Chapter 8, Problem 18QP

Interest Rate Risk Laurel, Inc., and Hardy Corp. both have 6.5 percent coupon bonds outstanding, with semiannual interest payments, and both are priced at par value. The Laurel, Inc., bond has 3 years to maturity, whereas the Hardy Corp. bond has 20 years to maturity. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of these bonds? If interest rates were to suddenly fall by 2 percent instead, what would the percentage change in the price of these bonds be then? Illustrate your answers by graphing bond prices versus YTM. What does this problem tell you about the interest rate risk of longer-term bonds?

Summary Introduction

To determine:The percentage change in the price of the bonds in the given situations and the graph showing the changes.

Coupon Rate:

The coupon rate refers to the rate at which interest is earned on the face value of a bond every year. This is the rate at which yield is funded from a fixed-income security.

Yield to Maturity:

The yield to maturity is the total yield or return which is derived from a bond until the time of the maturity. For this, it is assumed that the bond will be held until the maturity and would not be called.

Interest Rate Risk:

The interest rate risk refers to the risk which is associated with a bond because of the fluctuations of the interest rate. The value of the bond differs with the change in the interest rate.

Explanation of Solution

Given,

The yield to maturity of the bonds of both companies is 6.5%.

The bonds of L Company mature after 3 years.

The bonds of H Company mature after 20 years.

Calculation of the change in the price of bonds when the interest rate is increased by 2%:

The formula to calculate the change in price is,

$\text{Change}\text{in}\text{price}=\frac{\text{New}\text{price}-\text{Original}\text{price}}{\text{Original}\text{price}}$

For L company,

Substitute $947.47(refer working note) for the new price and $1,000 for the original pricein the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$947.47-\$1,000}{\$1,000}\\ =\frac{\left(\$52.53\right)}{\$1,000}\\ =\left(0.0525\right)\text{or}\left(5.25\%\right)\end{array}$

The change in bonds price is (5.25%).

For H company,

Substitute $809 (refer working note) for the new price and $1,000 for the original price in the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$809-\$1,000}{\$1,000}\\ =\frac{\left(\$191\right)}{\$1,000}\\ =\left(0.191\right)\text{or}\left(19.1\%\right)\end{array}$

The change in bonds price is (19.1%).

Calculation of the change in the price of bonds when the interest rate is decreased by 2%:

The formula to calculate the change in price is,

$\text{Change}\text{in}\text{price}=\frac{\text{New}\text{price}-\text{Original}\text{price}}{\text{Original}\text{price}}$

For L company,

Substitute $1.055.51 (refer working note) for the new price and $1,000 for the original price in the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$1,055.51-\$1,000}{\$1,000}\\ =\frac{\$55.51}{\$1,000}\\ =0.0555\text{or}5.55\%\end{array}$

The change in bonds price is 5.55%.

For H company,

Substitute $1,262.31 (refer working note) for the new price and $1,000 for the original price in the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$1,262.31-\$1,000}{\$1,000}\\ =\frac{\$262.31}{\$1,000}\\ =0.2623\text{or}26.23\%\end{array}$

The change in bonds price is 26.23%.

The graph showing the changes in the price is:

Fig 1

• The graph shows the change in prices of the L Company and H Company.
• The x-axisrepresents the yield to maturity.
• The y-axisrepresents the value of the bonds.
• The line in the graph shows the change in the prices of the bond with the increase and decrease in the yield to maturity.

Working note:

Calculation of thesemi-annual interest on bonds of L Company,

$\begin{array}{c}\text{Semi-annual}\text{interest}=\text{Face}\text{value}\times \text{Interest}\text{rate}\times \frac{1}{2}\\ =\$1,000\times 6.5\%\times \frac{1}{2}\\ =\$65\times \frac{1}{2}\\ =\$32.5\end{array}$

The semi-annual interest is $32.5.

Calculation of the semi-annual interest on bonds of H Company,

$\begin{array}{c}\text{Semi-annual}\text{interest}=\text{Face}\text{value}\times \text{Interest}\text{rate}\times \frac{1}{2}\\ =\$1,000\times 6.5\%\times \frac{1}{2}\\ =\$65\times \frac{1}{2}\\ =\$32.5\end{array}$

The semi-annual interest is $32.5.

