Chapter 22, Problem 31QP

a.

Summary Introduction

To determine: Present value of a risk-free bond.

Risk Free Bond:

It is a theoretical bond, which is free from the risk of repayment and that repays interest and principal completely at the time of maturity.

Explanation of Solution

Given,

Face value of the bond is $40,000.

Risk free rate is 6%.

Maturity is 5 years.

Formula to calculate present value of a risk free bond is,

$PV=FV{e}^{-RT}$

Where,

• PV is present value.
• FV is face value of the bond.
• R is risk free rate.
• T is maturity time.

Substitute $40,000 for FV, 6 for R and 5 for T in the above equation.

$\begin{array}{c}PV=\$40,000e^{-0.06\times 5}\\ =\$40,000\times 0.740818221\\ =\$29,632.73\end{array}$

Conclusion

Hence, the present value of risk free bond is $29,632.73.

b.

Summary Introduction

To determine: Value of a put option on company’s assets.

Put Option:

It is a future contract between two parties for the sale of an asset on or before a predetermined date.

Explanation of Solution

Given,

Exercise price is $40,000.

Risk free rate is 6%.

Maturity is 5 years.

Current value of company’s assets is $38,000.

Annual variance ( ${\sigma }_{}^2$) is 0.50.

Calculated Values,

$N{d}_1$ is 0.7827.

$N{d}_2$ is 0.3732.

Formula to calculate value of a call option is,

$C=SN\left(d_1\right)-E{e}^{-RT}N\left(d_2\right)$

Where,

• S is current value of assets of the company.
• E is the exercise price.
• R is risk free rate.
• T is maturity time.

Substitute $38,000 for S, 0.7827 for $N\left(d_1\right)$, $40,000 for E, 0.06 for R, 5 for T and 0.3732 for $N\left(d_2\right)$ in the above equation.

$\begin{array}{c}C=\left(\$38,000\times 0.7827\right)-\left(\$40,000e^{-0.06\times 5}\times 0.3732\right)\\ =\left(\$29,742.6\right)-\left(\$40,000\times 0.740818221\times 0.3732\right)\\ =\$29,742.6-\$11,049.42\\ =\$18,693.17\end{array}$

Formula to calculate put option is,

$C-P=S-E{e}^{-RT}$

Where,

• C is price of call option.
• P is price of put option.
• S is current value of assets of the company.
• E is the exercise price.
• R is risk free rate.
• T is maturity time.

Substitute $18,693.17 for C, $38,000 for S, $40,000 for E, 0.06 for R and 5 for T in the above equation.

$\begin{array}{c}18,693.17-P=\$38,000-\$40,{000}^{-0.06\times 5}\\ P=\$40,000\times 0.740181221+18,693.17-\$38,000\\ =\$48,300.42-\$38,000\\ =\$10,300.42\end{array}$

Working notes:

Calculation of $d_1$,

$\begin{array}{c}{d}_1=\frac{\text{In}\left(\frac{S}{E}\right)+\left(R+\frac{{\sigma }^2}{2}\right)T}{\sqrt{{\sigma }^{2}t}}\\ =\frac{\text{In}\left(\frac{\$38,000}{\$40,000}\right)+\left(0.06+\frac{{0.50}^2}{2}\right)\times 5}{\sqrt{{0.50}^2\times 5}}\\ =\frac{-0.051293294+0.185\times 5}{0.50\times \sqrt{5}}\\ =0.7815\end{array}$

Calculation of $N{d}_1$

Cumulative probability value of 0.78 is 0.2823.

Therefore,

$\begin{array}{c}N(0.78)=0.5+0.2823\\ =0.7823\end{array}$

Cumulative probability value of 0.79 is 0.2852.

Therefore,

$\begin{array}{c}N(0.79)=0.5+0.2852\\ =0.7852\end{array}$

Since, 0.7815 is 15 percent of the way between 0.78 and 0.79; it can be interpolated as,

$\begin{array}{c}N\left(0.7815\right)=N\left(0.78\right)+0.15\times \left(N\left(0.79\right)-N\left(0.78\right)\right)\\ =0.7823+0.15\times \left(0.7852-0.7823\right)\\ =0.7823+0.15\times 0.0029\\ =0.7827\end{array}$

Calculation of $d_2$

$\begin{array}{l}{d}_2=d_1-\sqrt{{\sigma }^{2}T}\\ =0.7815-\sqrt{{0.50}^2\times 5}\\ =0.7815-1.118\\ =-0.3365\end{array}$

Calculation of $N{d}_2$

Cumulative probability value of -0.33 is 0.1293.

