Bonds and Bond Derivatives, Second Edition

7

TIME VALUE WITH NONFLAT TERM STRUCTURE

The term structure of interest rates is the relationship between maturities (or term) and interest rates. Typically, longer-maturity debt has higher interest rates (a so-called rising yield curve), although the relationship between maturity and interest rates varies widely. In some cases, short-maturity interest rates are higher than long-maturity interest rates (a declining yield curve). Occasionally, interest rates rise over short maturities, reach a peak, and then decline for longer maturities (a humped yield curve). Declining yield curves and humped yield curves occur when interest rates are high by historical standards. This chapter discusses present value computations with nonflat term structures.

Spot Interest Rates

An interest rate that begins today and continues until some future date is frequently called a spot interest rate. We will denote spot interest rates by capital R.

Each spot interest rate has two subscripts. The first subscript denotes the point in time when the interest rate is observed. The second denotes how long the interest rate lasts. Recall that time 0 is now. Future points in time are denoted by time 1, time 2, etc. Thus, R_0,1 is observed now and runs for one period. R_0,2 is observed now and lasts two periods. R_0,n is observed now and lasts n periods. Figure 7.1 illustrates this.

In this chapter, we will stick to interest rates that are observed now, with a first subscript of 0. In later chapters, other cases are ...