Introduction to Mathematics of Finance
Dr. Tsang
Chapter 7
Bonds & fixed income
securities - II
1
Topics
• Duration & convexity
• Asset-Liability matching & immunization
• Yield Curve & Term structure of interest
rates
• Forward rates
• Discounted cash flow analysis
• Preferred stocks
2
Duration
• The average time taken by a bond, on a discounted
basis, to pay back the original investment.
• (Macaulay) Duration of a bond is the weighted
average of the maturity of individual cash flows, with
the weights being proportional to their present
values:
where CFt is the cash-flow at time t and PV is the
present value function of its argument, y is the YTM rate
and
3
Duration of a Zero-Coupon Bond
4
Duration of a coupon Bond
5
Duration changes as the coupons
are paid to the bondholder
The lever is no longer in balance as
the first coupon payment is removed
from the red lever and paid to the
bondholder
The fulcrum must now move to the right in
order to balance the lever again:
6
Modified Duration
• A bond’s interest rate risk can be
measured by its relative price change with
respect to a change in yield. This is called a
bond’s modified duration:
The term modified duration is partially due to its link with
the bond’s duration.
Modified duration is also called the volatility, a measure
of the sensitivity of the bond to interest rate changes.
7
Relationship between Modified Duration & Duration
B
Therefore we have
MD = D / (1+i)
dB/di = - DB/(1+i)
8
Duration is used to estimated change in
bond price due to change in yield
dB = - (MD) B dy
Change in
bond price
Change in yield
9
Example: Consider a 4-year T-note, with face value $100
and 7% coupon paid semi-annually, selling at $103.50,
yielding 6%.
in pay period
Duration is
Modified duration (volatility) is
D = (738.28)/103.50/2 = 3.57
MD = D/(1+y) = 3.57/1.03 = 3.46
10
Example(cont.): As the yield changes, the bond price also
changes
For small yield changes, pricing by MD is accurate.
For large yield changes, pricing by MD is inaccurate.
11
Example(cont.): Bond price is not a linear function of the
yield. For large yield changes, the effect of curvature (i.e.,
nonlinearity) becomes important.
12
Duration – graphic interpretation
13
What is
Convexity?
Convexity is the curvature of the bond price (per $)
as a function of the yield:
Therefore,
14
Convexity – graphic interpretation
15
(1+y)-2
16
Example(cont.): 4-year T-note as before (face value of
$100, 7% coupon, selling at $103.50, yielding 6%).
17
Duration of a portfolio of series of
bonds/casflows
18
Asset-Liability matching & immunization 1
Suppose an organization will make
investments (assets) so that funds will be
available to provide for future outgoing
payments (liabilities). It would like to make
sure that the present value of its assets less
the present value of its liabilities (net position)
is hedged against movements in the rates of
interest.
19
Asset-Liability matching and immunization 2
20
Asset-Liability matching and immunization 3
Theorem: If
21
Asset-Liability matching and immunization 4
From these we get:
which is easier to remember.
22
Immunization example 1
This immunization against small changes in i is
called Redington immunization.
Example: Liability payments of 100 each are due
to be paid in 2, 4 and 6 years from now. Asset
cashflow consists of A1 in 1 year and A5 in 5 years.
1) Find A1 and A5 to have asset cashflow immunize
the liability cashflow by matching present value
and duration at i0 = 10%.
2) Is this a Redington immunization?
23
Immunization example 2
Let v=1/1.1
A1 v + A5 v5 = 100(v2 + v4 + v6) = 207.39
A1 v + 5A5 v5 = 100(2v2 + 4v4 + 6v6) = 777.18
Solution: A1 = 229.44 and A5 = 71.14
We also find A1 v + 25A5 v5 > 100(4v2 + 16v4 + 36v6).
So this is Redington immunization.
24
Assignment 7.1
A corporation has a debt of $1 million that becomes due in 10 years
from now (t=0). What is the duration of this debt? It is decided that
the debt is to be paid off by purchasing a portfolio of bonds which
can be sold when the debt is due. The bonds in the portfolio all
have a face value of $100 and a 1 year period. They are described in
the following table.
[a] Find the current price and duration of each bond.
