Ch 16-09 Build a Model Solution							3/6/2001
Chapter 16. Solution to Ch 16-09 Build a Model
Elliott Athletics is trying to determine its optimal capital structure, which now consists of only debt
and common equity. The firm does not currently use preferred stock in its capital structure, and it
does not plan to do so in the future. To estimate how much its debt would cost at different debt
levels, the company's Treasury staff has consulted with investment bankers and, on the basis of
those discussions, has created the following table:
Debt/Assets	Equity/Assets	Debt/Equity	Debt	B-T Cost of
Ratio (wd)	Ratio (wc)	Ratio (D/E)	Rating	Debt (kd)
0	1	0.00	A	7.00%
0.2	0.8	0.25	BBB	8.00%
0.4	0.6	0.67	BB	10.00%
0.6	0.4	1.50	C	12.00%
0.8	0.2	4.00	D	15.00%
Elliott uses the CAPM to estimate its cost of common equity, ks. The company estimates that the
risk-free rate is 5 percent, the market risk premium is 6 percent, and its tax rate is 40 percent.
Elliott estimates that if it had no debt, its "unlevered" beta, BU, would be 1.2.
a. Based on this information, what is the firm's optimal capital structure, and what would the
weighted average cost of capital be at the optimal structure?
Solution to Part a:
Inputs provided in the problem:
Risk-free rate		5%
Market risk premium		6%
Unlevered beta		1.2
Tax rate		40%
Next, we construct a table (like that in the model) that evaluates WACC at different levels of debt.
The beta is found using the Hamada equation:
bL = bU [1+ (1-T)(D/E)]
In Excel format, here is the equation for bL with 10% debt:
bL = 1.2*[1+(1-$C$35)*C51] = 1.28.
Then, with bL, we can apply the CAPM equation to find ks, the cost of equity, and then we can find
the WACC.
A-T kd = kd(1-T)
ks = kRF + b(kM-kRF)
WACC = wd(kd)(1-T) + ws(ks).
Debt/Assets	Equity/Assets	Debt/Equity	A-T Cost of	Leveraged	Cost of		D/A at min
Ratio (wd)	Ratio (wc)	Ratio (D/E)	Debt (kd)	Beta	Equity	WACC	WACC
0.0	1.0	0.00	After tax cost of debt. 4.20%	1.20	12.200%	12.20%	Debt/Asset ratio at minimum WACC. Fill in cell G63 first, and then use IF function. 0
0.2	0.8	0.25	4.80%	1.38	13.28%	11.58%	0
0.4	0.6	0.67	6.00%	1.68	15.08%	11.45%	0.4
0.6	0.4	1.50	7.20%	2.28	18.68%	11.79%	0
0.8	0.2	4.00	9.00%	4.08	29.48%	13.10%	0
From the table, we see that the optimal capital structure consists of 40% debt and 60% equity.
Using Excel's Minimum function, we find the Min WACC to be:						11.45%
Using MAX, find the WACC minimizing D/A ratio:						MAX of column H. 40%
b. Plot a graph of the A-T cost of debt, the cost of equity, and the WACC versus (1) the Debt/Assets
ratio and (2) the Debt/Equity ratio.
Capital costs versus D/A Ratio.
The top graph is like the one in the textbook, because it uses the D/A ratio on the horizontal axis.
The bottom graph is a bit like MM showed in their original article in that the cost of equity is linear
and the WACC does not turn up sharply. It is not exactly like MM because it uses D/A rather than
D/V, and also because MM assumed that kd is constant whereas we assume the cost of debt rises
with leverage.
Note too that the minimum WACC is at the D/A and D/E levels indicated in the table, and also that
the WACC curve is very flat over a broad range of debt ratios, indicating that WACC is not
sensitive to debt over a broad range. This is important, as it demonstrates that management can use
a lot of discretion as to its capital structure, and that it is OK to alter the debt ratio to take
advantage of market conditions in the debt and equity markets, and to increase the debt ratio if
many good investment opportunities are available.
c. Would the optimal capital structure change if the unlevered beta changed? To answer this
question, do a sensitivity analysis of WACC on bU for different levels of bU.
Set up a data table where you find WACC at different values of bU. Then create graphs of WACC vs.
bu and Optimal Cap. Str. Vs bu.
		WACC at
		Optimal	Optimal
	Unleveraged	Cap. Str.	D/A Ratio
	Beta	11.45%	40%
	0	4.96%	20%
	0.6	8.27%	20%
	1.2	11.45%	40%
	1.6	13.46%	40%
	2.2	16.35%	60%
The first graph shows that WACC rises if the firm's unlevered beta rises. A higher bU means
more business risk, and risk raises the cost of capital.
The second graph shows the optimal capital structure rising with bU. This occurs because (1) the
cost of equity rises with bU, (2) in our example kd does not rise with bU, hence (3) higher bU's
penalize equity, hence (4) using more debt is especially advantageous at high bU values.
This result occurs because of the way we set up the problem. Realistically, a higher bU would
lead to a higher kd at all levels of bU. That would alter the relationship, possibly resulting in
no relationship between bU and the optimal capital structure.
The point of this part of the problem is to demonstrate that the inputs determine the outputs.
Note that MM assumed that firms could borrow at the riskless rate, regardless of how much
debt they used, and regardless of bU. However, they assumed that the cost of equity varied both with
bU and the amount of debt used. Others have modified the MM assumptions, but our problem
demonstrates that unless the input data are known for sure, which is never the case, we
cannot determine the optimal capital structure for sure. We can find one, but it might be wrong.