Chapter 16, 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 the profitability measure that computes the amount of dollar a firm creates with every dollar of shareholders equity.

Explanation of Solution

Given information:

Company R has no outstanding debt and its market value is $165,000. The EBIT (Earnings before Interest and Taxes) are expected to be $21,000 at normal economic conditions. If the economy condition is strong, then the EBIT will increase to 25% and if the economy enters into a recession, then it will decrease to 35%. The company has market-to-book value ratio of 1.0%.

Formula to calculate the ROE:

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

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$13,650}{\$165,000}\\ =\$0.0827\end{array}$

Hence, the ROE during recession period is 0.0827.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$21,000}{\$165,000}\\ =\$0.1273\end{array}$

Hence, the ROE during normal period is 0.1273.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{\$26,250}{\$165,000}\\ =\$0.1591\end{array}$

Hence, the ROE during expansion period is 0.1591.

Formula to calculate the percentage change in ROE:

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

Compute the percentage change in ROE for recession period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE @ normal period}}\times 100\\ =\frac{\$0.0827-\$0.1273}{\$0.1273}\times 100\\ =-35\end{array}$

Hence, the percentage change in ROE for recession period is -$35.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE @ normal period}}\times 100\\ =\frac{\$0.1591-\$0.1273}{\$0.1273}\times 100\\ =+25\end{array}$

Hence, the percentage change is ROE for expansion period is +25.

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

	Recession	Normal	Expansion
ROE	0.0827	0.1273	0.1591
%ΔROE	–35	0	25

b)

Summary Introduction

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

Introduction:

The ROE (Return on equity) is the 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 $60,000 with the rate of interest @7%. At present, the outstanding shares of $5,500 exist.

Formula to calculate the share price:

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

Compute the share price:

$\begin{array}{c}\text{Share price}=\frac{\text{Equity }}{\text{Shares outstanding}}\\ =\frac{\$165,000}{5,500}\\ =\$30\end{array}$

Hence, the price of the share is $30.

Formula to calculate the repurchased shares:

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

Compute the repurchased shares:

$\begin{array}{c}\text{Shares repurchased}=\frac{\text{Debt issued}}{\text{Share price}}\\ =\frac{\$60,000}{\$30}\\ =\$2,000\end{array}$

Hence, the repurchased shares are $2,000.

Formula to calculate the payment of interest:

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

Compute the payment of interest:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$60,000\times 0.07\\ =\$4,200\end{array}$

Hence, the payment of interest is $4,200.

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

	Recession	Normal	Expansion
EBIT	$13,650	$21,000	$26,250
Interest	4,200	4,200	4,200
NI	$9,450	$16,800	$22,050

Note:

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

Formula to calculate equity:

$\begin{array}{c}\text{Equity}=\text{Net income}-\text{Debt amount}\\ =\$165,000-\$60,000\\ =\$105,000\end{array}$

Compute the equity:

$\begin{array}{c}\text{Equity}=\text{Net income}-\text{Debt amount}\\ =\$165,000-\$60,000\\ =\$105,000\end{array}$

Hence, the equity is $105,000.

Formula to calculate the ROE:

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

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$9,450}{\$105,000}\\ =0.09\end{array}$

Hence, the ROE during recession period is 0.09.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$16,800}{\$105,000}\\ =0.16\end{array}$

Hence, the ROE during normal period is 0.16.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$22,050}{\$105,000}\\ =0.21\end{array}$

Hence, the ROE during expansion period is 0.21.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.09-\$0.16}{\$0.16}\times 100\\ =-43.75\end{array}$

Hence, the percentage change in ROE for recession period is -$43.75.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.21-\$0.16}{\$0.16}\times 100\\ =+31.25\end{array}$

Hence, the percentage change in ROE for expansion period is +31.25.

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	0.09	0.16	0.21
%ΔROE	–43.75	0	31.25

c)

Summary Introduction

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

Introduction:

The ROE (Return on equity) is the 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 taxes:

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

Compute taxes for three periods:

$\begin{array}{c}\text{Tax during recession}=\text{EBIT}\times \text{Tax rate}\\ =\$13,650\times 0.35\\ =\$4,778\end{array}$

Hence, the tax during recession is $4,778.

