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Chapter 16. Solution to Ch 16-07 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: Market Market Market Debt/Value Equity/Value Debt/Equity Ratio (wd) Ratio (wc) Ratio (D/S) 0 1 0.00 0.2 0.8 0.25 0.4 0.6 0.67 0.6 0.4 1.50 0.8 0.2 4.00

Debt Rating A BBB BB C D

B-T Cost of Debt (rd) 7.00% 8.00% 10.00% 12.00% 15.00%

Elliott uses the CAPM to estimate its cost of common equity, rs. 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 Market risk premium Unlevered beta Tax rate 5% 6% 1.2 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/S)] In Excel format, here is the equation for b L with 10% debt: bL = 1.2*[1+(1-$C$35)*C51] = 1.28. Then, with bL, we can apply the CAPM equation to find r s, the cost of equity, and then we can find the WACC.

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A-T rd = rd(1-T) rs = rRF + b(rM-rRF) WACC = wd(rd)(1-T) + ws(rs).

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Debt/Value Equity/Value Debt/Equity A-T Cost of Ratio (wd) Ratio (wc) Ratio (D/S) Debt (rd) 0.0 1.0 0.00 4.20% 0.2 0.8 0.25 4.80% 0.4 0.6 0.67 6.00% 0.6 0.4 1.50 7.20% 0.8 0.2 4.00 9.00%

Leveraged Beta 1.20 1.38 1.68 2.28 4.08

Cost of Equity 12.200% 13.28% 15.08% 18.68% 29.48%

D/A at min WACC WACC 12.20% 0 11.58% 0 11.45% 0.4 11.79% 0 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: 40% b. Plot a graph of the A-T cost of debt, the cost of equity, and the WACC versus the Debt/Value ratio.

Capital costs versus D/V Ratio.

Capital Costs Vs. D/V
35% 30% 25% 20% 15%
A-T Cost of Debt (rd) Cost of Equity Cost of WACC

5% 0% 0% 20% 40% 60% 80% 100%

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%

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17.0% 15.0% 13.0% 11.0% 9.0% 7.0% 5.0% 0 0.5 1 1.5 2

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Optimal Capital Structure Vs. bU
60% 50%
40% 30%

20% 10%






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 rd 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 rd 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.

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