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 dollars a firm creates with every dollar of a shareholder’s equity.

Explanation of Solution

Given information:

Company P has no debt outstanding and its market value is $180,000. The EBIT (earnings before interest and taxes) are expected to be $23,000 at normal economic conditions. If the economy condition is strong, then EBIT will increase to 20% and if the economy enters into a recession, then it will decrease to 30%. The company has the 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:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$16,100}{\$180,000}\\ =\$0.0894\end{array}$

Hence, the ROE during recession period is 0.0894.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$23,000}{\$180,000}\\ =\$0.1278\end{array}$

Hence, the ROE during normal period is 0.1278.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$27,600}{\$180,000}\\ =\$0.1533\end{array}$

Hence, the ROE during expansion period is 0.1533.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0894-\$0.1278}{\$0.1278}\times 100\\ =-30\end{array}$

Hence, the percentage change in ROE for recession period is -$30.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.1533-\$0.1278}{\$0.1278}\times 100\\ =+20\end{array}$

Hence, the percentage change is ROE for expansion period is +20.

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

	Recession	Normal	Expansion
ROE	0.0894	0.1278	0.1533
%ΔROE	–30	0	20

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 ROE, assuming that the company goes through a proposed recapitalization.

Introduction:

The ROE (return on equity) is the profitability measure that computes the amount of dollars a firm creates with every dollar of a shareholder’s equity.

Explanation of Solution

Given information:

The company is considering the debt issue of $75,000 with the rate of interest at7%. At present, the outstanding shares of $6,000 exist.

Formula to calculate the share price:

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

Compute the share price:

$\begin{array}{c}\text{Share price}=\frac{\text{Equity }}{\text{Shares outstanding}}\\ =\frac{\$180,000}{6,000}\\ =\$30\end{array}$

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:

$\begin{array}{c}\text{Shares repurchased}=\frac{\text{Debt issued}}{\text{Share price}}\\ =\frac{\$75,000}{\$30}\\ =\$2,500\end{array}$

Hence, the repurchased shares are $2,500.

Formula to calculate the payment of interest:

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

Compute the payment of interest:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$75,000\times 0.07\\ =\$5,250\end{array}$

Hence, the payment of interest is $5,250.

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

	Recession	Normal	Expansion
EBIT	$16,100	$23,000	$27,600
Interest	5,250	5,250	5,250
NI	$10,850	$17,750	$22,350

Note:

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

Formula to calculate equity:

$\text{Equity}=\text{Net income}-\text{Debt amount}$

Compute the equity:

$\begin{array}{c}\text{Equity}=\text{Net income}-\text{Debt amount}\\ =\$180,000-\$75,000\\ =\$105,000\end{array}$

Hence, the equity is $105,000.

Formula to calculate the ROE:

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

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$10,850}{\$105,000}\\ =0.10\end{array}$

Hence, the ROE during recession period is 0.10.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$23,000}{\$105,000}\\ =0.22\end{array}$

Hence, the ROE during normal period is 0.22.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$27,600}{\$105,000}\\ =0.26\end{array}$

Hence, the ROE during expansion period is 0.26.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.10-\$0.22}{\$0.22}\times 100\\ =-54.55\end{array}$

Hence, the percentage change in ROE for recession period is -$54.55.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.26-\$0.22}{\$0.22}\times 100\\ =+18.18\end{array}$

Hence, the percentage change in ROE for expansion period is +18.18.

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	0.10	0.22	0.26
%ΔROE	–54.55	0	+18.18

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 at35%.

Introduction:

The ROE (return on equity) is the profitability measure that computes the amount of dollars a firm creates with every dollar of a shareholder’s 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:

$\begin{array}{c}\text{Tax during recession}=\text{EBIT}\times \text{Tax rate}\\ =\$16,100\times 0.35\\ =\$5,635\end{array}$

Hence, the tax during recession is $5,635.

$\begin{array}{c}\text{Tax during normal}=\text{EBIT}\times \text{Tax rate}\\ =\$23,000\times 0.35\\ =\$8,050\end{array}$

Hence, the tax during normal period is $8,050.

$\begin{array}{c}\text{Tax during normal}=\text{EBIT}\times \text{Tax rate}\\ =\$27,600\times 0.35\\ =\$9,660\end{array}$

Hence, the tax during expansion is $9,660.

Formula to calculate the NI:

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

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession period}=\text{EBIT}-\text{Taxes}\\ =\$16,100-5,635\\ =\$10,465\end{array}$

Hence, the net income during recession is $10,465.

$\begin{array}{c}\text{NI during normal period}=\text{EBIT}-\text{Taxes}\\ =\$23,000-8,050\\ =\$14,950\end{array}$

Hence, the net income during normal period is $14,950.

$\begin{array}{c}\text{NI during expansion period}=\text{EBIT}-\text{Taxes}\\ =\$27,600-9,660\\ =\$17,940\end{array}$

Hence, the net income during expansion period is $17,940.

