The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 602 Wednesday, February 4, 2004 1:33 PM

602 The Mathematics of Financial Modeling and Investment Management

ences in the bonds’ coupon rates. Hence, it is necessary to develop more

accurate and reliable estimates of the term structure of interest rates. We

will show how this is done later. Basically, the approach consists of identi-

fying yields that apply to zero-coupon bonds and, therefore, eliminates

the problem of nonuniqueness in the yield-maturity relationship.

Limitations of Using the Yield to Value a Bond

The price of a bond is the present value of its cash flow. However, in our

illustrations and our discussion of the pricing of a bond above, we assume

that one interest rate should be used to discount all the bond’s cash flows.

The appropriate interest rate is the yield on a Treasury security, with the

same maturity as the bond, plus an appropriate risk premium or spread.

To illustrate the problem with using the Treasury yield curve to deter-

mine the appropriate yield at which to discount the cash flow of a bond,

consider the following two hypothetical 5-year Treasury bonds, A and B.

The difference between these two Treasury bonds is the coupon rate,

which is 12% for A and 3% for B. The cash flow for these two bonds per

$100 of par value for the 10 six-month periods to maturity would be:

Period Cash Flow for A Cash Flow for B

1–9 $6.00 $1.50

10 106.00 101.50

Because of the different cash flow patterns, it is not appropriate to

use the same interest rate to discount all cash flows. Instead, each cash

flow should be discounted at a unique interest rate that is appropriate

for the time period in which the cash flow will be received. But what

should be the interest rate for each period?

The correct way to think about bonds A and B in order to avoid arbi-

trage opportunities is not as bonds but as packages of cash flows. More

specifically, they are packages of zero-coupon instruments. Thus, the

interest earned is the difference between the maturity value and the price

paid. For example, bond A can be viewed as 10 zero-coupon instru-

ments: one with a maturity value of $6 maturing six months from now; a

second with a maturity value of $6 maturing one year from now; a third

with a maturity value of $6 maturing 1.5 years from now, and so on. The

final zero-coupon instrument matures 10 six-month periods from now

and has a maturity value of $106. Likewise, bond B can be viewed as 10

zero-coupon instruments: one with a maturity value of $1.50 maturing

six months from now; one with a maturity value of $1.50 maturing one

year from now; one with a maturity value of $1.50 maturing 1.5 years

from now, and so on. The final zero-coupon instrument matures 10 six-