The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 608 Wednesday, February 4, 2004 1:33 PM

608 The Mathematics of Financial Modeling and Investment Management

Letting $X denote the face amount of the 6-month Treasury security, z 1 is

one-half the bond-equivalent yield (BEY) of the theoretical 6-month spot

rate, and z 2 represents one-half the BEY of the theoretical 1-year spot

rate, then the investor will be indifferent toward the two alternatives if

X(1 + z 1 )(1 + f) = X(1 + z 2 )^2

where f is the 6-month forward rate six months from now. Solving, we get

( 1 + z 2 )^2

f = ----------------------– 1

( 1 + z 1 )

Doubling f gives the BEY for the 6-month forward rate six months

from now. In our illustration, f is 1.8% and therefore the 6-month for-

ward rate on a BEY basis is 3.6%.

We can generalize the 1-period forward rates as follows.^2 Let fn

denote the 1-period forward rate contract that will begin at time n.

Then f 0 is simply the current 1-period spot rate.

Exhibit 20.2 shows all of the 6-month (i.e., 1-period) forward rates

for the Treasury yield curve and corresponding spot rate curve shown in

Exhibit 20.1. The forward rates reported in Exhibit 20.2 are the annual-

ized rates on a bond-equivalent basis. The set of these forward rates is

called the short-term forward-rate curve.

The relationship between the n-period spot rate, the current 6-

month spot rate, and the 6-month forward rates is as follows:

zn = [(1 + z 1 ) (1 + f 1 ) (1 + f 2 ) ... (1 + fn–1)]1/n – 1

The discount function can be expressed in terms of forward rates as

follows:

1

Dn = ---------------------------------------------------------------------------------------------------------------------

[( 1 + z 1 )( 1 + f 1 )( 1 + f 2 )...( 1 + 1 fn – 1 )]

1 ⁄ n

• 1

Swap Curve

Instead of using a government spot rate curve, market participants are

more often using the swap curve or London Interbank Offered Rate

(LIBOR) curve for reasons described below. A swap curve is derived

(^2) We will generalize the notation later in this chapter when continuous time is used.