Chapter 22, Problem 24QP

Mergers and Equity as an Option Suppose Sunburn Sunscreen and Frostbite Thermalwear in the previous problems have decided to merge. Because the two companies have seasonal sales, the combined firm’s return on assets will have a standard deviation of 29 percent per year.

a. What is the combined value of equity in the two existing companies? The value of debt?

b. What is the value of the new firm’s equity? The value of debt?

c. What was the gain or Joss for shareholders? For bondholders?

d. What happened to shareholder value here?

Summary Introduction

To compute: Combined value of equity and debt for SS Company and FT Company on the basis of previous question.

Option Pricing:

Option pricing helps in determining correct or fair price in the market. It is the value of one share on the basis of which option is traded. Black-Scholes is one of the pricing methods. Further, equity is also used as an option.

Explanation of Solution

Given,

SS Company’s value of equity is $4,585.07.

SS Company’s value of debt is $17,114.93.

FT Company’s value of equity is $7,108.75.

FT Company’s value of debt is $20.091.25.

Formula to calculate combined value of equity is,

$\text{Value}\text{of}\text{equity}=\text{SS}\text{Company's}\text{equity}+\text{FT}\text{Company's}\text{equity}$

Substitute $4,585.07 as SS Company’s value of equity and $7,108.75 as FT Company’s value of equity.

$\begin{array}{c}\text{Value}\text{of}\text{equity}=\$4,585.07+\$7,108.75\\ =\$11,693.82\end{array}$

Formula to calculate combined value of debt is,

$\text{Value}\text{of}\text{debt}=\text{SS}\text{Company's}\text{debt}+\text{FT}\text{Company's}\text{debt}$

Substitute $17,114.93 as SS Company’s value of debt and $20,091.25 as FT Company’s value of debt.

$\begin{array}{c}\text{Value}\text{of}\text{debt}=\$17,114.93+\$20,091.25\\ =\$37,206.18\end{array}$

Hence, combined value of equity is $11,693.82 and combined value of debt is $37,206.18.

Summary Introduction

To compute: The value of the new firm’s equity and value of debt.

Explanation of Solution

Given,

Stock price is 48,900.

Exercise price is 45,000.

Risk free rate is 0.05.

Time to expire is 1 year.

Formula to calculate the value of equity by using Black Scholes model is,

$\text{Value}\text{of}\text{equity}=\text{SN}\left({\text{d}}_{\text{1}}\right)-{\text{Ee}}^{\text{-Rt}}\text{N}\left({\text{d}}_{\text{2}}\right)$

Where,

• S is stock price.
• E is exercise price.
• R is risk free rate.
• T is time to expire.

Substitute $48,900for S, $45,000 for E, 0.05for R, and 1for T.

$\begin{array}{c}\text{Value of equity}=\left(\$48,900\left(0.7257\right)\right)-\left(\$45,000e^{-0.05\times 1}\left(0.6217\right)\right)\\ =\$35,486.73-\left(\$45,000\times 0.9512294\left(0.6217\right)\right)\\ =\$8,874.66\end{array}$

Formula to calculate the value of debt is,

$\text{Value}\text{of}\text{debt}=\text{Value}\text{of}\text{firm}-\text{Value}\text{of}\text{equity}$

Substitute $48,900 as value of firm and $8,874.66 as value of equity.

$\begin{array}{c}\text{Value}\text{of}\text{debt}=\$48,900-\$8,874.66\\ =\$40,025.34\end{array}$

Working notes:

Calculation of stock price,

$\begin{array}{c}\text{Stock}\text{price}=\text{Stock}\text{price}\text{of}\text{SS}\text{Company}+\text{Stock}\text{price}\text{of}\text{FT}\text{Company}\\ =\$21,700+\$27200\\ =\$48,900\end{array}$

Calculation of exercise price,

$\begin{array}{c}\text{Exercise}\text{price}=\$20,000+\$25,000\\ =\$45,000\end{array}$

Calculation of $d_1$,

$\begin{array}{c}{\text{d}}_{\text{1}}=\frac{\text{In}\left(\frac{\text{S}}{\text{E}}\right)+\left(\text{R}+\frac{{\text{σ}}^{\text{2}}}{\text{2}}\right)\text{t}}{\sqrt{{\text{σ}}^{\text{2}}\text{t}}}\\ =\frac{\text{In}\left(\frac{\$48,900}{\$45,000}\right)+\left(0.05+\frac{{0.29}^2}{2}\right)\times 1}{\sqrt{{0.29}^2\times 1}}\\ =\frac{0.083115+\left(0.05+0.04205\right)}{0.29}\\ =0.6040\end{array}$

From normal distribution table ${\text{N(d}}_{\text{1}}\text{)}=0.7257$

Calculation of $d_2$

$\begin{array}{c}{\text{d}}_{\text{2}}={\text{d}}_{\text{1}}-\sqrt{{\text{σ}}^{\text{2}}\text{t}}\\ =0.6040-\sqrt{{0.29}^2\times 1}\\ =0.6040-0.29\\ =0.3140\end{array}$

From normal distribution table ${\text{N(d}}_2\text{)}=0.6217$

Hence, value of the new firm’s equity is $8,874.66 and value of debt is $40,025.34.

Summary Introduction

To compute: Gain or loss for shareholders and bondholders.

Explanation of Solution

Given,

Equity of new firm is $2,930.64.

Debt of new firm is $45,969.35.

Combined value of equity is $3,051.26.

Combined value of debt is $45,848.74.

Formula to calculate change in the value of equity is,

$\text{Change}\text{in}\text{the}\text{value}\text{of}\text{equity}=\text{Equity}\text{of}\text{new}\text{firm}-\text{Combined}\text{value}\text{of}\text{equity}$

Substitute $2,930.64as equity of new firm and $3,051.26as combined value of equity.

$\begin{array}{c}\text{Change}\text{in}\text{the}\text{value}\text{of}\text{equity}=\$2,930.64-\$3,051.26\\ =-\$120.62\end{array}$

Formula to calculate change in the value of debt is,

$\text{Change}\text{in}\text{the}\text{value}\text{of}\text{debt}=\text{Debt}\text{of}\text{new}\text{firm}-\text{Combined}\text{value}\text{of}\text{debt}$

Substitute $45,969.35as debt of new firm and $45,848.74as combined value of debt.

$\begin{array}{c}\text{Change}\text{in}\text{the}\text{value}\text{of}\text{debt}=\$45,969.35-\$45,848.74\\ =\$120.61\end{array}$

Hence, in case of firm’s equity there is a loss for shareholders and in case of firm’s debt there is a gain for bondholders.

Summary Introduction

To explain: Status of shareholder’s value.

Answer to Problem 24QP

Value of the shareholders has reduced due to reduction in the value of new firm’s equity.

Explanation of Solution

• On comparison of the new firm with the old pre-merger companies, standard deviation of the new company.
• Because of this situation equity holder’s value has also decreased.
• Loose amount by the shareholders directly goes in the pocket of bondholders.

Hence, core value of the shareholders has been decreased.