Question

The Neal Company wants to estimate next year's return on equity (ROE) under different financial leverage ratios. Neal's total capital is $11 million, it currently uses only common equity, it has no future plans to use preferred stock in its capital structure, and its federal-plus-state tax rate is 40%. The CFO has estimated next year's EBIT for three possible states of the world: $4.5 million with a 0.2 probability, $3.4 million with a 0.5 probability, and $0.6 million with a 0.3 probability. Calculate Neal's expected ROE, standard deviation, and coefficient of variation for each of the following debt-to-capital ratios. Do not round intermediate calculations. Round your answers to two decimal places.

Debt/Capital ratio is 0.

RÔE:	%
σ:	%
CV:

Debt/Capital ratio is 10%, interest rate is 9%.

RÔE:	%
σ:	%
CV:

Debt/Capital ratio is 50%, interest rate is 11%.

RÔE:	%
σ:	%
CV:

Debt/Capital ratio is 60%, interest rate is 14%.

RÔE:	%
σ:	%
CV:

Answer 1

ROE is net income divided by common equity capital. Additions of Debt / Capital ratio, interest rate & tax rate are largely to convert the EBIT numbers to Net Income. So we will begin with that first.

Case 1: Debt / Capital ratio is 0% and Interest Rate hence not applicable as the company holds no debt.

EBIT = $4.5 Million

Here EBIT = EBT as there is no interest on debt

EBT= $4.5 Million

Net Income = EBT * (1-Tax Rate) = 4.5 Million *(1-40%) = 2.7 Million

Similarly calculate for the other 2 EBITs provided in the question i.e. of $ 3.4 Million & of $ 600K

Case 2: Debt / Capital ratio is 10% and Interest Rate is 9%.

EBIT = $4.5 Million

As debt to capital ratio is 10% hence Debt = Capital *10%. However as capital = debt + equity, therefore the 11 million invested in equity will be 90% of the invested capital & hence total capital is 11Million / 0.9. Thus debt is equal to 11 Million*0.1 / 0.9 = 1.22 Million

Interest on debt = 9%*1.22 Million = $ 110K

EBT= EBIT - Interest = 4.5 Million - 110K = 4.39 Million

Net Income = EBT * (1-Tax Rate) = 4.39 Million *(1-40%) = 2.63 Million

Similarly calculate for the other 2 EBITs provided in the question i.e. of $ 3.4 Million & of $ 600K

Case 3: Debt / Capital ratio is 50% and Interest Rate is 11%.

EBIT = $4.5 Million

As debt to capital ratio is 50% hence Debt = Capital *50%. However as capital = debt + equity, therefore the 11 million invested in equity will be 50% of the invested capital & hence total capital is 11Million / 0.5. Thus debt is equal to 11 Million*0.5 / 0.5 = 11 Million

Interest on debt = 11%*11 Million = $ 1.21 Million

EBT= EBIT - Interest = 4.5 Million - 1.21 Million = 3.29 Million

Net Income = EBT * (1-Tax Rate) = 3.29 Million *(1-40%) = 1.97 Million

Similarly calculate for the other 2 EBITs provided in the question i.e. of $ 3.4 Million & of $ 600K

Case 4: Debt / Capital ratio is 60% and Interest Rate is 14%.

EBIT = $4.5 Million

As debt to capital ratio is 60% hence Debt = Capital *60%. However as capital = debt + equity, therefore the 11 million invested in equity will be 60% of the invested capital & hence total capital is 11Million / 0.6. Thus debt is equal to 11 Million*0.4 / 0.6 = 16.5 Million

Interest on debt = 14%*16.5 Million = $ 2.31 Million

EBT= EBIT - Interest = 4.5 Million - 2.31 Million = 2.19 Million

Net Income = EBT * (1-Tax Rate) = 2.19 Million *(1-40%) = 1.31 Million

Similarly calculate for the other 2 EBITs provided in the question i.e. of $ 3.4 Million & of $ 600K

Now incorporating the probabilistic scenarios for ROE & hence also calculated Standard deviation and COV (coefficient of variation)

ROE = Net Income / Equity Invested; As we have Net Income now across all the cases and all the various EBIT scenarios and also as Equity Invested is constant at $ 11 Million, we get the ROE for all cases.

To get the probabilistic ROE we just multiply each ROE with the corresponding probability as can be seen in the table below:

	Case 1			Case 2			Case 3			Case 4
Debt / Capital Ratio	0%	0%	0%	10%	10%	10%	50%	50%	50%	60%	60%	60%
Interest Rate	0%	0%	0%	9%	9%	9%	11%	11%	11%	14%	14%	14%
Debt	-	-	-	1,222,222	1,222,222	1,222,222	11,000,000	11,000,000	11,000,000	16,500,000	16,500,000	16,500,000
Interest on Debt	-	-	-	110,000	110,000	110,000	1,210,000	1,210,000	1,210,000	2,310,000	2,310,000	2,310,000
EBIT	4,500,000	3,400,000	600,000	4,500,000	3,400,000	600,000	4,500,000	3,400,000	600,000	4,500,000	3,400,000	600,000
EBT	4,500,000	3,400,000	600,000	4,390,000	3,290,000	490,000	3,290,000	2,190,000	(610,000)	2,190,000	1,090,000	(1,710,000)
Net Income	2,700,000	2,040,000	360,000	2,634,000	1,974,000	294,000	1,974,000	1,314,000	(366,000)	1,314,000	654,000	(1,026,000)
ROE (r)	24.55%	18.55%	3.27%	23.95%	17.95%	2.67%	17.95%	11.95%	-3.33%	11.95%	5.95%	-9.33%
Probability	20%	50%	30%	20%	50%	30%	20%	50%	30%	20%	50%	30%
Expected ROE - 1	4.91%	9.27%	0.98%	4.79%	8.97%	0.80%	3.59%	5.97%	-1.00%	2.39%	2.97%	-2.80%
Expected ROE - 2	15.16%			14.56%			8.56%			2.56%
(r-ExpectedROE2)	9.38%	3.38%	-11.89%	9.38%	3.38%	-11.89%	9.38%	3.38%	-11.89%	9.38%	3.38%	-11.89%
(r-ExpectedROE2)2	0.88%	0.11%	1.41%	0.88%	0.11%	1.41%	0.88%	0.11%	1.41%	0.88%	0.11%	1.41%
Variance	0.18%	0.06%	0.42%	0.18%	0.06%	0.42%	0.18%	0.06%	0.42%	0.18%	0.06%	0.42%
Std Dev	8.11%			8.11%			8.11%			8.11%
COV	0.53			0.56			0.95			3.16

In the table above:

Expected ROE - 1 = ROE* Probability

We then add it to get the effective ROE for the case. Similarly to calculate the standard deviation, we first take deviation of the ROE from the effective ROE, then we calculate Variance = Probability*(r-ExpectedROE2)²

Then we add up all the variances and take a square root of it to get the effective standard deviation.

COV = Standard deviation / Effective ROE

Hence for each of the cases are answers are as follows upto 2 decimal places:

	Case 1			Case 2			Case 3			Case 4
Debt / Capital Ratio	0%	0%	0%	10%	10%	10%	50%	50%	50%	60%	60%	60%
Interest Rate	0%	0%	0%	9%	9%	9%	11%	11%	11%	14%	14%	14%
ROE	15.16%			14.56%			8.56%			2.56%
Std Dev	8.11%			8.11%			8.11%			8.11%
COV	0.53			0.56			0.95			3.16