Suppose a 10-year, $1000 bond with an 8% coupon rate and semiannual coupons is trading for a price of $1034.74.a. What is the bond’s yield to maturity (expressed as an APR with semiannual compounding)?b. If the bond’s yield to maturity changes to 9% APR, what will the bond’s price be?
Question
Suppose a 10-year, 1034.74.a. What is the bond’s yield to maturity (expressed as an APR with semiannual compounding)?b. If the bond’s yield to maturity changes to 9% APR, what will the bond’s price be?
Solution
a. To calculate the bond's yield to maturity (YTM), we need to solve the following equation for YTM:
1034.74 = 40 * (1 - (1 + YTM/2)^-20) / (YTM/2) + 1000 / (1 + YTM/2)^20
This equation is derived from the formula for the present value of a bond, which is the sum of the present value of the coupon payments and the present value of the face value. The coupon payments are an annuity, so we use the formula for the present value of an annuity to calculate their present value. The face value is a single payment in the future, so we use the formula for the present value of a single future payment to calculate its present value.
This equation cannot be solved algebraically, so we need to use a financial calculator or a spreadsheet program to solve it. The solution is YTM = 7.5% APR with semiannual compounding.
b. If the bond's yield to maturity changes to 9% APR, we can calculate the bond's price using the formula for the present value of a bond:
Price = 40 * (1 - (1 + 0.09/2)^-20) / (0.09/2) + 1000 / (1 + 0.09/2)^20
Using a calculator or a spreadsheet program, we find that the price is approximately $981.92.
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