Question

Midland Oil has $1,000 par value bonds outstanding at 8 percent interest. The bonds will mature in 25 years. Compute the current price of the bonds if the present yield to maturity is

a.7 percent.

b.10 percent.

c.13 percent.

Short answer

a. $1,116.32

b. $818.16

c. $633.4

Step-by-step solution

Definition of Bonds

Bonds are defined as securities issued by a business that requires regular interest payment to its holder. These are debt securities and offer regular income to its holder.

Present value of bond under first case – 7% yield to maturity

$\begin{array}{rcl}\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{b}\mathit{o}\mathit{n}\mathit{d}& \mathbf{=}& \mathbf{[}\mathbf{C}\mathbf{o}\mathbf{u}\mathbf{p}\mathbf{o}\mathbf{n}\mathbf{}\mathbf{A}\mathbf{m}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{t}\mathbf{\times }\frac{\mathbf{1}\mathbf{-}{\displaystyle \frac{1}{{(1+Yieldrate)}^{Numberofyears}}}}{\mathbf{Y}\mathbf{i}\mathbf{e}\mathbf{l}\mathbf{d}\mathbf{}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{A}\mathbf{m}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{t}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{Y}\mathbf{i}\mathbf{e}\mathbf{l}\mathbf{d}\mathbf{}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}\mathbf{)}}^{\mathbf{N}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{y}\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{s}}}\mathbf{]}\\ & \mathbf{=}& \mathbf{[}\mathbf{\$}\mathbf{80}\mathbf{\times }\frac{\mathbf{1}\mathbf{-}{\displaystyle \frac{1}{{(1+0.07)}^{25}}}}{\mathbf{0}\mathbf{.}\mathbf{07}}\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{07}\mathbf{)}}^{\mathbf{25}}}\mathbf{]}\\ & \mathbf{=}& \mathbf{\$}\mathbf{932}\mathbf{.}\mathbf{32}\mathbf{+}\mathbf{\$}\mathbf{184}\\ & \mathbf{=}& \mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{116}\mathbf{.}\mathbf{32}\end{array}$

Present value of bond under the second case – 10% yield to maturity

$\begin{array}{rcl}\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{b}\mathit{o}\mathit{n}\mathit{d}& \mathbf{=}& \mathbf{[}\mathbf{Coupon}\mathbf{}\mathbf{Amount}\mathbf{\times }\frac{\mathbf{1}\mathbf{-}{\displaystyle \frac{1}{{(1+\mathrm{Yield}\mathrm{rate})}^{\mathrm{Number}\mathrm{of}\mathrm{years}}}}}{\mathbf{Yield}\mathbf{}\mathbf{rate}}\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{Amount}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{Yield}\mathbf{}\mathbf{rate}\mathbf{)}}^{\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{years}}}\mathbf{]}\\ & \mathbf{=}& \mathbf{[}\mathbf{\$}\mathbf{80}\mathbf{\times }\frac{\mathbf{1}\mathbf{-}{\displaystyle \frac{1}{{(1+0.01)}^{25}}}}{\mathbf{0}\mathbf{.}\mathbf{01}}\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{01}\mathbf{)}}^{\mathbf{25}}}\mathbf{]}\\ & \mathbf{=}& \mathbf{\$}\mathbf{726}\mathbf{.}\mathbf{16}\mathbf{+}\mathbf{\$}\mathbf{92}\\ & \mathbf{=}& \mathbf{\$}\mathbf{818}\mathbf{.}\mathbf{16}\end{array}$

Present value of bond under the third case – 13% yield to maturity

$\begin{array}{rcl}\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{b}\mathit{o}\mathit{n}\mathit{d}& \mathbf{=}& \mathbf{[}\mathbf{Coupon}\mathbf{}\mathbf{Amount}\mathbf{\times }\frac{\mathbf{1}\mathbf{-}{\displaystyle \frac{1}{{(1+\mathrm{Yield}\mathrm{rate})}^{\mathrm{Number}\mathrm{of}\mathrm{years}}}}}{\mathbf{Yield}\mathbf{}\mathbf{rate}}\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{Amount}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{Yield}\mathbf{}\mathbf{rate}\mathbf{)}}^{\mathbf{Number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{years}}}\mathbf{]}\\ & \mathbf{=}& \mathbf{[}\mathbf{\$}\mathbf{80}\mathbf{\times }\frac{\mathbf{1}\mathbf{-}{\displaystyle \frac{1}{{(1+0.13)}^{25}}}}{\mathbf{0}\mathbf{.}\mathbf{13}}\mathbf{]}\mathbf{+}\mathbf{[}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{13}\mathbf{)}}^{\mathbf{25}}}\mathbf{]}\\ & \mathbf{=}& \mathbf{\$}\mathbf{586}\mathbf{.}\mathbf{4}\mathbf{+}\mathbf{\$}\mathbf{47}\\ & \mathbf{=}& \mathbf{\$}\mathbf{633}\mathbf{.}\mathbf{4}\end{array}$