Chapter 16 in Class problems-Solutions-Final PDF

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FNCE 3P93 Corporate Finance II, Solutions for Chapter 16 In-class Practice Problems 1. You own 25% of Unique Vacations, Inc. You have decided to retire and want to sell your shares in this closely held, all equity firm. The other shareholders have agreed to have the firm borrow $1.5 million to purchase your 1,000 shares of stock. What is the total value of this firm today if you ignore taxes?

Price per share=

$ 1,500,000 =$ 1,500 1,000

Current no of shares=

1,000 =4,000 0.25

[ ( 4,000 −1000 )∗$ 1,500 ] +$ 1,500,000=$ 6,000,000∨simply 4,000∗$ 1,500=$ 6,000,000 2. Your firm has a debt-equity ratio of .75. Your pre-tax cost of debt is 8.5% and your required return on assets is 15%. What is your cost of equity if you ignore taxes?

r E=r A +( r A −r D)

D 0.75 =0.15+ ( 0.15− 0.07 ) =0.1988=19.88 % E 1

3. Bigelow, Inc. has a cost of equity of 13.56% and a pre-tax cost of debt of 7%. The required return on the assets is 11%. What is the firm’s debt-equity ratio based on MM Proposition II with no taxes?

r E=0.1356=r A + ( r A −r D )

D D D =0.11 + ( 0.11 −0.07) → =0.64 E E E

4. Gail’s Dance Studio is currently an all equity firm that has 80,000 shares of stock outstanding with a market price of $42 a share. The current cost of equity is 12% and the tax rate is 34%. Gail is considering adding $1 million of debt with a coupon rate of 8% to her capital structure. The debt will be sold at par value. What is the levered value of the equity?

V L =V U + D T C=(80,000∗$ 42)+ ( 1,000,000∗0.34 )=3,700,000 V L =Firm Value=V E +V D → VE = $3.7m − $1m = $2.7m 5. Hey Guys!, Inc. has debt with both a face and a market value of $3,000. This debt has a coupon rate of 7% and pays interest annually. The expected earnings before interest and taxes is $1,200, the tax rate is 34%, and the unlevered cost of capital is 12%. What is the firm’s cost of equity?

EBIT ( 1−T C )

1,200∗( 1−0.34 ) =$ 6,600 0.12 RU V L =V U + D T C=6,600+ (3,000∗0.34 )=$ 7,620 V U=

=

V L =Firm Value=V E +V D → V E=V L−V D =$ 7,620−$ 3,000=$ 4,620

r E=r A +( r A −r D)

D 3,000 ( 1−T c )=0.12+ ( 0.12 −0.07 ) 4,620 ∗(1−0.34 )=0.1414=14.14 % E

6. Bertha’s Boutique has 2,000 bonds outstanding with a face value of $1,000 each and a coupon rate of 9%. The interest is paid semi-annually. What is the amount of the annual interest tax shield if the tax rate is 34%?

Annual Intrerest tax Shield =D∗R D∗T c →(2000∗1,000)∗0.09∗0.34 =61,200 7. A firm has debt of $5,000, equity of $16,000, a leveraged value of $8,900, a cost of debt of 8%, a cost of equity of 12%, and a tax rate of 34%. What is the firm’s weighted average cost of capital?

WACC=

E 16,000 D 5,000 ∗8 %∗ ( 1−0.34 )=10.40 % r E + r D ( 1−T C )= ∗12 %+ 21,000 V V 21,000

8. If a firm is unlevered and has a cost of equity capital 12%, what would its cost of equity be if its debtequity ratio became 2? The expected cost of debt is 8%.

r E=r A +( r A −r D)

D 2 ∗(1−T c )=0.12 + ( 0.12− 0.08 ) =0.20=20 % E 1

9. A firm has zero debt in its capital structure. Its overall cost of capital is 9%. The firm is considering a new capital structure with 40% debt. The interest rate on the debt would be 4%. Assuming that the corporate tax rate is 34%, what would its cost of equity capital with the new capital structure be?

