by 0.045 square 1 - 1 by 1.045 to the power 20 because 10 years is the maturity.
Now, it is halfway payment that will become n become 20 + your n into a 20 into 100 - f – 4.5 that
is the coupon our half yearly coupon divided by your 0.045 it is 9% divided by 2 then divided by
1.045 to the power n + 1 that is your 21 divided by 100 that is the basically the price of the bond
because bond is issued at par. Then, you will get 13 approximately it will get 13 so this particular
is the duration in the half years duration in half years. Then your annualized duration will become
13 by 2 that are 6.5.
Duration in half years = =
4.5
20[100−(
)]
4.5
1
0.045
[1−
]+
2
20
21
(1.045)
0.045
(1.045)
100
Annualized return = 13/2 = 6.5
(Refer Slide Time: 27:32)
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= 13
So, then what are the properties of the duration? we have certain properties; lower the coupon,
greater the duration. longer the term to maturity, greater the duration. Other things remain same if,
you are comparing between the 2 bonds, lower the coupon rate greater will be the duration, longer
the term to maturity. Obviously, if the term to maturity will go, over more than duration will be
more.
For a zero coupon bond, this Macaulay’s duration is equal to the bonds term to maturity. anyway
there is no periodic coupons available. And the modified duration is n by 1 + y that yield what you
are calculating? that yield if you divide it with respect to that term to maturity or the duration, then
you can get it the modified duration.
Higher the yield to maturity, lower the duration that means the slope of the price will curve is
flatter at the height the slope of the price. Yield curve is generally flat at the higher yield to
maturity. So, these are the different observations or different properties, what you can get with
respect to the duration principle.
(Refer Slide Time: 28:56)
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So, if you know the bonds duration, you can formulate the bond portfolio strategy how? For
example, let a bond speculator who is anticipating a decrease in the interest rate across all the
maturity that means there is a downward shift of the yield curve. In that case, they could realize a
potentially greater expected return but also greater risk by purchasing a bond with a relatively large
duration.
Just now, we have seen but in contrast a bond portfolio manager who is expecting a parallel upward
shift in the yield curve they can go to for a defensive strategy. Defensive action against the possible
capital losses by reallocating their portfolio such that they will have a lower portfolio duration. So,
this is the way duration can be used as a bond portfolio strategy measure, that actually you have to
keep in the mind.
(Refer Slide Time: 30:12)
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So, the duration is also used as an estimate of the percentage change in the bonds price for a small
change in the interest rate. For example, there is a 10 year bond ,9% coupon which is selling at
100 rupees to yield 9% if the interest rate will increase by later 10 annual basis point, that means
from 9% to 9.1%. Then if you go by the using the modified duration equation if you find this bond
price will decrease approximately by 0.65%.
%∆𝑃𝑏 =
%∆𝑃𝑏 =
𝑑𝑝𝑏
𝑃0𝑏
𝑑𝑝𝑏
𝑃0𝑏
= [Modified Duration] dy
= [6.5] [0.0910-0.09] = 0.0065
So, that is 0.65% what you can get so that is very much close to the percentage change in the actual
percentage change at the bond price. Because, the bond price is you can calculate let, we are using
the same example then their coupon is 4.5 semiannually if you calculate the price of the bond that
will you will get 99.3524 and percentage change in the bond price will be 99.3524 - 100 divided
by 100 that is approximately 0.6476 %.
%∆𝑃𝑏 =
99.3524−100
100
Where: P= ∑20
𝑡=1
= -0.006476
4.50
(1.0455)𝑡
+
100
(1.0455)20
= 99.3524
But, if you are using the modified duration directly you got your duration modified duration is 6.5.
in the previous example,e directly you can put this percentage change in the price of the bond then
you get 65.65%, which is a very much close to this.
(Refer Slide Time: 31:46)
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But little bit, if you increase your interest rate change let 200 basis point let it has increased to 11%
in that case what you find? You find that the price will change by point you will get this .13 but in
true sense it is basically 11.95%. If you calculate the price of the bond you will get 88.0496 then
effectively it is 11.95%.
%∆𝑃𝑏 =
%∆𝑃𝑏 =
𝑑𝑝𝑏
𝑃0𝑏
= [6.5] [0.11-0.09] = 0.13
88.0496−100
100
20
Where: P= ∑𝑡=1
= -0.1195
4.50
𝑡 +
(1.055)
100
(1.055)20
= 88.0496
So, that means what we can conclude, if the interest rate will change by in a larger amount then
this modified duration may not help you or may not give the exact value of change in the bond
prices but in the small change that thing can be used.
(Refer Slide Time: 32:34)
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So, greater the change in yield the less accurate duration is estimating the approximate percentage
change in the price. So, using the duration to estimate the discrete changes does not capture the
asymmetrical gain and loss relation that characterize the price yield curve. For yield increases
duration over estimates the price change for you decreases duration underestimate this price
change.
(Refer Slide Time: 33:00)
So already, we have explained this portfolio duration here one example has been given. So here,
it is the coupon rate it is the maturity, it is the price these are the market value of this particular
bonds. So, this on the basis of the market value total market value the weights are calculated. So,
modified duration of each bond has been given then this multiplied by this weight, the weighted
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duration of each bond is calculated. Then finally basically we are calculating this portfolio duration
of that particular bond.
So, the portfolio has a modified variation of 5.5586 that means if yield on each of the bonds change
by 100 basis point; then the portfolio value will change approximately by 5.55886%. So, each
bonds weighted duration measures the bonds contribution to the overall portfolios duration. So,
bond a, contributes this percentage that is 2410 divided by this portfolio duration of 5.86 and bond
d contributes this that is the way basically we can calculate this or use this concept of the portfolio
derivation.
(Refer Slide Time: 34:17)
So, in the conclusion what we find? that the duration of a bond measures the weighted average
maturity of a bonds cash flow on the present value basis. Or it is also defined as a percentage
change in the bond’s price given a small change in the yield. Lower the coupon, greater the
duration. longer the term to maturity, greater the duration. For a zero coupon bond Macaulay’s
duration is equal to the bond’s term to maturity. And the modified duration is equal to the
Macaulay’s duration divided by 1 + yield.
(Refer Slide Time: 34:44)
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So, these are the references what you can go through. thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 05
Lecture No # 24
Bond Convexity
Welcome back! So, in the previous class, we discussed the concept of duration, which is
basically a weighted time measure of a particular bond, and as well as also, we have seen that
duration can be used as a price sensitivity measure. Due to the change in interest rate, how the
price of the bond is going to be changed, that also we can measure, we can judge from the
duration measure. So, today we will be discussing another concept related to that that is called
bond convexity.
(Refer Slide Time: 00:49)
So, whenever you talk about this bond convexity, we will be discussing certain concepts related
to that like, what is analyzed convexity, the concept of convexity bias, approximate duration, and
approximate convexity. These are the different approximate measures of duration and convexity.
Also, we can discuss this because there is a link between duration and convexity. So, what we
have seen in the previous class is that you see whenever the interest rate is changing a little bit in
a larger percentage or change in interest rate is relatively more.
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In that case, your duration was not able to give you an accurate figure in terms of the change in
the bond prices. So, that thing can be resolved through the use of the concept of convexity. So,
let us see what exactly convexity means, and how convexity is defined?
(Refer Slide Time: 02:00)
If you see the convexity, you see the convexity is basically a concept, that is very much relevant
in terms of the bond risk management strategy. because, the price yield curve is not a linear
curve, and it is convex. It is both. just now, we have seen already all of you know that the price
yield curve is not linear, but convex from below. So, the duration that we have discussed
basically measures the slope of the price yield curve at a given point.
That is the first order derivative of the bond price equation we have estimated and there we have
seen the duration is basically, a measure of the slope of the price yield curve. But, whenever we
come to the concept of convexity, it basically measures the change in the slope of the price of the
yield curve. The duration measures the slope of the pricing curve, but convexity measures the
change in the slope of the price yield curve, which means it basically discussed the second-order
derivatives.
The convexity is basically the second-order derivative of the price equation, with respect to the
change in interest rate. So, in a general sense convexity basically tries to measure, how the bowshaped price yield curve is? What is the curvature of the price yield curve? that basically is
reflected through the concept of convexity? So that is, why convexity has a lot of relevance from
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the investment point of view. Whenever anybody is investing in the bond market, convexity has
its own relevance, and convexity always is used in that particular context.
(Refer Slide Time: 04:13)
So, let us see in the mathematical sense how convexity is defined. Already, we have discussed
that convexity means that the slope of the price yield curve, gets smaller as you move down this
particular curve or the YTM is increasing. In the x-axis, we are taking YTM; in the price yield
curve. So, whenever convexity, you are trying to measure or you are trying to define, you will
find that the slope of the price yield curve, which is basically your delta.
Let we are representing dP by dy which basically gets smaller, as you move down the curve or
the YTM is increasing then what does it imply? It implies that for a given absolute change in the
yield, the percentage increase in the price will be greater in absolute value for the yield increase
than the percentage decrease in the price in absolute value for the yield decrease. You got this
point? let the interest rate has changed. So, whenever the yield is increasing, then what will
happen?
The price of the bond will change. So, the percentage increase in the price will be greater in
absolute value, whenever the yield is changing or increasing, than the percentage decrease in the
price in the absolute value for yield is decreasing.
(Refer Slide Time: 06:19)
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So, what basically here we are trying to see from here? You see already I told you, that convexity
is the change in the slope of the price yield curve divided by P. So, we have a price yield curve.
So, this is your price yield curve and these are the different slopes the dP by dy, dP by dy, dP by
dy. these are the different slopes and the slope is changing whenever you are moving your slope
is basically changing.
So, what is the implication you can draw from here? So, whenever the interest rate is increasing
or decreasing, if you are talking about whenever your yield is basically increasing you will find
that the percentage change in the price, or we can say the slope of the yield curve is not going to
be changed or is going to be changed in a different way in the lower zone than in the upper zone.
Is it clear?
(Refer Slide Time: 07:53)
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So, let us see what basically the use of that or what is the implication of that? So, for a bond
investor who is long in a bond, that means buying the bond. Its convexity basically suggests that
capital gain results from a decrease in rates. If the interest rate will decline, then obviously the
price will increase. So, the capital gain which is resulting from a decrease in rates will be greater
than the capital loss resulting from an increase in rates of the same absolute magnitude.
Let the interest rate will go down by 1% or go up by 1%. Then, whatever capital gain that person
will get in terms of the declining interest rate that will be more than, if the interest rate would
have increased by 1%, then will have lost in the market. Some capital loss would have occurred.
Then, if it will increase by 1%, then the capital gain that he or she will get, will be more than
what basically they would have lost.
If the interest rate would have been increased by 1%, that is what basically the convexity is
trying to imply to you. So, that is why bonds are the bond portfolio with greater convexity and
have a greater asymmetrical gain or loss relationship. The gain and loss relationship is very much
asymmetric in nature. So here, from the investment perspective, what you can say is all other
things equal greater the bonds convexity more the valuable the bond is from the investor
perspective, that actually you can keep in the mind.
(Refer Slide Time: 10:22)
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So let us see, that how the convexity is measured mathematically? already we have explained it.
Convexity is the change in the slope of the pricing curve for a small change in the yield. So, this
is basically the second-order derivative of the price equation. So, if you want to derive it, you can
take the derivative of the equation of dP by dy and divide the resulting equation by the current
price.
So, already we have this first-order derivative equation available to you. Then you take the
second derivative with respect to the yield. That is why that means you are I am talking about the
dP square by this d square P by dy square. Then, the convexity will be 1 by P0 into your t =1 to
N t into t +1 in bracket cash flow at the time t divided by 1+ y to the power t +2. Or if you
related the equation whatever we have used for measuring this duration in the previous session.
Convexity =
1
𝑃0 𝑏
[∑𝑁
𝑡=1
𝑡(𝑡+1)(𝐶𝐹)1
(1+𝑦)𝑡+2
]
If you are considering the convexity of a bond that pays a fixed coupon each period and principle
at the maturity, then the convexity will be your 2c by y cube into in bracket 1-1 by 1+ y to the
power N -2 CN divided by y square 1+ y to the power N +1+ N + N into n +1 F– C by y divided
by 1+ y to the power N +2. You can also derive it on your own, whenever you have the price
equation available to you.
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If you are considering the convexity of a bond that pays a fixed coupon each period and principle
at the maturity is
Convexity =
𝑐
𝑁(𝑁+1)[𝐹−( )]
2𝐶
1
2𝐶𝑁
𝑦
[1−
]− 2
+
𝑌3
(1+𝑌)𝑁
𝑦 (1+𝑦)𝑁+1
(1+𝑦)𝑁+2
𝑃0 𝑏
Take the first-order derivative with respect to the change in the yield and again, go for a second
or derivative with respect to that. Then, you can get this particular equation. Then, you divide it
with respect to the current price then you can calculate the convexity. Is it clear? So, this is the
second order derivative of the bond price equation with respect to the change in the interest rate.
(Refer Slide Time: 13:00)
So like duration, convexity also reflects the length of periods between the cash flows. It also
reflects the length of the periods between the cash flows and convexities also conventionally
always represented annually. So, the annualized convexity is basically found by dividing the
convexity measured in terms of n times payments per year by n square. In the duration of what
we have done? The convexity for bonds with or the duration of the bonds with n times payments
per year divided by n that we have done.
But whenever you are making the convexity analysis, this is convexity for the bond with n
payments per year divided by n, square. N payments per year mean, how many times the coupon
is paid in a particular year? So, if it is semiannual, then it is 2 to power 2. If it is quarterly, then
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like that your figure will be changed. So, let what is the convexity for the 10 year 9% coupon
bond with semiannual payments and bond is issued at par?
The same example that we are taking, is a semiannual coupon. So that is why 4.5 is the semiannual coupon payment, what you will be getting? So, then 2 into 4.5 divided by your y to the
power or 0.045 to the power 3 into 1-1 by 1.045 to the power 20-2 into 4.5 into 20 divided by
0.045 square into 1.045 to the power n +1 that means 21 plus your 20 n into n +1 that is 20 into
100-4.5 divided by 0.045 divided by 1.045 to the power 22 that is n + 2 that divided by 100. That
is the price of the bond.
Convexity (1/2 yrs) =
4.5
(20)(21)[100−(
)]
2(4.5)
1
2(4.5)(20)
.045
[1−
]−
+
3
20
2
21
22
(1.045)
0.045
(0.045) (1.045)
(1.045)
100
= 225.43
We are assuming that the bond is issued at par then your convexity in terms of half yearly, you
are getting 225.43. So, if you are calculating the analyzed convexity, then it is 225.43 divided by
n square and here n = 2 because it is a semiannual coupon payment bond. Then that will be
225.43 divided by 4 which will be 56.36 approximately. So, 56.36 is the annualized convexity of
this particular bond.
Annualized convexity =
225
22
= 56.36
(Refer Slide Time: 15:57)
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So, then what are the properties of this particular convexity or the use of this particular
convexity? First is already we have seen, that there is an asymmetrical gain or loss relation. Let
us visualize it, and how basically it will happen? The greater a bond’s convexity, the greater its
capital gains and smaller its capital losses for a given absolute change in the end. Let the y 0
here, y 0 was the initial YTM, and at that time this was the price.
Now, let the interest rate has gone down, and YTM has gone down from y 0 to y 1. Then what
we have observed? the price has gone up let this be P 0, then it becomes P 1. Let this same
interest rate has gone up with the same amount that y 0 to y 2. So, then what will happen? There
will be the price will go down and this is P 2. So now, what we have seen is that the gain that we
are getting is more than the loss, and this is your price yield curve.
Come to this is a little bit more convex in comparison to this. This is a little bit flatter in
comparison to this. The curvature of this curve is more than this curve. If you see the same
amount of interest rate chains, let this be your original price. This is P 1 and this is your P 2.
What we have seen here, the gain is more than the losses, here also gain is more than the losses.
But, here it is more in comparison to this, also it is more, that means it shows more the curvature,
more the valuable the bond is.
The asymmetrical gain and loss relations if you are trying to measure, then that is basically more
visible, more captured on the basis of the curvature of this particular income in this case. So, this
one is more than this. This one is also more than this. But here, it is the gain is much higher than
the loss in comparison to this. So that is the first property that you can draw, the greater a bond’s
convexity, greater its capital gains and the smaller the capital losses for absolute changes in the
yield and that also depends upon the convexity of that particular price yield curve.
(Refer Slide Time: 19:39)
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Second thing, convexity also can be used with duration or like duration to estimate the
percentage change in the bond price given a change in the yield. We have seen in the duration
case, also we can find out the percentage of a change in the price with respect to change in the
interest rate by using the concept of duration. But, unlike duration which can only provide a good
estimate when the yield change is small, whenever you are using the convexity, it allows for a
better estimate for the large head changes.
So, even if the interest rate will increase by a little bit more percentage, the percentage change in
the price can be measured through the use of the convexity measure which was not possible in
the case of the duration measure. So, if you want to derive the formula which is estimating the
percentage change in the price for a large change in the yield, basically you can go for using
Taylor’s expansion.
Here, I am not giving the complete Taylor's expansion. but this expansion basically will yield the
percentage change in the price of the bond is equal to your modified duration multiplied by the
change in the yield that is delta y+ 1 by 2 into the convexity into your delta y square.
1
%∆𝑃𝑏 = [Modified Duration] ∆𝑦 + 2[ Convexity] (∆𝑦)2
So, that is the formula for calculation of the change in the price, whenever there is a change in a
large change in the yield and that thing you are directly calculating using the concept of
convexity. Using the concept of convexity basically, you are trying to measure it.
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(Refer Slide Time: 21:54)
Let us take the example, how basically it happens. We will continue with our previous example
also. The estimated percentage change in the price of a 10-year, 9% coupon bond. let the interest
rate increase by 200 basis points that means 2%. Now, you use this particular formula. The
percentage change in the price of the bond will be your convexity is 6.5. In the previous case, we
have measured that is -6.5 into 0.02 then + 0.02 mean 2% increases in the delta y that is 2% plus
half of the 56.36 which is the modified duration.Yield curve into your delta y square, that is 0.02
whole square, that you got -0.1187.
1
%∆𝑃𝑏 = [-6.5] (0.02) + 2[ 56.36] (0.02)2 = -0.1187
That means -11.87% means the interest rate has gone up. So, because of that, the price will go
down. So, the change in the price of the bond percentage the price of the bond will be negative.
So, the 11.87% decrease is actually very much close to the actual degrees of or actual decrease
per change in the price of the bond, which is 11.95%.
Then the estimated 13% decrease was obtained using the duration measure. In the previous case
we have seen, that whenever we are 200 basis points or 2% interest rate we have changed, we got
a percentage change in the price will be 13% using the duration. But actually, it is 11.95% but
whenever we have used the convexity, we find it is 11.87% and actually it is 11.95%. But
whenever you are using the convexity, you are getting 11.87 that mean it is closure.
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So, this formula also results in non-symmetrical percentage increases and decreases. How? Let
the rate has decreased by 200 basis points. If the rate will decrease by 200 basis points, then the
percentage obviously the bond price will increase the percentage of the change in the bond price
will be positive. Then we got that is 14.13%, not the 13% that the duration measure basically
yields.
1
%∆𝑃𝑏 = [-6.5] (0.02) + 2[ 56.36] (0.02)2 = 0.1413
So here, what basically you are we are trying to find out? that convexity is a better measure in
terms of measuring the change in the price of the bond or percentage change in the price of the
bond, than the duration whenever the change in the interest rate is a little bit higher. Whenever
the yield change is a little bit higher, then you can use convexity for calculating the percentage
change in the bond price in comparison to the durations. So, that is the advantage that you can
get from here.
(Refer Slide Time: 26:00)
So, as the yield increases or decreases, the convexity of the bond decreases or increases. this is
basically referred to as positive convexity. That is, when the yield is increasing, the convexity of
the bond is decreasing. Whenever the yield is decreasing, the convexity of the bond is increasing.
This is basically called positive convexity. The concept is basically called positive convexity.
For a given yield and maturity lower the coupon, the greater the convexity, and for a given yield
and modified duration lower the coupon, the smaller the convexity. So, these are the different
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properties that basically you can consider you can always try to find out, whenever you are using
the concept of convexity.
(Refer Slide Time: 27:12)
So, the value of the convexity if you are trying to measure, already we say that the bonds having
more convexity is more valuable. Let you consider 2 similar bonds X and Y, which are trading at
the same yield let y 0 and have the same duration but with bond X having greater convexity than
the bond Y. For example, so if you assume that there is an efficient market, then investors always
would take the greater convexity of the X into their consideration and price it higher, and accept
a lower yield.
(Refer Slide Time: 27:59)
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But the question here is how much would the market pay for the convexity? So, the answer
basically depends on how much the market expects the rates to change? How this interest rate is
going to change? If the investor expects the yield to change very little, that means the market
with low-interest rate volatility. Then the advantage of X over Y is very small and in this market,
an investor would pay little for the convexity.
On the other hand, if the investor is expecting that change is very much significant which means
high-interest rate volatility, then the advantage of X over Y is more important. So, in this market
investors will pay more for the convexity and given that longer-term bonds have a greater
duration and therefore volatility. One would expect that yields on a long-term bond to be affected
more by convexity than the shorter bonds.
Given that the longer-term bonds of greater duration and also the volatility, one can expect the
yields on a long-term bond to be affected more by the convexity, than certain bonds. And the
influence of the convexity on the shape of the yield curve and the relation between the returns
and the duration is called as the convexity bias. That is basically referred to as convexity bias.
How the convexity is affecting the shape of the yield curve and the return relationship between
the duration and this return and the duration is basically called the convexity bias.
(Refer Slide Time: 30:23)
So, there are alternative duration and convexity formulas. Generally, we use that is we call the
approximate measures of the duration and complexity. So, duration and convexity can be also
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estimated by determining the price of the bond. When the yield increases by a small number of
basis points like 2 to 10 basis points generally and when the yield decreases by the same number
of basis points that is P – or P +.
When it is increased, then obviously the price will be + P plus, and when the yield decreases by
the same number of basis points that is P-. So, these measures are basically referred to as
approximate duration and approximate convexity. We have seen how the duration and convex it
are measured. But, approximately also if you know the price changes due to the change in the
yield then you can also calculate the approximate duration and approximate convexity.
So, the approximate duration is your P- change in the price due to whenever the price is
declining whenever the interest is increasing. Then it is or vice versa that is, P - v + divided by
your 2 into P 0 into delta y. P 0 is the initial price. So, in this case, you can calculate the
approximate duration and approximate convexities of your P++ P –2 P0. Whenever the interest
rate will increase or decrease you find out your P + and P -, then your initial price is P0 is -2 P0
divided by P0 into delta y square. That will give you the measure of the approximate convexity.
Approx. Duration =
𝑃− − 𝑃+
2(𝑃0 ) (∆𝑦)
Approx. Convexity =
𝑃+ + 𝑃−− 2𝑃0
(𝑃0 ) (∆𝑦)2
(Refer Slide Time: 32:40)
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So, what we have seen here is convexity is a measure of change in the slope of the price yield
curve. The greater the bond’s convexity, greater its capital gain and the smaller its capital losses
for absolute change in the yield, and the influence of convexity on the shape of the yield curve
generally is referred to as the convexity bias and given that the longer-term bonds of greater
duration and one would expect the yields on a long-term bond to be affected more by the
convexity than the short-term bonds.
And another thing also you can keep in the mind using convexity, you can also measure the
percentage change in the price of the bond. Whenever, the interest rate increases better,
whenever the change in the interest rate is a little bit higher. The duration was not able to be
approximately measured in a better way that can be always used the concept of convexity. So,
whenever we change in the yield is more convexity will give you a more accurate measure in
terms of the change in the price of the bond than the use of the duration.
(Refer Slide Time: 34:01)
So, these are the references what you can go through for today’s session. thank you.
373
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 05
Lecture No # 25
Default Risk
Welcome back! So, in the previous class, we discussed about two important concepts with
respect to the bond risk management strategy. one is your duration and other one is the convexity
and other most important risk also the investors always is exposed to whenever they try to invest
in the bond market, that is basically your default risk or the credit risk. So, the credit evaluation
models and other kind of advance techniques with respect to the credit risk that we will discuss
further.
But we will just introduce the concept of the credit risk and what exactly the credit risk is and
what are the basic concepts that are linked to that particular credit risk? That basically we can
discuss today.
(Refer Slide Time: 01:12)
So, using this concept of the credit risks which; is largely has a strong association with this bond
investment. We will discuss about broadly different concepts related to that like what exactly the
default risk or credit risk is? And the concept of the credit spread. And throughout this particular
discussion you will come across certain keywords.
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(Refer Slide Time: 01:39)
Like credit rating, the concept of default loss rate, recovery rate, the concept of credit watch then,
ratings transition matrix. So, these are the different concepts what generally we use whenever we
try to always discuss about the default risk or the credit risk in the bond investment.
(Refer Slide Time: 02:12)
Let us see that what exactly the default risk is? The default risk or the credit risk is basically a
kind of uncertainty. Which is arises or which basically always we are facing and because of this
the realized return may deviate from the expected return. Due to the failure of the issuer to meet
the contractual obligations specified in the bonds indenture provisions.
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So for any reason if the issuer is not able to pay the interest or the principal or any other thing
which are specified in the indenture provision, then that means the bond investor or bond holder
is exposed to the default risk. Like market risk it is a common type of risk always we face
whenever we invest in the bond market. So, whenever we invest in the bond market credit risk is
inevitable.
So that is why we always give lot of emphasis on the credit risk, whenever we try to invest in the
bond market or always we try to assess the credit risk whenever at any point of time we try to
invest in the bond market. So, then we will see that how somebody will come to know about the
credit risk as a bond investor, if I want to go and invest in the bond market. What are the things I
will look at or how i can get the information about the credit risk with respect to that particular
bond? So that is the fundamental question always we should try to answer.
(Refer Slide Time: 04:17)
So, because of this, in practical sense if you look at many large institutional investors have their
own credit analysis department. They evaluate the bond issues. Any corporate or any other bonds
which are issued in the market they look at that particular company they consider certain factors
into the constitution and they try to determine the abilities of the companies to meet their
contractual obligations.
Whether the company which issuing the bond in the future they will be really able to repay their
required cash flows to them or not in terms of the interest or in terms of the principal and all.
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They will evaluate and accordingly. They decide whether they will be interested to invest in that
particular bond or not but it is not possible for everybody. So, because of that some of the
institutional investors and if it is a retail or individual bond investor, they largely rely on the
rating companies.
The credit rating agencies, whose primary job is to rate this particular companies in terms of
their ability to fulfill the certain requirements in terms of the payment of the regular payment of
the cash flows to the bond holders so they rely upon them.
(Refer Slide Time: 05:53)
So, what are those credit rating agencies there are many credit rating agencies you might have
come across. We have in India if you look at, we have CRISIL, the credit rating and information
services of India limited, Who is a larger organization or very prominent organization who gives
the rating to the different companies? Investment information and credit rating agency limited
ICRA that is another body.
We have a CARE called credit analysis and research limited. We have also a kind of
organization called brickwork ratings India private Limited, and then we have SMERA, SME
rating agency of India limited. But CRISIL is more prominent or more popular people always
refer to the CRISIL’S ratings, whenever they try to invest in the market. That is in a common
word for common kind of organization always comes to our mind.
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If you look at the global players, we have Moody's, we have Standard and Poor, we have the
Fitch ratings. So, they are the rating agencies very popular rating agencies who, basically gives
the rating to the different bond investors to the different bonds. And whenever the bond investors
try to invest in this kind of companies or the companies who are issuing the bond, then they
always look at the ratings of these agencies.
(Refer Slide Time: 07:33)
So little bit let us see that, what are the different type of rating symbols? the CRISIL basically
gives, CRISIL has a different category, we have triple A rated bond which is called the highest
safety the safety level. Then, we have double A rated bond high safety, A rated bond adequately
safety, BBB moderately safety, BB inadequate safety, B is highly risky bonds, C is substantial
risk and D is default bonds.
So, these are the different rating symbols you can always find whenever you are trying to invest
in the bond market. In India these are the different ratings what CRISIL basically gives to the
different kind of bonds which are issued by the different companies.
(Refer Slide Time: 08:42)
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If you look at the Moody's and standard force rating symbol, they have classified it broadly in
four category; very high quality, high quality, speculative bonds and very poor bonds. Only the
symbols are little bit different more or less same, but only the symbol different rating agencies
use the different ratings. If you go by Moody's let Moody uses Aaa or Aa. But in the same case
standard and poor basically uses AAA and AA.
So, these types of bonds are very high quality bonds. That means regularly they are fulfilling all
those requirements, they are paying the interest periodically, they are paying the principals at the
time of requirement. So, all type of bond characteristics which are mentioned in the indentured
provision, they are completely fulfilled by this particular rated companies.
So, if you go by A and Baa by the Moody’s and A, BBB by the standard and poor, they are high
quality bonds but not like the AAA or the AA rated bonds. So, in that case they have certain
restrictions, certain things they may not fulfill but this is very minor. That is not very important
in that particular or very much prominent in terms of the investment point of view, but they are
considered as the high quality bonds.
But this Ba, B, Caa, Ca in terms of Moody's and BB, B, CCC, CC by standard & poor, these are
called the speculative bonds. And Caa, Ca, CCC, CC, these are highly speculative bonds and the
Ba, BB; theese are the least speculative bonds. And D is a default bond in same case what the
CRISIL uses for India. The D is basically a default bond. if any kind of rating any company is
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getting D, then that means it is a default company, who are not basically able to pay the interest
or principal adequately at a specific time period. So, these are called the D rated bonds.
(Refer Slide Time: 11:47)
So, then what are those factors generally companies look at whenever they give the ratings to or
the rating agencies always look at whenever they give the ratings? There are many factors and
that factors also vary on the basis of the types of the companies. Like if you are giving a rating to
a manufacturing company or you are giving a rating to a finance company about a bank, the
rating parameters are not different.
But more or less there are certain common variables or common kind of factors they consider
they go for a business analysis of the company. They look at the sales, the volatility of the sales,
period of time how they are basically utilizing their resources. So those things come under the
business analysis part. Then they do the financial analysis. Financial analysis means mostly it is
very important they consider the financial risk like what is the leverage ratio and they give more
importance to the interest coverage ratio.
Whether it is adequate enough the companies are really generating that much profit to fulfill their
debt requirements or interest payments whatever they really need. So financial analysis what
they do they evaluate the management, the corporate governance part and this intangible asset ,
what the company is holding, what is the company's vision, mission and all kinds of things. The
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management evaluation they do whether the company is properly audited or not who are the
auditors.
So, there are different aspects they look at also from the management perspective and
accordingly they decide whether the company is a good company or not. Who has the adequate
ability to fulfill their desired requirements? Geographical analysis, where the company is placed,
whether there is political stability and other things is quite good in that particular country or not
or particular region or not so those things they look at.
Competition like regulatory and competitive environment, they look at the market competition,
threats opportunities, which are available with respect to that particular business. And who are
those competitors, what is the, what kind of market it is whether it is a monopoly market or profit
market or monopolistic market. So those things basically they look at what is the scope of that
particular business, what the company is trying to do? whether the regulatory environment is
very much restrictive in that particular country or not?
So those things they look and also, they look at the fundamental analysis so they are basically
they look at the liquidity aspects of the company. And the asset liability management aspects of
the company. whether the companies are able to manage their assets and liabilities properly or
not? And whether the liquidity is properly maintained by the companies or not? So, those things
basically they look at and finally they give a rating score or they give a score to that particular
company and on the basis of that score the rating can be assigned to that.
So, these are the different factors what the company or the credit rating agencies always look at
whenever they assign any kind of rating to any kind of companies who are issuing the bonds.
(Refer Slide Time: 15:45)
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So, whenever you talk about the ratings, you see in terms of the ratings there are 2 types of the
bonds we can come across. One is bond with relatively low chance of default and another one is
bonds with relatively high chance of default. And bonds which are having the low chance of
default, these are called the investment grade bonds. If you talk about us the quality ratings of
Baa or BBB, because the symbols are varying across the rating agencies or higher.
So, these are basically the investment grade bonds and the bonds which are relatively high
chance of default generally called the non-investment grade bonds or we call them the
speculative grade bonds or junk bonds and where the ratings are below the Baa, or BBB. So,
Baa, BBB, A, AA, AAA so all those bonds are basically investment-grade bonds. Then below
that whichever bonds are available, these are basically called the non-investment grade bonds.
(Refer Slide Time: 17:19)
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So, here one thing always the rating agencies or the analysts who are trying to assess the
company's ability, they try to always understand or try to measure three things. They try to
understand from this one is your cumulative default rates, the unconditional probability rates and
the conditional probability rates. These are the 3 things always the rating agencies basically try to
measure. What do you mean by the cumulative default rate?
These are basically the default chance through time. what is the probability of default basically
through time? how the default rate is changing over the time, that is basically called the
cumulative default rates. Unconditional probability rate means, these are the probabilities of
default in a given year as viewed from time zero.
That means the unconditional probability of a bond defaulting during the year t is nothing but the
difference between the cumulative probabilities in the year t minus the cumulative probabilities
of default in the year t - 1. It is only the difference between the 2 different periods of the
cumulative default rates. And the conditional probability is it is basically the probability of a
default in a given year conditional on no prior defaults.
And the conditional probabilities of default generally known as the probability intensities. It is
generally known as the probability intensity. How we can measure it? this is basically equal to
the unconditional probability of default in time t as a proportion of the bonds probability of
survival at the beginning of the period.
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And how we calculate this bonds probability of survival? The bonds probability of survival is
nothing but it is 100 minus the cumulative probability of that particular year. The cumulative
probability of that, particular year that is basically called the bonds probability of survival. Let us
see take one example then it will be clearer for you.
(Refer Slide Time: 20:28)
Let there is a bond that is A rated bond. Here, we have given some hypothetical figures. The
cumulative probability, unconditional probability and conditional probability let these are the
years 1-year bond, 2-year bond, 3 years bond, 4 years bond, 5 years bond. Right? 1, 2, 3, 4, 5
these are the years. The cumulative probability of a bond which has the rating of A in the first
year it is 0.05, second year it is 0.16, third year it is 0.22, 4 year be 0.34, fifth year 0.45. these are
all hypothetical numbers which are given to you.
Then what is the unconditional probability that means if you are trying to interpret this
cumulative probability then it is 5% chance of defaulting after 1 year. Cumulative probability
basically measures that there is a 5% chance of defaulting after 1 year, there is 16% chance after
2 years. The bond is going to be default 22% chance after 3 years, 34% of chance after 4 years
and 45% of the chance after 5 years. so that is the way the cumulative probabilities are defined.
Now, you come to the unconditional probability. what is unconditional probability? If you see
the table let you are trying to find out unconditional probability of bond default during the year
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three. That is this basically the conditional probability. Sorry! this cumulative probability of the
year three that is 22% , the cumulative probability of the year 2 that is 16%, then 22% minus
16% that will be 6%. If you are trying to measure the unconditional probability of year 2 that is
16% minus 5% that is 11%.
That year 5, that is 45% minus 34%, that is 11%. is it clear? So, that is the way we call it the
unconditional probability of that particular bond. Then what is conditional probability? we have
said let you take the example, the probability that a bond will survive until the end of year 3 that
is how much that is 78%. That means how got 78%? that is 100 minus 22%, 22% chance this is
going to be defaulted, then the probability of survival is 78%, that is 100 minus 22%, that will
give you 78%.
And the probability that the bond basically will default, the A rated bond during default during
the year 4 conditional on no prior default that is basically 13.09% that means it is 6% divided by
78% that will give you 13.09%. So, that is the way the conditional probability is measured. you
can measure for other bonds like that. Not other bond like in the different other years what is the
conditional probability of default of that particular bond is it ok? That is the way basically the
three concepts always we use.
(Refer Slide Time: 24:41)
Now let us see the concept called the default loss rate. you see the holders of defaulted bonds
generally recover a percentage of their investment. It not that the completely one is defaulted
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they can recover something from that, so that is called basically recovery rate. So, your default
loss rate is nothing but the default rate(1 - recovery rate) .
For example, if the recovery rate from a defaulted bond is estimated to be 40% and bonds default
rate is 8% then what is the default loss rate? the default loss rate will be 8% (1 – 0.40), that will
be 4.8%. The default loss rate is very much important in this perspective, that actually you have
to keep in the mind.
(Refer Slide Time: 25:45)
Other thing also you have to keep in the mind. over the time the rating may change. there may be
a downgrading or upgrading of the bonds. The credit risk involves not only the uncertainty
related to actual default, but also the concern that the bonds default risk may increase leading to a
downgrade in its quality rating and decrease in the bond price. So, when the issuer is placed on a
credit watch, it is for a downgrade not an upgrade, that means you should watch that particular
credit position of the company.
And rating agencies generally accumulate the statistics on how the ratings change over the time
and this information is generally presented in the ratings transition matrix. And the matrix is
basically defined in terms of the quality ratings at the beginning of the year which is represented
in the column wise and ratings at the end of year which is represented in the row. We will see
that, how basically the rating transition matrix is presented?
(Refer Slide Time: 27:21)
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Let this is a rating transition matrix example. So, in the column if you see this column wise if
you see this is basically the ratings in the start of the year and row wise if you see these are the
ratings which are given in the end of the year. Right? So let you take example of a double a rated
bond. if you look at a double rated bond if you want to interpret this particular transition matrix.
Then what basically we have seen here there is a 94% of the chance or the 94% of the AA bonds
at the beginning of the year remained that way at the end of the year here also it was 94% here
also it 94%. Clear?
And 1% of the AA bonds in the beginning of the year are upgraded to AAA rated you know
AAA rated bond. And 4% of the AA bonds at the beginning of the year were downgraded to a by
the end of the year. And 0.5% has been downgraded to BBB, 0.5% to BB and so on. So, if you
add of all these it should be 100%. So, this part basically should be this row wise if you add it
then it should be 100%.
So, that is the way basically the rating transition matrix we represent and from there we also can
conclude that, whether the bonds should be put on the credit watch or not. And whether there is a
downgradation is happening with respect to the bond or not or how the bonds also can be
upgraded.
(Refer Slide Time: 29:52)
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So, another thing also you have to understand that is credit spread. So, the credit spread on a
bond can be thought of as investor’s expected loss from a principle from default. Let you
consider a portfolio of seven-year BBB bonds which is trading at 3% credit spread, that means
the 2% premium that the investors receive from the bonds portfolio. sorry it is 3%.
So, the 3% premium that investor received from the bonds portfolio represents their
compensation for an implied expected loss of 3% for a year of the principal from the defaulted
bonds that actually you have to keep in the mind. So, how you can use it if the spreads were 3%
and bond investor believed that the expected loss from the default and such bonds would be only
2% per year of the principle.
Then the bond investor would want more BBB rated bonds. Then, the driving the prices up and
the yield will down until the premium reflected a 1% spread. So, if the spread were 3% and bond
investor believed that the default loss and a portfolio of the BBB bond should be 4% per year,
then the demand and price of the, such bonds should decrease, increasing the yield to reflect the
credit spread of the 4%. So, these are the things basically what you can use in terms of the credit
spread.
(Refer Slide Time: 31:37)
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So, over a period of time the probability of default and the spreads will change depending on the
quality of the credit. Usually adverse economic conditions result in greater default probability
and credit spread. For corporate bonds it is like economic factors like recession, industry factors
like declining sales due to the competition, and some specific factors like investment and the
financing decisions.
And for other bonds like municipal bonds, adverse economic developments include the declining
the property values government deficits, unemployment and the use of the debt services or debt
reserves that also will increase or decrease the credit spread.
(Refer Slide Time: 32:29)
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So, event risk also can increase the credit spread if there is a natural disaster or an industrial
accident or there is a regulatory change any kind of new regulation or act has come. Or any
takeover activities merger activities have taken place or is there any corporate restructuring so
those things generally happen. So then also there is a possibility that the event risk can create this
default risk of that particular company and finally the bond can be defaulted.
(Refer Slide Time: 33:07)
So, what basically we discussed here? the credit risk is the uncertainty where the realized return
may deviate from the expected return because of the failure of the issuer to meet their contractual
obligations. Bond investors most rely on the rating agencies while they invest in the bond. And
credit spread of a bond can be thought of as a bond investors expected loss from the principal
from the default.
And default risk also may sense due to some external events like mergers and activities,
regulatory activities, natural calamities or industrial accidents and all these things. So, these are
the factors which can create also the credit risk of this, particular bonds in the market.
(Refer Slide Time: 33:57)
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So, these are the references what you can go through for today's discussion. thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 06
Lecture No # 26
Short – Term Government Bond Markets
Good morning and welcome back so after the discussion on the different characteristics of the
bond and as well as the bond risk we can; now discuss about the different types of the bond
markets which are prevailing in the different economies. Particularly we are talking about from
an emerging perspective like India, and as well as from a developed prospective or developed
economy perspective like U.S. Because the importance of the bond market is already we have
discussed.
But the question here is that, what are the particular type of securities, or particular type of bonds
and what are the trading procedures, and how the particular bonds are issues so all kinds of
things are quite important. Whenever we try to invest in the bond market or we try to minimize
our risk using the bond as one of the instruments in our portfolio.
So today we will be discussing about the short-term bond markets particularly the government
bond market. Then gradually we can move forward to the corporate bonds and the municipal
bonds then the other type of bonds which are also traded in the different type of markets as a part
of the debt market instruments.
(Refer Slide Time: 01:42)
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So today will be covering up two things; one is your treasury bills which is the most important
instrument in terms of the short term government securities, then the another one is the cash
management bill which is also used as one of the bond market instruments in India. So these are
the broad two concepts what we are going to cover in today's class.
(Refer Slide Time: 02:13)
Then you will come across certain keywords; like your strip bills, Treasury bill yield or the other
name of that thing is also we call it the banker’s discount yield which concept is used in the U.S
market. But Treasury bill yield is the most common concept always we come across are always
we use whenever we discuss about the treasury bills market.
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So before we discuss about the characteristics of the Treasury bill and as well as the how the
Treasury bill yield is calculated.
(Refer Slide Time: 02:53)
Let us see what is the importance of the government securities market, why you think that the
government securities market is quite important? If you look at, the actual figures particularly
from the Indian perspective. You will find that whenever we talk about the bond market, the
bond market is largely dominated by the government securities market. The corporate securities
market is not that developed.
Not only in India across the globe if you find that always the government securities market
importance is quite more in terms of the volume or in terms of the uses, in terms of the issuance
all kinds of things in comparison to the corporate bonds. But why that happens, you see first of
all; all of us know that the government security market is a type of market which is a risk free
market. Or we can say that whenever anybody invests in the government securities, then the
return is achieved.
So in that case whatever yield we can get or whatever pricing we try to determine with respect to
the government bond market that can be used as a benchmark for the rest of the market. So the
yield curve whatever we are deriving in the bond market particularly in the government
securities market that can be used as a benchmark yield for the other type of bonds, which are
issued in other segments like the corporate debt market and the other markets.
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So that’s why first of all whenever we price the security we can use this particular kind of yield
what basically we get it from the government securities market as a kind of benchmark. That is
the most important thing what basically you have to keep in the mind that is number one.
Number 2 already we have discussed about the open market operations to control the money
supply.
The central bank on behalf of the government tries to use the government securities or try to sell
or buy the government securities depending upon the requirements, and here if you see that the
most important kind of instrument as a part of the monetary policy we can say that. That is
basically, your buying and selling activities of the government securities because that is in the
control of the monetary authority.
So to increase or decrease the money supply they can take the help of the different instruments
which are available in the government securities market. And already I told you that it is totally
free of default risk or the credit risk; because government is the backup of this particular
security. Then the probability of default from the issuer perspective is completely nil in terms of
the government securities; so that’s why the reliance on this particular security is quite high.
And another thing you also keep in the mind it is a part of the SLR instruments. The short term
government securities are part of the SLR instruments. SLR means the statutory liquidity ratio;
what the commercial banks has to maintain. That means the certain amount of securities
whatever they have, they have to invest in the highly liquid instruments and also to some extent
we can say that the risk-free instruments.
And if you consider the risk-free instruments and in terms of liquidity always give high rating or
high weightage to the securities which are issued by the government. So that’s why the
government securities market is quite relevant quite important from the commercial bank
perspective also because; if any commercial bank wants to invest more in the government
securities they fulfill the regulatory requirement in terms of the SLR requirements. So that is
another importance what basically you can consider as a part of the government securities
market.
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The government security market also facilitates the public borrowings at a reasonable cost and
avoidance of the automatic monetization of the government deficit. Although this particular
automatic monetization of the government deficit has been abolished; since 1997 in India this
particular automatic monetization of government deficit has been abolished. Now if the
government wants to finance the deficit then they have to sell the government securities.
And whenever they are buying or selling the government securities in the market basically they
are dealing with the public borrowings. So they have to raise the money from the public or they
have to inject more money which can go to the public through the different financial
intermediaries, and those financial intermediaries are basically the commercial banks largely.
So that’s why any kind of borrowing activities which is happening in this particular market, or
any kind of activities which is related to the government borrowings that always we can say that
carry through, or carried through this government securities market only. So that’s why it
basically provides flexibility to the monetary authorities for the proper debt management. So the
debt management office of the central bank basically manages this operation, so lot of flexibility
you can get if your government securities market is quite developed.
So that is why a liquid government securities market is very much important from a country's
macro-economic policy perspective. So the monetary policy operations and as well as the public
debt or public borrowings what activities generally happen in the market that always down
through this government securities market; that’s why it is quite important from that perspective.
Then already I have discussed with you it basically plays a crucial role in the monetary policy
transmission mechanism. You know what monetary policy transmission mechanism is. You
know, whenever you talk about the monetary policy. Let me tell you what exactly this
transmission mechanism is, whenever you conduct this monetary policy, the monetary policy has
3 steps.
So first of all certain instruments are used and these instruments will have the impact on some
intermediary target; and finally it will affect the outcome or the target variable. So here what is
the instrument here the instrument is basically we talk about the open market operation in the
context of the government securities market, so buying and selling of the government securities.
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So that will have the impact on the money supply, so money supply is the intermediary target. So
if they want to reduce the money supply they will basically use a different instrument in terms of
the open market operation, or they want to increase the money supply they take the different
position in terms of the open market operation. So the money supply gets affected it will have the
impact on the outcome or the target variable; and the target variable is basically your inflation
and GDP growth.
So whenever the central bank on behalf of the government, selling or buying the government
securities it will have the impact on the money supply. So if the money supply gets affected then
this increase on the decrease of the money supply will have the impact on the inflation and the
GDP. So that is the one of the channels. So there are various monetary policy transmission
mechanism we have an equity price channel, we have exchange rate channel, we can say that
bank rate channel, or maybe the bank credit channel what we call it.
So there are different channels and this is the fundamental concept with respect to the process of
the monetary policy or monetary policy operating procedure. And here the government securities
markets play very significant role in terms of this particular operations. So that’s why the
government securities market we can say plays a very significant role in terms of the monetary
policy effectiveness; or monetary policy operations; or monetary policy transmission mechanism
for any country.
So these are some of the important issues what basically how to keep in the mind to establish the
relevance or the importance of the government securities market for an economy; and already we
have discussed that this market is quite developed in comparison to the other debt markets or the
bond markets which are prevailing in the system.
(Refer Slide Time: 13:56)
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Now let us see what are the different types of short-term securities the government basically
issues? Whenever you talk about the short-term government securities in India from the
emerging market perspective, already I told you that the most important instrument is the
treasury bills. And another instrument is basically cash management bills. If you talk about the
developed economy like U.S also they are also mostly relying upon the treasury bills.
So we can say that Treasury bill is the most important instrument in the short term government
securities market. So what basically we have to see or we have to discuss about the
characteristics of the treasury bills; how the yield of the Treasury bill is calculated. And further
in the other sessions we will discuss about the issuance procedures of the treasury bills; how the
treasury bills are issued, what is the mechanism basically we use whenever we try to issue this
treasury bills in the market.
(Refer Slide Time: 15:26)
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So now you see that what the characteristics of Treasury bill. Already we discussed about the
importance of the government securities market from the both short-term and long-term
perspective. You see already this is a highly liquid money market instrument, why we call it as a
money market instrument? Because, it is very short term in nature, the term to maturity of the
treasury bills is not very long.
So that’s why already all of you know that if you categorize the market in two parts in terms of
the maturity period we have a money market, we have a capital market. And money market
generally deals with the all those short-term securities; and this capital market generally deals
with the long term securities. So the treasury bills are a highly liquid money market instrument.
It has high liquidity, because it is very easy to convert the treasury bills into the cash.
Liquidity means how easily or how fast these particular assets can be converted into cash that
basically measures the degree of liquidity; that’s why the money market is highly liquid number
one. Number two already like other fixed income securities which are traded in the government
securities market it is also there is absence of the risk of default. The credit risk of the treasury
bills is basically zero.
Another importance is ready availability because there is a regular basis the monetary authorities
issues the treasury bills and when they are going to issue the treasury bills there is a calendar
basically they maintain, so there is a ready availability. So at any point of time the treasury bills
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can be available by the commercial banks, for the commercial banks, and other financial
intermediaries who are eligible to borrow the money from the central bank; and finally this come
to the public for the investment.
Assured yield: - the yield is basically very much assured in nature whenever we are investing in
the instrument like treasury bills; and it has low transaction cost. Investment in the treasury bills
in does not incur very high transaction cost that’s why it has a low transaction cost. And in the
beginning I told you it has the eligibility for inclusion in the SLR. Whenever the commercial
banks invest in the treasury bills this can be considered as a part of the SLR instruments.
So that’s why it has the eligibility to be part of or we can say that as a component of the SLR
instruments what the government basically defines. And finally it has negligible capital
depreciation, why basically it has negligible capital depreciation; because the treasury bills
basically are always issued at a discount and redeemed at par.
The treasury bills are issued at discount and redeem at par so because of this you will find that
you will not find much capital depreciation, if you are investing in the treasury bills. So that is
another thing basically you have to keep in the mind. So these are the different characteristics
you have to consider whenever we are investing in the instrument like treasury bills.
(Refer Slide Time: 19:24)
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Then let us see about the treasury bills in India. So whenever we talk about the treasury bills in
India already we know that what exactly the Treasury bill is; Treasury bill is a kind of finance
bill or a promissory note which is put out by the government of the country to meet the needs of
supplementary short-term finance. If there is any kind of deficit by the government, they can take
the help of the treasury bills to financing that deficit.
So on behalf of the government generally the central bank of the country issues these treasury
bills. If you talk about India the reserve bank of India basically issues this treasury bills. And
already all of you have the idea that treasury bills are zero coupon bonds because it does not pay
any interest in between. There is no periodic coupon involved whenever you are investing in the
treasury bills, so that’s why the treasury bills are zero coupon securities and pay no interest.
So it is basically issued at discount and redeems at par at the maturity, so generally the par value
of the security is it those 100 rupees. Let if it is issued at discount that means the issue price
should be less than the par value, so then, let it has been bought at a price of 97 and sold and the
maturity basically it will pay you that 100 rupees.
Broadly there are three types of treasury bills which are traded in Indian market, treasury bills
market we have 91 day treasury bills, 182 days treasury bills, and 364 treasury bills. So these
treasury bills are characterized or classified on the basis of the term to maturity. So these are the
3 types of treasury bills what you will find in the Indian short term government securities market.
(Refer Slide Time: 21:55)
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Then the question here is that, who are those investors. Generally the treasury bills are sold by
the central bank on behalf of the government of India. And who are investing, who are buying it
can be bought by the commercial banks, primary dealers which are basically identified by the
central bank; who are those primary dealers, then you have the state governments, provident
funds, other financial institutions, insurance companies, non-banking financial companies, FII or
foreign institutional investors also are eligible to invest in the treasury bills.
But there are certain kinds of restrictions, certain kind of guidelines for time to time the reserve
bank of India always formulates that they have to follow that. And NRI’s also can invest in the
treasury bills, so these are the different investors who are available who are eligible to invest in
the treasury bills market. And the treasury bills are available for a minimum amount of 25,000 or
a multiple of 25,000 that actually you have to keep in the mind.
And how the treasury bills are sold or it generally these are issued; these are basically sold or
issued through the auction process. We will discuss about the auction process further what
basically the auction process is, what are the different types of auction process, and how the yield
of that particular securities are decided through that; that basically we will discuss further,
So reserve bank of India conducts the weekly auctions for the treasury bills generally on
Wednesday and the settlement or the allocation generally is done on the t + 1 basis. Generally the
auctions are conducted in the Wednesday days and the final allocation journal is made on the t +
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1 basis. So this is about the investors who are basically are allowed to invest in the treasury bills
market in India.