$\begin{array}{c}\text{Tax during normal}=\text{EBIT}\times \text{Tax rate}\\ =\$21,000\times 0.35\\ =\$7,350\end{array}$

Hence, the tax during normal period is $7,350.

$\begin{array}{c}\text{Tax during normal}=\text{EBIT}\times \text{Tax rate}\\ =\$26,250\times 0.35\\ =\$9,188\end{array}$

Hence, the tax during expansion is $9,188.

Formula to calculate the NI (Net Income):

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

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession period}=\text{EBIT}-\text{Taxes}\\ =\$13,650-4,778\\ =\$8,872\end{array}$

Hence, the net income during recession is $8,872.

$\begin{array}{c}\text{NI during normal period}=\text{EBIT}-\text{Taxes}\\ =\$21,000-7,350\\ =\$13,650\end{array}$

Hence, the net income during normal period is $13,650.

$\begin{array}{c}\text{NI during expansion period}=\text{EBIT}-\text{Taxes}\\ =\$26,250-9,187.5\\ =\$17,063\end{array}$

Hence, the net income during expansion period is $17,063

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

	Recession	Normal	Expansion
EBIT	$13,650	$21,000	$26,250
Interest	0	0	0
Taxes	4,778	7,350	9,188
NI	$ 8,873	$13,650	$17,063

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:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$8,873}{\$165,000}\\ =0.0538\end{array}$

Hence, the ROE during recession period is 0.0538.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$13,650}{\$165,000}\\ =0.827\end{array}$

Hence, the ROE during recession period is 0.827.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$17,063}{\$165,000}\\ =0.1034\end{array}$

Hence, the ROE during expansion period is 0.1034.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0538-\$0.0827}{\$0.0827}\times 100\\ =-35\end{array}$

Hence, the percentage change in ROE for recession period is -$35.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.1034-\$0.0827}{\$0.0827}\times 100\\ =+25\end{array}$

Hence, the percentage change is ROE for expansion period is +25.

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	0.0538	0.0827	0.1034
%ΔROE	–35	0	25

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:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$60,000\times 0.07\\ =\$4,200\end{array}$

Hence, the payment of interest is $4,200.

Formula to calculate taxes:

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

Compute taxes for the three periods:

$\begin{array}{c}\text{Tax during recession period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(13,650-4,200\right)\times 0.35\\ =\$3,307.5\end{array}$

Hence, the tax during recession is $3,307.5.

$\begin{array}{c}\text{Tax during normal period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(21,000-4,200\right)\times 0.35\\ =\$5,880\end{array}$

Hence, the tax during normal period is $5,880.

$\begin{array}{c}\text{Tax during expansion period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(26,250-4,200\right)\times 0.35\\ =\$7,718\end{array}$

Hence, the tax during expansion period is $7,718.

Formula to calculate the NI (Net Income):

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

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$13,650-4,200-3,307.5\\ =\$6,143\end{array}$

Hence, the net income during recession is $6,143.

$\begin{array}{c}\text{NI during recession}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$21,000-4,200-5,880\\ =\$10,920\end{array}$

Hence, the net income during normal period is $10,920.

$\begin{array}{c}\text{NI during recession}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$26,250-4,200-7,717.5\\ =\$14,333\end{array}$

Hence, the net income during expansion period is $14,333.

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

	Recession	Normal	Expansion
EBIT	$13,650	$21,000	$26,250
Interest	4,200	4,200	4,200
Taxes	3,308	5,880	7,718
NI	$6,143	$10,920	$14,333

Formula to calculate the ROE:

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

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$6,143}{\$105,000}\\ =0.0585\end{array}$

Hence, the ROE during recession period is 0.0585.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$10,920}{\$105,000}\\ =0.1040\end{array}$

Hence, the ROE during normal period is 0.1040.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$14,333}{\$105,000}\\ =0.1365\end{array}$

Hence, the ROE during expansion period is 0.1365.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0585-\$0.1040}{\$0.1040}\times 100\\ =-43.75\end{array}$

Hence, the percentage change in ROE for recession period is -$43.75.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE @ normal period}}\times 100\\ =\frac{\$0.1365-\$0.1040}{\$0.1040}\times 100\\ =+31.25\end{array}$

Hence, the percentage change is ROE for expansion period is +31.25.

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	0.0585	0.104	0.1365
%ΔROE	–43.75	0	+31.25