Bond Valuation
Professor Thomas Chemmanur
Bond Valuation

A bond represents borrowing by firms from investors.

F  Face Value of the bond (sometimes known as "par
value"); this is the amount the company will pay back at the
maturity of the bond.

Ct  Coupon or interest payment at date t. Usually coupon
amounts are the same at each date, in which case we will
simply use C for coupon.
A coupon is often expressed as a percentage c of the face
value. This percentage is called "coupon rate". Then, C = c•F,
where c is expressed in decimals. Notice that the coupon rate
is specified as part of the bond (it is not something that
changes according to economic conditions).

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Bond Valuation

n  The number of periods to maturity.

r  The rate of return per period at which the bond cash flows
are to be discounted, determined by financial market conditions.
For the present, we will assume that cashflows at all dates are to
be discounted at the same rate. r changes as the general level of
interest rates in the economy changes.
The price that you are willing to pay for the bond is simply the
present value of all the cash flows that the bond entitles you to;
Thus,

C3
Cn  F
C1
C2
P


 ..... 
2
3
(1  r ) (1  r ) (1  r )
(1  r ) n
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Bond Valuation


Usually coupon payments are made semi-annually; this
means that the length of a period is a half-year, and r should
be in terms of return per half-year; n should also be in halfyears.
If the price of the bond is less than the face value, we say that
the bond is selling at a "discount"; if they are the same, the
bond is selling at par; if the price is higher than the face
value, it is selling at a "premium".
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Example

If Teletron Electronics has a bond outstanding that pays a $75
coupon semi-annually, what is its price? The annual interest
rate = 10%; 15 years to maturity.
r = 10/2 = 5%
n = 15*2 = 30 PERIODS
P = 75 (PVIFA 5%, 30) + 1000(PVIF 5%, 30)
= 75 (15.3725) + 1000(0.2314)
= $1384.34

There are several rules about bond pricing that can be kept in
mind.
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Some Rules of Bond Valuation
•
•
•
•
•
(i) WHEN c = r , THE BOND SELLS AT PAR
(ii) WHEN c > r, THE BOND SELLS AT A PREMIUM
(FOR n > 0); WHEN c < r, THE BOND SELLS AT A
DISCOUNT (FOR n > 0);
(iii) WHEN r  , PRICE FALLS; AND WHEN r  ,
PRICE INCREASES.
(iv) THE LONGER THE MATURITY, THE MORE
SENSITIVE THE BOND PRICE TO INTEREST RATE
CHANGES.
(v) AS n  0 (THE BOND APPROACHES
MATURITY), P  F (FACE VALUE); AT n = 0, P = F.
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Summary of Bond Valuation Rules
Bond Price, p
r<c
r=c
F
r>c
0
Time to Maturity, n
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Yield to Maturity



The yield to maturity of a bond is the rate of return we would
earn if we bought the bond at a price P and held it till it
matures.
That is, it is that rate of return r that solves the bond pricing
equation for a known P. Clearly YTM will be different for
different values of P. (Usually, yield to maturity is computed
and quoted in the financial press assuming semi-annual
compounding).
The assumption here is that you are able to re-invest all
coupons you receive also at the rate of return r.
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What determines the yield of bonds?



1. Real Interest Rate
2. Inflation Rate
3. Time to Maturity

Real Interest Rate ( rc )
Is defined as the interest rate that would prevail in a world
without inflation. It depends on:
 (a) Investor Preferences (current vs. future consumption)
 (b) Production Opportunities
 Inflation Rate (p)
 Inflation exists in most economies, measured by:
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What determines the yield of bonds?
PRICE OF A BASKET OF GOODS NEXT YEAR
(1 + p) =
PRICE OF SAME BASKET THIS YEAR
 p = (Ratio of PRICES) – 1
 E.g. Consumer Price Index

Relationship between real and nominal (money) interest rates.
(Fisher’s Theory)
 (1 + r) = (1 + rc) (1 + p)
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Term Structure of Interest Rates




Often, the discounting rate to be applied to a one year loan is
different from that for a two year loan, etc.
The relationship between the length of a loan (or, to talk in
terms of a bond, the time to maturity) and the rate of return you
can earn on a loan (the yield to maturity in the case of a bond) is
called the term-structure of interest rates or the yield curve.
To make this relationship clear, we will think in terms of 'spot
rates.'
A spot rate is the rate of return on a loan or a bond which has
only one cash flow to the investor: the investor makes a loan of
an amount P and gets back the amount F at the maturity of the
loan. Remember that this is exactly the same as investing in a
bond with a zero coupon rate, which has a current price P and
face value F.
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Term Structure of Interest Rates