(Refer Slide Time: 24:22)
Then how do Treasury bill yield is calculated, or what we call the Treasury bill rate? Already
what do you mean by this it is basically the rate of interest at which the bills are sold by the
reserve bank of India. So how this particular interest rate or the return from that particular
Treasury bill is calculated? So this is basically nothing but or the effective return on the Treasury
bill what basically we are trying to calculate it is nothing but the, or it is calculated on the basis
of the difference between the price at which they are sold and their redemption value.
At what price the particular security was sold to the investors and what is the redemption value
of that particular security that will basically decide, or that will basically the basis for the
calculation of the Treasury bill yield. So what is the formula here for treasury yield already you
might have seen that is this 100 rupees is the maturity value or the redemption value - p divided
by p into 365 by d into 100.
Yield= 100 – P
P
X
365
D
X
100
100 is multiplied to calculate or to convert it into the percentage, so p is the purchase price, and
your d is the actual number of days to the maturity divided by 365. We are talking about the day
count convention here that is actual number of days to matuity by 365.
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(Refer Slide Time: 26:15)
So if you if you take an example how it is calculate assume that; there is a 91 day treasury bill
which is issued at let 98.20; then what is the yield then your redemption price is 100 – 98.2
divided by 98.2 into your 365 by 91; d = 91 because term to maturity is 91 into 100 that is
7.352%.
Yield= 100 – 98.20 X 365 X 100 = 7.3521%
98.20
91
So in the market right after 41 days if the same treasury, bill is trading at a price of 99 then what
will be the yield?
Yield= 100 – 99 X 365 X 100 = 7.3737%
99
50
Yield will be already 41 days are gone so 50 days are remaining so if you are trying to calculate
the yield it will be 100 - 99 divided by 99 into 365 by 50. 91 - 40 that will be 50 into 100 that has
become 7.373%. So that is the way basically the yield is calculated and the day count convention
is actual by 365 days.
(Refer Slide Time: 27:28)
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Let us see about U.S markets some facts about the treasury bills market in U.S. The same thing
the treasury bills are short-term instruments sold on a pure discount basis in a multiple of 1000
dollar, the 1000 dollar is the par value of the treasury bills in the U.S market. And the interest on
the treasury bill is the difference between the face value and the price paid. And what are the
maturities available in the U.S market that is basically the 13 weeks or 91 days; or 26 weeks is
182 days.
These are the 2 major types of treasury bills which are traded in the U.S market and they are sold
weekly basis, but regularly they are basically sold to the investors in the weekly basis but
regularly. The treasury of the U.S market also sells some special type of t- bills on an irregular
basis that is called the strip bills. The strip bills, means this is a package of the treasury bills with
the different maturities in which the buyer basically agrees to buy the bills at their bills price for
several weeks. So that is a kind of special type of t- bills which are issued in the U.S market that
is called the strip bills.
(Refer Slide Time: 29:02)
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So in U.S whenever we calculate the treasury will yield it is always quoted as an annualized
discount yield and it is also called the banker’s discount yield. But here you if you see that is
basically little bit different is that is F – P0 divided by F into 360 by days to maturity. Here 30 by
360 day count convention is used whenever we are calculating the yield of the treasury bills in
the U.S market right.
ADY = F – P0
360
.
F
Days to Maturity
(Refer Slide Time: 29:39)
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So if you see this example let a 182 days bill sold at auction at 96.67 it will be quoted as a
discount rate of 6.5868%, how you got it that is 100 - 96.67 divide by 100 into 360 by 182 that is
0.065868 that means 6.5868%. So that is the only difference what you can observe between the
treasury bill yield calculation in the context of U.S and in the context of India.
ADY = 100 – 96.67
96.67
360
182
= 0.065868
(Refer Slide Time: 30:12)
So from this you can calculate also the price if you know the yield then the price also can be
calculated from this. Then your P0 is equal to 100 into 1 - 0.065868 into 182 by 360 that means
your days to the your what is the yield you are calculating that 1 - the yield that is r d into days to
the maturity that is 182 divided by 360 that exactly you get it 96.67.
P0 = 100 [1 – (0.65868) (182/360)] = 96.67
(Refer Slide Time: 30:48)
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So then the cash management bill; the cash management bills in India basically it is an
instrument in India this is issued to meet the temporary mismatches in the cash flow of the
government and to absorb the excess liquidity in the system. And the maturity period is basically
less than 91 days; that is 32 days or it can be 56 days or 86 days like that. So the tenure and the
notified amount and the date of issue of the cash management bills generally depend upon the
temporary cash requirement of the government.
When the government needs this they can use this cash management bill. And again it is issued
at discount and redeemed at par. And the settlement of the auction of this cash management bill
is basically on the basis of t + 1. So that is a basically a kind of temporary instrument, what
basically used by the reserve bank of India to match this kind of cash flow of the government,
whenever there is a requirement.
(Refer Slide Time: 32:04)
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So what basically we discussed today the government securities are quite important due to the
unique characteristics like risk of default, case and as well as the important instrument of the
monetary policy and all kinds of thing. And treasury bills and cash management bills are the
major components of the short-term government securities market in India.
For USA the treasury bills are the major constituents of the short term securities market and tbills or t bills yields are quoted on an annualized basis and it is also called the banker’s discount
yield. So these are the things what basically we have or discussed in today's class.
(Refer Slide Time: 32:48)
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So you can go through this, references to know more about or the more details about the treasury
bills market in India and the U.S.
Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 06
Lecture No # 27
Long – Term Government Bond Markets
Welcome back, so in the previous class, we discussed about the characteristics of a short-term
government securities market. So, in today's session will be discussing about the long-term
government bond markets which are operated in India and as well as U.S.
(Refer Slide Time: 00:40)
So, what basically we are going to discuss today which are those instruments, which are
available, particularly the long-term instruments which are available in the government securities
market in India. And which are those long-term government securities which are traded in the
U.S. market. Why we have taken India and U.S. because India is we have considered as a part of
the emerging economy and U.S. we have considered as a developed economy.
So, in this context we have basically, try to examine although there are many other countries
there are many other types of instruments, but we have considered largely to these two
economies for a comparative purpose.
(Refer Slide Time: 01:28)
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So, what are those keywords will you will or will discuss or will go through that, is the fixed rate
bonds, floating net bonds, inflation index bonds, sovereign gold bonds, treasury notes, treasury
bonds. So, these are the different type of instruments which are traded in this particular market,
and that vary from one particular economy to another economy. And there are certain basic
differences in terms of their characteristics whenever this kind of keywords, are used.
(Refer Slide Time: 02:06)
So let us see, that what is basically the situation with reference to the Indian government
securities market specifically the long-term government securities market. Already in the
previous class we have discussed that it is a kind of benchmark market and the yield what
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basically we calculate from this market that can be used as a reference point for yield calculation
in the other markets.
So that’s why it is very much important to understand these characteristics of the other long-term
securities which are traded in this particular segment and it is a dominant market. If you look at
the figures then government dated securities is a dominant market in India, in comparison to the
other long term corporate funds.
Let us see about the existing structure or the organization of the dated government securities
market in India, who are those players obviously, who is issuing this Reserve Bank of India. And
who are the major investors they are basically the commercial banks, primary dealers, other
institutional investors like the insurance companies and other financial institutions also they are
also are allowed to invest in this particular government securities market.
Foreign institutional investors also are allowed to participate in this particular market within a
particular quantitative limit which prescribed from time to time by the Reserve Bank of India.
FII’s that means are allowed to invest in this particular segment but there is some kind of limit
has been fixed for them because there is no such kind of unlimited positions they can take in this
particular segment.
And the corporate sector, also sometimes buy or sell the government securities to manage their
overall portfolio risk, so that is they can also invest in this particular government securities. So,
these are the different players which are always or major players who are basically participating
in this government securities market.
(Refer Slide Time: 04:40)
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So, whenever we characterize or classify this particular market in the different segments, on the
basis of the investors who are participating in this particular segment this market is classified
into different segments. What are those? Broadly this market is classified into 3 segments; one is
wholesale, then middle segment, then you have the retail segment.
And whenever you talk about the wholesale market with reference to the dated government
securities this basically comprises the players like banks, financial institutions, insurance
companies, primary dealers, mutual funds; so, these are the big investors. Their positions in the
market are quite large, so that is why we consider that market is a wholesale market segment.
Then we can see about the middle segment, in the middle segment we basically bring this
companies or the corporates the provident funds, NBFC and the cooperative banks. To some
extent cooperative banks are also allowed to participate in this particular segment, but their
volume of transactions is not at par with the other commercial banks or the primary dealers who
are basically the part of the wholesale segment. So, these are the major investors who comes
under the middle segment.
Then we have the retail segment these are basically consisting of the individuals mostly it is
consisting of the individuals and some non-institutional investors. Who are reasonably quite
small, size is quite small, so these are basically comes under the retail segment. So, this is the
way basically the markets are classified in India wholesale, middle and the retail; and these
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classifications are made on the basis of the investors who are participating in this particular
market.
(Refer Slide Time: 07:13)
Then let us see that how basically these particular investments they make. The holders or the
investors can maintain their securities in the dematerialized form either in 2 ways. So, like other
investment in the stock market and other markets this would have a demat account kind of thing,
or a dematerialized form only there is no physical form which is available everything is done
through in the dematerialized form.
But here whenever we talk about the stock market, we open the demat account with the different
stock brokerage firms, but whenever you talk about the dated government securities these are
basically carried out in the different way. Here what basically the process here the process is
basically the investors should have a SGL account that is called the subsidiary general ledger
account.
This would have an SGL account and that account will be opened with the Reserve Bank of
India. The Reserve Bank of India basically offers that account and that account only can be
opened by certain specific organizations, specific entities. That’s why this facility is available to
some select entities who can maintain their securities in the SGL accounts, maintained with this
public debt, offices of the reserve bank of India.
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Mostly the banks or the primary dealers they are basically eligible to open this SGL account and
the trading of the government securities or the transactions of the government securities are done
through this SGL account. Then there are some other investors, also they are allowed to invest
but they cannot directly invest with the Reserve Bank of India through this SGL account for
them they have a Gilt account.
Because the eligibility to open and maintain this SGL account with RBI is highly restricted, in
that case the investor has the option to open a Gilt account. And where they can open the Gilt
account, they can open this Gilt account with a bank or a primary dealer; who is eligible to open
this SGL account with RBI.
So, whenever you are investing in the government rated securities and you are not a bank or you
are not a primary dealer, so in that particular case you have to rely upon the particular entity who
can open this SGL account or who are eligible to the SGL account. And you will open a Gilt
account with them and on behalf of you that particular entity will help you to invest in this
particular dated government securities.
So that is the practical way the investment in the government securities are done in India. So here
the important thing is Gilt account and the SGL account. Through SGL account the commercial
banks and primary dealers do the transactions and the through the Gilt account the other
individuals or other kind of investors who are interested to invest in this market they can always
take the help of this. So, this is the way basically the investors can participate in this particular
segment.
(Refer Slide Time: 11:12)
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Then let us come to the different types of long-term securities which are traded in India. There
are different types of long-term securities are traded in India. You have a bond where the coupon
is fixed that is called fixed rate, you have floating rate bonds, you have zero coupon bonds, your
capital index bonds, your inflation index bonds, your bonds with embedded options.
You have a kind of special kind of instrument that is STRIPS that is separate trading of
registered interest and principal of securities, you have sovereign gold bonds, which is now
prevailed we have savings bond, we have state deployment loans. So, there are different type of
instruments let us see one by one what exactly these instruments are how these instruments are
defined.
(Refer Slide Time: 12:03)
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What do you mean by fixed rate bond, the fixed rate bond basically pays a fixed coupon over the
entire life of this particular bond. And the period of the coupon payments is semi-annually
generally, the semi-annual coupons will be paid at any point of time the coupon rate is not going
to be changed and the coupon will be paid up to the maturity. So, these are called the fixed rate
bonds, these are coupon barrier, all the data securities are coupon bearing bonds and the coupons
are fixed over the life of the bond.
Then we have a floating rate bond here whenever by name itself you can understand what do you
mean by the floating rate bond. The floating rate bonds basically pay the interest at a variable
coupon rate that is reset at a pre announced intervals. When this new coupon rate will come that
basically pre-announced periods are mentioned before and these particular dates are announced
before and accordingly the new coupon rate can be reset.
And how the coupon rates are decided, the coupon rates basically are decided in India if you look
at it can be as average of the implicit cut off rate of the last 182 days T-bill auctions. Every 6
months if the coupon is going to be changed then that thing is decided from 6-month treasury
bills 180 days means it is more or less a 6-month treasury bill. So, the last 3 years average of the
182 days treasury bill yield implicit cut off rate they consider to decide the coupon of that
particular period of this floating rate bond.
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Or it can be also equivalent to the weighted average yield of the last 182 days treasury bills + a
fixed spread because it is long term in nature, is relatively riskier than the short-term instruments
like treasury bills. So, because of that they can add some premium into that some spread into that
that is the addition of the weighted average yield of last 382 days T-bill auctions and accordingly
the coupon rate of that particular floating rate bond can be decided.
Or it can be also reset every 5 years or the prevailing 5 years government security yield as on last
working day prior to the commencement of each period of the 5 years. So, the 5 years bonds
which are traded in the market that yield also can be used to decide the coupons of that particular
floating rate bonds. So, any of the methods can be used to fix the coupon rate in the periodical
basis by the Reserve Bank of India for this dated government securities. So, this is about your
floating rate bonds.
(Refer Slide Time: 15:49)
Then we discussed about zero coupon bond that already you know that and let me tell you also in
India, the floating rate bonds are also not quite popular. People are more inclined to invest in the
fixed rate bonds. And the zero-coupon bonds where there is no coupon and it is issued at a
discount redeemed at par that is also not favored in the market. You will not find much zerocoupon bonds, long-term zero-coupon bonds which are issued by the government in the Indian
market, because there is no such demand.
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People are not that inclined to invest in that type of security, so because of that this issuer also is
not very much inclined to issue that kind of security, that so that is also not favored. India also
has started a bond called the capital indexed bond, so here what will happen the principal amount
is linked to the inflation index, the principal amount is only will be linked to the inflation index if
the inflation will be changed your principal amount also will be changed. So that is called the
capital index point that is also not quite popular in the Indian market.
And we have the inflation index bonds; here what is happening what is the difference between
the capital index bond and the inflation index bonds. In the case of inflation index bonds both
principle and coupon flows are protected against the inflation, both coupon and the principal
what is fixed for this particular bond they are protected against the inflation.
Reserve bank of India also issues some of the bonds which have some call and put features, all of
you know that there is a callable bond there is a puttable bond. If the bond has a call feature that
after a certain point of time the bond the issuer can call back that bond, that buy back that
particular bond from the investor right. So, there is a risk involved in that that is called the call
risk.
And like that the bond can have a put feature where, it will give the right to the bond investor to
sell that bond back to the issuer so there is a put risk also involved in that particular context. So
how the call risk and put risk is going to affect the total return of the bond or how they return
total return what you are expecting that is going to be deviated from the actual return what you
are getting that will be discussing further.
But there are certain bonds which are issued by the reserve bank of India, which can have the
call feature or the put feature. Then with the STRIPS, STRIPS mean it is a separate zero-coupon
bond created by breaking down the cash flows of a regular coupon paying government security.
So that means one regular coupon bearing government security can be converted into many zerocoupon bonds, so that is what basically we call it the STRIPS.
(Refer Slide Time: 19:36)
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Then we have some other bonds also recent some 2-3 years back we have in India, we have
started the sovereign gold bonds, here what happens here it is basically linked with the gold
price. These bonds generally are issued in different tranches and denominated in units of one
gram gold and multiple thereof; it pays a fixed coupon per annum and redeemed at a simple
average of closing price published by the India Bullion and Jewelers Association Limited of the
gold of the previous 3 business days from the date of repayment.
That means if you have invested in that particular type of bond periodically you will get a
coupon. But your value of money what you will be getting in the end that will be decided on the
basis of the market rate of the gold. So, there will be a capital appreciation if the gold price has
increased, there will be a depreciation if the gold price has declined. So that is what basically
what you can, that’s why the sovereign gold bonds are linked to the price of the gold which is
prevailed in the market.
So, then we have a savings bond it is generally specifically issued for the retail investors at par
for a minimum of 1000 face value and bears a floating rate of interest which is reset every 6
months basis. And generally, these are taxable whatever interest rate you get from this that is
taxable or sometimes also some bonds which are traded in the market these are non-taxable that
depends upon the government's policy.
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That what are the whether these are the taxable bonds or the non-taxable ones like if you look at
in U.S. market, we will discuss about more on that the municipal bonds are not taxable. Then we
have a state development loans; so here by name itself if you can charge the market borrowings
of the state government through the semi-annual coupon paying long-term bonds that is called
the state government bonds.
The state government also can go to the market for raising the money, so in that case the state
governments basically go for the market borrowing through the semi-annual coupon paying the
long-term bonds, so these are issued by the state government. So, these are the broadly different
type of bonds which are available in the long-term government securities market in India.
(Refer Slide Time: 22:43)
Let us see that what are those instruments which are available in U.S. U.S. also more or less the,
if you see the characteristics more or less, they are same not much differences. You will find
some names which are different from the bonds which are traded in the Indian market. In U.S we
call the treasury notes, treasury bill's name is same in both the cases which are short term but in
the long-term case we can call the treasury notes, treasury bonds, or treasury inflation-index
bonds and notes like that.
So only the nomenclature wise there is some differences but the characteristic size wise you will
not find much differences, in terms of the bonds which are traded in the Indian market and the
U.S market.
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(Refer Slide Time: 23:33)
So, treasury notes and bonds these are the coupon bearing bonds, these are the coupon bearing
bonds and the t-notes the treasury notes have the original maturity up to 10 years. But if you talk
about the treasury bonds, they have the maturities which can range between 10 to 30 years. So,
the maturity period of the treasury bonds is larger than the maturity period of the treasury notes.
In India also the term to maturity of the long-term debt securities can go up to 30 years.
So, both treasury notes and bonds they are sold in denominations of the 1000 dollar or more and
they both pay the semi-annual coupon interest. The coupons are paid basically semi-annually,
India also you might have observed the coupons are basically paid semi-annually. And the
whenever the treasury bills are issued the price or yield of that particular treasury bill is decided
through the auction process right.
So, when this particular treasury notes or bonds are auctioned, they are always quoted on the
yield basis, there is another one that is called the price basis we will discuss on that. But
generally, in U.S when the treasury notes and bonds are issued, they are basically always quoted
on the basis of the yield. The coupon rate is not set until the auction is completed, so the coupon
rate of that particular bond will be decided once the auction is completed.
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They will find that they will decide that what will be the yield of that particular bond accordingly
the pricing and the other type of activities will take place that basically you have to keep in the
mind.
(Refer Slide Time: 25:49)
Then we have treasury inflation index bonds what we call the TIPS, what do you mean by this
treasury inflation index bonds. So, each period coupon payment is equal to a specifics fixed rate
times on inflation adjusted principle, so at the maturity the bond basically pays the larger of the
inflation adjusted principle or the original par value.
Let us take this example suppose the central bank or the treasury issues a 3-year tips the treasury
inflation index bonds with a nominal principal of 1000 dollar and a semi-annual coupon rate of
4%. If there is no inflation in the ensuring 3 years then the tip will pay the bond holders 40
rupees semi-annually for 3 years right and 1000 rupees or 1000 dollar at the maturity.
If there is inflation which can be measured through the consumer price index what the U.S is
using, in India also we are our inflation is measured through the consumer price index or all the
calculations basically are based on the consumer price index. So, if there is an inflation which is
measured through the consumer price index, then the treasury will adjust this nominal principle
of the 1000 whatever you have paid.
424
Suppose the U.S experience an annual inflation rate of 3% or 1.5% semi-annual rate. For the first
semi-annual period of the particular bond, the inflation adjusted principle would be 1015, 1.5%
is the inflation. So, then the inflation adjusted principle will be 1000 you have to adjust it with
respect to the 1.5% inflation then 15 dollars will be added into that, then that is 1015 dollar.
And how much coupon the bond holder will receive, the bond holder will receive basically your
1015 into 0.04 instead of 1000 into 0.04, they will be getting the coupon on the basis of the
inflation adjusted principle that is 1015 into 0.04 that is 40.60 is it clear. So now what is
happening if there is any inflation which is going on; on the basis of the CPI then your coupon
payments also get affected by that, so that is called the treasury inflation index bonds, which is
also issued by the government in the U.S market.
(Refer Slide Time: 29:16)
So, we have a treasury strip in U.S also like in India we have; the treasury strip is formed by a
dealer who purchases a treasury bond or a treasury note, and then creates a 2 general type of
zero-coupon securities to sell the investors. There are 2 types of securities can be created one is
principle only security what we call it the PO (is also called the corpus, and denoted np when it is
quoted) in the market. And the PO security is a zero-discount bond that pays the treasury bonds
principle at its maturity.
And also, you can have an interest only securities IO (which is denoted, as I when quoted) in the
market these securities are again the zero-discount instrument with each paying a principal equal
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to the treasury bonds coupon and with a maturity coinciding with the bond’s coupon date, so that
is called the interest only securities. So that is why they have stripped that particular data security
like treasury bonds and treasury notes into different type of zero-coupon bonds and accordingly
these particular bonds are traded in the market.
(Refer Slide Time: 30:41)
So, what we have seen here the major players in the government securities market in India
include Reserve Bank of India, commercial banks, primary dealers and all. The investors in the
market can be classified into different segments like wholesale segment, middle segment, retail
segment. And the segments are decided on the basis of the investor’s characteristics. And the
different type of long-term government securities which are traded in the market these are
classified on the basis of coupon payments, maturity and as well as the other external conditions
like inflation and all.
(Refer Slide Time: 31:24)
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So, these are the references what you can go through for the more detailed discussion on this.
Thank you.
427
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 06
Lecture No # 28
Issuance and Trading Mechanism of Government Securities
Welcome back.
So, in the previous class we discussed about the different long-term securities which are traded in
the government securities market. And today we will be discussing about the issuance and
trading mechanism of the government securities. How the government securities are issued and
how the trading of government securities, are taking place in the market. Or what is the trading
mechanism of the government is equal to in the market that basically will be discussing today.
(Refer Slide Time: 00:48)
So, the major concepts what will be covering in today's session that is basically your yield-based
auction, price-based auction, competitive bidding, noncompetitive bidding, as well as some other
concepts also we will be discussing mostly this particular session will deal with the how the
particular government securities are issued and how they are traded in the secondary market.
What are the different ways the trading takes place and how the in the primary market the
securities are issued by the issuer.
(Refer Slide Time: 01:34)
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You see that there is different type of keywords again you will find that uniform price auction,
multiple price auction, negotiable dealing system, or negotiated dealing system, over the counter
market, stop out price so these are the different keywords that you will come across while
discussing these particular sessions. In this particular session these are the words we will come
across throughout the discussion whatever we are going to do today.
(Refer Slide Time: 02:12)
You see that already you know that like other markets, this market also has two segments one is
your primary segment another one is the secondary segment. Or there is a primary market for
this or there is a secondary market for this like other markets. Any market in the financial system
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whenever we see we have a primary market of that particular security and we have a secondary
market also for that security.
What do you mean by the primary market because, the primary market is basically nothing but
one of first time the securities are coming to the market there is a process they have to follow and
the securities should be listed in the different exchanges. Then finally and the buying and selling
mechanism will take place by the traders which are rebelled in the market. And accordingly, the
demand and supply forces of that particular security will decide the price, and because of this in
the secondary market the price fluctuation will take place, that already you have the idea.
So, whenever you talk about the government securities, if you look at the issuance mechanism of
the government securities, these are basically issued through the auction process. In the previous
session also, we have used this concept of auction the, these are issued through the auction
process. And what is auction? Auction is basically, a process of calling the bids with an objective
of arriving at a market price.
So, whenever we go for auction, we try to invite the bids and people will go for the bidding that
means they are going to demand for that particular security, and on the basis of the bids the
equilibrium price of that particular security can be determined. So, this is basically a process
which invites the bids, and the objective is to finally arrive at an equilibrium market price.
For the dated securities, or the long-term securities the Reserve Bank of India basically issues the
auction calendar in March and September for the next 6 months, with the consultation of the
Government of India. Every 6 months basis the calendar will be prepared, tentative calendar you
can keep in the mind; the tentative calendar will be prepared with the consultation of the
Government of India and accordingly the other investor will be ready for that.
For T-bills which are the short-term government security the RBI issues the calendar at quarter
end for the next quarter. One the quarter will be ended for the next quarter, what is the auction
calendar that basically they will disclose to the market by that the investors can invest in that
type of security. The cash management bills auctions are announced directly on the RBI website
without a calendar, why because the cash management builds auctions are not held regularly
because they depend upon the temporary cash requirements of the government.
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If there is a requirement of the government then this kind of bills will be issued or this kind of
cash management bills will be issued. There is no requirement there is no such kind of issuance
will take place, but the treasury bills and the government dated securities these are the regular
affairs; so, there is an auction calendar for this and on the basis of that auction calendar generally
they will try to issue those particular securities in the market.
So, the auctions are conducted by RBI on the t+ 1 settlement basis that means if the today
auction is over, then your credit and other settlements will be done by tomorrow; there is a t + 1
settlement basis on the there is an electronic platform, they have that is called the E-Kuber
electronic platform. So, the auctions are conducted by RBI on the t + 1 settlement basis on the ekuber electronic platform whatever they are using for the online transactions. So that is the way
basically the issuance mechanism of the government securities is started in the primary market.
(Refer Slide Time: 07:21)
So then if you see the auctions for the dated securities are normally conducted on Fridays, with
the underwriting options for the primary dealers on the preceding day; and with the settlement on
Monday because Saturday, Sunday is a holiday. Weekly auctions are held for 91 days, 182 days
and 364 days T-bills on Wednesdays that already I told you. And the state development loans
auctions are generally held on the Tuesdays, so different kind of security there are different dates
what the central bank has fixed.
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And accordingly, the auctions are basically held in that particular calendar. Recently also, not
recently some years back the Reserve Bank of India has started another segment in that particular
government securities part that is called the when issued market. If I correctly recall that, is I
think in 2005/ 06 this concept of when issued market has come.
So, the when issued market generally why it is set up, it is a part of the market development
where the traders can resell or redistribute their stocks which are likely to get allocated in the
upcoming auction. It is basically, done for better pricing, in this particular segment. So, the when
issued market also has to some extent helped for the development of the government securities
markets.
(Refer Slide Time: 9:23)
Here what we have discussed we have discussed that the auctions basically are carried out and
auctions are held to issue this government securities, whether it is short term or it is long term.
So, in true sense there are 2 types of auctions one is yield-based auction other one is the pricebased auction. The yield-based auction is generally conducted when a new government security
issued. So, the investor will go for bidding up to the 2 decimal points like 7.49%, 8.21% etc.;
how much yield they expect from this particular kind of security.
Then what this RBI or the issuer basically done, they arrange these bids in the ascending order
and accordingly the cutoff yield is arrived; and what is the cutoff yield? It is the yield
corresponding to the notified amount of that particular auction; and this cutoff yield is considered
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as a coupon for that particular security; it is considered as a coupon rate for that particular
security. Who are those successful bidders, successful bidders are those who basically have gone
for bidding at or below this cutoff yield.
Because the bids are arranged in the ascending order whoever has asked for the lower yield or
the yields which were decided which is corresponding to the notified amount, there will be the
successful bidders; and the bidders which are higher than the cutoff yield are rejected.
(Refer Slide Time: 11:41)
Let us take one example how basically it works. Let there is a government dated security right;
the government dated security the maturity date let us mentioned here sixth august 2024. The
coupon is it is determined on the basis of the cutoff yield auction date is late fifth august 2014,
10 years bond, the term to maturity is 10 years. Now auction settlement date is sixth august 2014
provided it should not be a holiday, notified amount is let 1,000 crores clear.
Then what has happened whenever this particular advertisement or this particular notification has
come then the investors like commercial banks, primary dealers and other people will go for
bidding. Somebody will go for bidding at let 8.5%, somebody will go for 8.25%, somebody, will
go for 8.2% like that, so here the bids are already arranged in the ascending order. Let we have
received 8 bids and the bids are varying the yield is varying between 8.19% to the 8.24%.
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Let the bank who was gone for bidding at a rate of 8.19% and they are asking that bank is asking
for 300 crores right, another bank who has quoted the yield 8.2 % he is asking 200 crores, third
bank is also asking for the 8.2% yield again asking for 250 crores like that. Then the 8.21% they
are asking 150 crores then 8.22% they are asking 100 crores then 8.3% that is 150 crores then
8.4% that is 100 crores.
So, if you see the fourth column it gives you the cumulative amount that is 300, 500, 300 + 200,
500, 500 + 250, 750, 750 +150, 900, 900 + 100, 1000 and then 1100, 1250, and 1350, because
government has notified for the 1000 crore. So now the demand is if you look at the cumulative
amount of bidding that is 300, 500, 700, 900, 1000 then 1100, 1250, 1350 but notified amount is
1000 so government can go up to 1000 crore.
So, when the 1000 crore is over up to the fifth bidder that is 8.22% that 8.22 % that 1000 crores
is adjusted, so that is why the cutoff yield has been decided as 8.22%. But now if you see that the
sixth bidder is also asking the fifth bidder is 8.22% asking 100 crores there it is 1000 crores is
exhausted, this is the notified amount now this is exhausted.
Sixth bidder the cutoff the yield is same with the fifth bidder, and they are also asking 100
crores, but now the amount is not available. So, in that case what will happen; in that case if you
see below since the bid number 6 also is at the same yield, then both the bid numbers 5 and 6
would get the allotment on the pro-rata basis, so that the notified amount is not exceeded. So
now the fifth one is getting 100 or asking for 100, getting 100; sixth one is also asking for 100.
So now what this RBI will do RBI will allow 50 crores to fifth bank and 50 crores to the sixth
bank, because they are asking same. Let 100 crores is remaining let this fifth bank is asking 200
crores and the sixth bank is asking 100 crores, so now accordingly they will divide the total 100
crores will be allotted, now the two third will go to the fifth bank, and one third will go to the
sixth bank.
So, like that the allocation will be made and the bid numbers 7 and 8 at rejected because the
cutoff yield is 8.22% so this pro-rata allocation will be done, whenever the cutoff yield is same,
or the bidding whatever they have done that at the same rate, so now the 8.23%, 8.24% are
rejected.
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(Refer Slide Time: 18:00)
Then the other type of auction is the price-based auction, so what is the price-based auction that
auction generally is conducted when the government of India reissues the securities which are
issued earlier right. So here the bidders basically will code the terms of the price for 100 of the
face value, and they will code the price they will bid in terms of the price like 100 to 101, 100 99 like that.
But in the case of yield we are arranging in the ascending order but in case of price we will
arrange it in the descending order. And the successful bidders are those who have a bid at or
above the cutoff price and bids which are below the cutoff price are rejected, in the same way.
You can see this example.
(Refer Slide Time: 19:00)
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Let the maturity date is 22 august 2014 the same example we have taken here auction date is 5th
august, notified amount is 1000 crores then the banks or the other primary dealers have gone for
the bidding. Let one whenever, already it is arranged in the descending order 100.31, .26, .25,
.21, .20, .20, .16, .15 and these are the amount of the bidding whatever they have asked for. So
here itself if you see that 1000 crore is get exhausted here, up to the fifth bidder.
The sixth bidder also has quoted the same price they are also asking 100 crores like the previous
case we have to give 50 crores to fifth bidder, and 50 crores to the sixth bidder. That is on the
basis of the pro-rata basis we have to allot, is it clear. So that is the way whenever the price is
more your yield will be less the implicit yield will be less. So, the price held relationship already
you know accordingly the yield and yield can be calculated from the price or the price can be
calculated from the yield and yield can be calculated from the price. So, depending upon the
quotes whatever the Reserve Bank of India asks for we can decide either of these two variables.
(Refer Slide Time: 20:46)
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But here one thing you have observed that for everybody we can give a yield of 8.22% or a price
of 100.20 so that’s why the auction can be uniform price-based auction or it can be also multiple
price-based auction. What does it mean that means whenever the cutoff yield or cutoff price is
calculated everybody will be allotted the fund on the basis of that yield or that price. Otherwise,
whatever quotes you have made on that basis your yield and price will be imposed on them.
So, like in case of uniform price-based auction all successful bidders are required to pay for the
allotted quantity of securities at the same rate; that is the cutoff rate what we have decided that is
8.22%. Irrespective of the rate quoted by them somebody has quoted 8.19, somebody has quoted
8.20, somebody has quoted 8.21, but everybody will get 8.22%. But if it is a multiple price-based
auction then what will happen, the successful bidders are required to pay for the allotted quantity
of securities at the respective price or the yield at which they have bid.
If somebody has quoted 8.22, you will be paying 8.22, he will be paid 8.22 somebody is quoting
8.19 then he will be paid 8.19 like that. The choice of multiple versus the uniform price-based
auction is generally made by RBI and that will basically be decided on the basis of the market
conditions.
They will try to examine the market conditions or expected conditions in the market and they
will decide that whether in the auction; they will go for a multiple price auction or they will go
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for uniform price auction. So that is another concept with respect to the auction mechanism that
actually you have to keep in the mind.
(Refer Slide Time: 23:18)
The auctions whatever we have discussed that auction process is a competitive bidding process
right, but we have also a non-competitive bidding. So here everybody is bidding somebody is
winning somebody is losing, somebody is successful somebody is not successful. But what
happens from the monetary authority perspective all the investors are not very much informative.
So, because of that what basically they have planned they have started a concept of the noncompetitive biddings. So, in case of competitive biddings who are basically should go for
competitive bidding, this is basically made by the highly informed investors or sophisticated
investors like banks, financial institutions, primary dealers, insurance companies, mutual funds
and so on. And the minimum bid amount is 10,000 and the multiple of 10,000.
Thereafter multiple bidding also is allowed in the case of competitive bidding, the multiple
biddings is also allowed the investor may put in several bids at various prices or the yield levels.
One bidder can also quote for the multiple prices whichever is the cutoff yield on that basis that
will be decided.
But for whom the non-competitive bidding is basically designed, the non-competitive bidding
generally is open to the individuals, HUF means Hindu Undivided Families, the regional rural
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banks, cooperative banks, companies, corporate provident funds or the trust. So, for them some
percentages of the funds are allotted, so that is basically down through complete non-competitive
bidding process.
So, under this scheme the eligible investors basically apply for a certain amount of securities in
an auction without mentioning a specific price or yield, so they will not quote any price. So, such
bidders are the non-competitive bidders are allotted the securities at the weighted average price
yield of the auction. So whatever cutoff yield or whatever average yield will observe in the
competitive bidding process that price basically will be applicable for them.
So, for them they will not participate that they will not quote any particular price or the yield
whenever they will ask for these particular securities, so that is called the non-competitive
building, so this is the auction mechanism.
(Refer Slide Time: 26:29)
So, whenever it is a part of the primary market then whenever it comes to the secondary market,
the secondary market trading basically takes place through a system called the negotiated dealing
system order matching NDS-OM platform. So, this is basically an anonymous screen based
electronic order matching module, the participants can trade anonymously by placing their orders
on the system or by accepting the orders already placed by the other participants.
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And this is operated by the Clear Corp Dealing System Limited which is a full owned subsidiary
of CCIL that Clear Corporation of India Limited which is established by RBI. So that particular
organization facilitates this negotiated dealing system order matching platform through which
the secondary market transactions take place. It can be also done through over the counter market
transactions basically undertaken between the market participants in the OTC or the telephone
markets.
And whatever transactions are done they have to be reported in the NDS platform negotiated link
system platform within 15 minutes after the deal is put through over the telephone. Recently they
have started a concert called NDS-OM-WEB in that platform this was launched to enable the
guilt account holders to control their orders and get access to real time live quotes which are
available on NDS-OM.
And also, we have proper stock exchanges through which the secondary market transactions
takes place. And these are introduced for the retail investors and they can participate in the
secondary government securities market through these stock exchanges like NSE and BSE. They
have a proper marketing platform through which the transactions can be carried out.
(Refer Slide Time: 28:39)
If you look at U.S more or less the process is same these are sold through sealed-bid auction
process, in which the treasury announces an issue and dealers and investors submit either a
competitive or a non-competitive bid. When with a competitive bid investor specifies the yield
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like annualized discounted yield for T -bills and annualized YTM for the treasury coupon issues,
and the quantity he or she wants.
With a non-competitive bid the investor specifies only the amount he or she wants and accept the
weighted average value is same. Whatever thing we follow the same thing is followed in U.S.
The bidders must file the tender forms with one of the Federal Reserve banks or the branches or
with the treasury bureau of the public debt that is their regulatory norm.
(Refer Slide Time: 29:35)
Each dealer submits either a competitive bid or non-competitive bids; bids are ranked from
lowest yield to the highest yield; or the highest price to the lowest price like India, that exhaust
the issue which includes both competitive and non-competitive bids. The last price is known as
this stop out price, what we call it here the cutoff yield, that here we call it the stop-out price.
And non-competitive bids basically get the average price. This is basically the treasury auction
process which prevails in the U.S market in the primary segment.
(Refer Slide Time: 30:15)
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In the secondary market mostly the treasury securities are part of the over-the-counter market,
then handled by the treasury security dealers. There is a 24 hours market and the major dealers
are in New York, London and Tokyo. The OTC market generally consists of both investors who
prefer to buy from dealers, instead of through the auctions, investors buying and selling under
outstanding securities, and the federal reserve which buy the T - bills and other treasury
securities as a part of the open market operations. So, all are the participants of the OTC market.
In addition to the primary and secondary market for the treasury securities there is also an
interdealer market in U.S., in which the primary and non-primary dealers trade billions of dollars
each day among themselves. The interdealer market functions through the government security
brokers that for a commission match dealers and other investors who wants to sell with those
who are interested to buy.
So that is a, they have a different segment they have that is called the inter-dealer market within
the dealers the transactions basically take place among the transactions take place. So, this is the
secondary market operation with respect to the U.S. market, this is the way these treasury
securities or government securities are traded in the secondary market in U.S.
(Refer Slide Time: 31:58)
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So, what we have discussed all government securities are traded both in primary and secondary
markets. So, in the primary market all these are issued through the auction process, the auction
can be yield-based auction or a price-based auction. And it can be also the multiple prices or the
uniform price-based auctions, and also the bidding can be competitive and the non-competitive.
And the secondary markets all the trades are all are traded so the over-the-counter market and the
stock exchanges, and the secondary market transactions in U.S. are handled by the treasury
security dealers. And there are many investors or many participants are involved in this
particular market from the different perspective both from the buying and the selling
perspectives. So, this is about the trading mechanism or the issuance mechanism of the
government securities in India and as well as USA.
(Refer Slide Time: 33:04)
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So, these are the references you can go through for today's session.
Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 06
Lecture No # 29
Short –Term Corporate Bond Markets
Good morning, so in previous class we discussed about the government securities market
particularly the how these securities are issued, what are those different types of instruments
which are available and we discussed about the government securities market with respect to
India as well as U.S. to make a comparative analysis we have considered one market from the
emerging economy and another market from the developed economy. Today’s discussion will be
based on corporate bond market. So, we will discuss about short-term bond market, then we can
move towards the long-term bond market.
(Refer Slide Time: 01:00)
So, the concepts, what we are going to cover in today's session that will be who are those players
who are playing the significant role in the corporate bond market. And particularly as we know
that the most important instrument which is traded in the corporate bond market that is
commercial paper. We will be concentrating more on that commercial paper, because the
commercial papers are traded both in the developed economy and as well as the emerging
economy like India.
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So, like our government securities market discussion will be again discussing about the
commercial paper market in India and the commercial paper market in U.S.
(Refer Slide Time: 01:50)
So, you will come across certain keywords what basically over the sessions will be using that is
debenture trustees, qualified institutional buyers, directly placed commercial paper, dealer placed
commercial paper, rolling over paper, rollover risk and all kinds of things. So, these are the some
of the keywords basically will get to know that whenever we are going to discuss about the
short-term securities market with respect to an emerging economy and as well as the developed
economy.
(Refer Slide Time: 02:29)
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So, whenever you talk about the corporate bond market you see that like your government
securities market, we have certain players who are playing basically the significant role in this
particular segment and who are those players. Obviously from the investment point of view we
have the issuers and we have the investors. So, who are those issuers and who are those
investors. You see if you look at the major players in the corporate bond market in India who
basically issues these commercial papers mostly the companies are basically issuing these
commercial papers right. And the non-banking financial companies they are also issuing these
commercial papers. Banks are not that kind of role they play in terms of these but if you see that
mostly the commercial papers issued by the companies.
Then we have the debenture trustees there is another player who basically play the role in that
case, these are basically secures any issue of debentures by corporate entities for the benefit of
the debenture investors. As you know that the debentures are not secured instruments, so in that
case there are certain development debenture trustees, who are basically approved by or the
registered with securities exchange board of India (SEBI) they basically play in this particular
context.
So, another player we have the qualified institutional buyers and whenever you talk about the
qualified institutional buyers, they are basically the institutional and high net worth individuals.
Who are basically allowed to private placement market that is a part of this corporate bond
market or we can say part of the primary market of the corporate bonds. You see these players
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whatever we are talking about these players basically related to the aggregate bond market it
includes the short-term and long-term bond.
The companies are issuing the commercial papers, debenture and all and the banks are issuing
the certificate of deposits and all these things, so that also you should keep in the mind. So,
whenever you talk about the qualified institutional buyers, who are those qualified institutional
buyers? The qualified institutional buyers are like commercial banks, insurance companies,
mutual funds, state industrial development corporation, pension funds and provident funds.
But the whenever you talk about the pension funds or provident funds, they should have a
minimum corpus of rupees 25 crores and the foreign institutional investors are the portfolio
investors who have registered with securities and exchange board of India that is SEBI. And
also, some venture capital funds which are also registered with SEBI, largely these are basically
considered as a qualified institutional buyer in the bond market or in the financial market.
Apart from these players these are the major players, apart from these players we have some
retail individual investors or we can say that retail investors, the small investors. The small
investors also can invest in the corporate bond market and they can apply for the investment of
the worth 2 lakhs or less. There is a certain limit for that but the small investors also are eligible
to invest in this particular market, but their total investment amount is limited.
And already what I told you that whenever you talk about the short-term corporate bond market
largely the commercial paper is the major short-term debt instrument, which are traded in the
corporate bond market in India and as well as also in the U.S market. So, the commercial papers
are the most used instrument in this particular segment.
So largely our discussion will be based upon the different aspects like the features of commercial
papers and what is the market for the commercial papers, how the commercial papers are traded
in the market, who can issue the commercial papers; all those things basically you will get to
know in this particular session.
(Refer Slide Time: 07:22)
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So, whenever you talk about the commercial papers, what exactly the commercial paper is?
Commercial paper is also an unsecured instrument, it is an unsecured instrument it is not backed
by any kind of assets. So, if you formally define this commercial paper, the commercial paper
basically is an unsecured money market instrument why we call it the money market instrument
because it is a short-term instrument, which is issued in the form of a promissory note by a
corporation with high credit ratings to finance its short-term needs. Whenever companies need
the short-term financing or they required short-term money for their operational activities they
can take the help of the commercial papers to raise the money. So, they can issue the commercial
papers to raise the money to fulfill their short-term requirements.
So, these instruments are issued in a wide range of denominations, it can be either discounted or
interest bearing. But if you look at the aggregate market in India you will not find a very
developed secondary market for this mostly the commercial papers are available in the primary
market. So, whenever the company needs a short-term financing, they can go and issue these
commercial papers and accordingly their financing requirements can be fulfilled.
So that actually you have to keep in the mind, we do not have a very mature or developed market
in terms of a secondary market with respect to these commercial papers. So, this is the way the
commercial paper is defined.
(Refer Slide Time: 09:22)
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Then we will see that what are those characteristics of the commercial paper or we can say that
the features of the commercial paper. Already we have discussed that the commercial papers can
be issued on a discount to the face value basis or it can also issue on a fixed interest basis. So,
both the options are available with respect to the commercial paper. So, these are basically
unsecured, negotiable and normally have a buyback facility if the issuer wants, they can buyback
those particular commercial papers from the investors.
And this is a short-term debt financing, this is a major source of short-term debt for the
companies and it is regarded as a highly safe, simple, flexible and quality liquid instrument.
Because it is the term to maturity of this instruments are quite small or it is a completely shortterm instrument then the liquidity of this particular instruments is also relatively higher. So, these
are also considered as a safe instrument and highly liquid instrument, that is what basically the
basic feature of the commercial papers.
(Refer Slide Time: 11:00)
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Then if you see that or if you look at the Indian market, if you little bit go towards the history or
historical development of the emergence of these commercial papers market in India. It was
introduced in 1990 with an objective of increasing the short-term financing for the corporates or
to fulfill the short-term requirements of the corporates. So that’s why it is a who can issue this
whenever we take the case that it is a safe instrument it should be issued by the highly rated
corporate borrowers and the basic objective is to diversify the sources of short-term borrowings.
And also, it will provide an additional instrument to the investors. The investors having the
short-term horizon period they can always rely upon this commercial paper as a part of their
investment alternatives. So subsequently the primary dealers and all India financial institutions
some specific institutions they are also permitted to issue the commercial paper to meet their
short-term funding requirements for their operations.
So that’s, why they got the permission from the regulatory authority they can also use the
commercial paper as a source of financing for them. So that’s why in general what we can say
the companies or the corporates, primary dealers and the all-India financial institutions are
eligible to issue the commercial paper in Indian market.
(Refer Slide Time: 12:54)
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Then if you see that who can issue these commercial papers, because all the companies are not
eligible to issue the commercial papers in the market. So, in the beginning we have discussed
that only the companies with high credit rating which are highly rated those companies are
eligible to issue the commercial papers. But with that we have certain restrictions certain kind of
criteria the companies have to fulfill whenever they will be issuing the commercial papers in the
market.
Who are those or what are those conditions first of all the tangible net worth of the company as
for the latest audited balance sheet should not be less than 4 crores. The tangible net worth of the
company should not be less than 4 crores as per the audited balance sheet that is number one
condition. And number 2 the commercial papers can be issued in the denomination of 5 lakhs or
the multiple of that, so the commercial papers denominations are always reflected with a value of
rupees 5 lakhs or the multiple of the 5 lakhs.
And the working capital the fund based working capital of the company should not be also less
than 4 crores. So, these are all basically the regulatory requirements for the company have to
fulfill whenever they will be eligible to issue the commercial papers in the market. So, every
issue of commercial papers including the renewal should be treated as a fresh issue.
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Even if somebody is renewing the issuance of the commercial paper that issue will be considered
as a fresh issue and there is no lock-in period for the commercial papers. Lock-in period concept
is not involved with respect to the commercial papers which are issued in the market.
(Refer Slide Time: 15:07)
So, then the CP what is the maturity period of the commercial papers. In India the commercial
papers can be issued for a maturity between the minimum period of 7 days and the maximum up
to 1 year from the date of issue. This is applicable since October 2004, this particular regulation
is implemented since October 2004, the minimum maturity period is 7 days and maximum it can
go up to 1 year.
And already we have discussed who can invest, we have discussed about who can issue, now we
are discussing who can invest. The individuals, banking companies, other companies or other
corporate bodies or we can say that nonresident Indians, foreign institutional investors all can
invest in the CP market. The CP market is open to all type of players who are actively
participating in the financial market.
So, whenever the particular issue is or particular commercial papers are issued, the total amount
of commercial paper which are the companies proposing to be issued should be raised within a
period of 2 weeks from the date on which the issuer opens this issue for this subscription.
Whenever the company has thought of raising that particular issue that total amount what they
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are proposing that should be issued within a period of 2 weeks that is the regulatory norm they
have to follow.
And the mandatory credit rating for the issuance of the commercial paper if you look at it should
be P-2 for the CRISIL and A-2 for the ICRA. Already we have discussed about the credit rating
concept. So, the companies which have this kind of rating they are basically eligible to issue the
commercial paper in the Indian market.
(Refer Slide Time: 17:30)
So, then we will see that what is the basic operational modality of the issuance of the commercial
paper. So, here we have a concept of the IPA and the IPA basically plays a significant role in this
particular context. And who can become this IPA, only a scheduled bank can act as an IPA for
the insurance of the commercial property. And commercial papers are actively traded in the OTC
market that is the over-the-counter market.
But these transactions; has to be reported in FIMMDA platform within the 15 minutes of the
trade and what is the objective of the reporting that trading issues in the FIMMDA platform to
disseminate the trade information to market participants. And once the information is
disseminated to all the market participants then we can assume that the transparency in the
market is going to be increasing.
But when the CP is held in a physical form the commercial paper shall present the instrument for
the payment to the issuer through the IPA. And when the commercial paper is held in the demat
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form the holder of the commercial paper will have to get redeemed through the depository and
receive payments from the IPA. So, IPA is the basically the intermediary organization or the
playing a mediating role between the issuer and the investors, so that is what basically what you
have to keep in the mind then.
(Refer Slide Time: 19:24)
What is the responsibility of this work? Whenever any company is issuing the commercial paper
what are those responsibility they should fulfill whenever they are issuing this commercial paper.
The first of all every issuer should appoint an IPA, so generally the banks are acting as the IPA.
And this is the basic objective of the issuer to disclose to all the financial information to all the
potential investors. The investors should know all the financial details about the company; this is
the standard practice what basically they have to follow.
So, all those data which are pertaining to this balance sheet or the cash flow statement of the
company that has to be disseminated to all those investors, potential investors, who are inclined
to invest in this particular market. So, after the exchange of the deal confirmation between the
investor and the issuer, the issuing company can issue the certificate to the investor or arrange
for crediting this commercial paper to the investors account with a depository.
So once this confirmation is done, the deal confirmation is completed. So, then this particular
credit process will go on and the commercial papers can be credited to their account. And that
thing will be done through the IPA’s.
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(Refer Slide Time: 21:16)
Then what is the responsibility of the IPA; The IPA should ensure that the issuer has the
minimum credit rating as stipulated by the RBI. That means just now we have discussed that the
companies who are eligible to issue this commercial paper they should have a minimum credit
rating or above. So, it is the job of the IPA should ensure that the company has the adequate
credit rating from the credit rating agency and they are eligible to issue these commercial papers.
And also, they should ensure that they are also eligible to mobilize this much amount of the fund
through this issuance of the commercial papers. And the amount which is mobilized through the
issuance of the commercial paper that should be within the quantum indicated by the credit
rating agency for the specified rating or it should be approved by the board of directors of this
particular company whichever is the lower that thing should be checked upon by this IPA.
So that is the basic role of the IPA to ensure that the particular company is within this particular
state regulated regulatory limit to issue these particular commercial papers in the market.
Another job is the IPA has to verify all the documents submitted by the issuer like the copy of
the board resolution, signatures of the authorized executives, who are basically authorized to sign
in that particular document and issue a certificate the documents are in order.
And it should also certify that this company has a valid agreement with the issuer. And the
certified copies of the original documents also should be verified by the IPA and it should be in
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the custody of the IPA only. So, all those necessary things basically IPA has to check. That
means the investor does not have to worry to check all those documents of the companies who
are issuing these particular commercial papers.
This is the job of the IPA to ensure that all those requirements are fulfilled by the companies for
issuance of these particular commercial papers. So, the IPA role is basically to give a confidence
to the investor and as well as checking the regulatory norms for this issuance of the commercial
papers in the market. So once those things are basically done then the investor has the confidence
to invest in that type of commercial papers, who are basically issued by these corporates who are
eligible to issue this particular instrument.
(Refer Slide Time: 24:37)
So, then the credit rating agency what they should do as per the SEBI for the credit rating
agencies, they have to always go for rating this commercial papers. So, they have to undertake
the rating of the capital market instrument which will be applicable to them for rating the CP.
And the credit rating agencies also have the discrimination to determine the validity period of the
rating. One rating is given maybe one company is using that rating for this issue.
But whether this particular rating is valid or not that also has to be checked out in a particular
period. And accordingly, the credit rating agency sell at the time of rating clearly indicate that
the date when the rating is due for the review. Because once the rating is given the rating is not
basically permanent, so we have to see that whether the rating is valid or not.
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So, the credit trading agencies can decide the validity period of the credit rating and the rating
agencies would have to closely monitor the ratings which are assigned to these issuers their track
record as at regular intervals. And they would be required to make its revision in the rating
through its publication and the website that actually they have to keep in the mind.
(Refer Slide Time: 26:18)
Now come to the commercial papers market in U.S, if you see that the commercial papers in the
market in U.S can be classified in 2 types either they are directly placed or they can be placed
through the dealer. That is so we have a directly placed market and we have a dealer placed
commercial papers; so directly place CP is sold directly by the issuing firm to the investors and
many of the issuers are basically the finance companies in U.S.
If you are talking about the dealer placed CP here the dealers or the agents who sell the issuer’s
paper on behalf of the issuers the dealers basically are selling that particular commercial papers
in the market.
