A bond with a face value of $8000 and a 2.7% interest rate (compounded semiannually) will mature in 20 years. What is a fair price to pay for the bond today?
A fair price to buy the bond at would be $_____.
(Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (1)
Calculate the semiannual interest rate: 2 2.7% = 0.0135 .
Calculate the number of semiannual periods: 20 × 2 = 40 .
Calculate the present value of the face value: ( 1 + 0.0135 ) 40 8000 ≈ 4904.12 .
Calculate the present value of the coupon payments: 108 × 0.0135 1 − ( 1 + 0.0135 ) − 40 ≈ 3096.24 .
Calculate the fair price: 4904.12 + 3096.24 = 8000.36 .
Explanation
Understanding the Problem We are given a bond with a face value of $8000, an annual interest rate of 2.7% (compounded semiannually), and a maturity of 20 years. We need to find the fair price to pay for this bond today. The fair price is the present value of all future cash flows, which include the face value at maturity and the coupon payments.
Calculating Semiannual Interest Rate and Periods First, we need to determine the semiannual interest rate and the number of semiannual periods. The annual interest rate is 2.7%, so the semiannual interest rate is: 2 2.7% = 1.35% = 0.0135 The bond matures in 20 years, so the number of semiannual periods is: 20 × 2 = 40
Calculating Present Value of Face Value Next, we calculate the present value of the face value, which is the value of the bond at maturity, discounted back to today. The formula for present value is: P V = ( 1 + r ) n F V where P V is the present value, F V is the future value (face value), r is the semiannual interest rate, and n is the number of semiannual periods. In this case: P V = ( 1 + 0.0135 ) 40 8000 = ( 1.0135 ) 40 8000 ≈ 1.63119 8000 ≈ 4904.12
Calculating Semiannual Coupon Payment Now, we calculate the semiannual coupon payment. The coupon payment is the interest payment made each period, which is the face value multiplied by the semiannual interest rate: C o u p o n = 8000 × 0.0135 = 108
Calculating Present Value of Coupon Payments We need to calculate the present value of the coupon payments. Since the coupon payments are made regularly over time, we can use the present value of an annuity formula: P V A = C o u p o n × r 1 − ( 1 + r ) − n where P V A is the present value of the annuity, C o u p o n is the coupon payment, r is the semiannual interest rate, and n is the number of semiannual periods. In this case: P V A = 108 × 0.0135 1 − ( 1 + 0.0135 ) − 40 = 108 × 0.0135 1 − ( 1.0135 ) − 40 ≈ 108 × 0.0135 1 − 0.61297 ≈ 108 × 0.0135 0.38703 ≈ 108 × 28.6689 ≈ 3096.24
Calculating Fair Price of the Bond Finally, we calculate the fair price of the bond by adding the present value of the face value and the present value of the coupon payments: F ai r P r i ce = P V F a ce Va l u e + P V C o u p o n s = 4904.12 + 3096.24 = 8000.36 Therefore, the fair price to pay for the bond today is approximately $8000.36.
Examples
Bonds are a common investment. Calculating the fair price of a bond helps investors determine if a bond is overvalued or undervalued in the market. This calculation ensures that investors make informed decisions about buying or selling bonds, maximizing their returns and managing their risk effectively. Understanding bond valuation is crucial for anyone involved in financial markets, from individual investors to large institutional fund managers.
