AFM 371 Practice Problem Set #2 Winter Suggested Solutions

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1 AFM 371 Practice Problem Set #2 Winter 2008 Suggested Solutions 1. Text Problems: 16.2 (a) The debt-equity ratio is the market value of debt divided by the market value of equity. In this case we have $10 million / $20 million, or 1/2. (b) The CAPM implies: r S = r f + β E(r M ) r f The weighted average cost of capital is then ( ) B r WACC = B + S ( 10 = = =.085. = ( S r B + ).05 + ) r S B + S ( ).085 Note: some students have a version of the textbook which gives different information for this question. In particular, the cost of debt is given as 14% instead of 5%. In this case we would have: ( ) ( ) r WACC = = (c) An implication of MM Proposition I (No Taxes) is that firm value is independent of capital structure, and so therefore the weighted average cost of capital is also independent of capital structure. Therefore the cost of capital for an otherwise identical all-equity firm should also be 7.33%. We can verify this using MM Proposition II (No Taxes): r S = r 0 + (B/S)(r 0 r B ).085 = r 0 + (1/2)(r 0.05).11 = (3/2)r 0 r 0 = Under Plan I, earnings after interest are $8,000.1($9,000) = $7,100. EPS is $7,100/800 shares = $8.875 per share. Under Plan II, earnings after interest are $8,000.1($13,500) = $6,650, and so EPS is $6,650/700 shares = $9.50 per share. In the all-equity case, EPS is $8,000/1,000 shares = $8.00 per share. Plan II has the highest EPS and the all-equity plan has the lowest Before the restructuring, the market value of the firm s equity was $5,000,000 (100,000 shares at $50 per share). Since $1,000,000 of debt was issued and the proceeds used to repurchase shares, the market value of the firm s equity after the restructuring is $4,000,000. Since 20,000 shares were repurchased, there are 80,000 shares left (which, at the price of $50 per share, gives the total market value of equity of $4,000,000). Since Ms. Hannon owned $10,000 worth of the firm s stock, this represented a fraction of.002 of the firm value ($10,000/$5 million) before the restructuring. She also borrowed $2,000 at the interest rate of 20%, 1

2 implying interest payments of $400 per year. Let X be the firm s earnings per year. Before the restructuring, Ms. Hannon s overall payoff was.002x $400. After the restructuring, the firm must pay interest of $200,000 per year (20% of $1 million). Also, Ms. Hannon s fractional holding of firm value is now.0025 ($10,000/$4 million). Her overall payoff now is therefore.0025(x $200,000) $400, or.0025x $900. In order to get back to her original payoff, she must sell.0005($4,000,000) = $2,000 of her shares. This leaves her with a payoff from her equity position in the firm of.002(x $200,000) =.002X $400, which is her original payoff. This implies that she must take the $2,000 proceeds from selling her shares and use the money to pay off her borrowing of $2,000. Her total position after all of this would be that she owns $8,000 worth of shares. Ms. Finney owned 1% of the firm s stock before the restructuring ($50,000 out of $5 million). She also lent $6,000, implying that she was receiving interest of $1,200 per year. This means that her overall payoff was.01x + $1,200. After the restructuring, she owns 1.25% of the firm s stock ($50,000 out of $4 million). Her overall payoff now is therefore.0125(x $200,000) + $1,200, or.0125x $1,300. In order to get back to her original payoff, she must sell.0025($4,000,000) = $10,000 of her shares. This leaves her with a payoff from her equity position in the firm of.01(x $200,000) =.01X $2,000. To get back to her overall original payoff, she must receive $3,200 worth of interest, implying that she must lend a total of $16,000 (i.e. she must take the $10,000 she gets from selling her shares and lend it out). Therefore, her total position would be that she owns $40,000 worth of the firm s shares and lends $16,000. Ms. Grace owned.004 of the firm value ($20,000/$5 million) before the restructuring, and had a payoff of.004x. After the structuring, she owns.005 of the firm s equity ($20,000/$4 million), and receives a payoff of.005(x $200,000) =.005X $1,000. In order to get back to her original payoff, she must sell.001($4,000,000) = $4,000 of her shares. This gives a payoff from her equity position of.004(x $200,000) =.004X $800. To get back to her overall original payoff, she must receive interest of $800, implying that she must lend $4,000 (i.e. her proceeds from selling shares). So her total position is that she owns $16,000 worth of stock and lends $4, (a) By MM Proposition II (No Taxes), the firm s weighted average cost of capital will not change with the substitution of debt for equity, i.e. it will remain at 18%. (b) Use MM Proposition II (No Taxes): (c) We have: r S = r 0 + (B/S)(r 0 r B ) =.18 + ($400,000/$1,600,000)(.18.10) =.20. ( ) ( ) B S r WACC = r B + r S B + S B + S =.2(.10) +.8(.20) =.18, which is consistent with part (a) (a) Prior to the announcement, the value of the firm s equity is $5 million (250,000 shares at $20 per share). Since it is all-equity, the firm s assets are also worth $5 million. This value is generated by perpetual earnings of $750,000 being discounted at the rate of 15%. The market value balance sheet is: Assets $5,000,000 Debt $0 Equity $5,000,000 Total assets $5,000,000 Total debt + equity $5,000,000 (b) 1. According to the efficient market hypothesis, the firm s stock price will change immediately to reflect the NPV of the transaction, which is NPV = $300,000 + $120,000/.15 = $500,000. 2

