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9. a. The rate of return earned will be the dividend yield. The company has debt, so it must make an
interest payment. The net income for the company is:
The investor will receive dividends in proportion to the percentage of the company’s shares he
owns. The total dividends received by the shareholder will be:
So the return the shareholder expects is:
b. To generate exactly the same cash flows in the other company, the shareholder needs to match
the capital structure of ABC. The shareholder should sell all shares in XYZ. This will net
$30,000. The shareholder should then borrow $30,000. This will create an interest cash flow of:
The investor should then use the proceeds of the stock sale and the loan to buy shares in ABC.
The investor will receive dividends in proportion to the percentage of the company’s share he
owns. The total dividends received by the shareholder will be:
The total cash flow for the shareholder will be:
The shareholder’s return in this case will be:
c. ABC is an all equity company, so:
To find the cost of equity for XYZ, we need to use M&M Proposition II, so:
d. To find the WACC for each company, we need to use the WACC equation:
And for XYZ, the WACC is:
When there are no corporate taxes, the cost of capital for the firm is unaffected by the capital
10. With no taxes, the value of an unlevered firm is the EBIT divided by the unlevered cost of equity, so:
11. If there are corporate taxes, the value of an unlevered firm is:
VU = EBIT(1 – tC) / RU
Using this relationship, we can find EBIT as:
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:
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:
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:
c. Using M&M Proposition II with taxes again, we get:
d. The WACC with 25 percent debt is:
And the WACC with 50 percent debt is:
14. a. The value of the unlevered firm is:
V = EBIT(1 – tC) / R0
b. The value of the levered firm is:
V = VU + tCB
15. We can find the cost of equity using M&M Proposition II with taxes. First, we need to find the
market value of equity, which is:
V = B + S
Now we can find the cost of equity, which is:
RS = R0 + (R0 – RB)(B/S)(1 – tC)
Using this cost of equity, the WACC for the firm after recapitalization is:
WACC = (S/V)RS + (B/V)RB(1 – tC)
When there are corporate taxes, the overall cost of capital for the firm declines the more highly
16. Since Unlevered is an all-equity firm, its value is equal to the market value of its outstanding shares.
Unlevered has 4.5 million shares of common stock outstanding, worth $78 per share. Therefore, the
value of Unlevered:
Modigliani-Miller Proposition I states that, in the absence of taxes, the value of a levered firm equals
the value of an otherwise identical unlevered firm. Since Levered is identical to Unlevered in every
The market value of Levered’s debt is $73 million. The value of a levered firm equals the market
value of its debt plus the market value of its equity. Therefore, the current market value of Levered
is:
The market value of Levered’s equity needs to be $351 million, $57.5 million higher than its current
Intermediate
17. 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:
Now we can find the value of the levered firm as:
Applying M&M Proposition I with taxes, the firm has increased its value by issuing debt. As long as
18. a. With no debt, we are finding the value of an unlevered firm, so:
b. The general expression for the value of a leveraged firm is:
VL = VU + tCB
If debt is 50 percent of VU, then D = (.50)VU, and we have:
And if debt is 100 percent of VU, then D = (1.0) VU, and we have:
c. According to M&M Proposition I with taxes:
VL = VU + tCB
With debt being 50 percent of the value of the levered firm, D must equal (.50)VL, so:
If the debt is 100 percent of the levered value, D must equal VL, so:
VL = VU + T[(1.0)(VL]
19. According to M&M Proposition I with taxes, the increase in the value of the company will be the
present value of the interest tax shield. Since the loan will be repaid in equal installments, we need to
find the loan interest and the interest tax shield each year. The loan schedule will be:
Year Loan Balance Interest Tax Shield
So, the increase in the value of the company is:
20. a. Since Alpha Corporation is an all-equity firm, its value is equal to the market value of its
outstanding shares. Alpha has 18,000 shares of common stock outstanding, worth $35 per
share, so the value of Alpha Corporation is:
b. Modigliani-Miller Proposition I states that in the absence of taxes, the value of a levered firm
equals the value of an otherwise identical unlevered firm. Since Beta Corporation is identical to
c. The value of a levered firm equals the market value of its debt plus the market value of its
equity. So, the value of Beta’s equity is:
d. The investor would need to invest 20 percent of the total market value of Alpha’s equity, which
is:
Beta has less equity outstanding, so to purchase 20 percent of Beta’s equity, the investor would
need:
e. Alpha has no interest payments, so the dollar return to an investor who owns 20 percent of the
company’s equity would be:
Beta Corporation has an interest payment due on its debt in the amount of:
So, the investor who owns 20 percent of the company would receive 20 percent of EBIT minus
the interest expense, or:
f. From part d, we know the initial cost of purchasing 20 percent of Alpha Corporation’s equity is
$126,000, but the cost to an investor of purchasing 20 percent of Beta Corporation’s equity is
only $109,000. In order to purchase $126,000 worth of Alpha’s equity using only $109,000 of
Notice that this amount exactly matches the dollar return to an investor who purchases 20
percent of Beta’s equity.
g. The equity of Beta Corporation is riskier. Beta must pay off its debt holders before its equity
21. a. A firm’s debt–equity ratio is the market value of the firm’s debt divided by the market value of
a firm’s equity. So, the debt–equity ratio of the company is:
b. We first need to calculate the cost of equity. To do this, we can use the CAPM, which gives us:
We need to remember that an assumption of the Modigliani-Miller theorem is that the company
debt is risk-free, so we can use the Treasury bill rate as the cost of debt for the company. In the
absence of taxes, a firm’s weighted average cost of capital is equal to:
c. According to Modigliani-Miller Proposition II with no taxes:
This is consistent with Modigliani-Miller’s proposition that, in the absence of taxes, the cost of
22. a. To purchase 5 percent of Knight’s equity, the investor would need:
And to purchase 5 percent of Veblen 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
earnings available to common stock holders at the end of each year. While Knight’s expected
Veblen will distribute all of its earnings to shareholders, so the shareholder will receive:
However, to have the same initial cost, the investor has borrowed $52,500 to invest in Veblen,
so interest must be paid on the borrowings. The net cash flow from the investment in Veblen
will be:
b. Both of the two strategies have the same initial cost. Since the dollar return to the investment in
Veblen is higher, all investors will choose to invest in Veblen over Knight. The process of
investors purchasing Veblen’s equity rather than Knight’s will cause the market value of
