Chapter 7, Problem 17QP

Summary Introduction

To determine: The percentage change in bond price.

Introduction:

A bond refers to the debt securities issued by the governments or corporations for raising 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 17QP

The percentage change in bond price is as follows:

Yield to maturity	Bond J	Bond K
4%	−19.59%	−16.04%
8%	25.78%	20.54%

The interest rate risk is high for a bond with lower coupon rate and low for a bond with high coupon rate.

Explanation of Solution

Given information:

There are two bonds namely Bond J and Bond K. Both the bonds mature after 15 years. The coupon rate of Bond J is 3 percent, and the coupon rate of Bond K is 9 percent. Both the bonds pay the coupon semiannually. The yield to maturity of the bonds is 6 percent.

Formulae:

The formula to calculate the bond value:

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

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.

The formula to calculate the percentage change in price:

$\text{Percentage change in price}=\frac{\text{New price}-\text{Initial price}}{\text{Initial price}}\times 100$

Compute the bond value of Bond J at present:

The coupon rate of Bond J is 3 percent, and its face value is $1,000. Hence, the annual coupon payment is $30 $(\$1,000\times 3\%)$. As the coupon payments are semiannual, the semiannual coupon payment is $15 $(\$30\text{divided}\text{by }2)$.

The yield to maturity is 6 percent. As the calculations are semiannual, the yield to maturity should also be semiannual. Hence, the semiannual yield to maturity is 3 percent $(6\%\text{divided}\text{by }2)$.

The maturity time which is remaining is 15 years. As the coupon payment is semiannual, the semiannual periods to maturity are 30 years $(15\text{years}\times 2)$. In other words, “t” equals to 30 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$15\times \frac{\left[1-\frac{1}{{\left(1+0.03\right)}^{30}}\right]}{0.03}+\frac{\$1,000}{{\left(1+0.03\right)}^{30}}\\ =\$15\times \frac{\left[1-\frac{1}{2.4272}\right]}{0.03}+\frac{\$1,000}{{\left(1+0.03\right)}^{30}}\end{array}$

$\begin{array}{c}=\$15\times \frac{\left[1-0.4119\right]}{0.03}+\frac{\$1,000}{2.4272}\\ =\$294.05+\$411.99\\ =\$706.04\end{array}$

Hence, the bond price of Bond J is $706.04 at present.

Compute the bond value of Bond K at present:

The coupon rate of Bond K is 9 percent, and its face value is $1,000. Hence, the annual coupon payment is $90 $(\$1,000\times 9\%)$. As the coupon payments are semiannual, the semiannual coupon payment is $45 $(\$90\text{divided}\text{by }2)$.

The yield to maturity is 6 percent. As the calculations are semiannual, the yield to maturity should also be semiannual. Hence, the semiannual yield to maturity is 3 percent $(6\%\text{divided}\text{by }2)$.

The maturity time which is remaining is 19 years. As the coupon payment is semiannual, the semiannual periods to maturity are 30 years $(15\text{years}\times 2)$. In other words, “t” equals to 30 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$45\times \frac{\left[1-\frac{1}{{\left(1+0.03\right)}^{30}}\right]}{0.03}+\frac{\$1,000}{{\left(1+0.03\right)}^{30}}\\ =\$45\times \frac{\left[1-\frac{1}{2.4272}\right]}{0.03}+\frac{\$1,000}{{\left(1+0.03\right)}^{30}}\\ =\$45\times \frac{\left[1-0.4119\right]}{0.03}+\frac{\$1,000}{2.4272}\end{array}$

$\begin{array}{c}=\$882.15+\$411.99\\ =\$1,294.14\end{array}$

Hence, the bond price of Bond K is $1,294.14 at present.

Compute the new interest rate (yield to maturity) when the interest rates rise:

Note: The percentage change in the bond value of Bond J and Bond K when 2 percent interest rate increases.

The interest rate refers to the yield to maturity of the bond. The initial yield to maturity of the bonds is 6 percent. If the interest rates rise by 2 percent, then the new interest rate or yield to maturity will be 8 percent $(6\text{ percent}+2\text{ percent})$.

Compute the bond value when the yield to maturity of Bond J rises to 8 percent:

The coupon rate of Bond J is 3 percent, and its face value is $1,000. Hence, the annual coupon payment is $30 $(\$1,000\times 3\%)$. As the coupon payments are semiannual, the semiannual coupon payment is $15 $(\$30\text{divided}\text{by }2)$.

The yield to maturity is 6 percent. As the calculations are semiannual, the yield to maturity should also be semiannual. Hence, the semiannual yield to maturity is 4 percent $(8\%\text{divided}\text{by }2)$.

