Chapter 5, Problem 6QP

Summary Introduction

To determine: The yield to maturity

Explanation of Solution

Compute the current price of the bond:

$\begin{array}{c}\text{Current price of the bond}=\text{Face value of the bond}\times \text{Rate at which the bonds sells}\\ =\yen 100,000\times 103.25\%\\ =\yen 103,250\end{array}$

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\yen 100,000\times 4.9\%\\ =\yen 4,900\end{array}$

Compute the yield to maturity of the bond as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \yen 103,250=\yen 4,900\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{18}}\right]}{r}+\frac{\yen 100,000}{{\left(1+r\right)}^{18}}\end{array}$…Equation (1)

Finding “r” in Equation (1) would give the yield to maturity. However, it is difficult to simplify the above the equation. Hence, the only method to solve for “r” is the trial and error method.

The first step in trial and error method is to identify the discount rate that needs to be used. The bond sells at a premium in the market if the market rates (Yield to maturity) are lower than the coupon rate. Similarly, the bond sells at a discount if the market rate (Yield to maturity) is greater than the coupon rate.

In the given information, the bond sells at a discount because the market value of the bond is less than its face value. Hence, substitute “r” with a rate that is more than the coupon rate until one obtains the bond value close to ¥1032,250.

The first attempt under the trial and error method using 4.5 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\yen 4,900\times \frac{\left[1-\frac{1}{{\left(1+0.045\right)}^{18}}\right]}{0.045}+\frac{\yen 100,000}{{\left(1+0.045\right)}^{18}}\\ =\yen 59,583.9598+\yen 45,280.0369\\ =\yen 104,863.99\end{array}$

The second attempt under the trial and error method using 4.6 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\yen 4,900\times \frac{\left[1-\frac{1}{{\left(1+0.046\right)}^{18}}\right]}{0.045}+\frac{\yen 100,000}{{\left(1+0.046\right)}^{18}}\\ =\yen 60,425.5597+\yen 44,507.1390\\ =\$104,932.6987\end{array}$

The third attempt under the trial and error method using 4.63 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\yen 4,900\times \frac{\left[1-\frac{1}{{\left(1+0.0463\right)}^{18}}\right]}{0.0463}+\frac{\yen 100,000}{{\left(1+0.0463\right)}^{18}}\\ =\yen 58,971.4527+\yen 44,277.9947\\ =\yen 103,249.45\end{array}$

Hence, the yield to maturity is 4.63%.