(Refer Slide Time: 27:08)
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So again, credit rating also comes into the picture. The issuers of the commercial paper in U.S
tend to have high credit ratings and there are some smaller and less known companies usually
issues CP with credit supports. They are eligible to issue the commercial purpose through credit
supports that is called the credit supported commercial papers. Or they should be backed by high
quality assets that are called the asset backed commercial paper.
Even if in Indian context it is unsecured instrument but in U.S if it is backed by certain highquality asset then some of the small companies can also issue the commercial papers. And the
credit supported commercial purpose if you talk about either they should have a letter of credit
by the banks that is called the LOC paper that means the bank will pay off the paper at maturity
if the issuer fails. The bank has given the guarantee that if the issuers fails then that particular
papers cash flow will be paid off by them.
And also, it can be credit supported through the surety bond, the insurance policy basically
which is written by the insurance company to protect a party against the loss or the violation of
the contract. So that also can play the role in this particular context. So, the CPs which are issued
in U.S they are basically maturity period if you talk about then there tend to be less than 270
days or the 90 days. So, the period basically is different varies between the Indian market and the
U.S market.
(Refer Slide Time: 28:59)
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So, the commercial paper market if you look at the CP’s basically who are the issuer, they often
pay off the commercial paper investors by selling the new CP, which is called the rolling over
paper. And the risk what the investors face is that the issuer will not be able to issue the new
paper at the maturity that is called the rollover risk. If this particular concept works then the
rollover risk basically comes into the picture.
So as a safeguard against the rollover risk the commercial paper is typically backed by a bank
credit line. Banks basically charge for a for providing this credit line which increases the cost of
the commercial paper. So, if it is backed by a bank credit line then to provide those services the
bank basically charges certain money. So that’s why it will increase the cost of the commercial
papers in the market.
(Refer Slide Time: 30:02)
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So, the secondary market in U.S also is quite small in terms of the commercial papers and the
commercial paper investors generally tend to hold their paper to the maturity. So, if this CP is
sold before the maturity, then it is usually back to the issuer or the dealer, the buyback concept
what basically we have discussed before.
So, a major investor of CP is the money market funds in U.S. Their investment in the
commercial paper generally governed by their SEC rule 2a-7 of the Investment Company Act of
1970 that basically limits the credit risk, this is basically for your information point of view. And
this act basically restricts the money market fund to investments in eligible paper defined by tier
1 and tier 2.
So, the eligible paper is what it is the papers which are basically 2 highest credit rating 1or 2
from at least 2 nationally owned rating agencies. And the tier 1 paper is eligible paper, that is
rated 1 by at least 2 of the rating agencies and tier 2 is eligible paper that is not tier 1. That is the
way basically the particular regulations are defined in the U.S market.
(Refer Slide Time: 31:24)
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So, there are certain factors which are affecting the commercial paper market development in
both the emerging and always as well as developed economy. One is credit quality of the issuer,
the liquidity of the market and what are those alternative sources which are available to finance
the short-term requirements of the companies and whether there is any proper adequate financial
market infrastructure is there for the development or for issuance of the commercial paper and
the working capital limits fixed by the banks for these particular companies. Because if they
want to finance their short-term requirements, they can do it through the bank financing. So
instead of that if they will go for this one of the commercial papers that will depend upon what is
the limit the bank has fixed for them. So, these are the different factors which are affecting the
commercial paper market development in the different economies.
(Refer Slide Time: 32:25)
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So, what basically we have discussed here the commercial papers are the short-term bonds issued
by the corporate sector. All these most of the commercial papers are issued in the discount and
redeemed at par. And all the companies are not eligible to issue the commercial papers they
should fulfill certain requirements for that. The secondary market for the CP is not very
developed. And both company specific and external factors generally determine the development
of the commercial paper market across the globe. So, these are the overall things what basically
we have discussed in today's class.
(Refer Slide Time: 33:04)
So, these are the references what you can go through for the detailed discussion on this.
Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 06
Lecture No # 30
Medium- and Long-Term Corporate Bond Markets
Welcome back, in the previous session we discussed about the certain government securities and
mostly and it is the commercial papers. So, today we are discussing about the other long term
and the medium-term securities which are traded in the corporate bond market and what are
those markets available for that and what kind of instruments are traded and all these things?
(Refer Slide Time: 00:44)
So, in today’s session we will be discussing about the largely long-term corporate bond markets
and more specifically the long-term corporate bond markets. If you see the one of the major
instruments that are basically debentures. So, there are the broad concepts what will be
discussing in the today’s session.
(Refer Slide Time: 01:08)
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So, you will come across many keywords I have just reported certain keywords like what
generally we use it called the plain vanilla bonds, floating red bonds, deferred coupon bonds,
masala bonds and perpetual. So those kind of what is basically what will be using in today’s
discussion. But there are many other types of bonds which are traded in the market that basically
you will get to know or we will be discussing over these particular sessions.
(Refer Slide Time: 01:46)
So, let us see what are those different types of long-term instruments which are traded in the
corporate bond market. You see that whenever we classify the instruments which are traded in
the corporate bond market generally the classification is based upon or may be the bonds are
classified on the basis of the coupon payments, on the basis on the currency, on the basis of the
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embedded options, which are involved with respect to that particular bond or it can be claim
based like whether it is secured or unsecured or there are some other specific special types of
bonds that also are headed in the market.
So, whenever we are classifying the bonds on the basis of the coupons we have the bonds like
plain vanilla bonds, floating red bonds, zero coupon bonds, caps and floor, inverse floater,
inflation indexed bond, step up bonds, default coupon bonds, deep discount bond and all these
things and the bonds which are classified on the basis of currency they are classified as the
foreign currency denominated bonds and the masala bonds and if you talk about the embedded
options is the criteria to classify the bonds.
Then, we have the callable bonds, puttable bonds, convertible bonds, warrants and all these
things. Then, if you have to classify on the basis of security or the claim then we have secured
bond and unsecured bond, subordinate bond, credit enhanced bond and all these things. Then
other bonds already I told you we have the perpetual, you have the additional tier 1 bonds,
additional tier 2 bonds, this green bonds, tax free bonds, asset linked bonds and all type of bonds
which are the specific bonds which are also the part of the long-term bond market in India and as
well as the other countries.
So, let us see one by one how those particular instruments are defined? And how basically the
characteristics are varying from one type of bond to another type of bond?
(Refer Slide Time: 04:01)
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So, let us start with the discussion on the coupon-based instrument in the corporate bond market.
Who are those or what are those instruments which are classified on the basis of the coupons?
So, whenever you talk about zero coupon bonds as all of you know that no coupon is paid and it
is always redeemed at par at the end of the term to maturity. So, it is issued at discount and
redeemed at par so this is about your zero-coupon bond.
That already all of you are very much aware about and then we have the most common used
word in the bond market that is called the plain vanilla bonds. So, what this plain vanilla bond is
all about. So, these bonds are generally issued to raise the medium and the long-term resources
and many bonds pay the interest annually the coupon payments are made annually. So, if you
talk about the U.S. corporate bonds particularly the coupon bearing bonds, they generally use the
day count convention of 30 by 360.
In India we have actual by actual that is actual by 365 days but in US generally they use this
convention of 30 by 360. So, most issuers of the debt securities select basically coupon rate that
makes the bond sell initially at par. That is the general observations that is not a thumb rule but
that is the general observations that basically what you can keep in the mind and in US if you
talk about generally 1000 dollar per value bond.
If you talk about and if you are using the 30 by 360-day count convention then if a bond whose
par value is 1000 dollar and coupon is 9% then they would pay the 90 dollar per year and its
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interest would occur at a rate of 7.5 dollar per month that is 90 by 12 and 0.25 dollar per day that
is 90 by 360. That is the concept what basically you have to keep in the mind whenever you talk
about the plain vanilla bond which is a coupon bearing bond.
(Refer Slide Time: 06:40)
So, then we have another bond that is called the floating rate bonds you remember that the plain
vanilla bond is quite popular in both US and as well as India. Mostly the investors are inclined to
invest in the plain vanilla bonds. Floating rate bonds are there but they are not that much popular
in both the markets. So, if you use this term floating rate here the floating rate bond or note or
floater generally, we call it the floater is generally used to define any bonds with an interest rate
that is adjusted periodically.
The interest payments or the coupon payments against these bonds are generally changed over a
period of time. Periodically there is some adjustment takes place with respect to the coupon
payments. So, this is linked to generally an index the value of that particular coupon the interest
rate is linked to an index and the coupon is reset mostly every half year and it is market revenue.
Technically a floating rate bond is basically defined as a debt instrument with the coupon based
on a short-term index let Treasury bill rate.
The treasury rate can be used as a kind of benchmark for deciding that interest rate for that
particular floating rate bonds. And it is reset more than once a year and an adjustable rate note or
variable rate note generally is defined as a debt security with its coupon based on a longer-term
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rate. So, there are 2 ways the rates are defined one can be based on the basis of the short-term
instruments or short-term index like treasury bill rate.
And other one is that the adjustment also can be made on the basis of another instrument which
is long term in nature that is called the adjustable-rate bond or the variable rate bond. But in US
it is called the adjustable rate note or the variable rate note. So, that is basically another floating
rate bond definition what basically or the concept what basically you have to understand.
(Refer Slide Time: 09:19)
And if you see that the floating rate bonds are issued where the coupon rates basically reset
periodically that is called the coupon reset date. And the reset is based on a reference rate plus a
quoted margin or spread so the reset date rate if you are talking about that how the rates are
basically changing over a period of time that is the reference rate plus a margin. And why there
is a margin it is basically the additional rate the issuer basically agrees to pay above the reference
rate.
And why basically they are ready to pay that because there is a credit risk involved in that so the
margin is basically adjusting the credit risk. So, there is a premium which is basically paid
against that so, that is basically we call it the margin or the spread. So, the reference rate already
we have discussed that may be 3 months Treasury bill or it can be live or in the international
market. You can use the LIBOR rate as a reference rate or you can also use the 3 months treasury
bill rate as a reference rate in that particular context.
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(Refer Slide Time: 10:40)
Then we have a concept a bond called inverse floater so what is the inverse floater? It is a type of
bond whose coupon rate has an inverse relationship to the benchmark rate. So, the inverse floater
basically adjusts its coupon payment as the interest rate changes whenever the interest rate
changes the inverse floater basically adjusts its coupon payment. It is also known as the inverse
floating rate note or the reverse floater the other name of the inverse floater is the inverse
floating rate note or the reverse floater.
So, that is another way basically this flexible interest rate is prevailed or flexible coupon rates are
prevailed for that particular type of bonds. Then another important bond if you see we have the
inflation index bond. So, these bonds are linked to the wholesale or the consumer price index.
That means the inflation is going to be changed over the time if the inflation is going to be
changed then this particular principal or the coupon value of that particular bond is going to be
changed. So, these bonds are basically protecting the investors from the inflation.
Then we have the caps and floors what are those bonds are? So here the interest rate cap is a type
of interest rate contract in which the buyer of the cap contract receives the payment at the end of
each period in which the interest rate exceeds the agreed rate. This is a kind of options which are
linked to that and accordingly the payment will be made to the bond investors. And the interest
rate floor is a contract where the buyer basically receives the payment at the end of each period
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in which the interest rate is below the agreed strike price or agreeable price which has been fixed
before.
So, if these conditions are satisfied then the payments of the interest rate will be made unless it
will not be made. So that is another type of bond which basically we observe in the market these
is called the caps and floors.
(Refer Slide Time: 13:30)
Then what are those other bonds other bonds basically is the step-up bonds what are those stepup bonds? Here in this type of bonds earlier coupon paid is very low let the maturity period is 7
years. So, in the beginning the coupons rates are low and slowly it will rise with time movement
towards the maturity of the bond. So, first year coupon rate it will further increase in the second
year it will further increase in the third year and so on. So, the coupon rate should be higher once
the bonds is approaching towards the maturity. So that is called the step-up bonds. So, that
means the coupon is slowly increasing over the time the coupon rate is changing and it is
increases slowly over the time.
Then we have another bond called the deferred coupon bonds, so what is deferred coupon bonds?
The deferred coupon bonds basically what these bonds do not pay any coupon at the beginning
but the coupon is paid out at a later date. That means the coupons are deferred in the beginning
they do not pay the coupons but in the later date they will be paying the coupon against those
particular bonds. So, that particular kind of characteristics will be defined in the beginning that
there is a possibility that the coupons can be deferred so that is called the deferred coupon bonds.
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Then we have a bond called the deep discount bonds what are those deep discount bonds? These
bonds generally sold at very high discount to the face value. These are the discounted bonds but
the discounts are very high whenever the bonds are sold and these bonds generally have low
coupon rate and the price is below the par.
And who are basically inclined to invest then only some investors particularly the pension funds
they find this deep discount bonds are attractive because they have less reinvestment risk in the
sense that is the interest-on-interest risk. Because the pension funds have the less amount of or
there less exposure towards the reinvestment risk, they always prefer this deep discount bonds
which are issued with a high discount in the beginning. So, that is basically called the deep
discount bonds.
(Refer Slide Time: 16:25)
Then we have certain bonds which are classified on the basis of the claims. You know that is a
broad classification of the corporate bonds we have a secured bond and we have an unsecured
bond. How to define a secured bond? By definition itself you can understand these bonds are
backed by a specific asset and the asset may have a claim over the general assets of the company.
So, this is basically defined as one of the bonds which have a lien giving the bondholder via the
trustees or the right to sell the pledged asset in order to pay the bondholders if the company
defaults.
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So, if the company is going to be defaulted if the company is going towards the liquidation, then
those assets can be utilized to repay the proceedings like your principal and the interest payments
to the bond investors. So, the secured bonds can be also classified in terms of the collaterals
which are pledged against that particular type of bond. What are the different types of collaterals
are used against that particular bond?
Accordingly, you can also differentiate among these secured bonds. Like if you have a mortgage
bond. Generally, the mortgages concept comes whenever the collaterals or the assets basically is
a fixed assets like the lands, property or buildings in that particular context, we are using the
word mortgage. So, if the particular company's real property and the buildings are basically used
to back up that particular kind of bond then we call it the mortgage bonds.
And we can have an also equipment trust bond, so this bond has a lien on specific equipment like
airplanes, trucks, computers and whatever assets the tangible assets the company has that
equipment can be used as the collateral against that. And a collateral trust bond it is a different
type of bond it is basically secured by the equity shares of a company's subsidiary or holdings of
other company stocks and bonds or the government securities or other financial claims what the
company has it is mostly backed by certain financial assets.
So, a company might secure its debt with personal property like your cash, liquid assets, account
receivables, all these things. And all these assets are generally short term in nature. That’s why
they are usually used as collateral for the short-term debt publications of the companies using
their cash or liquid assets as the backup asset against this particular bond. Those things are not
mostly used for the long-term corporate bonds these are mostly used for the short-term debt
obligations of the particular company. So, this is about your secured bond.
(Refer Slide Time: 20:32)
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But whenever you talk about the secured bonds always, we need to establish a priority of the
claims when more than 1 debt obligation is secured by the same asset. Let the company has
issued 2, 3 bonds and the same type of asset has been used as the backup asset or collateral
against that. Then who will get if there is any default or any kind of liquidation then what should
be the priority, who will get the proceedings first?
Then who will be getting the second? So, in that case we call it the concept of senior bonds,
junior bonds or the subordinate bonds. Then a closed end bond basically prohibits the asset
securing the bond from being used to secure any other asset. So, in that context if you see that
there are if there is a closed-end bond that means if 1 asset is already used as collateral for 1 type
of bond then further this particular asset cannot be used that is called the closed-end bond.
But the open-end bond allows the asset securing the bond to be used as collateral on other debt
also or rather bond also. So, that’s why whether this particular bond is a closed-end bond or it is
open-end bond that will decide that the same asset can be used for the multiple debt instruments
or not.
(Refer Slide Time: 22:08)
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Then we have the unsecure bond so these are basically the bonds or debts which are issued by
the companies without any specific assignments of the assets of the company or these particular
bonds are not backed by any of the assets of the company so these bonds are called debentures.
Mostly the debentures are the unsecured bonds, so when the companies issue a number of
debentures some may be issued that are subordinate to the others that is called the subordinate
debentures.
And debentures also are often issued with protective covenants; sometimes or the restrictions on
additional debt. So, debentures also can be sold with credit enhancements like you can get a kind
of guarantee from your credit from the third party like the banks. If you are going to be defaulted
then the bank is going to pay that is called the credit enhancement. That possibility is also there
whenever companies are issuing these debentures.
(Refer Slide Time: 23:29)
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So, we have another type of bond that is called the guaranteed bond. So, what is the guaranteed
bond? These bonds generally issued by one company and guaranteed by another economic
entity. So, these guarantees generally applies both the interest and the principal. So, if there is a
guarantee then the default risk of this particular bond shifts from the borrower to the financial
capacity of the insurer basically it is the kind of insurance.
If somebody is giving the guarantee then if this particular issuer is going to be defaulted then
who are the guarantors, they are basically going to fulfill the requirements so that is called the
guaranteed bonds. So, the guarantor could be the parent company, if the issue is basically coming
from the subsidiary company. The guarantor can be another company and there may be multiple
guarantors for 1 particular type of bond the guarantors also can be multiple guarantors. Not
necessary that one bond should be backed by a single guarantor it can be multiple guarantors.
For some corporate issues a financial institution also may provide the guarantee or the banks also
can provide the guarantee. That if there is a default then the bank is going to pay for that and
bank generally charges certain kind of commissions against that and that is a non interest income
what basically they can earn. So, these are called the guaranteed bonds.
(Refer Slide Time: 25:08)
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So, we have the foreign currency denominated bonds these bonds are generally issuable and
payable only in the foreign currency and sold in a foreign market to the foreign investors. And
another concept we have in India that is called the masala bonds. So, these are issued and
payable in Indian rupee and issued in a foreign market to the foreign investors that is the basic
difference between the foreign currencies denominated bonds and the masala bonds.
(Refer Slide Time: 25:40)
Then, we have the embedded options accordingly also we can classify the bonds we have a
callable bond. These bonds have a call back feature call back option for the issuer at any point of
time the bond can be bought back by the issuer itself. And then generally if the refinancing cost
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drops and the issuer wants to refinance the obligations then they can go for this kind of process,
they can go for calling back the bond.
We have a puttable bond here these bonds are having the sell back option for the investor, if the
market interest rate rises and the investor wants to reinvest the higher paying securities with the
same risk exposure that thing can be possible. Then, we have a convertible bond so it is the debt
security that makes the interest payments but can be converted into predetermined number of
equity shares of the issuing company these bonds can be converted into equity after a certain
point of time.
And the conversion can be done at a certain fixed dates during the bonds life and generally it is
the discrimination of the bond holder whether they want to convert it to the equity or not these
are called the convertible bonds.
(Refer Slide Time: 26:59)
Then, we have a warrant this is the kind of instrument which gives the holder the right to buy a
specific number of shares of the stock or another designated security at a specified price. So, a
warrant which is attached to the bond can only be exercised by the bond holder. Then we have
the income bonds which pay the interest only if earnings are sufficient, then the participating
bond so here the bonds so which can pay minimum rate plus an additional rate up to a specified
maximum if the earnings are sufficient.
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Then we have an extendable bond which gives the issuer or the investor to the option to extend
the securities maturity the maturity period also can be extended that is called the extendable
bonds. Then we have the credit sensitive bonds that are the bond which has a coupon that varies
inversely with the issuers credit ratings. So obviously that is called the credit sensitive bond if
the credit rating is going to be changed then the coupon rate also is going to be changed so that is
called the credit sensitive bond.
(Refer Slide Time: 28:13)
Then, we have the commodity linked bonds commodity derivatives market, commodity market is
quite prominent in today's context. So, there are certain bonds which are available with respect to
the commodities also which are traded in the market that is called the commodity linked bonds.
So, these bonds generally are defined in this way where the principal and coupons are tied to the
price of a particular commodity. Let like oil or any kind of commodity index so these are
basically defined as the commodity linked bond.
Then, we have voting bonds, which is basically available in US market mostly. It basically
provides the voting privilege to the bondholders and it is usually limited to certain issues or the
conditions. Then we have assumed bonds again it is applicable in US market. Here the bonds
whose obligations are taken over by another corporations generally that happens in the context of
mergers or the reorganizations of the company.
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So, if that thing will happen then other company who is taking over the company, they will take
care of that particular procedures or the payments of that particular security. That is called the
assumed bonds these are another type of bond which are also traded in the market.
(Refer Slide Time: 29:41)
There are some specific bonds if you look at, let perpetual means they do not have any maturity
date it will continue to pay the coupons during the life of the company. Then, we have AT1
bonds that is basically the additional tier 1 bonds these are type of unsecured perpetual bond that
banks issued to increase their capital base generally to meet the Basel-III norms after the
implementation of Basel-III.
Then AT2 bonds these are components of tier 2 capital of the banks. So, that is why the bank can
go for this. Then REIT that is basically the investment fund on the real estate companies that is a
special type of bond. Infrastructure investment trusts these bonds are issued to promote the
infrastructure sector in India. Then green bonds generally they are used to raise the money for
the climate and environmental projects if anybody and government or any corporate body are
basically going for this kind of projects, they can raise the money through this green bonds.
(Refer Slide Time: 30:50)
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Then we have PSU bonds mostly the PSU bonds means these are issued by the public sector
units and mostly the PSU bonds are sold in the private placement market. Sometimes the
investment bankers are roped in but mostly it is issued in the private investment market and these
are transferable and they are generally issued in the demat form. So, to attract more investors and
to increase the liquidity the issuers get their bonds rated by the rating agencies like CRISIL,
ICRA, CARE and all. And some of the issues also may be guaranteed by the central or the state
governments to enable them to get better rating this public sector unit basically is bonds are
issued in this fashion.
So, the PSU bonds are maturity period varies between 5 to 10 years the default rates are quite
negligible, so sometimes the PSU bonds can be classified into 2 types taxable and tax free. And
the major investors in the PSU bonds are banks, mutual funds, insurance companies, provident
funds and the individuals generally they are the major investor in the PSU bonds.
(Refer Slide Time: 32:15)
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So, there is a medium term note which is mostly popular in the US market, so this is a debt
instrument sold on a continuing basis to the investors who are allowed to choose from a group of
bonds from the same corporation but with the different maturity. So, these medium-term notes
growth generally can be attributed to the flexibility they provide in the type of securities they can
offer in the different maturity basically this is available.
So, from the same corporation that is why people are sometimes inclined to invest in that
whenever they find that their investment horizon period is matching with that particular maturity
period these are called the medium-term notes.
(Refer Slide Time: 33:01)
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So, what we discussed in today's session that the long-term bonds are classified on the basis of
coupons, claim, currency, embedded options and all and some of the other long-term bonds
which are traded in the market are perpetual, green bond, income bond, commodity linked bonds,
infrastructure bonds etc. And the most of the PSU bonds are sold in the private placement basis
and the major investors in the PSU bonds are banks, mutual funds, insurance companies,
provident funds and individuals. So, this is the overview of the medium and the long-term bond
market in India and as well as U.S.
(Refer Slide Time: 33:43)
So, these are the references you can go through for the comprehensive discussion on these.
Thank you
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social sciences
Indian Institute of Technology Kharagpur
Module # 07
Lecture # 31
Issuance and Trading Mechanism of Corporate Securities
So, in the previous class we discussed about the certain types of corporate bonds which are
issued in the Indian corporate bond market and as well as the US bond market. We discussed
both the short-term and long types of bonds that are issued in these two markets, which are
considered as one of developed markets and as well as the emerging market. Today, we will be
discussing this issuance and trading mechanism of corporate securities. How are those corporate
bonds issued in this particular market?
(Refer Slide Time: 00:56)
So, the concepts that will be discussed in today's class basically will deal with the 2 types of
markets which are existing with respect to corporate bonds. One is as you know that we have a
primary market, which means whenever the first-time particular bonds are issued and the other
type of market is the secondary market. Where once the bond has bonds listed in the market, how
does the trading of those kinds of securities are taking place?
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So, that’s why the trading mechanism of corporate securities in the secondary market and what is
the issuance procedure of the corporate bonds in the primary market. So, these are the 2 broad
things what we are going to discuss in today's class.
(Refer Slide Time: 01:55)
So, the keywords that you will come across in this particular class are like public issue, private
placement, merchant banker, right issue and underwriting. So, this kind of concept is what
generally we use in the trading of the corporate bonds in the different markets that basically you
will be coming across in today's session.
(Refer Slide Time: 02:28)
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So, let us see that first of all we have to explain a certain kind of process with respect to the
primary market. So, how are the corporate bonds are issued in the primary market in India? Then
we can move towards the issuance procedure of the corporate bonds in the US market. So,
whenever the corporate bonds are issued in the primary market there are 2 ways the bonds can be
issued. One is through the public offer, the other is the private placement.
So, there are 2 ways the bonds can be raised or one can be issued in the primary market of a
particular corporate bond market. So, what do you mean by the public issue? The public issue
means, basically we the companies go to the public to raise the funds by issuing the bonds. So, in
this case, what basically, we can say? That the company basically makes the invitation to the
public to subscribe those bonds what they are going or they are planning to issue and how
basically they issue those particular bonds through the prospectus.
Through a prospectus, the particular bonds are issued to the public or the companies go to the
public to raise the funds. But if you talk about private placement, private placement is basically
the offer of the sales of the debt by any of the issuers generally, always made to some select
group of people or the select group of institutions. So, it is not available or is not applicable to
the common public.
It is only applicable to some select group of investors and the companies decide to whom they
are going to issue that particular bond to raise the money or to raise the capital. So, that is the
basic difference between the public issue and the private placement. So, let us see that how the
corporate bond market issuance of the bonds is taking place through this public issue. Then we
can move towards the discussion on private placement.
(Refer Slide Time: 05:22)
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So, whenever you talk about the public issuance process. First of all, you see that the basic
objective of the public issue is the particular issues of the debt to be listed on the stock
exchanges. It was not there before in the stock exchanges; the trading was not taking place in the
stock exchanges. So, once the public issuance procedure will be over, the particular stocks can be
traded in the secondary market.
So, the basic objective of the public issue is basically that the particular debt should be listed on
the stock exchange. Then what happens here? The company basically goes for an offer document
and that offer document should be approved by the stock exchange, where it is to be listed. So, in
the stock market, you might have the idea that the initial prospectus or the offer document that
they make has to be approved by the securities and exchange board of India.
But in the case of the bonds, the approval is not necessary from the SEBI, but the approval
should be sort from the stock exchange, where the particular bond is going to be listed. So, the
first is the offer document should be approved by the stock exchange where it is to be listed.
Then what happens? The process basically will start; it is just like a book-building process what
we follow whenever we raise the IPOs in the stock market.
So, in the case of the bond market in India, the particular process will start with the appointment
of a merchant banker. Generally, the merchant bankers are the investment banks. So, once the
particular company who is issuing this particular bond will appoint the merchant banker. Then
the initial process will start. Then what will this merchant banker will do? The merchant banker
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basically will evaluate the characteristics of the particular bond and what are those things are
basically the evaluate.
First of all, they will see what the objective of the fundraising is. Why is the company basically
issuing the debt? How that particular money which will be raised through this particular debt
issuance are going to be utilized? So, what is the objective of the fundraising and the structure of
that particular instrument which will be decided or which will be evaluated on the basis of the
issuer’s financial conditions right.
That means they will look at the company's profitability, look at the cash flow position in a
particular company. Then accordingly they will evaluate whether the company is really in a
position to raise that particular money through that issuance of the bond and whether the
company is able to fulfill all the requirements from the investment perspective. That actually
they have to evaluate first.
Once the merchant banker is appointed, they will go through or they will analyze these aspects;
and how to define the structure of the instrument? Already all of you are aware of that. The
structure of instruments basically includes the type of redemption pattern, the option of the
interest payment that means whether the interest basically will be paid annually or it will be paid
semi-annually or it will be paid at the end of the maturity.
So, all kinds of options they will look at; the class of the investors. What kind of investors are
you targeting? Whether the investors you are targeting; the institutional investors or it is a
combination of institutional investors and the retail investors and all these things and as well as
the timing of the issue. When the particular company is going to issue that particular bond? So,
here what basically you have observed, there is a big role the merchant banker plays in terms of
the public issuance of the corporate bonds.
(Refer Slide Time: 10:34)
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So, once this evaluation process is done then what will happen next things that the issuer has to
obtain a required credit rating for that particular issue from the credit rating agency. That means,
depending upon the rating of that particular bond, the yield and the price of the particular bond
are generally determined. The particular rating will provide the riskiness of that particular bond.
How much credit exposure the particular bond has or how much credit risk is involved with
respect to that particular issue? So, that thing basically can be judged from the rating what the
particular issue is getting. So, all of you know that the rating agencies again will look at the
financial position of the company, then their debt repayment capacity and all types of things have
to be checked open.
Some of the factors are internal and some of the factors are external. So, after that the rating
agency will give the rating to that particular issue and once the rating will be received from the
credit rating agency, that rating will be mentioned in the offer document. What is the rating of
that particular bond then what will happen? The issuer has to appoint one or more debenture
trustees. In the previous class we have discussed that because most corporate bonds are
unsecured bonds.
So, in that place, the debenture trustees basically will play a significant role in more or less the
security or the guarantee we can get from the Debentures trustees. So, these Debentures trustees
are basically approved by the securities and exchange board of India. So, once the credit rating is
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received, then after that the issuer has to appoint one or more Debentures trustees in accordance
with the Companies Act.
Now, we are going through the Companies Act, 2013. On the basis of the Companies Act, 2013,
they will always look for appointing one or more Debentures trustees; then what will happen?
All these formalities are over, all the processes are over, then final approval will be sought from
the stock exchange and then it will be marketed to the investors. This particular instrument or
this particular issue will be marketed to the potential investors who are eligible to invest in that
type of issue.
So generally, the timeline also has been fixed. What is the timeline? The timeline is basically for
a listing of the debt securities is 6 days for all public issues of the debt securities. So, within 6
days that particular issue has to be listed on the stock exchange. So, all those kinds of bidding
process have to be over within those particular 6 days period. So, this is basically a common
public issue, more or less it is very much closely related to the book-building process what we
follow whenever we go for the issuance of the IPOs in the stock market.
Then we have another issue which also works in terms of the public issue, so that is called the
right issue. What is the right issue? The right issue is basically a kind of offers which are only
given to the holders of the issuer’s debt securities or who are those investors who are already
existing or they are already investing in that particular company, they are holding a certain
amount of bonds what the company has issued before. The right issues basically are related to
them.
The public issue whenever you talk about, there we may have a certain kind of concept like
anybody can also start the investing in that particular company for the first time. But whenever it
is a right issue, then those issues are only offered to the existing holders of the issuer's debt
securities. So, that is basically what we call the right issue.
(Refer Slide Time: 15:38)
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Then we have a private placement. What is the private placement process? Here the debt
securities are offered to select group of investors. Even if they are existing shareholders, if it is a
right issue maybe you can. It is only open to the existing shareholders which are there in the
company. But if it is a private placement, then the debts or the instrument what the particular
company is trying to raise or trying to issue in the particular market.
So, that is only applicable or only available to the select group of investors. So, the company will
decide to whom they want to basically sell that particular security or from whom they want to
raise the capital for their investments. So, what will happen in that case? All the necessary
information about the issue to be given to the private placement memorandum. There is a
concept called the private placement memorandum.
The company will prepare that memorandum and that all those information related to that
particular issue will be mentioned there and that information has to be provided to all those select
groups of investors, to whom the company is targeting and all the issuance has to be made
through the electronic book mechanism for transparency and the better price discovery. All those
kinds of activities which is going to happen with respect to this private placement process that
has to be always carried through or always will be carried out through this electronic book
mechanism.
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Because the electronic book mechanism will increase the transparency and the price discovery of
that particular bond will be better in that case. So, all the stock exchanges can act as the
electronic book providers that we call electronic book providers or in short, we call EBP and
they can provide the platform for placing the bids, ensuring safety, secrecy, integrity and at any
point of time the data can be retrieved.
The retrievability of the data is quite important in that case. That means the data is not missed
out. At any point in time, the data can be retrieved. So, the stock exchange has act as an
electronic book provider and they provide the platform for the bidding, maintaining all kinds of
things like safety, security, secrecy, integrity and all these things. Again, in the private placement
case also, the issuer has to obtain the credit rating from the credit rating agency which is required
or what they basically are eligible to get.
And after that, they file a disclosure document with the stock exchange. What we can say is the
rating they have and accordingly what should be the characteristics of the particular bond
instrument those things will be disclosed to the stock exchange. So, the rating is quite important
in the private placement case also.
(Refer Slide Time: 19:32)
Then all the participants in this particular private placement market have to require to enroll
themselves with the EBP’s prior to entering the bidding process. Before they participate in the
bidding process before they go for the bidding of all those debt instruments, they have to enroll
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themselves with the stock exchange or this electronic book provider. So, all those exchanges
provide the codes to their participants which can be used to participate in this EBP platform.
So generally, all those participants like all those investors who are inclined to participate in this
particular market will be given a particular code by the stock exchanges or by the EBP and they
can use that code for all types of transactions and all types of trading process. So, the minimum
bid is 10 lakhs or the multiple of the 10 lakhs thereafter. But if you look at the non-banking
financial companies (NBFC) and the housing finance companies (HFC) the minimum bid lot is 1
crore or and the multiples of 10 lakhs thereafter.
The minimum is 1 crore and after that, if you want to increase it can be 1 crore 10 lakhs, 1 crore
20 lakhs like that. So, there are some kinds of regulations with respect to the minimum amount
of investments what the investors are trying to make. So, that is basically your private placement
process, what generally we see in the Indian corporate bond market.
(Refer Slide Time: 21:33)
Then after that what happens? If the particular issues are reissued or in case of reissuance, the
issuer has to specify the ISIN number. What is the ISIN number? It is called the International
Securities Identification Number of the existing security whatever they are holding and the
accrued interest as on the day of paying for the reissued securities. That actually also they have
to specify and the bidding is done in the case of reinsurance on a clean price basis.
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Already, we have discussed that thing. In the beginning, you might have remembered, we have a
clean price and we have a dirty price or the full price what we call it. So, whenever the particular
private placements are carried out in the case of the reissuance process, the bidding has to be
done with a clean price only. So, the total price is your clean price plus the accrued interest. So,
that particular thing has to be kept in the mind whenever the reissuances are happening with
respect to the private placement market.
The particular information memorandum has to disclose the mode of bidding, either it is an open
mode or it is a closed mode or and the manner of allotment whether it is uniform. Already we
have discussed these things before in the auction process. Uniform price-based auction and
multiple price-based auctions and how the allotment is going to be made whether it is uniform
yield basis or multiple yield basis and the settlement cycle.
The memorandum has to disclose all the information with respect to the mode of the bidding, the
manner of allotment, the manner of settlement and the settlement cycle. All that information has
to be given there. Then in the case of open biddings, just now we said that close bidding or open
bidding. What do you mean by this open bidding? In the case of open bidding, the bid
information is like the bid value or build yield.
Except for the bidder’s name, the bidder’s name will not be mentioned there. These are available
to the market during the bidding window. So, that is called the open bidding process. But in case
of the closed bidding, the same information will be disclosed in the post-bidding process. So, in
the open bidding case, the bid value, bid rate and all these things will be available in the market
during the bidding window; whenever the bidding is going on, but in case of the closed bidding,
this information will only be available after the bidding is over. So, the cumulative bidding
amount is displayed to the eligible participants during the bidding process and finally, the issuer
has to disclose the estimated cutoff field to the EBP or the stock exchange at least 1 hour before
the opening of the bidding of the issue. So that is the regulation that basically we have to keep in
mind.
(Refer Slide Time: 25:49)
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So, the successful bids are determined by the EBP on yield-time priority basis and it will notify
the successful bidders about the total amount, details of the paying account, paying date and cutoff time and all these things. Once the bidding window closes, the stock exchange gives out the
aggregate volume data, which includes the yield/coupon/price, amount with over subscription,
total bids received, ratings and category of investor, all types of investors will be available, once
this bidding window is closes.
(Refer Slide Time: 26:36)
Then, if you look at the primary market of the corporate bond in the US more or less it is the
same process. We call it the public issue, but they call it an open market and we have a private
placement. The open markets in the US are handled through the investment bankers like the
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merchant bankers, who will either underwrite the issue, that means the investment banker agrees
to buy at a set price and then hopefully set at a higher one and they can also form an
underwriting syndicate, it’s not necessary to single underwriter will be there.
Maybe a syndicate of underwriters may be there in that case. They use their best effort to sell the
bonds at the best price and provide a stand-by underwriting agreement. So, underwriting
agreement means the particular issue is not subscribed then one particular company has to
subscribe to that particular issue. So, the way the company chooses to offer an issue to the public
that generally depends on the size of the issue and the risk of a price, decreases during the time
the issue is being sold. So, these are certain factors they will consider.
(Refer Slide Time: 28:16)
So, already I told you what you mean by the underwriting issues? The investment bank basically,
the underwrites the issue, buying the securities at an agreed-upon price and selling it in the
market at a higher price. The agreement generally is referred to as the firm commitment. So, the
difference in the prices is basically known as the price spread or the floatation cost.
Approximately, it is around 1% in the US market. But there is a risk involved that is called the
underwriting risk.
If the rates increase, then obviously the bond price will decrease in the market between the times
the investment banker buys the bond and sells them. So, that risks the underwriter takes or the
investment banker takes. So, that is called the underwriting issues.
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(Refer Slide Time: 29:11)
So, like India they prepare the indenture or the agreement, register themselves with the securities
and exchange commission in the US, issue the preliminary prospectus, what they call the red
herring and form the selling group like investment bankers like a wholesaler and the other
dealers and sell this particular issue through this selling group. The selling group basically plays
the role in that particular case.
(Refer Slide Time: 29:46)
So, they have a rule called Rule 415. So, that is called the shelf registration rule. So, this
basically allows a firm to register on inventory of securities of a particular type for up to 2 years.
The firm can sell the securities whenever it wishes during that time and always the securities
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remain on the shelf. It is basically to reduce this cost and this tedious process which generally we
follow for these ones.
So, to minimize the cost a company planning to finance a number of projects over a period of
time could register a large issue and sell the parts of the issue at different times. So, that is the
benefit what the issuer can get through this particular role.
(Refer Slide Time: 30:44)
Coming back to the private placement process in the US, it is through the private placement
market they can do it. They are sold through direct negotiation with the buyer, then the privately
placed bonds usually have fewer restrictive covenants, than publicly issued ones, and they are
more tailor made to both buyers and seller's perspective. But, one of the disadvantages of the
privately placed bonds was the lack of marketability due to the absence of the active secondary
market for that.
So, under the SEC Act of 1933 the firms only offer securities privately to investors who are
deemed or who are basically known as sophisticated investors like insurance companies, pension
funds, banks and all these things. Anybody cannot be a participant in the private placement
market.
(Refer Slide Time: 31:59)
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Then further in 1991, they have adopted the 144A rule and under this rule what basically will
happen, the issuer could sell the unregistered securities to one or more investment bankers who
could resell those securities to the qualified institutional buyers. And the qualified institutional
buyers can sell freely with each other that have not been registered. So, this particular rule
eliminated some of the restrictions on the secondary trading of the privately placed bonds and
created a secondary market for that. And the privately placed bonds are exempt from the SEC
registration, because they do not involve a public offering.
(Refer Slide Time: 32:53)
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So, this rule creates a secondary market and many privately placed issues are for the smaller and
less known companies also have started participating in this particular market and this market
competes with the corporate bank loan market in the US.
(Refer Slide Time: 33:20)
So, coming back to the secondary market, the secondary market bonds basically if you talk
about, they are generally done through the OTC market and the lot size of trading is around 1
lakh. And all the OTC trades transactions have to be reported within 15 minutes of the deal
closure on the reporting platform of the stock exchange which can be consolidated and
disseminated on the FIMMDA website. And the corporate bond trades are settled on delivery
versus payment basis at participant level through the clearing corporation of the stock exchange
that is in India.
(Refer Slide Time: 34:04)
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If you look at the secondary bond market in the US that is also dominated by the OTC market
and here the OTC market has been conducted by dealers, who take the positions in bonds to
fulfill the buy and sell orders. Today the market is more transparent due to a trading reporting
and compliance engine that is called trace and there has been a transaction from traditional
trading towards electronic trading.
So, many corporate bonds, if you look at them, are held by the investors up to maturity; that’s
why the secondary market for some bonds can be relatively thin. Because they are holding the
bonds up to the maturity there are no transactions before the maturity or less amount of
transactions are happening before the maturity.
(Refer Slide Time: 34:53)
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So, they have a multiple dealer system, multi dealer system which allows the customers to
execute trades from multiple courses. The system will display the best bids or offer prices of
those posted by all dealers and the participating dealers usually act as the principal in these
particular transactions. They can also have a single dealer system; this system basically allows
the investors to execute the transactions directly with the specific dealers.
Then they have a cross matching system that brings dealers and institutional investors in an
electronic trading network and that provides the real time period cross matching. An auction
system allows the market participants to conduct the electronic auction of securities in early
stages. So, these are the different process or different systems which work in the secondary bond
market corporate bond market in the US.
(Refer Slide Time: 35:52)
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So, what basically we have discussed here, that the corporate bonds are sold through public offer
and the private placement basis. In the US also bonds are sold either through open market or
private placement. Open market sales are handled through the investment bankers and the
principal secondary market for the corporate bond is the OTC market. But; that market is not
very developed because most of the bonds basically the investor holds up to the maturity.
So, these are the references that you can go through for a detailed idea on this. Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 07
Lecture No # 32
Municipal Bonds – I
Welcome back!
So, in the previous session we discussed about the issuance and trading mechanism of corporate
in both India and the US market. So, another most important type of bond or highly popular bond
that we always look at that is called the municipal bonds. The municipal bonds are not that much
prominent in the Indian market but that is quite prominent in the developed market like the US.
So, let us discuss certain kind of characteristics or the features of the municipal bonds and what
is the advantages and uses of municipal bonds? And why those bonds are quite popular in those
type of market and what is the use of that? So those things are basically what we are going to do
in today’s session.
(Refer Slide Time: 01:16)
So here in today's session we will be covering instruments like municipal anticipation notes.
These are the different types of bonds which the municipalities or the different entities in the US
market generally issue. We have the general obligation bonds and the revenue Bonds. These are
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the major types of bonds which are issued in the US market like in the form of the municipal
bonds.
(Refer Slide Time: 01:51)
So, we will come across many keywords. Some of the keywords are highlighted here. We have
equivalent taxable yield, tax-anticipation notes, revenue anticipation notes, Bond anticipation
notes, unlimited tax general obligation bonds, limited tax general obligation bonds, doublebarreled bonds. So, there are different types of bonds what basically we come across whenever
we discuss the issue with respect to the municipal bonds.
(Refer Slide Time: 02:25)
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So, what do mean by a municipal bond? What is the meaning of that? Because you see
government issues the bond, we call it the government securities or the government bond or
treasury securities. The corporate sector issues the bonds, we call them the corporate bonds. So,
like that the municipal and other local governments also sell the bills, notes that mean the longterm bonds to finance their long-term projects and the short-term deficits resulting from their
current operations.
The issuance of the municipal bonds has certain objectives. They have typical and clear-cut
objectives that why these particular bonds are issued. State and local governments, if you look at,
they undertake many long-term capital projects that is the basic job of this kind of entity. They
go for constructing the school buildings, highways, water treatment facilities, airports, hospitals,
the inner-city housing and infrastructures.
All kinds of things which are basically very much required for the economic growth and creation
of the job like employment opportunities. So how basically those kinds of projects financed? So,
the financing of all those projects can be carried out through the issuance of the municipal bonds.
So, to finance the long-term investments generally these kinds of institutions sell 2 types of notes
or the bonds. That is called the general obligation bonds and the revenue bond.
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And the state and local government also have short-term cash flow needs which are created from
the differences in their expenditure and revenue what they are generating or the time gap
between the projects begin when the permanent financing supporting from the project is set. For
example, if somebody has constructed a road; the road will generate certain revenue through toll
tax. But once the particular project is started immediately this particular revenue cannot be
expected.
Once the project is completed then this kind of revenue generation will be possible. So that’s
why there is a gap. So, because of this they finance some of these kinds of activities through this
issuance of the anticipation notes. They are called the short-term anticipation notes. There are
certain capitals there is certain money through that. So, what we have understood that municipal
bonds have a clear-cut objective that why the particular bonds are issued with some long-run and
short-term objectives.
(Refer Slide Time: 05:57)
Then if you look at the features of the municipal bonds that is very interesting. The major
importance of the benefits or the advantages of the municipal bond is the tax-exempt status.
Interest but not the capital gain is exempted from the federal income taxes. Right whatever
interest payment you are receiving periodically from these bonds they are basically exempted
from the tax. The states also exempt their own securities from state income taxes.
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So that’s why the municipal bonds are very attractive to the individuals and corporations in
higher income tax brackets and the investment funds whose clients are in the higher brackets.
Because they get a tax rebate and the tax-exempt status is involved with respect to that. This
market is one of the biggest markets in the US and quite a prominent market in the US.
(Refer Slide Time: 07:17)
So, in these cases what basically we have seen how these taxes are going to affect the return
from that particular bond? Let an investor is in the 35% tax bracket so that person would be
indifferent with other factors that remain the same are equal to a fully taxed bond yielding 10%
and tax-exempt bond yielding 6.5%. Alternatively, you can say if the yield on a tax-exempt bond
is 6.5% then the investor’s equivalent taxable yield on a fully taxable bond is 10%.
Equivalent Taxable Yield = Tax Exempt Yield
( 1 - Tax rate)
=
0.065 = 0.10
1 – 0.35
So how can we calculate the equivalent taxable yield that is tax exempt yield divided by 1 minus
tax rate. So here we have taken the tax-exempt yield to 6.5% and tax rate is 35% then your
equivalent taxable yield is 10%. That is 65% divided by 1 - 35%. That will give you 10%. So
that is the biggest advantage that we get the tax rebate against these interest payments that we are
receiving against these municipal bonds.
(Refer Slide Time: 08:47)
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So, if the federal tax rate increases, then the tax-except feature of the municipal securities
becomes more valuable. So, a tax increase or decrease would cause the demand and price of the
municipal bonds to increase and the yield to fall. If the tax face increases then the demand and
price of the municipal bond will increase and the yields are going to fall. So, the possibility of
changes in the tax rate, in turn, exposes the municipal investors to the additional risk that is
called the tax risk.
(Refer Slide Time: 09:46)
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So, there are some taxable municipal bonds also. There are some taxable municipal bonds, in
which the interest is subject to federal income taxes. Federal income taxes and these taxable
municipals or the municipal bonds in short, we use it as the municipals they sell at a lower price
and higher yield than the comparable tax-exempt municipals. What kind of bonds are and what is
the objective of issuing that type of the bond?
Generally, the types of taxable municipalities include those which are used to finance certain
projects like the sport facilities or private investors housing initiatives. In which there are federal
restrictions which limit or prohibit financing with tax-exempt municipal bonds. So, there are
certain kinds of activities what the particular municipalities or local governments carry out? So,
against that particular kind of project whatever bonds are issued they are basically taxable. So,
like your sports facilities and private investors and housing initiatives.
(Refer Slide Time: 11:26)
Already I told you there are major 3 types of municipal bonds. The municipal anticipation notes
are short-term and the general obligation bonds and the revenue bonds are the long-term.
(Refer Slide Time: 11:45)
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And there are different types of municipal anticipation notes. So, these short-term or
intermediate notes or the bonds are issued to finance their operations or the projects in which the
future revenue is anticipated. In which the future revenue is expected or anticipated. What are
those types? We have a tax-anticipation notes short we call it the TANs you have a revenue
anticipation notes or RANs, bond anticipation notes, grant-anticipation notes. tax-exempt
commercial paper, tax-exempt floating rate obligations.
So, these are the short-term notes which are issued by the municipals and the state bodies local
bodies to fulfill certain financing requirements.
(Refer Slide Time: 12:47)
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So, what we mean by these tax anticipation notes or revenue anticipation notes. These are
generally used to cover the regular recurring government expenses before taxes and other
anticipated revenues are received. And the BANs are used as temporary financing or
construction financing for the long-term projects. Like roads, university buildings and libraries
and all these things, with the principal paid from the proceeds from the sale of a long-term
municipal bond.
The tax-exempt CP has a maturity which is ranging from 30 days to 270 days. And generally, it
is secured with a bank letter of credit or line of credit or a purchase agreement in which the bank
basically agrees to buy the bond if the issuer fails. That was called LOC which was given by the
bank. It is the agreement in which there is assurance, if there is a failure of a repayment then the
particular bank is going to fulfill that particular requirement. That is basically we call it the taxexempt CP and it is backed by a certain kind of bank letter of credit.
(Refer Slide Time: 14:40)
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So, the general obligation bonds if you talk about already, we told that; this is the long-term
instrument or long-term bond which is issued by the municipalities or the local bodies. So, these
are long-term and intermediate term bonds generally they are secured by the issuing
governments general taxing power. So how are these particular bonds being; secured? They are
secured by the government's tax power.
If the general obligation bonds issued by the state and large municipal governments have a
number of tax revenue sources and unlimited tax power, they are referred to as unlimited tax
general obligation bonds. Generally, if the particular entity has a lot of tax revenue sources and
they have unlimited tax power then if those entities are issuing the bonds, then we call it the
unlimited tax general obligation bonds.
And generally, they are backed by full faith and credit of this issuer. Because these issuer’s credit
worthiness is quite high and they generally the public have the full faith or the investors have the
full faith on these particular bodies. Because of their volume, because of their size and as well as
because of their taxing power and they have also the number of tax revenue sources. So there, it
is generally backed by the full faith and the credit of the issuer.
But if the general obligation bonds which are issued by the smaller municipalities whose
revenues are limited like property tax or who have the statutory limits on the tax rate that the
issuer may always levy to finance the debt are known as the limited tax going obligation bonds.
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Because those organizations, those entities have not that much ability to raise the revenue in
terms of the tax. And they do not have that much power also which can increase their tax
revenue at the time of requirements. So, these are called the limited tax general obligation bonds.
(Refer Slide Time: 17:59)
So, whenever we talk about the general obligation bonds the bonds agreement or the indenture is
generally accompanied by 2 things. One is your official statement and the other one is the legal
opinion. What is the official statement? Official statement is basically a document that is more or
less similar to the prospectus for a stock or a corporate bond just in the previous class we
discussed about this public issuance of the corporate bonds.
So, it is almost similar in that. It will generally provide the return risk and the other
characteristics of the issue and also provides the information about these words. What are those
information which are there in the official statement? This information are generally the amount
of the issue, credit rating, the names of the underwriters, who are the selling group, sources of
the payments, sources and uses of the fund statement, financial statements, debt service and
notice of any pending legislation and the bond insurance.
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All kinds of details about the issuer and about that particular issue will be mentioned in the
official statement and that official statement is nothing but it is a similar instrument or similar
document like the prospectors.
(Refer Slide Time: 19:58)
The other one is the legal opinion and what is the legal opinion? This is another document that
generally will provide the legal issues related to the bonds collateral. What is the security this
particular organization has used for these ones of the bond? And what is the legal aspects or legal
issues related to that collateral, priority of the claims? All kinds of things basically they will get
from the legal opinion document.
So, whenever we evaluate the credit worthiness of the general obligation bond. The investors
need to review the legal opinion to determine certain things. What were they were trying to
determine from the legal opinion? The extent of the state or local governments unlimited taxing
power, how much power they have in terms of levying the tax, possible statutory or the
constitutional limitations on the judiciary the taxing power, whether they are under the control of
this particular taxing power or not it is under their control or not whether they want to increase or
decrease the tax. That jurisdiction is very much important from that perspective. The priority of
the claims on the general bonds, the bondholder’s redress in case of a default, any statutory or
the constitutional questions that could be problematic.
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So, all kinds of things have to be checked upon or can be determined from this legal document.
So, if you look at these particular entities who are issuing this bond, they have a lot of power or
they are kind of under their jurisdiction this tax and all these things can be changed. Then what
will happen is that the investor will have more confidence in that particular type of issuer.
(Refer Slide Time: 22:37)
Then we have the revenue bonds. So, what are those revenue bonds? Revenue bonds are
generally the municipal securities paid by the revenues generated from the public or quasi-public
projects. Proceeds from a specific tax or by specific assessment on an existing tax. Some
revenue bonds are backed by the specific revenue sources as well as the general revenues. They
are called the double-barreled bonds. They are called the double-barreled bonds and there are
certain bonds which are basically dedicated tax brackets or dedicated tax backed revenue bonds.
So, these bonds generally are paid from the dedicated revenues like your tobacco settlement,
lottery or specific fee from which the particular money will be paid or the payment of those
bonds should be possible only from specific sources. So, some revenue bonds are secured by and
paid from more than one revenue source not necessary that every bond has a single revenue
source. Some revenue bonds are secured and paid from more than one revenue sources that also
we can observe certain times.
