Chapter 13, Problem 3QP

a)

Summary Introduction

To calculate: The return on equity for the three economic scenarios before any issue of debt and compute the percentage changes in return on equity.

Introduction:

The ROE (Return on equity) is a profitability measure that computes the amount of dollar a firm creates with every dollar of shareholders equity.

Explanation of Solution

Given information:

Company K has no outstanding debt and its market value is $125,000. The EBIT (Earnings before interest and taxes) are expected to be $10,400 at normal economic conditions. If the economy condition is strong, then EBIT will increase to 20% and if the economy enters into recession, then it will decrease to 35%. The outstanding shares are $6,250.

$\text{ROE}=\frac{\text{NI (Net income)}}{\text{Market value}}$

Compute the ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$6,760}{\$125,000}\\ =0.05408\text{ or 5}\text{.41}\%\end{array}$

Hence, the ROE during recession period is 5.41%.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$10,400}{\$125,000}\\ =0.0832\text{ or 8}\text{.32}\%\end{array}$

Hence, the ROE during normal period is 8.32%.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$12,480}{\$125,000}\\ =0.0998\text{ or 9}\text{.98}\%\end{array}$

Hence, the ROE during expansion period is 9.98%.

Formula to calculate the percentage change in ROE:

$\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100$

Compute the percentage change in ROE for recession period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{0.0540-0.0832}{0.0832}\times 100\\ =-35.\text{ 096}\%\end{array}$

Hence, the percentage change in ROE for recession period is -$35.096%.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0998-\$0.0832}{\$0.0832}\times 100\\ =+19.95\end{array}$

Hence, the percentage change is ROE for expansion period is +19.95%.

Table showing the ROE for the three possible periods of economy under the present capital structure with no taxes:

	Recession	Normal	Expansion
ROE	5.40%	8.32%	9.98%
%ΔROE	–35.096%	0	19.95%

b)

Summary Introduction

To calculate: The return on equity for the three economic scenarios before any issue of debt and compute the percentage changes in return on equity assuming that the company goes through a proposed recapitalization.

Introduction:

The ROE (Return on equity) is a profitability measure that computes the amount of dollar a firm creates with every dollar of shareholders equity.

Explanation of Solution

Given information:

The company is considering the debt issue of $42,000 with the rate of interest at 6%. At present, the outstanding shares of $6,250 exist.

Formula to calculate the share price:

$\text{Share price}=\frac{\text{Equity }}{\text{Shares outstanding}}$

Compute the share price:

$\begin{array}{c}\text{Share price}=\frac{\text{Equity }}{\text{Shares outstanding}}\\ =\frac{\$125,000}{6,250}\\ =\$20\end{array}$

Hence, the price of the share is $20.

Formula to calculate the repurchased shares:

$\text{Shares repurchased}=\frac{\text{Debt issued}}{\text{Share price}}$

Compute the repurchased shares:

$\begin{array}{c}\text{Shares repurchased}=\frac{\text{Debt issued}}{\text{Share price}}\\ =\frac{\$42,000}{\$20}\\ =\$2,100\end{array}$

Hence, the repurchased shares are $2,100.

Formula to calculate the payment of interest:

$\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}$

Compute the payment of interest:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$42,000\times 0.06\\ =\$2,520\end{array}$

Hence, the payment of interest is $2,520.

Table showing the income statement for the three possible periods of economy under the planned recapitalization:

	Recession	Normal	Expansion
EBIT	$6,760	$10,400	$12,480
Interest	2,520	2,520	2,520
NI	$4,240	$7,880	$9,960

Note:

• The net income is computed by subtracting the interest from the EBIT.

Formula to calculate the equity:

$\text{Equity}=\text{Market value}-\text{Debt amount}$

Compute the equity:

$\begin{array}{c}\text{Equity}=\text{Market value}-\text{Debt amount}\\ =\$125,000-\$42,000\\ =\$83,000\end{array}$

Hence, the equity is $83,000.

