Chapter 14, Problem 7P

Summary Introduction

To identify: The expected return on equity, standard deviation, and coefficient of variance from the given situation.

Introduction:

Return on Equity:

The return, which is generated on the equity that is invested by the stockholders is known as return on equity.

Explanation of Solution

The capital ratio at 0% and none interest rate.

Compute the expected return on equity.

State-1

Compute the net income.

Given,

The probability is 0.2.

The EBIT is $4,200,000.

Formula to calculate the net income,

$\text{Net}\text{income}=\left(\text{EBIT}-\text{I}\right)\times \left(1-\text{T}\right)$

Where,

• EBIT is earning before interest and tax.
• I is interest.
• T is tax rate.

Substitute $4,200,000 for EBIT, 0 for I and 0.40 for T.

$\begin{array}{c}\text{Net}\text{income}=\left(\$4,200,000-0\right)\times \left(1-0.40\right)\\ =\$4,200,000\times 0.60\\ =\$2,520,000\end{array}$

The net income of state 1 is $2,520,000.

Compute the return on equity of state 1.

The net income is $2,520,000. (Calculated above)

The equity is $14,000,000. (Given)

Formula to calculate the return on equity,

$\text{Return}\text{on}\text{equity}=\frac{\text{Net}\text{income}}{\text{Equity}}\times 100$

Substitute $2,520,000 for net income and $14,000,000 for equity.

$\begin{array}{c}\text{Return}\text{on}\text{equity}=\frac{\$2,520,000}{\$14,000,000}\times 100\\ =0.18\times 100\\ =18\%\end{array}$ (1)

The return on equity of state 1 is 18%.

State-2

Compute the net income.

Given,

The probability is 0.5.

The EBIT is $2,800,000.

Formula to calculate the net income,

$\text{Net}\text{income}=\left(\text{EBIT}-\text{I}\right)\times \left(1-\text{T}\right)$

Where,

• EBIT is earning before interest and tax.
• I is interest.
• T is tax rate.

Substitute $2,800,000 for EBIT, 0 for I and 0.40 for T.

$\begin{array}{c}\text{Net}\text{income}=\left(\$2,800,000-0\right)\times \left(1-0.40\right)\\ =\$2,800,000\times 0.60\\ =\$1,680,000\end{array}$

The net income of state 2 is $1,680,000.

Compute the return on equity of state 2.

The net income is $1,680,000. (Calculated above)

The equity is $14,000,000. (Given)

Formula to calculate the return on equity,

$\text{Return}\text{on}\text{equity}=\frac{\text{Net}\text{income}}{\text{Equity}}\times 100$

Substitute $1,680,000 for net income and $14,000,000 for equity.

$\begin{array}{c}\text{Return}\text{on}\text{equity}=\frac{\$1,680,000}{\$14,000,000}\times 100\\ =0.12\times 100\\ =12\%\end{array}$ (2)

The return on equity on state 2 is 12%.

State-3

Compute the net income.

Given,

The probability is 0.3.

The EBIT is $700,000.

Formula to calculate the net income,

$\text{Net}\text{income}=\left(\text{EBIT}-\text{I}\right)\times \left(1-\text{T}\right)$

Where,

• EBIT is earning before interest and tax.
• I is interest.
• T is tax rate.

Substitute $700,000 for EBIT, 0 for I and 0.40 for T.

$\begin{array}{c}\text{Net}\text{income}=\left(\$700,000-0\right)\times \left(1-0.40\right)\\ =\$700,000\times 0.60\\ =\$420,000\end{array}$

The net income of state 3 is $420,000.

Compute the return on equity of state 3.

The net income is $420,000. (Calculated above)

The equity is $14,000,000. (Given)

Formula to calculate the return on equity,

$\text{Return}\text{on}\text{equity}=\frac{\text{Net}\text{income}}{\text{Equity}}\times 100$

Substitute $420,000 for net income and $14,000,000 for equity.

$\begin{array}{c}\text{Return}\text{on}\text{equity}=\frac{\$420,000}{\$14,000,000}\times 100\\ =0.03\times 100\\ =3\%\end{array}$ (3)

The return on equity on state 3 is 3%.

Compute the expected return on equity of 3 states.

The return on equity of state 1 is18%. (Calculated in equation (1))

The return on equity of state 2 is 12%. (Calculated in equation (2))

The return on equity of state 3 is 3%. (Calculated in equation (3))

The probability of state 1 is 0.2. (Given)

The probability of state 2 is 0.5. (Given)

The probability of state 3 is 0.3. (Given)

Formula to calculate the expected return on earnings,

$\text{Expected}\text{return}\text{on}\text{earnings}={\displaystyle \sum \text{Probability}\times \text{Return}\text{on}\text{earnings}}$

Substitute 0.2, 0.5 and 0.3 for probability and 18%, 12% and 3% for return on earnings.

