Question

Suppose a risk-free bond has a face value of \(250,000\) with a maturity date four years from now. The bond also gives coupon payments of \(8,000\) at the end of each of the next four years. a. What will this bond sell for if the risk-free lending rate in the economy is 4 percent? b. What will this bond sell for if the risk-free lending rate is 5 percent? c. What is the relationship between the bond's price and the level of interest rates in the economy in this exercise?

Short answer

a. The bond will sell for the present value of its future cash flows at 4 percent interest rate: \(P = \frac{8000*(1-(1+0.04)^{-4})}{0.04} + \frac{250000}{(1+0.04)^4}\) b. The bond will sell for the present value of its future cash flows at 5 percent interest rate: \(P = \frac{8000*(1-(1+0.05)^{-4})}{0.05} + \frac{250000}{(1+0.05)^4}\) c. The bond's price decreases as the level of interest rates in the economy increases.

Step-by-step solution

Understand the bond attributes

This bond has a face value of \(250,000\), it matures in 4 years, and it gives out coupon payments of \(8,000\) at the end of each year.

Bond Pricing for 4 percent interest rate

The price of the bond can be calculated using the bond pricing formula which is the present value of the bond's future cash flows. The formula is \[ P = \frac{C*(1-(1+r)^{-n})}{r} + \frac{FV}{(1+r)^n} \] where: \(P\) is the price of the bond, \(C\) is the annual coupon payment, \(r\) is the interest rate, \(n\) is the number of years until maturity, \(FV\) is the face value of the bond. So in this case, \(P = \frac{8000*(1-(1+0.04)^{-4})}{0.04} + \frac{250000}{(1+0.04)^4} \]

Bond Pricing for 5 percent interest rate

Applying the same bond pricing formula with the new interest rate of 5 percent gives: \(P = \frac{8000*(1-(1+0.05)^{-4})}{0.05} + \frac{250000}{(1+0.05)^4}\)

Relationship between the bond's price and the level of interest rates

As can be seen from the above calculations, as the interest rate increases, the price of the bond decreases. This is because as the interest rate increases, the present value of the bond's future cash flows reduces, thereby reducing the price of the bond. This is the relationship between the bond's price and the level of interest rates in the economy in this exercise.

Conclusion

Bonds are priced based on their future cash flows which include coupon payments and face value at maturity. The interest rate in the economy plays a key part in pricing these cash flows. As interest rates increase, the price that individuals are willing to pay for these future cash flows decreases, therefore decreasing the price of the bond. Conversely, when interest rates decrease, the price of bonds will increase.