3 (c) The market value of the firm s equity will rise by this amount to $5.5 million. The price per share will be $5.5 million/250,000 or $ The market value balance sheet is: Old assets $5,000,000 Debt $0 NPV $500,000 Equity $5,500,000 Total assets $5,500,000 Total debt + equity $5,500, Since $300,000 must be raised, the number of shares issued will need to be $300,000/$22 or 13, The firm will receive $300,000 in cash after the equity issue, which will increase assets by $300,000. The market value of equity will be $5.8 million (263, shares at $22 per share). The market value balance sheet will be: Old assets $5,000,000 Debt $0 Cash $300,000 Equity $5,800,000 NPV $500,000 Total assets $5,800,000 Total debt + equity $5,800, When the firm makes the purchase, it will pay $300,000 in cash and receive the PV of its competitor s facilities, which is $120,000/.15 or $800,000. The market value balance sheet will be: Old assets $5,000,000 Debt $0 PV (new assets) $800,000 Equity $5,000,000 Total assets $5,800,000 Total debt + equity $5,800, The firm s old assets generate $750,000 of earnings per year, and the new assets produce $120,000 per year, giving total earnings of $870,000 per year. The expected return is this amount divided by the market value of equity of $5.8 million, or 15%. 7. The firm s weighted average cost of capital after the buyout will be 15% (since the firm will still be all-equity financed and the expected rate of return on its equity is 15%). 1. By the efficient market hypothesis, the market value of the firm s equity will immediately rise to reflect the NPV of the transaction. The market value balance sheet will be: Old assets $5,000,000 Debt $0 NPV $500,000 Equity $5,000,000 Total assets $5,500,000 Total debt + equity $5,500, The firm will receive $300,000 in cash after the debt issue. Its market value balance sheet will be: Old assets $5,000,000 Debt $300,000 NPV $500,000 Equity $5,500,000 Cash $300,000 Total assets $5,800,000 Total debt + equity $5,800, After the purchase the market value balance sheet will be: Old assets $5,000,000 Debt $300,000 PV (new assets) $800,000 Equity $5,500,000 Total assets $5,800,000 Total debt + equity $5,800, The firm s old assets produce earnings of $750,000 per year, and the new assets generate earnings of $120,000 per year. However, the firm must pay interest of $30,000 per year, so the cash flow to equity holders is $840,000 per year. Since the total market value of equity is $5.5 million, the expected rate of return is $840,000/$5.5 million, or 15.27%. This can also be calculated from MM Proposition II (No Taxes): r S = r 0 + (B/S)(r 0 r B ) =.15 + (.3/5.5)(.15.10) = By MM Proposition I (No Taxes), we know that the value of the firm and the weighted average 3

