Question

Kilgore Natural Gas has a $1,000 par value bond outstanding that pays 9 percent annual interest. The current yield to maturity on such bonds in the market is 12 percent. Compute the price of the bonds for these maturity dates:

a. 30 years.

b. 15 years.

c. 1 year.

Short answer

1. Bond price for 30 years is $758.78
2. Bond price for 15 years is $795.60
3. Bond price for 1 year is $973.22

Step-by-step solution

Computing the coupon. 

$\begin{array}{rcl}\mathit{A}\mathit{n}\mathit{n}\mathit{u}\mathit{a}\mathit{l}\mathit{C}\mathit{o}\mathit{u}\mathit{p}\mathit{o}\mathit{n}& \mathbf{=}& \mathit{P}\mathit{a}\mathit{r}\mathit{V}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{\times }\mathit{C}\mathit{o}\mathit{u}\mathit{p}\mathit{o}\mathit{n}\mathit{R}\mathit{a}\mathit{t}\mathit{e}\\ & \mathbf{=}& \mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{\times }\mathbf{9}\mathbf{\%}\\ & \mathbf{=}& \mathbf{\$}\mathbf{90}\\ & & \end{array}$

Computing the price of the bonds for 30 years.

• Par value of bond (P) is $1,000.
• Yield to maturity (r) is 12%.
• Years to maturity (n) is 30.
• $\begin{array}{rcl}\mathit{B}\mathit{o}\mathit{n}\mathit{d}\mathit{P}\mathit{r}\mathit{i}\mathit{c}\mathit{e}& \mathbf{=}& \mathit{C}\mathit{o}\mathit{u}\mathit{p}\mathit{o}\mathit{n}\mathbf{\times }\frac{\left[1-\frac{1}{{\left(1+r\right)}^n}\right]}{\mathbf{r}}\mathbf{+}\frac{\mathbf{P}}{{\left(1+r\right)}^{\mathbf{n}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{90}\mathbf{\times }\frac{\left[1-\frac{1}{{\left(1+0.12\right)}^{30}}\right]}{\mathbf{0}\mathbf{.}\mathbf{12}}\mathbf{+}\frac{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{000}}{{\left(1+0.12\right)}^{\mathbf{30}}}\\ & \mathbf{=}& \mathbf{\$}\mathbf{90}\mathbf{\times }\mathbf{8}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{33}\mathbf{.}\mathbf{38}\\ & \mathbf{=}& \mathbf{\$}\mathbf{758}\mathbf{.}\mathbf{78}\\ & & \end{array}$

Computing the price of the bonds for 15 years.

• Par value of bond (P) is $1,000.
• Yield to maturity (r) is 12%.
• Years to maturity (n) is 15.
• $$\begin{gathered} \mathbf{BondPrice}\mathbf{=}\mathbf{Coupon}\mathbf{\times }\frac{\left[1-\frac{1}{{\left(1+r\right)}^n}\right]}{\mathbf{r}}\mathbf{+}\frac{\mathbf{P}}{{\left(1+r\right)}^{\mathbf{n}}} \\ \mathbf{=}\mathbf{\$}\mathbf{90}\mathbf{\times }\frac{\left[1-\frac{1}{{\left(1+0.12\right)}^{15}}\right]}{\mathbf{0}\mathbf{.}\mathbf{12}}\mathbf{+}\frac{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{000}}{{\left(1+0.12\right)}^{\mathbf{15}}} \\ \mathbf{=}\mathbf{\$}\mathbf{90}\mathbf{\times }\mathbf{6}\mathbf{.}\mathbf{81}\mathbf{+}\mathbf{\$}\mathbf{182}\mathbf{.}\mathbf{70} \\ \mathbf{=}\mathbf{\$}\mathbf{795}\mathbf{.}\mathbf{60} \end{gathered}$$
  

Computing the price of the bonds for 1 years.

• Par value of bond (P) is $1,000.
• Yield to maturity (r) is 14%.
• Years to maturity (n) is 18.

$$\begin{gathered} \mathit{B}\mathit{o}\mathit{n}\mathit{d}\mathit{P}\mathit{r}\mathit{i}\mathit{c}\mathit{e}\mathbf{=}\mathit{C}\mathit{o}\mathit{u}\mathit{p}\mathit{o}\mathit{n}\mathbf{\times }\frac{\left[1-\frac{1}{{\left(1+r\right)}^n}\right]}{\mathbf{r}}\mathbf{+}\frac{\mathbf{P}}{{\left(1+r\right)}^{\mathbf{n}}} \\ \mathbf{=}\mathbf{\$}\mathbf{90}\mathbf{\times }\frac{\left[1-\frac{1}{{\left(1+0.12\right)}^1}\right]}{\mathbf{0}\mathbf{.}\mathbf{12}}\mathbf{+}\frac{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{000}}{{\left(1+0.1200\right)}^{\mathbf{1}}} \\ \mathbf{=}\mathbf{\$}\mathbf{90}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{8929}\mathbf{+}\mathbf{892}\mathbf{.}\mathbf{86} \\ \mathbf{=}\mathbf{\$}\mathbf{973}\mathbf{.}\mathbf{22} \end{gathered}$$