Chapter 6, Problem 18QP

Bond Price Movements. Bond X is a premium bond making semiannual payments. The bond has a coupon rate of 8.5 percent, a YTM of 7 percent, and has 13 years to maturity. Bond Y is a discount bond making semiannual payments. This bond has a coupon rate of 7 percent, a YTM of 8.5 percent, and also has 13 years to maturity. What are the prices of these bonds today assuming both bonds have a $1,000 par value? If interest rates remain unchanged, what do you expect the prices of these bonds to be in one year? In three years? In eight years? In 12 years? In 13 years? What’s going on here? Illustrate your answers by graphing bond prices versus time to maturity.

Summary Introduction

To determine: The bond’s price at different periods.

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, the investor receives the coupons every year until the date of maturity.

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 18QP

The price of the bond at different periods is as follows:

Time to maturity (Years)	Bond X	Bond Y
13	$1,126.6776	$883.3285
12	$1,120.4378	$888.5195
10	$1,106.5930	$900.2923
5	$1,062.3745	$939.9184
1	$1,014.2477	$985.9048
0	$1,000.0000	$1,000.0000

Explanation of Solution

Given information:

Bond X is selling at a premium. The coupon rate of Bond X is 8.5 percent and its yield to maturity is 7 percent. The bond will mature in 13 years. Bond Y is selling at a discount. The coupon rate of Bond Y is 7 percent and its yield to maturity is 8.5 percent. The bond will mature in 13 years. Both the bonds make semiannual coupon payments. Assume that the face value of bonds is $1,000.

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

Compute the bond price of Bond X at different maturities:

Compute the annual coupon payment of Bond X:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 8.5\%\\ =\$85\end{array}$

Hence, the annual coupon payment of Bond X is $85.

The bond value or the price of Bond X at present:

The bond pays the coupons semiannually. The annual coupon payment is $85. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $42.50$\left(\$85÷2\right)$.

Secondly, the remaining time to maturity is 13 years. As the coupon payment is semiannual, the semiannual periods to maturity are 26$\left(13\text{ years}\times 2\right)$. In other words, “t” equals to 26 6-month periods.

Thirdly, the yield to maturity is 7 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 3.50 percent $\left(7\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$42.50\times \frac{\left[1-\frac{1}{{\left(1+0.035\right)}^{26}}\right]}{0.035}+\frac{\$1,000}{{\left(1+0.035\right)}^{26}}\\ =\$717.8399+\$408.8378\\ =\$1,126.6776\end{array}$

Hence, the current price of the bond is $1,126.6776.

The bond value or the price of Bond X after one year:

The bond pays the coupons semiannually. The annual coupon payment is $85. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $42.50$\left(\$85÷2\right)$.

Secondly, the remaining time to maturity is 12 years after one year from now. As the coupon payment is semiannual, the semiannual periods to maturity are 24$\left(12\text{ years}\times 2\right)$. In other words, “t” equals to 24 6-month periods.

Thirdly, the yield to maturity is 7 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 3.50 percent $\left(7\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$42.50\times \frac{\left[1-\frac{1}{{\left(1+0.035\right)}^{24}}\right]}{0.035}+\frac{\$1,000}{{\left(1+0.035\right)}^{24}}\\ =\$682.4806+\$437.9571\\ =\$1,120.4378\end{array}$

Hence, the price of the bond will be $1,120.4378 after one year.

The bond value or the price of Bond X after 3 years:

The bond pays the coupons semiannually. The annual coupon payment is $85. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $42.50$\left(\$85÷2\right)$.

Secondly, the remaining time to maturity is 10 years after three years from now. As the coupon payment is semiannual, the semiannual periods to maturity are 20$\left(10\text{ years}\times 2\right)$. In other words, “t” equals to 20 6-month periods.

Thirdly, the yield to maturity is 7 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 3.50 percent $\left(7\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$42.50\times \frac{\left[1-\frac{1}{{\left(1+0.035\right)}^{20}}\right]}{0.035}+\frac{\$1,000}{{\left(1+0.035\right)}^{20}}\\ =\$604.0271+\$502.5658\\ =\$1,106.5930\end{array}$

Hence, the price of the bond will be $1,106.5930 after three years.

The bond value or the price of Bond X after eight years:

The bond pays the coupons semiannually. The annual coupon payment is $85. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $42.50$\left(\$85÷2\right)$.

Secondly, the remaining time to maturity is 5 years after eight years from now. As the coupon payment is semiannual, the semiannual periods to maturity are 10$\left(5\text{ years}\times 2\right)$. In other words, “t” equals to 10 6-month periods.

Thirdly, the yield to maturity is 7 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 3.50 percent $\left(7\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$42.50\times \frac{\left[1-\frac{1}{{\left(1+0.035\right)}^{10}}\right]}{0.035}+\frac{\$1,000}{{\left(1+0.035\right)}^{10}}\\ =\$353.4557+\$708.9188\\ =\$1,062.3745\end{array}$

Hence, the price of the bond will be $1,062.3745 after eight years.

The bond value or the price of Bond X after twelve years:

The bond pays the coupons semiannually. The annual coupon payment is $85. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $42.50$\left(\$85÷2\right)$.

Secondly, the remaining time to maturity is one year after twelve years from now. As the coupon payment is semiannual, the semiannual periods to maturity are two $\left(\text{1 year}\times 2\right)$. In other words, “t” equals to two 6-month periods.

Thirdly, the yield to maturity is 7 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 3.50 percent $\left(7\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$42.50\times \frac{\left[1-\frac{1}{{\left(1+0.035\right)}^2}\right]}{0.035}+\frac{\$1,000}{{\left(1+0.035\right)}^2}\\ =\$80.7370+\$933.5107\\ =\$1,014.2477\end{array}$

Hence, the price of the bond will be $1,014.2477 after twelve years.

