The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 607 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 607

D =^1

n ----------------------

( 1 + z )

n

n

The reason for describing the term structure in terms of the discount

function is that bond prices can be expressed in an easy way in terms of

it. The price of a bond is simply the sum of the products of the cash flow

expected from the bond at time t and the discount function for time t.

That is, for a bond with a maturity n and a cash flow of C for periods

1,...,n–1 and maturity value of M, the price is

n – 1

∑ DtCD+ n (CM+ )

t – 1

Forward Rates

In addition to spot rates and discount functions to describe the term

structure, there is another important analytical concept that can be used

to describe the term structure: forward rates. Forward rates can be

derived from the Treasury yield curve by using arbitrage arguments, just

as we did for spot rates.

To illustrate the process of obtaining 6-month forward rates, we will

use the yield curve and corresponding spot rate curve from Exhibit 20.1.

For this construction, we will use a very simple arbitrage: If two invest-

ments have the same cash flows and have the same risk, they should have

the same value.

Consider an investor who has a 1-year investment horizon and is

faced with the following two alternatives:

■ Alternative 1. Buy a 1-year Treasury security

■ Alternative 2. Buy a 6-month Treasury security and, when it matures in

six months, buy another 6-month Treasury security

The investor will be indifferent toward the two alternatives if they

produce the same return over the 1-year investment horizon. The investor

knows the spot rate on the 6-month Treasury security and the 1-year

Treasury security. However, he does not know what yield will be available

on a 6-month Treasury security that will be purchased six months from

now. That is, he does not know the 6-month forward rate six months

from now. Given the spot rates for the 6-month Treasury security and the

1-year Treasury security, the forward rate on a 6-month Treasury security

is the rate that equalizes the dollar return between the two alternatives.