Chapter 16, Problem 3QP

ROE and Leverage Suppose the company in Problem 1 has a market-to-book ratio of 1.0.

1. a. Calculate return on equity, ROE, und.er each of the three economic scenarios before any debt is issued. Also calculate the percentage changes in ROE for economic expansion and recession, assuming no taxes.
2. b. Repeat part (a) assuming the firm goes through with the proposed recapitalization.
3. c. Repeat parts (a) and (b) of this problem assuming the firm has a tax rate of 35 percent

Summary Introduction

To determine: The return on equity and change in return on equity for the given scenario.

Introduction:

Return on equity is the evaluation of profitability that calculates how many amounts of profits the firm generates with each dollar of shareholder’s equity. The return on equity is computed by dividing the net income to the number of shareholders.

Explanation of Solution

Calculate the return on equity:

The market-to-book ratio is 1.0. The company’s total equity is equal to the market value of equity. The market value of company is $295,000.

The net income of the recession is $13,800, normal is $23,000 and for expansion it is $28,750.

Recession:

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity}}\\ =\frac{\$13,800}{\$295,000}\\ =0.467\mathrm{or}4.68\%\end{array}$

Therefore, the return on equity for recession is 4.68%.

Normal:

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity}}\\ =\frac{\$23,000}{\$295,000}\\ =0.0779\mathrm{or}7.79\%\end{array}$

Therefore, the return on equity for normal is 7.79%.

Expansion:

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity}}\\ =\frac{\$28,750}{\$295,000}\\ =0.0974\mathrm{or}9.75\%\end{array}$

Therefore, the return on equity for expansion is 9.75%.

Calculate percentage change in the return on equity:

Recession:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{recession}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(4.68\%-7.80\%\right)}{7.80\%}\\ =\frac{-0.0312}{0.078}\\ =-0.4\text{or}40\%\end{array}$

Therefore, the percentage change in ROE is 40%.

Normal:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{recession}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(7.80\%-7.80\%\right)}{7.80\%}\\ =\frac{0\%}{0\%}\\ =0\%\end{array}$

Therefore, the percentage change in return on equity is 0%.

Expansion:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{recession}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(9.75\%-7.80\%\right)}{7.80\%}\\ =\frac{1.95\%}{7.80\%}\\ =0.25\text{or}25\%\end{array}$

Therefore, the percentage change in return on equity is 25%.

Summary Introduction

To determine: The return on equity and change in the percentage of equity.

Explanation of Solution

If the company carries out a proposed recapitalization, the new equity value is

calculated as follows.

Calculate the equity:

It is given that the debt issued value is $85,000.

$\begin{array}{c}\text{Equity}=\text{Total equity}-\text{Debt issued }\\ =\text{\$295,000}-\text{\$88,500}\\ \text{=\$206,500}\end{array}$

Therefore, the total equity value is $206,500.

Calculate the return on equity:

Recession:

The net income of the company is $6,720 and the total equity value is $206,500.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$6,720}{\$206,500}\\ =0.325\text{or}\text{3}\text{.25\%}\end{array}$

Therefore, the return on equity value is 3.25%.

Normal:

The net income of the company is $15,920 and the total equity value is $206,500.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$15,920}{\$206,500}\\ =0.077\text{or}\text{7}\text{.70\%}\end{array}$

Therefore, the return on equity value is 7.70%.

Expansion:

The net income of the company is $21,670 and the total equity value is $206,500.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$21,670}{\$206,500}\\ =0.1049\text{or}10.\text{49\%}\end{array}$

Therefore, the return on equity value is 10.49%.

Calculate percentage change in the return on equity:

Recession:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{recession}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(3.25\%-7.71\%\right)}{7.71\%}\\ =\frac{-4.46\%}{7.71\%}\\ =-0.5784\text{or}57.79\%\end{array}$

Therefore, the percentage change in return of equity value is 57.79%.