[b] What is the mix of bond 1 & 2 in the portfolio so that the
corporation can meet its obligation regardless of small fluctuations
of market yield?
25
MATLAB--Financial Toolbox
Bnddury, bnddurp -- Bond duration given yield/price (SIA compliant)
[ModDuration, YearDuration, PerDuration] =
bnddury(Yield, CouponRate, Settle, Maturity, Period, Basis,
EndMonthRule, IssueDate, FirstCouponDate, LastCouponDate, StartDate,
Face)
[ModDuration, YearDuration, PerDuration] =
bnddurp(Price, CouponRate, Settle, Maturity, Period, Basis,
EndMonthRule, IssueDate, FirstCouponDate, LastCouponDate, StartDate,
Face)
SIA=Securities Industry Association
26
Example:
modified duration in years
Macaulay duration in years
Macaulay duration reported on a
semiannual bond basis
27
Yield Curve & Term structure of interest rates
•
Investor prefer short term investments, so must be coerced into buying
longer term rates with higher interest rates.
• A graphical depiction of the relationship between the
yield on bonds of the same credit quality, but different
maturities is known as the yield curve.
• Term structure of interest rates is the relation between
yield to maturity (YTM of zero coupon securities with
same credit quality) and their maturities.
•
Yield-to-maturity on zero-coupon securities for different maturities is
also called the spot rate for that maturity. Therefore, term structure of
interest rate may also be defined as the pattern of spot rates for different
maturities.
28
How to Construct the Term Structure of
Interest Rates?
• The yield on Treasury securities is a
benchmark for determining the yield
curve on non-Treasury securities.
Consequently, all market participants
are interested in the relationship
between yield and maturity for
Treasury securities.
29
Term Structure of Interest Rates
Normally, long rates
are higher than
short rates.
30
1953
1954
1955
1956
1957
1958
1960
1961
1962
1963
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1965
1967
1968
1969
1970
1971
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1981
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1985
1986
1988
1989
1990
1991
1992
1993
1995
1996
1997
1998
1999
2000
2002
2003
2004
2005
2006
2007
Why do yields vary over time?
US yields 1953-2008
18.00
16.00
14.00
Sometimes, short rates
are higher. The yield
curve becomes inverted.
12.00
10.00
8.00
6.00
4.00
2.00
0.00
3m T-bill
10y T-bill
31
1953
1954
1955
1956
1958
1959
1960
1961
1963
1964
1965
1966
1968
1969
1970
1971
1973
1974
1975
1976
1978
1979
1980
1981
1983
1984
1985
1986
1988
1989
1990
1991
1993
1994
1995
1996
1998
1999
2000
2001
2003
2004
2005
2006
2008
Why do yield spreads vary over
time?
US yield spread 1953-2008
5.00
4.00
3.00
2.00
1.00
0.00
-1.00
-2.00
-3.00
-4.00
Sometimes, short rates
are higher. The yield
curve becomes inverted.
10y-3m
32
Term Structure of Riskless Interest Rates
• U.S. Treasury Strips (zero-coupon bonds), May 3, 2004:
Maturity
Bid
Asked
Ask Yield
Nov. 2004
99.12
99.12
1.21%
Nov. 2005
96.31
96.31
2.02%
Nov. 2006
93.17
93.17
2.66%
Nov. 2007
89.22
89.22
3.11%
33
Spot Interest Rates
• Spot interest rate, rt, is the (annualized)
interest rate for a transaction between
today, 0, and a future date, t.
–rt is for payments only on date t.
–rt is the “average” rate of interest between
now and date t.
–rt is different for each different date t.
34
Forward interest rates
• are rates for a transaction between two
future dates, for instance, t1 and t2.
• For a forward transaction to borrow money
in the future:
– Terms of transaction is agreed on today, t = 0
– Loan is received on a future date t1
– Repayment of the loan occurs on date t2.
• Future spot rates can be different from
current corresponding forward rates.
35
Forward Rates
• Forward rates of interest are implicit in the term structure of interest
rates
t=0
1
r1
2
4…
3
1f2
r2
2f3
r3
3f4
• Note the notation: 3f4 means “the forward rate from period 3 to period
4.”
• When the beginning subscript is omitted, it is understood that the
forward rate is for one period only: 3f4 = f4 .