$\begin{array}{c}\text{Expected}\text{return}\text{on}\text{earnings}=\left(0.2\times 18\%\right)+\left(0.5\times 12\%\right)+\left(0.3\times 3\%\right)\\ =3.60\%+6\%+0.90\%\\ =10.50\%\end{array}$

The expected return on earnings is 10.50%.

Compute the standard deviation.

State	Probability	Return on Equity (ROE)	Expected Return on Equity (EROE)	Deviation $\left(\left(\text{ROE}-\text{EROE}\right)^2\right)\times \text{P}$
1	0.2	18%	10.50%	11.25%
2	0.5	12%	10.50%	1.125%
3	0.3	3%	10.50%	16.88%
	Variance			29.25%

Table (1)

The variance is 29.25%. (Calculated above)

Formula to calculate the standard deviation,

$\text{Standard}\text{deviation}=\sqrt{\text{Variance}}$

Substitute 29.25% for variance.

$\begin{array}{c}\text{Standard}\text{deviation}=\sqrt{29.25\%}\\ =5.408\%\end{array}$

The standard deviation is 5.408%.

Compute the coefficient of deviation.

The standard deviation is 5.408%.

The expected return on equity is 10.50%.

Formula to calculate the coefficient of variance,

$\text{Coefficient}\text{of}\text{variance}=\frac{\text{Standard}\text{deviation}}{\text{EROE}}$

Where,

• EORE is expected return on equity.

Substitute 5.408% for standard deviation and 10.25% for EROE.

$\begin{array}{c}\text{Coefficient}\text{of}\text{variance}=\frac{5.408\%}{10.50\%}\\ =0.515\end{array}$

The coefficient of variance is 0.515.

The capital ratio at 10% and interest rateis 9%.

Compute the expected return on equity

Statement to show the computation of return on equity of each state

Table (2)

Compute expected return on equity of 3 states.

The return on equity of state 1 is 19.40%. (Calculated above)

The return on equity of state 2 is 12.73%. (Calculated above)

The return on equity of state 3 is 2.73%. (Calculated above)

The probability of state 1 is 0.2. (Given)

The probability of state 2 is 0.5. (Given)

The probability of state 3 is 0.3. (Given)

Formula to calculate the expected return on earnings,

$\text{Expected}\text{return}\text{on}\text{earnings}={\displaystyle \sum \text{Probability}\times \text{Return}\text{on}\text{earnings}}$

Substitute 0.2, 0.5 and 0.3 for probability and 19.40%, 12.73% and 2.73% for return on earnings.

$\begin{array}{c}\text{Expected}\text{return}\text{on}\text{earnings}=\left(0.2\times 19.40\%\right)+\left(0.5\times 12.73\%\right)+\left(0.3\times 2.73\%\right)\\ =3.88\%+6.37\%+0.82\%\\ =11.07\%\end{array}$

The expected return on earnings is 11.07%.

Compute the standard deviation

State	Probability	Return on Equity (ROE)	Expected Return on Equity (EROE)	Deviation $\left(\left(\text{ROE}-\text{EROE}\right)^2\right)\times \text{P}$
1	0.2	19.40%	11.07%	13.88%
2	0.5	12.73%	11.07%	1.378%
3	0.3	2.73%	11.07%	20.87%
	Variance			36.128%

Table 3

The variance is 36.128%. (Calculated above)

Formula to calculate the standard deviation,

$\text{Standard}\text{deviation}=\sqrt{\text{Variance}}$

Substitute 36.128% for variance.

$\begin{array}{c}\text{Standard}\text{deviation}=\sqrt{36.128\%}\\ =6.01\%\end{array}$

The standard deviation is 6.01%.

Compute the coefficient of deviation.

The standard deviation is 6.01%.

The expected return on equity is 11.07%.

Formula to calculate the coefficient of variance,

$\text{Coefficient}\text{of}\text{variance}=\frac{\text{Standard}\text{deviation}}{\text{EROE}}$

Where,

• EORE is expected return on equity.

Substitute 6.01% for standard deviation and 11.07% for EROE.

$\begin{array}{c}\text{Coefficient}\text{of}\text{variance}=\frac{6.01\%}{11.07\%}\\ =0.543\end{array}$

The coefficient of variance is 0.543.