Calculation of the price of bonds if yield to maturity is 8.5%:

For L Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{4.25\%,6}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{4.25\%,6}\right)\end{array}\right]\\ =\left(\$32.5\times 5.199\right)+\left(\$1,000\times 0.779\right)\\ =\$168.97+\$779\\ =\$947.97\end{array}$

The value of the bond is $947.97.

For H Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{4.25\%,40}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{4.25\%,40}\right)\end{array}\right]\\ =\left(\$32.5\times 19.077\right)+\left(\$1,000\times 0.189\right)\\ =\$620+\$189\\ =\$809\end{array}$

The value of the bond is $809.

Calculation of the price of bonds if yield to maturity is 4.5%:

For L Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{2.25\%,6}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{2.25\%,6}\right)\end{array}\right]\\ =\left(\$32.5\times 5.554\right)+\left(\$1,000\times 0.875\right)\\ =\$180.51+\$875\\ =\$1,055.51\end{array}$

The value of the bond is $1,055.51.

For H Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{2.25\%,40}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{2.25\%,40}\right)\end{array}\right]\\ =\left(\$32.5\times 26.194\right)+\left(\$1,000\times 0.411\right)\\ =\$851.31+\$411\\ =\$1,262.31\end{array}$

The value of the bond is $1,262.31.

Calculation of ${\text{PVIFA}}_{4.25\%,6}$,

$\begin{array}{c}{\text{PVIFA}}_{4.25\%,6}=\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ =\left[\frac{1-\frac{1}{{\left(1+0.0425\right)}^6}}{0.0425}\right]\\ =5.199\end{array}$

Calculation of ${\text{PVIF}}_{4.25\%,6}$,

$\begin{array}{c}{\text{PVIF}}_{4.25\%,6}=\frac{1}{{\left(1+r\right)}^n}\\ =\frac{1}{{\left(1+0.0425\right)}^6}\\ =0.779\end{array}$

Calculation of ${\text{PVIFA}}_{4.25\%,40}$,

$\begin{array}{c}{\text{PVIFA}}_{4.25\%,40}=\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ =\left[\frac{1-\frac{1}{{\left(1+0.0425\right)}^{40}}}{0.0425}\right]\\ =19.077\end{array}$

Calculation of ${\text{PVIF}}_{4.25\%,40}$,

$\begin{array}{c}{\text{PVIF}}_{4.25\%,40}=\frac{1}{{\left(1+r\right)}^n}\\ =\frac{1}{{\left(1+0.0425\right)}^{40}}\\ =0.189\end{array}$

Calculation of ${\text{PVIFA}}_{2.25\%,6}$,

$\begin{array}{c}{\text{PVIFA}}_{2.25\%,6}=\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ =\left[\frac{1-\frac{1}{{\left(1+0.0225\right)}^6}}{0.0225}\right]\\ =5.554\end{array}$

Calculation of ${\text{PVIF}}_{2.25\%,6}$,

$\begin{array}{c}{\text{PVIF}}_{2.25\%,6}=\frac{1}{{\left(1+r\right)}^n}\\ =\frac{1}{{\left(1+0.0225\right)}^6}\\ =7.875\end{array}$

Calculation of ${\text{PVIFA}}_{2.25\%,40}$,

$\begin{array}{c}{\text{PVIFA}}_{2.25\%,40}=\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ =\left[\frac{1-\frac{1}{{\left(1+0.0225\right)}^{40}}}{0.0225}\right]\\ =26.194\end{array}$

Calculation of ${\text{PVIF}}_{2.25\%,40}$,

$\begin{array}{c}{\text{PVIF}}_{2.25\%,40}=\frac{1}{{\left(1+r\right)}^n}\\ =\frac{1}{{\left(1+0.0225\right)}^{40}}\\ =0.411\end{array}$

Conclusion

Thus, the percentage change in price when the rate is increased by 2% for L Company is 5.25%and for H Company is19.1%. The percentage change in price when the rate is decreased by 2% for L Company is5.55% and for H Company is 26.23%.