Yields and Forward Rates on Coupon Bonds
Suppose that we observe an 8% coupon bond making semiannual payments with 1 year until maturity, selling at $986.10, and a 10% coupon bond, also with a year until maturity, selling at $1,004.78. To infer the short rates for the next two 6 month periods, we first attempt to find the present value of each coupon payment taken individually, that is, treated as a mini–zero coupon bond. Call d1 the present value of $1 to be received in half a year and d2 the present value of a dollar to be received in 1 year. (The d stands for discounted values; therefore, d1 = 1/(1 + r1), where r1 is the short rate for the first 6 month period.) Then our two bonds must satisfy the simultaneous equations
986.10 = d1 X 40 + d2 = 1,040
1,004.78 + d1 X 50 + d2 = 1,050
In each equation the bond’s price is set equal to the discounted value of all of its remaining cash flows. Solving these equations we find that d1 = .95694 and d2 = .91137. Thus if r1 is the short rate for the first 6 month period, then d1 = 1/(1 + r1) = .95694, so that r1 = .045, and d2 = 1/[(1 + r1)(1 + f2)] = 1/[(1.045)(1 + f2)] = .91137, so that f2 = .05. Thus the two short rates are shown to be 4.5% for the first half year period and 5% for the second.