## Explanations of the Term Structure

Figure 5A.1 showed one of many possible relationships between the spot rate and maturity. We now want to explore the relationship in more detail. We begin by defining a new term, the forward rate. Next, we relate this forward rate to future interest rates. Finally, we consider alternative theories of the term structure.

Definition of Forward Rate Earlier in this appendix, we developed a two-year example where the spot rate over the first year is 8 percent and the spot rate over the two years is 10 percent. Here, an individual investing \$1 in a two-year zero-coupon bond would have \$1 X (1.10)2 in two years.

In order to pursue our discussion, it is worthwhile to rewrite16

when rounding is performed after four digits.

12.04 percent is equal to

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Chapter 5 How to Value Bonds and Stocks 133

■ Figure 5A.2 Breakdown of a Two-Year Spot Rate into a One-Year Spot Rate and Forward Rate over the Second Year

Date Date Date

With a two-year spot rate of 10 percent, investor in two-year bond receives \$1.21 at date 2.

This is the same return as if investor received the spot rate of 8 percent over the first year and 12.04-percent return over the second year.

Because both the one-year spot rate and the two-year spot rate are known at date 0, the forward rate over the second year can be calculated at date 0.

Equation (A.3) tells us something important about the relationship between one- and two-year rates. When an individual invests in a two-year zero-coupon bond yielding 10 percent, his wealth at the end of two years is the same as if he received an 8-percent return over the first year and a 12.04-percent return over the second year. This hypothetical rate over the second year, 12.04 percent, is called theforward rate. Thus, we can think of an investor with a two-year zero-coupon bond as getting the one-year spot rate of 8 percent and locking in 12.04 percent over the second year. This relationship is presented in Figure 5A.2.

More generally, if we are given spot rates, r1 and r2, we can always determine the forward rate, f2, such that:

We solve for f2, yielding:

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