The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 599 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 599

For the terminal value to be P(1 + y)n, each of the coupon payments

must be reinvested until maturity at an interest rate equal to the yield to

maturity. If the coupon payment is semiannual, then each semiannual

payment must be reinvested at the yield y.

Clearly, as the equation indicates, the investor will realize the yield to

maturity that is calculated at the time of purchase only if (1) all the cou-

pon payments can be reinvested at the yield to maturity, and (2) the bond

is held to maturity. With respect to the first assumption, the risk that an

investor faces is that future interest rates at which the coupon can be rein-

vested will be less than the yield to maturity at the time the bond is pur-

chased. This risk is referred to as reinvestment risk. And if the bond is not

held to maturity, it may have to be sold for less than its purchase price,

resulting in a return that is less than the yield to maturity. The risk that a

bond will have to be sold at a loss is referred to as interest rate risk.

Our focus in this section has been on coupon-bearing bonds. In the

special case of a bond that produces only one cash flow, the maturity value,

the yield to maturity does measure the rate at which the initial investment

rises. We can see this if we substitute zero for the coupon payments in the

last equation. As explained in Chapter 3, bonds that do not make coupon

payments are called zero-coupon bonds. The advantage of these bonds is

that they do not expose the investor to reinvestment risk. Zero-coupon

bonds play a key role in the valuation process as explained later.

THE TERM STRUCTURE OF THE INTEREST RATES AND THE

YIELD CURVE

The relationship between the yield on bonds of the same credit quality

but different maturities is generically referred to as the term structure of

interest rates. The graphical depiction of the term structure of interest

rates is called the yield curve.

There are different yield measures that can be used to construct the

yield curve. As we will see in this chapter, the alternative yield measures

that can be used are (1) the yield to maturity on a country’s benchmark

government bonds; (2) the spot rate; (3) the forward rates; and (4) and

the swap rate. We will explain the last three yield measures later in this

chapter. Market participants typically construct yield curves from the

market prices and yields in the government bond market of a country or

from swap rates. As we will see, the other two rates—spot rates and for-

ward rates—are derived from market information.

In the United States it is the U.S. Treasury securities market and the

resulting yield curve is referred to as the Treasury yield curve. Two rea-