Download Chapter 8

Valuing Bonds
Bond Cash Flows, Prices, and Yields

Bond Terminology

Face Value


Coupon Rate


Notional amount used to compute the interest payments
Determines the amount of each coupon payment, expressed as an
annual percentage rate
Coupon Payment
CPN 
Coupon Rate  Face Value
Number of Coupon Payments per Year
Zero-Coupon Bonds

Zero-Coupon Bond (Zero)

Does not make coupon payments

Always sells at a discount (a price lower than face value), so
they are also called pure discount bonds

Treasury Bills are U.S. government zero-coupon bonds with
a maturity of up to one year.
Zero-Coupon Bonds

Suppose that a one-year, risk-free, zero-coupon bond with
a $100,000 face value has an initial price of $96,618.36.
The cash flows would be:

Although the bond pays no “interest,” your compensation is
the difference between the initial price and the face value.
Zero-Coupon Bonds

Yield to Maturity

The discount rate that sets the present value of the promised bond
payments equal to the current market price of the bond.

Price of a Zero-Coupon bond
FV
P 
(1  YTM n ) n
Zero-Coupon Bonds

Yield to Maturity

For the one-year zero coupon bond:
96,618.36 
1  YTM1

100,000
(1  YTM 1 )
100,000

96,618.36
Thus, the YTM is 3.5%.
 1.035
Example

Suppose the following three zero-coupon bonds are
trading at the prices shown. What is the yield to maturity
for each bond?
Maturity
1 Year
2 Years
3 years
Price
980.39
942.60
901.94
Example

The general formula is found by re-arranging the present
value equation:
FV
P
(1  YTM n ) n

1
 FV 
 YTM n  
 1
 P 
n
The specific solutions are:
1 Year – YTM
$1,000/$980.39 -1
2.0%
2 Years – YTM
($1,000/$942.60)1/2 - 1
3.0%
3 Years – YTM
($1,000/$901.94)1/3 - 1
3.5%
Zero-Coupon Bonds

Risk-Free Interest Rates

A default-free zero-coupon bond that matures on date n provides
a risk-free return over that time period equal to the yield to
maturity. Thus, the Law of One Price guarantees that the risk-free
interest rate equals the yield to maturity on such a bond.

Risk-Free Interest Rate (risk free zero coupon yield or spot rate)
with Maturity n
rn  YTM n

The zero coupon yield curve is a plot of the spot rates for
different maturities.

Things will not be this simple when we consider coupon bonds.
Coupon Bonds

Yield to Maturity

The YTM is the single discount rate that equates the present
value of the bond’s remaining cash flows to its current price.

Yield to Maturity of a Coupon Bond

1 
1
FV
P  CPN 

1 
N 
y 
(1  y ) 
(1  y) N
YTM – Example

On 9/1/95, PG&E bonds with a maturity date of
3/01/25 and a coupon rate of 7.25% were selling
for 92.847% of par, or $928.47 per $1,000 of face
value. What is their YTM?

Semiannual coupon payment = 0.0725*1000/2 =
$36.25.
Number of semiannual periods to maturity
= 30*2 – 1 = 59.

YTM - Example
 1

1
1000
$928.47 = $36.25


59
59 
r
/
2
(
r
/
2
)(
1

r
/
2
)
(1
+
r
/
2
)





r/2 can only be found by trial and error. However, calculators
and spread sheets have algorithms to speed up the search.
Searching reveals that r/2 = 3.939% or a stated annual rate
(YTM) of r = 7.878%.
This is an effective annual rate of:
(1.03939)  1  8.03%
2
Example



Suppose that now the yield to maturity for the PG&E
bonds has changed to 7% (on a stated annual basis, APR,
with semi-annual compounding).
What must be the current price of the bond?
Note that the bond itself has not changed (assume for
simplicity that the change occurs instantaneously) only
the pricing.
Example

We find the price using the present value of the bond
payments discounted at the YTM.
 1

1
1000
Price = $36.25


59
59 
.
07
/
2
(.
07
/
2
)(
1

.
07
/
2
)
(1
+
.
07
/
2
)


 1

1
1000
$1031.02  $36.25


59
59 
.
07
/
2
(.
07
/
2
)(
1

.
07
/
2
)

 (1 + .07 / 2 )

Does it make sense that the price changed like this?
Dynamic Behavior of Bond Prices

Discount


Par


A bond is selling at a discount if the price is less than the face
value.
A bond is selling at par if the price is equal to the
face value.
Premium

A bond is selling at a premium if the price is greater than the
face value.
Discounts and Premiums

If a coupon bond trades at a discount, an investor will earn a
return both from receiving the coupons and from receiving a
face value that exceeds the price paid for the bond.


If a coupon bond trades at a premium it will
earn a return from receiving the coupons but
this return will be diminished by receiving a face value less
than the price paid for the bond.


