EXAMPLE 3 Pricing a Bond Using a Binomial Tree Using the interest rate tree below, find the correct price for a three-year, annual-pay bond with a coupon rate of 5%. EXHIBIT 8 Three-Year Binomial Interest Rate Tree Time 0 Time 1 Time 2 8.0% 5.0% 2.0% 6.0% 3.0% 4.0% Solution: Calculating the bond's value includes being careful with the timing of cash flows. A three-year bond pays coupons and returns principal at the end of each year. When we state an annual interest rate, that rate is effective as of the beginning of that year. EXHIBIT 9 Three-Year Binomial Tree Time 0 Time 1 Time 2 Time 3 C =5 R V = 100 C =5 R C =5 R C = 5 R R V = 100 R C=5 V = R C =5 R R C =5 V = 100 C = 5 R R R C = Cash Flow (% of par) C =5 R = One-Year Interest Rate (%) V = Value of Bond's Future Cash Flows (% of par) V = 100 No matter what level interest rates move to at Time 3, the cash flow from a three- year bond at Time 3 will be the same: par plus a final coupon payment. In addition, a coupon payment will be made at Time 2. Consequently, Time 2 values will be: Time 0 Time 1 Time 2 0.5 x [(105/1.08 + 105/1.08)] + 5 = 102.2222 0.5 x [(105/1.06 + 105/1.06)] +5 = 104.0566 0.5 x [(105/1.04 + 105/1.04)] +5 = 105.9615 Time 1 values will be the average of Time 2 discounted plus the coupon payment: Time 0 Time 1 0.5 x [(102.2222/1.05 + 104.0566/1.05)] + 5 = 103.2280 0.5 x [(104.0566/1.03 + 105.9615/1.03)] + 5 = 106.9506 Finally, we bring the price back to Time 0. Because no time has elapsed, there is no coupon payment at Time 0, making the Time 0 value the average of the Time 1 values discounted to today: Time 0 0.5 x [(103.2280/1.02 + 106.9506/1.02)] = 103.0287