Question

Suppose a risk-free bond has a face value of \(100,000\) with a maturity date three years from now. The bond also gives coupon payments of \(5,000\) at the end of each of the next three years. What will this bond sell for if the annual interest rate for risk-free lending in the economy is a. 5 percent? b. 10 percent?

Short answer

In the first case, with an interest rate of 5%, the bond will sell for approximately $114,833.21. In the second case, with an interest rate of 10%, the bond will sell for approximately $102,531.27.

Step-by-step solution

Understanding the Exercise

First, there are three coupon payments of \(5,000\) each and a face value of \(100,000\), all of which will be received in the future. To find how much the bond will sell for today, we need to calculate the present value of these amounts. And to do this, we need to apply the interest rates given in the questions: a. 5%, and b. 10%.

- Determine the Present Value with 5% interest rate

On the first case, we will apply the formula of present value \((PV)\) which is: \[PV=\frac{C} {(1+r)^n}+\frac{C} {(1+r)^{n+1}}+...\] for the coupon payments and \[\frac{FV} {(1+r)^n}\] for the face value. The annual coupon \(C\) is $5000, the face value \(FV\) is $100,000 and the annual interest rate \(r\) is 5 percent, so \(r=0.05\). Substituting: \[PV=\frac{5000}{(1+0.05)^1} + \frac{5000} {(1+0.05)^2} +\frac{5000} {(1+0.05)^3} + \frac{100000} {(1+0.05)^3}\] After solving, the present value is approximately $114833.21.

- Determine the Present Value with 10% interest rate

Next, for the 10% interest rate, the same calculations are made, just the rate \(r\) is exchanged to 0.10. Therefore: \[PV=\frac{5000}{(1+0.10)^1} + \frac{5000} {(1+0.10)^2} +\frac{5000} {(1+0.10)^3} + \frac{100000} {(1+0.10)^3}\] The present value now is approximately $102531.27. Thus, the bond will sell for a lower price if the interest rate is higher, if all other factors remain equal. This is due to the concept of discounting, where future payments are worth less in today's terms when the interest rate is higher.