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ANALYSIS OF
FIXED INCOME SECURITIES
Valuation of Fixed Income Securities
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VALUATION
Valuation is the process of determining the fair value of a financial asset. The fair value
of an asset is its “price”. Often times the market also used yield and yield spread as a
relative measure of value.
DEFINITIONS OF FIXED INCOME SECURITY TERMINOLOGIES
Fixed Income Security is a financial obligation of an entity that promises to pay a
specified sum of money at a specified future dates. The entity that promises to pay is
called the issuer. Fixed Income Securities fall into two broad categories:
1. Debt obligations – here the issuer is the borrower and the lender is the investor or
creditor. The issuer promises to pay interest and principal at maturity at agreed
future dates e.g. bonds, mortgage backed securities, asset backed securities and
bank loans
2. Preferred Stock - represents an ownership interest in a corporation with fixed
dividend payments. Dividend is however paid from after tax profit and is payable
only when there is profit but in preference to the ordinary stock holders
Indenture & Covenants – are the terms in terms of rights of the holders and the
promises of the issuer. These terms are usually managed by a Trustee on behalf of all
bondholders. As part of the indenture there are affirmative and negative covenants.
Term to Maturity is the number of years over which the issuer promises to meet the
obligation. The term to maturity is the same as the tenor only on the date the security is
issued, thereafter the term to maturity is lower than the tenor as time elapses.
Par Value is the amount the security promises to repay at maturity i.e. the principal,
face value, redemption value and maturity value. The practice is to quote the price of
a security as a percentage of its Par Value i.e. the Par Value is deemed to be 100.
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When a security is trading at below its Par Value, it is said to be trading at a Discount.
When it is the converse i.e. it is trading above its Par Value, then it is said to be trading
at a Premium.
Coupon rate is the interest rate that the issuer agrees to pay each year. Also, called
the nominal rate. The actual monetary amount paid annually is called the Coupon. Not
all fixed income securities promises to pay a coupon. When a security promises to pay
only one fixed amount, it is called a Zero-Coupon instrument.
BASIC PRINCIPLES OF VALUATION
The Price of any financial instrument is equal to the present value of the expected cash
flows from the instrument. There are 3 (three) basic steps to valuation ·
Step 1:
Estimate of the expected cash flows
·
·
periodic coupon payments to maturity date
par (or maturity) value at maturity
Estimate of the appropriate discount rate on comparable
·
Step 2:
·
bonds in the market i.e. non-callable bonds of the same credit quality and same
maturity.
Step 3:
Calculation of the Present Value (PV) of the expected cash
flows in Step 1 using the discount rate(s) in Step 2.
STEP 1: ESTIMATE THE EXPECTED CASH FLOWS
Cash flow is simply the cash that is expected to be received in the future from an
investment whether it is interest or principal. The expected cash flows for some
instruments are easy to compute, for others, the task is more difficult e.g.:
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1. When the issuer or investor has the option to change the contractual date of the
repayment of the principal e.g. callable/putable bonds, mortgage backed securities,
asset backed securities, etc.
2. The coupon payment is reset periodically based on a formula that depends on some
value or values for reference rates, prices or exchange rates e.g. floating rate
securities
3. The investor has a choice to convert or exchange the security into common shares
e.g. convertible bonds and exchangeable bonds
STEP 2: DETERMINING THE APPROPRIATE RATE(S)
The minimum rate an investor should use is that on a default-free cash flow (i.e.
Treasuries) with the same maturity. The premium over this rate should reflect the
additional risks of the specific instrument which the investor takes. Consequently, the
required rate is the rate for financial instruments with comparable risk or alternative (or
substitute) investments for each of the tenors of the expected cash flows.
STEP 3: DISCOUNTING THE CASH FLOWS
In general, the price of a bond is given by
P
=
C
+
C +
1+ r
(1+ r)2
or:
P
where:
=
Σ C
+
t
(1+ r)
C
P
n
r
M
t
=
=
=
=
=
=
C
+ ……+ C
+
M
3
n
(1+ r)
(1+ r)
(1+ r)n
M
(1+ r)n
coupon payments usually semiannual
price of the investment
number of periods
periodic interest rate
maturity value
time period when the payment is received
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This valuation model supports the use of a different discount rate for each cash flow
otherwise we may use the Annuity Concept.
A short-cut formula for computing the value of a security when a single discount rate is
used is to compute the present value of the Annuity (the interest payments) and then
adding the Present Value of the maturity value.
