Spot Yields and Coupon-Bond Prices

As noted in Chapter 2, a Treasury bond can be seen as a bundle of individual zero-coupon securities, each maturing on one of the bonds cash flow payment dates. In this view, the Treasury s price is the sum of the present values of all the constituent zero-coupon bond yields. Assume that the spot rates for the relevant maturities—ri,r2,r, --rfj—can be observed. If a bond pays a semiannual coupon computed at an annual rate of C from period 1 to period N, its present value can be derived using equation (17.7). 

Equation (17.7) differs from the conventional redemption yield formula in that every cash flow is discounted, not by a single rate, but by the zero-coupon rate corresponding to the maturity period of the cash flow. To apply this equation, the zero-coupon-rate term structure must be known. These rates, however, are not always readily observable. Treasury prices, on the other hand, are and can be used to derive implied spot interest rates. (Although in the market the terms are used interchangeably, from this point on, zero coupon will be used only of observable rates and spot only of derived ones.) To see how the derivation works, consider the 10 hypothetical U.S. Treasuries whose maturities, prices, and yields are shown in FIGURE 17.9. Assume that the yield curve is positive and that the securities settlement date—March 1, 1999—is a coupon date, so none of them has accrued interest. 

The first bond matures in precisely six months and thus has no intermediate cash flow before redemption. It can therefore be treated as a zero-coupon bond, and its yield of 6 percent taken as the six-month spot rate. Using this, the one-year spot rate can be derived from the price of a 

FIGURE l/.a MATURITY DATE III nypnineii YEARS TO MATURITY w.H irHHsnriHS COUPON (%) YIELD TO MATURITY PRICE 

The combined present values of these cash flows is given by equation (17.8). 

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