The bond value or the price of Bond X after thirteen years:

The thirteenth year is the year of maturity for Bond X. In this year, the bondholder will receive the face value of the bond. Hence, the price of the bond will be $1,000 after thirteen years.

Compute the bond price of Bond Y at different maturities:

Compute the annual coupon payment of Bond Y:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 7\%\\ =\$70\end{array}$

Hence, the annual coupon payment of Bond Y is $70.

The bond value or the price of Bond Y at present:

The bond pays the coupons semiannually. The annual coupon payment is $70. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $35$\left(\$70÷2\right)$.

Secondly, the remaining time to maturity is 13 years. As the coupon payment is semiannual, the semiannual periods to maturity are 26$\left(13\text{ years}\times 2\right)$. In other words, “t” equals to 26 6-month periods.

Thirdly, the yield to maturity is 8.5 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 4.25 percent $\left(8.5\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$35\times \frac{\left[1-\frac{1}{{\left(1+0.0425\right)}^{26}}\right]}{0.0425}+\frac{\$1,000}{{\left(1+0.0425\right)}^{26}}\\ =\$544.4669+\$338.8616\\ =\$883.3285\end{array}$

Hence, the current price of the bond is $883.3285.

The bond value or the price of Bond Y after one year:

The bond pays the coupons semiannually. The annual coupon payment is $70. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $35$\left(\$70÷2\right)$.

Secondly, the remaining time to maturity is 12 years after one year from now. As the coupon payment is semiannual, the semiannual periods to maturity are 24$\left(12\text{ years}\times 2\right)$. In other words, “t” equals to 24 6-month periods.

Thirdly, the yield to maturity is 8.5 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 4.25 percent $\left(8.5\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$35\times \frac{\left[1-\frac{1}{{\left(1+0.0425\right)}^{24}}\right]}{0.0425}+\frac{\$1,000}{{\left(1+0.0425\right)}^{24}}\\ =\$520.2426+\$368.2769\\ =\$888.5195\end{array}$

Hence, the price of the bond is $888.5195 after one year.

The bond value or the price of Bond Y after three years:

The bond pays the coupons semiannually. The annual coupon payment is $70. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $35$\left(\$70÷2\right)$.

Secondly, the remaining time to maturity is 10 years after three years from now. As the coupon payment is semiannual, the semiannual periods to maturity are 20$\left(10\text{ years}\times 2\right)$. In other words, “t” equals to 20 6-month periods.

Thirdly, the yield to maturity is 8.5 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 4.25 percent $\left(8.5\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$35\times \frac{\left[1-\frac{1}{{\left(1+0.0425\right)}^{20}}\right]}{0.0425}+\frac{\$1,000}{{\left(1+0.0425\right)}^{20}}\\ =\$465.3028+\$434.9895\\ =\$900.2923\end{array}$

Hence, the price of the bond is $900.2923 after three years.

The bond value or the price of Bond Y after eight years:

The bond pays the coupons semiannually. The annual coupon payment is $70. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $35$\left(\$70÷2\right)$.

Secondly, the remaining time to maturity is 5 years after three years from now. As the coupon payment is semiannual, the semiannual periods to maturity are 10$\left(5\text{ years}\times 2\right)$. In other words, “t” equals to 10 6-month periods.

Thirdly, the yield to maturity is 8.5 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 4.25 percent $\left(8.5\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$35\times \frac{\left[1-\frac{1}{{\left(1+0.0425\right)}^{10}}\right]}{0.0425}+\frac{\$1,000}{{\left(1+0.0425\right)}^{10}}\\ =\$280.3811+\$659.5373\\ =\$939.9184\end{array}$

Hence, the price of the bond is $939.9184 after eight years.

The bond value or the price of Bond Y after twelve years:

The bond pays the coupons semiannually. The annual coupon payment is $70. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $35$\left(\$70÷2\right)$.

Secondly, the remaining time to maturity is one year after twelve years from now. As the coupon payment is semiannual, the semiannual periods to maturity are two $\left(1\text{ year}\times 2\right)$. In other words, “t” equals to two 6-month periods.

Thirdly, the yield to maturity is 8.5 percent per year. As the calculations are semiannual, the yield to maturity must also be semiannual. Hence, the semiannual or 6-month yield to maturity is 4.25 percent $\left(8.5\%÷2\right)$.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$35\times \frac{\left[1-\frac{1}{{\left(1+0.0425\right)}^2}\right]}{0.0425}+\frac{\$1,000}{{\left(1+0.0425\right)}^2}\\ =\$65.7776+\$920.1272\\ =\$985.9048\end{array}$

Hence, the price of the bond is $985.9048 after twelve years.

The bond value or the price of Bond Y after thirteen years:

The thirteenth year is the year of maturity for Bond Y. In this year, the bondholder will receive the face value of the bond. Hence, the price of the bond will be $1,000 after thirteen years.

Table indicating the bond prices of Bond X and Bond Y at different maturities:

Table 1

Time to maturity (Years)	Bond X	Bond Y
13	$1,126.6776	$883.3285
12	$1,120.4378	$888.5195
10	$1,106.5930	$900.2923
5	$1,062.3745	$939.9184
1	$1,014.2477	$985.9048
0	$1,000.0000	$1,000.0000

Graphical representation of the bond prices of Bond X and Bond Y from Table 1:

Explanation of the graph:

The graph indicates a “pull to par” effect on the prices of the bonds. The face value of both the bonds is $1,000. Although Bond X is at a premium and Bond Y is at a discount, both the bonds will reach their par values at the time of maturity. The effect of reaching the face value or par value from a discount or premium is known as “pull to par”.