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So that’s why this is little bit different from the general obligation bonds and both are issued for
the long-term project financing but not necessary that the objective of these bonds or issuance of
these bonds are the same. So, the revenue bonds are generally issued with some specific
objective and they are backed by certain specific revenue sources. But the general obligation
bonds may not always be backed by some specific revenue source. So that is the difference what
basically, you can observe between these 2.
(Refer Slide Time: 25:26)
So, the question here is that about the revenue bonds whenever we discuss the revenue bonds if
you look at the important provisions that are delineated in the indenture and the legal opinion.
Which include what are those whether the issuer can increase the tax or users free underlying the
revenue source. Whoever is issuing the bond whether they have the power to increase the tax or
they can increase the fee.
Whether the issuer can incur additional debts secured by the revenue of the project or under what
conditions the new debt can be incurred. How the revenues of the project are to be directed like
net revenue, structured revenue bond in which the bond holders are paid after operating expenses
but before other expenses. And gross revenue structured bonds in which the bondholders are paid
first.
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So, the net revenue structured revenue bond and the gross revenue structure bond these are the
difference in terms of when the payment will be made. So, in the first case or in case of net
revenue structured revenue bond the bond holders are paid after operating expenses but before
the other expenses. But whenever you talk about the gross revenue structure bond there the bond
holders are paid first in comparison to the other stakeholders.
Another thing also they look at whether there is any additional collateral or guarantees which are
linked to that particular bond. Whether the particular bonds are insured or is there any other
guarantor of that particular bond. That also can be looked upon whenever we discuss about this
revenue bonds.
(Refer Slide Time: 27:42)
Then the next question is that revenue bonds are important sources of funding for a number of
public projects. And those projects are basically roads, breeze tunnels, airports, hospitals, power
generating facilities, water treatment plants, municipal buildings, university buildings,
educational programs, inner-city housing development and student loan programs. So, all types
of funding are done through the issuance of these revenue bonds; so, this has quite relevance,
quite important in the US market in terms of financing this kind of activity.
(Refer Slide Time: 28:30)
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So now the question is that from where the money comes to pay the interest and the principals. If
you look at the revenue bonds the revenues which are or which has to be generated to pay the
interest and the principal, these are basically usually project specific. These are project specific
and what are those kinds or types of the income or the revenue; what they are generating; what
are those; they can be in terms of tolls, they can be in terms of rent, user charges, the Earmarked
revenues from the fees, specific taxes. All kinds of things are basically the sources of revenue for
payment of the interest and the principal. And there are many types of revenue bonds. There are
n numbers of revenue bonds which are traded in the market. That we will discuss further. But
these are the major sources of revenue. What the municipality basically uses to pay their interest
and the principal payments.
(Refer Slide Time: 29:57)
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So, what basically we discussed today, the municipals and other local governments sell the bills,
notes and bonds to finance their long term and short-term projects. So, there are 3 types of
municipal bonds: - municipal anticipation notes, general obligation bonds and revenue bonds.
And the common feature of the municipal securities is their tax-exempt status. The
characteristics and the purpose of issuance of the municipal bond vary across the types of the
bonds.
The revenues used to pay the interest and principal payments on those bonds are usually project
specific. So, these are the things what we have discussed in today's session.
(Refer Slide Time: 30:49)
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So, you can go through these references for the detailed discussion on this.
Thank you.
522
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 07
Lecture No # 33
Municipal Bonds – II
Welcome back!
So, in the previous class we started the discussion on the municipal bonds. There we have seen
that municipal bonds are a kind of specific bonds which are issued for a specific purpose. And
the revenues are generated through the different kinds of fees, taxes and as well as the tolls and
all these things. And it has a lot of relevance in terms of the economic growth process and we
have seen that there are 3 types of bonds.
So, one is your anticipation notes then we have the general obligation bonds; then we have the
revenue bonds. And we have discussed about the general obligation bond and the anticipation
notes and we started the discussion on the revenue bonds. Then will carry out our discussion
with respect to the revenue bonds because the revenue bonds are generally issued for some
specific reasons.
(Refer Slide Time: 01:20)
So today will be discussing about; the types of revenue bonds. And there are some special types
of municipal bonds also issued by the local bodies. What are those and also the different aspects
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of the municipal bond market. So, there are 3 concepts what we will be discussing in today’s
session.
(Refer Slide Time: 01:50)
Let us see that what are those keyword’s, there are many keywords will come across there are
some words have been highlighted here. We have highway revenue bonds, water and Sewer
revenue bonds, airport revenue bonds and industrial development bonds. Then there are certain
specific types of or the special features like municipal floaters and there is a kind of serial
municipal bonds which are also issued in that particular case.
That also you will come across in that particular case or context of the municipal bonds. Let us
discuss one by one. What are those basic features of the different types of revenue bonds which
are issued in the municipal bond market.
(Refer Slide Time: 02:40)
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One is that if they call it the highway revenue bonds the name of the bond is the highway
revenue bond. And these bonds or we can say that revenue bonds. These are used to finance the
highway systems and their related infrastructures like bridges, tunnels and all these things. And
there is a huge expenditure incurred in that particular context. And it is quite long term in nature.
There is a long-term financing is required for these highway revenue bonds.
So, these are basically paid by the toll charges, license fees, auto registration fees and all these
things which are the revenue sources from these projects. Once the projects are completed the
revenue generation will start and the bondholders will be getting their coupons and as well as the
other cash flows from this kind of source. So, these particular bonds are paid by the toll charges,
license fees, auto registration fees etc. So, the major source of the revenue of this bond is these.
Then we have the water and sewer revenue bonds. Water and sewer revenue bonds basically
what? They are basically issued to finance the building of water treatment plants, pumping
stations and the sewers. The sewerage system also developed through this. Generally, the local
governments or the special bond authorities who are involved in this kind of projects generally
usually issue them.
These bonds are usually paid by the user charges because we are using the water for that we pay.
We are using this other facility with respect to that water sewerage facility, in other facilities. For
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that these are certain fees and from these fees or the user charges, the interest and the principal of
this particular bond are paid.
Then we have the type bond called lease rental bonds. What is the lease rental bond? These
bonds are used to finance the construction of the public office buildings, stadiums, university
facilities and the purchase of computers and other types of capital equipment. And how the
particular bond is paid? How the bondholders are paid? They are generally paid by the rents
generated from the users, tuition fees, annual appropriation of the general fund and the stadium
receipts.
Because they have made the stadium if any kind of activities happen in the stadium for that they
charge certain rent. And some other kind of receipts they also make day to day activities which
generally carry out in that and from that the particular money will be paid to the bondholders. So
that is called the lease rental bonds.
(Refer Slide Time: 06:42)
Then we have the hospital revenue bonds. You see in the US almost all types of infrastructure
are financed through these kinds of bonds. So, these bonds are used to build or expand the
hospitals, to purchase the medical equipment and so on which are related to the health sector.
And the fees are the major source of the revenue. The medical expenses what these individuals
make whenever they try to access this particular service from that money will be paid to
bondholders.
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There is a specific bond also in the US market they offer that is called the airport revenue bonds.
And that airport revenue bond generally used the construction, expansion and improvement of
the airports. And these bonds are secured by the lease with the major airlines for the use of
terminals or by the revenues obtained from the landing fees and the fuel fees paid by the airlines
and the concession fees paid by the terminal store users.
All types of activities which are linked to this airport business like the terminal fees; use of the
terminals for that they pay, landing fees, fuel fees, terminal stores whenever the different airlines
industry use those terminal stores. And also, the different kinds of commercial activities which
happen in the airport and those particular kinds of activities also involve certain kinds of revenue
generation. They give the rents from that generally money is paid to the bondholders.
Then we have the industrial development bonds. State and local governments also sell the
industrial development or the revenue bonds to finance the expansion of the area’s industrial
base or to attract new industries. So, these bonds have been generally used to finance industrial
parks, electric-generating plants like that. So basically, for the industrialization for the expansion
of the industry the money which is raised through these bonds are basically used. That is called
the industrial development bonds. So those bonds are generally used for making more industries
within that particular area.
(Refer Slide Time: 10:13)
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Then we have in the US they have called the lottery bonds. What is your lottery bonds? These
are basically secured by expected future lottery revenue. Mostly they are used to finance the
construction of new school facilities. These bonds are basically issued to construct the new
school facilities in the US.
Then they have the pollution control bonds. These bonds are used to help the companies or the
corporations to purchase the pollution control equipment. A municipal government will buy the
equipment through the sale of these bonds and lease this equipment to the different companies.
And from the leasing generally they raise the revenue and the money will be paid back to the
bondholders or the investors. Because the corporations are not eligible to issue the municipal
bonds and if they will issue, they will be considered as a corporate bond.
And they do not have any tax-exempt status. But if the government revenues or government
entities are going to issue that particular bond then it will be considered as a municipal bond.
And finally, some tax exemption this investor can get so that’s why they can attract more
investors with respect to that, so the equipment will be bought by the government or the local
bodies. Then it will be given to the different companies who are involved in this kind of activity
and finally they get some money from them.
Then they have resource recovery revenue bonds. These bonds are generally financing the
resource recovery operations that convert their solid waste into commercially recoverable
products and landfill residue. It is a very important bond because solid waste management is
quite important because that problem is rising across the globe. So, to make a particular thing
more viable or to remove that particular kind of problem they always were trying to use that
solid waste to some commercially viable products.
And the revenue bonds basically help in that particular record. So, the revenues which are from
these operations generally come from the fees for delivering the garbage or sales from the
products which are generated. Once the products are generated, they will be sold in the market
from that they will get certain revenue and also for delivering the garbage to a certain place they
charge certain money and from that they try to pay to the bondholders. That is called the resource
recovery revenue bonds.
(Refer Slide Time: 13:57)
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Then they have the public power revenue bonds. These bonds are generally used to finance the
construction of electricity-generating power plants and the distribution systems. They may be
issued to finance the construction of one or several power plants. So, in that case we call it joint
power financing.
Sports Complex and Convention Center bonds, these bonds are issued as permanent financing of
sports stadiums and convention centers. These bonds may be lease-rental bonds paid for by the
rental income from the facility or they may be paid from the revenue generated from outside
revenue sources like the local hotel taxes or city or the country taxes. So whatever rents and
whatever kind of other sources; revenue sources they have through which the bond holders
basically will be paid off.
Then they have life-care revenue bonds. These bonds are issued by the state and local
development agencies to finance the construction of long-term residential care facilities for the
elderly people. And these are generally managed by the non-profit agencies and the religious
groups. And what is the revenue source? The revenues basically supporting these bonds are
generated from the lease rentals and the lump-sum payment made by the residents.
That is called the life care revenue bonds. So, to provide a certain kind of social security and
shelter to the elderly people and this kind of projects generally carried out by the non-profit
organizations. And the revenues which are generated generally are generated through the lease
and as well as the lump-sum payments which are made by the residents.
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(Refer Slide Time: 16:35)
Then we have multi-family mortgage revenue bonds. There are many revenue bonds the U.S
market always offers. So, in-case of the multi-family mortgage revenue bonds. These bonds are
generally used to finance the multi-family structures for low-income families and the senior
citizens. And they can have single family mortgage revenue bonds. These bonds are used to
secure mortgages on single-family homes which are insured by the Federal Housing
Administration, Veteran Affairs or private mortgage insurance.
Then they have a section 8 bond. These bonds are generally issued to finance the low- and
middle-income rental housing. Some middle income or the low-income group want some houses
on rent, and some of the local bodies try to build these kinds of buildings and the houses and they
give on rent. And the particular bond will pay from the rents whatever they will raise from these
group of the people who are staying in that particular house.
(Refer Slide Time: 18:17)
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Then we have college and university revenue bonds. The Education bonds which are quite
popular in the U.S. These bonds are used like a permanent financing of the college buildings,
libraries, classroom buildings, dormitories, all these things. And other university capital projects
like computers and networks and all these things. And from where the revenue is generated? The
revenue is basically generated through the tuition fee and the rental fees and there is a special fee
required.
So, these also sometimes fall under the lease-rental bonds what we have already discussed.
Then we have the student-loan revenue bonds. These are generally issued by state government
agencies with the proceeds used to support the loan to the college students and how basically it
will be paid and what is the source of revenue? These proceeds from these bonds generally are
used to purchase the federally guaranteed student loan made by the local banks. And once the
loan will be repaid then accordingly the bond holders will get their money in terms of the
coupons and other types of cash flows.
Then they also issue a bond called the tax allocation bonds. So, these bonds generally are issued
to finance the office and property development in low-income areas and the bond holders are
paid from property taxes that are expected to increase from improved real estate values. That is
called the tax allocation bonds.
(Refer Slide Time: 20:40)
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Then there are certain municipal bonds having the special feature. Already I told you that there
are serial bonds, there is an insured bond, secured bonds, legally binding bonds, municipal
floaters, municipal put options or put bonds, municipals with warrants, mini-bonds, and stripped
municipals bonds. So, these are the different other types of bonds for the municipalities or other
local bodies that can also issue for some specific reason or specific purpose.
(Refer Slide Time: 21:30)
So, what do we mean by this? Whenever we talk about the serialization. What does it mean?
This particular thing refers to the breaking up of bond issues into different maturities. For
example, to finance a 40-million-dollar convention center, a particular government local
government might sell a serial issue with 4 types of securities, each with a face value of 10
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million, but with one maturity in 5 years, another one is 10 years, third one is 15 years and the
other one is 20 years.
The maturity of the bonds has been divided that is called the serial issues. The municipalities
also sell the zero-coupon bonds where the generally the particular bond will be matured at the
end it will be issued at discount and redeemed at the bar. There are no periodic payments are
involved in that particular case and it is also very convenient for them. Because they have a
reasonable period of time to generate certain resources or revenues from the activities what they
are going to carry out.
The municipal governments and authorities also sell a variation of a zero-coupon bond. That is
known as the municipal multiplier or the compound interest bond accretion or the compound
interest bond. These bonds pay the coupon interest which is not distributed. It is not given to the
bondholder. But rather reinvested to the bond’s maturity. Are you getting the point? The coupons
are there and the cash flow is involved but it will not be given to the bondholder.
Instead of that this particular coupon amount will be reinvested in the market and whenever there
is a maturity the total amount will be repaid to with the reinvestment of the coupons, the money
will be paid to the bondholder. So that’s why that particular characteristic makes this particular
bond similar to a zero-coupon bond. That is called the municipal multiplier or the compound
interest bond.
(Refer Slide Time: 24:38)
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Then there are some municipal bonds are insured. The insurers basically write the insurance
policies in which they agree to pay the interest and principal to the bondholders when the issuer
fails to do so. That means the bond is insured and the municipalities is basically paying the
insurance premium. It is not the bond holder who is going to pay the insurance premium. It is the
bond issuer like the local bodies, local government who are going to basically pay this insurance
premium.
So here it is more or less a kind of secured kind of instrument and here the security is the
insurance and the letter of credit backed by the municipal bonds. These are secured by the letter
of credit from the commercial bank. The commercial banks can give this kind of services kind of
guarantee through this letter of credit. And if this issuer fails to pay the bond holder, then the
bank is going to pay for that.
Then we have refunded bonds. These bonds are secured by an escrow fund. And what is the
escrow fund? This basically consists of high-quality securities like treasury bonds, federal
agency bonds and all these things. And there are also refunded municipal bonds backed by an
escrow fund consisting of a mix of treasury bonds and non- treasury bonds such as the municipal
bonds. Another municipal bond also can be a part of that particular escrow fund that is called the
refunded bonds; these are backed by the high-quality funds.
Then the legally binding bonds. These types of bonds generally take the form of an obligation to
withhold or provide the government aid to pay any defaulted debt issue. The government can
help them out to pay the interest and other kinds of cash flows if there is any default. So that is
called the legally binding bonds. That provision will be there in the indenture in the beginning
and using that particular provision the government may be ready to pay that. That is called the
legally binding bonds.
(Refer Slide Time: 27:39)
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Then we have municipal floaters. Here the municipal bonds with floating rate. This is basically
the coupons are not fixed, coupons are floating. And that floating rate is generally linked to a
reference rate like the other corporate bonds and to whom they are linked. They are linked to the
treasury bills rate or the libor rate (London Inter Bank Offer Rate) or also they can be linked to
the return what we are getting from a municipal bond index which is traded in the market. So, the
base rate or the reference rate can be linked to either Treasury bill rate or the libor or it is the
municipal bond index.
Municipal bonds also can have a put feature that means that bonds can be cashed at a specific
value before the maturity if the investor wants. Municipal bonds also can have the warrants that
means that allows the holder to buy additional bonds at set prices.
Then they have mini bonds. They generally have low denomination issues like 100-dollar, 5,000
dollar and one thousand dollars like that per value that are sold directly to the public without the
investment bankers’ interference. Then we have the stripped municipals their interest only or
principal only stripped municipals that part we have already discussed whenever we discuss
about the corporate bonds and the government securities. The municipal bonds can also be
stripped bonds.
(Refer Slide Time: 29:44)
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If you look at the market, in the primary market the sale of the general obligation bonds and
revenue bonds are handled by the investment banker or a syndicate of commercial banks and
dealers who can underwrite this issue and resell them in the open market. Traditionally, the
selection of an underwriter or syndicate was done on a competitive bid basis, with many states
requiring the general obligation bonds to be marketed with the competing bids.
Because of complexities with municipal bonds, more underwriters are being selected through the
negotiation. And the information on upcoming municipal bond sales can be found in the bond
buyer. And the bond buyer is basically a publication of the municipal bond industry. The book or
online subscription to its website generally provides information about the future bond sales and
the results from the recent sales.
(Refer Slide Time: 30:51)
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In the secondary market municipal bonds are primarily traded in the OTC market through the
municipal bond dealers. And the local banks and regional brokerage firms often handle the issues
of smaller municipalities generally call it the local credits. Why the larger investment companies
and municipal bond departments of larger banks handle the issue of the larger governments that
are called the general names.
Many dealers in the secondary market make bids and ask quotes in terms of the yield to maturity
or yield to call. The information on trading and prices can be always obtained from the Securities
Industry and Financial Market Association (SIFMA) via their website.
(Refer Slide Time: 31:45)
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So, what basically we discussed that the revenue bonds are classified on the basis of their uses
and the sources of payments of these bonds and they also vary across the types of the revenue
bonds. The sale of the general obligation bonds and some revenue bonds are generally handled
by the investment bankers in the primary market and in the secondary market the municipal
bonds are primarily traded in the OTC market.
(Refer Slide Time: 32:14)
So, these are the references you can go through.
Thank you.
538
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 07
Lecture No # 34
Debt Securities of Commercial Banks
Welcome back. So, in the previous class we discussed about the different types of securities
which are issued by the corporates and as well as the governments. Then we discussed about
certain securities which are issued by the municipalities or the local bodies, local government
bodies. And all those instruments which are issued by the municipals or generally we call the
municipal bonds are issued basically for some specific purpose.
And the revenue or the use of those kinds of securities vary on the basis of the nature of the
instrument which is issued in that particular market. Then today we will discuss the certain
securities which are issued by the commercial banks or the debt instruments which are issued by
the commercial banks.
(Refer Slide Time: 01:14)
So, as you know that the commercial bank is one of the most important financial entities in the
financial system and they generally play the significant role in terms of issuance of certain kinds
of long-term and short-term securities. Particularly the debt instruments for financing the
539
working capital requirements of the companies. And also, the financing of some long-term
investments for the companies.
So, if you look at the different types of instruments which are issued mostly if you talk about the
bond type of instruments or the characteristics; which are very much compatible with the bond.
These are basically the certificate of deposits, bank notes and the banker’s acceptances. And as
you know that the banks are always known for providing the loans to the corporate sector for
their short-term and long-term needs.
(Refer Slide Time: 02:19)
So here you will come across a certain kind of keywords like non-negotiable certificate of
deposits, negotiable certificate of deposits, prime certificate of deposits, non-prime certificate of
deposits like that. Then you have certain certificates of deposits like jumbo CDS, floating rate
CDS, bull and beer CDS, leveraged loans and all kinds of keywords basically we will come
across while discussing the different types of instruments which are issued by the commercial
banks.
(Refer Slide Time: 02:56)
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So let us discuss this largely the debt instruments, the debt instruments if you talk about from the
commercial banking perspective; that already told you these are mostly the certificate of deposits
bank notes, banker's acceptances and the bank loans.
(Refer Slide Time: 03:16)
So first we can start with the certificate of deposit. What do you mean by a certificate of deposit?
If you recall that whenever you talk about the short-term instrument of the companies issues
generally those are called the commercial papers. These are also the short-term instruments and
those short-term instruments the company basically rises to finance their short-term needs like
your working capital financing.
541
So here the certificate of deposit is also the same type of instrument which are issued by the
banks to fulfill certain kinds of short-term requirements. So, what exactly the certificate of
deposit means it is a kind of certificate which is issued by the bank or any financial institutions,
certifying that the specified some of the funds has been deposited at the issuing depository
institutions.
And who are basically issuing this kind of instrument? These instruments are basically issued by
the banks and some of the other thrift institutions which are operating in the particular financial
system. And generally, those CDs are finance their loans and generally they offer these CDs to
finance their loans and the investments. So, the CDs is also like other financial instruments, they
have a specified maturity date and interest rate and can be either negotiable or non-negotiable.
So, what do you mean by the non-negotiable CDs; the non-negotiable CDs basically are the
deposits that cannot be transferred and typically must be held to maturity. And often a penalty is
applied for early withdrawal if anybody wants to withdraw that particular deposit beforehand
then some penalty will be imposed on them. And they can be issued in any denomination. So, the
non-negotiable CDs are held up to maturity largely. And why basically it is held up to maturity
because if there is an early withdrawal then the penalty will be imposed upon them.
The negotiable CDs are higher denomination certificates typically issued to the different
institutions and this can be sold to the other investors also in the secondary market. Both
negotiable and non-negotiable CDs; Sorry the negotiable CDs basically can be invested in the
secondary market but the non-negotiable CDs are not.
So, in that case there is a proper market for the negotiable CDs but particularly the secondary
market but for the non-negotiable CDs secondary market is basically not existing.
(Refer Slide Time: 06:24)
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So here if you look at the different features of the certificate of deposits in India. So here already
what we have discussed is that the CDs can be issued by the scheduled commercial banks
excluding this RRBs (Regional Rural Banks) and also the local area banks. So, they are not
eligible to issue the certificate of deposits but it can be issued by some of the all-India financial
institutions.
All India financial institutions means they are basically established for some specific purpose for
example NABARD; NABARD is all India financial institutions like that there are some of the
all-India financial institutions who are eligible to also do the issuance of this particular certificate
of deposits. And banks have the freedom to issue the certificate of deposit depending upon their
funding requirements but the minimum amount of CDs should be 1 lakh.
The minimum amount of CDs would be 1 lakh that means the minimum deposit that could be
accepted from a single subscriber should not be less than 1 lakh; minimum deposit should be 1
lakh rupees. And it can be a multiple of 1 lakh thereafter so the minimum amount of deposits
against this particular CDs has to be 1 lakh rupees.
And who are those investors who are basically buying these securities or holding these securities.
If you look at this investment perspective generally these investors can be individuals, it can be
the corporations, it can be the trust, it can be the other kind of funds, NRIs so all kinds of entities
are eligible to invest in the CDs. So, the CD market is almost open to all types of individuals
which are existing within that particular system.
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And what is the maturity period of the CDs in India? The maturity period of CDs issued by the
banks should not be less than 7 days and not more than 1 year. So that means the maturity period
of certificates of deposits in India which are issued by the banks generally vary from 7 days to 1
year. But if it is issued by the all-India financial institutions then they can issue the CDs for a
period not less than 1 year but not exceeding 3 years.
That means the maturity period of the CDs which are issued by the other financial institutions
like all India financial institutions there the maturity period is 1 year to 3 years. And CDs
generally are issued at discount and redeemed at the par. So, CDs may be issued at a discount on
the face value. So that is another characteristic also you can add or another feature you can add
whenever you talk about the feature of the certificate of deposits in India. These are the different
features that we see whenever certificates of deposits are invested in the Indian market.
(Refer Slide Time: 09:55)
So, another thing also if you see all CDs are subject to the cash reserve ratio and the statutory
liquidity ratio requirement. All CDs are subject to the CRR and SLR requirement on the issue
price of the CDs. But one thing the banks and the financial institutions cannot grant the loans
against the CDs or also cannot buy back their own CDs before the maturity. That means if you
are using or you are trying to use CD as a collateral for getting the loan from any kind of bank
then it is not allowed or the same bank who has issued the CDs, they cannot basically buy back
their own CDs before the maturity.
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So, CDs in India generally freely transferable by endorsement and the delivery. Time to time the
guidelines for the issuance of CDs with respect to the maturity or size or denomination can be
modified and the regulatory body of this kind of process is the reserve bank of India. Mutual
funds also are allowed to invest in CDs with certain limits but which are fixed by the securities
indexing board of India (SEBI).
There also can have some components they can invest in the certificate of deposits but with the
limit whatever has been stipulated by the savings so these are some of the features of the
certificate of deposits with respect to the Indian market.
(Refer Slide Time: 11:52)
Let us discuss something about the features of certificate of deposits in the US market. For a
comparative reason we have discussed the CDs in the Indian market now we come to the CDs in
the US market. Is there any difference or not? If you look at the maturity, the maturities on the
negotiable CDs generally range from 3 to 18 months in the US market. But it has been observed
that most of the CDs which are issued in the US market have a maturity period either 6 months
or less but it can range from 3 months to 18 months.
Little bit of the maturity period with reference to India if you look at our banks it varies from 7
days to 1 year; in the US market the maturity period is relatively different. The CDs whose
maturity period is greater than 1 year are called the term CDs in the US markets. The CDs issued
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with the maturity greater than 1 year are called then only the term certificate of deposits in the
context of the US.
CDs are basically the interest-bearing notes usually sold at their face value with the principal and
interest generally paid at the maturity. If the CD is less than 1 year and the payment will be
semiannually if it is a term CDS. So, if the maturity period is less than 1 year then obviously all
the payments will be made in the end for that particular maturity but if it is more than 1 year then
the semi-annual interest payment can be possible in that particular context.
If you look at the general trend of the yield of the CDs, yields of the CDs generally are larger
banks, generally if you look at yields for the CDs of the larger banks. Generally, they call it the
prime CDs tend to be lower than the CDs which are issued by the smaller banks which are called
the non-prime CDs. So, because the larger banks are considered as the most secure banks or they
have from the resource point of view or resource allocation point of view and the utilization
point of view they have the advantage over the smaller banks.
So that’s why the CDs which are issued by the large banks are called the prime CDs and the CDs
which are issued by the small banks are called the non-prime CDs. But obviously because of the
less risk and the high security the yields of the CDs of the larger banks is lesser than yields of the
CDs of the smaller banks, so that is another thing what we can observe from the US context.
(Refer Slide Time: 15:07)
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Further also, if you see that the minimum denomination of the negotiable CDs which are traded
in the US market is 1 lakh dollar ($100,000 dollar), in India we have 1 lakh rupees and they have
1 lakh dollar. With the average denomination being 1 million and there are also the called the
jumbo CDs and by name itself there the face value of these particular CDs is 10 million dollar
and more.
So, if the face value of any kind of CDs is going beyond these 10 million dollars and more, then
the US market they call it the jumbo CDs. Like other bank savings and the loan deposits the
federal deposit insurance corporation that FDIC generally they go for insuring the CDs up to the
1 lakh dollar against any kind of default. So, there is a limit that has been fixed: the FDIC
provides the insurance up to the 1 lakh dollar against any kind of default risk.
If there is any default for the payment of the principal or the interest in the future so to cover up
that particular default risk, they can go up to insurance of 1 lakh dollar. Most negotiable CDs has
been observed in the US market to have denominations exceeding this 1 lakh dollar and therefore
there are some default risks involved in that. Because there is coverage up to 1 lakh but if the
CDs which are issued whose values face values are more than 1 lakh dollar, so because of this
some amount of default risk is involved with respect to that.
So, if you look at the yield on CDs the return on CDs generally reflects the risk of the swing
financial institutions and the maturity of the CDs, the yield generally depends upon that the risk
and as well as the maturity. So, the CD yields tend to exceed the rates on the treasury and the
short-term federal agency instruments. That means the yield rate of the certificate of deposits are
more than the yield of the treasury bills and the other short-term instruments which are issued by
the federal agency in the US.
What is the reason, the reason is basically it is considered as the certificate deposits are the
riskier instrument than the instruments like treasury bills and the other short-term instrument
issued by the federal agency. So more the risk then the risk premium should be added already
extensively we have discussed about the term structure theories. There you might have seen that
even though with the same term to maturity the return may be different because of the risk. Or
maybe it is also kind of we can say that with the different maturity obviously the interest rates
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will vary. The long-term interest rates are more than the short-term interest rate because of the
risk involved in that part already we have discussed.
(Refer Slide Time: 18:19)
So, the rates that banks pay on the CDs are quoted on a 360-day basis in the US everything they
mostly there are 2 ways. Basically, we go to the day count convention. One is your actual by
actual; actual by 365 another one is 30 by 360. So, whenever the rates we calculate against the
CDs they generally quoted on a 360-day basis instead of 365 days.
What is the example; for example, if you see an investor buying a one-million-dollar CD which
is maturing in 180 days and paying a 6% interest then they will receive 1,030,000 dollars from
the bank at the maturity. So how we calculate this is 1,000,000 dollars + 1 +0.06 into 180 by
360. So here we are basically considering the basis of a 360 days, 30 by 360-day count
convention so effectively they will be getting 1,030,000.
$1,000,000 [ 1 + 0.06 (180/360)] = $1,030,000
What if we look at the quoting of the CD trading in the secondary market if you look at that
aspect is little bit different than the rate. How it is different if you look at here, the quotes of CDs
trading in the secondary market are made in terms of the bank’s discount yield. So, if the CD is
yielding 6% while trading in the secondary market then what is the rate of that rate will be
calculated in this way that is your 1,030,000 – 1,000,000 divided by 1,030,000 into 360 by 180.
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Rate = $ 1,030,000 - $ 1,000,000 (360/180) = 0.058 = 5.8%
$1,030,000
So effectively the yield basically the discount yield what we are calculating in this case is 5.8%.
So, if the CD which is trading at, if the CD is yielding at 6% then if they will be traded in the
market then the rate will be quoted as 5.8%. So that is the way the secondary market quotes are
basically reported with respect to the CD of the maturity of 180 days in this particular example.
(Refer Slide Time: 21:02)
So then other things if you look at with respect to the US market that the money market funds
banks, bank trust department, state and local governments, foreign governments, central banks
and corporations who are the major investors in CDs. And some smaller negotiable CDs are held
by the smaller investors also. But the CDs market if you look at the investment point of view
then mostly the investments are made by the very large organizations like banks and the
corporations and all.
So, if you look at the general features of the CDs are generally held up to the maturity generally;
the investors hold their CDs until the maturity period. So once somebody is holding the CD up to
maturity then obviously the secondary market for these instruments is not active like the treasury
bills market. So, what basically does it mean because if nobody is going to redeem that particular
CDs before the maturity or they are not going to sell that security before the maturity, then
obviously the secondary market trading is not happening in that particular context.
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For example, if you are holding a bond up to 10 years maturity, if you are selling the bond at the
end of the 5 years and 6 years then only you are taking the help of the secondary market. And
accordingly, the total return and all these things are going to be changed but if you are holding
up to the maturity obviously your return what you are expecting that you will be getting that total
return will be the same.
And in that particular context the availability of the secondary market is not possible. So that is
basically the general characteristics what we have observed with respect to the certificate of
deposits market in the US or as well as in India. The existence of an active secondary market
means that if the yield curve were positively sloped and did not change an investor could earn a
higher rate higher than either shorter- or longer-term cd.
By buying the longer-term CD and selling it later in the secondary market at the higher price
which is associated with the lower rate on the short-term maturity that is possible if you have a
proper active secondary market with respect to that instrument. But that is not prevailed or that is
not happening with respect to the certificate of deposits market.
(Refer Slide Time: 23:54)
Then if you see that let if you talk about one example how basically this is possible with respect
to the secondary market. Let there is a 6-month CD which is yielding 5%, then what is the p or
price; price is equal to 100 by 1.05 to the power 0.5 so you will get a price of 97.59. Now let 1
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year CD be yielding 6% then what is the price then price is equal to 100 by 1.06 you will be
getting 94.3396.
Then in that case what the investor can do the investor; can buy the one-year CD which is at a
price of 94.3396, hold it for 6 months, keep it with him for 6 months and sell it for 97.59
assuming that the yield curve did not change. Then what will happen obviously his annualized
yield will be 7% how the annualized will be 7% because it will be 97.59 divided by your 94.3396
to the power 1 by 0.5 – 1, 1 by n that basically -1 that will be giving you 7%.
6-month CDs yielded 5%: P = 100/ (1.05)0.5 = 97.59
1-year CD yielded 6%: P = 100/1.06 = 94.3396
R = (97.59/94.3396)1/0.5 – 1 = .07 or 7%
So, this strategy any investor can adopt if you have an active secondary market which is
prevailing in this particular system. So, this strategy is generally referred to as the trading down
the yield curve then we call it trading down the yield curve. But that sometimes is not possible if
you do not have an active secondary market with respect to that instrument.
(Refer Slide Time: 25:52)
Then we have some special features of CDs like floating rate CDs in the US market particularly
the maturity of this floating rate CDs will range from 18 months to 5 years and the coupon rates
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are reset periodically to the rate on a comparable CD rate or the London interbank offer rate or
the libor rate. Already the floating rate bonds calculation we have discussed before and how the
rates are reset on a periodical basis on the basis of a benchmark generally the rates are reset.
So here the benchmark is libor or it can be a comparable CD which is available in the market
with respect to that maturity. You can have a bull and beer CDs these rates generally the rates on
these CDs are tied with the stock market indices so on the basis of the return that we get from the
popular stock market indices accordingly the rates are decided for these CDs. The rising rate
CDs longer term CDs with gradually increasing the rates more the maturity period more the
return.
Then you have the forward CDs here are the contracts to buy the CDs in the future. So, these are
the some of the CDs which are available in the US market, what we call the CDs with some
special features which are traded in the or which are available in the US market.
(Refer Slide Time: 27:22)
Then another instrument what they basically provide that is called the banknote. So, in the US
the bank notes are similar to the medium-term loans; they are generally sold as a program
consisting of a number of notes with different maturities, generally they are typically ranging
from 1 to 5 years. And offered either continuously or maybe there can be a discontinuity between
that.
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So, bank notes are usually sold to the institutions in high denominations. The value can range
from 5 million dollar to 25 million dollars with the total offering ranging from 50 million to 1
billion dollars. Banks also sell the bank notes through the international syndicates as a part of the
euro capital market the banks also sell this type of notes through some kind of international
agencies or international bodies or international associations or some international syndicates.
(Refer Slide Time: 28:29)
Then we have another instrument that is called the banker’s acceptance. So, these are basically
the time drafts or the postdated cheques you can say that are drawn on a bank and guaranteed by
the bank. The guarantee basically improves the credit quality of the bank making the banker’s
acceptances marketable. In the secondary market the banker’s acceptances are traded as a zerocoupon bond with the face value equal to the payment order and with the maturity between the
30 days to 270 days. With the bank guarantee they are considered the prime quality instruments
with relatively low yields.
(Refer Slide Time: 29:17)
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So let us see that how the banker’s acceptance basically works. You see this example, consider a
case where let there is a US oil refinery that wants to import 2 million barrels of crude oil let 50
dollars for barrel that means 2 million barrels that means 100 million dollars. From an oil
producer in Brazil, assume this is the hypothetical example: suppose the Brazilian oil exporter
wants to be paid before the shipping, whereas the US importer wants the crude oil before the
payment right.
So here there is a trust factor works here, so then how this particular transaction can be facilitated
So to facilitate the transaction suppose they agreed to finance the sale with a banker’s
acceptance. In which in their particular banker’s acceptance the US importer's bank will
guarantee 100-million-dollar payment 60 days from the shipment. So, with this understanding
what will happen, the US oil importer would obtain a letter of credit LOC from his bank.
And let the LOC would say that the bank would pay this exporter let 4 billion dollars if the US
importer failed to do so. This is the hypothetical number that we have taken. Then the LOC
would then be sent by the US bank to the Brazilian bank of the exporter then once this Brazilian
bank receives this LOC, they will notify the oil exporter who would then ship these 2 million
barrels of the crude oil.
So, the oil exporter would then present this shipping document to the Brazilian bank and receive
the present value of the 100 million local currencies from the bank itself. So that is a bank
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basically looks as a guarantor in that particular context that increases the value of that particular
agreement, so that is basically what we call it the banker’s acceptance.
(Refer Slide Time: 31:44)
So, what will happen is the Brazilian bank would then present the draft to the US bank who
would stamp this acceptance on it and create this banker’s acceptance. And the US importer
would sign the note and receive the shipping document and at this point the Brazilian bank is the
holder of the banker’s acceptance. And the bank can hold this banker’s acceptance as investment
or sell it to the American bank as a price equal to the present value of 100 million dollars.
So, the Brazilian bank opts for the latter thing so then the US bank holds the banker’s acceptance
and can either return it or sell it to an investor like the money market fund or the banker’s
acceptance dealer. If all goes well then at maturity the oil importer will present the shipping
document to the shipping company to obtain the 2 million barrels of crude oil as well as deposit
100 million funds in his bank.
Whoever is holding the banker’s acceptance on the due date will present it to the US importers
bank to be paid whether it is a money market fund or whether it is the Brazilian bank. So, the use
of the banker’s acceptance to finance the transactions is generally known as the acceptance
financing and the banks that create this banker’s acceptance are known as the accepting banks.
So, this is another instrument what the banks generally provide as a part of the debt financing.
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(Refer Slide Time: 33:25)
Then bank loans that all of you are very much aware about this; the commercial banks loan if
you look at there are many classifications. On the basis of the different types of housing loan,
you have the personal loan or education loan, you have the different types of loans in the Indian
context you can look at. But there are some classifications of the US market also do, they can
classify this particular thing as investment grade loans or the leveraged loans.
The investment grade loans mean the loans are given to the borrowers with investment grade
ratings, they should have the investment grade ratings. They usually take the form of revolving
line of credit with no maturity and with the bank setting a maximum amount that can company
can borrow. But the leveraged loans are loans to corporations with below the investment grade
ratings they usually have a maturity and a floating rate and in contrast to investment grade bonds
they are often sold to the institutional investors.
Some of these kinds of leveraged loans are also packaged and securitized as collateralized debt
obligations that what will be discussed in the future sessions. So, this is another classification
you can also observe and this is another instrument what the bank can provide as a part of the
debt financing.
(Refer Slide Time: 34:45)
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So, what basically we have discussed certificates of deposit. This is a kind of institution where
the banks provide the short-term financing or they raise the short-term financing through that
particular issuance. And banks have the freedom to issue the CDs depending upon their
requirements and the bank notes, banker acceptances and bank loans are the other major source
of debt financing with respect to the commercial banks. And there are some differences we have
observed between the certificate of deposit in India and as well as the US market.
(Refer Slide Time: 35:25)
So, these are the references what you can see for the detailed discussion on this.
Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social sciences
Indian Institute of Technology Kharagpur
Module No # 07
Lecture No # 35
International bonds
Welcome back. So, in the previous class we discussed about the different debt instruments which
are issued by commercial banks. Largely, we have discussed about the certificate of deposits,
banker’s acceptances, bank notes and all these things and also the different types of loans what
the banks provide. Today we will be discussing international bonds. Generally, there are some
kinds of bonds that are issued by international agencies or what we call the international bonds.
(Refer Slide Time: 00:53)
So, today we will be discussing largely the different types of international bonds, which are
available in the financial system and what are the different types of risks we face, whenever we
are investing in these kinds of bonds. Particularly, whenever we have the concept like foreign
exchange involved in between, then obviously we are exposed to a certain type of risk and what
are those types of risks we generally always face with respect to that? That also will be covering
up today's discussion.
(Refer Slide Time: 01:30)
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If you look at there are certain keywords we will come across. Let there be different types of
bonds, we see in the international market. We have Samurai bonds, we have Yankee bonds, we
have Bulldog bonds, we have European currency unity, Cross-border risk, we have Sovereign
risk. These are the keywords we will come across while discussing about the international bonds
in this particular session.
(Refer Slide Time: 01:57)
So, let us see what are those different types of foreign debt securities which are traded or which
are available for the investments. You look at the different types of foreign debt securities. We
have a type of foreign debt security particularly, that is called the foreign bond. There is a euro
bond, there is some global bond and that is the euro currency deposits. So, these are the different
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types of foreign debt securities, which are available in the market. So, let us see what are those
technical differences between these types of instruments? So, let us see that what exactly the
foreign bond is.
(Refer Slide Time: 02:56)
So, whenever you talk about the foreign bond of a country's foreign bond market if you see, it is
a market in which the bonds of the issuers are not domiciled in that country, where it is sold and
traded. For example, the bonds of German companies issued in Japan or traded in the Japanese
secondary market would be a part of the Japanese foreign bond market. The bonds of a German
company issued in Japan or traded on Japan's secondary market would be part of the Japanese
foreign bond market. So, that particular point we call it is a foreign bond for the Japan.
(Refer Slide Time: 04:03)
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So, then in the context of this foreign bond if you look at the different types of features. Typical
features you will observe, whenever we categorize a foreign bond market or there is a foreign
bond. Foreign bonds are sold in the currency of the local economy. So, if in the Japanese market
the German company is issuing the bond, then those things are sold in the Japanese currency. So,
that’s why the foreign bonds are sold in the currency of the local economy.
Obviously, they are subject to the regulations which are governed in that particular country or in
those particular regulatory bodies which are existing or present in that particular country and
what is the objective of the foreign bonds. Foreign bonds basically provide foreign companies’
access to the funds what they want to use to finance their operations in the country where they
are selling the bonds or where they are operating.
So, they get some kind of funds they raise certain kinds of funds from different economies or
different countries where they are operating. So, that’s why we call it they are called foreign
bonds, but they are issued in the local currency.
(Refer Slide Time: 05:36)
So, there are certain names if you look at, what are those foreign bonds? Let any, if you took the
example of the foreign bonds in Japan are called the Samurai bonds. Foreign bonds in the United
Kingdom are called Bulldog bonds. Foreign bonds in the US are called Yankee bonds. Foreign
bonds in Spain are called the Matador bonds. So, like those different names you can observe with
respect to the different countries, whenever the foreign bonds are issued.
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The samurai bonds and the Yankee bonds are quite popular names in the international bond
market. So, these are the different types of bonds foreign bonds in the Netherlands, Rembrandt
bonds. So, these are the different names has been already established names are there, whenever
we try to explain foreign bonds or the international debt market instruments. So, these are some
of the examples that have been reported here.
(Refer Slide Time: 06:43)
Then we have another major instrument. We look at that as the Eurobonds; it is quite popular
across the developed economies. You see whenever you talk about the bond market in India; we
have certain kinds of problems with respect to the bond market. Because the bond market is not
that developed in that sense we are emerging, but still there are a lot of things that have to be
done. It is not that way developed like the stock markets in India.
So, if you look at the international bond market we are not participating in that way, but the other
developed countries like US and UK and Germany, Japan, and all the countries. They are the
largest contributors of the international bond market. So, Euro bonds if you look at by name, if
you look at this it is not like that it is only the bonds which are available in Europe or maybe
some European countries like that.
The Euro bond is basically issued outside the country in which the currency is denominated and
they are usually sold in a number of countries. For example, to raise the funds or to finance its
equal operations a US Company might sell a bond denominated in British pounds that is
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possible. To finance its UK operations let the US company has an operation in the UK market
and there to finance that activity in the UK market. They can sell a bond which is denominated in
the British found in the UK market.
Generally, these Eurobonds are issued by Multinationals. The Multinational corporations
generally finance many of their global operations by selling Euro bonds because they have the
existence of almost more than one country. So, to raise the finance they generally take the help
of euro bonds. So, the euro bonds generally are a source of intermediate and long-term financing
of the Sovereign governments and international institutions like the World Bank.
So, through this, the financing can be raised. The Euro bonds are a major source or major
instrument through which this long-term and intermediate term financing can be always made
through the use of the euro bonds.
(Refer Slide Time: 09:32)
So, that’s why there is a euro bond market we have. The Euro bond market if you look at
generally it is handled through an international syndicate which consists of multinational banks,
brokers and dealers. So, any company or a government if they want to issue a Euro bond will
usually contact this Multinational bank and who will basically form a syndicate of other banks,
dealers and brokers from different countries.
Through that, this particular bond can be sold to the different economies and to the different
investors. So, the members of the syndicate generally agree to underwrite a portion of the issue
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which they will usually sell to the other banks, brokers and dealers. Underwriting services, you
know that they have agreed open to buy certain particular issues if it is not subscribed. But if
there is a demand for that particular security, then that is not required.
But if they will underwrite that particular kind of issue, then they have the obligation to invest in
that particular security or they can buy a certain kind of type of security from that and finally,
they can always sell that particular security to the other banks, brokers and the dealers.
(Refer Slide Time: 11:13)
So, in the Eurobond market if you look at the market makers generally handled the secondary
market for the euro bonds. Generally, these sophisticated informed investors or informed market
participants, they generally call the market makers. So, here the market makers generally handle
the secondary market for the euro bonds. So, many of the market makers are the same dealers
that are a part of the syndicate that helped underwrite the issue and many belong to the
association of international bond dealers.
There is an association they have that is called the association of international bond dealers and
who are basically have the major responsibility for the secondary market transactions of the
Eurobond? Automatically they are also part of the underwriting services and they also provide
these underwriting services for the issuance of this Euro bonds. So, any investor who wants to
buy or sell an existing Eurobond can contact several market makers in the international OTC
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market, that over-the-counter market to get several bid-ask quotes and from that, they can select
the best one.
So, they can collect information about the bid price and tax price from the different kinds of
dealers from the different market makers and finally, they can decide that which one they want to
choose and which is the best one for them. So, although most secondary trading of the Euro
bonds generally occurs in the OTC market, many euro bonds are listed also in the organized
stock exchanges like Luxembourg, London and Zurich some of the euro bonds are listed in these
stock exchanges.
These listings generally are done to accommodate the investors from the countries that prohibit
institutional investors from acquiring securities that are not listed. So, the organized stock
exchange where the Euro bonds are basically traded. Those particular stock exchanges are
existing, only to facilitate or to accommodate the investors from the countries that prohibit the
institutional investors from acquiring securities that are not listed.
(Refer Slide Time: 14:16)
So, what is the feature of a Euro bond? What are the basic features of Eurobond? If you look at a
general plain vanilla bond how it looks like? It basically has a face value, it has a coupon, it pays
an annual coupon or it is a semiannual coupon. Like that it is a term to maturity and all like that
generic plain vanilla euro bond, which also pays an annual fixed interest and has a long-term
maturity and there are a number of different currencies in which Euro bonds are sold.
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But the major currency denominations if you look at this are the US dollar, Japanese Yen and
Euro. So, these are the 3 currencies on which euro bonds are basically dominated. Some of the
cases the euro bonds also have been valued in terms of the portfolio of the currencies like
European currency units (ECU) and many bonds are unregistered and issued as bearer bonds.
Already we discussed what is the difference between the resisted bond and the bearer bond? So,
most of the Euro bonds are the unregistered and they are generally issued as the bearer bonds.
(Refer Slide Time: 15:44)
So, they have fewer protective covenants, making them an attractive financing instrument for the
companies. But, some amount of risk is involved in terms of investment in the Euro bonds. Euro
bonds differ in terms of their default risk and are rated in terms of quality ratings. The maturity
of the Euro bonds also varies. They have intermediate terms like 2 to 10 years. Generally, they
are called the Euro notes and they also have a long-term maturity like 10 to 30 years.
They are called euro bonds. So, whenever the maturity period is varying between 2 to 10 years,
we call the Euro notes. Whenever the maturity period is between 10 to 30 years, we call it the
Euro bonds. There are some short-term instruments also available in this particular segment.
These are called the short-term Euro paper or Euro medium-term notes. So, these are some of the
instruments also which are available on the basis of the maturity period.
The name of these particular instruments gets changed and that’s why they like your commercial
paper and all which are short-term instruments in the corporate bond market. They can have a
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short-term Euro paper or the Euro medium-term notes which are also traded in this particular
segment.
(Refer Slide Time: 17:09)
So, the dual-currency Euro bonds pay the coupon interest in one currency and principal in
another. If there is a Dual currency euro bond, then they will pay the interest in one currency and
the principle in another currency. Then the option currency Euro bond offers investors a choice
of currency. For example, a sterling or Canadian dollar bond gives the holder the right to receive
the interest in principle in either of the currencies.
So, the number of Euro bonds also have some special conversion features like one type of
convertible is a dual currency bond which allows the bondholder to convert the bond into stocks
or another bond that is denominated in another currency. There is some flexibility involved with
respect to some of the typical Eurobonds, which are traded in the market.
So, like you have the Dual currency of the sun, you have the options for getting the interest in
one currency and getting principle in another currency and all these things. So, there is a
possibility that you can also convert them into stocks and other instruments.
(Refer Slide Time: 18:24)
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So, some of the warrants sold with Euro bonds include or which basically gives the holder the
right to buy back. The warrants feature is like that buyback stock, additional bonds, currency or
gold. There are also some floating rate Euro bonds with the rates generally tied with the libor and
floaters with the red cap, whatever benchmark they are using that is called the floaters.
The Eurobond market also has issued zero discount bonds. That particular type of Euro bond is
also available which is considered a typical characteristic of a zero-coupon bond. That is also
available in this Eurobond market.
(Refer Slide Time: 19:20)
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So, then another type of bond we have is called the Global bond. What is the Global bond? The
global bond is a bond it is both a foreign bond and a Euro bond. Both the characteristics have to
be satisfied to consider one bond as a Global bond. That means it is issued and traded as a
foreign bond and also it is sold through a Euro bond syndicate as a Euro bond. It is issued and
traded in a foreign bond which means being registered in a particular country and also it is sold
through a Euro bond syndicate as a Euro bond. If both these things have been satisfied then we
call it a Global bond.
(Refer Slide Time: 20:09)
Then whenever we are talking about either this foreign bond or Euro bond or the Global bond
whatever it may be. Generally, we are exposed to some degree of risk. What are those basic risks
we face? Mostly we face the risk we call the cross-border risk and the sovereign risk. What do
you mean by the cross-border risk? This risk basically comes due to either any kind of political
changes in that particular country, where the bond is traded or some social and economic
conditions have been changed.
So, if the bonds are issued in that particular country and obviously the country's dynamics can
change with respect to other external factors, then obviously the performance of those particular
bonds also gets affected. So, the risk which is arising due to the political, social and economic
conditions in the countries where the bonds are issued and where the company is incorporated.
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So, in that case, we are exposed to a certain amount of risk and that risk in the aggregate sense
we call the cross-border risk and another type of broad risk we can also face, that is called the
sovereign risk and what do we mean by the sovereign risk? It is the risk that the government is
unable or unwilling to serve its state, due to some political changes. You can have any kind of
situation that can arise within that particular economy. There may be some kind of political
unrest, some political instability, that can also prevail in that particular system.
So, at that particular point of time, we are generally exposed to this sovereign risk. So, these are
the 2 broader types of risk, generally what we face. Whenever we try to invest in the foreign
bond market.
(Refer Slide Time: 22:17)
So, if you look at there are certain factors which should be considered whenever we are going to
invest in the foreign bond or anybody wants to invest in the foreign bond market and then they
have to look at certain things. First of all, they have to look at the size and diversification of the
country's exports. So, the countries that specialize in exporting only a few products may be more
successful always, there is a probability that if the exports are commodities are limited.
There is always a suspicion that they can go into a recession. The probability of facing a
recession is quite high. If this size of the export is relatively less. Then the political stability, the
strength of the legal system, the amount of unemployment and the distribution of wealth in
aggregate, we call the stability of the system. How strong is your judiciary system? The
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macroeconomic factors like unemployment, growth, income distribution, and all these things are
sometimes also responsible factors to destabilize the economy.
So, wherever these particular bonds are issued there is a possibility that the company may not
perform in that particular context because of this kind of dynamics. So, in that case, the political
stability will be hampered or the political system may be unstable. So, in that case, the
company's bond performance also gets hampered. In the history of meeting the debt obligations,
you can also look at the history, how this debt obligation history looks like in that particular
country or a particular kind of company.