4 cost of capital must still be 15%. We can verify this as follows: ( ) ( ) B S r WACC = r B + r S B + S B + S = (.3/5.8)(.10) + (5.5/5.8)(.1527) = (a) False. A reduction in leverage will decrease both the risk of the stock and its expected return. MM state that, in the absence of taxes, these two effects exactly cancel each other out and leave the price of the stock and the overall firm value unchanged. (b) False. MM Proposition II (No Taxes) states that the expected return on a firm s equity is positively related to the firm s debt-equity ratio (r S = r 0 +(B/S)(r 0 r B )). Therefore, any increase in the amount of debt in a firm s capital structure will increase the expected return on a firm s equity (a) The total market value of the firm s equity is $10 million, so buying 1% of it costs $100,000. If Michael borrows 20% of this amount, it will cost him $80,000 (net of debt) to buy 1% of the firm s equity. If he borrows 40%, it will cost him $60,000 (net of debt). If he borrows 60%, it will cost him $40,000 (net of debt). (b) With a 1% stake in the firm s equity, Michael is entitled to 1% of the firm s annual earnings of $1.5 million, or $15,000. If Michael borrows 20% of the cost of the investment, he must make interest payments of $2,000 per year (10% of $20,000). His expected return is then $13,000/$80,000 or 16.25%. If he borrows 40%, his interest payments are $4,000 and his expected return is $11,000/$60,000 or 18.33%. If he borrows 60%, his interest payments would be $6,000 and so his expected return would be $9,000/$40,000 or 22.5% (a) Use MM Proposition I (Corporate Taxes): (b) We have: (a) We have: V L = V U + T C B $1,700,000 = V U + (.34)($500,000) V U = $1,530,000. (b) Use MM Proposition I (Corporate Taxes): EBIT $306,000 Interest $50,000 Pre-tax earnings $256,000 Taxes (34%) $87,040 After-tax earnings $168,960 V U = EBIT(1 T C) = $2,500,000(.66) = $8,250,000. r 0.20 V L = V U + T C B = $8,250, ($600,000) = $8,454,000. (c) Since interest payments are tax deductible, debt lowers the firm s taxable income and creates a tax shield for the firm. This tax shield increases the value of the firm. (d) The MM assumptions in the case of corporate taxes are: (i) there are no personal taxes; (ii) there are no costs of financial distress; (iii) all cash flows are perpetuities (for simplicity); (iv) there are no transaction costs; (v) individuals and firms can borrow at the same rate; and (vi) there is no information asymmetry (a) The expected return on a firm s equity is the ratio of annual after tax earnings to the market value of the firm s equity. Here annual after tax earnings are $1,500, = $900,000, and the market value of the firm is $10 million, implying an expected return of 9%. (Of course, the market value of $10 million is just the perpetual after tax earnings of $900,000 divided by the discount rate of 9%.) 4

5 (b) The market value balance sheet is: Assets $10,000,000 Debt $0 Equity $10,000,000 Total assets $10,000,000 Total debt + equity $10,000,000 Since the firm has 500,000 shares outstanding, the price is $20 per share ($10,000,000/500,000). (c) When the debt issue is announced, the value of the firm will increase by the present value of the debt tax shield. Since the debt is perpetual, this is just T C B, or.40($2,000,000) = $800,!000. Since the firm has not yet actually issued the debt, the value of the firm s equity rises to $10.8 million. The market value balance sheet is: Old assets $10,000,000 Debt $0 PV(tax shield) $800,000 Equity $10,800,000 Total assets $10,800,000 Total debt plus equity $10,800,000 (d) The market price of the firm s stock is now $10,800,000/500,000 or $21.60 per share. (e) The number of shares repurchased is the size of the debt issue divided by the market price per share, i.e. $2 million divided by $21.60, which works out to be 92, This leaves 407, shares outstanding. (f) The 407, shares left are worth $21.60 each, so the total market value of equity is $8.8 million. The market value balance sheet is: Old assets $10,000,000 Debt $2,000,000 PV(tax shield) $800,000 Equity $8,800,000 Total assets $10,800,000 Total debt plus equity $10,800,000 (g) Use MM Proposition II (Corporate Taxes): r S = r 0 + (B/S)(1 T C )(r 0 r B ) =.09 + (2/8.8)(.60)(.09.06) = (a) When there are corporate taxes, the weighted average cost of capital is: ( ) ( ) B S r WACC = (1 T C )r B + r S. B + S B + S This depends on the debt-value ratio and the equity-value ratio. We are only given the debt-equity ratio of 2.5. However: B B = 2.5 B = 2.5S S Therefore, using the weighted average cost of capital: B + S = 2.5S 3.5S =.7143 S B + S = S 3.5S = =.7143(.65)(.10) r S.2857r S =.1036 (b) Use MM Proposition II (Corporate Taxes): r S = r S = r 0 + (1 T C )(B/S)(r 0 r B ).3625 = r (2.5)(r 0.10).525 = 2.625r 0 r 0 =.20. 5