Normal:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{normal}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(7.71\%-7.71\%\right)}{7.71\%}\\ =0\text{or}0\%\end{array}$

Therefore, the percentage change in return of equity value is 0%.

Expansion:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{normal}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(10.49\%-7.71\%\right)}{7.71\%}\\ =\frac{2.78\%}{7.71\%}\\ =0.360\text{or}36.05\%\end{array}$

Therefore, the percentage change in return of equity value is 36.05%.

Summary Introduction

To determine: The return on equity and change in return on equity for the given scenario.

Explanation of Solution

If there are corporate taxes and the firm maintains its current capital structure, the return on equity is.

Determine the return on equity:

Recession:

The net income value is $8,970 and total equity value is $295,000.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$8,970}{\$295,000}\\ =0.0304\text{or}\text{3}\text{.04\%}\end{array}$

Therefore, the return on equity is 3.04%.

Normal:

The net income value is $14,950 and total equity value is $295,000.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$14,950}{\$295,000}\\ =0.0506\text{or}\text{5}\text{.07\%}\end{array}$

Therefore, the return on equity is 5.07%.

Expansion:

The net income value is $18,688 and total equity value is $295,000.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$18,688}{\$295,000}\\ =0.0633\text{or}\text{6}\text{.33\%}\end{array}$

Therefore, the return on equity is 6.33%.

Calculate percentage change in the return on equity:

Recession:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{recession}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(3.04\%-5.07\%\right)}{5.07\%}\\ =\frac{-2.03\%}{5.07\%}\\ =-0.400\text{or}40.00\%\end{array}$

Therefore, the percentage change in return of equity value is 40.00%.

Normal:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{normal}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(5.07\%-5.07\%\right)}{5.07\%}\\ =0\text{or}0\%\end{array}$

Therefore, the percentage change in return of equity value is 0%.

Expansion:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\left({\text{Return on equity}}_{\text{expansion}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(6.33\%-5.07\%\right)}{5.07\%}\\ =0.248\text{or}25\%\end{array}$

Therefore, the percentage change in return of equity value is 25%.

Calculate the return on equity with debts:

Recession:

The net income value is $4,368 and total equity value is $206,500.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$4,368}{\$206,500}\\ =0.0211\text{or}\text{2}\text{.12\%}\end{array}$

Therefore, the return on equity is 2.12%.

Normal:

The net income value is $10,348 and total equity value is $206,500.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$10,348}{\$206,500}\\ =0.05011\text{or}\text{5}\text{.01\%}\end{array}$

Therefore, the return on equity is 5.01%.

Expansion:

The net income value is $14,086 and total equity value is $206,500.

$\begin{array}{c}\text{Return on equity=}\frac{\text{Net income}}{\text{Total equity value}}\\ =\frac{\$14,086}{\$206,500}\\ =0.06821\text{or}\text{6}\text{.82\%}\end{array}$

Therefore, the return on equity is 6.82%.

Calculate percentage change in the return on equity:

Recession:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{recession}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(2.12\%-5.01\%\right)}{5.01\%}\\ =\frac{-2.89\%}{5.01\%}\\ =-0.5768\text{or}57.58\%\end{array}$

Therefore, the percentage change in return of equity value is 57.58%.

Normal:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{normal}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(5.01\%-5.01\%\right)}{5.01\%}\\ =0\text{or}0\%\end{array}$

Therefore, the percentage change in return of equity value is 0%.

Expansion:

$\begin{array}{c}\text{Percentage change in ROE=}\frac{\left({\text{Return on equity}}_{\text{expansion}}-{\text{Return on equity}}_{\text{normal}}\right)}{{\text{Return on equity}}_{\text{normal}}}\\ =\frac{\left(6.82\%-5.01\%\right)}{5.01\%}\\ =\frac{1.81\%}{5.01\%}\\ =0.3612\text{or}36.12\%\end{array}$

Therefore, the percentage change in return of equity value is 36.12%,