ECO359H5 : Practice problem--it's very helpful ~
ECO359/2011īīUniversityīofīTorontoī
PracticeīProblemsī3ī īīīīīīīīīīī DepartmentīofīEconomicsī
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ī
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. ModiglianiāMiller 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.
ModiglianiāMiller 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 firmās debt is
$25,000 (B). The value of Betaās equity is S= VL ā B = $100,000 ā $25,000 = $75,000. Therefore,
the market value of Beta Corporationās equity (S) is $75,000.
d. Since the market value of Alpha Corporationās equity is $100,000, it will cost $20,000 (= 0.20*
$100,000) to purchase 20% of the firmās equity.
Since the market value of Beta Corporationās equity is $75,000, it will cost $15,000 (= 0.20*
$75,000) to purchase 20% of the firmās 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 firmās 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 Betaās 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 firmās earnings available to equity holders is $31,750 (=$35,000 ā $3,250). The dollar return to
an investor who owns 20% of the firmās equity is $6,350 (= 0.20*$31,750).
f. The initial cost of purchasing 20% of Alpha Corporationās equity is $20,000, but the cost to an
investor of purchasing 20% of Beta Corporationās equity is only $15,000 (see part d).
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ECO359/2011īīUniversityīofīTorontoī
PracticeīProblemsī3ī īīīīīīīīīīī DepartmentīofīEconomicsī
2ī
ī
In order to purchase $20,000 worth of Alphaās 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 Alphaās 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 Betaās equity.
Strategy Summary:
1. Borrow $5,000 at 13%.
2. Purchase 20% of Alphaās stock for a net cost of $15,000 (= $20,000 ā $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 firmās earnings. If the firm does not do particularly well, all of the firmās earnings
may be needed to repay its debt holders, and equity holders will receive nothing.
Question 16.2
a. A firmās debtāequity ratio is the market value of the firmās debt divided by the market value of a
firmās equity. The market value of Acetateās debt $9 million, and the market value of Acetateās
equity is $30 million.
Debt-Equity Ratio = Market Value of Debt/Market Value of Equity = $9M/$30M = 0.03. Therefore,
Acetateās DebtāEquity Ratio is 30%.
b. By CAPM, the cost of Acetateās equity is:
rS = rf + īS[E(rm) ā rf] = 0.07 + 0.85( 0.21- 0.07) = 0.189. The cost of Acetateās equity (rS) is 18.9%.
Assume a cost of debt of 14% (missing in the statement of the problem).
Acetateās 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, Acetateās weighted average cost of capital is 17.77%.
c. According to ModiglianiāMiller Proposition II (No Taxes):
rS = r0 + (B/S)(r0 ā rB). Thus, 0.189= r0 + (9/30)(r0 ā 0.14). Solving for r0 = 0.1777. Therefore, the
cost of capital for an otherwise identical allāequity firm is 17.77%. This is consistent with
ModiglianiāMillerās proposition that, in the absence of taxes, the cost of capital for an allāequity
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*
