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 debt outstanding and its market value is $194,775. The EBIT (Earnings before interest and taxes) are expected to be $13,800 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 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 the ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$8,970}{\$194,775}\\ =0.0461\text{ or }4.61\%\end{array}$

Hence, the ROE during recession period is 4.61.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$13,800}{\$194,775}\\ =0.0709\text{ or }7.09\%\end{array}$

Hence, the ROE during normal period is 7.09%.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$16,560}{\$194,775}\\ =0.0850\text{ or }8.50\%\end{array}$

Hence, the ROE during expansion period is 8.50%.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{0.0461-0.0709}{0.0709}\times 100\\ =-35\%\end{array}$

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

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0850-\$0.0709}{\$0.0709}\times 100\\ =+20\end{array}$

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	4.61%	7.09%	8.50%
%ΔROE	–35%	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 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 $39,750 with the rate of interest @6%. At present, the outstanding shares of $7,350 exist.

Formula to calculate the share price:

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

Compute the share price:

$\begin{array}{c}\text{Share price}=\frac{\text{Equity }}{\text{Shares outstanding}}\\ =\frac{\$194,775}{7,350}\\ =\$26.5\end{array}$

Hence, the price of the share is $26.5.

Formula to calculate the repurchased shares:

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

Compute the repurchased shares:

$\begin{array}{c}\text{Shares repurchased}=\frac{\text{Debt issued}}{\text{Share price}}\\ =\frac{\$39,750}{\$26.5}\\ =\$1,500\end{array}$

Hence, the repurchased shares are $1,500.

Formula to calculate the payment of interest:

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

Compute the payment of interest:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$39,750\times 0.06\\ =\$2,385\end{array}$

Hence, the payment of interest is $2,385.

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

	Recession	Normal	Expansion
EBIT	$8,970	$13,800	$16,560
Interest	2,385	2,385	2,385
NI	$6,585	$11,415	$14,175

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:

$\begin{array}{c}\text{Equity}=\text{Market value}-\text{Debt amount}\\ =\$194,775-\$39,750\\ =\$155,025\end{array}$

Hence, the equity is $155,025.

Formula to calculate the ROE:

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

Compute ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$6,585}{\$155,025}\\ =0.0425\text{ or}4.25\%\end{array}$

Hence, the ROE during recession period is 0.0425.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$11,415}{\$155,025}\\ =0.0736\text{ or }7.36\%\end{array}$

Hence, the ROE during normal period is 0.0736.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$14,175}{\$155,025}\\ =0.0914\text{or 9}\text{.14\%}\end{array}$

Hence, the ROE during expansion period is 0.0914.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0425-\$0.0736}{\$0.0736}\times 100\\ =-42.31\%\end{array}$

Hence, the percentage change in ROE for recession period is -$42.31.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.0914-\$0.0736}{\$0.0736}\times 100\\ =+24.18\end{array}$

Hence, the percentage change in ROE for expansion period is +24.18.

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	4.25%	7.36%	9.14%
%ΔROE	–42.31%	0	+24.18%

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 @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

Given information:

Company K has no debt outstanding and its market value is $194,775. The EBIT (Earnings before interest and taxes) are expected to be $13,800 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 company has the market to book value ratio of 1.0%.

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:

$\begin{array}{c}\text{Tax during recession}=\text{EBIT}\times \text{Tax rate}\\ =\$8,970\times 0.35\\ =\$3,139.5\end{array}$

Hence, the tax during recession is $3,139.5.

$\begin{array}{c}\text{Tax during normal}=\text{EBIT}\times \text{Tax rate}\\ =\$13,800\times 0.35\\ =\$4,830\end{array}$

Hence, the tax during normal period is $4,830.

$\begin{array}{c}\text{Tax during expansion}=\text{EBIT}\times \text{Tax rate}\\ =\$16,560\times 0.35\\ =\$5,796\end{array}$

Hence, the tax during expansion is $5,796.

Formula to calculate the NI (Net Income):

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

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession period}=\text{EBIT}-\text{Taxes}\\ =\$8,970-\$3,139.5\\ =\$5,830.5\end{array}$

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

$\begin{array}{c}\text{NI during normal period}=\text{EBIT}-\text{Taxes}\\ =\$13,800-4,830\\ =\$8,970\end{array}$

Hence, the net income during normal period is $8,970.