r E=r A +( r A −r D )

( 0.09− 0.04) ∗0.4 D ∗( 1− 0.34 )=0.112=11.2 % ∗ ( 1−T c ) =0.09+ 0.6 E

10. Consider two firms, U and L, both with $100,000 in assets. Firm U is unlevered, and firm L has $40,000 of debt that pays 5% interest. There are two investors, Mike and Steve, who own 20% of firm L each. Mike believes that leverage works in his favor. Steve says that this is an illusion, and that with the possibility of borrowing on his own account at 5% interest, he can replicate Mike's payout from firm L. Given a level of operating income of $5000, show the specific strategy that Steve has in mind. Mike is entitled to 0.2 ($5000 - $2000) = $600  0.2* (Operating income-Interest expense) Steve liquidates his position (20% of L’s equity) and receives 20% * (100,000-40,000) = $12,000 Steve borrows $8,000 at 5% interest (20% of Firm L's Debt) and pays $400 in interest. Steve then purchases 20% of Firm U and pays 20% * (100,000) = $20,000 Steve is now entitled to receive 20% * ($5000) = $1000 from firm U’s earnings. Steve's total payout is $1000 - $400 = $600, or Mike's payout.

11. The Nantucket Nugget is unlevered and is valued at $640,000. Nantucket is currently deciding whether including debt in its capital structure would increase its value. The current cost of equity is 12%. Under consideration is issuing $300,000 in new debt with an 8% interest rate. Nantucket would repurchase $300,000 of stock with the proceeds of the debt issue. There are currently 32,000 shares outstanding and effective marginal tax bracket is zero. What will Nantucket's new WACC be?

New Firm Value: $640,000 + (.0) ($300,000) = $640,000 Capital Structure = D + E = 300,000 + 340,000 rE = 0.12 + (300/340)*(0.12 - 0.08) = 0.12 + 0.0353 = 0.1553 = 15.53% WACC = (300/640)*(0.08) + (340/640)*(0.1553) = 0.0375 + 0.0825 = 0.12 = 12% The value of the firm stays at $640,000 (MM I), the cost of levered equity rises to 15.53% and the WACC remains at 12%. 12. (WACC) The Nantucket Nugget is unlevered and is valued at $640,000. Nantucket is currently deciding whether including debt in its capital structure would increase its value. The current of cost of equity is 12%. Under consideration is issuing $300,000 in new debt with an 8% interest rate. Nantucket would repurchase $300,000 of stock with the proceeds of the debt issue. There are currently 32,000 shares outstanding and its effective marginal tax bracket is 34%. What will Nantucket's new WACC be? New Firm Value: $640,000 + (.34) ($300,000) = $742,000 Capital Structure = D + E = $300,000 + $442,000 rE = 0.12 + (300/442)*(0.12 - 0.08)*(1-0.34) = 0.12 + 0.0179 = 0.1379 = 13.79% WACC = (300/742)*(0.08)*(1-0.34) + (442/742)*(0.1379) = 0.0213 + 0.0821 = 0.1034 = 10.34% The value of the firm increases to $742,000 (From Value of the Tax Shield), increasing the relative weight of equity and the cost of levered equity rises to 13.79% and the WACC falls to at 10.34% consistent with the increase in firm value. 13. (BREAK-EVEN EPS EBIT) A company’s capital structure is made up of 200,000 common shares and $1,000,000 debt at 12 percent interest. The company’s tax rate is 50 percent. An additional $500,000 has to be raised, and the following financing alternatives are available: Common shares: The company can sell additional shares at $10 a share. Hence, 50,000 new shares would have to be issued. Debt: Debt can be issued at 12 percent, requiring interest payments of $60,000. (Note that debt is not used to repurchase shares here.) Compute EPS as a function of EBIT for both alternatives and derive the break-even point. Solution: For the break-even EBIT, EPS values under the “shares” and “debt” capital structures should be equal. Earnings per share for each alternative is computed as follows.

