Chapter 3, Problem 30PS

Prices and yields If a bond’s yield to maturity does not change, the return on the bond each year will be equal to the yield to maturity. Confirm this with a simple example of a four-year bond selling at a premium to face value. Now do the same for a four-year bond selling at a discount. For convenience, assume annual coupon payments.

Summary Introduction

To discuss: Illustrate on return on bond equals yield to maturity (YTM).

Explanation of Solution

4-year bond selling at a premium to face value and the 3% coupon bond is 2%.

$\begin{array}{c}\text{PV}=\left(0.03\times \$1,000\right)\times \left(\begin{array}{l}\left(\frac{1}{\text{0}.02}\right)\\ –\left[\frac{1}{0.02\times {\left(1+0.02\right)}^4}\right]\end{array}\right)+\left[\frac{\$1,000}{{\left(1+\text{0}.02\right)}^4}\right]\\ =\$30\times \left(\left(\$50–\left[\$46.19\right]\right)\right)+\left[\frac{\$1,000}{1.0824}\right]\\ =\$30\times \left(\$3.81\right)+\$923.87\\ =\$1,038.17\end{array}$

If the yield to maturity remain same, 1 year later the bond will sell as follows:

$\begin{array}{c}\text{PV}=\left(0.03\times \$1,000\right)\times \left(\begin{array}{l}\left(\frac{1}{\text{0}.02}\right)\\ –\left[\frac{1}{0.02\times {\left(1+0.02\right)}^3}\right]\end{array}\right)+\left[\frac{\$1,000}{{\left(1+\text{0}.02\right)}^3}\right]\\ =\$30\times \left(\left(\$50–\$47.125\right)\right)+\left[\frac{\$1,000}{1.061208}\right]\\ =\$30\times \left(\$2.875\right)+\$942.322\\ =\$1,028.57\end{array}$

Calculation of interest rate:

$\begin{array}{c}r=\left(\$30+\left[\frac{\$1,028.84-\$1,038.08}{\$1,038.08}\right]\right)\\ =1.999\%or2\%\end{array}$

Thus, the interest rate equals yield to maturity.

Summary Introduction

To discuss: Illustrate on return on bond equals yield to maturity (YTM).

Explanation of Solution

4-year bond selling at discount to face value and the 3% coupon bond is 4%.

$\begin{array}{c}\text{PV}=\left(0.03\times \$1,000\right)\times \left(\begin{array}{l}\left(\frac{1}{\text{0}.04}\right)\\ –\left[\frac{1}{0.04\times {\left(1+0.04\right)}^4}\right]\end{array}\right)+\left[\frac{\$1,000}{{\left(1+\text{0}.04\right)}^4}\right]\\ 0.046799\\ =\$30\times \left(\left(25–\left[\$21.368\right]\right)\right)+\left[\frac{\$1,000}{1.1699}\right]\\ =\$30\times \left(\$3.632\right)+\$854.77\\ =\$963.73\end{array}$

If the yield to maturity remain same, 1 year later the bond will sell as follows:

$\begin{array}{c}\text{PV}=\left(0.03\times \$1,000\right)\times \left(\begin{array}{l}\left(\frac{1}{\text{0}.04}\right)\\ –\left[\frac{1}{0.04\times {\left(1+0.04\right)}^3}\right]\end{array}\right)+\left[\frac{\$1,000}{1.1249}\right]\\ =\$30\times \left(\left(\$25–\$22.22\right)\right)+\left[\frac{\$1,000}{1.1249}\right]\\ =\$30\times \left(\$2.78\right)+\$888.97\\ =\$972.37\end{array}$

Calculation of interest rate:

$\begin{array}{c}r=\left(\$30+\left[\frac{\$972.25-\$963.70}{\$963.70}\right]\right)\\ =0.040or4\%\end{array}$

Thus, the interest rate equals yield to maturity.