Chapter 6, Problem 22QP

Summary Introduction

To determine: The yield to maturity and the current yield

Introduction:

Yields refer to the return on the investment made by an investor. A bond yield refers to the return earned by the investor on the bond, if he or she holds the bond until the bond matures. It is also known as the yield to maturity.

The current yield of the bond is the annual coupon of the bond divided by the bond’s price.

Answer to Problem 22QP

The yield to maturity of the bond is 11.76 percent. The current yield on the bond is 11.29 percent.

Explanation of Solution

Given information:

The following is the information regarding the bond issued by Company IoU. The piece of information is available in today’s newspaper dated April 15, 2013. Assume that the face value of the bond is $1,000 and the coupon payments are semiannual.

Table 1

Company Ticker	Coupon	Maturity	Last price	Last yield	EST Volume (000's)
IoU (IoU)	9.75	April 15, 2027	86.352	??	1,827

The formula to calculate annual coupon payment:

$\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}$

The formula to calculate the current price:

$\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}$

The formula to calculate the current yield:

$\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}}$

The formula to calculate the yield to maturity:

$\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}$

Where,

“C” refers to the coupon paid per period

“F” refers to the face value paid at maturity

“r” refers to the yield to maturity

“t” refers to the periods to maturity

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 9.75\%\\ =\$97.5\end{array}$

Hence, the annual coupon payment is $97.5.

Compute the current price of the bond:

The face value of the bond is $1,000. The last price is 86.352% of the face value of the bond as per Table 1.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{86.352}{100}\\ =\$863.52\end{array}$

Hence, the current price of the bond is $863.52.

Compute the current yield:

$\begin{array}{c}\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}}\\ =\frac{\$97.5}{\$863.52}\\ =0.1129\text{ or 11}\text{.29}\%\end{array}$

Hence, the current yield is 11.29 percent.

Compute the semiannual yield to maturity of the bond as follows:

The bond pays the coupons semiannually (Assumption). The annual coupon payment is $97.5. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $48.75 $\left(\$97.5÷2\right)$.

The remaining time to maturity is 14 years $(2027-2013)$. As the coupon payment is semiannual, the semiannual periods to maturity are 28 $\left(14\text{ years}\times 2\right)$. In other words, “t” equals to 28 6-month periods.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$863.52=\$48.75\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{28}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{28}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual 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 lower than its face value. Hence, substitute “r” with a rate that is greater than the coupon rate until one obtains the bond value close to $916.45.

The coupon rate of 9.75 percent is an annual rate. The semiannual coupon rate is 4.875 percent $\left(\text{9}\text{.75 percent}÷2\right)$. The trial rate should be above 4.875 percent.

The attempt under the trial and error method using 5.881 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$48.75\times \frac{\left[1-\frac{1}{{\left(1+0.05881\right)}^{28}}\right]}{0.05881}+\frac{\$1,000}{{\left(1+0.05881\right)}^{28}}\\ =\$661.5934+\$201.8808\\ =\$863.47\end{array}$

The current price of the bond is $863.47, when “r” is 5.881 percent. This value is close to 863.52. Hence, 5.881 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =5.881\%\times 2\\ =11.76\%\end{array}$

Hence, the yield to maturity is 11.76 percent.