6 (c) With a debt-equity ratio of 0.75, the firm s cost of equity capital would be: r S =.20 + (.65)(.75)(.20.10) = Given B/S =.75, then B =.75S, and so B/(B+S) =.75S/1.75S =.4286 and S/(B+S) = S/1.75S = This gives: r WACC =.4286(.65)(.10) (.24875) = With a debt-equity ratio of 1.5, then r S =.20 + (.65)(1.5)(.20.10) = Given B/S = 1.5, then B = 1.5S, and so B/(B + S) = 1.5S/2.5S =.6 and S/(B + S) = S/2.5S =.4. This gives: r WACC =.6(.65)(.10) +.4(.2975) = (a) To maximize the overall value of the firm, debt should be used to finance the $100 million purchase. Since interest payments are tax deductible, debt will reduce the firm s taxable income, creating a tax shield that will increase the overall value of the firm. (b) Since there are 15 million shares worth $32.50 per share, the total market value of the firm s equity is $487.5 million. As there is no debt, this is the overall value of the firm. The market value balance sheet is: Assets $487,500,000 Debt $0 Equity $487,500,000 Total assets $487,500,000 Total debt + equity $487,500,000 (c) 1. Since this is an all-equity firm, the discount rate to be used is the firm s unlevered cost of equity capital of 12.5%. We have: NPV = $100,000,000 + $25,000,000(.6).125 = $20,000, In an efficient market, the value of the firm will increase by the NPV of $20 million as soon as the purchase is announced. The market value balance sheet becomes: Old assets $487,500,000 Debt $0 NPV new assets $20,000,000 Equity $507,500,000 Total assets $507,500,000 Total debt + equity $507,500,000 The firm s share price after the announcement will be $33.83 ($507.5 million divided by 15 million shares). The number of shares which the firm must issue is the amount to be raised ($100 million) divided by the share price ($33.83), or 2,955, The firm will receive $100 million from the share issuance. This will increase the firm s assets and equity by $100 million, giving a market value balance sheet of: Old assets $487,500,000 Debt $0 Cash $100,000,000 Equity $607,500,000 NPV of project $20,000,000 Total assets $607,500,000 Total debt + equity $607,500,000 Since the firm issues 2,955,956 new shares, the total number of shares outstanding increases to 17,955,956. The share price is $607,500,000 divided by 17,955,956, or $ The present value of the project is $15,000,000/.125 or $120,000,000. The market value balance sheet becomes: Old assets $487,500,000 Debt $0 PV of project $120,000,000 Equity $607,500,000 Total assets $607,500,000 Total debt + equity $607,500,000 6

7 (d) 1. By MM Proposition I (Corporate Taxes), if the purchase is financed with perpetual debt, the firm will be worth: V L = V U + T C B = $607,500, ($100,000,000) = $647,500, After the debt issue, the value of the firm will rise by the NPV of the project, the cash received, and the PV of the tax shield. The market value balance sheet becomes: Old assets $487,500,000 Debt $100,000,000 Cash $100,000,000 Equity $547,500,000 NPV of project $20,000,000 PV tax shield $40,000,000 Total assets $647,500,000 Total debt + equity $647,500,000 After the purchase, the market value balance sheet would be: Old assets $487,500,000 Debt $100,000,000 PV of project $120,000,000 Equity $547,500,000 PV tax shield $40,000,000 Total assets $647,500,000 Total debt + equity $647,500,000 The price per share of the firm s stock is the total equity value of $547.5 million divided by the number of shares outstanding (15 million), or $ (e) Under equity financing, the price per share is $ Under debt financing, it is $ Therefore the price per share is maximized with debt financing (a) In a recession, the firm generates $100 million, but it has a required debt payment of $150 million. In this case the shareholders get nothing. In a boom, the firm generates $250 million, so the shareholders get $100 million (after debt is repaid). Therefore the value of equity is:.60($100,000,000) +.40($0) 1.12 = $53.57 million. (b) The promised rate of return on the firm s debt is the face value of debt divided by the market value of debt, minus one. In this case, we have (150/108.93) 1 = 37.70%. (c) We have: V = B + S = $ $53.57 = $162.5 million. (d) In a recession, the debt holders will receive $100 million (if there are no bankruptcy costs). In a boom, they will receive $150 million. The value of debt is therefore:.60($150,000,000) +.40($100,000,000) 1.12 = $ million. (e) Let X be the expected payoff after bankruptcy costs in a recession. We have: (.60)$ X $ = 1.12 $122 = $ X X = $32/.40 = $80, so the expected payoff is $80 million. (f) The expected bankruptcy costs if a recession happens are the amount the firm has ($100 million) less the expected payoff to the debt holders ($80 million), i.e. $20 million (a) In the case of Steinberg, the shareholders will receive $1.25 million if the expansion continues and $50,000 if a recession happens (these values being the firm s earnings in the two states of the economy 7