$\begin{array}{c}\text{NI during expansion period}=\text{EBIT}-\text{Taxes}\\ =\$16,560-5,796\\ =\$10,764\end{array}$

Hence, the net income during expansion period is $10,764.

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

	Recession	Normal	Expansion
EBIT	$8,970	$13,800	$7,590
Interest	0	0	0
Taxes	3,139.5	4,830	5,796
NI	$5,830.5	$8,970	$10,764

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:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$5,830.5}{\$194,775}\\ =2.99\%\end{array}$

Hence, the ROE during recession period is 2.99%.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$8,970}{\$194,775}\\ =4.61\%\end{array}$

Hence, the ROE during recession period is 4.61%.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Market value}}\\ =\frac{\$10,764}{\$194,775}\\ =5.53\%\end{array}$

Hence, the ROE during expansion period is 5.53%.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.029934539-\$0.046053138}{\$0.046053138}\times 100\\ =-35\end{array}$

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

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.055263765-\$0.046053138}{\$0.046053138}\times 100\\ =+20\end{array}$

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	2.99%	4.61%	5.53%
%ΔROE	–35%	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:

$\begin{array}{c}\text{Interest payment}=\text{Debt issued}\times \text{ Rate of interest}\\ =\$39,750\times 0.06\\ =\$2,385\end{array}$

Hence, the payment of interest is $2,385.

Formula to calculate the taxes:

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

Compute the taxes for the three periods:

$\begin{array}{c}\text{Tax during recession period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(8,970-2,385\right)\times 0.35\\ =\$2,304.75\end{array}$

Hence, the tax during recession period is $2,304.75.

$\begin{array}{c}\text{Tax during normal period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(13,800-2,385\right)\times 0.35\\ =\$3,995.25\end{array}$

Hence, the tax during normal period is $3,995.25.

$\begin{array}{c}\text{Tax during expansion period}=\left(\text{EBIT}-\text{Interest}\right)\times \text{Tax rate}\\ =\left(16,560-2,385\right)\times 0.35\\ =\$4,961.25\end{array}$

Hence, the tax during expansion period is $4,961.25.

Formula to calculate the NI (Net Income):

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

Compute NI for three periods:

$\begin{array}{c}\text{NI during recession}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$8,970-\$2,385-\$2,304.75\\ =\$4,280.25\end{array}$

Hence, the net income during recession is $4,280.25.

$\begin{array}{c}\text{NI during normal}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$13,800-\$2,385-\$3,995.25\\ =\$7,419.75\end{array}$

Hence, the net income during normal period is $7,419.75.

$\begin{array}{c}\text{NI during expansion}=\text{EBIT}-\text{Interest}-\text{Taxes}\\ =\$16,560-\$2,385-\$4,961.25\\ =\$9,213.75\end{array}$

Hence, the net income during expansion period is $9,213.75

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

	Recession	Normal	Expansion
EBIT	$8,970	$13,800	$16,560
Interest	$2,385	$2,385	$2,385
Taxes	$2,304.75	$3,995.25	$4,961.25
NI	$4,280.25	$7,419.75	$9,213.75

Formula to calculate the ROE:

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

Compute the ROE:

$\begin{array}{c}\text{ROE at recession period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$4,280.25}{\$155,025}\\ =2.76\%\end{array}$

Hence, the ROE during recession period is 2.76%.

$\begin{array}{c}\text{ROE at normal period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$7,419.75}{\$155,025}\\ =4.79\%\end{array}$

Hence, the ROE during normal period is 4.79%.

$\begin{array}{c}\text{ROE at expansion period}=\frac{\text{NI (Net income)}}{\text{Equity}}\\ =\frac{\$9,213.75}{\$155,025}\\ =5.94\%\end{array}$

Hence, the ROE during expansion period is 5.94%.

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:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE at normal period}}\times 100\\ =\frac{\$0.027610062-\$0.047861635}{\$0.047861635}\times 100\\ =-42.31\%\end{array}$

Hence, the percentage change in ROE for recession period is -$42.31%.

Compute the percentage change in ROE for expansion period:

$\begin{array}{c}\text{Percentage change in ROE}=\frac{\text{Changes in ROE}}{\text{ROE @ normal period}}\times 100\\ =\frac{\$0.059433962-\$0.047861635}{\$0.047861635}\times 100\\ =+24.18\end{array}$

Hence, the percentage change is ROE for expansion period is +24.18.

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	2.76%	4.79%	5.94%
%ΔROE	–42.31%	0	+24.18%