Chapter 7, Problem 29QP

Summary Introduction

To determine: The interrelationship between the different bond yields.

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 29QP

The price of Bond P at present is $1,179.17, the price of Bond P after one year is $1,161.78, the current yield is 8.48 percent, and the capital gains yield is −1.47%.

The price of Bond D at present is $820.88, the price of Bond D after one year is $838.38, the current yield is 4.77 percent, and the capital gains yield is 2.18 percent.

Explanation of Solution

The interrelationship between the different types of bond yields:

The current yield of premium bond is higher than the discount bond. The capital gains yield on premium bonds is lower than the capital gains yield on discount bonds. However, both the bonds will yield 7 percent return.

Explanation:

Given information:

Bond P sells at a premium. Its coupon rate is 10 percent. Bond D sells at a discount, and its coupon rate is 4 percent. Both the bonds will mature in 8 years, have 7 percent yield to maturity, and make annual coupon payments. Assume the face 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,

The formula to calculate the current yield:

$\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}}$

The formula to calculate the capital gains yield:

$\text{Capital gains yield}=\frac{\text{New price}-\text{Original price}}{\text{Original price}}$

Compute the annual coupon payment of Bond P:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 10\%\\ =\$100\end{array}$

Hence, the annual coupon payment is $100.

Compute the current price of Bond P as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$100\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^8}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^8}\\ =\$100\times \frac{\left[1-\frac{1}{1.7181}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^8}\\ =\$100\times \frac{\left[1-0.5820\right]}{0.07}+\frac{\$1,000}{1.7181}\end{array}$

$\begin{array}{c}=\$597.14+\$582.03\\ =\$1,179.17\end{array}$

Hence, the current price of Bond P is $1,179.17.

Compute the price of Bond P after one year as follows:

After one year, the maturity period is 7 years. Hence, “t” is equal to 7.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$100\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^7}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^7}\\ =\$100\times \frac{\left[1-\frac{1}{1.6057}\right]}{0.07}+\frac{\$1,000}{1.6057}\\ =\$100\times \frac{\left[1-0.6227\right]}{0.07}+\frac{\$1,000}{1.6057}\end{array}$

$\begin{array}{c}=\$539+\$622.78\\ =\$1,161.78\end{array}$

Hence, the price of Bond P after one year is $1,161.78.

Compute the current yield:

$\begin{array}{c}\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}}\\ =\frac{\$100}{\$1,179.17}\\ =0.0848\text{ or }8.48\%\end{array}$

Hence, the current yield is 8.48%.

Compute the capital gains yield:

$\begin{array}{c}\text{Capital gains yield}=\frac{\text{New price}-\text{Original price}}{\text{Original price}}\\ =\frac{\$1,161.78-\$1,179.17}{\$1,179.17}\\ =-0.0147\text{or}-1.47\%\end{array}$

Hence, the capital gains yield is −1.47%.

Compute the annual coupon payment of Bond D:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 4\%\\ =\$40\end{array}$

Hence, the annual coupon payment is $40.

Compute the current price of Bond D as follows:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$40\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^8}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^8}\\ =\$40\times \frac{\left[1-\frac{1}{1.7181}\right]}{0.07}+\frac{\$1,000}{1.7181}\\ =\$40\times \frac{\left[1-0.5820\right]}{0.07}+\frac{\$1,000}{1.7181}\end{array}$

$\begin{array}{c}=\$238.85+\$582.03\\ =\$820.88\end{array}$

Hence, the current price of Bond D is $820.88.

Compute the price of Bond D after one year as follows:

After one year, the maturity period is 7 years. Hence, “t” is equal to 7.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$40\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^7}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^7}\\ =\$40\times \frac{\left[1-\frac{1}{1.6057}\right]}{0.07}+\frac{\$1,000}{1.6057}\\ =\$40\times \frac{\left[1-0.6227\right]}{0.07}+\frac{\$1,000}{1.6057}\end{array}$

$\begin{array}{c}=\$215.6+\$622.78\\ =\$838.38\end{array}$

Hence, the price of Bond D after one year is $838.38.

Compute the current yield:

$\begin{array}{c}\text{Current yield}=\frac{\text{Annual coupon payment}}{\text{Current price of the bond}}\\ =\frac{\$40}{\$838.38}\\ =0.0477\text{ or }4.77\%\end{array}$

Hence, the current yield is 4.77%.

Compute the capital gains yield:

$\begin{array}{c}\text{Capital gains yield}=\frac{\text{New price}-\text{Original price}}{\text{Original price}}\\ =\frac{\$838.83-\$820.88}{\$820.88}\\ =0.0218\text{ or 2}\text{.18}\%\end{array}$

Hence, the capital gains yield is 2.18 %.