36
Example problem
As the CFO of a multinational corporation, you
expect to repatriate $10 M from a foreign
subsidiary in 1 year, which will be used to pay
dividends 1 year later. Not knowing the interest
rates 1 year in the future, you would like to lock
into a lending rate one year from now for a
period of one year. What should you do?
The current interest rates are
37
Solution:
Strategy:
1. Borrow $9.524 M (=10/1.05) now for one year at 5%
2. Invest the proceeds $9.524 M for two years at 7%.
Outcome (in million dollars):
The net effect is you locked in a 1-year lending rate 1 year
from now at 9.04%, which is the forward rate for year 2.
38
We can deduce Forward Rates from Observed
spot rates
(1  y n )
(1  f n ) 
n 1
(1  y n 1 )
n
fn = one-year forward rate for period from n-1 to n
yn = yield for a security with a maturity of n, from 0 to n
39
Spot and forward rates
40
Assignment 7.2
Suppose that discount bond prices are as follows:
A customer would like to have a forward contract to borrow
$20 M three years from now for one year. Can you (as a
banker) quote a rate for this forward loan?
f4 = 8.51%.
41
Discounted cash flow analysis (1)
• So far, we have analyzed the present value of a
regular series of payments, such as coupon
bonds, annuities…
• This approach can be extended to any pattern
of cash returns.
• By obtaining the present value of any pattern
of future payments, we are performing a
discounted cash flow analysis.
42
Discounted cash flow analysis (2)
• Let Ck represent contributions by an investor that
are made at time k.
• Let Rk represent returns to the investor at time k.
• These returns may also be considered as
withdrawals.
• Since contributions and returns are equivalent
concepts seen from the opposite sides of the
transaction. So we have Ck = -Rk.
43
Discounted cash flow analysis (3)
• So contributions may be referred to as cash-flowsin while returns may be referred to as cash-flowsout.
• Note that in certain situations, contribution and
return happened at the same time. If that happens,
Ck (or Rk) will be the difference between the
contributions and the returns.
• Sometimes we referred to that as net cash-flows.
44
Example 7.1(1)
• A 10-year investment project requires an initial
investment of $1,000 and subsequent beginningof-year payments of $100 for the following 9
years.
• The project is expected to produce 5 annual
returns of $600 commencing 6 years after the
initial investment.
• Find the IRR of this project.
45
Example 7.1(2)
• We have the following cash-flows:
C0 = 1,000 = R0
C1 = C2 =  = C5 = 100 = R1 = R2 =  = R5
C6 = C7 =  = C9 = 100 – 600
= 500 = R6 = R7 =  = R9
C10 = 600 = R10
46
Example 7.1(3)
• Given an investment (interest) rate i, the (net)
present value of the returns is determined as follows:
47
Example 7.1(4)
The problem is then reduced to determining the value of i
for which PV0 = 0.
• This value of i is called the yield rate of investment:
– If the yield rate is higher than expected (cost of capital), then
the investment is attractive.
– If the yield is lower than expected (cost of capital), then the
investment is not attractive.
– If the yield rate is the same as expected (cost of capital), then
the investment is acceptable.
Often we have to solve the equation numerically.
48
Example 7.2
• Suppose there are two options for investment: Option A credits
9% effective for the first five years and a certain rate i for the
second five years, while Option B credits 8% effective for ten
years. Find i for which the two options have the same yield rate.
• The equation of value is
• Or
• Thus if the investor expects prevailing interest greater than 7.01%
at the end of five years, than option A is a better choice.
• Otherwise, option B is a better choice.
49
Example 7.3
• Consider a transaction in which a person makes
payments of $250 immediately and $330 at the end of
two years in exchange for a payment in return of $575
at the end of one year.
• An equation for this transaction is
250 (1 + i)2 + 330 = 575 (1 + i).
• Or
50(1 + i)2  115 (1 + i) + 66 = 0.
• Factoring we obtain [10(1 + i) – 11] [5(1 + i) – 6] = 0.
• So the yield rate is either 10% or 20%.
50
Multiple or unique yield rate(s)
• Often when solving for the yield rate, we may be left
with a polynomial equation.