Therefore,

$\begin{array}{c}N(-0.33)=0.5-0.1293\\ =0.3707\end{array}$

Cumulative probability value of -0.34 is 0.1331.

Therefore,

$\begin{array}{c}N(-0.34)=0.5-1331\\ =0.3669\end{array}$

Since, -0.3365 is 65 percent of the way between -0.33 and -0.34; it can be interpolated as,

$\begin{array}{c}N\left(-0.3365\right)=N\left(-0.33\right)+0.65\times \left(N\left(-0.33\right)-N\left(-0.34\right)\right)\\ =0.3707+0.65\times \left(0.3707-0.3669\right)\\ =0.3707+0.65\times 0.0038\\ =0.3732\end{array}$

Conclusion

Hence, the price of call option is $18,693.17 and put option is $10,300.42.

c.

Summary Introduction

To determine: Value of company’s debt and continuously compounded yield on company’s debt.

Explanation of Solution

Calculated values,

Value of risk free bond is $29,632.73.

Value of put option is $10,300.42.

Formula to calculate value of firm’s debt is,

$\text{Value of firm's debt}=\text{Value of risk free bond}-\text{value of put option}$

Substitute $29,632.73 for value of risk free bond and $10,300.42 for value of put option n the above equation.

$\begin{array}{c}\text{Value of firm's debt}=\$29,632.73-\$10,300.42\\ =\$19,332.31\end{array}$

Calculated values,

Present value of debt is $19,332.31.

Face value of bond is $40,000.

Time to maturity is 5 years.

Formula to calculate continuously compounded yield on company’s debt is,

$\text{Present value of firm's debt}=\text{Face value of bond}\times e^{-RT}$

Substitute $19,332.31 for present value of debt, $40,000 for face value of bond and 5 for T in the above equation.

$\begin{array}{c}\$19,332.31=\$40,000\times e^{-R5}\\ \text{In}\left(\frac{\$19,332.31}{\$40,000}\right)=-R5\\ -0.72710166=-R5\\ R=0.1454\text{ or 14}\text{.54\%}\end{array}$

Conclusion

Hence, the value of firm’s debt is $19,332.31 and continuously compounded yield on company’s debt is 14.54%.

(d)

Summary Introduction

To determine: Value of firm’s debt and continuously compounded yield on company’s debt after the restructuring of assets.

Explanation of Solution

Given,

Exercise price is $40,000.

Risk free rate is 6%.

Maturity is 5 years.

Current value of company’s assets is $38,000.

Annual variance ( ${\sigma }_{}^2$) is 0.60.

Calculated Values,

$N{d}_1$ is 0.8040.

$N{d}_2$ is 0.4854.

Formula to calculate value of a call option is,

$C=SN\left(d_1\right)-E{e}^{-RT}N\left(d_2\right)$

Where,

S is current value of assets of the company.

E is the exercise price.

R is risk free rate.

T is maturity time.

Substitute $38,000 for S, 0.8040 for $N\left(d_1\right)$, $40,000 for E, 0.06 for R, 5 for T and 0.4854 for $N\left(d_2\right)$ in the above equation.

$\begin{array}{c}C=\$38,000\times 0.8040-\$40,000e^{-0.06\times 5}\times 0.4854\\ =\$30,552-\$40,000\times 0.740818221\times 0.4854\\ =\$30,552-\$14,371.36\\ =\$16,180.64\end{array}$

Formula to calculate put option is,

$C-P=S-E{e}^{-RT}$

Where,

• C is price of call option.
• P is price of put option.
• S is current value of assets of the company.
• E is the exercise price.
• R is risk free rate.
• T is maturity time.

Substitute $16,180.64 for C, $38,000 for S, $40,000 for E, 0.06 for R and 5 for T in the above equation.

$\begin{array}{c}16,180.64-P=\$38,000-\$40,{000}^{-0.06\times 5}\\ P=\left(\$40,000\times 0.740181221\right)+16,180.64-\$38,000\\ =\$45,787.89-\$38,000\\ =\$7,787.89\end{array}$

Calculated values,

Value of risk free bond is $29,632.73.

Value of put option is $7,787.89.

Formula to calculate value of firm’s debt is,

$\text{Value of firm's debt}=\text{Value of risk free bond}-\text{value of put option}$

Substitute $29,632.73 for value of risk free bond and $7,787.89 for value of put option n the above equation.

$\begin{array}{c}\text{Value of firm's debt}=\text{\$29,632}\text{.73}-\text{\$7,787}\text{.89}\\ =\$21,844.8\text{4}\end{array}$

Calculated values,

Present value of debt is 21,844.84.

Face value of bond is $40,000.