8 less the amount owed to debt holders of $750,000). The debt holders will receive $750,000 in either case. Thus:.80($1,250,000) +.20($50,000) Equity value: 1.15 Debt value: $750,000 = $652, = $878,261 In the case of Dietrich, it shareholders will receive $1 million if the expansion continues and nothing if a recession happens (since its earnings in a recession are less than the amount owed to the debt holders). The debt holders get paid $1 million in an expansion and $800,000 in a recession, so:.80($1,000,000) +.20($0) Equity value: = $695, ($1,000, ($800,000 Debt value: = $834, (b) The value of Steinberg is $878,261 + $652,174 = $1,530,435. The value of Dietrich is $695,652 + $834,783 = $1,530,435. (c) The CEO is wrong. The risk of bankruptcy does not affect a firm s value. It is the actual costs of bankruptcy that decrease a firm s value (and in this problem it is assumed that there are no costs of bankruptcy) The statement is false. If a firm has debt, it might be advantageous to stockholders for the firm to undertake risky projects, even those with negative net present values. This incentive results from the fact that most of the risk of failure is borne by bondholders. Therefore, value is transferred from the bondholders to the stockholders by undertaking risky projects, even if the projects have negative NPVs The firm should issue equity to finance the project. The tax-loss carry-forwards make the firm s effective tax rate zero. Therefore, the firm will not benefit from the tax shield that debt provides. Moreover, since the firm already has a moderate amount of debt in its capital structure, additional debt will likely increase the probability that the firm will face financial distress. As long as there are costs associated with financial distress, the firm should issue equity to finance the project (a) Low risk project:.50($500) +.50($700) = $600. High risk project:.50($100) +.50($800) = $450. The low risk project maximizes the value of the firm. (b) Low risk project:.50($0) +.50($200) = $100. High risk project:.50($0) +.50($300) = $150. (c) Risk-neutral investors prefer the strategy with the highest expected value. The firm s shareholderd prefer the high risk project since it gives the highest expected value of equity. (d) In order to make the stockholders indifferent between the low risk project and the high risk project, the bondholders will need to raise the required debt payment so that the expected value of equity is the same for both projects. Let X be the debt payment if the high risk project is chosen. Then $100 =.50($0) +.50($800 X) X = $600. Therefore the bondholders should raise the required debt payment by $100 if the high risk project is selected (a) 1. If the firm continues with all-equity financing, its value remains at its unlevered level of V U. To evaluate the change under debt financing, use MM Proposition I (Corporate and Personal Taxes): V L = V U + 1 (1 T C)(1 T S ) B = V U + 1 (.60)(.70) $13,500,000 (.70) = V U + $5,400,000. Since the value of the firm rises by $5.4 million under debt financing, this is the preferred choice. 8

9 (b) 2. Under the all-equity plan, corporate taxes paid by the firm are $1.2 million. Taxes paid by investors are 30% of the distributed net income of $1.8 million, or $540,000. Therefore, total taxes paid by the firm and its investors are $1.74 million. Under the levered plan, corporate taxes are $660,000, and taxes paid by investors are 30% of interest of $1.35 million (i.e. $405,000) and 30% of the distributed net income of $990,000, or $297,000. Therefore, total taxes paid are $1.362 million. Therefore, total tax revenue is higher in the all-equity financing case. 3. In the all-equity case: In the debt financing case: V U = EBIT(1 T C)(1 T S ) = $3,000,000(.60)(.70) = $6,300,000. r 0.20 V L = V U + 1 (1 T C)(1 T S ) = $6,300,000 + = $11,700, (.60)(.70) (.70) B $13,500, In the all-equity case, investors receive $1.8 million before tax, and they pay a tax rate of 20% on this, leaving an after tax annual cash flow of $1,440,000. Under debt financing, equity investors receive $990,000 before tax, and they keep 80% of this, leaving $792,000 after tax. 2. In the all-equity case, there is no debt, so after tax cash flow to debtholders is zero. Under debt financing, debt holders receive interest of $1,350,000 before tax, but they only keep 45% of this after tax, i.e. $607, (a) In their no tax model, MM assume that T C, T B, and C(B) are all zero. Under these assumptions, V L = V U, signifying that the capital structure of a firm has no effect on its value. There is no optimal debt-equity ratio. (b) In their model with corporate taxes, MM assume that T C > 0 and both T B and C(B) are zero. Under these assumptions, V L = V U + T C B (this is assuming perpetual debt to simplify the present value of the debt tax shield), implying that raising the amount of debt in a firm s capital structure will increase the overall value of the firm. This model implies that the debt-equity ratio of every firm should be infinite. (c) Under the assumptions that costs of financial distress are zero and the tax rate on equity distributions T S is also zero, MM Proposition I (Corporate and Personal Taxes) implies that: V L = V U + 1 (1 T C) B = V U $1,000, = V U + $175,000, so the firm value will increase by $175,000. (d) Since the firm has large tax-loss carry-forwards, the effective corporate tax rate T C is zero. Therefore: 1 V L = V U + 1 B = V U $1.80 = V U $0.25, so the firm value would decrease by $0.25 if it were to issue $1 of perpetual debt rather than equity. 9

