Chapter 6, Problem 25QP

Summary Introduction

To determine: The yield to maturity

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.

Answer to Problem 25QP

The yield to maturity of the bond is 5.29 percent.

Explanation of Solution

Given information:

The present time is May 2013. The face value of the bond is $1,000. Assume that the coupon payments are semiannual. The following is the information regarding the Treasury bond that matures in May 2023:

Rate	Maturity	Bid	Asked	Chg	Ask yield
5.850	May 23	104.1762	104.3850	+0.4513	??

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 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:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 5.85\%\\ =\$58.5\end{array}$

Hence, the annual coupon payment is $58.5.

Compute the current price of the bond:

The face value of the bond is $1,000. The last asked price is 104.3850% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{104.3850}{100}\\ =\$1,043.85\end{array}$

Hence, the current price of the bond is $1,043.85.

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

The bond pays the coupons semiannually (Assumption). The annual coupon payment is $58.5. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $29.25 $\left(\$58.5÷2\right)$.

The remaining time to maturity is 10 years $(2023-2013)$. As the coupon payment is semiannual, the semiannual periods to maturity are 20 $\left(10\text{ years}\times 2\right)$. In other words, “t” equals to 20 6-month periods.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,043.85=\$29.25\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{20}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{20}}\end{array}$ 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 higher than the coupon rate.

In the given information, the bond sells at a premium because the market value of the bond is higher than its face value. Hence, substitute “r” with a rate that is lower than the coupon rate until one obtains the bond value close to $1,043.85.

The coupon rate of 5.85 percent is an annual rate. The semiannual coupon rate is 2.925 percent $\left(5.85\text{ percent}÷2\right)$. The trial rate should be below 2.925 percent.

The attempt under the trial and error method using 2.645 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$29.25\times \frac{\left[1-\frac{1}{{\left(1+0.02645\right)}^{20}}\right]}{0.02645}+\frac{\$1,000}{{\left(1+0.02645\right)}^{20}}\\ =\$449.799+\$593.2585\\ =\$1,043.05\end{array}$

The current price of the bond is $1,043.88, when “r” is 2.645 percent. This value is close to 1,043.85. Hence, 2.645 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =2.645\%\times 2\\ =5.29\%\end{array}$

Hence, the yield to maturity is 5.29 percent.