Chapter 5, Problem 26QP

Summary Introduction

To determine: The interrelationship between different bond yields

Explanation of Solution

Compute the current price of Bond P as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$84\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^8}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^8}\\ =\$1,083.60\end{array}$

Compute the price of Bond P after one year as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$84\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^7}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^7}\\ =\$1,075.45\end{array}$

Compute the current yield:

$\begin{array}{c}\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}\left(\text{Ask price}\right)}\\ =\frac{\$84}{\$1,083.60}\\ =0.0775\text{ or 7}\text{.75}\%\end{array}$

Hence, the current yield is 7.75%.

Compute the capital gains yield:

$\begin{array}{c}\text{Capital gains yield}=\frac{\text{New price}-\text{Original price}}{\text{Original price}}\\ =\frac{\$1,075.45-\$1,083.60}{\$1,083.60}\\ =\left(0.0075\right)\text{ or }\left(0.75\%\right)\end{array}$

Compute the current price of Bond D as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$56\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^8}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^8}\\ =\$916.40\end{array}$

Compute the price of Bond D after one year as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$56\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^7}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^7}\\ =\$924.55\end{array}$

Compute the current yield:

$\begin{array}{c}\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}\left(\text{Ask price}\right)}\\ =\frac{\$56}{\$916.40}\\ =0.0611\text{ or 6}\text{.11}\%\end{array}$

Compute the capital gains yield:

$\begin{array}{c}\text{Capital gains yield}=\frac{\text{New price}-\text{Original price}}{\text{Original price}}\\ =\frac{\$924.55-\$916.40}{\$916.40}\\ =0.0089\text{ or 0}\text{.89}\%\end{array}$

Hence, the capital gains yield is 0.89 percent.