So, that also can help you to decide that whether you should invest in that particular bond or
foreign bond or not. The balance of payments ratio contains the total debt to export ratio, that
also you can look at because that generally gives you the idea of how much exposure or how
much kind of obligations the particular country has with the rest of the economy or the rest of the
world. Inflation growth in gross domestic product or the GDP, interest rates; these are all factors
that also have to be considered where the particular companies are operating or wherever these
particular bonds are issued.
That is also quite important in that particular context. So, if you look at mostly what you are
trying to explain here, we are trying to explain overall if you see the macroeconomic
fundamentals and the political stability. Macroeconomic fundamentals and political stability are
the 2 major factors we have to consider whenever any bond investor wants to invest in that
particular foreign bond.
So, macroeconomic factors include the GDP, inflation and all these things; interest rates and all
and political stability includes we can say that stability of the government, strength of the legal
system, distribution of wealth and all these things are relatively subjective in nature. But that
also, you can always look at a proper assessment of that before deciding whether you should go
invest in that particular foreign bond or not.
(Refer Slide Time: 26:15)
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Then we have a Euro currency market. So, this is a kind of money market. The euro currency
market is the money market equivalent of the euro bond market. Here, in this market, the funds
are generally the maturity period of the particular funds are relatively less and they are
intermediated, deposited or loaned outside the country of the currency in which the funds are
denominated.
For the large investor the market offers 2 types of instruments one is the Euro currency CDs and
the second is the primary deposits. The Euro currency CDs are denominated as currencies
outside the currency's country. The Maturity of Euro currency CDs ranges from one day to
several years, with the most common maturity being 1, 3, 6 and 12 months. The rates on CDs are
usually negotiated between the investor and the bank and they can be either fixed or the floating.
The rates frequently quoted relative to the LIBOR rate or in terms of the Singapore Interbank
offer rate.
If it is available in the Asian market, Asian dollar CDs are quoted in terms of the Singapore
interbank offer rate that is called this SIBOR. The rates on the Eurodollar CDs are typically
higher than the rates on comparable domestic CDs. Because of the relatively high risk, then the
primary deposits are time deposits with negotiated rates and short-term maturities. So, these are
the basically the characteristics of the instruments of the euro currency market.
(Refer Slide Time: 28:08)
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So, already we have discussed certain factors with respect to the investment point. But if you
look at the investment in general in the international debt market, we look at the political risk and
the exchange rate risk. Political risk is defined as the uncertainty of converting the international
securing holdings because of unexpected changes in the country's laws or customs. So, for
investors, political risk can result from the government placing restrictions on currency
conversion, freezing the assets, imposing taxes and all these things.
Because of government policies, some regulations and political risks are always greater in
developing countries with unstable political reasons. Generally, that happens many times and
exchange rate risk is basically the uncertainty over the change in the exchange rate that generally
happens every day. And when an investor purchases a foreign currency or foreign security, then
he or she usually has to buy the foreign currency.
And when he or she receives income and principal from the security or sells it she or he usually
has to convert the process back to her own currency and that conversion whenever they do that
time the exchange rate plays a significant role. So, when a corporation basically sells security or
borrows the funds in a currency that must be converted to finance the operations in a different
country. Then the company is subject to the exchange rate risk.
So, both from the issuer’s point of view and from the investor’s point of view, they are always
exposed to exchange rate risk. That is a common risk always; we face whenever we operate in
the international debt market. So, this is the major risk although there is another type of risk;
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common risk also involved. But these are the different risks that we generally come across
whenever we try to discuss the international debt markets.
(Refer Slide Time: 30:17)
So, what we have discussed is that foreign bonds generally are sold in the currency of the local
currency. Foreign bonds generally provide the access to the funds to finance the operations of
foreign companies in other countries, wherever they are operating. Multinational corporations
generally finance their operations or investments, whenever they do Global operations by selling
the Euro bond.
And market makers generally handle the secondary market for Euro bonds. And investors in the
international debt market generally face various types of risks like cross-border risks, Sovereign
risks, foreign exchange risks, political risk and all. So, these are a common risk what we face.
(Refer Slide Time: 31:00)
So, these are the references you can go through for the detailed discussion. Thank you.
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Management of fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 08
Lecture No # 36
Bonds with Embedded Options
Welcome back, in the previous session we discussed about the international bond market so
largely we covered of many bond markets that includes the bonds issued by the government,
bonds issued by the corporations, bond issued by the municipalities, bonds issued by the
banks or the intermediaries that we can say and the bonds also issued by the international
agencies. So, then we can discuss about certain things in today's session. There are certain
bonds which have some typical embedded options like some bonds have some call features,
some put features, some bonds have some sinking fund provisions.
So, there are different types of typical options can be or kind of embedded options can be
involved with respect to that. So, in that case how the valuation of those bonds are done
number 1.Number 2 is there any typical risk involved with respect to that type of bond
whenever they have this kind of extra features are linked to that. So those things basically the
prime focus of these particular sessions, what we are going to discuss at least for another 4 or
5 sessions will be discussing about these issues with some typical examples taking some
hypothetical examples we can discuss that.
(Refer Slide Time:01:51)
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So todaywe will be discussing about certain concepts like the call risk and the valuation of
the callable bonds. We can start the valuation of callable bonds but we will carry on this
particular discussion. Mostly the session will cover of what we can say that what are those
kinds of embedded options are involved in the bonds and how those particular options is
basically affecting the valuation and the risk of that particular bond?
(Refer Slide Time:02.29)
So, we will come across certain concepts like optional adjusted spread, price compression,
and binomial interest rate tree. So, these are the different keywords that you will come across
in today's session.
(Refer Slide Time:02:45)
So, let us first see that what are those embedded options generally we have? So many bonds
have the embedded options features like call features, they can have a put features, they can
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have a sinking fund provision and all these things. So, what does it mean by the call feature?
The call feature means it gives the issuer the right to buy back the bond from the bondholder
after certain point of time. According to this indenture provision this time period will be
mentioned. So, after that particular period the bond issuer can buy back that particular bond
from the bond holder.
So, what is the put feature? Put feature is basically here it this particular feature gives the
bondholder (the investor) the right to sell the bond back to the issuer. This first one is called
the callable bonds; second one is called the puttable bonds. The inclusion of this kind of
features increases this call risk, put risk etc. And because of this the evaluation of these bonds
is relatively more difficult in comparison to the evaluation of the common bonds whatever we
have discussed.
(Refer Slide Time:04:13)
So, then what is the risk involved in that? Let we take the one example that is the call feature.
So, if you take the call feature then we call it if any bond has a call feature, we generally face
a typical risk that is called the call risk. So, what is the call risk? The call risk is basically a
type of uncertainty that the realized return will deviate from the expected return because the
issuer calls the bond. Forcing the investor to reinvest in a market with lower rates and when
the call feature will work or will be feasible.
The call feature basically will be active whenever the issuer will find the interest rate in the
market is low. Whenever the interest rate in the market is low in that particular point of time
it will be beneficial for the issuer to buy back that particular bond at a call price which has
been fixed before right.So, some callable bonds can be called at any time that depends upon
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the agreement or the indentured provision which has been provided which has been
mentioned beforehand.
Whereas for some others; the call is deferred for a certain period giving the investor
protection during this deferment period. And some bonds as a part of the sinking fund
arrangements are retired over the life of the bond usually with the issuer having the choice of
purchasing the bond directly at a market price or calling the bonds at a specified call price.
That type of provisions also available in some of the bonds not in all the type of bonds but
mostly whenever you talk about it is a callable bond.
That means the bond can be called back at any point of time or after a certain point of time
with a particular price and that price is generally, we call it the call price. But if there is a call
feature, then your realized return may be different from the expected return what you are
trying to get. From the investment perspective your realized return will be different from the
expected return what basically you are expecting from that particular bond.
(Refer Slide Time:06:59)
So, let us see that what happens that? When a bond is called the investors realized rate of
return is affected in 2 ways. Already for a normal bond we have discussed that first. Since the
call price is generally above the bonds face value. Generally, the call price is fixed above this
bond space value the actual rate of return the investor earns for the period from the purchase
of the bond to its call generally is greater than the yield on the bond at the time it was
purchased. There is a price effect. So here we call it the price effect. Because as per the
provision the call price is more than the face value of the bond then obviously even if the
bond will be called back. Then the return should be more than the return what basically or
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yield whatever they have utilized to purchase the bond at the time of purchasing the bond,
what was the yield if the call price is more than obviously, they can expect more return from
that.
But if the investor originally bought the bond because its maturity basically is matching with
the investor’s horizon date then the investor will be facing with the disadvantage of
reinvesting the call and the investment proceeds are the lower market rates. Now whatever
proceeds they will get that process they have to reinvest in the market at a lower rate right.
The money they have got so that money has to be reinvested in the market at a lower rate.
So, in that case they will be facing the reinvestment risk. So, let the bonds maturity period
was 10 years the call features were possible after 3 years. So, after 3 years if the interest rate
has come down even if the face value of the bond was 1000 and the call price was at 1100
whatever money they will get after 3 years that money if the investors horizon period is 10
years or let8 years.
In that case they are basically always exposed to more reinvestment risk. What this money
they are going to use? How that particular money is they are going to use? what they will do
with that particular process that is basically the risk what always they are facing. So that is
called the reinvestment risk and if the reinvestment risk is dominating the first effect that is
the price effect, then what will happen?
The rate of return over the investor’s horizon will be lower than the promised yield to
maturity when the bond was bought. Let the promised yield to mass rate was 10%, if the
reinvestment effect or the reinvestment risk is exceeding this price risk or price effect so here
the price risk is low because the bond call price is relatively higher. But if the interest rate is
quite low and the coupon or the investors horizon period is not matching with their maturity
period then what will happen that?
In this kind of case the yield or the return what they are going to get that will be lower than
the return what they are expected to get whenever they have bought the bond.
(Refer Slide Time:10:57)
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So let us consider one example this if you take this example let an investor with a 10-year
horizon that investor’s horizon period is 10 years and they purchase a 10-year maturity bond
with the 10% coupon at its far value of 1000.It is bond which is issued at let par. And the
coupon payments paid annually and the bonds can be callable at a price of 1100 so you
assume that the yield curve for such bonds is let fall at 10% and it remains that way for the
first 3 years the investor holds the bond and at the end of the 3 years, you assume that the
yield curve is shifting down to 8% the interest rate has gone down to 8% it was 10% then it
has gone down to 8% and that time let the issuer has gone for calling the bonds. Whenever
the interest rate does after 3years the interest rate has gone down from 10% to 8%, you
assume that the issuer is now calling the bond. So, in that case what will be the investor’s
total return for this 3 years period?
You see investors horizon period is 10 years now after 3 years the market interest rate has
gone down from 10% to 8% that time the issuer was ready to buy back that particular bond
has gone for this utilizing this call option. So, in that case you return what you are expecting
that is a 10%.Now whether you are getting the 10% or more than 10% or less than 10% in
this particular scenario that is basically our objective to find out. So, let us see what will
happen in that case?
(Refer Slide Time: 13:07)
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1431
TR3= [1000]1/3-1= 0.1269=12.69%
TR10= [
1431(1.08)7 1/3
1000
2452.48
] = TR3= [ 1000 ]1/3-1= 0.09386
TR10= [(1.1269)3 (1.08)7 ]1/10-1 =0.09386
You see at the end of 3 years the investor’s cash flow if you look at or the cash value how
much the investor will have. If you see in the first year the 100rupees is the coupon. They
got these 100 rupees and whatever 100 rupees they have received they can reinvest it in the
market at a rate of 10%.So, in the end of the 3 years that 100 rupees will become 121 rupees
and in the end of the second year they will get another 100 rupees.
So that 100 rupees can be reinvested for another one year at a rate of 10%. They will be
getting 110 rupees right. So, end of the 3 years how much they will get? Now the issuer has
called back the bond utilizes the call option if they will they have exercise the call option then
the cash flow will be their coupon 100 rupees then 1100 is the call price then you will get
1200 rupees at the end of the 3 years.
Then the total cash what you will be receiving at the end of the 3 years you are of this 3 here
these 2 includes your coupon as well as the reinvestment of the coupons for these 2 years that
is 121. And 110 and end of the 3 years you will be getting 1200, so total you will get 1431.In
the end of the 3 years that1431 rupees you will be getting and at what price you have bought
the bond? You have bought the bond at the price of 1000 rupees the bond was issued at par.
So, at the end of the 3 years if you the particular issuer has gone for the buy back this
particular bond or as exercise this call option your total return will be your 1431 divided by
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1000 to the power 1 by 3 - 1 that will be 12.69%.You will be getting a return of 12.69%
which is more than your promise deal that is the 10%. So, for the call period the investor
earns a rate of return greater than the initial yield to maturity that is the 10%.
But the investor’s horizon period was 10 years the investors horizon period was not 3 years
he was forced to sell that particular bond to the issuer. So now if the investor’s horizon period
is 10 years, then the investor has to reinvest that 1431 cash whatever they have received for 7
more years yes, or no? They have to invest it for another 7 years. Because the horizon period
is 10 years and that time the market interest rate is low that is 8%.
So then if you assume that the investor reinvests all coupons to the horizon date at 8% then
your rupees 1431 will grow at an 8% annual rate to get an amount of 2452.48 that you get it
1431 into 1.08 to the power 7 at the end of the tenth year. So, if you calculate the total return
for the 10 years period then how will you calculate? So here we have calculated the return at
the end of 3 year that is the TR3.
Now we have to calculate your TR10 the total return at the end of the 10 years. Now how
much you got that is your 8% interest in the market then your value will become 1431into
1.08 to the power 7 that is your 2452.48. Then your purchase price was 1000 divided by
(1000) to the power 1 by n -1 here n =10 years. Then 1 by 10 - 1 effectively you get 9.386%.
It is your 9.386%.
But initial promise deal to maturity was 10% in the third year you are getting a return which
is more than the initially into maturity. But effectively whenever you calculate it for the 10
years period your return has come down to 9.386% is it clear? So, in that case what basically
here also you can calculate here that means what you say that? The reinvestment risk or the
reinvestment effect what basically we have observed that basically has the impact on the
return expected or the realized return what you are trying to get from that particular kind of
features.
So that’s why we are always exposed to a call risk. Total return also you can calculate in this
way. That is1.1269 to the power 3 into 1.08 to the power 7 whole to the power 1 by n -1 that
means 1 by 10-1 that also give you 9.386% that is the way also you can calculate it.So now
the return has come down that means there is a risk involved in that.
(Refer Slide Time: 19:16)
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So now with the total return of 9.386% which is less than the initial YTM, the second effect
of reinvesting in a market with lower rates dominate this first effect of a call price greater
than the bond face value. One face value 1000 call price was 1100 there whatever extra return
you are getting that is basically dominated by the reinvestment effect. So, what this example
basically shows the bonds are callable are subject to the uncertainty that the actual rate will
be less than the investors expected YTM.
Because of this call risk, there is usually a lower market demand and a price for callable
bonds than the non-callable bonds which generally result in a higher rate of return or interest
premium on callable bonds over the non-callable bonds. They generally give more return or
more premium than the non-callable bonds.
(Refer Slide Time:20:25)
583
So, this size of the premium generally depends on the investors and the borrowers’
expectations about the interest rate. When the interest rates are high and expected to fall then
the bonds are more likely to be called. So, in a period of high interest rate, a relatively low
demand and higher rate will be on the callable and the non-callable bonds respectively. In
contrast if you expecting that interest rates are low and it is expected to rise then we expect
effect of call provisions on interest rate to be negligible generally not find much differences
between these 2.
The difference in yield between a bond with option features such as call option or sinking
fund call or the put option or any kind of features whatever they have. And an identical
option free bond generally is called the option adjusted spread (OAS). The bonds having the
option features the yield whatever you are getting and the bonds having the no features the
option free bond that difference or that spread is called the option adjusted spread.
(Refer Slide Time: 21:45)
So, the call features generally put the limitations on the price yield curve. So, at the rate
where the bond could be called the price yield curve generally becomes flatter with the price
equal to the call price. This limitations on the price generally called are the price
compression. If you see this figure then AA shows this price yield curve for the non-callable
bond but your AA prime shows this yield curve for the callable bond. So, they are basically
there is a concept we use that is called the price compression.
(Refer Slide Time: 22:29)
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So, if you look at this figure the AA is shown for a non-callable bond and this curve is
negatively sloped and convex from below. But if you look at the AA prime that represents the
price yield curve for the comparable callable bond and this call basically flattens out and
becomes concave at the rate that is Y*. And here Y*represents a threshold rate that basically
corresponds to a bond price equal or approximately equal, to the call price.
Since the callable bonds should likely to be called if rates at Y*or less then you would not
expect the investors to pay price for such bond greater than the call price. So, the price yield
curve for the callable bonds would tend to flatten out at the Y*. So that then the curve
basically becomes concave in that particular point or particular context.
(Refer Slide Time: 23:38)
Then we are talking about the valuation of the callable bonds how generally we do the
valuation of the callable bonds? So here if you see the valuation of the callable bonds
585
depends upon the interest rate possible change in the interest rate whether the interest rate
will decline or not. If the interest rate will decline then that basically will lead to the calling
back the bond unless the issuer is not based very much profitable for the issuer or not
lucrative for the issuer to call back the bond.
So, if it is called then the bonds cash flow patterns would be different than if rates increased
and the bonds was not called. If it is called then the cash flow pattern will be different and if
the bond the particular is not called then the cash flow pattern will be different. So given this
kind of uncertainty of the bonds cash flow the valuation of callable bonds and other bonds
with some embedded options is more difficult than valuation of option free bond.
One of the approaches what generally, we can use for evaluation of the callable bonds to
incorporate the interest rate fluctuations or interested volatility by using the binomial interest
rate tree. The binomial interest rate tree model can be used to understand the probable
changes of the interest rate and accordingly we have to decide that how this particular value
of that particular bond can be determined? So binomial interest rate tree is a one of the
approaches which can be used for that.So, what exactly this binomial interest rate model is?
(Refer Slide Time: 25:53)
The binomial interest rate model basically assumes the spot rate of a given maturity they
follow a binomial process and each period where in each period it has either a high or the
lower rate. Let you assume a one period risk free spot rate let that is represented as S follows
a process in which in each period the rate is equal to a proportion u times its beginning of the
period value or a proportion d times its initial value, where u is greater than d. That means
what does it mean in after one period there would be 2 possibilities.
586
Let this value was S0 in the initial period either it can go up to uS0 or it can go up to dS0. If it
is going up then it is uS0 if it is going down it is the dS0. So, if the proportions u and d are
constant over the different periods then obviously after 2 periods that will be Suu that means it
will be u2S0 and it can be Sud that means udS0or Sdd it may be d2S0. So, this kind of
distribution basically what you can look at.
(Refer Slide Time: 27:48 )
So let me show you that how this distribution basically looks like so this is called the
binomial interest rate model. We start with S0 it can go up to Su means this is u into S0 or it
can go to Sd which is d S0. So, this again have 2 options after this it can either increase or it
can decline. If it is again increasing then Suu that means u2S0or it can be uS0 into d or udS0
or it can further decline you can go for d2S0.
And again, if you see that another period Suuu then it is u3S0or it may be u2dS0or it may be
Sudd that is the ud2S0 or it may be Sddd that is the d3S0is it okay? So that is the way the
distribution basically looks like.
(Refer Slide Time: 28:51)
587
Let us take one example let the current spot rate is 10%u you have taken 1.1 downward
parameter you have taken 0.95, then what are the possible one period rates after one period?
What are the 3 possible 1 period rates after 2 periods? And what are those 4 possible 1 period
rates after the 3 periods? So that is the question.
(Refer Slide Time: 29:17)
So whatever thing we have discussed if you want to measure this you start with the 10% u
=1.1d = 0.95.So then 10% has become if it will increase then it is 1.1 into 10 become 11% or
it will be0.95 into 10 it will be 9.5%. Further from 11% to it can go again increase by 1.1 it
can become 12.1% that is 11 into 1.1 that will give you 12.1% or it can also come
down.So,11 into 0.95 it can become 10.45% again if it is further going down then again 9.5
into 0.95 that will give you 9.025%
588
So then from 12.1 to it can go up to here it is 12.1% multiplied by 1.1 that will give you this
rate or it can decline so this will be 12.1 into 0.95 that will give you this rate and again if it is
10.45.This can again go down that means 10.45 into 0.95 or it can be it can also become this
8.574% if it is further decline. Like that the distribution this 12.1 become 13.1 or 12.1 can
become 11.49 or it can also from here; if it is increasing also, it can be also 10.45 into 1.1.
So, there are different ways this particular binomial tree model can work out and, in that case,
the different interest rate scenario can be built upon. So once this different interest rate
scenario will be built upon accordingly the valuation of the callable bonds can be made. So,
we will discuss extensively about the 2 period, 3 period callable bonds, puttable bonds,
sinking fund provision bonds and all these things in the subsequent sessions.
(Refer Slide Time: 31:29)
So, what we have discussed today about the call risk which is the risk which makes the
uncertainty between the realized return can be deviated from the actual return, what expected
return the investor is trying to get. Call features put the limitations on the price yield curve
and these limitations on the price generally called as the price compression. And binomial
interest rate approach generally used one of the approaches which can be used for the
valuation of the callable bonds.
So, future will see with the different examples how exactly the evaluation of the callable
bonds, puttable bonds and the sinking fund provision bonds are generally made.
(Refer Slide Time: 32:14)
589
So, these are the references what you can see. Thank you.
590
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 08
Lecture No # 37
Binomial Valuation of Callable Bond - I
Welcome back. So, in the previous class we have introduced the concept of the binomial tree
model. Basically, whenever it is assumed that the interest rate may change in the future or the
yield curve is going to be changed. And in that particular context how the valuation of the
different type of bonds which have certain kind of embedded features or embedded options can
be evaluated. Or how the evaluation of those bonds can be done with some kind of embedded
options with this concept of volatility of the interest rate.
In that particular context we just introduced the binomial tree of the interest rate movements in a
particular point of time. Then accordingly we can use that particular probable interest rate which
is going to prevail in the market, and accordingly the valuation of that particular callable bonds
or the bonds having the call features can be estimated.
(Refer Slide Time: 01:27)
So let us see that what are those concepts, we are going to cover in today's class. We will be
covering 2 things one is the valuation of the 2-period option free bonds and the valuation of 2
591
periods collable bonds. So, what is the basic difference between the valuations of the 2 periods of
an option free bond.
That means if the interest rate is going to be changed and that interest rate probably, interest rate
in the 2 periods first we have to find out using the binomial interest rate introspective model.
And from there we can use that particular interest rate for the valuation of this particular bonds.
(Refer Slide Time: 02:17)
So, the different keywords what will be using for today's class or you will come across. These
are basically the call options, call price, this spot rates, the intrinsic value all these things. These
are the things basically what you will get to know or basically will be used in today's sessions.
(Refer Slide Time: 02:43)
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So let us see that how this thing is basically going to work, you just recall in the previous class
that what basically the binomial interest rate tree we have constructed. This is the hypothetical
example whatever we have taken we have started with the initial spot rate that is 10% and here
we have taken your u = 1.1 and d = 0.95. So here this either the interest rate can go up or the
interest rate can go down.
So, if the interest rate will go up it can become 11% so that is nothing but the 10% into 1.1, and
it is nothing but this 10% into 0.95; so, they can either become 11% or become 9.5%. Next
period it is the one period data the next period from 11% to it can go up to 12.1% it is nothing
but this 11% into 1.1. And again, it can also come down to 10.45% the 10.45% is nothing but
11% into 0.95, or it can also go from 9.5% to one point multiplied by 1.1.
So, there is a node here that this node basically gives you the figure that is 10.45%; or again it
can further go down that will be 9.5 into 0.95 that will be 9.025%. Further again you can go for
this is the first period nodes, and this is your second periods nodes, and this is your third period
nodes. So, third period again the 12.1% can become 13.31 that is 12.1 into 1.1, or it can go down
also up to 11.495% that basically you get that 12.1 into 0.95; or it may be 10.45 into 1.1 that will
become 11.495%.
Or it can go also 10.45 to 9.927% it can go down then 10.45 into 0.95 that you can get here, or it
can also 9.025% can also reach here, if you multiply 1.1 you can reach also 9.927%. Or it can
again further go down that d cube S 0 that may be 8.574%. So, like that you can also expand this
particular interest rate tree for the next period so either, it can from 13.31 to it can go up to 13.31
into 1.1 or it can go down also 13.31 to 0.95.
So, u is equal to 1.1 and d is equal to 0.95 that is what we have assumed, or we have considered.
Further we will see that how this u and d can be estimated, but now you can assume that that u =
1.1 and d = 0.95. So, keep this particular figure into your mind then we are going to use this
particular numbers for evaluation of a hypothetical callable bonds. And the bonds also which do
not have any call features, and what is the difference between the valuation of that callable bond
or the valuation of the bonds which have no call options.
(Refer Slide Time: 06:45)
593
So then here we have taken this example, so given this possible one period spot rates suppose we
want to value a bond with the following features. Let you are considering a bond and the bond
basically matures in 2 periods that is the first assumption we have taken the bond is going to be
matured in two periods. The bond has no default risk this is the second assumption what we have
taken.
And the bond let us first case the bond has no embedded option features, let that is option free
bond, it has no call option feature, or boot option feature or any sinking fund provisions, no other
options are available, the bond has no embedded options features. And you assume that the bond
pays a coupon of 8% in each period, and the bond pays rupees 100 the principal at the maturity,
the par value of the particular bond is 100 rupees.
Coupon is 8% the par value the bond is 100 rupees and this is option free bond. That is the
assumption what basically we have taken and it has no default risk and the bond matures in the 2
periods, that is more important that actually you have to keep in the mind.
(Refer Slide Time: 08:12)
594
So, in that case what basically we can see first of all; let us see that how the valuation of a 2period option free bond can be carried out. Since the bond has no default risk or the call risk; the
only risk the investor is basically is assuming that is the market risk. And why basically market
risk because the interest rate is going to be changed. So, considering that market risk we are
trying to see that on the basis of the binomial interest rate tree whatever we have considered this
interest rate scenario or the change in the interest rate in the different periods how it is going to
play the role for the evaluation of this particular bond.
So, at that time the original 2 period bond will have one period to maturity where there is a
certain payoff of 1.08. But we do not know whether the one period rate will be 11% or 9.5%. It
was started from 10% so either it can go up to 11% or it can go down to 9.5% that is not clear to
us. So, if the rate will be 11% then what will be the value of the bond, then it will be the Bu
represents if the interest rate is going to be increasing and it will reach 11% from the 10%.
Then the value of the bond will be your 108 the cash flow what you are getting in the end of the
period because 8 rupees 8% is the coupon and per value is bond is 100. That we have assumed
then 108 divided by 1.11 that is 97.297, so if the interest rate will go down from 10% to 9.5%
then the worth of the bond or the value of the bond will be 108 divided by 1.095 that is 98.630
right.
Bu = 108/1.11 = 97.297
Bd = 108/1.095 = 98.630
595
(Refer Slide Time: 10:28)
So now if you see that given the 2 possible values in the period one that what we have taken the
2 values one is your with respect to the 11% another one is with respect to the 9.5%. So given
this particular 2 possible values in the period one, the current value of the 2-period bond can be
found by calculating the present value of the bonds expected cash flow in the period one.
Whatever probable cash flows you are getting in the period one that probable cash flow can be
used to calculate the value of this particular bond in the present time.
B0 = pr [Bu + C] + (1-pr) [Bd +C]
1+S0
So, in that case the expected cash flow you have in one case it is 97.297 and other case it is
98.630. If the bond the interest rate is going to be 11% then the value will be 97.297; and if the
interest rate is 9.5% then it will be 98.630. So let assume there is an equal probability of the one
period spot rate being higher that is 0.5 probability is 50% or it can be lower that is one also 0.5;
(1 – pr) that is 0.5.
And probability of cash flows in the 2 different periods will be is 50% and the value of the bond
whenever the interest rate is going to be up that is 97.297. And whenever it is going down that
will be your 98.630 that we have estimated we have calculated in the previous slide.
So then how we can calculate the value of for the particular bond in the present time. So, your
probability that is the 0.5 whatever we have considered into your Bu which represents whenever
596
the interest rate is going to be up what is the cash flow or the value of the bond in that particular
point of time that is 97.297. And you get your coupon that is basically your 8 so 97.297 + 8 +(1 –
pr) into your Bd; Bd represents the value of the bond of the interest rate is going to be down so
that is your 98.630 + your coupon that is 8.
So, this is the expected value what basically you are getting in the current period. You are going
rolling down. There is a backward calculation what we are making. So, cash flow what we have
calculated on the basis of the change in interest rate in the first period. Then we are basically
discounting it with respect to the interest rate which is going to be prevailed or which is already
prevailing in the market in the current period.
Bo = 0.5 [97.297 + 8] + 0.5 [98.630 + 8] = 96.330
1.10
In the current period the interest rate we have assumed we have started from the interest rate of
10%. So, the cash flow what you get 0.5 into 97.297 + 8 + 0.5 into 98.630 + 8 divided by 1.1 this
1.1 means this is 1 + r this r is the 10%, so then we get it 96.330. So that is the current value of
this particular bond if the interest rate either going to be up or down then these are the cash flows
what you are getting. Then if you discount it with respect to the current interest rate that is 10%
then you get the value of the bond in the current period.
(Refer Slide Time: 14:20)
597
So now what basically here we have seen we have just summarized these calculations whatever
we have made. If you see that we started with this we have assumed this coupon is 8% means
100 rupees is the face value. That’s why we have assumed coupon is the amount is 8 rupees face
value is 100 rupees your current interest rate is 10%, u = 1.1 d = 0.95 right. So, what has
happened, if interest rate was 10% it can go up to 11% or it can go down up to 9.5%.
And your cash flow what basically you are getting here it is basically Buu if you consider the 8
rupees is the coupon 100 rupees is the first value that is 108 every cases it will be 108. Now what
basically we have done we have discounted that 108 with respect to the expected interest rate
which is going to be prevailed in this node that is 11%. So, 108 divided by 1.1 you got 97.297, if
assume interest rate has gone down then it will become 9.5% then your value of the bond will be
98.630, now these are the expected cash flows what you are getting.
Bu = 108/1.1 = 97.297
Bd = 108/1.095 = 98.630
Now what you are doing you discount it with respect to the current interest rate that is basically
10% then you can get the current value of the bond that is 96.330 so that is the current value of
that particular bond in the current time. So, this is what basically what we get it for an option free
one whenever you are going for a violation of a 2-period option free bond.
(Refer Slide Time: 16:20)
598
So let us see that how basically this thing will be prevailed or this thing will work for valuation
of a 2-period callable bond. Suppose you assume that there is a 2-period bond 8% is the coupon
rate it has a call feature. It has a call feature which allows this particular issuer to buy back that
particular bond at a call price of 98. So here what basically you are doing using this binomial tree
approach this call option can be incorporated into the valuation of the bond by determining at
each node in period one; whether or not the issuer would exercise his right to call. Depending
upon the price of the particular bond expected price of the bond in that particular period, you can
decide that whether the issuer will go for calling the bond or not if it is profitable, then they will
go for utilizing this call option if it is not profitable then they will not exercise. So, the issuer will
find it profitable to exercise whenever the bond price is above the call price, whenever the bond
price in the market will be more than the call price.
Then only the, it will be profitable for the issuer to call back the bond assuming no transaction or
the holding cost you assume that. If your bond price in the market will be more than this call
price, whatever has been fixed before then there is a possibility that the issuer can exercise that
particular option.
(Refer Slide Time: 18:23)
599
So now let us see what exactly happens in this case. So, since the bond is only exercised when
the call price is less than the bond value. Then the value of the call option in period one is what it
is the minimum of its call price or it is a binomial value. So, your BtC (t is the time period) will
be your minimum of Bt or the call price. So, what basically you are doing rolling the 2 callable
bond values in period 1 that what basically we are getting that is 97.297 and 98 to the present.
BtC = Min [Bt, CP]
First period if the interest rate has gone up then what was the value of the bond, we have
calculated that is 97.297 and the call price we have fixed that is 98, right. Then we can obtain the
current price of that particular bond that is your 0.5 into 97.297 + 8 + 0.5 into 98 + 8 and you can
discount it with respect to the 10% you get the value of the 96.044. So here what basically we are
trying to say that when the bond call option will be exercise. The call option will be exercised
whenever the market value will be more than the call price otherwise not.
So, in the lower interested scenario, the value of the bond was 98 point something 98.630, so in
that case the call option will be exercised. But in the first case the call option will not be
exercised because the 97.297, is less than the 98 rupees. So that’s why instead of using the
98.630 as the cash flow we have used 93 is the cash flow which is the call price. So now the
value of the call option has become 96.044.
B0C = 0.5 [97.297 + 8] + 0.5 [98 + 8] = 96.044
1.10
600
(Refer Slide Time: 20:59)
So now let us see that what exactly we have done here you see this particular binomial tree then
it will be clear for you. This part already you know the coupon is 8, f = 100 call price we have
taken 98, then S0 is 10 % you can write it 0.10, u = 1.1 d is equal to it is also 0.95 right. So here
what basically we have done in the previous case we have done that Bu = 97.297 and your Bd =
98.630.
Then your BuC is the minimum of Bu or CP, so minimum of Bu and CP means it is 97.297 or 98
whichever is the lower we have considered that that is 97.297. And here in this case you have
minimum of the Bd and the CP and that is Bd = 98.630 and CP = 98. So, we have considered 98
so then the cash flow in the first period has become 97.297 and the 98.
BuC = Min [Bu, CP] = Min [97.297, 98] = 97.297
BdC = Min [Bd, CP] = Min [98.630, 98] = 98
So, then the expected cash flow in the end of this first period will be you get this price +
obviously the coupon will be there then it will be 97.297 + 8 and it will be 98 + 8 and probability
we have considered it has been 50-50 % chance that interest rate can be high interest rate it can
be low. So, then it will be 0.5 into 97.297 + 8 + 0.5 into 98 + 8 divided by 1.1 that will give you
96.044.
601
So little bit higher than the value of the option free bond, whatever; we have considered in the
previous case. So, this is the way basically the valuation of 2 period callable bond can be
calculated.
(Refer Slide Time: 23:14)
There are other methods also so here what we have seen before let us see. The bonds, embedded
call option lowers the value of the bond from 96.330 to the 96.044 sorry it has lowered the value
of the bond from 96.330 to 96.044. And at each of the nodes in period 1 the value of the callable
bond is determined by selecting the minimum of the binomial bond value or the call price. Then
rolling the callable bond value to the current period that is the process basically what we have
followed for calculation or for valuation of the bond in the current context right.
(Refer Slide Time: 24:08)
602
So, then what basically here we have seen when the one period spot rate is 9.5% in period 1, and
the bond is priced at 98.360. It is profitable for the issuer to call the bond and the price of the
bond in this case would be 98 that is the call price. When the one period spot rate is 11% in
period 1, it is not profitable for the issuer to exercise the call and the price of the bond in this
case remains as 97.297, anyway that call option is not going to be exercised.
So, in the 9.5% case the issuer could buy the bond back at 98 financed by issuing one period
bond at 9.5% interest and one period later the issuer would owe 98 into 1.095 that is 107. 31, so
that basically represents the savings of 0.69 that is 108 – 107.31 that is 0.69. And the present
value of that particular savings in period 1 will be 0.69 divide by 1.095 that is 0.63. And that
0.63 is nothing but the difference between the bond price and the call price.
The bond price was 98.630 one about the interest rate has come down to 9.5 and your call price
is 98 and you get a difference of 0.63. So, this is the process how this particular saving is
basically working out in this case that is what basically what we are trying to explain through
this.
(Refer Slide Time: 26:10)
603
So, the other aspect of the valuation of the callable bonds also there within this particular same
framework. What actually you can do instead of using a price constraint at each node. The price
of the callable bond also can be alternatively found by determining the value of the call option at
each node. And then subtract that particular value from the non-callable bonds value.
You have the non-callable bonds value without any call feature already you have calculated. If
you can calculate the value of the call option and you subtract that value from the non-callable
bonds value, then also you can calculate the value of this particular callable bond. In this 2period case what basically we are considering the values of the call option are equal to their
intrinsic value or the exercise value.
BtC = BtNC - VtC
So, the intrinsic value is nothing but is the maximum of the BtNC; NC means it is the non-call
option; non-callable feature – C P or 0; C P means it is the call price. So, your VtC the means it is
the value of the call option at each node, that is nothing but the maximum of either BtNC that
value of the bond without any call option - the call price or zero.
VtC = Max [BtNC – CP,0]
(Refer Slide Time: 28:07)
604
So, in our example if you see the 2 possible call values in the period 1 are either 0 or 0.63 and
two corresponding callable bond values are either 97.297 and 98. So then the value of this
particular call option in the current period is equal to the present value of the expected call value
in the period 1. So how you can calculate this call premium that is 0.5 into 0 if it is not basically
exercise then the value will be 0 into 0.5 into 0.63 divided by 1.1 that will give you 0.2864 that is
the 0.2864.
V0C = 0.5[0] + 0.5[0.630] = 0.2864
1.10
So now what you can do you can subtract the call value of 0.2864 from the non-callable bond
value that is 96.330 we have calculated before for the non-callable bond value is 96.330. So,
96.330 - the value of the call option whatever you have calculated that is 0.2864 then we can
obtain this value of the callable bond. That is basically another method another process through
which our alternative approach through which you can also calculate the value of the callable
bond.
(Refer Slide Time: 29:54)
605
So, this is the example same thing what basically we have done S0 = 10% non-callable bond
value is this then here it can go up to 11 or it could go to 9.5. So non-callable bond value in this
period is 97.279 it is 98.630 then your VuC is the maximum between these 2 and here it is 97.297
– 98, 0. So obviously this will be minus then the value will be basically 0 the 0 is bigger than
this, because the value is minus. So, then your BuC will be 97.297 - 0 that is 97.297.
VuC = Max [Bu – CP,0] = Max [97.297 – 98,0] = 0
BuC = BuNC - VuC = 97.297 – 0 = 97.297
Second case it will be 98.630 – 98, 0 zero that is 0.630, then this will be 98.630 – 0.630 that is
98. So, then we have used that particular value here it is 0 here it is 0.630, then 0 into 0.5 into
0.630 divided by 1.1 that is what you got the value of the call of call option. Then you take the
difference 96.330 – 0.2864 that will give you 96.044 that is basically the, another approach
through which the value of the call option can be calculated.
VdC = Max [Bd – CP,0] = Max [98.630 – 98,0] = 0.630
BdC = BdNC – VdC = 98.630 – 0.630 = 98
V0C = 0.5[0] + 0.5[0.630] = 0.2864
1.10
B0C = B0NC – V0C = 96.330 – 0.2864 = 96.044
(Refer Slide Time: 31:24)
606
So, what basically we discussed here what is the conclusion here. The issuer basically finds it
profitable to exercise the call option whenever the bond price is above the call price. As the bond
is only exercised when the call price is less than the bond value, the value of the callable bond in
period 1 is the minimum of its called price or its binomial value. And the value of the call option
is difference between the value of the non-callable bond and the value of the call option at each
node. That is the thing what basically we have observed here.
(Refer Slide Time: 32:01)
So, these are the references you can see.
Thank you.
607
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social sciences
Indian Institute of Technology Kharagpur
Module No # 07
Lecture No # 38
Binomial Valuation of Callable Bonds – II
So, in the previous session, we discussed the value of a callable bond, particularly the 2-period
case we have considered using the Binomial tree model.
(Refer Slide Time: 00:38)
And today we will continue with this particular discussion and we will be considering or we can
extend this particular example whatever we have considered in the previous session to a 3-period
model. So, today we will see how we can use this binomial tree approach for the valuation of the
3-period option free bonds and the valuation of 3 period callable bonds. Conceptual
understanding of whatever we have discussed in the previous class and the example of whatever
we are taking, the same example we are going to continue for the next period.
And we will see that how it is different a little bit different. This 3-period model is different from
the 2-period model.
(Refer Slide Time: 01:22)
608
And we will come across certain kinds of keywords like upper node, lower node, exercise value
and all these things for this particular discussion. And that thing is applicable for any kind of
callable options or any kind of portable options whenever you are considering.
(Refer Slide Time: 01:42)
So, again the same example we are considering; so, keep that particularly interesting scenario in
your mind, which we have started with 10% it can go up to 11 and go down to 9.5. 11 can go up
to 12.1 or can go down to 10.45 or 9.5 can go up to 10.45 or it can go down to 9.025. So, all
these scenarios it is the continuation of the example, whatever we have considered before that
same example we are continuing.
609
So, this is the interest rate tree that we have considered the same interest rate tree we are
considering for the 3-period model. Because of the 2-period model we have already discussed in
the previous class. So, we keep this particular tree in your mind.
(Refer Slide Time: 02:39)
Let us see that what we need? Whenever we go for a valuation of a 3-period option-free bond,
then we need a 2-period interest rate tree. If we want to value a 3-period bond then we need a 2period interest rate tree and suppose we want to value a 3 period 9% coupon bond with no
default risk or option features. This bond has no option feature and there is no default risk
involved in that only the market risk is taken into consideration.
So, market risk basically can exist in the 2 periods. Period 3, where there are 3 possible spot
rates, and period 2 where there are 2 possible spot rates, that is basically, what you can consider.
(Refer Slide Time: 03:55)
610
Let us see, what happens in that case? What are the steps you have to follow for the valuation of
a 3-period option-free bond? First, determine the 3 possible values of the bond in period 2, given
the 3 possible spot rates and the bond’s certain cash flow next period. You have to consider 3
possible values of the bond in period 2, given the 3 possible spot rates and bond certain cash
flows which are the next period.
So, what are those 3 possible values? It can be Buu or it can be Bud or it can be Bdd. It starts from
the B0 it can be it can go Bu then it can go to Buu or here it is Bd. So, from Bu to it can come down
also Bud and further also it can come down to Bdd. So, these are the 3 values what we need in
period number 2. So, what is the coupon? Coupon we have considered that is 9% of your per
value is equal to 100.
Then here the cash flow will be in the end would be 109 you have to discount it with respect to
the interest rate of 12.1% because this value will be 12.1. It starts with 10, 11, 9.5, then further it
can go up to 12.1 or it can go down to 10.45 or it can further go down to 9.025. So, we have used
these 12.1% one values you can calculate that is 109 by 1.121, that will be 97.2346. You can get
your Bud at a rate of 10.45% that is 109 by 1.1045 that will give you 98.6872 or it can be Bdd that
will be 109 divided by 1.09025 that will give you 99.9771.
Buu = 109/1.121 = 97.2346
Bud = 109/1.1045 = 98.6872
Bdd = 109/1.09025 = 99.9771
611
So, you got the 3 possible values in the 3 different nodes. Then what can you do? Given these
values basically, we next roll these trees to the first period and determine the 2 possible values as
the present value of the expected cash flow in the period 2. So, we know this 50-50% chance we
are considering. The value can become 97.2346 or it can become 98.6872. So, in the first period,
the interest rate is 11% or 9.5%.
So, the cash flow what we get that is Buu which is 97.2346 or 98.6872 or 99.9771 then what we
have done? First 2 cases, we have considered 0.5 into 97.2346+9+0.5 into 98.6872+9 divided by
1.11 because in this period, this interest rate can prevail at 11%. So, you got the value 96.3612 or
it can be it will be Bu we are considering this 11%. So, it can be also Bd, that will be your 0.5 into
98.6872 +9 + 0.5 into 99. 9771 + 9 divided by 1.095 at the rate of 9.5% that you can get 98.9335.
Bu = 0.5[97.2346 + 9] + 0.5[98.6872 + 9] = 96.3612
1.11
Bd = 0.5[98.6872 + 9] + 0.5[99.9771 + 9] = 98.9335
1.095
So finally, you got the 2 values in period 1 this is the period one's value. So, now using the bond
values in period 1, we roll the tree to the current period and finally, you can determine the value
of the bond. So, then what will be the value of the bond in the current period? That B 0 = 0.5 into
96.3612 + 9 + 0.5 into 98.9335 + 9 divided by 1.1 that will give you 96.9521. Then the value of
the bond has become 96 that is what the current period price, current period value.
B0 = 0.5[96.3612 + 9] + 0.5[98.9335 + 9] = 96.9521
1.10
(Refer Slide Time: 10:15)
612
So, if you want to show this then how it will basically, how the calculation really, we have done
in this case? You see we are going backward, rolling back your 9% is the coupon and 100 is the
par value. So, in the end, the cash flow will be basically 109 every cases it will be 109 and the
second-period interest rate already we have seen the 10% can go up to 11%. 11% can go up to
12.1% then we calculated your value of the bond in this particular period which is 109 divided
by 1.121 and you got 97.2346.
Buu = 109/1.121 = 97.2346
Here the interest rate either from 11% to it has gone up to 12.1 or can go down to 10.45 or from
9.52 also it can go up to 10.45. So, you can calculate your Bud that is 109 divided by 1.1045, you
got 98.6872 in this period of this node. Here, again if the further it is going down you get 109
divided by 1.09025 that will be 99.9771. Now, we have 3 nodes and the cash flow what basically
we are calculating in this period. Either it can become 97.2346 or it can become 98.6872 or it
will become 99.9771.
Bud = 109/1.1045 = 98.6872
Bdd = 109/1.1.09025 = 99.9771
So, these are basically 2 alternatives and they can also be 2 alternatives. So, if you consider these
2 alternatives the probability can be 97 or 98. Then you get 0.5 into 97.2346 + 9 + 0.5 into
613
98.6872 + 9 and you are discounting it in this period interest rate that is 11%. Then you get
96.3612 or it can also be the 98.6872 and 99.9771. So, then you can discount it with respect to
9.5% and you get 98.9335.
Bu = 0.5[97.2346 + 9] + 0.5[98.6872 + 9] = 96.3612
1.11
Bd = 0.5[98.6872 + 9] + 0.5[99.9771 + 9] = 98.9335
1.095
Now, these 2 values you got in the first period, this is the second period. So, then what has
happened in the current period, what do you get? You got the 2 cash flows 96.3612, 98.9335.
Then you can discount it with respect to the current period’s interest rate that is 10%. Then your
probability is 0.5. We are considering 50%, 0.5 into 96.3612 + 9 + 0.5 into 98.9335 + 9. You can
discount it with respect to 10% you got 96.9521 that is the current period value. So, that is
basically a 3-period option-free bond.
B0 = 0.5[96.3612 + 9] + 0.5[98.9335 + 9] = 96.9521
1.10
(Refer Slide Time: 13:44)
Now, you go to discuss about the valuation of a 3-period callable bond a little bit different. First
compare each of the non-callable bond values with the call period, in period 2. Then take the
minimum of the 2 as a callable bond value. Next roll the callable bond values from period 2 to
614
period 1 and determine the 2 bond values at each node as the present value of the expected cash
flow. And then for each case, select the minimum of the value calculated or the call price.
Finally, roll these 2 callable bond values to the current period and determine the callable bond
price as the present value of the period once expected cash flows. Logic whatever we have used
for the 2-period model we are just expanding that particular logic to the 3-period callable bond.
(Refer Slide Time: 14:50)
Now, let will continue with this particular example. Let this binomial tree value of the 3-period
bond we are trying to do and the coupon is 9%, the bond has a call feature and the call price is
98. So, at the 2 lower nodes in period 2 the bond would be called at 98 and therefore the callable
bond price would be 98. But at the top node in period 2 the bond price of 97.2346 will prevail.
So, rolling this period two's price to period one, the present value of the expected cash flows is
what 96.0516 are the 11% spot rate and 97.7169 are the 9.5% rate. Since neither of these values
is less than the call price of 98 each represents the callable bond value at that node. Rolling
period 1’s two values to the current period, we can obtain the value of the 3-period callable bond.
But how are these values calculated? How we have received or how we got these particular
values?
(Refer Slide Time: 16:27)
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If you see these things, it will be clearer for you here the coupon is 9% or 9 rupees on 100 rupees
F = S, CP = 98, S0 = 10%, u = 1.1, d = 0.95. That; we are considering starting with 10%
becoming 11%, 12.1% and now what we have done? This 109 is the cash flow in this period. So,
we have discounted it to calculate the Buu, 109 divided by 1.121, we got 97.2346. Now, what is
the BuuC? That is basically the minimum of Buu, CP, that already you know.
Buu = 109/1.121 = 97.2346
BuuC = Min [Buu, CP] = Min [97.2346, 98] = 97.2346
Then it will be 97.2346 and 98. So, now 97.2346 less than 98. So, the price remains at 97.2346
only because the price of the bond is less than the call price. In the second case, here if you see
what you got 98.6872 and your call price is 98. So, now the call price is less than this. Then this
value will prevail. The third node, so is node 1, node 2, and node 3, and the third node becomes
99.9771. The call price is 98 that’s why 98 will prevail.
Bud = 109/1.1045 = 98.6872
BudC = Min [Bud, CP] = Min [98.6872, 98] = 98
Bdd = 109/1.09025 = 99.9771
BddC = Min [Bdd, CP] = Min [99.9771, 98] = 98
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Now, what have we considered in this period? 97.2346 and 98, then 98 and 98 there are 2
alternatives. So, the cash flows. So, that’s why the Bu in the first period, if you are calculating
then it is 0.5 into 97.2346 + 9 + 0.5 into 98 + 9 divided by 1.11; 11% of the discount rate. We got
96.0516. So now in the second case, we got 98 and 98. These are the cash flows and as well as a
coupon that is 9 and 9, we got 97.7196 then this has become 97.7169.
Bu = 0.5[97.2346 + 9] + 0.5 [98 + 9] = 96.0516
1.11
BuC = Min [Bu, CP] = Min [96.0516, 98] = 96.0516
Bd = 0.5[98 + 9] + 0.5 [98 + 9] = 97.7169
1.095
BdC = Min [Bd, CP] = Min [97.7169, 98] = 97.7169
So, now these 2 cash flows whatever you have considered, these 2 cash flows are less than 98.
So, both the prices will prevail. So, this is a minimum of 96.0516 and 98 that will be 96.0516.
Here also it will be 97.7169. So, now this cash flow we are using or discounting is rolling back to
the current interest rate of that is 10%. So, then your 96.0516+9 into 0.5 + 0.5 into 97.7169 + 9
divided by 1.1 you got 96.258.
B0C = 0.5[96.0516+ 9] + 0.5 [97.7169 + 9] = 96.258
1.10
(Refer Slide Time: 19:54)
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So now, what we have observed here, is that the value of the call option is different because
actual cash flow depends upon the market price of the bond or the value of the bond at that node
and the call price. So, there is an alternative approach. Whatever approach we have discussed for
the 2-period bond to some extent, we can also use it for these 3 periods collaborative bonds also.
The alternative approach is to go for a valuation of the callable bond.
First of all, you can find out the call option at each node. Then subtract that call option value
from the non-callable bond value. That we have discussed in the previous case. That is called
basically the intrinsic value of the exercising value. So, what is the basic difference here if you
see? The difference from our previous 2 period case, when there are 3 periods or more, we need
to take into account that prior to maturity, the bond issuer to have 2 choices: either the investor
can exercise the option or the investor can hold it for another period.
Then let the value of the holding let VH, if you want to calculate it, be nothing but the present
value of the expected cash value in the next period. So, that means it is basically V H will be your
probability of this VuC + (1 – pr) of the VdC divided by 1 + S. Let us see how this particular
concept is working in this case?
VH = Pr VuC + (1 – pr) VdC
1+S
(Refer Slide Time: 22:13)
Let in period 2, the value of holding is zero at all 3 nodes. Since, next period is maturity, where it
is too late to call. The issuer, though, would find it profitable to exercise in 2 of the 3 cases,
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where the call price is lower than the bond values. The 3 possible callable bond values in period
2, if you want to calculate that is your 97.2346 that is the value of the non-callable bonds minus
the maximum of 97.32346 - 98 and 0.
This will be a minus then minus comma zero, then obviously this 97.2346 - 0 that will be
97.2346. This will be 98.6872 minus the maximum of this and here also maximum will be this
will be basically what, how much will be 0.6872, the 98.6872 – 0.6872 that will be 98. Here also,
it is more than this 99.9771 minus this that will be giving you 98. In all these cases, you got your
3 possible callable bond values in period 2.
Then what will you do? In period 1 at the lower node, the non-callable bond price is greater than
the call price. So, the non-callable bond price is 98.9335 and the callable bond value is 98 then
the intrinsic value what you got that is basically 0.9335. So, if you want to calculate the value of
holding the call if you want to hold the call option. Then what is the value of holding the call?
0.5 into the maximum of 98.6872 – 98, 0 + 0.5 into the maximum of 98.9771 – 98, 0 divided by
1.095 in that lower node that interest rate is 9.5%.
VH = 0.5[Max [98.6872 – 98,0]] + 0.5 [ Max [99.9771 – 98,0]] = 1.2166
1.095
Then you got that is 1.2166, so the issuer would find it more valuable to defer the exercise 1
period. So, as a result, the value of the call option is 1.2166 and the value of the callable bond
will become 97.7169 how? Because the value of the non-colorable bond is 98.9335 and the value
of the particular call option becomes 1.2166 then this minus this will give you 97.7169. So, here
is basically what we are trying to say. This is a little bit different in the sense that there is a
concept of the holding period; it can be whole.