10 17.12 (a) If the firm retires all of its debt, it will become an unlevered firm with: V U = EBIT(1 T C)(1 T S ) = $1,100,000(.65)(.90) = $3,217,500. r 0.20 (b) Use MM Proposition II (Corporate and Personal Taxes): V L = V U + 1 (1 T C)(1 T S ) B = $3,217, (.65)(.90) $2,000, = $3,657, (a) We have: (.10)$1,000 + (.40)$2,000 + (.50)$4,200 V U = = $15, (b) 1. MM Proposition I (No Taxes) implies that V L = V U, so firm value will remain at $15, Since the total value is $15,000, and this is 50% debt and 50% equity, value of the firm s equity is $7, Use MM Proposition II (No Taxes): r S = r 0 + (B/S)(r 0 r B ) =.20 + (1)(.20.11) = Without taxes, the weighted average cost of capital is: ( ) ( ) B S r WACC = r B + r S =.50(.11) +.50(.29) =.20. B + S B + S (c) 1. Taxes will reduce the value of the firm because the government becomes a claimant on the firm s assets. The size of the pie does not change, but there is less available for the firm s stockholders and bondholders. 2. We have: V L = V U + T C B = EBIT(1 T C) + T C B r 0 = $3,000(.60) +.40($7,500).20 = $12,000. (d) We have: V L = V U + 1 (1 T C)(1 T S ) = EBIT(1 T C)(1 T S ) + r 0 = $3,000(.60)(.85).20 = $7,650 + $1,615 = $9, B 1 (1 T C)(1 T S ) 1 (.60)(.85).65 $7,500 B 10

11 17.14 (a) The president is correct when he claims that common stock is the cheapest form of financing, as 9.5% is the lowest rate that the firm can obtain. The lowest after-tax rate on debt that is available for the firm is (.60)(.17) = 10.2%. Mr. Daniels is incorrect. If there are personal taxes, the increase in firm value is not T C B, but rather 1 (1 T C)(1 T S ) B. Ms. Henderson is also incorrect. Using the expression above, if corporate bonds are issued, the change in firm value is: V = 1 (.60)(1) $100,000,000 (.85) = $29.41 million, while if pollution control bonds are issued: V = 1 (.60)(1) $100,000,000 = $40 million. (1) Thus, Ms. Henderson s claim that the debt choice does not matter is incorrect. (b) The firm should not be indifferent about its financing choice. Issuing pollution control bonds adds value of $40 million, and so is the best choice. Issuing corporate bonds adds value of $29.41 million, and so is the second best choice. The worst option is issuing equity, which does not add any value. It is important to note that this analysis only implies that debt adds value relative to equity. The overall value of the firm is likely to fall with any of the three financing choices since pollution control equipment represents a cash outflow (a) Use the following general expression for the value of a levered firm with both corporate and personal taxes and present value of financial distress costs C(B) is: V L = V U + 1 (1 T C)(1 T S ) B C(B) = EBIT(1 T C)(1 T S ) + r 0 = $800,000(.65)(1) (1 T C)(1 T S ) 1 (.65)(1) (.85) = $5,200,000 + $282,353 $60,000 = $5,422,353. B C(B) $1,200,200.05($1,200,000) (b) The value of firm without debt would be $5,200,000 and its value with debt is $5,422,353, so the added value of the firm s debt is $222, There are several reasons why firms may choose legal bankruptcy over a private workout: (i) it may be less expensive (although this is not usually the case); (ii) equity investors can use legal bankruptcy to hold out and hope that the court violates strict priority; (iii) a complicated capital structure makes private workouts more difficult to arrange; and (iv) conflicts of interest between creditors, equity investors, and the firm s management can make private workouts impossible The proceeds from the liquidation should be distributed as follows: 11