The present value of an annuity is equal to –
1 Annuity payment
X
1
(1 + r)n
r
PRESENT VALUE PROPERTIES
A bond that matures in 4 years time has a coupon rate of 10% with an annual interest
payment frequency.
a.
Assuming applicable discount rate for similar security is 8%, what is the price of
the bond today?
b.
If the discount rate is changed to 12%, what will the price be?
Year
Cash Flow
Present value Present value
@ 8%
@ 12%
1
N10
9.2593
8.9286
2
10
8.5734
7.9719
3
10
7.9383
7.1178
4
110
80.8533
69.9070
106.6243
93.9253
Price
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NOTE:
· TIMING: The longer the term to maturity, the lower the value of the cash flow today
· DISCOUNT RATE: The higher the discount rate, the lower the security value
· RELATIONSHIP OF COUPON RATE, DISCOUNT RATE & PRICE RELATIVE TO
PAR:
Coupon rate = yield required; price = par value
Coupon rate < yield required; price < par value
Coupon rate > yield required; price > par value
· CHANGE IN VALUE AS TIME MOVES CLOSE TO MATURITY: A bond’s valueo Decreases over time if the bond is selling at a premium
o Increases over time when the bond is selling at a discount
o Is unchanged if the bond is selling at par value
o At maturity is equal to its Par Value
VALUATION USING MULTIPLE DISCOUNT RATES
The proper way to value the cash flows of any security is to use a different discount rate
that is unique to the time period in which the cash flow will be received.
Suppose we have a 4-year 10% coupon bond with annual interest payment and the
appropriate discount rates are as follows:
Year 1
Year 2
Year 3
Year 4
6.8%
7.2%
7.6%
8.0%
The present value or price for this instrument today is:
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Cash flow
Discount
rate
Discounted
value
Year 1
10
6.8%
9.3633
Year 2
10
7.2%
8.7018
Year 3
10
7.6%
8.0272
Year 4
110
8.0%
80.8533
Price
106.9456
Practice: What is the value of a 5-year 7% coupon bond assuming the payments are
received annually and the discount rates for each year are as follow:
Year 1
Year 2
Year 3
Year 4
Year 5
3.5%
3.9%
4.2%
4.5%
5.0%
VALUING SEMIANNUAL CASH FLOWS
For semi-annual cash flows, the computation remains the same except that both the
cash flow and discount rate are adjusted to half of the annual. A period is a six-month
period i.e. the number of periods will double.
VALUING ZERO-COUPON BOND
A zero coupon bond has only one cash flow – the maturity value. The value of a zerocoupon bond that matures n years from now is
Maturity value
(1 + i)n x 2
Where i is the semi-annual discount rate
n is the number of years to maturity
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Note: The number of periods is n X 2 for consistency with the pricing of coupon bearing
bonds in the market.
Exercise:
Complete the following table for a 10 year zero coupon bond with a maturity value of
$1,000 for each of the following annual discount rates.
Annual
rate
Semiannual
rate
Price
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
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COMPLICATIONS IN VALUING FIXED INCOME SECURITIES
1.
Next Coupon Payment in less than six months
For coupon paying securities, a complication arises when we try to price the instrument
between coupon payments. This is because one of the cash flows (the immediate)
encompasses two components as shown below:
Interest earned by seller
Last coupon
payment date
Interest earned by buyer
Settlement
date
Next coupon
payment date
Interest earned by the seller is the interest accrued between the last coupon payment
and the settlement date. This is called Accrued interest. The buyer must compensate
the seller for this and recovers it from the next interest payment.
The Price determined using the present value concept is called the Full or Dirty Price
is that is the amount paid by the investor. From the Full Price, the accrued interest is
deducted to get the Clean Price which is true price of the bond.