Max [IV, VH] = Max [0.8394, 1.2166] = 1,2166
BdC = BdC - VdC = 98.9335 – 1.2166 = 97.7169
(Refer Slide Time: 26:08)
619
So, then what we have seen in the third step in period 1 at the upper node, the price of the noncallable bond is 96.3612 and the exercise value is 0. Then the value of the call option, in this
case, is equal to its holding value if we calculated 0.3095 how we have calculated? That is the
0.5 into a maximum of 97.2346 - 98 and 0 and the 0.5 into a maximum of 98.6872 – 98, 0
divided by 1.11. 11% is the interest rate at the upper node then you got 0.3095.
VH = 0.5[Max [97.2346 – 98,0]] + 0.5 [ Max [98.6872 – 98,0]] = 0.3095
1.11
So, then the value of the callable bond is 96.3612 which is the value of the non-callable bond
minus your 0.3095 which is basically your 96.0517.
BuC = BuNC – VuC = 96.3612 – 0.3095 = 96.0517
(Refer Slide Time: 27:22)
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So, if you are rolling these 2 periods of possible options values of 0.3095 and 1.2166 in period 1
to the current period. Then we can find the current value of the option and the same callable
bond value using the first approach. So, now you got 0.5 into 0.3095 + 0.5 into 1.2166 divided
by 1.1 that is basically giving you 0.6937. So, then your value of the call option is the value of
the non-callable bond minus the value of the call option, that is your value of the non-callable
bond has become 96.9521, that already we have calculated - 0.6937 that will give you 96.2584.
V0C = 0.5[0.3095] + 0.5[1.2166] = 0.6937
1.10
B0C = B0NC - V0C = 96.9521 – 0.6937 = 96.2584
So, this is basically another approach what you can use for the calculation of the 3-period
callable bond. This is another alternative approach that also can be used.
(Refer Slide Time: 28:44)
621
So, what basically we have discussed if you see the summary of this? Then what basically we
have done here? Other things we have all these things already it is clear to you and now we have
calculated the non-callable bond value is this. Here it is basically this, here it is this, this and this.
These are the non-callable bond values then what we have done? The intrinsic value we have
calculated the intrinsic value basically in this case is if you see that it is 0 then the B uc that
becomes is the same with this.
Suu = 12.1%, BuuNC = 97.2346
IV = Max [BuuNC – CP,0] = Max [97.2346 – 98, 0] = 0
VuuC = IV = 0
BuuC = BuuNC - VuuC = 97.2346 – 0 = 97.2346
Sud = 10.45%, BudNC = 98.6872
IV = Max [BudNC – CP,0] = Max [98.6872 – 98, 0] = 0.6872
VudC = IV = 0.6872
BudC = BudNC - VudC = 98.6872 – 0.6872 = 98
Sdd = 9.025%, BddNC = 99.9771
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IV = Max [BddNC – CP,0] = Max [99.9771 – 98, 0] = 1.9771
VddC = IV = 1.9771
BddC = BddNC – VddC = 99.9771 – 1.9771 = 98
Here it is 0.6872 which is why this value becomes this minus this which will be 98 here also if
you see that it is 1.9771. So, this minus this would give you 98. So, now what has happened? We
have considered these values here. So, now we got this 96.3612 value. So, here it is either 0 or it
is what 0.6872. So, here we have gone 0.5 into 0 + 0.5 into 0.6872 divided by 1.11. We got
0.3095. So, with the maximum of 0 and 0.3095 in the second case, we got 0.
Su = 11%, BuNC = 96.3612
VH = 0.5(0) + 0.5(0.6872) = 0.3095
1.11
IV = Max [BuNC – CP,0] = Max [96.3612 – 98, 0] = 0
VuC = Max [VH, IV] = Max [0.3095, 0] = 0.3095
BuC = BuNC - VuC = 96.3612 – 0.3095 = 96.0516
So, either of these 2 basically if you consider 0 and this you got 0.3095. So, the value of this
particular call option in this node will be this minus this. That will be 96.0516. In the second
case, what have we done? That will be your 1.2166 and here it is 0.9335, then it has become
1.2166 then this 98.9335 – 1.2166 that will give you 97.7169. So, now we again roll back to the
current period.
Sd =9.5%, BdNC = 98.9335
VH = 0.5(0.6872) + 0.5(1.9771) = 1.2166
1.095
IV = Max [BdNC – CP,0] = Max [98.9335 – 98, 0] = 0.9335
VdC = Max [VH, IV] = Max [1.2166, 0.9335] = 1.2166
623
BdC = BdNC – VdC = 98.9335 – 1.2166 = 97.7169
The current period is 0.3095 this is the call option here in this case the call option is 1.2166. So,
then it is 0.5 into 0.3095 + 0.5 into 1.2166 divided by 1.1 that will give you 0.6937. So, in this
period the value of the non-callable bond is this. So, 96.9521 – 0.6937 will give you 96.2584. So,
that is the call option value, in this case, is 96.2584. So, that is the alternative approach of the
summary, whatever we have discussed that just we have presented in one case.
S0 =9.5%, B0NC = 96.9521
V0C = VH = 0.5(0.3095) + 0.5(1.2166) = 0.6937
1.10
B0C = B0NC – V0C = 96.9521 – 0.6937 = 96.2584
(Refer Slide Time: 31:57)
So, what basically we have discussed here is that the value of the 3 period callable bonds can be
determined by discounting the present value of the expected cash flows in each node in each
period. For each case, the minimum of the value calculation or the call price is selected and the
alternative approach for the valuation of callable bonds is to determine the call option at each
node and subtract the value from the non-callable value. That is another approach that also can
be followed.
624
So, these are the references. Thank you
625
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology, Kharagpur
Module No # 08
Lecture No # 39
Binomial Valuation of Puttable and Sinking-Fund Bonds
Welcome back, so in the previous class we discussed the valuation of a callable bond which has
used the 3 period binomial tree models. Today we will be discussing the other embedded options
which are available that is one of the things is that there can be a put feature or there can be
sinking fund provisions.
(Refer Slide Time: 00:47)
So, let us see what those things what we are going to discuss in today's class that is your
binomial valuation of a puttable bond and the binomial valuation of a sinking fund bond. The
bonds having the sinking fund provisions. How that valuation can be made? How the valuation
of those kinds of bonds also can be made? so these are the 2 things that we are going to do in
today's session.
(Refer Slide Time: 01:21)
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So, the keywords mostly will be using the major keywords that are the put option and the sinking
fund.
(Refer Slide Time: 01:26)
So, again will be using the same interest rate tree model the binomial interest rate model starts
with the 10%. It can go up to the first period 11% or 9.5% or it can go up to 12.1, 10.45, 9.025
further it can go up to 13.31 or go down to 11.49 or it can reach to 9.927 or 8.574. So, these are
the different interesting scenarios that we have built up in the beginning. That same example or
same tree we are going to use for valuation of the bonds having the put options or the put
features and the sinking fund provisions.
(Refer Slide Time: 02:16)
627
You know what do you mean by a puttable bond. In the case of callable bond what we have seen
there is a call feature involved and the investor whenever they are holding these callable bonds if
the issuer wants at any point of time they can buy back or call back the bond that is called the
call features and the call price is decided from the beginning. Wherever you go for a puttable
bond the puttable bond means it gives the bond investor or the bond holder the right to sell the
bond back to the issuer at a specified exercise price or we call it the put price.
Generally, the price put price is less than the face value of the bond. So, in contrast to the
callable bonds the puttable bonds basically benefit the holder the risk is not with the holder the
risk is with the issuer. So, in this case if you observe if the price of the bond decreases below the
exercise price; that means whenever the interest rate is quite high then the bond holder can sell
the bond back to the issuer at the exercise price exercise by means that the pre agreed price what
already decided between the issuer and the bond holder.
So, if the price will go down below then it is profitable for the bond holder to sell back that
particular bond to the issuer. From the bondholder’s perspective generally the put option gives a
hedge against the decrease in the bond’s price. So, if the rate is decreasing in the market, then the
bond holder benefits from the resulting higher bond prices. If the rate is increasing then the bond
holder can exercise that option giving some kind of downward protection to them.
628
It gives a kind of protection to them that, below that if the price is going down in the market,
they can exercise that particular option and that agreed upon price they can use it to sell the bond
back to the issuer so that is the basic concept of the puttable bond.
(Refer Slide Time: 04:54)
So, now if you see that whenever you go for the binomial valuation of the puttable bond it
already, we have seen that it gives the bondholder the right to exercise. So, the price of a puttable
bond is nothing but it is equal to the price of the identical non-puttable bond plus the value of the
put option. In the previous case what we have seen is the value of the identical non-callable bond
minus the call option.
B0P = B0NP + V0P
Here in this case what we have seen is the price of the non-puttable bond plus the value of the
put option. As the bond holder generally or will find it profitable to exercise whenever the put
price basically exceeds the bond price. It is profitable for them whenever the price of the bond in
the market is lower and the put option price is basically higher. Then the value of a put option
can be found using the binomial approach by comparing the bond price at each node with the put
price and selecting the maximum of these 2 which one is the maximum.
So, the BtP, P means it is the put that can be either the bond price or the time t or it is the put
price which one is the maximum that actually we have to see. So, if the price of the bond in the
629
market is very low or below this put price then the bondholder can exercise that particular option
so because of that the BtP is equal to the maximum of the Bt or the PP.
BtP = Max [Bt, PP]
(Refer Slide Time: 06:56)
So now the same binomial value also can be found by determining the value of the put option at
each node. And, then pricing this puttable bond as the value of the identical non puttable bonds
plus the value of the put option. If you are using the second approach then the value of the put
option also will be the maximum of either its intrinsic value or exercise value or it is the holding
value; the holding value concept that we have already discussed in the previous session.
IV = Max [PP – Bt, 0]
It is nothing but the present value of the expected put values in the next period that also you can
use. Because the option may not be exercise, they can hold that option that a particular option is
not necessary either they will exercise or it will not be exercise there is option of holding that
particular option also for the next period. So that’s why there is holding value, we are trying to
use it in this particular case.
(Refer Slide Time: 08:02)
630
So, now suppose let us take the example suppose there is a 3 period 9% option free bond. And
here the put price we have fixed it 97 in the period 1 or 2. So here we have 2 periods 1 is period
one another 1 is period two that already we have seen in the previous example also for the
valuation of a 3-period bond callable or puttable we need at least the 2 periods data or 2 period
information by that we can roll back that particular information and find out the value of the
current period.
So, in the period 2 what can be possible using the 2 period trees of 1 period spot rate and the
corresponding bond values for the option free bond. We can start at period 2 and investigates
each of the nodes to determine if there is an advantage for the holder to exercise that particular
option or not. First, we have to see whether the bond holder will exercise that particular option or
it will not exercise. If in all 3 of the cases in period 2 the bond price exceeds this exercise price.
So, there is no possible exercise advantage in this period and each of the possible prices of the
puttable bond are equal to their non-puttable values and the values of each of the put options are
0.
(Refer Slide Time: 09:55)
631
How it is possible we will see that thing because already we have calculated the value of the
without any options accordingly, we are concluding that thing. So let us come to the period 1
period 1 we have 2 nodes one is upper node one is lower node. In period 1 it is profitable for the
holder to exercise when the spot rate is 11%. In the first case it is 10% to 11%. It can so at that
node the value of the non-puttable bond is 96.3612 and the put price is 97 that we have
considered.
BuP = Max [96.3612, 97] = 97
So, the value of the puttable bond is its exercise price of 97. It is the maximum of these 2 that is
96.3612 and 97 so ninety-seven is the maximum so we will consider that. So, the puttable bond
price of 97 also you can find or you can calculate by subtracting the value of the put option from
the price of the non-puttable bond. So, the value of the put option at this node is how much, it is
the maximum of the intrinsic value or the exercise value and the V H.
VuP = Max [IV, VH] = Max [97 – 96.3612, 0] = 0.6388
BuP = BuNP + VuP = 96.3612 + 0.6388 = 97
So, its intrinsic value is basically between 97- 96.3612 and 0. And here obviously this value will
be zero 0.6388 so it will be 0.6388. So, then the value of the put option will be the value of the
non-put options bond divided by the value of the put option so that is 96.3612 plus this that will
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give you 97. So, either you can use or this one or you can use this one both the cases you are
getting that the bond will be exercised and the put price is 97 that will be considered in both the
cases.
(Refer Slide Time: 12:20)
So, in the lower node it is not profitable to exercise nor is there any holding value of the put
option since there is no exercise advantage in the period 2. So, in the lower node the non-puttable
bond price prevails so then current period if you want to calculate rolling the 2 puttable bond
values in the period 1 to the present. So, then it will be the first case it is 97 + 9 second case it is
98.9338 + 9 it will not be exercised because the value is more than 97. So, the same price will
prevent + 9 and you discount it with respect to 10% you get this value of the bond that is
97.2425.
B0P = 0.5 (97 + 9) + 0.5 (98.9335 + 9) = 97.2425
1.10
(Refer Slide Time: 13:20)
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Let us see the alternative approach. What is the alternative approach? The current value also can
be obtained using the alternative approach by computing the present value of the expected put
option values in the period 1. Then, adding that thing to the current value of the non-puttable
bond. With possible exercise values of 0.6388 and 0 in period 1 the current value of the put
option will be 50-50% chance 0.5 into 0.6388+ 0.5 into 0 divided by this that will give you
0.2904. So, in that period the value of the non-puttable bonds is 96.9521+ your 0.2904 that will
give you also the value of the bonds having the put options.
V0P = 0.5 (0.6388) + 0.5 (0) = 0.2904
1.10
B0P = B0NP + V0P = 96.9521 + 0.2904 = 97.2424
(Refer Slide Time: 14:17)
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So, let us see these things. This particular figure will give you the clear idea we are rolling back
from the second period to the first period. This is basically your second period. This is second
period data and this is your first period and this is your current period. So basically, here we start
with 109 every case 109 divided by 1.121 that is 12.1% if the interest rate got 97.2346 then your
BuuP will be the maximum of this Bu and the put price that is 97.2346 and 97 then it is 97.2346
that price will prevail because it is more than the put price it will not be exercised so this price
will prevail. Second case if you see also, you discount it with 10.45% you got this value then if
you see between this and this; this value is higher then again, this value will prevail that’s why
we have seen that, in the second period in all the nodes the bond will not be exercised the put
option will not be going to be exercised by the bondholder.
Suu = u2, S0 = 12.1%
Buu = 109/1.121 = 97.2346
BuuP = Max [Buu, PP] = Max [97.2346, 97] = 97.2346
Sud = ud, S0 = 10.45%
Bud = 109/1.1045 = 98.6872
BudP = Max [Bud, PP] = Max [98.6872, 97] = 98.6872
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Sdd = d2, S0 = 9.025%
Bdd = 109/1.09025 = 99.9771
BddP = Max [Bdd, PP] = Max [99.9771, 97] = 99.9771
Third case third node you see the price of the bond has become 99.9771 compare it with 97
obviously this is higher so this price will prevail. Come to the first period, in the first period what
is basically happening we got this there is a 50% chance it can become this, 50% chance it can
become this. So, we calculate this expected value of that 0.5 into this + 9 + 0.5 into this + 9 that
is the coupon the cash flow.
Su = u, S0 = 11%
Bu = 0.5(97.2346 + 9) + 0.5(98.6872 + 9) = 96.3612
1.11
BuP = Max [Bu, PP] = Max [96.3612, 97] = 97
This cash flow and this cash flow +9 obviously is the coupon what you are going to receive then
if you discount it with respect to that period's discount rate that is 11% you got 96.3612 now if
you compare it with your put option price that is 97, then 97 is more than this. So now 97 is more
than 96.3612 so then this price will prevail. That is 97. Second case if you see now your cash
flow is 98.6872 and your 99.9771.
Sd = d, S0 = 9.5%
Bd = 0.5(98.6872 + 9) + 0.5(99.9771 + 9) = 98.9335
1.095
BdP = Max [Bd, PP] = Max [98.9335, 97] = 98.9335
So, 0.5 into 98.6872 + 9 + 0.5 into 99.9771 + 9 divided by discount rate is 9.5% you got
98.9335. And if you compare this and this, this is higher so we retain this 98.9335. So now you
got the cash flow of 97 another one is 98.9335 now you come back to the current period so it is
discount rate is 10%; 0.5 into 97 here in this case we have considered the 97 + 9 + 0.5 into
98.9335 +9 divided by 1.1 that will give you 97.2425.
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S0 = 910%
B0 = 0.5(97 + 9) + 0.5(98.9335 + 9) = 97.2425
1.10
That is what the value of the current period price of that particular bond is, if you are considering
this the put option features into the considerations. So, this is for a binomial valuation of a 3period puttable bond.
(Refer Slide Time: 18:31)
Then if you see that alternative case, what is going to happen? In the alternative case, what we
have seen we have to find out the intrinsic value. The intrinsic value you see if you minutely
observe here, we got this value is 97.2346 and the intrinsic value become this maximum of the
put price minus this and 0 then 97 - 97.2346 that value is minus and 0 then that’s why the value
will be 0 then obviously the value of the BuuP that value of the bond having the put option that
become 97.2346 + 0 that will become this.
Suu = 12.1%, BuuNP = 97.2346
IV = Max [PP - BuuNP,0] = Max [97 - 97.2346, 0] = 0
VuuP = IV = 0
BuuP = BuuNP + VuuP = 97.2346 + 0 = 97.2346
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Sud = 10.45%, BudNP = 98.6872
IV = Max [PP - BudNP,0] = Max [97 – 98.6872, 0] = 0
VudP = IV = 0
BudP = BudNP + VudP = 98.6872 + 0 = 98.6872
Sdd = 9.025%, BddNP = 99.9771
IV = Max [PP – BddNP,0] = Max [97 – 99.9771, 0] = 0
VddP = IV = 0
BddP = BddNP + VddP = 99.9771 + 0 = 99.9771
Second case, also if you see it is 98.6872 so 97 minus this, 0 this will also 0 then the value will
remain the same 98.6872. Third case also 0 the value will remain the same 99.9771; now what
has happened that come back to here, the intrinsic value is 0 here it is 0 then it becomes 0 then
you are between this 0 and this 97 - 96.3612 if you get the value; this value 0.6388. So that’s
why this value of the put option becomes 0.6388 and this is 96.633612 + 0.6388 that 97 is the
value of the bond having the put option.
Su = 11%, BuNP = 96.3612
VH = 0.5(0) + 0.5(0) = 0
1.11
IV = Max [PP - BuNP,0] = Max [97 – 96.3612, 0] = 0.6388
VuP = Max [VH, IV] = Max [0, 0.6388] = 0.6388
BuP = BuNP + VuP = 96.3612 + 0.6388 = 97
Sd = 9.5%, BdNP = 98.9335
VH = 0.5(0) + 0.5(0) = 0
1.095
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IV = Max [PP – BdNP,0] = Max [97 – 98.9335, 0] = 0
VdP = Max [VH, IV] = Max [0, 0] = 0
BdP = BdNP + VdP = 98.9335 + 0 = 98.9335
Second case, also you have seen that the intrinsic value becomes 0 so because of this the value of
the bond of having the put option will remain same with this. Then in this case what we have
considered that is here it is 0.6388 and here also it is basically 0 then we have discounted it with
respect to the 10% then you got 0.2904 then this value of the option bond having the put option
will become your 96.9521 + 0.2904 then it will become 97.2425 this is also another approach
what you can use for valuation of this.
S0 = 10%, B0NP = 96.9521
V0P = 0.5(0.6388) + 0.5(0) = 0.2904
1.10
B0P = B0NP + V0P = 96.9521 + 0.2904 = 97.2425
(Refer Slide Time: 21:08)
Then we have another feature also the bond can have that is called the sinking fund bonds. So,
the sinking fund arrangements generally what, here the issuer has the option to buy some of the
bonds back either at their market price or at a call price. So, in this case the issuer basically
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makes the scheduled payments into a fund or buys off a certain proportion of the bond issue in
each period.
Often when the sinking fund agreement specifies the orderly retirement of the issue, the issuer is
given an option of either purchasing the bonds in the market or calling the bonds at a specified
call price. So that is basically what we call the sinking-fund provisions of the bond so that can be
also another embedded option can be prevailed with respect to certain types of bonds.
(Refer Slide Time: 22:25)
So, the call option embedded in the sinking fund agreement makes the sinking fund valuable to
the issuer. So, if the interest rates are relatively high then the issuer will be able to buy back the
requisite amount of the bond at a relatively low market price. If rates are low and the bond price
high though also, the issuer will be able to buy back the bond on the call option at the call price.
So, the sinking fund bonds or bonds having the sinking fund provisions with this type of call
provision should trade at a lower price than the identical non-sinking fund bonds. Lot of risk is
involved from the investor point of view if you are having the sinking fund provisions.
(Refer Slide Time: 23:29)
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So, in this case how we can use our binomial valuation model for a sinking fund bond; bonds
having the sinking fund provisions. The sinking fund arrangement generally requires an
immediate exercise or bond purchase at a specified sinking fund date. So, the possible values of
the sinking fund’s call feature at those rates trades are equal to their intrinsic values. Generally,
this differs from the valuation of a standard callable bond where a holding value is also
considered in determining the value of the call options that we have already discussed in the
previous class.
So, the callable bonds with the sinking fund provision are relatively difficult for the valuation in
comparison to the bonds having only the call features or only the call options.
(Refer Slide Time: 24:37)
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Suppose a company issues 21 million rupees 3 period bonds with the sinking fund obligations
requiring: The issuer sink rupees 7 million of the face value after the first period and 7 million
after the second period assume. And the issuer has the option of either buying the bonds in the
market or calling them at a call price of 98. Assume the same interest rate tree and one value
which is characterizing the 3 period valuations.
Whatever we have discussed coupon is 9% what already we have discussed previously without
its sinking fund agreement. But now whenever you are bringing the sinking fund agreement the
issuer has 2 options at the end of the period 1 the issuer can buy this 7 million worth of the bond
either at rupees 98 or at the bonds market rise. At the end of the period 2 the issuer has another
option to buy 7 million worth of the bond either at 98 or the market price.
(Refer Slide Time: 26:08)
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So, then what basically will happen in this case. So, already we know that 10% has become 11%
or 9.5%. So, the value of a non-sinking fund in this case will be 96.3612; so, what is the intrinsic
value, the maximum of this minus call price and 0 or otherwise 0. So, this is 96.3612 – 98, 0 then
obviously this will be minus then this will be 0. Second case the price becomes 98.9335 which is
obviously more than the call price. So that’s why it will be 98 call prices of the 90 so that’s why
this will become the 0.9335.
Su = 11%
BuNSF = 96.3612
IV = Max [BuNSF – CP, 0] = Max [96.3612 – 98, 0] = 0
Sd = 9.5%
BdNSF = 98.9335
IV = Max [BdNSF – CP, 0] = Max [98.9335 – 98, 0] = 0.9335
So, now what you have to do here; the intrinsic value is 0, here the intrinsic value is 0.9335. So
now we are discounting it here 0.5 into 0 + 0.5 into 0.9335 divided by 1.1 that will give you your
0.4243.
S0 = 10%
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V0SF(1) = 0.5(0) + 0.5(0.9335) = 0.4243
1.10
(Refer Slide Time: 27:40)
Then what basically here we have seen, second case whenever you go for let you are calling back
the bond in the period 2 and sinking fund call price is 98. So again, you already know this value
of the bond without any sinking fund provisions NSF non-sinking fund that is 97.2346 then the
maximum of this is 0 between this and this here it is 98.6872 between this and this, it is 0.6872 is
the maximum. Here in this case, it is 1.9771 is the maximum.
Suu = 12.1%, BuuNSF = 97.2346
IV = Max [BuuNSF – Cp, 0] = Max [97.2346 – 98, 0] = 0
Sud = 10.45%, BudNSF = 98.6872
IV = Max [BudNSF – Cp, 0] = Max [98.6872 – 98, 0] = 0.6872
Sdd = 9.025%, BuuNSF = 99.9771
IV = Max [BddNSF – Cp, 0] = Max [99.9771 – 98, 0] = 1.9771
Then you take these values here it is the value of the non-sinking front bonds has already we
have calculated that is 96.3612 then you take 0 here 0.6872 here multiply this discount it with
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respect to 1.11 you got 0.3095. Here in this case, you take this, take this you got 1.2166 then take
these 2 values here and again discount it with respect to 1.1or 10%. So, 0.5 into 0.3095 + 0.5 into
1.2166 that will give you 0.6937 in the second period.
Su = 11%, BuNSF = 96.3612
VuSF(2) = 0.5(0) + 0.5(0.6872) = 0.3095
1.11
Sd = 9.5%, BdNSF = 98.9335
VdSF(2) = 0.5(0.6872) + 0.5(1.9771) = 1.2166
1.095
S0 = 10%, B0NSF = 96.9521
V0SF(2) = 0.5(0.3095) + 0.5(1.2166) = 0.6937
1.10
(Refer Slide Time: 29:20)
So, in this case what we have seen is that since each option represents one by third of the issue
then the value of the bond's sinking fund option is 0.3727 and the value of the sinking fund bond
is 96.5794 for 100 rupees face value. So how you got it that basically here 1 by 3 into 0.4243 + 1
by 3 into 0.6937 that becomes 0.3727. So, your 96.9521 - 0.3727 that would give you 96.5794.
V0SF = (1/3)(0.4243) + (1/3)(0.6937) = 0.3727
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B0SF = B0NSF - V0SF = 96.9521 – 0.3727 = 96.5794
And the issue value will be your 96.5794 divided by 100 into your 21 million was the value of
the face value of the particular bond if you multiply this you got this. So, the total value of this
particular bond will be this much. That is the way basically we can use the valuation of the
binomial tree for the bond having the sinking fund provisions.
Issue Value = 96.5794 * 2100000 = Rs. 2028674
100
(Refer Slide Time: 30:40)
So, what basically we have discussed; the puttable bond basically gives the holder the right to
sell back the bond at a particular price. And generally, it is considered as a hedge against the
decrease in the bond prices and the price of a puttable bond is equal to the price of an identical
non puttable bond plus the value of the put option. Sinking fund arrangements generally in which
the issuer has the option to buy some of the bond back either at their market price or at a call
price.
And the call option embedded in the sinking fund agreement makes the sinking fund valuable to
the issuer perspective.
(Refer Slide Time: 31:20)
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So, these are the references what you can see, thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Module No # 08
Lecture No # 40
Binomial Valuation of Convertible Bonds
Good morning and welcome back. So, we were discussing about the valuation of the different
bonds which have the embedded options. And we discussed about the bonds having the call
feature, bonds having the put features, and as well as the sinking fund provisions. And there we
have seen that how the call features or the put features or the sinking fund provisions are going to
change the value of that particular bond.
Mostly we have used the binomial interest rate tree model for the valuation of this kind of
securities. So today we will be discussing about another type of bonds which have some
embedded options like the convertible features. So that is why we will see that how the binomial
evaluation model is going to help us for valuation of the convertible bonds.
(Refer Slide Time: 01:12)
So, we will be discussing about 2 things here valuation of the convertible bonds and as well as
valuation of the callable convertible bonds. So, if the convertible bond also has the call feature,
then we will see that how the valuation of those bonds generally carried out.
(Refer Slide Time: 01:34)
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So, you will come across certain keywords during this particular session. You have the
conversion ratio, you have the conversion value, straight debt value, conversion price, and all
these things. So, these are the some of the keywords that you will come across while discussing
about this particular topic.
(Refer Slide Time: 01:56)
So let us understand first what do we mean by a convertible bond; the convertible bond is
basically nothing but it particularly kind of a bond where the bond can be converted into equity
after a specified point of time. So that’s why we can say that it gives the holder the right to
convert the bond into a specified number of stocks. The specified number of stocks or specified
number of shares of stock can be converted from the bond.
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So, if you are holding a bond instrument after certain point of time you have the right to convert
that particular bond into equity, so that is basically what we call it the convertible bond. From the
investor point of view the convertible bonds basically offers the potential for a high rate of
return, if the company does well and its stock price basically increases. And also, there are some
downside risks involved in the convertible bond and there are some kinds of we can say that
downside protection also you can get if you are holding a convertible bond.
And what kind of protection basically at what time the protection you can get? The protection
basically you can get whenever the stock price declines. So, if you look at the convertible bonds,
they are usually callable in nature, where the convertible bondholder has the right to convert the
bond to the stock if the issuer does the call. If the issuer is going for utilizing this call option,
then the convertible bondholder has the right to convert that particular bond into the stock. So,
from the investor point of view, it is a kind of advantage what the investor can get.
(Refer Slide Time: 04:05)
So, there are certain terms generally we come across while discussing about the convertible
bond. The first one is the conversion ratio, so what the conversion ratio means? it is basically
nothing but the number of stocks or number of shares of the stock which can be converted when
the bond is tender for the conversion. Then we have the conversion value, the conversion value is
nothing but it is the conversion or convertible bonds value as a stock.
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So, at a point of time the conversion value is nothing but it is the conversion ratio times the
market price of the stock. So, whatever the conversion ratio is there multiplied by the price of the
stock at that particular point of time that will give you the conversion value.
CVt = (CR)PtS
Then also we can use another term while discussing about the convertible bond that is called the
straight debt value, it is basically nothing but the convertible bonds value as a non-convertible
bond. So, this value is generally obtained by discounting the convertibles cash flow or
convertible bonds cash flow, by the discount rate on a comparable non-convertible bond. Then
we have the conversion price; the conversion price is nothing but it is the bonds par value
divided by the conversion ratio. Let the conversion ratio is 10 the par value is 1000, then your
conversion price will be 100. So, these are the different types of terms generally we come across
whenever we discuss about the convertible bonds.
(Refer Slide Time: 05:58)
So then, whenever you talk about the convertible bond, generally whenever we look at the
market the arbitrators generally always ensure that the minimum price of a convertible bond is
the greater of either of its the straight debt value or its conversion value. So that means it is you
have to look at the maximum of the conversion value or the straight debt value that basically we
have to consider in that particular case.
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Min BtCB = Max [CVt , SDVt]
So that means if a convertible bond is priced below its conversion value, then what the
arbitrageurs can do; the arbitrageurs can buy it that particular bond, converts that particular bond
into the stock then sell the stock in the market to earn a risk less profit. And generally, the
arbitrageurs seeking such opportunities whenever the arbitrageur will go for this kind of
operations that will push the prices of the convertible up until it is equal to its conversion value
right.
So, if the convertible bond is priced below the conversion value, then arbitrageur can buy this
convert that particular bond, into the stock and sell it in the stock market. And in that process, he
can create the arbitrage opportunity or he can generate some riskless profit and, in this process,
whenever the process will go on then obviously the demand for these activities will push the
prices of the convertibles, and it will reach particular level which will be equal to the value of the
equal to the conversion value.
So, if a convertible is selling below this SDV, then the arbitrageur could profit by buying the
convertibles and selling it as a regular bond instead of converting into the stock they can create
the profit by selling it as a regular bond. So, this is the way the mechanism generally works in
the market.
(Refer Slide Time: 08:42)
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Then if the convertible is callable, let the convertible bond is callable, then the call price at which
the issuer can redeem that particular bond generally places a maximum limit on the convertibles,
or the convertible bonds. What does it mean, means the issuer will find it profitable to buy back
the convertible bond once its price is equal to the call price.
So, buying back the bond generally what will happen once they will go for buy backing the bond
it will basically freeze the company to sell the new stock or the bonds at prices higher than the
stock, or the straight debt values which are generally involved or associated with these
convertibles. So, the maximum price of a convertible bond is the call price.
And the actual price that a convertible will trade generally will be at a premium above its
minimum value what is below its maximum value. That basically is the general observations
always we see whenever we are dealing with the convertible bonds.
(Refer Slide Time: 10:25)
Let us see that how the convertible bonds valuation is taking place through this binomial models.
The valuation of a convertible bond which has some embedded call option is more difficult than
the valuation of a bond with just one option feature, either it is a convertible feature or it is a call
feature. In comparison to the valuation of those kind of bonds, the bonds having a call feature
and as well as a convertible feature is relatively difficult, the valuation of these kind of bonds is
relatively difficult in nature.
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Then whenever we are going for a valuation of the convertible bonds what generally we should
look at; first of all, we have to look at the interested patterns. How this interest rate is going to
move number one. And because the stock is involved in this particular process it is very much
important to understand what is the random pattern of the stock prices, how the stock prices is
moving, how the fluctuation of the prices is happening, that also we have to look at.
Then obviously the correlation between the interest rate and the stock prices that also has to be
considered, we have to consider that thing also. So, these particular considerations or items are
very much required whenever we are going for the valuation of the convertible bonds.
(Refer Slide Time: 12:15)
Now let us see that how this particular evaluation is done let us start with an example. Let there
is a three-period 10% convertible bond, 10% means it is the coupon is 10%, face value is 1000
rupees and the conversion ratio is 10. That means the convertible into 10 shares of the underlying
company stock the conversion ratio is 10. So here you are taking certain assumptions, then what
are those assumptions you are taking: the bond has no call option and no default risk.
And current yield curve is flat and it is remaining at rate 5% and the yield curve is not going to
change it will remain at 5% in this particular 3 years that means we are assuming no market risk
involved. So, the convertible bonds underlying stock price let we are assuming follows a
binomial process. So where in each period it has an equal chance, it can either increase which is
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the value will be equal to u times its initial value, or it is decreased to d times the initial value,
that u and d concept that already we have discussed extensively.
And in continuation with that value whatever we have considered while discussing about the
binomial interest rating model, we have assumed that u = 1.1 and d = 1/1.1. It is generally
happens for the large samples that will see further it will be 0.9091. And q that we are assuming
the probability of stock increasing in one period let that is 0.5 there is a 50% chance it may
increase there is a 50% it may decreasing let the current stock price is 92 rupees ok.
So, at the maturity that means the end of the period of 3 or at end of the 3 years, the bond holder
will have a coupon which will be 100 rupees and will either convert the bond into stock or
receive the principal of 1000 rupees which is the par value of the bond right. So, these are the
data or the assumption what basically we have considered to understand the binomial valuation
of the convertible bond. Using this particular data or using this particular assumption, let us see
how that particular valuation basically goes on.
(Refer Slide Time: 15:11)
If you see this particular example whatever basically we have taken here, what we have taken the
stock price is 92, u is 1.1 d is 0.9091, spot rate is 5%, convertible bonds coupon is 100 and par
value or the face value is 1000, conversion ratio is 10 that we have taken. Now what basically
will happen whenever it was basically consider in this particular context, how this interest rate or
the price can move.
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Let the price was 92 then the conversion value is 92 into 10 that is 920, then either it can increase
up to a multiplier of 1.1 or it can decline up to a multiplier of 0.9091. So, in that case the price
can become 101.2 or it can come down to 83.64 further. So obviously the conversion value will
change that will be multiplied by 10 that will be 1012 it will be 836.40. Further it can go up to
111.32, again this(101.2) multiplied by 1.1 or it can decline also from this to 92 or again here it
can reach up to 92 or it can decline to 76.03.
Further if you are considering again there is a chance to increase, then it can go to next period in
the third period it can reach here the conversion value will become this, or it can reach here
conversion value become this, or it can reach here conversion value will become this, or it can
reach here the conversion value will become this right. So then here we will see that what
basically happens here, so let us see that here the conversion value has become 1244.50 your
face value of the bond is 1000 coupon is 100.
So now if you are calculating this the convertible bonds value it should be maximum of the CV
or F + C, that means it will be 1000 + 100 that is 1100 or it will be 1244.50, whichever is the
maximum. It is either maximum of the face value or the conversion value: the conversion value
is 1244.50, face value is 1000. Then it has become basically you have to consider 1244.50 + your
coupon what you are getting so that means 1224.50 + 100 so this value basically we have to
consider.
PuuuS = 122.45
CV = 1224.50
F = 1000
C = 100
BuuuCB = Max [CV, F] + C = Max [1224.50, 1000] + 100 = 1324.50
Second case also we have seen that it is more than this face value that’s why this value of the
bond will be this, in this case if you see compare the conversion value with the face value, the
face value is higher that is why we will get 1100 in this case also will be getting 1100, whichever
is the maximum that basically we have already discussed.
PuudS = 101.20
CV = 1012
F = 1000
C = 100
BuudCB = Max [CV, F] + C = Max [1012, 1000] + 100 = 1112
PuddS = 83.64
CV = 836.4
F = 1000
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C = 100
BuddCB = Max [CV, F] + C = Max [836.4, 1000] + 100 = 1100
PdddS = 69.12
CV = 691.2
F = 1000
C = 100
BdddCB = Max [CV, F] + C = Max [691.2, 1000] + 100 = 1100
Then we will see that how this particular price of the bond is calculated in this node in the
second period. In the second period if you see already, we have taken that probability is 50 % the
q = 0.5 that basically we have considered, then it will be 0.5 into 1324.5 + 0.5 into 1112 and you
have to discount it at a rate of 5 % that is 1.05. So, in the first node in the second period the value
of the bond can be 1160.24. So, if you consider between this conversion value and this this value
whatever you are calculating then you will get your conversion value is 1113.20 that is 113.2
into 10. So now this one is the more than this; that’s why the value we have to consider this.
PuuS = 111.32
CV = 1113.20
Suu = 5%
Buu = 0.5(1324.50) + 0.5(1112) = 1160.24
1.05
BuuCB = Max [CV, Buu] = Max [1113.20, 1160.24 ] = 1160.24
Second case if you look at it is 920 and it is 1053.33 that basically this value and this value we
have considered and accordingly we have calculated this then we have to consider this.
PudS = 92
CV = 920
Sud = 5%
Bud = 0.5(1112) + 0.5(1100) = 1053.33
1.05
BudCB = Max [CV, Bud] = Max [920, 1053.33] = 1053.33
Third case again the if you look at your conversion value is this but your bond value in this case
is this that’s why we have to consider this then value will be this.
PddS = 76.03
CV = 760.30
Sdd = 5%
Bdd = 1100 = 1047.62
1.05
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BddCB = Max [CV, Bdd] = Max [760.30, 1047.62] = 1047.62
So now you consider these values to calculate the price of the bond in the first period, so we have
taken 0.5 into 1160.24 + 100 + 0.5 into 1053.33 then you discount it you get this value. Then you
get this one is the maximum of this then you consider this.
PuS = 101.20
CV = 1012
Su = 5%
Bu = 0.5(1160.24 + 100) + 0.5(1053.33 + 100) = 1149.32
1.05
BuCB = Max [CV, Bu] = Max [1012, 1149.32] = 1149.32
Then again, the same thing you can get this value 1095.69.
PdS = 83.64
CV = 836.4
Sd = 5%
Bd = 0.5(1053.33 + 100) + 0.5(1047.62 + 100) = 1095.69
1.05
BdCB = Max [CV, Bd] = Max [836.4, 1095.69] = 1095.69
Then finally you use this particular thing here you get this total value of the bond is 1164.29. So
here what basically we have seen that the particular value of this convertible bond has become
1164.29, considering these assumptions whatever we have taken.
P0S = 92
CV = 920
S0 = 5%
B0 = 0.5(1149.32 + 100) + 0.5(1095.69 + 100) = 1164.29
1.05
B0CB = Max [CV, B0] = Max [920, 1164.29] = 1164.29
(Refer Slide Time: 21:34)
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Let us already we have explained this I will just give you some cases in period 3 whenever the
stock price is 122.45 this value has become this 1324.5 whenever in the period 3 at price of
this(101.20) the value has become this(1120) and period 3 this(83.64 & 69,12) value has become
this(1100) like that. Already we have explained this particular process in the exhibit whatever we
have just now explained. So, like that in period 2 the value of the convertible bond is equal to the
maximum for either the present value of the bonds expected value at maturity or its conversion
value.
So here the at all three stock prices the present value of the bonds expected values next period
are greater than the bonds conversion value. So that’s why we have considered this 1324.50 then
1112 then we get this(1160.24) then finally we have to consider this(1160.24) value.
(Refer Slide Time: 22:37)
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Then it will go on then in the period 1 at each bond the possible bond values in period 1
generally generated by rolling the 3 convertible bonds value in period 2 to the period 1, also
exceed their conversion values. So finally, we obtain a convertible bond value that is 1164.29
that already we have discussed.
(Refer Slide Time: 23:08)
So now what basically we can do we have to see if the particular convertible bond has a call
feature, then what will happen. So, with a callable convertible bond the issuer will find it
profitable to the call the convertible prior to the maturity, whenever the price of the convertible is
greater than the call price that already you know. That if the price of the convertible bond is
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greater than the call price then the issuer will find it profitable to call back the bond or utilize this
call option.
So, when the convertible bondholder is faced with a call then what he or she usually do she has
the choice either tendering the bond at the call price or converting into the stock, that only we
have explained or we have discussed before. So, since the issuer will call whenever the call price
basically exceeds the convertible bonds price in effect forcing the holder to convert it will force
the bond investor to convert.
So, by doing this the issuer basically takes away the bond holders value of the holding the
convertible and forcing the convertible bond price equal to its the conversion value, that
basically can happen. So, in that process we will see that how that particular mechanism works.
(Refer Slide Time: 24:47)
Let us take the example in this case we have already considered this values stock price is 92, u =
1.1, d = 0.9091, spot rate 5%, coupon is 100, face value is 1000, conversion ratio 10 and we have
taken the callable at call price of 1100; that means the call price we have considered 1100.
Already I have discussed with you that the price can reach here from this(92) to this(101.20) it
can reach here(83.64) or it can reach in the second period, it can reach these 3 figures(111.32, 92,
76.03).
Then in the third period it can reach the four figures(122.45, 101.20, 83.64, 69.12). The
conversion price is mentioned(1224.50), here is the conversion price(1012), so this is the
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conversion price(836.4)(691.2), so all the conversion prices are mentioned here. So, the
maximum price is here(1324.50), this is here(1112), it is here(1100), it is here(1100). Now what
will happen that, if you are calculating this, then how basically the value of the convertible bond
with a call feature can be calculated.
So here the value is 1160.24. But here whenever we have said that it is more than the call price
then the call option will be exercised. So, if the call option will be exercised, then the value will
become that 1113.20 ok, whichever is the maximum between the conversion value and the
minimum of the call price and the value of the call convertible bond. So, the value of the
minimum of this means it is basically the it is the 1100 this part conversion value is 1113.20.
Then if you take the maximum of this 2 then it is 1113.20. Like that here will find that it will be
1053.33, here it will be 1047.62.
CV = 1113.20
CP = 1100
Buu = 0.5(1324.50) + 0.5(1112) = 1160.24
1.05
BuuCB = Max [CV, Min[Buu , CP]] = 1113.20
CV = 920
CP = 1100
Bud = 0.5(1112) + 0.5(1100) = 1053.33
1.05
BudCB = Max [CV, Min[Bud , CP]] = 1053.33
CV = 760
CP = 1100
Bdd = 1100 = 1047.62
1.05
BddCB = Max [CV, Min[Bdd , CP]] = 1047.62
Then again if you discount it using this, particular values then it will become 1126.29, it will be
1095.69. Then finally we have considered here it will be 1100, it will be 1095.69. Then if you go
for utilizing these values for the valuation then you will get 1140.80 right. So now the value of
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the bonds having the convertible bonds having the call feature the value has been changed, in
comparison to only the bonds having the convertible features.
PuS = 101.20
CV = 1012
Su = 5%
CP=1100
Bu = 0.5(1113.20 + 100) + 0.5(1053.33 + 100) = 1126.92
1.05
BuCB = Max [CV, Min[Bu , CP]] = 1100
PdS = 83.64
CV = 836.4
Sd = 5%
CP=1100
Bd = 0.5(1053.33 + 100) + 0.5(1047.62 + 100) = 1095.69
1.05
BdCB = Max [CV, Min[Bd , CP]] = 1095.69
P0S = 92
CV = 920
S0 = 5%
B0CB = 0.5(1100 + 100) + 0.5(1095.69 + 100) = 1140.80
1.05
(Refer Slide Time: 28:05)
So, this is the explanation that already I have explained with you or we have discussed just now;
so, in the different nodes how this particular value is going to be changed.
(Refer Slide Time: 28:20)
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And the different period at the different nodes, then finally we have we find that it is 1140.80 and
the callable convertible bond value is less than the non-callable convertible bond value of
1164.29.
(Refer Slide Time: 28:40)
Then we have another kind of options here the binomial valuation of callable convertible bonds
with the different interest rates, assume this interest rate is changing here we have taken 5%. So
let you assume that the interest rate is changing so in that case what will happen. So first we use
the correlation or the regression analysis to estimate the relationship between the stock price and
the spot rate.
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And given the binomial model of the spot rates identify the corresponding stock prices. Then
given a binomial model of the stock prices identify the corresponding spot rates.
(Refer Slide Time: 29:20)
So, for example let you have run a regression, and you find your equation like this ok, between
the stock and the spot rate. Then let using this equation the spot rate associated with the stock
prices from the 3-period tree will be this. These are the stock prices and these are the spot rates
you have estimated from this particular equation.
St = 0.16 – 0.001PtS
(Refer Slide Time: 29:48)
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Then what you can do you can use that particular interest rates in the different stages other things
basically, you are keeping the same if you see this one. But here what has happened we have our
interest rate which was there previously 5% in this period it is 4.87 and this node it is 6.8, it is
8.4, it is 5.9, it is 7.64. So now your discount rate has been changed. So, if your discount rate has
been changed, then automatically you will find that the prices also are going to be changed, so
all these prices are going to be changed.
CV = 1113.20
CP = 1100
Suu = 4.87%
Buu = 0.5(1324.50) + 0.5(1112) = 1161.68
1.0487
BuuCB = Max [CV, Min[Buu , CP]] = 1113.20
CV = 920
CP = 1100
Sud = 6.8%
Bud = 0.5(1112) + 0.5(1100) = 1035.58
1.068
BudCB = Max [CV, Min[Bud , CP]] = 1035.58
CV = 760.30
CP = 1100
Sdd = 8.4%
Bdd = 1100 = 1014.76
1.084
BddCB = Max [CV, Min[Bdd , CP]] = 1014.76
PuS = 101.20
CV = 1012
Su = 5.9%
CP=1100
Bu = 0.5(1113.20 + 100) + 0.5(1035.58 + 100) = 1108.96
1.059
BuCB = Max [CV, Min[Bu , CP]] = 1100
PdS = 83.64
CV = 836.4
Sd = 7.64%
Bd = 0.5(1035.58 + 100) + 0.5(1014.76 + 100) = 1045.31
1.0764
BdCB = Max [CV, Min[Bd , CP]] = 1045.31
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CP=1100
P0S = 92
CV = 920
S0 = 6.8%
B0CB = 0.5(1100 + 100) + 0.5(1045.31 + 100) = 1097.99
1.068
Then finally you will find that this price here is 1097.99, this is the basically the value. So now
what basically we have observed that whenever the discount rate will change automatically the
value of the bond is also going to be changed.
(Refer Slide Time: 31:03)
So, what this exhibit basically shows the binomial tree of a stock prices along with their current
spending spot rates given the spot rate and the stock price relation and the convertible bond value
has changed to 1097.99. So, the value is lower than the previous case in which we assumed a
constant yield curve of the 5%. So now on our interest rate also we are changing automatically
your value of the bond is also going to be changed.
(Refer Slide Time: 31:36)
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So, what we have discussed that a convertible bond gives the holder the right to convert the bond
into a specified number of stocks. Convertible bonds are usually callable and the convertible
bond holder usually have the right to convert the bond to stock. If the issuer is going for a call,
then the arbitrageur always ensures that the minimum price of a convertible bond is greater of
either its straight debt value or its conversion value.
So, with a callable convertible bond the issuer will find it profitable to call the convertible prior
to the maturity whenever the price of the convertible bond is greater than the call price. And
whenever you have the different features either call features or the differential interest rate
features. Then you will find that the value of the bonds are going to be changed.
(Refer Slide Time: 32:29)
So, these are the references you can go through thank you.
668
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology – Kharagpur
Lecture – 41
Estimation of Binomial Trees – I
Welcome back. So, in the previous sessions we discussed about how the binomial models or
binomial interest rate tree models are going to help us for the valuation of the bonds and as
well as the valuation of the bonds having the different kind of embedded options like the call
feature, sinking fund provisions, convertible bonds having the call feature and all these
things.
But the question here is, there are two issues generally comes to our mind. there we have
taken that the interest rate can change in one year that either in one period it can go up to a
particular amount or it can go down to a particular amount, but if you change this interest rate
in a short horizon or frequently, we want to see that how the interest rate movements are
happening?
And how they are going to affect the value of the bond that will be more realistic in nature
that is number one and number two basically you have to keep in the mind we have
considered that u value is 1.1, d value is something 0.95 or 0.90. So, how this u and d are
calculated? what is the basic understanding for the estimation of that u and d. So, these are
the two important questions always we should keep in the mind.
And what are those methods are generally used to calculate this value of u and d. So, in
today’s class basically we will start this estimation of the binomial trees.
(Refer Slide Time: 01:57)
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So, here first thing we will discuss the subdivision of the binomial trees, that means we are
frequently changing this interest rate number one. Number two, we have to also see what are
those different approaches or different methods are used to estimate the binomial interest rate
movements? Mostly our botheration is our objective is to find out the values of u and d. So,
this is what the basic objectives of the two densities.
(Refer Slide Time: 02:39)
So, in this process you will understand or you will come across certain keywords like
binomial process, binomial distribution and all these things.
(Refer Slide Time: 02:49)
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So, let us first discuss about the subdivision of the binomial tree. So, if you look at the
binomial model whatever we have used or we have formulated for the valuation of these
particular bonds it will look more realistic, when we subdivide the periods to maturity into a
number of sub periods that means we are increasing the number of periods and by that your
nodes are going to be changed. in the different periods the nodes are basically going to be
changed.
So, when the number of sub periods increase, the length of each period become smaller
making the assumptions that the spot rate will either increase or decrease more logically. that
means we are assuming that the spot rates are going to be changed frequently and the number
of possible rates at maturity also is going to increase which again add some kind of realism to
the model.
It will be more realistic in that particular sense because the interest rate can change
frequently. So, that also bring certain kind of realism into our analysis. Now, let us see that
how basically it will look?
(Refer Slide Time: 04:51)
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So, whenever we are using a three-period binomial tree what basically we have discussed in
the previous sessions? There we have seen that the length of each period is one year.
Suppose, we want to do a valuation of a bond using a six-period tree with the length of each
period being six months. Let we are assuming the length of each period is six months then it
will be automatically a six-period tree.
Then if you are going to do that then, what are those changes or the adjustments we are going
to make? how we are going to make the adjustments in terms of the inputs which are used for
this particular analysis? First of all divide this one year spot rates in the annual coupon by
two because we have assumed that the length of each period is six months. So, we have to
divide the one period spot rates and the annual coupon by two.
Accordingly, adjust the u and d parameters to reflect the changes over the six-month period
instead of one year. U and d parameters whatever you are considering to find out the value of
that particular price of the asset or the value of that particular asset either in the increasing
scenario or in the decreasing scenario that also we are going to adjust it. To define this
binomial tree of spot rates for five periods each with a length of six months.
So, in general if you want to see let h = the length of the period in years and n = the number
of periods of length h defining the maturity of the bond, where your n = maturity in years
divided by h. So, let us see how basically let us take some examples how this thing is
basically going to be changed or how the numbers are going to be basically differ?
(Refer Slide Time: 07:33)
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So, here if you see for a three year bond let you are going to evaluate that bond over quarterly
basis. If you are going to do the evaluation on the basis of the quarterly basis then h = your 1
by 4 of the year then maturity will be n that is your h what basically we have considered in
years maturity is 3 then it will be 3 divided by 1 by 4 that means it will be 12 periods the n =
12.Then how many binomial interest rate tree will be there that already you know that is n – 1
that means 12 – 1 that will be 11 periods.
For a three-year bond evaluated over quarterly periods:
h = ¼ of a year
maturity=n= (3 years)/ (1/4 years) = 12 periods
Binomial interest rates tree with n-1 = 12-1=11 periods
So, if you are going for monthly basis like that h = 1 by 12 then your n = 3 by 1 by 12 then it
will be 36 then 36 – 1= 35 period binomial tree of the spot rates have to be formulated.
For a three-year bond evaluated over monthly periods:
h = 1/12 of a year
maturity=n= (3 years)/ (1/12 years) = 36 periods
Binomial interest rates tree with n-1 = 36-1=35 periods
So, if you are going for weekly like that the figures will change h = 1 by 52.Then the maturity
will be 3 by 1 by 52 that means it is 1 by 56 periods then you have to formulate 155 period
binomial tree of the spot rates.