The capital ratio at 50% and none interest rate is 11%.

Compute the expected return on equity

Statement to show the computation of return on equity of each state

Table (4)

Compute expected return on equity of 3 states.

The return on equity of state 1 is 29.40%. (Calculated above)

The return on equity of state 2 is 17.40%. (Calculated above)

The return on equity of state 3 is $-0.60\%$. (Calculated above)

The probability of state 1 is 0.2. (Given)

The probability of state 2 is 0.5. (Given)

The probability of state 3 is 0.3. (Given)

Formula to calculate the expected return on earnings,

$\text{Expected}\text{return}\text{on}\text{earnings}={\displaystyle \sum \text{Probability}\times \text{Return}\text{on}\text{earnings}}$

Substitute 0.2, 0.5 and 0.3 for probability and 29.40%, 17.40% and $-0.60\%$ for return on earnings.

$\begin{array}{c}\text{Expected}\text{return}\text{on}\text{earnings}=\left(0.2\times 29.40\%\right)+\left(0.5\times 17.40\%\right)+\left(0.3\times -0.60\%\right)\\ =5.88\%+8.70\%-0.18\%\\ =14.40\%\end{array}$

The expected return on earnings is 14.40%.

Compute the standard deviation.

State	Probability	Return on Equity (ROE)	Expected Return on Equity (EROE)	Deviation $\left(\left(\text{ROE}-\text{EROE}\right)^2\right)\times \text{P}$
1	0.2	29.40%	14.40%	45%
2	0.5	17.40%	14.40%	4.5%
3	0.3	$-0.60\%$	14.40%	67.5%
	Variance			117%

Table (5)

The variance is 117%. (Calculated above)

Formula to calculate the standard deviation,

$\text{Standard}\text{deviation}=\sqrt{\text{Variance}}$

Substitute 117% for variance.

$\begin{array}{c}\text{Standard}\text{deviation}=\sqrt{117\%}\\ =10.82\%\end{array}$

The standard deviation is 10.82%.

Compute the coefficient of deviation.

The standard deviation is 10.82%.

The expected return on equity is 14.40%.

Formula to calculate the coefficient of variance,

$\text{Coefficient}\text{of}\text{variance}=\frac{\text{Standard}\text{deviation}}{\text{EROE}}$

Where,

• EORE is expected return on equity.

Substitute 10.82% for standard deviation and 14.40% for EROE.

$\begin{array}{c}\text{Coefficient}\text{of}\text{variance}=\frac{10.82\%}{14.40\%}\\ =0.751\end{array}$

The coefficient of variance is 0.751.

The capital ratio at 60% and none interest rate is 14%.

Compute the expected return on equity.

Statement to show the computation of return on equity of each state,

Table (6)

Computation of expected return on equity of 3 states

The return on equity of state 1 is 32.40%. (Calculated above)

The return on equity of state 2 is 17.40%. (Calculated above)

The return on equity of state 3 is $-5.10\%$. (Calculated above)

The probability of state 1 is 0.2. (Given)

The probability of state 2 is 0.5. (Given)

The probability of state 3 is 0.3. (Given)

Formula to calculate the expected return on earnings,

$\text{Expected}\text{return}\text{on}\text{earnings}={\displaystyle \sum \text{Probability}\times \text{Return}\text{on}\text{earnings}}$

Substitute 0.2, 0.5 and 0.3 for probability and 32.40%, 17.40% and $-5.10\%$ for return on earnings.

$\begin{array}{c}\text{Expected}\text{return}\text{on}\text{earnings}=\left(0.2\times 32.40\%\right)+\left(0.5\times 17.40\%\right)+\left(0.3\times -5.10\%\right)\\ =6.48\%+8.70\%-1.53\%\\ =13.65\%\end{array}$

The expected return on earnings is 13.65%.

Compute the standard deviation.

State	Probability	Return on Equity (ROE)	Expected Return on Equity (EROE)	Deviation $\left(\left(\text{ROE}-\text{EROE}\right)^2\right)\times \text{P}$
1	0.2	32.40%	13.65%	70.31%
2	0.5	17.40%	13.65%	7.031%
3	0.3	$-5.10\%$	13.65%	105.47%
	Variance			182.8%

Table (7)

The variance is 182.8%. (Calculated above)

Formula to calculate the standard deviation,

$\text{Standard}\text{deviation}=\sqrt{\text{Variance}}$

Substitute 182.8% for variance.

$\begin{array}{c}\text{Standard}\text{deviation}=\sqrt{182.8\%}\\ =13.521\%\end{array}$

The standard deviation is 13.521%.

