Final answer:
To determine how much to pay for a 20-year bond with a coupon rate of 8% and semiannual payments, if you desire a 10.71% effective annual interest rate, convert that to a semiannual rate and calculate the present value of all future cash flows.
Step-by-step explanation:
The value of a bond can be calculated using the present value of its future cash flows, which include semiannual coupon payments and the par value at maturity. For a 20-year bond with a $1,000 par value and an 8% coupon rate paid semiannually, each semiannual payment would be $40 (0.08 × $1,000 ÷ 2). If you require an effective annual interest rate of 10.71%, it's necessary to convert this to a semiannual rate because the bond makes semiannual payments. The semiannual rate is 5.2145%, calculated by taking the square root of (1 + the effective annual rate) - 1. To find the present value of the bond, sum the present value of all the semiannual coupon payments plus the present value of the par value. To calculate each present value, you use the formula PV = FV / (1 + r)n, where FV is the future value of the payment, r is the semiannual discount rate, and n is the number of periods until the payment.
The calculation would yield the present value of the bond, which is the price you should be willing to pay for the bond.