• The problem with polynomial equations is that they
are liable to have several roots.
• In Example 7.3, we have two possible answers for the
yield rate.
• It is important in practice to be able to tell whether
there will be a unique yield rate.
51
Descartes rule of signs
• Descartes stated his rule on signs in La Géométrie, but
did not give a proof.
• He said that the maximum number of positive roots of
the equation f (x) = 0, where f is a polynomial, is the
number of changes in sign of the coefficients when
terms of f is written in descending order of degrees.
• This became known as Descartes' rule of signs.
• But this rule does not guarantee the existence of
positive root.
52
Theorem: existence of an IRR
)
53
Theorem: existence of an IRR
54
Example: IRR
• You purchased a bond for $800 5-years ago and
sold the bond today for $1200. The bond paid an
annual 10% coupon. What is his realized rate of
return?
n
• PV = S [CFt / (1+r)t]
t=0
• $800 = [$100/(1+r) + $100/(1+r)2 + $100/(1+r)3 +
$100/(1+r)4 + $100/(1+r)5] + [$1200/(1+r)5]
• To solve, you need use a “trial and error”
numerical approach. You plug in numbers until
you find the rate of return that solves the equation.
• The realized rate of return on this bond is found to
be 18.46%.
55
Rewrite the previous equation as
P(x) = -800 + 100(x+x^2+x^3+x^4) + 1300x^5
Can you show the existence of positive real
root?
Solution: P(0) < 0 & P(1)>0
56
MATLAB Financial Toolbox
bndyield --Yield to maturity for a fixed income security (SIA
compliant)
Syntax:
Yield = bndyield(Price, CouponRate, Settle, Maturity, Period, Basis,
EndMonthRule, IssueDate, FirstCouponDate, LastCouponDate,
StartDate, Face)
Same example in - using the “bndprice” function
57
The default face value of the bond is 100 in bndyield.
We have to rescale the equation in the IRR example.
Price=100*800/1200=66.6666666667
CouponRate=100/1200=0.08333333333
The corrected coupon rate
Period changed to 2
58
Uniqueness of yield rate (1)
• How can we tell if we will get a unique yield rate?
• To do that, we make use of Descartes’ rule of signs.
• If there are m changes going from cash-flows-in
to cash-flows-out (or vice versa), then there are a
maximum of m positive solutions to the
polynomial equation.
• So if there is only one change, then there is only
one yield rate.
• Note that Example 7.3 had 2 sign changes.
59
Uniqueness of yield rate (2)
• A unique yield rate will always be produced if all cash
flows in one direction are made before the cash flow in the
other direction, and continues for the rest of the investment
period.
• In the 10-year investment project of Example 7.1, there
was always a cash contribution in the first 5 years, and
then there are cash return for years 6 to 10.
• In the 10-year project above, a unique yield rate exists.
• It is possible for no yield rate to exist, because multiple
yield rates may be all imaginary.
60
Example 7.4
• What is the yield rate on a transaction in which a
person makes payments of $100 now and $101 at
the end of two years, in exchange for a payment of
$200 at the end of one year?
• An equation of value is:
100 (1 + i)2 + 101 = 200 (1 + i),
• Or
100 i2 =  1.
• So in this case, the yield rates are all imaginary!
61
Assignment 7.3
Tom bought a house for $200,000 and rented it out to
receive a steady rental income of $1,000/month. He
sold the house 6 years later to net $250,000 after
deducting all expenses. What is the annualized
return rate he got from the investment?
Set up the correct equation and use MATLAB to get the answer.
62
Assignment 7.4
Mary bought a 10 year bond with face value $100 &
a coupon rate of 6% at market price to yield 6.75%.
She sold it 7years later when its market yield rate
was 5%. What is the annualized return rate she got
from the investment?
63
Preferred stock & perpetual bonds
Preferred stock is a security that pays a
constant level of payments (dividends)
periodically.
The value of a preferred stock is equivalent to
the value for a perpetual annuity.
A perpetuity occurs when a cash flow is expected to be
received (or paid) every time period through infinity.
Examples include perpetual bonds (paying fixed interest
with no maturity date) and preferred stock (which pays a
fixed dividend payment).
64
Theoretical Value of Preferred stock
65
66
End !
67