If a bond trades at a discount, its yield to maturity will exceed its
coupon rate.
If a bond trades at a premium, its coupon rate will exceed its yield to
maturity.
Most coupon bonds have a coupon rate so
that the bonds will initially trade at, or very
close to, par.
Time and Bond Prices

Holding all other things constant, a bond’s yield to
maturity will not change over time.


Over time the price of the bond will change given a constant
yield to maturity.
Holding all other things constant, the price of a discount
or a premium bond will move towards par value over
time.
The Effect of Time on Bond Prices
Interest Rate Changes and Bond Prices

The sensitivity of a bond’s price to changes in interest
rates is measured by the bond’s duration.

Bonds with high durations are highly sensitive to interest rate
changes.

Bonds with low durations are less sensitive to interest rate
changes.

A bond’s term or time to maturity is a major determinant of
duration. All else equal, the longer the term the more sensitive
to changes in the interest rates is bond’s price.

However, that is not the whole story:
Problem

Consider a 15-year zero-coupon bond and a 30-year
coupon bond with 10% annual coupons. By what
percentage will the price of each bond change if its yield
to maturity increases from 5% to 6%?
Solution

First consider the price of each bond at the two yields.
Yield to Maturity
5%
6%



15-Year Zero
30-Year Coupon
100
 $48.10
15
1.05
10 
1  100
1


 $176.86

30 
30
.05  1.05  1.05
100
 $41.73
15
1.06
10 
1  100

 $155.06
1 
30 
30
.06  1.06  1.06
Now note that the price of the 15-year zero changes by -13.2% and
the price of the 30-year coupon bond changes by only – 12.3%.
Even though the coupon bond has a longer maturity it is less sensitive
to rate changes because of its large coupons.
There is an inverse relation between interest rates and bond prices
The Yield Curve and Bond Arbitrage

Using the Law of One Price and the yields of default-free
zero-coupon bonds, one can determine the price and
yield of any other default-free bond.

The yield curve provides sufficient information
to evaluate all such bonds.
Replicating a Coupon Bond

Replicating a three-year $1000 bond that
pays 10% annual coupon using three zerocoupon bonds:
Replicating a Coupon Bond

Yields and Prices (per $100 Face) for Zero Coupon Bonds
Maturity
1 year
2 years
3 years
4 years
YTM
3.50%
4.00%
4.50%
4.75%
Price
$96.62
$92.45
$87.63
$83.06
Replicating a Coupon Bond

By the Law of One Price, the three-year coupon bond must
trade for a price of $1153.
Value a Coupon Bond Using Zero-Coupon
Yields

The price of a coupon bond must equal the present value
of its coupon payments and face value.

Price of a Coupon Bond
PV  PV (Bond Cash Flows)
CPN
CPN



2
1  YTM 1
(1  YTM 2 )
CPN  FV

(1  YTM n ) n
100
100
100  1000
P 


 $1153
2
3
1.035
1.04
1.045
Coupon Bond Yields

Knowing the price we can of course find the yield to
maturity of the coupon bond.
100
100
100  1000
P  1153 


2
(1  y)
(1  y)
(1  y)3
100
100
100  1000
P 


 $1153
2
3
1.0444
1.0444
1.0444

Compare this to the zero-coupon yields.
Corporate Bonds

Corporate Bonds


Issued by corporations
Credit Risk

Risk of default

Investors pay less for bonds with credit risk
than they would for an otherwise identical
default-free bond.

The yield of bonds with credit risk will therefore be
higher than that of otherwise identical default-free bonds.
Corporate Bond Yields

No Default

Consider a 1-year, zero coupon Treasury Bill with a YTM of 4%.

What is the price?
1000
1000
P 

 $961.54
1  YTM 1
1.04
Corporate Bond Yields

Certain Default

Suppose that bond issuer will pay 90% of the obligation.

What is the price?
900
900
P 

 $865.38
1  YTM 1
1.04
Corporate Bond Yields

Certain Default

When computing the yield to maturity for a bond with certain
default, the promised rather than the actual cash flows are
used.
FV
1000
YTM 
 1 
 1  15.56%
P
865.38
900
 1.04
865.38

The yield to maturity of a certain default bond is not equal to
the expected return of investing in the bond. The yield to
maturity will always be higher than the expected return of
investing in the bond.
Corporate Bond Yields

Risk of Default

Consider a one-year, $1000, zero-coupon bond issued.
Assume that the bond payoffs are uncertain.

There is a 50% chance that the bond will repay its face value in full
and a 50% chance that the bond will default
and you will receive $900. Thus, you would expect to receive $950.

Because of the uncertainty, the discount rate is 5.1%.
Corporate Bond Yields

Risk of Default

The price of the bond will be
P 

950
 $903.90
1.051
The yield to maturity will be
FV
1000
YTM 
 1 
 1  .1063
P
903.90
Corporate Bond Yields

Risk of Default

A bond’s expected return will be less than the yield to maturity
if there is a risk of default.

A higher yield to maturity does not necessarily imply that a
bond’s expected return is higher.