Chapter 7, Problem 30QP

a)

Summary Introduction

To determine: The yield to maturity of the bond

Introduction:

A bond refers to the debt securities issued by governments or corporations to raise capital. The borrower does not return the face value until maturity. However, until the date of maturity, the investor gets the coupons, every year.

Bond price or bond value refers to the present value of the future cash inflows of the bond after discounting at the required rate of return.

Answer to Problem 30QP

The yield to maturity of the bond is 7.40 percent.

Explanation of Solution

Given information:

Person X buys a bond at $1,060. The coupon rate of the bond is 8 percent. The bond will mature in 19 years. The assumed par value of the bond is $1,000.

Formulae:

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:

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

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:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 8\%\\ =\$80\end{array}$

Hence, the annual coupon payment is $80.

Compute the yield to maturity of the bond as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,060=\$80\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{19}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{19}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the yield to maturity. However, it is difficult to simplify the above equation. Hence, the only method that can be used to solve “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 which is 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 larger 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 which must be less than the coupon rate until one obtains the bond value close to $1,060. The coupon rate is 8 percent. The trial rate should be below 8 percent.

The attempt under the trial and error method using 7.40 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$80\times \frac{\left[1-\frac{1}{{\left(1+0.0740\right)}^{19}}\right]}{0.0740}+\frac{\$1,000}{{\left(1+0.0741\right)}^{19}}\\ =\$80\times \frac{\left[1-\frac{1}{3.8822}\right]}{0.0740}+\frac{\$1,000}{3.8822}\\ =\$80\times \frac{\left[1-0.2575\right]}{0.0740}+\frac{\$1,000}{3.8822}\end{array}$

$\begin{array}{c}=\$802.70+\$257.58\\ =\$1,060.28\end{array}$

The current price of the bond is $1,060.28, when “r” is 7.40 percent. This value is more accurate. Hence, 7.40 percent is the yield to maturity.

b)

Summary Introduction

To determine: The selling price of the bond after two years, the holding period yield, and the reason why the holding period yield is different from the yield to maturity given in the solution of Part A.

Introduction:

A bond refers to the debt securities issued by governments or corporations to raise capital. The borrower does not return the face value until maturity. However, until the date of maturity, the investor gets the coupons, every year.

Bond price or bond value refers to the present value of the future cash inflows of the bond after discounting at the required rate of return.

Answer to Problem 30QP

The selling price of the bond is $1,162.95. The holding period yield is 12.12 percent.

Explanation of Solution

The reason why the holding period yield is different from the yield to maturity given in Part A of the solution:

The yield to maturity while buying the bond was 7.40 percent. The holding period yield is 12.12 percent. The holding period yield is higher than the yield to maturity. It is because the price of the bond rises when the yield falls, and the yield to maturity is fallen by 1 percent.

Explanation:

Given information:

Person X will sell the bond in 2 years. The yield to maturity after two years is reduced by 1 percent. Hence, the current yield to maturity is 6.40 percent $(7.40\%-1.00\%)$.

Compute the selling price of the bond after two years as follows:

After two years, the maturity period is 17 years. Hence, “t” is equal to 17.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$80\times \frac{\left[1-\frac{1}{{\left(1+0.064\right)}^{17}}\right]}{0.064}+\frac{\$1,000}{{\left(1+0.064\right)}^{17}}\\ =\$80\times \frac{\left[1-\frac{1}{2.8708}\right]}{0.064}+\frac{\$1,000}{{\left(1+0.064\right)}^{17}}\\ =\$80\times \frac{\left[1-0.3483\right]}{0.064}+\frac{\$1,000}{2.8708}\end{array}$

$\begin{array}{c}=\$814.62+\$348.33\\ =1,162.95\end{array}$

Hence, the selling price of the bond is $1,162.95.

Compute the holding period yield:

The bond value at the beginning was 1,060. The bond value at the end is $1,162.95. As the holding period is 2 years, “t” is equal to 2 years. Solve “r” to find the holding period yield.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$1,060=\$80\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^2}\right]}{r}+\frac{\$1,162.95}{{\left(1+r\right)}^r}\end{array}$

Finding “r” in Equation (1) would give the yield to maturity. However, it is difficult to simplify the above equation. Hence, the only method that can be used 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 higher than the coupon rate until one obtains the bond value close to $1,060. The coupon rate is 8 percent. The trial rate should be above 8 percent.

The attempt under the trial and error method using 12.12 percent as “r”:

The current price of the bond is $1060.14, when “r” is 12.12 percent. This value is more accurate. Hence, 12.12 percent is the holding period yield.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$80\times \frac{\left[1-\frac{1}{{\left(1+0.1212\right)}^2}\right]}{0.1212}+\frac{\$1,162.95}{{\left(1+0.12\right)}^2}\\ =\$80\times \frac{\left[1-\frac{1}{1.2570}\right]}{0.1212}+\frac{\$1,162.95}{1.2570}\\ =\$80\times \frac{\left[1-0.7955\right]}{0.1212}+\frac{\$1,162.95}{1.2570}\end{array}$

$\begin{array}{c}=\$80\times \frac{\left[0.2045\right]}{0.1212}+\$925.17\\ =\$80\times 1.6872+\$925.17\\ =\$134.97+\$925.17\\ =\$1,060.14\end{array}$

Hence, the bond value is $1,060.14 at the rate of 12.12%.