12 Prior claim Cash received under liquidation Trade credit $2,000 $2,000 Secured notes $2,000 $2,000 Senior debenture $6,000 $6,000 Junior debenture $2,000 $0 Equity $(2,000) $ There are many possible reorganization plans that could work. Here is one possibility: Assets Claims Going concern value $15,300 Senior debenture $10,000 Junior debenture $5,000 Equity $300 The holders of mortgage bonds would receive senior debentures in equal amounts. The holders of senior debentures would receive junior debentures at cents to the dollar. The holders of the junior debentures would receive equity at 12.5 cents to the dollar The mortgage bonds are secured by the buildings and would receive the $7.5 billion proceeds from the sale of the buildings. The remaining $4.5 billion claim would be included with unsecured creditors, who collectively would share the residual on a proportional basis: Claims Proposed Distribution ($ millions) ($ millions) Admin. costs and other $900 $900 Mortgage bonds $12,000 $9,023 Subordinated debentures $15,000 $5,077 The amount remaining after administrative costs, other expenses, and secured claims is $15 billion less $7.5 billion less $900 million, or $6.6 billion. The claims of the unsecured creditors are $4.5 billion (remaining amount owed on mortgage bonds) plus $15 billion (subordinated debentures), a total of $19.5 billion. Mortgage bondholders receive (4.5/19.5) 6.6 or $1.523 billion (which, when added to the $7.5 billion from the sale of buildings, gives a total of $9.023 billion). Holders of subordinated debentures receive (15/19.5) 6.6 or $5.077 billion There are many potential reorganization plans. One example: Assets Claims Going concern value $22.5 billion Debentures $ billion Equity $ billion The holders of mortgage bonds would receive debentures for $ billion and the holders of subordinated debentures would receive equity worth $ billion. In both cases the value is greater than they would have received in the liquidation. 2. Suppose an investor buys 5 shares at a cost of $50 (i.e. $10 per share), and also lends out $50 at the given interest rate of 10%. The overall position of this investor would be as follows: Expected State Value Probability EPS $0.00 $0.50 $2.00 $3.50 $4.00 $2.00 Earnings on 5 shares $0.00 $2.50 $10.00 $17.50 $20.00 $10.00 Interest (10% on $50) $5.00 $5.00 $5.00 $5.00 $5.00 $5.00 Dollar returns $5.00 $7.50 $15.00 $22.50 $25.00 $15.00 Percentage returns 5% 7.5% 15% 22.5% 25% 15% (on $100 invested) 12

13 Note that the percentage returns across the various states are identical to the ROE obtained if the firm stays with the all-equity financing strategy, as shown in the table on slide 12 in the notes. 3. (a) False. An increase in leverage will increase both the risk of the equity and its expected return. MM Proposition I shows that these two effects exactly offset each other, leaving the price of the stock unchanged. (b) False. Borrowing has some benefits, such as the debt tax shield and reduced agency costs of equity (i.e. added discipline to management). It has costs as well, such as costs of financial distress, agency costs of debt, and loss of financial flexibility. If the marginal benefit of adding debt exceeds its marginal cost, then firms should borrow. In this case, borrowing will reduce the cost of capital for the whole firm, generally through a higher portion of debt which has a cost which is lower than that of equity. (Note: There are several other ways to address this question. For example, one can argue on the basis of MM Proposition I (No Taxes) that the firm s cost of capital is not affected by the debt-equity ratio. As the debt-equity ratio falls, it is true that the cost of equity falls, but a smaller proportion of the firm is being financed by the lower cost of debt. The overall effect is to leave the firm s cost of capital unchanged. Another way to address this question would be to use MM Proposition II (Corporate Taxes). In this case, the cost of capital is decreasing in the amount of debt. As the firm borrows more, the stock becomes riskier and so its expected return must rise. However, firm value increases with debt because of the debt tax shield, meaning that the cost of capital for the firm is smaller. In either case, if you are making an argument based on one of the MM propositions, you should clearly state your assumptions.) (c) False. First, firms with more volatile earnings will have higher probability of incurring financial distress, so they will tend to have lower leverage. Second, this statement involves the tradeoff theory, not the pecking order theory, because higher default probability affects the cost of debt and thus the tradeoff between its marginal benefits and marginal costs. 4. (a) Firm value: Share price: $140. (b) Firm value: Share price: $155. V U = $30,000,000(.7).15 = $140,000,000. V L = V U + T C B = $140,000, ($50,000,000) = $155,000,000. (c) Shares repurchased: $50,000,000/$155 = 322, shares. (d) The upfront issuing costs are $500,000 (1% of $50 million). Since the debt is perpetual, these costs can be amortized over 5 years. This means that the firm can take a tax deduction of $100,000 per year at the end of each of the next 5 years. The overall present value of the issuing costs is then: $500, ($100,000)A 5.10 = $386,276. Then firm value would be reduced by the present value of these costs, i.e. V L = $155,000,000 $386,276 = $154,613,723. The price per share would be $ The net amount borrowed would be 99% of $50 million (since 1% is lost upfront as a result of issuance costs), which is $49.5 million. The number of shares repurchased is $49.5 million divided by the share price of $154.61, or 320, (e) Since the bonds are issued at par with annual interest payments, the interest rate is equal to the yield. Let the spread be X. Then the interest coverage ratio will be: Interest coverage ratio = $30,000,000 $50,000,000(.04 + X) 13

