Chapter 5
How to Value Bonds and Stocks
5A1
The Term Structure of Interest Rates, Spot Rates,
and Yield to Maturity
In the main body of this chapter, we have assumed that the interest rate is constant over all
future periods. In reality, interest rates vary through time. This occurs primarily because
inF
ation rates are expected to differ through time.
To illustrate, we consider two zero coupon bonds. Bond
A
is a oneyear bond and bond
B
is a twoyear bond. Both have face values of $1,000. The oneyear interest rate,
r
1
, is 8 per
cent. The twoyear interest rate,
r
2
, is 10 percent. These two rates of interest are
examples
of
spot rates.
Perhaps this inequality in interest rates occurs because inF
ation is expected
to be higher over the second year than over the ±
rst year. The two bonds are
depicted in the
following time chart:
2
8%
Bond
A
$1,000
10%
Bond
B
$1,000
1
0
Year 1
Year 2
We can easily calculate the present value for bond
A
and bond
B
as follows:
P
V
A
5
$925.93
5
$1,000
______
1.08
PV
B
5
$826.45
5
$1,000
______
(1.10)
2
Of course, if PV
A
and PV
B
were observable and the spot rates were not, we could determine
the spot rates using the PV formula, because:
PV
A
5
$925.93
5
$1,000
_______
(1
1
r
1
)
→
r
1
5
8%
and:
PV
B
5
$826.45
5
$1,000
________
(1
1
r
2
)
2
→
r
2
5
10%
Now we can see how the prices of more complicated bonds are determined. Try to do the
next example. It illustrates the difference between spot rates and yields to maturity.
Appendix 5A
www.mhhe.com/rwj
EXAMPLE 5A.1
On the Spot
Given the spot rates
r
1
equals 8 percent and
r
2
equals 10 percent, what should a
5 percent coupon, twoyear bond cost? The cash f ows
C
1
and
C
2
are illustrated in the Following time
chart:
2
8%
$50
10%
$1,050
1
0
Year 1
Year 2
The bond can be viewed as a portFolio oF zero coupon bonds with one and twoyear maturities.
ThereFore:
PV
5
$50
________
1
1
0.08
1
$1,050
__________
(1
1
0.10)
2
5
$914.06
(A.1)
(
continued
)
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Part II
Valuation and Capital Budgeting
Graphing the Term Structure
The
term structure
describes the relationship of spot
rates with different maturities. Figure 5A.1 graphs a particular term structure. In
Figure 5A.1
the spot rates are increasing with longer maturities—that is,
r
3
.
r
2
.
r
1
.
Graphing the
term structure is easy if we can observe spot rates. Unfortunately this can be done only if
there are enough zero coupon government bonds.
A given term structure, such as that in Figure 5A.1, exists for only a moment in time—
say 10:00 a.m., July 30, 2006. Interest rates are likely to change in the next minute, so that
a different (though quite similar) term structure would exist at 10:01 a.m.
We now want to calculate a single rate for the bond.
We do this by solving for
y
in the following
equation:
$914.06
5
$50
_____
1
1
y
1
$1,050
_______
(1
1
y)
2
(A.2)
In Equation A.2,
y
equals 9.95 percent. As mentioned in the chapter, we call
y
the
yield to maturity
on the bond. Solving for
y
for a multiyear bond is generally done by means of trial and error.
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 Spring '09
 MARYHARDY
 Interest Rates, Spot rate, oneyear spot rate

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