The maturity time which is remaining is 15 years. As the coupon payment is semiannual, the semiannual periods to maturity are 30 years $(15\text{years}\times 2)$. In other words, “t” equals to 30 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$15\times \frac{\left[1-\frac{1}{{\left(1+0.04\right)}^{30}}\right]}{0.04}+\frac{\$1,000}{{\left(1+0.04\right)}^{30}}\\ =\$15\times \frac{\left[1-\frac{1}{3.2433}\right]}{0.04}+\frac{\$1,000}{{\left(1+0.04\right)}^{30}}\\ =\$15\times \frac{\left[1-0.3083\right]}{0.04}+\frac{\$1,000}{3.2433}\end{array}$

$\begin{array}{c}=\$259.38+\$308.32\\ =\$567.7\end{array}$

Hence, the bond price of Bond J will be $567.7 when the interest rises to 8 percent.

Compute the percentage change in the price of Bond J when the interest rates rise to 8 percent:

The new price after the increase in interest rate is $567.7. The initial price of the bond was $706.04.

$\begin{array}{c}\text{Percentage change in price}=\frac{\text{New price}-\text{Initial price}}{\text{Initial price}}\times 100\\ =\frac{\$567.7-\$706.04}{\$706.04}\times 100\\ =\frac{-138.34}{\$706.04}\times 100\\ =-19.59\%\end{array}$

Hence, the percentage decrease in the price of Bond J is −19.59% when 8 percent of interest rate increases.

Compute the bond value when the yield to maturity of Bond K rises to 8 percent:

The coupon rate of Bond K is 9 percent, and its face value is $1,000. Hence, the annual coupon payment is $90 $(\$1,000\times 9\%)$. As the coupon payments are semiannual, the semiannual coupon payment is $45 $(\$90\text{divided}\text{by }2)$.

The yield to maturity is 8 percent. As the calculations are semiannual, the yield to maturity should also be semiannual. Hence, the semiannual yield to maturity is 4 percent $(8\%\text{divided}\text{by }2)$.

The maturity time which is remaining is 15 years. As the coupon payment is semiannual, the semiannual periods to maturity are 30 years $(15\text{ years}\times 2)$. In other words, “t” equals to 30 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$45\times \frac{\left[1-\frac{1}{{\left(1+0.04\right)}^{30}}\right]}{0.04}+\frac{\$1,000}{{\left(1+0.04\right)}^{30}}\\ =\$45\times \frac{\left[1-\frac{1}{3.2433}\right]}{0.04}+\frac{\$1,000}{{\left(1+0.04\right)}^{30}}\\ =\$45\times \frac{\left[1-0.3083\right]}{0.04}+\frac{\$1,000}{3.2433}\end{array}$

$\begin{array}{c}=\$778.16+\$308.32\\ =\$1,086.48\end{array}$

Hence, the bond price of Bond K will be $1,086.48 when the interest rises to 8 percent.

Compute the percentage change in the price of Bond K when the interest rates rise to 8 percent:

The new price after the increase in interest rate is $1,086.48. The initial price of the bond was $1,294.14.

$\begin{array}{c}\text{Percentage change in price}=\frac{\text{New price}-\text{Initial price}}{\text{Initial price}}\times 100\\ =\frac{\$1,086.48-\$1,294.14}{\$1,294.14}\times 100\\ =\frac{-207.66}{\$1,294.14}\times 100\\ =-16.04\%\end{array}$

Hence, the percentage decrease in the price of Bond K is −16.04% when the interest rates rise to 8 percent.

Compute the new interest rate (yield to maturity) when the interest rates decline:

Note: The percentage change in the bond value of Bond J and Bond K when the interest rates decline by 2 percent

The interest rate refers to the yield to maturity of the bond. The initial yield to maturity of the bonds is 6 percent. If the interest rates decline by 2 percent, then the new interest rate or yield to maturity will be 4 percent $(6\text{ percent}-2\text{ percent})$.

Compute the bond value when the yield to maturity of Bond J declines to 4 percent:

The coupon rate of Bond J is 3 percent, and its face value is $1,000. Hence, the annual coupon payment is $30 $(\$1,000\times 3\%)$. As the coupon payments are semiannual, the semiannual coupon payment is $15 $(\$30\text{divided}\text{by }2)$.

The yield to maturity is 4 percent. As the calculations are semiannual, the yield to maturity should also be semiannual. Hence, the semiannual yield to maturity is 2 percent $(4\%\text{divided}\text{by }2)$.