For a three-year bond evaluated over weekly periods:
h = 1/52 of a year
maturity=n= (3 years)/ (1/52 years) = 156 periods
Binomial interest rates tree with n-1 = 156-1=155 periods
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So, like that once your period analysis will change or your h will change automatically your
number of trees of the spot rates also going to be changed.
(Refer Slide Time: 09:21)
So, then let us see how this particular thing is going to be changed. Let you are going to have
a 5 period tree of the spot rates with our previous example we have started from the 10%. So,
it is 10% let u = 1.0488 d = 0.9747 then like that it will reach in six months it will reach
because u and d also has to be adjusted because in one year it will be 1.1 and it will be 0.95
accordingly it will reduce in the first case and second case it will increase.
So, therefore your u become 1.0488 and here your u has become 1.0488 and d become
0.9747. So, accordingly your interest rate movements figure also going to be changed. It will
become 10.488 in the first 6 months or it has come down to 9.747 further it can increase up to
11 or decline up to 10.223 or again it is 9.5. These are the numbers what we get whenever we
have consider one period.
Further, it can increase up to these and decline up to 10.721 or from here it can decline up to
9.964 or it can further decline up to 9.25, but if you look at these figures the 12.1, 10.45,
9.016, these are the figures what we are getting whenever we are considering the one year
period. Like that further it can go up to this or go down to this or it can reach here, it can
reach here, reach here, reach here, reach here.
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Like that the number of nodes are going to be changed and the number of periods are going to
be changed because these things looks more realistic in nature the reason is that the spot rates
or the interest rate can fluctuate frequently. So, as you go on increasing by your number of
periods, number of nodes then this validity of the model or the applicability of the model is
going to be better, going to be more realistic in nature.
So, this is what the 5-period tree of the spot rates will look like if you are considering the six
months h = 1 by 2 years. So, that means you will have the 1, 2, 3, 4, 5 periods will be there it
will be generally n – 1. n = 6 then your number of periods will be n – 1, that will be 5.
(Refer Slide Time: 12:19)
But in the beginning, I discussed with you the most important thing is how this u and d is
formulated. We have assumed certain u and d in the one-year period analysis. we have taken
this is 1.1 and 0.95, but how that u and d is calculated? So, there are generally two
approaches we use to estimate this binomial interest rate movement to find out this u and d.
One is u we call it the equilibrium model, through the equilibrium model we find out the u
and d.
And further we can use the calibration model which basically helps us to exactly find out this
or to estimate this binomial interest rate tree movement. So, this equilibrium model generally
developed by Rendelman and Bartter (1980). further again, Cox and Ingersoll and Ross
(1985) and the calibration model generally is also called the arbitrage free model. It is
developed by Black, Derman and Toy (1990), Ho and Lee (1986), Heath, Jarrow and Morton
(1992).
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So, these are the people who basically try to develop this model to estimate this binomial
interest rate movements or more specifically to find out the value of u and d.
(Refer Slide Time: 14:03)
Let us first before we go for this, we can start with this u and d estimation approach; that is
the equilibrium model. What this model basically does? This particular approach try to solve
the value of u and d which equate the mean and variance of the distribution of the logarithmic
return to their empirical values. Why it is considered the log of the returns? Generally,
whenever we go for analyzing certain things in statistical way in the financial market we
assume that the distribution should follow a normal distribution.
But whenever we go for the calculation of the returns or trying to decide this interest rate
trees that may not follow the normal distribution. So, the investors or the analyst basically try
to find out the value of the u and d which is trying to equate the mean and variance of the
distribution of the logarithmic return to their empirical values. So, the bond values which are
obtained using this technique are then compared to their market prices.
To determine if there is any mispricing that basically they want to determine after estimating
this value of a u and d. So, in general, if you see the models value and the market prices
generally differ. So, if it is different than the additional assumptions regarding the risk
premium must be made to explain the market prices, why it is different because of the risk
concept?
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So, that is why the additional risk premium has to be added to that particular interest rate that
is what basically what we call it the equilibrium model. How that particular model basically
looks like? what basically it explains let us see?
(Refer Slide Time: 16:43)
How, you can determine this u and d. Mathematically, if you want to solve this u and d,
which make this statistical characteristic of a binomial distribution of the stock’s logarithmic
returns equal to the characteristics estimated value. So, that is what the basic notion of the
equilibrium model. So, before that what basically we have to do to? understand this particular
approach, to understand this model.
Let us first examine the probability distribution that characterizes a binomial process. What is
the probability distribution which characterizes a binomial process? So, whenever you talk
about the probability distribution of a particular data, then how this binomial process
basically looks here or how basically it is going to explain, how it can be derived?
(Refer Slide Time: 17:55)
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So, here we are trying to derive a binomial process of the spot rates and this distribution
generally is not normally distributed because the left side of the distribution has a limit at 0
because interest rate cannot be less than 0. If you are depositing the money, nobody will
deduct some kind of interest rate from your money to keep the money. it can be 0 you may
not get anything, but it cannot be 0 because we do not have negative spot rates.
So, because of this but whenever we are going for a statistical analysis if you are assuming a
binomial distribution then some values can be 0 also. So, that is why the distribution of the
spot rate can be converted into a distribution of logarithmic returns. Let we are a ssuming a g
n, then what is the g n? g n = the ln of S n by S 0 and it can be 1, 2 any numbers the initial
spot rate whatever we are considering and one of the period spot rates let S n = S 1.
So, g 1 = ln S 1 by S 0. So, the distribution of the logarithmic returns can take on the negative
values and can be normally distributed if the probability of the spot rate increasing in one
period which is generally, we consider q is 0.5; q is basically the probability value.
(Refer Slide Time: 19:57)
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So, now let us see what exactly how it explains how the binomial process can be explained?
Let when n = 1. There are two possible spot rates and the logarithmic returns that means your
g u. if you want to calculate that means, whenever the interest rate or the spot rate is going to
be up your g = what? Your ln S u divided by S 0 and what is your ln S u that is nothing, but
your ln u S 0 divided by S 0 S u is nothing, but u into S 0.
So, S 0 S 0 can be cancelled out you can find out ln u. So, ln u means let we have to consider
u = 1.1 then ln 1.1 = 0.095 that is what basically we have mentioned here. The same thing if
you are going for the lower side or the downward side, then it is ln d S 0 by S 0, then it will
be ln d then ln 0.95 then it will be – 0.0513 like that for other periods also you can look at
this.
When n=1, there are two possible spot rates and logarithmic returns:
gu = ln
gd = ln
𝑢𝑆0
𝑆0
𝑑𝑆0
𝑆0
= ln (u) = ln(1.1) = 0.095
= ln (d) = ln(0.95) = -0.0513
(Refer Slide Time: 21:33)
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If n = 2 then there are three possible spot rates and the logarithmic returns then your g double
u = ln u square S 0 by S 0 which is nothing, but ln u square then ln u square means it is ln 1.1
square that is 0.1906 then ln g u d = ln u d S 0 divided by S 0 then it will be ln u d then ln u d
means it is ln 1.1 into 0.95 that is 0.044 like g d d if you are calculating then ln d square S 0
by S 0 that is ln d square then it will be – 0.1026.
When n=2, there are three possible spot rates and logarithmic returns:
guu = ln
gud = ln
gdd = ln
𝑢2𝑆0
𝑆0
𝑢𝑑𝑆0
𝑆0
𝑑2𝑆0
𝑆0
= ln (u2) = ln(1.12) = 0.1906
= ln (ud) = ln[ (1.1)(0.95)] = 0.044
= ln (d2) = ln(0.952) = -0.1026
(Refer Slide Time: 22:19)
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So like that if your n = 3 there are four possible spot rates in the logarithmic returns. One is
your ln u cube ln u square d ln u d square and ln d cube. So, like that you can get these values
like this. So, already we said that the binomial process; so here there is the title is it is
basically not the convertible bond it is the binomial process the example whatever we are
basically considering there is a typo error here.
So, here your g, ddd you can find out – 0.1539. What basically we have seen here the
binomial process if you are explaining in 1 over n = 1 you find out these values. Here also it
is binomial process explanations here also it is binomial process explanations. So, there are
some type errors please keep that thing in mind. So, here we have seen that these are
basically the values what we are getting whenever we are taking the log of this particular
values.
When n=3, there are four possible spot rates and logarithmic returns:
guuu = ln
guud = ln
gudd = ln
gddd = ln
𝑢3𝑆0
𝑆0
= ln (u3) = ln(1.13) = 0.2859
𝑢2𝑑𝑆0
𝑆0
𝑢𝑑2𝑆0
𝑆0
𝑑3𝑆0
𝑆0
= ln (u2d) = ln[ (1.12)(0.95)] = 0.1393
= ln (ud2) = ln [(1.1)(0.952)] = -0.0073
= ln (d3) = ln(0.953) = -0.1539
(Refer Slide Time: 24:04)
Then what we have seen here, whenever n = 1 there are two possible rates and logarithmic
returns 1 over n = 2 there are three possible rates in the logarithmic returns. When n = 3 there
are four possible rates in the logarithmic returns.
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(Refer Slide Time: 24:22)
So, now what basically we can do? You can generalize this. So, the probability of attaining
any of these rates is equal to the probability of the spot rate increasing j times in n periods
which can be represented as P n j. For example, the probability of attaining spot rate of
10.45% whatever binomial tree we have just explained before in period 2 is equal to the
probability of the spot rate increasing one time that is j = 1 in two periods that is n = 2 that
means it can be represented as p 21.
In a binomial process, the probability is
𝑛!
Pnj = (𝑛−𝑗)!𝐽! 𝑞 𝑗 (1 − 𝑞)𝑛−𝑗
So, p 21 is basically represented in this particular case. So, in a binomial process this
probability is P n j = your n factorial divided by your n – j factorial into j factorial into q to
the power j into 1 – q to the power n – j where q is the probability where q is basically the
probability that we are considering. So, now if you have a binomial process and you are
taking the log of that it follows a normal distribution then what basically we can do?
(Refer Slide Time: 26:07)
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We can see that how this binomial distribution looks like and how this mean and variance of
that particular distribution can be calculated? In this example, if you look at after two periods
the probability of spot rate is equal to 9.53% and your p 11 is 0.5. Your probability of the
spot rate = 9.53% is basically 0.5 that means p 11 = 0.5. If you are going to get -5.13% the
probability is also 0.5.
So, using this binomial probability the expected value and the variance of the logarithmic
return after one period can be calculate that is E g 1 = 0.5 into 0953 + 0.5 into – 0.0513 that is
0.022 and your V g 1 = 0.25 into 0.0953 – 0.022 square + 0.5 into – 0.513 – 0.22 square that
is 0.0054 and what do you mean by this 022? that basically mean this basically your variance.
So, this is the actual data deviation from the mean the square of that multiplied by the
probability + the probability – the actual spot rate – this mean that will give you the variance.
the expected value and variance after one period are:
E(g1) = 0.5(0.0953) + 0.5(-0.0513) = 0.022
V(g1) = 0.25(0.0953-0.022)2 + 0.5( -0.0513-0.022)2 = 0.0054
It is after one period. If you are going for the expected value and variance after two periods
then again you can calculate this on the basis of the rates what you are getting that will be
0.044 and the variance will be 0.0108.
the expected value and variance after two periods are:
E(g2) = 0.25(0.1906) + 0.25(-0.1026) = 0.044
V(g2) = 0.25(0.1906-0.044)2 + 0.5( 0.044-0.044)2 + 0.25(-0.1026-0.044)2 = 0.0108
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So, the expected value after three periods if you see that will become 0.066 and it will be
0.0162 after four periods it will be 0.088 and 0.0216. So, that means if you see that it is
basically n times this value whatever you are getting in the first period.
(Refer Slide Time: 29:00)
So, that means just now if you look at just now whatever calculations I have shown you this
thing has been reflected in one picture. You start with 0.1 it has become 0.11 so here because
you have assumed u = 1. 1, d = 0.95, q = 0.5 so p 11 = 0.5 so g u = this here your g d = this
and so on. Everywhere we have calculated this. So, here your probability has become 0.25 it
is 0.5 it is 0.25 and so on.
So, these are the expected mean value and the variance figures what basically here reflected.
So, now on that basis what is the property we got?
(Refer Slide Time: 29:55)
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In examining each distribution’s mean and variance note that as the number of periods
increases the expected value and variance also increase by a multiplicative factor such that
your expected g n n = the period will be n E g 1 and V g n = n V g 1. So, the parameter values
after n periods are equal to the parameter values for the one period time the number of
periods and the expected value and the variance are equal to n into q ln u + 1 – q ln d.And if
you are going for the variance your V gn = nq 1 – q into ln u – d whole square. So, this is
your mean this is basically your variance.
The expected value and the variance are equal to
E(gn) = n[ q ln u + (1-q) ln d]
V(gn) = nq(1-q) [ln (u/d)]2
(Refer Slide Time: 30:58)
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So, what basically we have discussed here? the binomial model becomes more realistic when
we subdivide the periods to maturity, into a number of sub periods. There are two general
approaches to estimate the binomial interest rate movements. One is your equilibrium model
and the calibration model and the equilibrium model basically solves for the u and d values
which equate the mean and variance of the distribution of the logarithmic return to their
empirical values.
(Refer Slide Time: 31:29)
This is the reference you can go through. Thank you.
686
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology – Kharagpur
Lecture – 42
Estimation of Binomial Trees – II
So, in the previous class we discussed about the subdivision of the binomial tree which looks
more realistic in nature and also we started the discussion on the different approaches which
are used for the valuation of the u and d. We will continue with that particular discussion.
(Refer Slide Time: 00:39)
So, today we will be discussing how the u and d value we can solve, we can get and the
another concept is basically the estimation of the annualized mean and variance of the log
returns that what we have introduced because we have introduced the concept of the normal
distributions of this particular data and we talked about this binomial process and the
binomial distribution.
Then, accordingly we will see that on the basis of the equilibrium model how this u and d is
calculated and as well as how the estimation of the annualized mean and variance of the
returns also calculated particularly the log returns can be calculated.
(Refer Slide Time: 01:26)
687
So, annualized mean and annualized variance these are the major keywords what we are
going to in particular session.
(Refer Slide Time: 01:36)
So, first of all come back to our discussion which is continued in this particular session; that
is basically solving the values of u and d and already, we have seen that we have assumed a
binomial distribution and the formula for estimating the u and d generally we can find by
solving for u and d values, that makes the expected value and variance of the binomial
distribution of the logarithmic return of the spot rates equal to their respective estimated
parameters under the assumptions that the distribution is normal.
Very minutely you observe this. We can solve these values of u and d. how? We can only
formulate the formula or we can find out the formula for the value of the u and d by solving
688
the particular concept or particular thing where the expected or the values of u and d which
make the expected value and variance of the binomial distribution of the logarithmic returns
of the spot rates equal to their respective estimated parameters values.
And here our assumption is there is a normal distribution. So, if you are assuming the mu e
and the V e be the estimated mean and variance of the logarithmic return of the spot rates for
the periodic fall in length to n periods then our objective is to solve for the u and d values
which satisfy the following equations. What are those equations. So, here your n E g 1 here
your n E g 1 = n q ln u + 1 – q ln d.
nE(g1) = n[ q ln u + (1-q) ln d]= µe
nV(g1) = nq(1-q) [ln (u/d)]2 = Ve
So, q is basically the probability and g 1 is the log returns that already you know and 1 – q =
the 1 – probability and u already you know the d also you know. So, if you assume that the
mu e and V e is the estimated mean and variance of the logarithmic return of the spot rates
for a period equal in length to n periods then we can find out the value of u and d which
basically satisfy these conditions. So, if you have these values with us then from there the u
and d values can be calculated.
(Refer Slide Time: 04:55)
So, let us see that let q = 0.5, then the formula values for u and d that satisfy the two
equations will be u = e to the power root of V e by n + mu e divided by n.
u = e√𝑉𝑒 /𝑛 + µe/ n
d = e√𝑉𝑒 /𝑛 + µe/ n
689
What do you mean by the mu e? The mu is basically the mean and V is basically the variance
and d = e – the root of V e by n the square root of the V e by n + your mu e by n. V is
nothing, but the variance of the returns and your mu is basically the mean of the returns.
So, if from the previous example we have calculated this mu e and V e. Let, if you find that
the expected value and variance of the logarithmic returns are mu = this and V = this for a
period equal in length and n = 2 because these values we get whenever the two period
analysis we are doing then you will find that u = e to the power your V = 0.0108 then it is
square root of 0.0108 divided by 2 + your 0.044 by 2 that will be 1.1. Here it is e to the power
– of this that will give you 0.95.
u = e√0.0108/2 + 0.044/ 2 = 1.1
d = e-√0.0108/2 + 0.044/ 2 = 0.95
So, now the question is that we have derived this particular formula with the assumptions that
q = 0.5 and mu = this and V = this.
(Refer Slide Time: 06:58)
If you are assuming that your n the number of observations are quite large, if you are
assuming or if you feel that the number of observations are quite large, then as the n increases
by statistical properties, when the n increases the mean term in the exponent goes to 0 quicker
than the square root term. As the n increasing, the mean value in the exponent goes to 0 and it
is reaching 0 quicker than the square root term.
690
So, in statistics or in a mathematical sense if you look at for large n, for example, n is greater
than or equal to 30 generally we consider, but generally it should be quite large in number.
The mean term’s impact on u and d is negligible. The mean term on u and d is negligible. So,
that is why this is your mu values what just now we have taken that your mu e divided by n
and your mu e divided by n these two terms the role of these terms becomes negligible in that
particular case.
And this formula becomes u = e to the power square root of V e by n and d = e to the power –
the square root of the V e by n that is nothing, but the 1 by n that generally happens for the
large samples. So, if your n is quite large then your mean term becomes 0 the mean term in
the exponent basically goes to 0 and only this variance term will remain so that is why the
formula becomes this e to the power square root of V e by n and e to the power – this is
square root of V e by n that is nothing, but the 1 by u.
u and d can be estimated as:
u = e√𝑉𝑒 /𝑛 + µe/ n
d = e-√𝑉𝑒 /𝑛 + µe/ n
(Refer Slide Time: 09:23)
So, then we have a concept of the annualized mean and variance in this case. So, the mu e
and V e; these two terms mu e and V e these are the mean and variance for the length of time
= bonds maturity’s. Mean and variance for a length of time equals to the bonds maturity.
Generally, whenever we deal with this bond markets for the fixed income securities market,
the annualized mean and variance generally are used.
691
And how the annualized mean and variance are found or are obtained that is generally always
we find by multiplying the estimated mean and variance of a given length. Let, for example,
you have taken month by the number of periods of that length in a year that means if you are
talking about the month is the frequency then you have to multiply with a 12. For example, if
half yearly data is used, then we simply multiply those estimates by 2 to obtain the
annualized parameters.
If it is let quarterly then you multiply it by 4. So, like that the annualized mean and variance
can be calculated.
(Refer Slide Time: 11:04)
So, when the annualized mean and variance are used, then these parameters must be
multiplied by the proportion h, as the time of the period being analyzed expressed as a
proportion of a year, and n is not needed since h defines the length defines the length of tree’s
period. So, in that case you can write u = e to the power h into V e A + h into mu e A and d =
e to the power – the square root of h V e A + h mu e A. U = e to the power the square root of
h V e A + h mu e A and d = e to the power – the square root of h V e A + h mu e A that is the
thing basically what you have to keep in the mind.
u = e√ℎ𝑉𝑒 𝐴 /𝑛 + hµeA
d = e-√ℎ𝑉𝑒 𝐴 /𝑛 + hµeA
692
(Refer Slide Time: 12:17)
So, let us see that if the annualized mean and variance of a logarithmic return of one year spot
rates that are 0.044 and 0.0108 and we wanted to evaluate a three-year bond with 6 months
period. Let h = 1 by 2 of the year then what we should do? Then we would use basically a 6period tree to value the bond that means n = your 3 divided by 1 by 2 that is the 6 period and
u and d basically you can calculate from that case then, what will be the value of the d and u?
The value of u will be e to the power square root of h = here h = 1 by 2 then your V is
basically 0.0108 and your mu e is basically your 0.044. So, now you are putting that values in
this formula that is 1 by 2 multiplied by your 0.0108 the square root of that plus h into this is
basically your mu e 044. So, then we get 1.1 then if you are going for d; d = e to the power –
of the square root of h this is h this is basically your V e A. So, this is basically your V e A
this is mu e A. So, then it is 1 by 2 into 0.044 that you get 0.95.
u = e√0.0108/2 + 0.044/ 2 = 1.1
d = e-√0.0108/2 + 0.044/ 2 = 0.95
So, like that if you make the length of the period let h = 1 by 12 then we would value the
three year bond with a 36 period tree and your u will be h = 1 by 12 into 0.0108 plus your 1
by 12 into 0.044 that will become this and d = will become this.
u = e√0.0108/12 + 0.044/ 12 = 1.03424
d = e-√0.0108/12 + 0.044/ 12 = 0.9740
693
So, this is the way basically the annualized mean and variance can be calculated that actually
you have to keep in the mind.
(Refer Slide Time: 15:11)
The annualized standard deviation if you are considering, that cannot be obtained simply by
multiplying the quarterly standard deviation by four. If you are dealing with the standard
deviation, then by multiplying only the number with that particular value will not give you
that. We must first multiply the quarterly variance by 4 and then the square root of the
resulting annualized variance has to be considered that actually after picking the standard
deviation then you multiply by 4 that will basically will be incorrect.
What basically you have to do first of all you have to find out the quarterly variance and then
first multiply the quarterly variance by 4 to make it annualized and then take the square root
of the resulting annualized variance to find out the annualized standard deviation. that is why
there is some technicality involved in that with respect to the calculations the standard
deviation or annualized standard deviation that actually you have to keep in the mind.
(Refer Slide Time: 16:22)
694
So, the simplest estimate of the mu e and V e is nothing, but the average mean and variance
computed from the historical sample of the spot rates. First of all how you calculate the mu e
and V? The simplest way is basically to calculate the average mean and variance from the
sample of the spot rates wherever you have. You take the sample of the spot rates from there
you find out the mean and variance of that.
From there you can use it for calculation of your u and d. So, let, for example, how basically
this can be used or this generally is used by the analyst or by investor to find out the value of
u and d in that particular consideration. Let us see that what basically happens in that
particular case? So, let we have a data for historical quarterly one year spot rates over 13
quarters are available.
Then from there you can calculate the 12 logarithmic returns by taking the natural log of the
ratio of the spot rates in one period to the rate in the previous period that is basically your S t
by S t – 1 that means your S 1 by S 0 or S 2 by S 1 and so on that S 3 by S 2 like that you can
calculate these 12 logarithmic returns value from this 13-quarter pattern. After that whenever
you have this data available with you, you can calculate this average mean and variance of
that particular series which can be used for calculations of your u and d.
(Refer Slide Time: 18:21)
695
So, let this is your data this is your different quarters 1 to 13. Let these are the spot rates
which are given in the percentage term. So, you have the S t by S t – 1 so obviously first
period you will not have any data because previous data is not available that is why we lose
one observation in that particular case, that means the 12 logarithmic returns can be consider
for our particular analysis.
So, this is basically S t by S t – 1, then after that, what basically we are doing? that we can go
for calculating the log of this S t by S t – 1. then from there you find out your mean here your
mu e = 0 that means the mean = 0, the mu = 0 and then the deviation from the mean you have
to consider in this case and after that you take the square of that. So, after you calculate the
square of that you take the summation of that; that is basically the summation of all.
So, from rounding wise whenever we do the calculation in X n issue just ignore that you just
find out the process into that; that is why some deviation you may find because we have just
calculated using the excel. that is why some kind of deviations you will find in terms of the
numbers. So, the total value you got 0.0463 so then your value will be your exactly the value
0.046297 we have rounded up to 0.0463.
So, then your variance of that particular series will be your 0.046297 divided by n – 1 where
n = 12 whatever we have considered and that is 11 then you got the value is 0.004209. The
variance basically in this case was basically 0.004209 that is what basically what you got. So,
now what has happened you got the mean value, you got the variance value and from there
you can use that particular value for the further calculations.
696
(Refer Slide Time: 21:11)
So, just now we have just given this how it is calculated the historical quarterly logarithmic
mean and return mean return and variance is nothing, but this is the summation of t = 1 to 12
g t divided by 12 that total summation has gone 0, 0 divided by 12 that has become 0 then V
is t = 1 to 12 into g t – mu e the square of that summation of the square of this divided by this
is basically n – 1 then n = 12. So, n = 11 then you got 0.004209.
µe =
∑12
𝑡=1 𝑔𝑡
Ve = =
12
=
0
12
=0
2
∑12
𝑡=1[𝑔𝑡− µ𝑡 ]
11
=
0.046297
11
= 0.004209
So, that is basically the V e or the variance then you multiply the historical quarterly mean
and variance by 4. So, these are the quarterly mean and variance then once you calculate your
quarterly mean and variance then what basically you can do? You multiply this with 4 then
you get this annualized mean and variance from this. So, obviously your quarterly mean was
0.
So, this is your quarterly mean so this is already 0. So, obviously your annualized mean also
will be equal to 0, but quarterly variance is 0.004209. So, the annualized variance shall be 4
into this so this you will get 0.016836. This is basically your annualized variance. So, now
what you can do using your quarterly data whenever you calculate this particular values?
these are the quarterly mean and variance.
697
And from the quarterly mean and variance we are basically getting this annualized mean and
variance by multiplying by 4. So, now these particular values can be utilized for the
calculations of the u and d. So, this is the simplest approach for the calculations of that. So, if
you are in general case if you are calculating this mean and variance and you have the
probability values then you have the probability of the spot rates in the different periods.
There also you can calculate your mean and variance or annualized mean and variance, but in
case of other data is not available then the simplest way of calculating this thing is by
considering the historical data for the particular spot rates then take the log of that and from
this whatever frequency data is available you can utilize it to calculate this mean and variance
then you multiply with the respective numbers on the basis of quarterly data if you are using
then you multiply with 4.
If you are using monthly data then you multiply with 12 like that then finally your annualized
mean and variance can be calculated from that.
(Refer Slide Time: 25:03)
So, from that this estimated annualized mean and variance you can find out your u and d.
Once we determine the number of periods to subdivide, that is why the concept of
subdivision of the periods is very much important in this particular case. So, if you have the h
is let 1 or one period in this case then this is your u, this is your d using that particular
formula we are finding it out that what is the formula you are using that u = what is the
formula you are using?
698
This is the formula what basically you are using, but if you are going by a large number then
the different formula, but this is h into your variance what basically you are getting
annualized variance plus h into this one already we have discussed that. So, from this
generally we can use this particular formula here then we can calculate this u value and the d
value.
So, if your h = 1 by 4 then accordingly this will be because in the square root this is h into
your V e + h into your mu e. This is what basically what we are considering this is u e to the
power this. If you are calculating d = e to the power – of this into V e A square root of this +
your h into mu e A. Just now I have shown you this is the formula what basically you are
using it for calculating this u and d.
And on the basis of the h your u and d values are basically changing. So, given this estimated
annualized mean and variance u and d can be estimated once we determine the number of
periods to subdivide that means you are finding out your h which is basically nothing about
this. So, then this is the way the u and d estimation can be that then further we will see that
how the calibration model is the extension of that or what are the different other approaches
have been considered in that particular case.
And in the different scenario, how this particular value of the bond or the kind of binomial
process and distributions are going to be utilized for finding out or estimating this binomial
tree?
(Refer Slide Time: 28:17)
699
So, binomial interest rate tree generally generated using u and d estimation approach is
constrained to have an end of the period distribution with a mean and variance that matches
the analyst’s estimated mean and variance. So, the binomial tree generated from the u and d
estimate is not constrained to yield a bond price that matches its equilibrium price that means
what price we obtained by discounting the bond’s cash flow by the spot rates.
And generally the investors or the analysts using such models to make the additional
assumptions to about the risk premium in order to explain the bonds equilibrium price
because of the risk involved in that case. but in contrast to this model, the calibration models
are constrained to match the current term structure of the spot rates which generally yield the
bond prices that are equal to the equilibrium values.
That we will see in detail that what exactly happens in terms of the calibration model and
how basically it is different from the equilibrium model for the estimation of the binomial
tree interest rate fluctuations particularly?
(Refer Slide Time: 29:35)
So, what basically we discussed in this session that the formula for u and d can be find out by
solving the expected which make the expected value and variance of the binomial distribution
of logarithmic return of the spot rates equal to their respective estimated parameters with the
assumptions of the normal distribution and the annualized mean and variance generally are
obtained by multiplying the estimated mean and variance for a given length by the number of
periods of that length in a year.
700
That means it is monthly multiply 12, if it is quarterly multiply 4, if it is 6 monthly then you
multiply 2 that is the way the annualized mean and variance can be calculated.
(Refer Slide Time: 30:24)
These are the references you can see. Thank you.
701
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology – Kharagpur
Lecture – 43
Estimation of Binomial Trees – III
Good morning and welcome back. So, in the previous class, we started the discussion on the
estimation of the binomial trees. So, we have discussed about how to find out the values of u
& d and there we have two approaches. One approach is your equilibrium model approach
where we are trying to solve this value of u & d on the basis of certain assumptions and the
other model is basically the calibration model.
(Refer Slide Time: 00:48)
So, today we will be discussing about the calibration model and mostly we will see that what
are those different conditions and different methods or different ways the pricing of or may
be the estimation of the binomial trees they are carried out through this calibration model.
(Refer Slide Time: 01:11)
702
And in this particular context you will come across two different major keywords, what we
will be using in this particular session, that is basically your variability conditions and the
price conditions. So, these are the two conditions which basically come under the calibration
model.
(Refer Slide Time: 01:35)
So, let us see that, what exactly the calibration model is all about. So, calibration model
basically generates a binomial tree by finding the spot rates of the different periods or the
different kind of notes. So, here whenever we are trying to use this calibration model to find
out the spot rates two things you are always try to or you are trying to always keep in the
mind or this particular model can work only in two conditions.
703
One is your variability condition, other one is the price conditions. So, what this variability
condition is basically talks about that or basically says that or basically tries to explain the
relationship between the upper and lower spot rates; the variability condition which basically
governs the relation between the upper and lower spot rates and another one is the price
condition.
So, here the price condition basically is what? Here, the value of the bond generally is found
from the tree which is consistent with the equilibrium bond price given the current spot yield
curve. If the current spot rate is known to you then and accordingly, you can find out that
whether the price condition is satisfied or not. So, here the bond value generally is obtained
from the tree which is consistent with the equilibrium bond price.
So, therefore the calibration model is based on the two conditions and the binomial tree
basically it can be generated following these two conditions. One is variability condition,
other one is the price condition. So, we will see that what exactly the variability condition is
and what is this price condition?
(Refer Slide Time: 03:47)
So, whenever you talk about the variability condition if you back in our previous session
what basically we have discussed? that we are trying to find out the value of u and d and how
we are trying to find out the value of u and d? We are basically trying to first calculate the log
value of the returns or the logarithmic return of the spot rates and we are also assuming that
this particular logarithmic returns are following the normal distribution.
704
So, following this normal distribution we have tried to solve the value of the u and d that is
what basically what we have discussed in the previous class. So, whenever we are trying to
use that particular kind of concept what basically we have used. So, here one assumption
basically on the basis of that particular resolution we are basically trying to explain that the
relationship between S u and S d.
So, if you are finding the value of S d how you are trying to find out the S u? So, here if you
see that the value of S u = S d e to the power 2 into root of the V e by n.
2
Su = 𝑆𝑑 𝑒 √𝑉𝑒 /𝑛
V e basically the variance of this particular returns. So, here how basically we get this? Why
you are telling that your S u = S d e to the power 2 into root of V e by n, how basically we
derive this?
(Refer Slide Time: 05:41)
So, here if you assume what basically we have derived or what basically over the binomial
tree concept we are trying to explain that your S u is nothing, but u into S 0. Your S u = u into
S 0 and your S d = d into S 0. So, by default what basically you can get? S u by u = S 0 = S d
by d then with this some kind of algebraic manipulation you will find S u = S d into u by d
and already you know what is the value of u? u = e to the power root of V e by n + mu e
divided by n that we have discussed in the previous class and your d = e to the power – root
of V e by n + mu e by n. So, if you are assuming a large number of observations in the log
normal distribution case our particular formula is coming to u = e to the power root of V e by
n and d = e to the power – root of V e by n that basically we have discussed in the previous
class.
705
So, from this what basically you have seen that your S d S u = S d e to the power root of your
V e by n + mu e divided by n divided by e to the power – root of V e by n + your mu e
divided by n. Now, let me e divided by n is basically we may not consider that part then
automatically it will become S d e to the power root of V e by n divided by e to the power –
root of V e by n.
Then if you say then it says nothing, but S d into e to the power root of V e by n and this is e
to the power – root of V e by n means that is basically what we can say divided by 1 by e to
the power root of V e by n. So, then automatically it will be S d into e to the power root of V
e by n into e to the power root of V e by n. So, finally what basically you are getting S d 2
root V e by n e to the power 2 into root of V e by n.
So, here basically what we are trying to say that this is basically your what we get that this
formula is basically derived that S u = S d e to the power 2 root of V e by n. So, now what
basically we got? So, if you are trying to convert it in terms of the annualized variance then
you will find your S u = S d into e to the power 2 into h V e A; V e basically annualized
variance h into V e by n.
Variability Condition:
Given: Su = uS 0
Sd = dS 0
𝑆
𝑆
Therefore: 𝑢𝑢 = 𝑆0 = 𝑑𝑑
𝑢
Su = 𝑆𝑑 𝑑
Substituting the equations for u and d, we obtain:
𝑒 √𝑉𝑒 ⁄𝑛+𝑢𝑒 ⁄𝑛
2
Su = 𝑆𝑑 −√𝑉𝑒⁄𝑛+𝑢𝑒 ⁄𝑛 = 𝑆𝑑 𝑒 √𝑉𝑒 /𝑛
𝑒
2
In terms of the annualized Variance : Su = 𝑆𝑑 𝑒
√ℎ𝑉0 𝐴
And h and n all those kind of concepts that already we have discussed. So, that means here
what basically we are trying to say? that we are trying to basically see that, how this S u and
S d relationship can be established?
(Refer Slide Time: 10:09)
706
Then we will see that how this particular formula whatever we have derived that is going to
help us to find out the actual values. So, let you take a kind of case in this particular context.
Let the lower rate is 9.5% that means your S d = 9.5% and your annualized variance is let
0.0054. The annualized variance is 0.0054 and S d = 9.5% then what will happen let the
upper rate you are trying to find out for one period and your h = 1.
2
Su = 9.5%𝑒 √.0054 = 0.11 = 11%
So, in this case if you are using that particular formula just now whatever we have just now
discussed that S u = S d into e to the power 2 into root of 0.0054. So, now if you put those
particular values here then we will find that your S u will become 11%.
So, here what basically we can say if the current one year spot rate is 10% then this upper and
lower rate should be consistent with the upward and downward parameters of u and d.
The parameter value of u and d we have consider 1.1 and 0.95. if you go back, whenever you
started the discussion about the binomial tree. Now, whatever we have seen that if the current
one year spot rate is 10%, then using this particular formula we can find out your u = 1.1 and
d = 0.95. This variability condition would result in a binomial tree identical to the example
whatever we have explained before that is what basically what we are trying to see here.
(Refer Slide Time: 12:25)
707
So, then let us see that what is the price condition? So, this is the variability condition and
what is the price condition? The price condition says that the bond value what you are trying
to obtain from the tree that is equal to equilibrium bond price given the current spot yield
curve. So, then if you want to generate a tree which basically satisfies this price condition
then what basically you have to do?
You have to basically solve for or you have to find out a lower spot rate which basically
along with the variability relations yields a bond price that is equal to equilibrium bond price.
So, whenever you are taking that condition that the value of a bond obtained from the tree is
equal to the equilibrium bond price given the current spot yield curve, then what basically we
have to do here to generate a tree which satisfies the price condition requires solving for the
lower spot rate that along with the variability relation yields a bond price that is equal to the
equilibrium bond price. that is what basically what you have to do, then how basically exactly
we can do that? Let us see these things.
(Refer Slide Time: 13:55)
708
Suppose, this is a kind of example or hypothetical case you can see. Suppose, a current spot
yield curve has the following one, two, three year spot rates this is given that your one year
spot rate is 10%, second year spot rate is 10.12238% and third year spot rate is 10.24488%
and you assume the estimated annualized logarithmic mean and variance the mean is
0.048167 and the variance is 0.0054.
So, whenever you are in the first case you are trying to use this u and d approach, then your
one period tree of length one year would have up and down parameters then what will be the
value of u and what will be the value of d? Just now, we are trying to explain that you know
the formula for the u and d; u = e to the power your root of just now if you say that actual
formula it will be your e to the power 2 into root of V e by n plus your mu e divided by n that
is what basically what we have seen.
So, here in this case if you see period is 1. So, therefore you can say that your that will be
your e to the power your h into the annualized volatility that is 0.0054 and your mu e is
basically mu A = 0.048167 then using this formula what basically we find that is basically
1.12936 that u value = 1.12936. In the same way if you are trying to d you are finding 0.975.
So, here what basically we observe? Give a one period spot rate that is 10% the binomial tree
possible spot rate will be your S u = 11.2936% and your S d = 9.75%.
u = 𝑒 √(1) 0.0054 + (1) 0.048167 = 1.12936
d = 𝑒 −√(1) 0.0054 + (1) 0.048167 = 0.975
709
But these rates are not consistent with the existing term structure whatever we have assumed.
In our example, whatever thing we have considered these rates are basically are not
consistent with that because this particular u and d values are basically different from the u
and d values whatever we have taken in the previous case.
(Refer Slide Time: 17:03)
So, then what basically we have to do? Let us take one example in this case. let there is a zero
coupon bond and we want to evaluate that zero coupon bond which term to maturity is two
years and the face value of that zero coupon bond is Rs. 1 and we are trying to evaluate or
trying to find out the value of the bond using the binomial tree, then how we can do this? You
go back or you recall that particular thing whatever we have discussed using the valuation of
zero coupon bond using the binomial tree.
Your B 0 will be 0.5 B u + 0.5 B d here we have taken the probability is 50%, 50% it can
increase, 50% it can decline. So, that basically we have to discount it with respect to the spot
rate which is prevailed in that particular period. So, what we have seen here using this u and d
whatever we have considered in this case, we find that the yields of the particular kind of
bond whatever we are considering that can increase up to 11.29% or it can decline up to
9.75% from the 10%.
And the current spot rate is one year spot rate is 10%. So, then it will be 0.5 into 1 by Rs. 1 is
the face value of the bond that is 1 by 1.112936 + 0.5 into 1 by 0.0975 we are trying to find
out the present value of the particular bond in these two different conditions then you
710
multiply with that particular probability then we are discounting it with respect to 10% to find
out the present value of that particular bond.
Then you find that the present rate so this is the rate whatever we have just now calculated.
So, B 0.5, 0.5 is basically the value of q or the probability value. So, then what basically we
got? We got 0.82258 that is basically the value of the zero coupon bond. So, now you have
the two year spot rate already given to you. So, if you are going to that is basically
10.12238% that in the beginning we have taken that particular hypothesis or that example.
Then your value in the market in that particular point of time that should be 1 by 1 + y 2
square and that will give you 1 by 1.1012238 whole square that value will become 8246.
B0 =
0.5𝐵𝑢 + 0.5𝐵𝑑
1+𝑆0
1
=
1
]
1.0975
0.5[ 1/1.112936+ 0.5[
1.10
= 0.82258
1
B0M = (1+𝑦 )2 = (1.1012238)2 = 0.8246
2
Now, this value whatever you have calculated and this value you have calculated that is not
same, that is basically not equal. So, then what basically we can say in that particular context.
(Refer Slide Time: 20:34)
That if you are going for a calibration of the binomial tree to 2 period zero coupon bond with
a face value of 1. What basically just now we have seen? the face value is Rs. 1 that means B
uu = 1, B ud also 1, B dd also 1. Then. what we have done your S u? basically we have to
find out your S u = your 0.0975 e to the power 2 into 0.0054 and period one basically one we
are taking here.
711
So, that is why that does not matter to that particular value then we find this one. Then, if you
discount it with respect to this rate then you are finding the value of this. Then, if you are
going to do this S d = this then your value of the bond in the lower node will be 0.9111617.
Now, again we are using these values here 0.5 into 0.898524 + 0.5 into 9111617 and you are
discounting at a rate of 10% which is the one year spot rate then you find 0.82258.
But in the market the value of that zero coupon bond is 0.8246. So, this is what basically is
not matching in this particular case.
(Refer Slide Time: 21:55)
So, then what we can do the tree what basically generated from our estimates of u and d is not
consistent with the current interest rate structure. So, to make our tree consistent with the
term structure we need to find out the S d value which will satisfy this condition that means S
u = S d e to the power 2 into root of h V e A; V e A is nothing, but the annualized volatility
annualized variance. So, here that will become S d into e to the power 2 into 1 into 0.0054.
So, if you use this particular formula then the value of this to your bond obtained from the
tree these will be equal to exactly 0.8246.
2
Su = 𝑆𝑑 𝑒
√ℎ𝑉0 𝐴
2
= 𝑆𝑑 𝑒 √(1) 0.0054
(Refer Slide Time: 22:54)
712
So, mathematically whenever we are trying to find out the value of S d that should satisfy this
condition that 1 by 1 + y 2 whole square = 0.5 B u + 0.5 B d divided by 1 + S 0. How to find
out S d that is what we are discussing then it is nothing, but 0.5 into 1 by 1 + S u + 0.5 into 1
by 1 + S d. So, then if you see that then what we got you are putting this particular values it is
S u is nothing, but S d into e to the power 2 into h V e A.
1
(1+𝑦2 )2
=
0.5𝐵𝑢 + 0.5𝐵𝑑
1+𝑆0
1
1
]+ 0.5[1/(
]
1+𝑠𝑢
1+𝑠𝑑
0.5 [
1
(1+𝑦2 )2
=
1+𝑆0
0.5[
1
(1+𝑦2 )2
=
1+𝑆𝑑 𝑒
1
1
] + 0.5[1/(
]
2
1+𝑠𝑑
𝐴
√ℎ𝑉0
1+𝑆0
Then, if we put this particular values then you will find out in terms of our example what we
can find that 1 by 1.1012238 whole square = 0.5 into 1 by 1 + S d e to the power 2 into
square root of 0.0054 + 0.5 into 1 by 1 + S d. So, here basically what we got that you
discount it with respect to 10% then if you solve this equation then you find out your S d =
exactly 9.5% which is basically consistent with our term structure. Is it clear?
1
(1.1012238)2
0.5[
=
1
1
] + 0.5[1/(
]
2
1+𝑠𝑑
√0.0054
1+𝑆𝑑 𝑒
1.10
= 9.5%
So, that is what basically what we are trying to find out. So, once you find out your S d then it
will be easy for you to find out the other values then finally the value of that particular bond
can be calculated.
(Refer Slide Time: 24:53)
713
So, then what we find here our S d = 9.5% we have a binomial tree of one year spot rates of S
u that should be 11% and S d will be 9.5% which basically satisfy our variability condition
and the price condition both the condition will be satisfied in this case. So, the rate is
consistent with the estimated volatility of 0.0054 and the current yield curve with one year
and two year spot rate of 10% and 10.12238% that is basically what we call it the price
condition.
(Refer Slide Time: 25:37)
Now, we see that in the calibration model how this particular value is exactly matching. You
start with 1 here B uu = 1 and B ud = 1, B dd = 1. So, now you find out your S d that is 9.5%
and your S u = 0.095 e to the power 2 into square root of 0.0054 that you got 11%. So,
automatically your B u = 0.9009 and your B d = 1 by 1.095 that will be 0.913242. So, now
you go back to our formula 0.5 into 0.9009 + 0.5 into 913242 you will find the value 0.8246.
714
And your market value using the spot rate whatever we are considering that is basically
0.8246 that is what basically in the beginning we have achieved. So, that is the current spot
rate in that particular period that is basically y 2 that is 8246. So, it is basically perfectly
matching.
(Refer Slide Time: 27:05)
So, then the lower rate of 9.5% represents a decline from the current rate of 10% which
basically tend to expect in a binomial process and this thing basically has happened because
we have calibrated the binomial tree to a relatively flat yield curve. So, if we had calibrated
the tree to a positively sloped yield curve, then it is possible that both rates in the next period
could be greater than the current rate the 10% to 11% or it is 9.5% instead of that it can go up
to more than 11% or it can go up to also more than 10% in the lower node.
So, both rates in the next period could be greater than the current rate although the upper rate
will be upper than the lower tree. So, if you are calibrated this tree for a negatively sloped
curve then it is possible that both the rates in the next period could be lower than the current
that is also possible in that particular context.
(Refer Slide Time: 28:15)
715
So, if you are let assuming a positively sloped curve. So, in the positively sloped curve in the
same example if you see let your B uu = 1, ud = 1 dd = 1. So, in that case what basically we
are trying to find out. Let we are finding that S d value we are assuming your 10.20066% so
B d = this and S u you are trying to find out using this value then you got 0.118156 that
11.8156% then B u = this.
So, now if you put these values here then you will find that your B 0 = your 8189% because
here what basically where we are assuming the two year spot rate is 10.5% instead of
10.12238%. So, then if you discount it with respect to 10.5% you will also find that 8189 is
the value of the bond which is matching here and why it is basically matching because this
particular rate is based on the rates whatever we have consider here that is 10.5% and how it
is derived that already we have discussed.
Mathematically if you want to derive that S d by using the two year spot rate which is
available in that particular point of time, then what basically you can find? you are using your
binomial tree model the value whatever you are finding the price basically what you are
finding that will be perfectly equal to the value what basically in the market you are going to
calculate or you are going to find out using this that particular current spot yield available in
that particular point of time. So, this 0.8189 will perfectly match.
So, this is basically can happen with respect to a positively sloped yield curve like that you
can also assume a particular rate. Let two year spot rate will be lower than the current spot
rate then accordingly these particular values can be changed then exactly you may find out
716
the value of the bond using the tree the calibration model or the market yield curve what you
are trying to use it both of them will be equal.
(Refer Slide Time: 30:49)
So, what basically we have discussed here. The calibration model basically generates the
binomial tree by finding the spot rates which satisfy two conditions. One is variability
condition and another one is the price condition. The variability condition basically governs
the relationship between the upper and lower spot rate that means the lower spot rate is
available how to find out the upper rate.
And the price condition basically tells the bond value what basically we find from the tree
that should be consistent with the equilibrium bond price given in the current spot rate that is
what just now we have seen both the cases the particular two rates are same.
(Refer Slide Time: 31:31)
717
So, these are the references you can go through. Thank you.
718
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology – Kharagpur
Lecture – 44
Estimation of Binomial Trees – IV
Welcome back. So, in the previous class we started the discussion on the calibration model
and there we have seen that how the variability condition and the price condition works and
in that particular context the value of the bond what we are estimating using the binomial tree
and the equilibrium price on the basis of the current spot yield curve that will be equal to
same.
(Refer Slide Time: 00:42)
So, in today’s class or in this particular session we will be discussing certain things that how
the variability and price condition work for the multiple periods because the one period
example we have taken in the previous case and how the calibration model can work for the
zero coupon, coupon and the callable bonds. So, these are the things basically what we will
be discussing in today’s session.
Mostly you will see that how this thing will work for the bonds having the coupon features
and also have the call options.
(Refer Slide Time: 01:22)
719
We will come across certain keywords that growing the binomial tree, equilibrium price,
arbitrage free price. So, these are some of the keywords that we will come across while
discussing about this particular topic in today’s session.
(Refer Slide Time: 01:43)
So, let us see that what will be the variability condition if the period is 2; if the period is 2
because one period case we have considered in the previous class and today’s class if we see
that let there is a 2 period model then how that particular condition will be and how we are
trying to find out the value of that. So, given the estimated one year spot rates after one
period of 9.5% and 11% we can now move to the second period and try to determine the trees
three possible spot rates using the similar methodology.
720
So, the variability condition what basically we have discussed in the previous class follows
the same for as the one period like S ud = S dd e to the power 2 into root of h into V e A V is
nothing, but the annualized variance of that particular log returns then S uu is nothing, but S
ud into e to the power 2 into root of h V e A and that basically is nothing, but S dd e to the
power 4 into root of h V e A.
2
Sud = 𝑆𝑑𝑑 𝑒
√ℎ𝑉0 𝐴
2
Suu = 𝑆𝑢𝑑 𝑒
√ℎ𝑉0 𝐴
4
= 𝑆𝑑𝑑 𝑒
√ℎ𝑉0 𝐴
That derivation that already we have discussed in the previous class. So, if you are increasing
the number of periods accordingly this particular formula are going to be changed.
(Refer Slide Time: 03:31)
So, now let us see that how this price condition basically works here? Already you know that
the price condition generally require that the binomial value of a three year zero coupon bond
should be equal to the equilibrium price. In the previous case, we have taken the two year
zero coupon bond and here we are talking about a three year zero coupon bond that will be
equal to the equilibrium price.
So, analogous to the one period case how this condition can be found, how we can find out
this condition? So, to find this condition and how we can find this condition? We can find it
by solving the lower rates S dd previous case we are trying to find out S d. So, here we try to
find out S dd along with the variability conditions and the rates for S u and S d what basically
we have obtained previously.
721
So, now how basically with the same example if you are trying to see then if you go for a
calibration model or using the calibration model if you are trying to try these particular trees
then you will find out the value of a three year zero coupon bond will be exactly equal to the
price on a three year zero coupon bond in the market which is yielding the 10.24488% that is
the assumption we have taken from the beginning y 1 = 10% and y 2 = 10.12238% and y 3 =
10.24488% that is the example we have started and this is the continuation of that.
So, now if you see that then how basically the particular mechanism works, how that process
basically works in this particular context.
(Refer Slide Time: 05:34)
So, now let us see. First of all our objective is to find out the value of S dd that is the lowest
node that basically first we have to find out then how basically we can find out the value of S
dd. To find out the value of S dd we have to basically see this particular equation should be
satisfied and what is this equation that this will be basically your 0.5 B u + 0.5 B d divided by
1 + S 0. That should be basically = 1 + y 2 to the power 3.
So, in this case what basically we have seen the three year spot rate is basically what in the
beginning we have taken that is your three year zero coupon bond yield rate is 10.24488%.
So, that means it is your 1 by 1.1024488 to the power 3. So, again if you observe here that
this is basically you see that your 0.5 B u + 0.5 B d divided by 1 + S 0.
So, in this case what just now we have seen that 0.5 B u + 0.5 B d first of all you have to find
out B u then the B d then your whenever you are trying to find out B u it is nothing, but your
722
0.5 B uu + 0.5 B ud divided by 1 + S u and here it is B d = 0.5 B ud + 0.5 B dd divided by 1 +
S d. We are going basically backward and whole divided by you have to discount it with
respect to the current rate that is basically S 0.
So, we have S 0, we have S 1, you have S 2. So, these three rates are available in the market.
So, now what basically we are doing. So, whenever you are finding S d that these rate is
giving you 10.24488%. So, that means your 1 by 1.1024488 to the power 3 = that means we
are representing that y 0, this is y 1, this is let your y 2 for example. So, now we are writing
0.5 into 0.5 1 by 1 + S dd e to the power 4 into root of 0.0054 that already we have derived in
the previous slide + 0.5 into 1 by 1 + S dd e to the power 2 into the square root of 0.0054.
Divided by your rate is 11% you have to discount it with respect to 11% then 1.11 + your
0.55 into B ud is nothing, but your S dd into e to the power 2 into square root of 0054 that
already we have seen here and it is 0.5 1 by 1 + S dd that we have to find out. Now, you got
all these values our objective is to find out the S dd which will basically satisfy these
conditions.
So, now if you are going to solve this particular equation then we will find that your S dd will
become 9.025%. So, there is a trial and error approach, iterative approach you can follow.
You find that in this rate of S d this particular equation will be satisfied. Now, you find your
S dd = 9.025%. So, if your S dd value you got now you got everything you will find out you
S d, you will find out S ud, you will find out your S u, you will find out S uu.
1
(1+𝑦2 )3
=
0.5𝐵𝑢 + 0.5𝐵𝑑
1+𝑆0
0.5[
1
(1+𝑦2 )3
=
1
(1.1024488)3
0.5𝐵𝑢𝑢 +0.5𝐵𝑢𝑑
0.5𝐵𝑢𝑑 +0.5𝐵𝑑𝑑
]+0.5[
]
1+𝑆𝑢
1+𝑆𝑑
1+𝑆0
0.5[
4
2
2
0.5[1/(1+ 𝑠𝑑𝑑 𝑒 √0.0054 ]+ 0.5[1/(1+ 𝑠𝑑𝑑 𝑒 √0.0054 ]
0.5[1/(1+ 𝑠𝑑𝑑 𝑒 √0.0054 ]+ 0.5[1/(1+ 𝑠𝑑𝑑 )]
+ ]+0.5[
]
1.11
1.095
=
1.10
All kinds of things you can find out because that is calibrated on the basis of the S dd value
which is the lowest value.
