The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 605 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 605

to the cash flow. Using $100 as par, the cash flow for the 1.5-year cou-

pon Treasury is $1.75 for the first two 6-month periods and $101.75 in

1.5 years when the bond matures. Letting zt represent one-half the

annualized spot rate for period t, then the absence of arbitrage requires

that the present value of the three cash flows when discounted at the

spot rates equal the market price, $100 in our illustration. That is,

1.75 1.75 101.75

----------------------+ ----------------------+ ---------------------- =^100

( 1 + z 1 )

1

( 1 + z 2 )

2

( 1 + z 3 )

3

Since the 6-month spot rate and 1-year spot rate are 3.0% and

3.3%, respectively, we know that: z 1 = 0.015 and z 2 = 0.0165. Substi-

tuting these spot rates into the above equation and solving for z 3 , we

obtain 1.7527%. Doubling this yield, we obtain the bond-equivalent

yield of 3.5053%, which is the theoretical 1.5-year spot rate. That rate

is the spot rate that the market would apply to a 1.5-year zero-coupon

Treasury security if, in fact, such a security existed.

Given the theoretical 1.5-year spot rate, we can obtain the theoreti-

cal 2-year spot rate. The cash flows for the 2-year coupon Treasury

security follows from Exhibit 20.1. Since the annual coupon rate is

3.9%, the cash flow for the first three periods is $1.95 and the cash flow

for the fourth period is $101.95. Given the spot rate for the first three

periods (z 1 = 0.015, z 2 = 0.0165, and z 3 = 0.017527), the 4-period spot

rate is then found by solving the following equation:

----------------------1.95 + -------------------------1.95 + --------------------------------1.95 + ----------------------101.95 = 100

(1.015)

1

(1.0165)

2

(1.017527)

3

( 1 + z 4 )

4

The value for z 4 is 0.019582 or 1.9582%. Doubling this yield, we obtain

the theoretical 2-year spot rate bond-equivalent yield of 3.9164%.

One can follow this approach sequentially to derive the theoretical

2.5-year spot rate from the calculated values of z 1 , z 2 , z 3 , and z 4 , and the

price and coupon of the 2.5-year bond in Exhibit 20.1. Further, one

could derive theoretical spot rates for the remaining 15 half-yearly rates.

The spot rates thus obtained are shown in the last column of

Exhibit 20.1. They represent the term structure of Treasury spot rates

for maturities up to 10 years.

In practice, yields for interim maturities are not readily available for

government bond markets. Hence, to construct a continuous spot rate

curve requires the use of a methodology described later in this chapter.