Thus the one-year spot rate is the yield to maturity on zero
coupon bonds (sometimes referred to as 'pure discount' bonds)
of one year maturity. Let us denote the one year spot rate by r1.
Then, the price of a one-year maturity zero-coupon bond with
face value F is given by:
F
P01 
(1  r1 )

Similarly, the two-year spot rate r2 is the rate of return on a twoyear loan, or equivalently, the yield to maturity on a two year
zero-coupon bond. Then, the price of a two-year zero coupon
bond is,
F
P02 
(1  r ) 2
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Term Structure of Interest Rates






We can write down similar relationships between the prices of
a three year pure discount bond, a four year zero coupon bond
etc., and the corresponding spot rates r3, r4, r5 etc.
Now, if r1 < r2 < r3 ...etc., the term structure is uniformly
upward sloping, which is very often the case.
On the other hand, if r1 > r2 > r3... the term structure is
uniformly downward sloping.
The term structure is 'flat' if r1 = r2 = r3 ...etc. The termstructure can also take other shapes as well.
With all the spot rates, we can price any
riskless bond with cash flows of any
magnitude and at arbitrary points in
time.
This is because we can show that all cash
flows of similar riskiness occurring at 13
the same point in time should be
Term Structure of Interest Rates

Thus, consider a bond with coupons C1, C2, C3, ...Cn
occurring at time periods t=1, 2, 3, ...n. In addition, let the
face value of the bond be F. The price of the bond is then
given by,
Cn  F
C1
C2
P

 .... 
2
(1  r1 ) (1  r2 )
(1  rn ) n
YTM
TYPICAL YIELD CURVES
0
Time to Maturity, n
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Example


Compute the spot rates given the following zero coupon bond
prices : P01 = $900; P02 = $820;
P03 = $725 (Assume $1000 face values).
P01 = 900 =
1000
1000
 r=
-1
(1 + r1 )
900
1000
1000
P02 = 820 =
 r2 =
-1
2
(1 + r2 )
820
1000
1000
3
P03 = 725 =
 r3 =
-1
3
(1 + r3 )
725
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Decomposing a Zero Coupon Bond

Decomposing a zero coupon bond into a portfolio of zero
coupon bonds:
 1. If the yield curve is upward sloping, the yield on a coupon
bond will be lower than on a zero coupon bond (why?)
 2. If the yield curve is downward sloping, the yield on a
coupon bond will be higher than on a zero-coupon bond.
 3. The key is to realize that the yield on coupon bonds are
weighted averages of the corresponding spot rates.
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The Law of One Price



What happens if the bond price is different from the price
computed by the above formula? We can show that in that
case, 'the law of one price' is violated, and there are
opportunities for making arbitrage profits which can be taken
advantage of by investors, and the price of the bond will be
driven back to the above price.
law of one price: securities (or portfolios of securities) which
have the same riskiness, and which entitle the holder to the
same stream of cash flows should have the same current price.
Remember that whenever there are arbitrage opportunities, the
security markets cannot be in equilibrium: thus the
equilibrium price of the above bond should be given by the
above formula.
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Problem

Is there an arbitrage opportunity?
BOND
A
B
C

CASH FLOW AT DATE:
1
2
80
1080
1100
120
1120
PRICE
982
880
1010
Let us try to replicate bond C with bonds A and B:
nA (80) + nB (1100) = 120
(1)
nA (1080) + nB (0) = 1120
(2)
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Problem

From (2)

Substituting in (1),
1120
nA =
= 1.037
1080
120 - 1.037(80)
nB =
= 0.03367
1100


So, by the law of one price,
PC = nA (PA) + nB (PB)
= 1.037 (982) + 0.03367 (880)
= 1047.96  1010
Therefore, an arbitrage opportunity exists. Buy bond C and
sell and equivalent portfolio of bonds A and B.
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Valuing Perpetual Bonds



If a bond pays a coupon C each period for ever (no face
value), its market value (at a discounting rate r) is, P = C/r,
since the coupon stream forms a perpetual annuity.
Also called a consol bond
Example:
C = $50/yr
r = 12% = 0.12
P = 50/0.12 = $416.67
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Valuing Preferred Stock


“Preferred stock” usually promise a fixed payment forever
(however, unlike corporate bonds, the company can miss
payments without going bankrupt).
Thus, they can be treated as a perpetuity: P = C/r. Here C
denotes the "preferred dividend".
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