The maturity time which is remaining is 15 years. As the coupon payment is semiannual, the semiannual periods to maturity are 30 years $(15\text{ years}\times 2)$. In other words, “t” equals to 30 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$15\times \frac{\left[1-\frac{1}{{\left(1+0.02\right)}^{30}}\right]}{0.02}+\frac{\$1,000}{{\left(1+0.02\right)}^{30}}\\ =\$15\times \frac{\left[1-\frac{1}{1.8113}\right]}{0.02}+\frac{\$1,000}{{\left(1+0.02\right)}^{30}}\\ =\$15\times \frac{\left[1-0.5520\right]}{0.02}+\frac{\$1,000}{1.8113}\end{array}$

$\begin{array}{c}=\$336+\$552.08\\ =\$888.08\end{array}$

Hence, the bond price of Bond J will be $888.08 when the interest declines to 4 percent.

Compute the percentage change in the price of Bond J when the interest rates decline to 4 percent:

The new price after the increase in interest rate is $888.08. The initial price of the bond was $706.04.

$\begin{array}{c}\text{Percentage change in price}=\frac{\text{New price}-\text{Initial price}}{\text{Initial price}}\times 100\\ =\frac{\$888.08-\$706.04}{\$706.04}\times 100\\ =\frac{\$182.04}{\$706.04}\times 100\\ =25.78\%\end{array}$

Hence, the percentage increase in the price of Bond J is 25.78% when the interest rates decline to 4 percent.

Compute the bond value when the yield to maturity of Bond K declines to 4 percent:

The coupon rate of Bond K is 9 percent, and its face value is $1,000. Hence, the annual coupon payment is $90 $(\$1,000\times 9\%)$. As the coupon payments are semiannual, the semiannual coupon payment is $45 $(\$90\text{divided}\text{by }2)$.

The yield to maturity is 4 percent. As the calculations are semiannual, the yield to maturity should also be semiannual. Hence, the semiannual yield to maturity is 2 percent $(4\%\text{divided}\text{by }2)$.

The maturity time which is remaining is 15 years. As the coupon payment is semiannual, the semiannual periods to maturity are 30 years $(15\text{years}\times 2)$. In other words, “t” equals to 30 years.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$45\times \frac{\left[1-\frac{1}{{\left(1+0.02\right)}^{30}}\right]}{0.02}+\frac{\$1,000}{{\left(1+0.02\right)}^{30}}\\ =\$45\times \frac{\left[1-\frac{1}{1.8113}\right]}{0.02}+\frac{\$1,000}{{\left(1+0.02\right)}^{30}}\\ =\$45\times \frac{\left[1-0.5520\right]}{0.02}+\frac{\$1,000}{1.8113}\end{array}$

$\begin{array}{c}=\$1,008+\$552.08\\ =\$1,560.08\end{array}$

Hence, the bond price of Bond K will be $1,560.08 when the interest declines to 4 percent.

Compute the percentage change in the price of Bond K when the interest rates decline to 4 percent:

The new price after the increase in interest rate is $1,560.08. The initial price of the bond was $1,294.14.

$\begin{array}{c}\text{Percentage change in price}=\frac{\text{New price}-\text{Initial price}}{\text{Initial price}}\times 100\\ =\frac{\$1,560.08-\$1,294.14}{\$1,294.14}\times 100\\ =\frac{\$265.94}{\$1,294.14}\times 100\\ =20.54\%\end{array}$

Hence, the percentage increase in the price of Bond K is 20.54% when the interest rates decline to 4 percent.

A summary of the bond prices and yield to maturity of Bond J and Bond K:

Table 1

Yield to maturity	Bond J	Bond K
4%	$888.08	$1,560.08
6%	$706.04	$1,294.14
8%	$567.7	$1,086.48

The interest rate risk faced by lower coupon bonds:

The interest risk refers to the fluctuations in the bond value because of the interest rate changes. The bonds with lower coupons face a higher risk than the bonds with lower coupons.

In the above solution, the coupon rate of Bond J is 3 percent. It is lower than Bond K. As the coupon rate of Bond J is lower, the price fluctuations are higher. Bond J −19.59% lost percent when the interest rates increased and gained 25.78% percent when the interest rates declined.

On the contrary, the fluctuations in Bond K’s value were less when compared to Bond J because they had a coupon rate of 9 percent. Bond K lost only −16.04% percent when the interest rates increased and gained only 20.54% percent when the interest rates declined. The fluctuation was lower than Bond J.

Hence, the interest risk is high if the bond has a lower coupon rate.