Compute the coefficient of deviation.

The standard deviation is 13.521%.

The expected return on equity is 13.65%.

Formula to calculate the coefficient of variance,

$\text{Coefficient}\text{of}\text{variance}=\frac{\text{Standard}\text{deviation}}{\text{EROE}}$

Where,

• EORE is expected return on equity.

Substitute 13.521% for standard deviation and 13.65% for EROE.

$\begin{array}{c}\text{Coefficient}\text{of}\text{variance}=\frac{13.521\%}{13.65\%}\\ =0.99\end{array}$

The coefficient of variance is 0.99.

Working note:

Compute the value of debt and equity at capital ratio of 10% and interest rate of 9%.

Given,

Thetotal capital structure is 14 million.

The capital structure is 10.

Compute the debt value,

$\begin{array}{c}\text{Debt}=\frac{\text{Capital}\times 10}{100}\\ =\frac{\$14,000,000\times 10}{100}\\ =\$1,400,000\end{array}$

The debt is $1,400,000.

Compute the equity

The total capital is $14,000,000. (Given)

The debt is $1,400,000. (Calculated)

Compute the equity value,

$\begin{array}{c}\text{Equity}=\text{Total}\text{capital}-\text{Debt}\\ \text{=\$14,000,000-\$1,400,000}\\ \text{=\$12,600,000}\end{array}$

The equity is $12,600,000.

Compute the interest on debt.

The interest rate is 9% or 0.09. (Given)

The debt is $1,400,000. (Calculated)

Compute the interest on debt,

$\begin{array}{c}\text{Interest}=\text{Debt}\times \text{Interest}\text{rate}\\ =\$1,400,000\times 0.09\\ =\$126,000\end{array}$

The interest on debt is $126,000.

Compute the value of debt and equity at capital ratio of 50% and interest rate of 11%.

Given,

The total capital structure is 14 million.

The capital structure is 50.

Compute the debt value,

$\begin{array}{c}\text{Debt}=\frac{\text{Capital}\times 50}{100}\\ =\frac{\$14,000,000\times 50}{100}\\ =\$7,000,000\end{array}$

The debt is $7,000,000.

Compute the equity.

The total capital is $14,000,000. (Given)

The debt is $7,000,000. (Calculated)

Compute the equity value,

$\begin{array}{c}\text{Equity}=\text{Total}\text{capital}-\text{Debt}\\ =\$14,000,000-\$7,000,000\\ =\$7,000,000\end{array}$

The equity is $7,000,000.

Compute the interest on debt.

The interest rate is 11% or 0.11. (Given)

The debt is $7,000,000. (Calculated)

Compute the interest on debt,

$\begin{array}{c}\text{Interest}=\text{Debt}\times \text{Interest}\text{rate}\\ =\$7,000,000\times 0.11\\ =\$770,000\end{array}$

The interest on debt is $770,000.

Compute the value of debt and equity at capital ratio of 60% and interest rate of 14%.

Given,

The total capital structure is 14 million.

The capital structure is 60.

Compute the debt value,

$\begin{array}{c}\text{Debt}=\frac{\text{Capital}\times 60}{100}\\ =\frac{\$14,000,000\times 60}{100}\\ =\$8,400,000\end{array}$

The debt is $8,400,000.

Compute the equity.

The total capital is $14,000,000. (Given)

The debt is $8,400,000. (Calculated)

Compute the equity value,

$\begin{array}{c}\text{Equity}=\text{Total}\text{capital}-\text{Debt}\\ =\$14,000,000-\$8,400,000\\ =\$5,600,000\end{array}$

The equity is $5,600,000.

Compute the interest on debt.

The interest rate is 11% or 0.11. (Given)

The debt is $8,400,000. (Calculated)

Compute the interest on debt,

$\begin{array}{c}\text{Interest}=\text{Debt}\times \text{Interest}\text{rate}\\ =\$8,400,000\times 0.14\\ =\$1,176,000\end{array}$

The interest on debt is $1,176,000.

Conclusion

Hence, the ROE, standard deviation, and coefficient of varianceat 0% capital ratio are 10.50%, 5.408% and 0.515.

The ROE, standard deviation, and coefficient of variance at 10% capital ratio and 9% interest are 11.07%, 6.01% and 0.543.

The ROE, standard deviation, and coefficient of variance at 50% capital ratio and 11% interest are 14.40%, 10.82% and 0.751.

The ROE, standard deviation, and coefficient of variance at 60% capital ratio and 14% interest are 13.65%, 13.52% and 0.099.