Question

Suppose the interest rate is 10 percent. What is the value of a coupon bond that pays \(80 per year for each of the next five years and then makes a principal repayment of \)1000 in the sixth year? Repeat for an interest rate of 15 percent.

Short answer

With an interest rate of 10%, the bond value will be $867.73, and with an interest rate of 15%, the bond value will be $700.50.

Step-by-step solution

Bond value when the rate of interest is 10%

The bond value will be the present value of the payment received for six years.

The present value of the coupons is calculated below:

$$\begin{gathered} FV=\$80 \\ r=\frac{10}{100} \\ =0.1 \\ n=1,2,3,4,5 \\ PV=FV{\left(1+r\right)}^{-1}+FV{\left(1+r\right)}^{-2}+FV{\left(1+r\right)}^{-3}+F{\left(1+r\right)}^{-4}+FV{\left(1+r\right)}^{-5} \\ =80{\left(1+0.1\right)}^{-1}+80{\left(1+0.1\right)}^{-2}+80{\left(1+0.1\right)}^{-3}+80{\left(1+0.1\right)}^{-4}+80{\left(1+0.1\right)}^{-5} \\ =80\left({\left(1+0.1\right)}^{-1}+{\left(1+0.1\right)}^{-2}+{\left(1+0.1\right)}^{-3}+{\left(1+0.1\right)}^{-4}+{\left(1+0.1\right)}^{-5}\right) \\ =80\left(0.9091+0.8264+0.7513+0.6830+0.6209\right) \\ =80\times 3.7907 \\ =\$303.26 \end{gathered}$$

The present value of the final payment is calculated below:

role="math" localid="1644554872741" $$\begin{gathered} FV=\$1,000 \\ r=0.15 \\ n=6 \\ PV=1,000{\left(1+0.15\right)}^{-6} \\ =1,000\times 0.56447 \\ =\$564.47 \end{gathered}$$

The bond value will be $867.73 (=$303.26 + $564.47).

Bond value when the rate of interest is 15%

The present value of the coupons is calculated below:
$$\begin{gathered} FV=\$80 \\ r=\frac{15}{100} \\ =0.15 \\ n=1,2,3,4,5 \\ PV=FV{\left(1+r\right)}^{-1}+FV{\left(1+r\right)}^{-2}+FV{\left(1+r\right)}^{-3}+F{\left(1+r\right)}^{-4}+FV{\left(1+r\right)}^{-5} \\ =80{\left(1+0.15\right)}^{-1}+80{\left(1+0.15\right)}^{-2}+80{\left(1+0.15\right)}^{-3}+80{\left(1+0.15\right)}^{-4}+80{\left(1+0.15\right)}^{-5} \\ =80\left({\left(1+0.15\right)}^{-1}+{\left(1+0.15\right)}^{-2}+{\left(1+0.15\right)}^{-3}+{\left(1+0.15\right)}^{-4}+{\left(1+0.15\right)}^{-5}\right) \\ =80\left(0.8696+0.7561+0.6575+0.5718+0.4972\right) \\ =80\times 3.3522 \\ =\$268.18 \end{gathered}$$

The present value of the final payment is calculated below:

$$\begin{gathered} FV=\$1,000 \\ r=0.15 \\ n=6 \\ PV=1,000{\left(1+0.15\right)}^{-6} \\ =1,000\times 0.43233 \\ =\$432.33 \end{gathered}$$

The bond value will be $700.50 (=$268.18 + $432.33).