Chapter 6, Problem 15QP

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 15QP

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

Time to maturity (Years)	Bond X	Bond Y
13	$1,167.1529	$850.2619
12	$1,158.8537	$856.7855
10	$1,140.4715	$871.6468
5	$1,082.004	$922.2068
1	$1,018.6915	$981.6514
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 9 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 9 percent. The bond will mature in 13 years. Both the bonds make annual 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 bonds’ current price:

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

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:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 9\%\\ =\$90\end{array}$

Hence, the annual coupon payment of Bond X is $90.

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

The bond pays the coupons annually. The annual coupon payment is $90. However, the bondholder will receive the same in one installment. Hence, annual coupon payment is $90.

Secondly, the remaining time to maturity is 13 years. As the coupon payment is annual.

Thirdly, the yield to maturity is 7 percent per year. As the calculations are annual, the yield to maturity must also be annual.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$90\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^{13}}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^{13}}\\ =\$752.1885+\$414.9644\\ =\$1,167.1529\end{array}$

Hence, the current price of the bond is $1,167.1529.

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

The bond pays the coupons annually. The annual coupon payment is $90. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $90

Secondly, the remaining time to maturity is 12 years after one year from now. As the coupon payment is annual, the annual periods to maturity is12 years.

Thirdly, the yield to maturity is 7 percent per year. As the calculations is annual, the yield to maturity must also be annual. Hence, the annual yield to maturity is 7 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$90\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^{12}}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^{12}}\\ =\$714.8418+\$444.0119\\ =\$1,158.8537\end{array}$

Hence, the price of the bond will be $1,158.8537 after one year.

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

The bond pays the coupons annually. The annual coupon payment is $90. However, the bondholder will receive the same in one installment. Hence, annual coupon payment is $90.

Secondly, the remaining time to maturity is 10 years after three years from now. As the coupon payment is annual, the annual periods to maturity is 10 years

Thirdly, the yield to maturity is 7 percent per year. As the calculations is annual, the yield to maturity is 7 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$90\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^{10}}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^{10}}\\ =\$632.1223+\$508.3492\\ =\$1,140.4715\end{array}$

Hence, the price of the bond will be $1,140.4715 after three years.

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

The bond pays the coupons annually. The annual coupon payment is $90. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $90

Secondly, the remaining time to maturity is 5 years after eight years from now. As the coupon payment is annual, the annual periods to maturity is 5 years.

Thirdly, the yield to maturity is 7 percent per year. As the calculations is annual, the yield to maturity is 7 percent

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$90\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^5}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^5}\\ =\$369.0178+\$712.9862\\ =\$1,082.004\end{array}$

Hence, the price of the bond will be $1,082.004 after eight years.

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

The bond pays the coupons annually. The annual coupon payment is $90. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $90

Secondly, the remaining time to maturity is 1 years after twelve years from now. As the coupon payment is annual, the annual periods to maturity is 1 years.

Thirdly, the yield to maturity is 7 percent per year. As the calculations is annual, the yield to maturity is 7 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$90\times \frac{\left[1-\frac{1}{{\left(1+0.07\right)}^1}\right]}{0.07}+\frac{\$1,000}{{\left(1+0.07\right)}^1}\\ =\$84.1121+\$934.5794\\ =\$1,018.6915\end{array}$

Hence, the price of the bond will be $1,018.6915 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 annually. The annual coupon payment is $70. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $70.

Secondly, the remaining time to maturity is 13 years after twelve years from now. As the coupon payment is annual, the annual periods to maturity is 13 years

Thirdly, the yield to maturity is 9 percent per year. As the calculations is annual, the yield to maturity is 9 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$70\times \frac{\left[1-\frac{1}{{\left(1+0.09\right)}^{13}}\right]}{0.09}+\frac{\$1,000}{{\left(1+0.09\right)}^{13}}\\ =\$524.0833+\$326.1786\\ =\$850.2619\end{array}$

Hence, the current price of the bond is $850.2619.

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

The bond pays the coupons annually. The annual coupon payment is $70. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $70.

Secondly, the remaining time to maturity is 12 years after twelve years from now. As the coupon payment is annual, the annual periods to maturity is 12 years.

Thirdly, the yield to maturity is 9 percent per year. As the calculations is annual, the yield to maturity is 9 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$70\times \frac{\left[1-\frac{1}{{\left(1+0.09\right)}^{12}}\right]}{0.09}+\frac{\$1,000}{{\left(1+0.09\right)}^{12}}\\ =\$501.2508+\$355.5347\\ =\$856.7855\end{array}$

Hence, the price of the bond is $856.7855 after one year.

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

The bond pays the coupons annually. The annual coupon payment is $70. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $70.

Secondly, the remaining time to maturity is 10 years after twelve years from now. As the coupon payment is annual, the annual periods to maturity is 10 years.

Thirdly, the yield to maturity is 9 percent per year. As the calculations is annual, the yield to maturity is 9 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$70\times \frac{\left[1-\frac{1}{{\left(1+0.09\right)}^{10}}\right]}{0.09}+\frac{\$1,000}{{\left(1+0.09\right)}^{10}}\\ =\$449.2360+\$422.4108\\ =\$871.6468\end{array}$

Hence, the price of the bond is $871.6468 after three years.

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

The bond pays the coupons annually. The annual coupon payment is $70. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $70.

Secondly, the remaining time to maturity is 5 years after twelve years from now. As the coupon payment is annual, the annual periods to maturity is 5 years.

Thirdly, the yield to maturity is 9 percent per year. As the calculations is annual, the yield to maturity is 9 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$70\times \frac{\left[1-\frac{1}{{\left(1+0.09\right)}^5}\right]}{0.09}+\frac{\$1,000}{{\left(1+0.09\right)}^5}\\ =\$272.2755+\$649.9313\\ =\$922.2068\end{array}$

Hence, the price of the bond is $922.2068 after eight years.

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

The bond pays the coupons annually. The annual coupon payment is $70. However, the bondholder will receive the same is one installment. Hence, annual coupon payment is $70.

Secondly, the remaining time to maturity is 5 years after twelve years from now. As the coupon payment is annual, the annual periods to maturity is 5 years.

Thirdly, the yield to maturity is 9 percent per year. As the calculations is annual, the yield to maturity is 9 percent.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$70\times \frac{\left[1-\frac{1}{{\left(1+0.09\right)}^1}\right]}{0.09}+\frac{\$1,000}{{\left(1+0.09\right)}^1}\\ =\$64.2202+\$917.4312\\ =\$981.6514\end{array}$

Hence, the price of the bond is $981.6514 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	$1,167.1529	$850.2619
12	$1,158.8536	$856.7855
10	$1,140.4715	$871.6468
5	$1,082.004	$922.2068
1	$1,018.6915	$981.6514
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”.