14 If we try X =.03, we would have annual interest payments of $3.5 million (7% of $50 million). This would imply an interest coverage ratio of = 8.57, which is greater than 5, and hence inconsistent with the given rating table. However, if we try X =.01, we have annual interest payments of $2.5 million (5% of $50 million). This leads to an interest coverage ratio of 12, which is consistent with the given rating table. Therefore, the new firm value is the previous value of $155 million reduced by the present value of the costs of bankruptcy, i.e. The share price is then $ $155,000, $155,000,000 = $154,612, (a) Consider the position of the investor before the capital structure change. If the firm s income X is less than the debt obligation of $100, the investor gets zero. If X exceeds $100, the investor gets 10% of the difference: X $100 X > $100 10% of firm s equity 0.10(X $100) Now consider the position of the investor after the capital structure change, assuming that the investor does not sell any shares. The firm now has a total debt obligation of $300. If X $300, the investor will receive nothing. Conversely, if X > $300, the investor will receive 20% of the income in excess of $300. This is clearly different from the original situation prior to the new debt issue. However, suppose the investor buys 10% of the new debt, financing this by selling back half of the shares and thus restoring the equity stake to 10%. If X $100, the new debt holders get nothing. If X is between $100 and $300, the new debt holders receive the excess of the amount over $100. If X > $300, the new debt holders are paid in full and the shareholders receive the excess amount over $300. This means that the investor has effectively undone the capital structure change: X $100 $100 < X $300 $300 < X 10% of firm s equity (X $300) 10% of new debt 0.10(X $100).10(200) Total 0.10(X $100).10(X $100) Note that the investor s total position is identical to what it was before the change. If X $100, the investor gets nothing; if X > $100, the investor gets 10% of the excess over $100. (b) Assuming no taxes or costs of financial distress, the overall effect on the value of the firm is zero: investors can achieve the same pattern of returns both before and after a new issue of subordinated debt at the same cost. 6. (a) We have: NPV A =.5($120) +.5($120) $100 = $20 million NPV B =.5($40) +.5($180) $100 = $10 million. (b) With naive lenders, the payoff to equity holders with project A is: The payoff to equity holders with project B is:.5($120 $100) +.5($120 $100) = $20 million..5($0) +.5($80) = $40 million. The shareholders will prefer the risky project B. Given that B is chosen, the naive lenders will receive an expected payoff of:.5($40) +.5($100) = $70 million. 14

15 (c) Sophisticated lenders will realize that equity holders have an incentive to invest in the riskier project. For the $100 million loan, they will require a promised future payment of $160 million. In this case they recover their investment since their expected payoff is: The expected payoff to equity holders is then:.5($40) +.5($160) = $100 million..5($0) +.5($20) = $10 million. The equity holders expected payoff is zero if the safe project A is chosen because in neither state of the economy will the debt obligation be met. Therefore, the equity holders will take project B despite its lower NPV. If the equity holders had been able to commit to taking project A, they would have created $20 million in value instead of $10 million. 7. $8 million must be allocated to the unpaid wages, taxes, and legal fees, levaing $47 million to be allocated to the various security holders. Of this, $20 million from the sale of the property will go to the secured debt holders, leaving $27 million for the balance of their claim ($4 million) and the senior debenture holders. Since the senior debenture holders are owed $30 million and the secured debt holders are owed $4 million, the $27 million needs to be split across security holders who are owed $34 million. Therefore, secured debt holders should receive 4/34 of the $27 million, or $3.176 million, and senior debenture holders should receive 30/34 of the $27 million, or $ million. Summarizing: Unpaid wages, taxes, and legal fees $8 million Secured debt $ million Senior debentures $ million Junior debentures $0 Common stock $0 Total $55 million 15