ECO359H5 : Practice problem--it's very helpful ~
ECO359/2011UniversityofToronto
PracticeProblems3 DepartmentofEconomics
1
ECO 359 Practice Problems 3
Solutions to selected Textbook questions
Question 16.1
a. Since Alpha Corporation is an all-equity firm, its value is equal to the market value of its outstanding
shares. Alpha has 5,000 shares of common stock outstanding, worth $20 per share.
Therefore, the value of Alpha Corporation is $100,000 (= 5,000 shares * $20 per share).
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 Alpha
Corporation in every way except its capital structure and neither firm pays taxes, the value of the two
firms should be equal.
Modigliani–Miller Proposition I (No Taxes): VL =VU.
Alpha Corporation, an unlevered firm, is worth $100,000 = VU.
Therefore, the value of Beta Corporation (VL) is $100,000.
c. The value of a levered firm equals the market value of its debt plus the market value of its equity. VL
= B + S. The value of Beta Corporation is $100,000 (VL), and the market value of the firm’s debt is
$25,000 (B). The value of Beta’s equity is S= VL – B = $100,000 – $25,000 = $75,000. Therefore,
the market value of Beta Corporation’s equity (S) is $75,000.
d. Since the market value of Alpha Corporation’s equity is $100,000, it will cost $20,000 (= 0.20*
$100,000) to purchase 20% of the firm’s equity.
Since the market value of Beta Corporation’s equity is $75,000, it will cost $15,000 (= 0.20*
$75,000) to purchase 20% of the firm’s equity.
e. Since Alpha Corporation expects to earn $35,000 this year and owes no interest payments, the dollar
return to an investor who owns 20% of the firm’s equity is expected to be $7,000 (= 0.20*$35,000)
over the next year.
While Beta Corporation also expects to earn $35,000 before interest this year, it must pay 13%
interest on its debt. Since the market value of Beta’s debt at the beginning of the year is $25,000,
Beta must pay $3,250 (= 0.13*$25,000) in interest at the end of the year. Therefore, the amount of
the firm’s earnings available to equity holders is $31,750 (=$35,000 – $3,250). The dollar return to
an investor who owns 20% of the firm’s equity is $6,350 (= 0.20*$31,750).
f. The initial cost of purchasing 20% of Alpha Corporation’s equity is $20,000, but the cost to an
investor of purchasing 20% of Beta Corporation’s equity is only $15,000 (see part d).
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ECO359/2011UniversityofToronto
PracticeProblems3 DepartmentofEconomics
2
In order to purchase $20,000 worth of Alpha’s equity using only $15,000 of his own money, the
investor must borrow $5,000 to cover the difference. The investor must pay 13% interest on his
borrowings at the end of the year.
Since the investor now owns 20% of Alpha’s equity, the dollar return on his equity investment at the
end of the year is $7,000 (= 0.20*$35,000). However, since he borrowed $5,000 at 13% per annum,
he must pay $650 (= 0.13* $5,000) at the end of the year.
Therefore, the cash flow to the investor at the end of the year is $6,350 (= $7,000 - $650). Notice
that this amount exactly matches the dollar return to an investor who purchases 20% of Beta’s equity.
Strategy Summary:
1. Borrow $5,000 at 13%.
2. Purchase 20% of Alpha’s stock for a net cost of $15,000 (= $20,000 – $5,000 borrowed).
g. The equity of Beta Corporation is riskier. Beta must pay off its debt holders before its equity holders
receive any of the firm’s earnings. If the firm does not do particularly well, all of the firm’s earnings
may be needed to repay its debt holders, and equity holders will receive nothing.
Question 16.2
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. The market value of Acetate’s debt $9 million, and the market value of Acetate’s
equity is $30 million.
Debt-Equity Ratio = Market Value of Debt/Market Value of Equity = $9M/$30M = 0.03. Therefore,
Acetate’s Debt–Equity Ratio is 30%.
b. By CAPM, the cost of Acetate’s equity is:
rS = rf + S[E(rm) – rf] = 0.07 + 0.85( 0.21- 0.07) = 0.189. The cost of Acetate’s equity (rS) is 18.9%.
Assume a cost of debt of 14% (missing in the statement of the problem).
Acetate’s weighted average cost of capital equals:
rwacc = {B / (B+S)} rB + {S / (B+S)}rS = ($9M/$39M)(0.14) + ($30M/$39M)(0.189) = (0.23)(0.14) +
(.77)(0.189) = 0.1777. Therefore, Acetate’s weighted average cost of capital is 17.77%.
c. According to Modigliani–Miller Proposition II (No Taxes):
rS = r0 + (B/S)(r0 – rB). Thus, 0.189= r0 + (9/30)(r0 – 0.14). Solving for r0 = 0.1777. Therefore, the
cost of capital for an otherwise identical all–equity firm is 17.77%. This is consistent with
Modigliani–Miller’s proposition that, in the absence of taxes, the cost of capital for an all–equity
firm is equal to the weighted average cost of capital of an otherwise identical levered firm.
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Document Summary
Question 16. 1: since alpha corporation is an all-equity firm, its value is equal to the market value of its outstanding shares. Alpha has 5,000 shares of common stock outstanding, worth per share. Corporation in every way except its capital structure and neither firm pays taxes, the value of the two firms should be equal. Alpha corporation, an unlevered firm, is worth ,000 = vu. Therefore, the value of beta corporation (vl) is ,000: the value of a levered firm equals the market value of its debt plus the market value of its equity. = b + s. the value of beta corporation is ,000 (vl), and the market value of the firm s debt is. The value of beta s equity is s= vl b = ,000 ,000 = ,000. Therefore, the market value of beta corporation s equity (s) is ,000: since the market value of alpha corporation s equity is ,000, it will cost ,000 (= 0. 20*
