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11. If there are corporate taxes, the value of an unlevered firm is:
VU = EBIT(1 – TC)/RU
The WACC remains at 8.4 percent. Due to taxes, EBIT for an all-equity firm would have to be higher
for the firm to still be worth $16.3 million.
12. a. With the information provided, we can use the equation for calculating WACC to find the cost
of equity. The equation for WACC is:
The company has a debt-equity ratio of 1.25, which implies the weight of debt is 1.25/2.25, and
b. To find the unlevered cost of equity we need to use M&M Proposition II with taxes, so:
c. To find the cost of equity under different capital structures, we can again use M&M Proposition
II with taxes. With a debt-equity ratio of 2, the cost of equity is:
RE = RU + (RU – RD)(D/E)(1 – TC)
With a debt-equity ratio of 1.0, the cost of equity is:
And with a debt-equity ratio of 0, the cost of equity is:
13. a. For an all-equity financed company:
b. To find the cost of equity for the company with leverage we need to use M&M Proposition II
with taxes, so:
RE = RU + (RU – RD)(D/E)(1 – TC)
c. Using M&M Proposition II with taxes again, we get:
RE = RU + (RU – RD)(D/E)(1 – TC)
d. The WACC with 25 percent debt is:
WACC = (E/V)RE + (D/V)RD(1 – TC)
And the WACC with 50 percent debt is:
WACC = (E/V)RE + (D/V)RD(1 – TC)
14. a. The value of the unlevered firm is:
VU = EBIT(1 – TC)/RU
b. The value of the levered firm is:
VL = VU + TCD
15. We can find the cost of equity using M&M Proposition II with taxes. Doing so, we find:
RE = RU + (RU – RD)(D/E)(1 – TC)
Using this cost of equity, the WACC for the firm after recapitalization is:
When there are corporate taxes, the overall cost of capital for the firm declines the more highly
leveraged is the firm’s capital structure. This is M&M Proposition I with taxes.
Intermediate
16. To find the value of the levered firm we first need to find the value of an unlevered firm. So, the value
of the unlevered firm is:
VU = EBIT(1 – TC)/RU
Now we can find the value of the levered firm as:
VL = VU + TCD
Applying M&M Proposition I with taxes, the firm has increased its value by issuing debt. As long as
M&M Proposition I holds, that is, there are no bankruptcy costs and so forth, then the company
should continue to increase its debt/equity ratio to maximize the value of the firm.
17. a. With no debt, we are finding the value of an unlevered firm, so:
VU = EBIT(1 – TC)/R0
b. The general expression for the value of a leveraged firm is:
VL = VU + TCD
If debt is 50 percent of VU, then D = (.50)VU, and we have:
VL = VU + TC[(.50)VU]
And if debt is 100 percent of VU, then D = (1.0)VU, and we have:
VL = VU + TC[(1.0)VU]
c. According to M&M Proposition I with taxes:
VL = VU + TCD
With debt being 50 percent of the value of the levered firm, D must equal (.50)VL, so:
VL = VU + TC[(.50)VL]
If the debt is 100 percent of the levered value, D must equal VL, so:
VL = VU + TC[(1.0)VL]
18. a. To purchase 5 percent of Knight’s equity, the investor would need:
And to purchase 5 percent of Day without borrowing would require:
In order to compare dollar returns, the initial net cost of both positions should be the same.
Therefore, the investor will need to borrow the difference between the two amounts, or:
An investor who owns 5 percent of Knight’s equity will be entitled to 5 percent of the firm’s
Day will distribute all of its earnings to shareholders, so the shareholder will receive:
However, to have the same initial cost, the investor has borrowed $32,500 to invest in Day, so
interest must be paid on the borrowings. The net cash flow from the investment in Day will be:
For the same initial cost, the investment in Day produces a higher dollar return.
b. Both of the strategies have the same initial cost. Since the dollar return to the investment in Day
is higher, all investors will choose to invest in Day over Knight. The process of investors
Challenge
19. M&M Proposition II states:
RE = RU + (RU – RD)(D/E)(1 – TC)
And the equation for WACC is:
WACC = (E/V)RE + (D/V)RD(1 – TC)
Substituting the M&M Proposition II equation into the equation for WACC, we get:
Rearranging and reducing the equation, we get:
WACC = RU[(E/V) + (E/V)(D/E)(1 – TC)] + RD(1 – TC)[(D/V) – (E/V)(D/E)]
20. The return on equity is net income divided by equity. Net income can be expressed as:
NI = (EBIT – RDD)(1 – TC)
So, ROE is:
RE = (EBIT – RDD)(1 – TC)/E
Now we can rearrange and substitute as follows to arrive at M&M Proposition II with taxes:
RE = [EBIT(1 – TC)/E] – [RD(D/E)(1 – TC)]
21. M&M Proposition II, with no taxes is:
Note that we use the risk-free rate as the return on debt. This is an important assumption of M&M
Proposition II. The CAPM to calculate the cost of equity is expressed as:
We can rewrite the CAPM to express the return on an unlevered company as:
We can now substitute the CAPM for an unlevered company into M&M Proposition II. Doing so and
rearranging the terms we get:
RE = A(RM – Rf) + Rf + [A(RM – Rf) + Rf – Rf](D/E)
Now we set this equation equal to the CAPM equation to calculate the cost of equity and reduce:
E(RM – Rf) + Rf = (1 + D/E)A(RM – Rf) + Rf
22. Using the equation we derived in Problem 21:
The equity beta for the respective asset betas is:
Debt-Equity Ratio Equity Beta
0 1(1 + 0) = 1
The equity risk to the shareholder is composed of both business and financial risk. Even if the assets