In computing the Full price of a bond, the formula is Present valuet
Where:
w period
=
=
Valuation of Fixed Income Securities
expected cash flow
(1 + I )t -1+w
days btw settlement and next coupon date
no of days in coupon period
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Illustration:
A 5-year semi-annual 10% coupon security is being sold on a day that is 78 days
between the settlement date and the next coupon date. There are 182 days in the
coupon period.
w
Period 1:
Period 2:
Period 3:
Period 4:
=
78/182 = 0.4286 periods
Present Value1 =
5
(1.04)
0.4286
(1.04)
.1.4286
(1.04)
.2.4286
Present Value2 =
5
Present Value3 =
5
Present Value4 =
5
= 4.9167
= 4.7276
= 4.5457
= 4.3709
.3.4286
Period 5:
Present Value5
(1.04)
=
105
(1.04)
= 88.2584
.4.4286
Full Price is the sum of the PV of the cash flows, which is 106.8192
The Accrued Interest (AI) is computed as Semi-annual coupon payment X (1 – w)
AI
=
Clean Price =
N5
X
(1 – 0.4286) =
N2.8571
N106.8192 - 2.8571 = N103.9621
Practice:
Suppose a bond is purchased between coupon periods. The days between settlement
date and the next coupon period is 58. There are 183 days in the coupon period.
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Suppose that the bond purchased has a coupon rate of 7% and there are 10
semiannual coupon payments remaining.
a.
b.
What is the full price for this bond if a 5% annual discount rate is used?
What is the Clean Price?
DAY COUNT CONVENTIONS
Different day-year (interest basis) conventions apply to instruments in different markets.
Those of us who engage in international capital markets, should be familiar with the
conventions:
Eurodollar (LIBOR) market
Actual/360
Eurobond market (AIBD)
360/360
US Treasury/ Ghanaian Money Market Actual/365
Nigeria Money Market (NMM)
Actual/Actual
The numerator depicts the number of days interest is to be earned while the
denominator depicts the number of days in a year in the relevant market.
Example:
A coupon bearing US Treasury security whose previous coupon payment was March 1
and the next is September 1. Suppose this Treasury security is purchased with a
settlement date of July 17th.
Days between Settlement and Next Coupon Dates is actual days from July 17th to
September 1st
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US Treasury
July 17 to July 31
August
September 1
14 days
31 days
1 day
46 days
Eurobond
13 days
30 days
1
44 days
The number of days in the coupon period is computed as follows
March
April
May
June
July
August
September
Total
2.
30 days
30
31
30
31
31
1
184 days
29 days
30
30
30
30
30
1
180 days
Cash flows are unknown
This problem occurs largely with callable or putable bonds. With these bonds, the cash
flows will depend on the current level of interest rate relative to the coupon rate. The
valuation of these securities is done using valuation models such as the Binomial
model (which seeks to price the embedded interest rate options) and the Monte Carlo
Simulation Model (which is used for mortgaged backed securities and certain types of
asset backed securities. This prices both embedded interest rate options and
prepayment options)
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VALUATION OF FIXED INCOME SECURITIES
Exercises
1. What is the price of a 5-year security that pays 7% annually assuming a discount
rate of 5%.
2. A 5-year amortizing security with a par value of N10,000 and a coupon rate of 5%
has an expected cash flow of N2,309.75 per year assuming there are no
prepayments. The annual cash flow includes interest and principal repayment. What
is the present value of this amortizing security assuming a discount rate of 6%.
3. What will be the initial offering price for the following bonds:
a. A 15-year zero coupon bond, assuming a discount rate of 12%
b. A 20-year zero coupon bond, assuming a discount rate of 10%
4. An 8.4% coupon bond issued by the State of Indiana sells for $1,000. The income
on this bond is tax exempt. What coupon rate on a corporate bond selling at its par
$1,000 value would produce the same after-tax return to the investor as the
municipal bond if the investor is in the:
a. 15% marginal tax bracket
b. 25% marginal tax bracket
c. 35% marginal tax bracket
5.
At what price would you expect a bond with the following characteristics to trade in
the market assuming the expected yield for this quality of bond is 10%
Par Value
Coupon rate
$1,000
8%
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Payment frequency annually
Term to maturity
10 years
6. What is the value of a 5-year 5.85% coupon bond if the appropriate discount rate for
each cash flow is as follows:
Year
Discount rate
1
5.90%
2
6.40%
3
6.60%
4
6.90%
5
7.30%
7. Suppose a bond is selling between coupon periods. The days between the
settlement date and the next coupon period is 115. There are 183 days in the
coupon period. Suppose the bond has a coupon rate of 7.4% and there are 10
semiannual coupon payments remaining.
a. What is the dirty price of the bond?
b. What is the Accrued Interest?
c. What is the clean price?
8. What is the value of a zero coupon bond that matures in 20 years, has a maturity of
$1 million and is selling to yield 7.6%
9. What factors will you take into consideration in valuing bonds?
10. What are the factors that complicate bond pricing and why?
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