Formula to calculate the ROE:

$\text{ROE}=\frac{\text{NI (Net income)}}{\text{Equity}}$

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$4,240}{\$83,000}\\ =0.05108\text{ or5}\text{.11}\%\end{array}$

Hence, the ROE during recession period is 0.05108.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$7,880}{\$83,000}\\ =0.0949\text{ or 9}\text{.49}\%\end{array}$

Hence, the ROE during normal period is 0.0949.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$9,960}{\$83,000}\\ =0.12\text{or 12\%}\end{array}$

Hence, the ROE during expansion period is 0.12.

Formula to calculate the percentage change in ROE:

$\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100$

Compute the percentage change in ROE for recession period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0510-\$0.0949}{\$0.0949}\times 100\\ =-46.25\%\end{array}$

Hence, the percentage change in ROE for recession period is -$46.25.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.12-\$0.0949}{\$0.0949}\times 100\\ =+26.45\end{array}$

Hence, the percentage change in ROE for expansion period is +26.45.

Table showing the ROE and the percentage changes in ROE for the three possible periods of economy under the present capital structure with no taxes:

	Recession	Normal	Expansion
ROE	5.10%	9.49%	12%
%ΔROE	–46.25%	0	+26.45%

c)

Summary Introduction

To calculate: The return on equity for the three economic scenarios before any issue of debt and compute the percentage changes in return on equity with the rate of tax at 35%.

Introduction:

The ROE (Return on equity) is a profitability measure that computes the amount of dollar a firm creates with every dollar of shareholders equity.

Explanation of Solution

If a firm maintains its present capital structure with the corporate taxes, then the ROE is as follows:

Formula to calculate the taxes:

$\text{Taxes}=\text{EBIT}\times \text{Tax rate}$

Compute the taxes for three periods:

$\begin{array}{c}\text{Tax during recession}=\text{EBIT}\times \text{Tax rate}\\ =\$6,760\times 0.35\\ =\$2,366\end{array}$

Hence, the tax during recession is $2,366.

$\begin{array}{c}\text{Tax during normal}=\text{EBIT}\times \text{Tax rate}\\ =\$10,400\times 0.35\\ =\$3,640\end{array}$

Hence, the tax during normal period is $3,640.

$\begin{array}{c}\text{Tax during expansion}=\text{EBIT}\times \text{Tax rate}\\ =\$12,480\times 0.35\\ =\$4,368\end{array}$

Hence, the tax during expansion is $4,368.

Formula to calculate the NI (Net Income):

$\text{NI}=\text{EBIT}-\text{Taxes}$

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession period}=\text{EBIT}-\text{Taxes}\\ =\$6,760-\$2,366\\ =\$4,394\end{array}$

Hence, the net income during recession is $4,394.

$\begin{array}{c}\text{NI during normal period}=\text{EBIT}-\text{Taxes}\\ =\$10400-3640\\ =\$6,760\end{array}$

Hence, the net income during normal period is $6,760.

$\begin{array}{c}\text{NI during expansion period}=\text{EBIT}-\text{Taxes}\\ =\$12,480-4,368\\ =\$8,112\end{array}$

Hence, the net income during expansion period is $8,112.

Table showing the income statement for the three possible periods of economy with the EPS and percentage change in EPS:

	Recession	Normal	Expansion
EBIT	$6,760	$10,400	$12,480
Interest	0	0	0
Taxes	2,366	3,640	4,368
NI	$4,394	$6,760	$8,112

Note:

• The net income is computed by subtracting the interest and taxes from the EBIT.

Formula to calculate the ROE:

$\text{ROE}=\frac{\text{NI (Net income)}}{\text{Market value}}$

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$4,394}{\$125,000}\\ =3.51\%\end{array}$

Hence, the ROE during recession period is 3.51%.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$6,760}{\$125,000}\\ =5.41\%\end{array}$

Hence, the ROE during recession period is 5.41%.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$8,112}{\$125,000}\\ =6.49\%\end{array}$

Hence, the ROE during expansion period is 6.49%.

