Chapter 8, Problem 19QP

Interest Rate Risk The Faulk Corp. has a 6 percent coupon bond outstanding. The Gonas Company has a 14 percent bond outstanding. Both bonds have 12 years to maturity, make semiannual payments, and have a YTM of 10 percent. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of these bonds? What if interest rates suddenly fall by 2 percent instead? What does this problem tell you about the interest rate risk of lower coupon bonds?

Summary Introduction

To determine: The percentage change in the price of the bonds.

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

Solution:

Given,

The yield to maturity of the bonds of both companies is 10%.

The bonds of both the company mature after 12 years.

The F Company has a 6% coupon rate outstanding.

The G Company has a 14% coupon rate outstanding.

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 F company,

Substitute $623.53 (refer working note) for the new price and $724 for the original price in the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$623.53-\$724}{\$724}\\ =\frac{\left(\$100.47\right)}{\$724}\\ =\left(0.1387\right)\text{or}\left(13.87\%\right)\end{array}$

The change in bonds price is (13.87%).

For G company,

Substitute $1,125.57 (refer working note) for the new price and $1,276 for the original price in the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$1,125.57-\$1,276}{\$1,276}\\ =\frac{\left(\$150.43\right)}{\$1,276}\\ =\left(0.1179\right)\text{or}\left(11.79\%\right)\end{array}$

The change in bonds price is (11.79%).

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 F company,

Substitute $847.41 (refer working note) for the new price and $724 for the original price in the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$847.41-\$724}{\$724}\\ =\frac{\$123.41}{\$724}\\ =0.1704\text{or}17.04\%\end{array}$

The change in bonds price is 17.04%.

For G company,

Substitute $1,457.29 (refer working note) for the new price and $1,276 for the original price in the above formula.

$\begin{array}{c}\text{Change}\text{in}\text{price}=\frac{\$1,457.29-\$1,276}{\$1,276}\\ =\frac{\$181.29}{\$1,276}\\ =0.1420\text{or}14.21\%\end{array}$

The change in bonds price is 14.21%.

Working note:

Calculation of the semi-annual interest on bonds of F Company:

$\begin{array}{c}\text{Semi-annual}\text{interest}=\text{Face}\text{value}\times \text{Interest}\text{rate}\times \frac{1}{2}\\ =\$1,000\times 6\%\times \frac{1}{2}\\ =\$60\times \frac{1}{2}\\ =\$30\end{array}$

The semi-annual interest is $30.

Calculation of the semi-annual interest on bonds of G Company:

$\begin{array}{c}\text{Semi-annual}\text{interest}=\text{Face}\text{value}\times \text{Interest}\text{rate}\times \frac{1}{2}\\ =\$1,000\times 14\%\times \frac{1}{2}\\ =\$140\times \frac{1}{2}\\ =\$70\end{array}$

The semi-annual interest is $70.

Calculation of the price of bonds if yield to maturity is 10%:

For F Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{5\%,24}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{5\%,24}\right)\end{array}\right]\\ =\left(\$30\times 13.8\right)+\left(\$1,000\times 0.310\right)\\ =\$414+\$310\\ =\$724\end{array}$

The value of the bond is $724.

For G Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{5\%,24}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{5\%,24}\right)\end{array}\right]\\ =\left(\$70\times 13.8\right)+\left(\$1,000\times 0.310\right)\\ =\$966+\$310\\ =\$1,276\end{array}$

The value of the bond is $1,276.

Calculation of the price of bonds if yield to maturity is 12%:

For F Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{6\%,24}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{6\%,24}\right)\end{array}\right]\\ =\left(\$30\times 12.551\right)+\left(\$1,000\times 0.247\right)\\ =\$376.53+\$247\\ =\$623.53\end{array}$

The value of the bond is $623.53.

For G Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{6\%,24}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{6\%,24}\right)\end{array}\right]\\ =\left(\$70\times 12.551\right)+\left(\$1,000\times 0.247\right)\\ =\$878.57+\$247\\ =\$1,125.57\end{array}$

The value of the bond is $1,125.57.

Calculation of the price of bonds if yield to maturity is 8%:

For F Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{4\%,24}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{4\%,24}\right)\end{array}\right]\\ =\left(\$30\times 15.247\right)+\left(\$1,000\times 0.390\right)\\ =\$457.41+\$390\\ =\$847.41\end{array}$

The value of the bond is $847.41.

For G Company,

$\begin{array}{c}\text{Value}\text{of}\text{bond}=\left[\begin{array}{c}\left(\text{Interest}\text{value}\times {\text{PVIFA}}_{4\%,24}\right)\\ +\left(\text{Principal}\text{value}\times {\text{PVIF}}_{4\%,24}\right)\end{array}\right]\\ =\left(\$70\times 15.247\right)+\left(\$1,000\times 0.390\right)\\ =\$1,067.29+\$390\\ =\$1,457.29\end{array}$

The value of the bond is $1,457.29.

Calculation of ${\text{PVIFA}}_{5\%,24}$:

$\begin{array}{c}{\text{PVIFA}}_{5\%,24}=\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ =\left[\frac{1-\frac{1}{{\left(1+0.05\right)}^{24}}}{0.05}\right]\\ =13.8\end{array}$

Calculation of ${\text{PVIF}}_{5\%,24}$:

$\begin{array}{c}{\text{PVIF}}_{5\%,24}=\left[\frac{1}{{\left(1+r\right)}^n}\right]\\ =\left[\frac{1}{{\left(1+0.05\right)}^{24}}\right]\\ =0.310\end{array}$

Calculation of ${\text{PVIFA}}_{6\%,24}$:

$\begin{array}{c}{\text{PVIFA}}_{6\%,24}=\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ =\left[\frac{1-\frac{1}{{\left(1+0.06\right)}^{24}}}{0.06}\right]\\ =12.551\end{array}$

Calculation of ${\text{PVIF}}_{6\%,24}$:

$\begin{array}{c}{\text{PVIF}}_{6\%,24}=\left[\frac{1}{\left(1+r\right)}\right]\\ =\left[\frac{1}{{\left(1+0.06\right)}^{24}}\right]\\ =0.247\end{array}$

Calculation of ${\text{PVIFA}}_{4\%,24}$:

$\begin{array}{c}{\text{PVIFA}}_{4\%,24}=\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ =\left[\frac{1-\frac{1}{{\left(1+0.04\right)}^{24}}}{0.04}\right]\\ =15.247\end{array}$

Calculation of ${\text{PVIF}}_{4\%,24}$:

$\begin{array}{c}{\text{PVIF}}_{4\%,24}=\left[\frac{1}{\left(1+r\right)}\right]\\ =\left[\frac{1}{{\left(1+0.04\right)}^{24}}\right]\\ =0.3901\end{array}$

Conclusion

Thus, the percentage change in price when the rate isincreased by 2% for F Company is (13.87%) and for G Company is (11.79%). The percentage change in price when the rate is decreased by 2% for F Company is 17.04% and for G Company is 14.21%.