(Refer Slide Time: 10:38)
723
So, now let us see that how that particular model will look like it is for a three period zero
coupon bond. We have taken the face value of 1. We are continuing with the previous
example your face value is basically 1. So, now you got your S dd from that equation you got
your S dd that is 9.025%. So, then you can find out your B dd where B dd will become 1 by
1.09025 that you got 0.9172208 you got your B dd.
If you got your B dd now if S dd you got you will find S ud. Your S ud is nothing, but
9.025% into e to the power 2 into square root of 0.0054. So, you got 10.45% you see it is
exactly same whatever we have derived our binomial tree from the beginning. So, now what
basically you can do? You can find out your B ud; your B ud has become 0.9053871. So, you
got S ud, you got your B ud.
Then like that you can find out your S double u that is again based upon this 9.025 into e to
the power 4 into square root of 0054 then you got 12.1% this is exactly also matching then
your B uu become 0.1 by 1.121 that basically you get 0.8920607. Now, basically what you
can do? Again roll back so use this particular values now you have three nodes from there
you can find out the B u and B d.
So, your S u is basically 11% that already you know S d = 9.5% that basically we know. Now
your 0.5 into 0.8920607 + 0.5 into 0.9053871 that you have to discount it with respect to 11%
then that will be 1.11 then you got this particular value then again 0.5 into this value + 0.5
into this value like 0.9053871 + 0.5 into 9172208 divided by your discount rate is 9.5% that
means 1.095. So, you got 0.832241.
724
So, now again these two values again roll back to the current period; current period spot rate
we have assumed 10%. So, then it will be 0.5 into 0.809661 + your 0.5 into 0.832241 divided
by 1.10 because that has to be discounted with the current spot rate that is 10% that one year
spot rate that is 10% then you got 0.7463, but if you are going to use the current yield in that
particular point of time then you get that value 0.7463.
That 10.24488% was the rate then it will be 0.7463. So, now what basically we have seen that
it is also 0.7463 it is also 0.7463 that is the way the particular pricing our prices are matching.
Now, we will see that how these particular models can be used or how the particular binomial
tree can be obtained.
(Refer Slide Time: 15:22)
So, the 2 period binomial tree generally is obtained by the combining the upper and lower
rates found for the first period with the three rates found for the second period and this yields
whatever we are finding these yields this will be this these yields basically a tree which is
basically trying to find out is consistent with the estimated variability condition and with the
current term structure of the spot rates.
(Refer Slide Time: 16:10)
725
And if you see that we have started from here you recall in the beginning we have started
from here that whenever the current spot rate was 10% it can up to 11% your u = we have
taken 1.1 and here your d you have started with 0.95. So, S u become 11%, S d become 9.5%
then your S uu become u square S 0 that is your 12.1% and your S ud that is u d S 0 that is
10.45% and S dd become 9.025%.
So, whenever this calibration model is imposed satisfying the variability and price conditions
then the binomial tree whatever we have estimated that is basically consistent with this rates
whatever we have assumed before that is what we have observed in this particular case.
(Refer Slide Time: 17:34)
Now, let us see that you can expand this particular three all these trees for the next, next to
next periods. So, to grow the tree we continue with the same process, for example, to obtain
726
the four rates in period 3 we solve for the S ddd first and after that the other rates should be
calculated on the basis of that formula whatever we can derive from there and along with the
spot rates which is found previously for period 1 and 2.
And the variability relations which basically will yield a value for 4 year this pure discount
bond or the zero coupon bond that is equal to the equilibrium price. So, you can also expand
this particular period N number of periods and accordingly you can find out the lowest one
and from the lowest one which will satisfy this particular condition then accordingly you can
find out or you can estimate these particular trees.
(Refer Slide Time: 18:41)
So, now we will see that how we can do these things for a coupon bond. We have discussed
about the zero coupon bond now we will come back to the valuation of the coupon bond. So,
one of the features of using the calibrated tree to determine the bond value is that the tree will
basically yield the prices that equal to the bonds equilibrium price that already we have seen.
So, let you have taken this example let the one year discount rate is 10%, two years
10.12238% the same example we are taking three years basically 10.24488%.
Another is the coupon bond and the coupon rate is 9% let the face value of the bond is Rs.
100 then what will be the value of the bond? So, Rs. 9 is the coupon in the first year so 9 by
1.1 again Rs. 9 in the second year then it will be 1.102238 square then in the end of the third
year we will be getting your face value of the bond that is Rs. 100 and Rs. 9 is the coupon
that means 109 divided by 1.1024488 to the power 3.
727
Then the value of the bond has become 96.9521. So, the value of a three year 9% option free
bond using the tree if you want to derive then that should be equal to 9.9521.
B3M =
9
1.10
+
9
(1.1012238)2
+
109
(1.1024488)3
= 96.9521
So, let us see whether it is happening or not.
(Refer Slide Time: 20:29)
So, let us go for a calibration model for the valuation of the coupon bond. So, the price of the
bond in the end the face value will be Rs. 100 and Rs. 9 is basically the coupon. So, here we
have assumed your coupon is 9 per year, face value is 100, S 0 is 10%, u = 1.1, d = 0.95.
These are the given. So, the cash flow what we will be getting at the end of the three periods
that is 109, 109, 109, 109. So, these are the cash flows what you are getting in the end.
So, if you are getting this cash flow in the end then from there we can start it. First of all we
have to find out the S dd. The S dd is basically we find 9.025% then your B dd has become
109 divided by 1.09025 that has become 99.9771 then you got your S dd and now you can
find out your S ud; S ud is 10.45%. So, your B ud will become 98.6872. So, S uu that you got
12.1%, u square S 0 that is 12.1% then B uu become 97.2346.
Now, you use this particular value roll back to this period and whenever you rollback to this
period we will find that 0.5 here the S u is 11%, S d = 9.5% that already we know. So, your B
u become 0.5 into this + Rs. 9 is the coupon that what you will be getting then 0.5 into this +
Rs. 9 is the coupon end of this period you will be getting Rs. 9 is the coupon then you
discount it with respect to this 11%.
728
So, if you discount it with respect to that 11% you will be getting 96.3612. So, then in this
node you have to use the values these two nodes this node and this node to find out the value
of this node. So, then here this value is 98.6872, here the value is 99.9771 so then your 0.5
into 96.6872 + 9 + 0.5 into 99.9771 + 9 and you discount it with respect to 9.5% so you got
your 98.9335 then you rollback to this previous period.
So, if you roll back to this previous period then what you will find S 0 = your 10% which was
given in before we started from here then values whatever you are getting here end of the
period one period you will get Rs. 9 with a coupon then price is cash flow here it is 96.3612 +
9 + 0.5 into 98.9335 + 9 and you discount it with respect to 10% then you got 96.9521 and
just now whenever you use the current spot rates in this case we are finding that 9 by 1.1 + 9
by 1.1012238 square + 109 divide by 1.1024488 to the power cube you also find 96.9521.
So, using this particular tree also you will find that is 96.9521. So, now the rates or the price
of the coupon bond whatever we have derived using this calibration model, using the
binomial tree that is 96.9521 that is exactly equal to the equilibrium price in the model. So,
that is what basically what we are trying to find out in this particular context.
(Refer Slide Time: 25:25)
Then let us see what will happen if there is a call feature. If there is a call feature then what is
going to happen. So, let the call price is 98 C = 9, F = 100, CP = 98 that means this is the call
price S 0 = 10%, u = 1.1, d = 0.95. So, then what will happen these are the final cash flows
729
109, 109 and 109. Your S dd is 9.025% so if you discount it your B dd with respect to this
cash flow this will be 97.9771.
And the call price is 98 and you know that your B dd c will be the minimum of these value or
the call price value B dd or CP. So, now which is the minimum? The call price is the
minimum so that is why the cash flow will be 98. Here, if you see the value whatever you
calculate that is 98.6872 109 divided by 1.1045. Then you got 98.6872 the minimum of the
call price and this is 98.
In this case you will find that the value is 97.2346, call price is 98 so that is why minimum is
this one that is why the cash flow is this. So, now what you can do use this cash flows for
discounting in this particular period S u is 11% that we find so 0.5 into 97.2346 + 9 + 0.5 into
98 + 9 and you will discount it up to 11% that is 1.11 then you got 96.0516. So, between this
call price and this value whatever you have derived.
The 96.0516 is the minimum that is why we have considered this. Second case also whenever
you find this value and this value you discount it with respect to the 9.5% that means 0.5 into
98 + 9 + 0.5 into 98 + 9 then you get this value of the 97.7169 then the minimum of these two
will become 97.7169 then you rollback here discount rate is 10%, use these two values
96.0516 and 97.7169 + 9 + 9 discount it up to 10% you got 96.2584.
Now, the price has declined why it has declined because the cash flow has changed and
wherever is the minimum the call price is the minimum or the price whatever you are
calculating that node that is the minimum that we have to consider. So, because of this the
value of the particular bond in the case in the end (()) (29:00). So, this is the calibration
model what we can use for the valuation of a callable bond.
(Refer Slide Time: 29:12)
730
So, there are certain features one of the features of the calibration model is that it prices a
bond equals to the equilibrium price and bonds equilibrium price is basically called as the
arbitrage free price. So, if the market does not price the bond at its equilibrium value then the
arbitrages should be able to always realize there is less return either by buying the bond,
stripping into number of zero coupon bonds and selling them or buying a portfolio of the zero
coupon or the zero discount bond bundling them into a coupon bond and selling it that can be
possible.
(Refer Slide Time: 29:52)
So, in general security can be valued by arbitrage by pricing it to equal to the value of its
replicating portfolio that means a portfolio or constructed so that it has the same cash flows.
So, the replicating portfolio of the coupon bond is the portfolio of the zero discount bond of
731
zero coupon bond and another feature of the calibration model is that it yields prices on
option free bonds that are arbitrage free.
And in addition to that satisfying an arbitrage free condition on option free bonds the
calibration model also values the bonds embedded options as arbitrage free prices. So, these
are the certain features what basically we can find out from this particular case.
(Refer Slide Time: 30:40)
So, what we have discussed in this session that one of the features of using a calibrated tree to
determine the bonds value is that tree will yield the prices which is equal to bonds
equilibrium price and bonds equilibrium price is an arbitrage free price. So, if the market
does not price the bond at its equilibrium value then the arbitrages always would be able to
realize a risk less return or the risk free return.
(Refer Slide Time: 31:11)
732
So, these are the references. Thank you.
733
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Sciences
Indian Institute of Technology – Kharagpur
Lecture – 45
Estimation of Binomial Trees – V
Welcome back. So, in the previous class we discussed about how we can estimate the
binomial trees for the zero coupon bonds having the multiple periods and as well as the
binomial tree estimation for the coupon bearing bonds and as well as the bonds having the
call features. The coupon bearing bonds, but they are having the call feature. So, these are the
three things we have discussed using the calibration model how the binomial interest rate
trees can be formulated or can be estimated.
(Refer Slide Time: 00:54)
So, in this session we will be discussing about the concept called the option adjusted spread;
the option adjusted spread analysis, the concept of effective duration and effective convexity.
So, these are the different concepts what we are going to discuss in today’s session.
(Refer Slide Time: 01:20)
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And then we will come across the words like already I told you option adjusted spread or the
path analysis. So, these kinds of words you will find in this particular class.
(Refer Slide Time: 01:38)
So, let us see that what do we mean by this option adjusted spread. Already we discussed that
particular concept before also, but here what we are trying to see that the binomial interest
rate tree also can be used to estimate this option adjusted spread. So, option spread is nothing,
but the yield to maturity of a bonds having a option feature let call option feature minus the
yield to maturity of a bonds having no options features or option features.
So, that is basically we call it the option adjusted or option spread generally we call it the
option spread. So, whenever we try to calculate this option spread how we calculate? The
simplest way of calculating the option spread is first to calculate this yield to maturity for a
735
bond with an option given the bonds value as determined by the binomial tree and then
subtract the particular yield from the yield to maturity otherwise identical option free bond.
So, using the binomial tree for a callable bonds or bonds having the option feature we find
out the yield to maturity then you find out the yield to maturity of the bonds having no call
feature or no option feature then take a difference between them you will find out the option
spread. So, in our previous example whatever we have used if you see that we have three
year 9% coupon bearing bonds which has a call feature.
And the call price was 98 if you recall so there what we have seen that whenever you find out
the price of that option free bond we find the price is 96.9521. So, if 96.9521 is the price of
that particular option free bond then if you want to solve this equation that price is equal to
your cash flow divided by 1 + discount rate then you can find out your YTM that already
extensively we have discussed whenever we are trying to calculate this yield into maturity.
9
9
109
Option-Free Bond: 96.9521= 1+𝑌𝑇𝑀 + (1+𝑌𝑇𝑀)2 + (1+𝑌𝑇𝑀)3
= YTM NC= 10.2306%
9
9
109
Callable Bond: 96.2584= 1+𝑌𝑇𝑀 + (1+𝑌𝑇𝑀)2 + (1+𝑌𝑇𝑀)3
= YTM C= 10.51832%
Option spread= YTM c – YTM NC = 10.51832-10.2306= 0.28772%
That is the option spread; that is the simplest way to calculate the option spread, but there is
some issues here. There are some kind of difficulties, some kind of complexity always we
can face whenever we are trying to find out the option spread in this way.
So, in that context we have find out what is that complexity and why that particular kind of
difficulty we face whenever we calculate the option spread in this particular way.
(Refer Slide Time: 06:07)
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So, here what is the problem? The problem is using this approach to estimate the spread is
that not all of the possible cash flows of the callable bonds are considered. Just now we have
seen that we have taken the cash flow if you go back to our previous slide we have taken
99109 where the cash flow whatever we have considered, but in true sense if you see in three
of the four interest rate scenarios, for example, the bond could be called, but the cash flow
patterns can be 99109 or it can be also 9 and 107 because whenever the bond will be
exercised or bond will not be exercised accordingly your cash flow will get (()) (07:14)
because the bond call price was 98 bond coupon is 9.
So, wherever it will be exercised the cash flow will be 98 + 9 that will be 107 and some cases
it will be 109. So, here in this case if you observe because the cash flows are varying the
straight forward way of discounting the bonds using those cash flows are not feasible, are not
practically applicable in this particular context. So, therefore to address that particular
problem we have an alternative approach that is called the option adjusted spread analysis
option adjusted spread that actually you have to keep in the mind.
So, the basic objective of the option adjusted spread analysis is what? We have to solve for
the option spread let we are assuming the option spread is k then we have to find out that k
which makes the average of the present value of the bonds cash flow from all of the possible
interest rate paths equal to bond’s market price and the same concept with the yield to
maturity.
737
But the extra k we have to find out this option spread k which makes the average of the
present value of the bonds cash flow from all of the possible interest rate path equal to the
bonds market price. So, k basically is the option spread we have to find out which spread
basically will satisfy this condition that is what is our objective. So, how we do this or how
we can do this?
(Refer Slide Time: 09:22)
So, you see there are different steps we have to follow. So, in the first step of the option
spread analysis or option adjusted spread analysis what we do specify the cash flows and
spots rates for the each path. So, whatever example we have taken that is the three year bond
valued with a two period binomial interest rate tree and there are four possible paths three
year bond, 2 period binomial interest rate tree and four possible paths.
So, in these case what basically we have seen here. So, your time period is 0, 1, 2, 3. So, here
if you see this there are cash flows in the different periods 10% was the initial discount rate
obviously there is no cash flow then the discount rate has become either 9.5 or 9.025% and
the cash flow either it will be 9, it will be 107 that is 98 + 9. So, this is for 1 period. Another
period the rates can be 10% or another path we can say these are the different paths basically.
So, this is the first, this is the second, this is the third, this is the fourth; so these are the
different paths. So, in the first path the cash flows are 9, 107 and the discount rates are 10%,
9.5%, 9.025%. Second path if you see your cash flow is 9 or 107 and discount rates are 10%,
9.5%, 10.45% which is in the node if you see that your 10% either become 9.5% or it can
become 10.45 so this is the second path.
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Third it can be 10%, 11%, 10.45% so this is another path and cash flow is 9 or 107 and in the
fourth path if you observe the discount rates are 10%, 11% or 12.1% and the cash flows can
be 9, can be 9 can be 109. So, first step you specify the cash flows, rates and path. So, these
are the four paths what we have seen. So, after finding these paths what is the next step what
we are going to do.
(Refer Slide Time: 13:09)
The next step is basically find out or determine the appropriate two year spot rates and three
year spot rates to discount the cash flows. So, how we can find out the two year spot rate and
three year spot rate go back to the term structure interest rate theory. So, according to that
these are nothing, but the geometric mean of the different spot rates what basically we have
estimated from this particular tree.
Path 1
Y1= 0.10
Y2= [(1.10) (1.095)]1/2-1= 0.097497
Y3= [(1.10) (1.095) (1.09025)]1/3-1= 0.095076
Path 2
Y1= 0.10
Y2= [(1.10) (1.095)]1/2-1= 0.097497
Y3= [(1.10) (1.095) (1.1045)]1/3-1= 0.099826
Path 3
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Y1= 0.10
Y2= [(1.10) (1.11)]1/2-1= 0.104989
Y3= [(1.10) (1.11) (1.1045)]1/3-1= 0.104826
Path 4
Y1= 0.10
Y2= [(1.10) (1.095)]1/2-1= 0.097497
Y3= [(1.10) (1.095) (1.121)]1/3-1= 0.110300
So, now in the second step we got the appropriate two year, three year spot rates which will
be used to discount the cash flows. Now, what is the next step we are going to do.
(Refer Slide Time: 17:45)
In the next step basically what we do we have to solve the k. In the beginning we have taken
that assumption. We have to solve for the k with basic which makes the average present value
of the path equal to the callable bond market price. So, now the rates are available with you,
cash flow is available with you then you can find out the market price, how we can find out
the market price in the different path.
9
[
107
1+0.10+𝑘
9
+ (1+0.097497+𝑘)2 ]
107
+ [1+0.10+𝑘 + (1+0.097497+𝑘)2 ]
9
107
9
+ (1+0.104989+𝑘)2 ]
B0M = (1/4) + [1+0.10+𝑘 + (1+0.104989+𝑘)2 ]
+[
1+0.10+𝑘
107
109
{
+ (1+0.110300+𝑘)3
}
740
So, you find out the values in 4 paths take a average of that; that will give you the market
value of that particular bond which will be equal to we have to understand that where the if
you solve this which makes the average present value of the path equal to the callable bond
market price; market price already you have. So, you solve that particular equation which k
basically will equalize this cash flow with the market price of the callable bond that is 96
point something whatever we have estimated before.
So, from there whatever k basically we will find that is the option spread. So, that is more
realistic and practical way of calculation of the option spread instead of directly using that
particular cash flows in finding out that particular path. So, that is basically called the option
adjusted spread analysis that can be also used for valuation of other types of bonds like
mortgage backed securities and other things that we will discuss further.
(Refer Slide Time: 22:58)
So, if the market price is equal to the binomial value we obtained using the calibration model
then the option spread k = 0. So, this reflects that we have calibrated the tree to the yield
curve and have considered all of the possibilities, but in practice we do not expect the market
price always equal to the binomial value. So, if the market price is below the binomial value
then the k will be positive that is basically what we call it the option adjusted spread analysis.
(Refer Slide Time: 23:46)
741
Then the option features of a bond can also have significant impact on the bonds duration and
as well as convexity. The option features of a bond may also have the impact on the bonds
duration and as well as the complexity, how? For example if you see when a bond has a call
option if the bond has a call option then let the interest rate has declined. So, in that case it
can lead to an early call the call option feature can be exercised.
Then what will happen which will reduce the life of the bond, let the maturity period of the
bond is 10 years and the bond has been called back up to 6 years then automatically the bonds
life will reduce the term to maturity of the bond has been reduced, if the maturity period will
reduce then it will basically decline the value of the division it will lower the duration value,
but if the interest rates increase then it will lengthen the expected life of the bond then
causing its duration to increase.
So, that is why the call feature can have the impact or any kind of option feature can have the
impact on the duration of the particular bond and as well as the convexity of the particular
bond. So, that generally we call it the effective duration.
(Refer Slide Time: 25:41)
742
So, what is duration? We remember what do you mean by the duration? Duration is again a
time measure and that generally is calculated by giving certain weights then how the weights
are given so if you recall your duration of a bond is nothing, but let summation t = 1 to M
your t into your cash flow t divided by the price of the bond if you recall these things. So, we
find out this and this we basically giving the weight on the basis of the cash flow whatever
we consider in that particular case.
So, there are certain other properties of the duration you might have recall it is basically your
present value of the cash flow divided by the price of the bond and this is your t time. So, this
is basically the weighted average of the bonds time period and the average is basically
generally given on the basis of the proportion of the present value of the cash flow in that
period with respect to the value of the bond.
So, in that case it is basically the weighted average maturity of the bonds cash flow on the
present value basis that is the way we define the duration and you recall also the duration can
be used as a price sensitivity measure. So, if you are using the duration as a price sensitivity
measure generally we calculate this is what we call it the Macaulay’s duration and if you are
using this duration as a price sensitivity measure.
𝐵 −𝐵
+
Effective Duration = 2(𝐵− ) (∆𝑦)
0
𝐵− −𝐵+ −2𝐵0
Effective Convexity= (𝐵 ) (∆𝑦)2
0
743
So, in that case if you recall we also discussed this part the effective variation is nothing, but
you can find out the B – you can find B + then you have the B 0 then delta y (()) (29:00)
change in the yield then you can find out the effective duration so B – means it is a price
which is associated with small decrease in the rates and B + means price basically which is
associated with the small increase in the interest rates.
So delta y means change in the interest rate. So, that is the way basically the effective
duration can be calculated. So, if there is a features then how the bond price is going to be
changed due to the increase in the interest rate or due to the decrease in the interest rate
accordingly the effective duration can be estimated. These are the approximate formula for
finding out the duration also that part we have discussed already.
(Refer Slide Time: 30:28)
So, the binomial tree calibrated to the yield curve can used to estimate your B 0, B – and B +.
B 0 can be defined by the current yield curve and the calibrated tree, B – can be estimated by
allowing for a small equal decrease in each of the yield curve rates then using the tree
calibrated to the new rates to find the price and B + can be estimated by a similar way by
allowing for a small equal increase in the yield curve states.
And then estimating the bond price using the tree calibrated to these higher rates. So, the
binomial tree model can be used to find out the B 0 B – and B + and then accordingly your
effective duration and convexity can be calculated.
(Refer Slide Time: 31:19)
744
So, what basically we have discussed in this session the objective of the option adjusted
spread is basically to solve for the option spread which makes the average of the present
value of the bond cash flow from all of the possible interest rate paths will be equal to the
bonds market price and the option features of a bond can have a significant impact on the
bond duration as well as its convexity. So, these are the things what basically we have
discussed in today’s session.
(Refer Slide Time: 31:52)
So, these are the references. Thank you.
745
Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Lecture - 46
Bond Investment Strategies - I
Welcome back. So, in the previous class we discussed about the different aspects of the
bonds which are the embedded options. There we have discussed about the valuation pricing
and all, so today we will be discussing about the different type of bond investment strategies.
So, what are the different kind of strategy the investor adopts whenever they try to invest in
the bond market to manage their risk.
(Refer Slide Time: 00:52)
So, in this particular session will be covering of the one particular types of the strategy that is
called the active bond investment strategy. Then, within that active bond investment strategy
one of the strategies is basically the interest rate anticipation strategy. So, largely the today
session will be covering of the different methods or different kind of approaches which are
generally, the investors use while investing in the bond.
And they are trying to adopt the active investment strategies and mostly it will deal with the
different kind of methods which are related to the interest rate anticipations.
(Refer Slide Time: 01:42)
746
So, while going through this particular session you will be coming across different type of
keywords like rate anticipation swap, yield curve shifts and the concepts like bullet strategy,
barbell strategy, bond laddering strategy or bond ladder. So, these kinds of words basically
will come across whenever we discuss about the active bond investment strategy and mostly
which is related to interest rate anticipation strategy.
(Refer Slide Time: 02:21)
So, here you see whenever you talk about the different types of the bond strategies. What are
those different kinds of strategies generally the investors adapt? Whether it is bond or it is a
stock or any other instrument whatever instruments we use, generally broadly we have two
types of strategies that are investors adopt, one is active other one is the passive. So, if you
are adopting both you are basically adopting the both type of strategy for your analysis.
747
Like some part is related to passive part and some part is related to active part, then we can
call it the hybrid strategy. So, we have then broadly can categorize it in terms of active
strategy, passive strategy and the hybrid strategy. So, if you are going to use both active and
passive strategies to some extent in your investment management then, we can call it the
hybrid strategy. So, in that case what basically we can see what do we mean by this active
strategy.
Active strategy is basically nothing but here this strategy basically involves taking these
different active positions in the bond investments. And the basic objective is basically getting
some abnormal return or the investors always try to beat the market. So, if you are having a
benchmark kind of index which is available in the system likes bond market indices. Then,
whatever return the bond market index is giving.
So, if the investors objective is to get more return what the bond market index is giving then,
we can say that the investors are basically trying to adopt the active strategy. So, that’s why
they always attempt to outperform the passive benchmark portfolio on a risk adjusted basis.
So, over the period of time frequently they change their positions the investors basically
change their positions.
And the basic objective is to get more return what the particular benchmark portfolio is
providing. So, that is basically in general term we call it the active strategy but, whenever you
talk about the passive strategy here generally if the investor is adopting a passive approach.
Here, the no change in the position is necessary once the bonds are selected, that means the
investor basically follows the market.
So, they try to always replicate the market or the market in the sense we are referring to the
bond market index which are already prevailed in the market. So, whatever return the bond
market index is giving the investors are always trying to get that return. So, this is a strategy
of holding a portfolio of securities without attempting to outperform other investors through
superior market forecasting.
So, in this case the investors do not change their positions frequently they generally hold on
to their particular portfolio for a reasonable period of time and their basic objective is to
follow this particular market or what the index is giving whatever return the index is giving
they always try to get that return or whatever return the particular bond market index is
providing. Then, already what we have discussed that the active strategy and passive strategy.
748
If you are combining these two then, or some part basically we can follow the from the active
point of view and some part you can follow from the passive point of view that means both
the features are involved in terms of the hybrid strategy. So, the particular strategy which are
both active and passive features we can say that these are a part of the hybrid strategy. So,
these are the different type of bond strategies for the bond investors always use while
investing in the bond market.
So, today we will largely focus on the active strategy part and gradually we can move
towards the passive and the hybrid strategy.
(Refer Slide Time: 07:07)
So, let us see what are the different types of active strategies which are available in the
system or in the market? So, if you see that broadly you can categorize them into four, one is
your interest rate anticipation strategies then you have the credit strategies then fundamental
valuation strategies then different type of swap strategies. These are the different type of
active strategies for the bond investor always adopt while investing on market and with a
basic objective of to maximize the return or to minimize the risk.
So, again within these four strategies whatever just now we have mentioned here in this case,
one by one will discuss what is the typical process generally, the investors follow while
adopting this strategy and also largely this session will cover of the interest rate anticipation
strategies then gradually, we can move towards the other strategies.
(Refer Slide Time: 08:17)
749
So, whenever you talk about the interest rate anticipation strategy. So, here it is basically a
kind of strategy where the investors try to select the bond or the bond portfolio with specific
durations based on the interest rate expectations. How the interest rate is going to change?
How the interest rate is going to move? That is the basic criteria to decide that particular bond
portfolio.
On the basis of the change in the interest rate or expectations about the change in interest rate
the investors basically try to make that particular strategy. So, that’s why these strategies are
basically based on the interest rate expectations and these shifts in the yield curve. How the
yield curve is shifting? And yield curve will shift whenever there is a change in the interest
rate.So, depending upon the interested expectations and change in the interest rate.
And accordingly, the movement of the yield curve the investors basically decide the strategy
that where to invest and whether they should invest in the long-term bond or they should
invest in the short-term bond or they should invest in both. If they are investing in both then
which one, they should buy which one they should sell. So, these kinds of questions generally
always investors try to answer through the prediction of the interest rate.
So, interest rate expectations are the major factors to decide that whether they should invest
in what kind of bond they should invest.
(Refer Slide Time: 10:04)
750
So, here we have one particular strategy called the rate anticipation swap. The rate
anticipation strategy is basically a kind of strategy where the investors simultaneously selling
and buying the bonds with different durations or different maturity.
In this case the investor is basically taking both the positions at a particular point of time
simultaneously they are buying the bond and they are selling the bond but the term to
maturity or the duration of this particular bonds will be different. That basically they will do
on the basis of the reduction of interested rates. How the interest rate is going to move? That
basically will decide them that whether they should invest in that particular bond or not.
And even if they are investing then which bond they should buy and which bond basically
they should sell.So, that is basically called the rate anticipation swap.
(Refer Slide Time: 11:14)
So, here let us understand the basic notion of the rate expectations approach. Here if you see
that generally the rate expectation approach it is one of the most aggressive strategy or the
active strategy for the investors. And in this particular approach the investors basically
continuously shift the maturities of the securities in line with the current forecasts of the
interest rate and the economy.
How this particular interest rate is going to be changed and how the economic conditions are
going to be changed accordingly they basically shift the maturity of their securities. So, in
this case if you see that they will call for shifting of investment towards the short end of
maturity spectrum, when the interest rates are expected to rise and they will move towards the
long-term end when the interest rates are expected to fall.
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So, what is the logic in that, what is the basic understanding in this particular context? So, if
they are expecting that the interest rates are expected to rise or they are assuming that the
interest rate is going to rise in the future. So, then what they basically do they shift their
investment towards the shorter maturity bonds or if they are expecting that the interest rates
are going to fall then they are basically going for the long-term bonds because, in the longerterm period they can get more return from that.
So, that’s why if you look at this particular figure these are the term to maturity or the
duration these are the maturity. So, if you see basically the percentage of the value of all the
securities where they are holding in the portfolio. If they are expecting that the interest rates
are going to rise then what basically they will do, they will go towards holding more
securities which are the short-term maturity bond.
And if they are expecting that the interest rate is going to fall in the future or in the long run
then they will go towards the long-term maturity bonds. So, in that particular context within
that particular process this particular strategy will offer a high potential for large capital gains
but also increases the chances of substantial loss. If the interest rate will not basically move
as per the expectations, then the investors basically may incur the losses.
But there is also high chance of high return if they will follow this kind of strategy in the
market.So, because of this we can call this strategy is one of the riskiest strategies or it is one
of the most aggressive strategies in terms of the bond investments.
(Refer Slide Time: 14:47)
752
So, let us explain it in this particular way, let when interest rates are expected to decrease
across all maturities assume across all maturities the interest rates are expected to decrease.
So, how basically the interest rate anticipation swap is working in that particular case. So,
what is the strategy the investor can adopt? The investor basically can adopt the strategy of
increasing the portfolios duration.
Portfolios maturity period basically they can increase lengthening the portfolios duration. So,
in this case what the portfolio manager can do or the bond investor can do? They can sell
their lower duration bonds or the bonds having the lower duration or the lower maturity and
by the high duration ones.So, that means what is the logic here in this case if they are
expecting that the interest rate is going to be down, that means the rate is down.
Then in that case what will happen the price will increase across all the maturities. So, which
will imply the long high duration bonds means buying, buying this long duration bonds and
short means it is basically selling, selling the low duration bonds. Even if in their portfolio
there are some bonds which are the shorter-term maturity or short-term duration, they can sell
that and they will buy the bonds having the longer-term maturity.
So, already what we have discussed the portfolio’s value would be more sensitive to interest
rate changes which provide the greater upside gains in values if the rates decrease but also
greater losses in value if the rates decline. As per expectation if the interest rate will not
decline then there is a high chance of the losses in that particular case.
(Refer Slide Time: 16:59)
753
In the same way, if you see that the interest rate is expected to increase across all maturities.
Then, what this investor can do or the portfolio manager can do they can always reduce the
portfolios duration. So, to reduce the portfolios duration they can sell the higher duration
bonds and buy the lower duration bonds. So, the expected interest rate is going to up then the
price will be down across all maturities.
Which basically implies the selling the high duration bonds and buying the low duration
bonds that is what we have mentioned here. So, the objective is to preserve the value of the
fund, what basically they are allocating for the bond investment. That is what basically what
we can think of or we can see about when the interest rates are expected to increase across all
the maturities.
(Refer Slide Time: 18:05)
So, another way also the rate anticipation swap can work using the concept of the cushion
bonds. What is this cushion bond? One of the ways to reduce the bonds portfolio duration is
to selling the longer duration bonds and buy the cushion bonds. So, generally the longer
duration bonds what the investor is selling, possibly they should be option free they should
not have any kind of embedded options, but whenever you talk about the buying of the
cushion bonds.
What do you mean by the cushion bond? The cushion bond is basically a callable bond with a
coupon that is above the current market interest rate. You can look for a bond which is
callable and the coupon rate is above the market interest rate. So, that’s why the coupon bond
has the following features it should be high coupon yield and with it is embedded call option
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a market price that is lower than a comparable non-callable bond in a low interest rate
environment. These are the two features what you can think of about holding a cushion bond.
(Refer Slide Time: 19:35)
So, how this cushion bond concept works? Let us take one example, suppose a manager with
a portfolio consisting of 10-year 10% option free bond, which is let valued at rupees 113.42
per Rs.100 par and the objective is to yield 8%.
In the market, there are comparable 10 years 12% coupon bonds which are callable at a price
of 110, that are trading at a price close to their call price that is let called the cushion bond.
So, what the manager can do? If the manager expects that the interest rates are going to
increase then, he could cushion the negative price impact on the bonds value. Selling the
option free bond and buying this higher coupon callable bond that is basically the cushion
bond.
So, the swap of the existing bond for the cushion bond generally provides an immediate gain
of the 3.42 rupees per 100 rupees that means (113.42- 110) =3.42.
And a higher coupon income in the future: 12% instead of the 10%. So, that is basically we
call it the concept of interest rate anticipation swap using the cushion bonds.
(Refer Slide Time: 21:22)
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So, a callable bond has a lower duration than a non-callable bond with the same maturity and
the coupon rate and the 10-year cushion bond with it call feature and higher coupon rate has
relatively lower duration than the 10-year option free bond. So, if in this example if you look
at the swap of the cushion bonds for this option free bonds represents a switch of longer
duration bonds for the shorter ones, that is basically nothing but a rate anticipation swap.
We are swapping with a cushion bond with an option free bond and generally, in that
particular process we are switching of a longer duration bond for a certain bond. That is what
basically what we can think of in this particular context.
(Refer Slide Time: 22:22)
So, then we can also see the yield curve shift. So, some of the rate interest rate anticipation
strategies are based on the forecasting the types of the yield curve shift and then
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implementing on appropriate strategy to profit from the forecasting. So, there are already we
have discussed this part there are three types of yield curve shift can occur. There may be a
parallel shift there may be shift with some twists and there can be a kind of humpedness in
that particular shifting. So, there are three ways the yield curves can be shifted.
(Refer Slide Time: 23:04)
So, whenever you feel that the interest rates on all maturities sends by the same percentage,
the interest rates of all maturity change by the same percentage or the basis points then we
can say that that is a parallel set.
(Refer Slide Time: 23:22)
If you find that a twist in that case, that means a twist is a non-parallel set either a flattening
or a steepening in the yield curve? Whenever there is a flattening of the yield curve then you
will find that the spread between the long term and short-term rates decline. Yield to maturity
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what you are getting from a long-term bond and the yield to maturity from a short-term bond
that difference will basically decline in this case.
But if you are assuming a steepening yield curve then what will find the spread between the
long term and short-term rates basically increases. So, your yield curve basically will shift in
this way, this is the way basically it is shifting from these two this will be shifting in this way.
So, that is why the YTM of a long-term security and YTM of a short-term security if you take
a difference of these two, then basically it is increasing.
(Refer Slide Time: 24:33)
But whenever you look at the humpedness, a shift with humpedness basically is a nonparallel shift again in which the short term and long-term interest rate change basically,
always more in magnitude than the medium term on the intermediate rates. So, there are two
things we can observe here one is positive butterfly other one is the negative butterfly. So, in
case of positive butterfly there is an increase in both short-term and long-term rates relative to
the intermediate rates.
And if you look at the negative butterfly, there is a decrease in both short term and long-term
rates relative to the intermediate rates. That is what basically we can observe in terms of the
yield curve or the humpedness curve what we can observe in the market.
(Refer Slide Time: 25:30)
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So, it will look like this, if you look at the positive butterfly the short term and long-term
rates change more than the intermediate. In this case negative butterfly if you look at the
intermediate rate change more than the short term and long-term rates that is basically, we
call it the negative butterfly. That is for the humpedness curve case we can observe this.
(Refer Slide Time: 25:55)
So, whenever we look at the yield curve shift strategy generally look at the three types of
strategy one is bullet strategy, barbell strategy and ladder strategy or we call it also the bond
laddering. What do you mean by the bullet strategy? Bullet strategy is generally formed by
constructing a bond portfolio which will be concentrated in one maturity area, one maturity
domain.
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For example, a bullet strategy consisting of a portfolio of the long-term bonds could be
formed if there is an expectation of a downward shift of the yield curve with a twist that the
long-term rates are expected to decrease more than the short-term rates that means largely the
investor will hold the long-term security bonds. Mostly, the bonds down to maturity or the
duration will be quite long.
If they will go for a barbell strategy that means, what do you mean by this if the investor
expects a simple downward parallel shift in the yield curve, then the bullet strategy with
longer duration bond will yield get greater return in case of the bullet strategy. So, if the
investor expects a simple downward parallel shift in the yield curve, then a bullet strategy
with longer duration bond yields the greater return.
Then, we have the barbell strategy what do you mean by the barbell strategy? Generally, it is
formed with investments which are concentrated in both short term and long-term bonds and
these strategies generally will be profitable, if any investor is forecasting a negative butterfly
yield curve shift. Where the intermediate interest rates are more than the short term and longterm rates and the barbell strategy will work better in that particular context.
Then, we have the ladder strategy. The ladder strategy is basically formed with equally
allocated investments in each maturity group that means equally you are allocating the same
amount of the funds of different bonds having the different maturity. So, this is what
basically call it the bond laddering.
(Refer Slide Time: 28:37)
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Let us see what do you mean by the barbell strategy, barbell strategy already we have
discussed that here the investing institutions or the investors basically place most of the funds
in short term portfolio of highly liquid securities at one extreme and a long-term portfolio of
the bonds in other extreme and minimal investment in the intermediate maturities.So, the
short-term portfolio will provide the liquidity and the long-term portfolio basically will
generate the income.
So, the barbell strategy is basically works in this particular direction.So, the half or a
substantial percentage of the portfolio will be always put either in the short-term bonds or in
the long-termbonds.So, we can always keep or always you can put the maximum amount of
the funds in short term securities or we can keep it in the long-term securities that is what
basically is a part of the barbell strategy.
(Refer Slide Time: 29:42)
Then, we have the ladder strategy or also we can call it the spaced maturity policy or spaced
majority strategy. So, here the equal percentages of the funds are allotted or allocated to the
different bonds having the different maturity. To invest in equal proportions of securities in
each of several maturity intervals until the maximum acceptable maturity is reached. So, here
let we have taken up to 5 years maturity, so 20% of the portfolio should be invested in one
year maturity bonds another 20% in 2 year another 20% in 3 year like that.
So, this strategy does not maximize the investment income, but it has an advantage of
reducing the income fluctuations. If there is a disparity among the interest rate fluctuations,
then somewhere whatever gain you are getting that can be adjusted with the losses or
whatever way you have incurred the losses that can be adjusted with the gains.
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(Refer Slide Time: 30:53)
So, what we have discussed that the active strategies basically attempt to outperform a
passive in smart portfolio on a risk adjusted basis and active bond investment strategies are
generally based on the interest rate anticipations, credit quality of a bond, fundamental
valuation and swaps and all these things that will be discussing in the forthcoming sessions.
And the rate anticipation strategies are basically selecting the bonds or the bond portfolio
with a specific duration.
Which are based on the interest rate expectations and forecasting the type of the yield of shift
and yield curve shift strategies include the bullet strategy,barbell strategy and the bond
laddering.
(Refer Slide Time: 31:40)
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So, these are the references.
Thank you.
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Management of Fixed Income Securities
Prof. Jitendra Mahakud
Department of Humanities and Social Science
Indian Institute of Technology, Kharagpur
Lecture - 47
Bond Investment Strategies - II
So, in the previous class we discussed about the bond investment strategy and largely we started
the discussion about the active bond investment strategy. There we discussed about the concepts
related to the interest rate anticipation strategy. So, we will continue with that particular
discussion in today's session also.
(Refer Slide Time: 00:41)
So, in this particular session we will come across the different concept like your horizon
analysis, the credit strategies, fundamental analysis of corporate credit. Although, extensively,
we will not discuss that part, we will just introduce that concept then largely it will be discussed
in the next session.
(Refer Slide Time: 01:07)
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Then we have the keywords what basically you will be finding in this particular session like
quality swap, sector rotation, fundamental analysis and all these things. So, here if you see in the
previous class, we discussed about the interest rate expectations or the rate anticipation swaps.
There are different types of strategies we have discussed like bullet strategy, barbell strategy then
we have the bond laddering and all these things.
And those things basically will depend upon whatever way the yield curve is shifting. How the
yield curve is shifting accordingly these particular strategies can be adopted to manage the risk.
(Refer Slide Time: 02:05)
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But what is the way, how to understand that how to find out the mechanism that how this
particular strategy or which strategy is going to work better in that particular context. So, the
correct yield curve strategies generally depend on the forecasting of the yield curve shifting. So,
one approach to use in identifying the appropriate strategies basically is horizon analysis. If you
recall the horizon analysis already, we have discussed in the one of the previous sessions
whatever we have discussed.
I will just again go through that horizon analysis to understand that how this particular strategy
or this particular analysis will help us to use a particular approach which would be beneficial for
the investors to maximize their return. So, horizon analysis is nothing but a total return analysis.
So, here it involves the determination of the possible total returns from different yield curve
strategies given different yield curve shifts.
We have to make the different scenarios and accordingly we will see that if the interest rate will
change then how the total return of the bonds are going to be changed. Then accordingly we can
decide that which particular strategy will be beneficial for the investors on the basis of their
objectives. So, that is basically what the horizon analysis is trying to help you.
(Refer Slide Time: 03:56)
So, let us see that how this horizon analysis or on the basis of horizon analysis we can decide that
what kind of strategies should be adopted.
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Case1: Let we have taken three bonds in this case. We have assumed three bonds. So, one bond
is let 5year maturity; coupon is 6%; bond is selling at par that means the yield is also 6%; and
this bond has duration of 4.46.
Case-2: Then another bond which has 11 years maturity; 6% is the coupon it is also selling at
par; and the duration is given 8.36.
Case-3: Another bond 20 years maturity; 6% selling at par; and duration is 12.16.
You assume that the yield curve is currently flat at 6%. So, in the previous session we have
discussed about the two different strategies one is barbell another one is the bullet. In the barbell
strategy what basically we are doing? We are investing one extreme some amounts of funds are
allocated to one particular extreme that means in terms of the maturity. And another 50% is
basically invested in another extreme and that is the bonds are decided on the basis of the term to
maturity or the duration.
So, let here Case-1 is a short-term bond, Case-3 is a long-term bond and Case-2 is relatively long
but we can consider this is a medium-term bond, you can assume that.
So, here let whenever you go for barbell strategy, we have invested 50% in bond A and 50% in
the bond C; 50% in the short-term bond and 50% in the quite long-term bond. Then we have the
bullet strategy where let you have invested all 100% in the bond B.
You assume two types of yield curve shift one year later that is the assumption what we are
taking.

One is there is a parallel shift and this shift can take place from -200 basis point to +200
basis point that means minus 2% to plus 2%.

Second condition we are assuming a flattening shift here the yield curve shifts basically
characterized by flattening in which each change in bond B that means the intermediate
bond; the bond A increases 25 basis point more and bond C decreases by 25 basis point
less.
Mathematically, this can be written as:
∆YA = ∆Y B + 25basis points
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And ∆YC = ∆Y B - 25basis points
That’s why we are assuming a flattening shifting of the yield curve. Then we will see in the
different scenarios how the total return of this particular portfolio is going to be changed.
(Refer Slide Time: 08:20)
So, now you come to this particular condition where we are assuming the first case, we are
assuming a parallel shift.
Change in basis points: We have started with the 200-basis point change that means plus 2% then
1.5%,1% 0.5, 0.25, 0 then it is again -25 basis point,-50 basis point that means - 0.5,-0.25,-1, 1.5and – 2%.
So, if you are calculating the value of A, value of B, value of C as well as return of A, return of
B and return of C.
And you have two types of strategy one is barbell strategy and we have the bullet strategy.
Situation 1: whenever the interest rate will change by 200 basis point or 2% if you are going by
the barbell strategy then you are getting your return is basically declined by - 6.92%. But in case
of bullet strategy, it is -7.42%.
Situation 2: like that if it is 150 basis points, if you are going by the barbell strategy then you are
getting your return is -3.98. But in case of bullet strategy, the return is -4.30.
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So, like that if you look at Case-11 when change in basis point is -200 basis points; the interest
rate will decline by 2% in terms of barbell strategy, your return is 22.76 and in terms of bullet
strategy it is 22.22.
If you make a difference between these two; the return what you are getting from the barbell
strategy and return what you are getting from the bullet strategy this is basically the difference
between these two i.e. the barbell strategy returns minus the bullet strategy return. So, here what
we have observed, the bullet strategy return basically the return what we get it from the
investment of the bond B because 100% investments are made on that.
If you look at this duration because this is a case of barbell strategy that is investment in 50%
bond A and 50% in bond C.
So, here if you see that this is also your barbell portfolio.
The return, you are getting from the barbell strategy is:
Bullet return= 0.5 (Bond return for A) + 0.5 (Bond return for C)
The barbell portfolio also has duration of 8.31. How you got it?
Bullet duration= 0.5 (4.46) + (0.5) 12.16 = 8.31.
So, that is approximately equal to the duration of the bond B.
If you look at the bond B, that is the duration is 8.36 but here in this case what we get the bonds
duration is basically 8.31. So, this is approximately same as the duration of the bond B.
So, you will not find much difference in terms of the return differences between these two types
of strategy if you look in that case.
Because, all 100% investment is bond B in terms of the bullet strategy. This is also barbell
portfolio these are the duration of 8.31 and the bullet strategies if you are adopting also there the
duration is also 8.31.
(Refer Slide Time: 12:56)
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So, now what basically we can see that what we can observe in this case for different parallel
shifts in the yield curve, there is not much difference in the returns on the bullet portfolio and the
barbell portfolio. And why this happens? Because this is due to both having the same duration,
one is 8.36 and whenever you go for the barbell strategy this is 8.31.
If one were expecting a significant downward shift in the yield curve then the bond C with the
largest duration would give you the greatest gains. That already you know in the properties of the
duration that part already we have discussed.
If one were expecting a significant downward shift in the yield curve then bond C with the
largest duration would give you the greatest gains and if one were expecting a significant upward
shift in the yield curve then the bond A with the lowest duration would give you the minimum
loss. So, what basically we have observed in this particular example.
The returns basically are consistent with duration as a measure of the bond’s price sensitivity to
the interest rate changes. Duration generally is a measure of price sensitivity with respect to the
interest rate change. So, here also we have seen not much difference we have observed between
these two strategies because generally the duration of the bond B is your 8.36 and in barbell
strategy, duration is 8.31. So, the barbell return is what? 50-50%.
So, your barbell return has been giving you, of 22.76 or (-6.92) depending upon the fluctuations
of the interest rate.
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Barbell return= (0.5) (bond return of A) + 0.5 (bond return of C). So, that is what basically we
get the barbell return and this also you got your return of the bond or the duration of the bond A
is 4.46.
So, duration of the bond= 0.5 (4.46) + (0.5) 12.16 = 8.31
That is basically your barbell portfolio duration, weighted average of the duration of all those
bonds. So, therefore what basically just now we have seen that these returns are basically
consistent with the duration as a measure of bond’s price sensitivity to the interest rate changes.
(Refer Slide Time: 16:16)
Then let us see about the case of the flattening shifting of the yield curve. If you look at here
whenever there is a positive 2% change of the interest rate or 200 basis point. So, your barbell is
giving minus 6.26, your bullet is giving minus 7.42, the difference is 1.16. In case of the minus
2% or minus 200 basis point, you will find the barbell is giving 24.25 and your bullet strategy is
giving 22.22.
So, here your already we have seen that your ∆YA = ∆Y B + 25basis points and ∆YC =
∆YB - 25basis points
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So, what we have observed here? In contrast to the parallel shift, there are differences between
the barbell and bullet portfolios. When the yield curve shift has a twist, it has a twist even though
they have the same durations. Just now, we have seen, they have the same durations one is 8.31
barbell portfolio duration and other one is 8.36.
But in this case what we have seen that whenever there is a twist in terms of the shifting of the
yield curve even if the durations are same we have observed that there is a huge return difference
between these two. What basically it implies? What basically it basically shows? Even if the
durations are same, if the portfolio shift is basically not parallel or portfolio shift has a twist then
there is a possibility that you will find the huge differences in terms of the two different
strategies.
The return differences you can observe and that return differences is coming because of the
shifting of the yield curve.
(Refer Slide Time: 18:43)
So, now in this case the different strategies are giving the different returns. Then we are coming
back to the credit strategies. So, in the context of the credit strategy, the first one is the quality
swap. So, what do you mean by this quality swap?
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The quality swap is basically strategy of moving from one quality group to another in
anticipation of a change in the economic conditions. Quality in the sense we are talking about the
quality of the bonds.
One quality group means one particular rated bond to another rated bond on the basis of the
economic fluctuations or change in the economic conditions.
Another one, the credit analysis so here we have to analyse the all type of bonds which are
generally traded in the market like your corporate bonds, municipal bonds, foreign bonds to
identify the potential changes in the default risk.
What are the potential changes in the default risk and that information can be used to identify the
bonds to include or the exclude in a bond portfolio or the bond investment strategy. Basically, we
have to analyse the all the aspects, all the features or the fundamental characteristics of the
particular type of bonds on the basis of the different features and accordingly we can decide that
whether that particular bond is suitable for the investment or not.
(Refer Slide Time: 20:44)
Let us first discuss about the quality swap then we can move towards the credit analysis. So,
what do you mean by the quality swap? The quality swap is nothing but it is a strategy of going
long and short in bonds with high or low quality rating based on the expectation of change in the
economic states or economic conditions. What does it mean? It means that let you are expecting
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that there is going to be recession. Then you long in high quality bonds that means you (long
means you buy) buy the high-quality bonds and short means it is basically sell. You sell the
locality bonds that mean you make a swap of the low-quality bonds with the high-quality bonds
if you are expecting that there is going to be recession.
But if you are expecting there is an expansion in the future then you can buy the low-quality
bonds and sell the high-quality bonds.
So, that is the strategy what basically we can adopt whenever you talk about the quality swap.
(Refer Slide Time: 22:48)
So, whenever we are adopting the quality swap, generally what we do? What is the process?
what is the basic kind of mechanism we follow whenever the strategy is called the quality swap
generally used by the investors. Quality swaps generally involve a sector rotation. What do you
mean by the sector rotation? Generally, in case of sector rotation more funds are allotted to a
specific quality sector in anticipation of a change in the economic conditions.
Expecting how the economic fluctuations are going to happen, how this condition is going to be
prevailed, we generally follow a sector rotation strategy. Here more funds generally allotted to a
specific sector or the bonds issued by the companies which belong to a particular sector.
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For example, a fund manager generally expects a recession which is also accompanied by flight
to safety in which the demand for high quality bonds will increase and the demand for low
quality bonds should decline.
So, in that case if they want to generate some profit (some extra return). For this expectation, the
manager could change the allocation of our bond fund by selling some of the low-quality bonds
and buying more high-quality bonds. They can sell some of the low-quality bonds and buy some
high-quality bonds where the ratings are relatively better.
Suppose reverse thing is happening, the fund manager is expecting that there is going to be
economic expansion. Then in this case what they are going to do? In this case, the demand for
the low-quality bonds will increase and the demand for high quality bonds will decline. So, to
profit from or generate return from this particular expectation the manager could change the
allocation of the bond by selling more high-quality bonds and buying more low-quality bonds in
their portfolio.
In that particular process basically, they can generate certain profit out of this. So, that is called
the quality swap. The quality swaps can also be constructed to profit from anticipated changes in
the yield spread between the quality sectors. Spread means the yield difference between t