Chapter 6, Problem 18QP

Summary Introduction

To determine: The current yield, the yield to maturity, and the effective annual 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. Effective annual yield refers to the actual earnings that the investor earns from the bond investments.

Answer to Problem 18QP

The current yield on the bond is 7.73 percent. The yield to maturity of the bond is 7.78 percent. The effective annual yield is 7.93 percent.

Explanation of Solution

Given information:

Company P issued bonds that have a coupon rate of 7.5 percent. The remaining life of the bonds is 22 years. The present market value of the bond is 97 percent of the face value. The bonds pay coupons semiannually. Assume that the face value of the bonds is $1,000.

Formula:

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 of the bond:

$\text{Current price of the bond}=\text{Face value of the bond}\times \text{Rate at which the bonds sells}$

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

The formula to calculate the effective yield:

$\text{Effective annual yield}={\left(1+r\right)}^t-1$

Where,

“r” refers to the yield to maturity

“t” refers to the periods to maturity

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 7.5\%\\ =\$75\end{array}$

Hence, the annual coupon payment is $75.

Compute the current price of the bond:

$\begin{array}{c}\text{Current price of the bond}=\text{Face value of the bond}\times \text{Rate at which the bonds sells}\\ =\$1,000\times 97\%\\ =\$1,000\times 0.97\\ =\$970\end{array}$

Hence, the current price of the bond is $970.

Compute the current yield:

$\begin{array}{c}\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}\left(\text{Ask price}\right)}\\ =\frac{\$75}{\$970}\\ =0.0773\text{ or 7}\text{.73}\%\end{array}$

Hence, the current yield is 7.73%.

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

The bond pays the coupons semiannually. The annual coupon payment is $75. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $37.50 $\left(\$75÷2\right)$.

The remaining time to maturity is 22 years. As the coupon payment is semiannual, the semiannual periods to maturity are 44 $\left(22\text{ years}\times 2\right)$. In other words, “t” equals to 44 6-month periods.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$970=\$37.50\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{44}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{44}}\end{array}$ 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 $970.

The coupon rate of 7.5 percent is an annual rate. The semiannual coupon rate is 3.75 percent $\left(\text{7}\text{.5 percent}÷2\right)$. The trial rate should be above 3.75 percent.

The attempt under the trial and error method using 3.89 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$37.5\times \frac{\left[1-\frac{1}{{\left(1+0.03894\right)}^{44}}\right]}{0.03894}+\frac{\$1,000}{{\left(1+0.03894\right)}^{44}}\\ =\$783.6892+\$186.2170\\ =\$969.90\end{array}$

The current price of the bond is $969.90, when “r” is 3.89 percent. This value is more close to $970. Hence, 3.89 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =3.89\%\times 2\\ =7.78\%\end{array}$

Hence, the yield to maturity is 7.78 percent.

Compute the effective annual yield:

The semiannual yield to maturity (r) is 3.89 percent, and the semiannual periods (t) in one year are “two”.

$\begin{array}{c}\text{Effective annual yield}={\left(1+r\right)}^t-1\\ ={\left(1+0.0389\right)}^2-1\\ =1.0793-1\\ =0.0793\text{ or 7}\text{.93}\%\end{array}$

Hence, the effective annual yield is 7.93 percent.