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

	Recession	Normal	Expansion
EBIT	$16,100	$23,000	$27,600
Interest	0	0	0
Taxes	5,635	8,050	9,660
NI	$10,465	$14,950	$17,940

Note:

• The NI 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:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$16,100}{\$180,000}\\ =0.0894\end{array}$

Hence, the ROE during recession period is 0.0894.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$23,000}{\$180,000}\\ =0.1278\end{array}$

Hence, the ROE during recession period is 0.1278.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$27,600}{\$180,000}\\ =0.1533\end{array}$

Hence, the ROE during expansion period is 0.1533.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0894-\$0.1278}{\$0.1278}\times 100\\ =-30\end{array}$

Hence, the percentage change in ROE for recession period is -$30.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.1533-\$0.1278}{\$0.1278}\times 100\\ =+20\end{array}$

Hence, the percentage change is ROE for expansion period is +20.

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	0.0894	0.1278	0.1533
%ΔROE	–30	0	+20

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:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$75,000\times 0.07\\ =\$5,250\end{array}$

Hence, the payment of interest is $5,250.

Formula to calculate taxes:

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

Compute taxes for the three periods:

$\begin{array}{c}\text{Tax during recession period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(16,100-5,250\right)\times 0.35\\ =\$3,797.5\end{array}$

Hence, the tax during recession is $3,797.5.

$\begin{array}{c}\text{Tax during normal period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(23,000-5,250\right)\times 0.35\\ =\$6,212.5\end{array}$

Hence, the tax during normal period is $6,212.5.

$\begin{array}{c}\text{Tax during expansion period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(27,600-5,250\right)\times 0.35\\ =\$7,822.5\end{array}$

Hence, the tax during expansion period is $7,822.5.

Formula to calculate the NI:

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

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$16,100-5,250-3,797.5\\ =\$7,052.5\end{array}$

Hence, the NI during recession is $6,143.

$\begin{array}{c}\text{NI during normal}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$23,000-5,250-6,212.5\\ =\$11,537.5\end{array}$

Hence, the NI during normal period is $11,537.5.

$\begin{array}{c}\text{NI during recession}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$27,600-5,250-7,822.5\\ =\$14,527.5\end{array}$

Hence, the NI during expansion period is $14,527.5.

Formula to calculate EPS:

$\text{EPS}=\frac{\text{NI (Net income)}}{\text{Outstanding shares}}$

Compute EPS:

$\begin{array}{c}\text{EPS at recession period}=\frac{\text{Net income}}{\text{Outstanding shares}}\\ =\frac{\$7,052.5}{\$3,500}\\ =\$2.02\end{array}$

Hence, the EPS at recession period is $2.02.

$\begin{array}{c}\text{EPS at normal period}=\frac{\text{Net income}}{\text{Outstanding shares}}\\ =\frac{\$11,537.5}{\$3,500}\\ =\$3.30\end{array}$

Hence, the EPS at normal period is $3.30.

$\begin{array}{c}\text{EPS at expansion period}=\frac{\text{Net income}}{\text{Outstanding shares}}\\ =\frac{\$14,527.5}{\$3,500}\\ =\$4.15\end{array}$

Hence, the EPS at expansion period is $4.15.

Note: After recapitalization, $2,500 was recovered from the total outstanding shares of $6,000. Now, the shares outstanding is $6,000-$2,500=$3,500.

Formula to calculate the percentage change in EPS:

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

Compute the percentage change in EPS for recession period:

$\begin{array}{c}\text{Percentage change in EPS}=\frac{\text{Changes in EPS}}{\text{EPS at normal period}}\times 100\\ =\frac{\$2.02-\$3.30}{\$3.30}\times 100\\ =-\$38.79\end{array}$

Hence, the percentage change in EPS for recession period is -$38.79.

Compute the percentage change in EPS for expansion period:

$\begin{array}{c}\text{Percentage change in EPS}=\frac{\text{Changes in EPS}}{\text{EPS at normal period}}\times 100\\ =\frac{\$4.15-\$3.30}{\$3.30}\times 100\\ =+25.76\end{array}$

Hence, the percentage change is EPS for expansion period is 25.76.

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

	Recession	Normal	Expansion
EBIT	$16,100	$23,000	$27,600
Interest	5,250	5,250	5,250
Taxes	3,797.5	6,212.5	7,822.5
NI	$7,052.5	$11,537.5	$14,527.5
EPS	$2.02	$3.30	$4.15
%EPS	–38.79	NIL	+25.76

Formula to calculate the ROE:

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

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$7,052.5}{\$105,000}\\ =0.0672\end{array}$

Hence, the ROE during recession period is 0.0672.

$\begin{array}{c}\text{ROE}\text{at}\text{normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$11,537.5}{\$105,000}\\ =0.1099\end{array}$

Hence, the ROE during normal period is 0.1099.

$\begin{array}{c}\text{ROE at}\text{expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$14,527.5}{\$105,000}\\ =0.1384\end{array}$

Hence, the ROE during expansion period is 0.1384.

Formula to calculate the percentage change in ROE:

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

Compute the percentage change in ROE for recession period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0672-\$0.1099}{\$0.1099}\times 100\\ =-38.85\end{array}$

Hence, the percentage change in ROE for recession period is -$38.85.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.1384-\$0.1099}{\$0.1099}\times 100\\ =+25.93\end{array}$

Hence, the percentage change is ROE for expansion period is +25.93.

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	0.0672	0.1099	0.1384
%ΔROE	-$38.85	0	+25.93