Formula to calculate the percentage change in ROE:

$\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100$

Compute the percentage change in ROE for recession period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0315-\$0.0541}{\$0.0541}\times 100\\ =-41.77\end{array}$

Hence, the percentage change in ROE for recession period is -$41.77.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0649-\$0.0541}{\$0.0541}\times 100\\ =+19.96\end{array}$

Hence, the percentage change is ROE for expansion period is +19.96.

Table showing the ROE and the percentage changes in ROE for the three possible periods of economy under the present capital structure with corporate taxes:

	Recession	Normal	Expansion
ROE	3.15%	5.41%	6.49%
%ΔROE	–41.77%	0	19.96%

If a firm undertakes the planned recapitalization with the corporate taxes, then the ROE is as follows:

Formula to calculate the payment of interest:

$\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}$

Compute the payment of interest:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$42,000\times 0.06\\ =\$2,520\end{array}$

Hence, the payment of interest is $2,520.

Formula to calculate the taxes:

$\text{Taxes}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}$

Compute the taxes for the three periods:

$\begin{array}{c}\text{Tax during recession period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(6,760-2,520\right)\times 0.35\\ =\$1,484\end{array}$

Hence, the tax during recession period is $1,484.

$\begin{array}{c}\text{Tax during normal period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(10,400-2,520\right)\times 0.35\\ =\$2,758\end{array}$

Hence, the tax during normal period is $2,758.

$\begin{array}{c}\text{Tax during expansion period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(12,480-2,520\right)\times 0.35\\ =\$3,486\end{array}$

Hence, the tax during expansion period is $3,486.

Formula to calculate the NI (Net Income):

$\text{NI}=\text{EBIT}-\text{Taxes}$

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$6,760-\$2,520-\$1,484\\ =\$2,756\end{array}$

Hence, the net income during recession is $2,756.

$\begin{array}{c}\text{NI during normal}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$10400-\$2,520-\$2,758\\ =\$5,122\end{array}$

Hence, the net income during normal period is $5,122.

$\begin{array}{c}\text{NI during expansion}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$12,480-\$2,520-\$3,486\\ =\$6,474\end{array}$

Hence, the net income during expansion period is $6,474

Table showing the income statement for the three possible periods of economy under the planned recapitalization:

	Recession	Normal	Expansion
EBIT	$6,760	$10,400	$12,480
Interest	$2,385	$2,385	$2,385
Taxes	$1,484	$2,758	$3,486
NI	$2,756	$5,122	$6,474

Formula to calculate the ROE:

$\text{ROE}=\frac{\text{NI (Net income)}}{\text{Equity}}$

Compute the ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$2,756}{\$83,000}\\ =3.32\%\end{array}$

Hence, the ROE during recession period is 3.32%.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$5,122}{\$83,000}\\ =6.17\%\end{array}$

Hence, the ROE during normal period is 6.17%.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$6,474}{\$83,000}\\ =7.8\%\end{array}$

Hence, the ROE during expansion period is 7.8%.

Formula to calculate the percentage change in ROE:

$\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100$

Compute the percentage change in ROE for recession period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0332-\$0.0617}{\$0.0617}\times 100\\ =-46.19\%\end{array}$

Hence, the percentage change in ROE for recession period is -$46.19%.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.078-\$0.0617}{\$0.0617}\times 100\\ =+26.42\end{array}$

Hence, the percentage change is ROE for expansion period is +26.42.

Table showing the ROE and the percentage changes in ROE for the three possible periods of economy under the present capital structure with corporate taxes:

	Recession	Normal	Expansion
ROE	3.32%	6.17%	7.8%
%ΔROE	–46.19%	0	+26.41%