Time to maturity is 5 years.

Formula to calculate continuously compounded yield on company’s debt is,

$\text{Present value of firm's debt}=\text{Face value of bond}\times e^{-RT}$

Substitute $21,844.84 for present value of debt, $40,000 for face value of bond and 5 for T in the above equation.

$\begin{array}{c}\$21,844.84=\$40,000\times e^{-R5}\\ \text{In}\left(\frac{\$21,844.84}{\$40,000}\right)=-R5\\ -0.60491471=-R5\\ R=0.1209\text{ or 12}\text{.09\%}\end{array}$

Working note:

Calculation of $d_1$,

$\begin{array}{c}{d}_1=\frac{\text{In}\left(\frac{S}{E}\right)+\left(R+\frac{{\sigma }^2}{2}\right)T}{\sqrt{{\sigma }^{2}t}}\\ =\frac{\text{In}\left(\frac{\$38,000}{\$40,000}\right)+\left(0.06+\frac{{0.60}^2}{2}\right)\times 5}{\sqrt{{0.50}^2\times 5}}\\ =\frac{-0.051293294+0.240\times 5}{0.50\times \sqrt{5}}\\ =0.8562\end{array}$

Calculation of $N{d}_1$

Cumulative probability value of 0.85 is 0.3023.

Therefore,

$\begin{array}{c}N(0.85)=0.5+0.3023\\ =0.8023\end{array}$

Cumulative probability value of 0.86 is 0.3051.

Therefore,

$\begin{array}{c}N(0.86)=0.5+0.3051\\ =0.8051\end{array}$

Since, 0.8562 is 62 percent of the way between 0.85 and 0.86; it can be interpolated as,

$\begin{array}{c}N\left(0.8562\right)=N\left(0.85\right)+0.62\times \left(N\left(0.86\right)-N\left(0.85\right)\right)\\ =0.8023+0.62\times \left(0.8051-0.8023\right)\\ =0.8023+0.62\times 0.0028\\ =0.8040\end{array}$

Calculation of $d_2$

$\begin{array}{l}{d}_2=d_1-\sqrt{{\sigma }^{2}T}\\ =0.8562-\sqrt{{0.60}^2\times 5}\\ =0.8562-1.34164\\ =-0.4854\end{array}$

Calculation of $N{d}_2$

Cumulative probability value of -0.48 is 0.1844.

Therefore,

$\begin{array}{c}N(-0.48)=0.5-0.1844\\ =0.3156\end{array}$

Cumulative probability value of -0.49 is 0.1879.

Therefore,

$\begin{array}{c}N(-0.49)=0.5-1879\\ =0.3121\end{array}$

Since, -0.4854 is 54 percent of the way between -0.48 and -0.49; it can be interpolated as,

$\begin{array}{c}N\left(-0.4854\right)=N\left(-0.48\right)+0.54\times \left(N\left(-0.48\right)-N\left(-0.49\right)\right)\\ =0.3156+0.54\times \left(0.3156-0.3121\right)\\ =0.3156+0.54\times 0.0035\\ =0.3175\end{array}$

Conclusion

Hence, the price of call option is $16,180.64, value of put option is $7,787.89, value of firm’s debt is $21,844.84 and continuously compounded yield on company’s debt is 12.09%.

(e)

Summary Introduction

To determine: Change in the value of equity and debt after restructure.

Explanation of Solution

Calculated values,

Value of equity before restructure is $18,693.17.

Value of equity after restructure is $16,180.64.

Value of debt before restructure is $19,332.31.

Value of debt after restructure is $21,844.84.

Formula to calculate change in equity value is,

$\text{Change in equity value}=\left(\begin{array}{c}\text{Value of equity before restructure}\\ -\text{Value of equity after restructure}\end{array}\right)$

Substitute $18,693.17 for value of equity before restructure and 16,180.64 for value of equity after restructure.

$\begin{array}{c}\text{Change in equity value}=\$18,693.17-\$16,180.64\\ =\$2512.53\end{array}$

Formula to calculate change in value of debt is,

$\text{Change in value of debt}=\left(\begin{array}{c}\text{Value of debt before restructure}\\ -\text{Value of debt after restructure}\end{array}\right)$

Substitute $19,332.31 for value of debt before restructure and 21,844.84 for value of debt after restructure.

$\begin{array}{c}\text{Change in value of debt}=\$19,332.31-\$21,844.84\\ =-\$2512.53\end{array}$

Conclusion

Thus, after the restructuring of assets the value of debt has increased by $2512.53 and the value of equity had decreased by the same amount and hence, the restructuring is favorable for bond holders than the equity shareholders.