EPSshares=

EPSdebt=

(EBIT −Interest )(1−T c ) [ EBIT − (1,000,000∗0.12 ) ]∗( 1−0.5) = 200,000 + 50,000 ¿ of shares outstanding

(EBIT −Interest )(1−T c ) { EBIT− [ ( 1,000,000∗0.12 ) +60,000 ] }∗( 1−0.5 ) = 200,000 ¿ of shares outstanding

EPSshares=EPS debt

→

[ EBIT −( 1,000,000∗0.12 ) ]∗0.5 {EBIT −[ ( 1,000,000∗0.12 ) +60,000 ]}∗0.5 250,000

=

200,000

( EBIT −120,000 )∗0.5 ( EBIT−180,000 )∗0.5 = →EBIT =$ 420,000 250,000 200,000 Note that the numerator represents the after-tax earnings, while the denominator is simply the number of common shares outstanding. To find the break-even point, equate the EPS for the two alternatives and solve for the EBIT. If EBIT is below $420,000, common shares would be favoured, but if EBIT is above $420,000 debt would be favoured. 14. Homemade Leverage Example: • • • •

You have 150 shares of an all-equity firm. There are 9,000 shares outstanding. EBIT = $40,000 and P0 = $42 Company is considering a new capital structure with 40% debt at 8% interest rate. Ignore taxes.

a) Calculate the current and proposed capital structures. Current Cap. Structure: Assets Debt (0) 378,000 Equity 9000*42=378,000

Proposed Cap. Structure: Assets Debt (378000*0.4) = 151,200 378,000 Equity 226,800 Firm will use debt in order to repurchase shares.

151200 =3600 shares will be repurchased 42 b) What’s the CF to you (as a S/H of 150 shares) under both capital structures?

Current (unlevered) EBIT = $40,000 Int = 0 Tax = 0

Proposed (levered) EBIT = $40,000 Int = (151,200*0.08) = $12,096 Tax = 0

40,000 ¿ = =$ 4.444444 ¿ of shares 9000 40,000 −12,096 ¿ =$ 5.167407 = 9000−3600 ¿ of shares

EPS =

EPS =

CF to you  150 shares * 4.44 EPS = $666.667

CF to you  150 shares * 5.17 EPS = $775.11

c) Assume that the firm keeps the current all-equity capital structure, and you’d like to have the proposed one. Can you use the homemade leverage to replicate the payoff of proposed capital structure? CFunl = $666.667  You have this.

CFunl = $775.11  You’d like to have this.

Increase leverage  You own stock of unlevered firm + Borrow & purchase more = Levered CF. You need to replicate the capital structure of the levered firm in your personal portfolio. Debt = 151,200 Equity = $226,800 

Debt 151,200 2 = = Equity 226,800 3

 this D/E ratio will be applied in your own portfolio.

Personal equity = 150 *42 = $6300

Personal debt = 6300 *

2 =4200 3

So, you need to borrow $4,200 and pay 8% interest for this loan just like the firm. You will purchase additional stocks with this loan 

4200 =100 shares will be purchased . 42

Now, you own 150 + 100 = 250 shares of the unlevered firm and $4,200 of personal debt. CF to you  250 * 4.444444 = $1111.11 Interest on debt  4,200 * 0.08 = $336 Net CF to you  1110 – 336 = $775.11 You have successfully replicated the payoff of the investor who invests in the levered firm by staying in the unlevered firm and mimicking the levered firm’s capital structure in your personal portfolio. d) Now assume that you own 150 shares in the company that follows the proposed capital structure, that is levered and following a D/E ratio of 2/3. Can you use the homemade leverage to replicate the payoff of current capital structure (unlevered firm’s capital structure)? CFunl = $666.667  You’d like to have this.

CFunl = $775.11  You have this.

Decrease leverage in your personal portfolio  You own stock of levered firm + Sell & Lend = UnLevered CF. You need to replicate the capital structure of the unlevered firm in your personal portfolio.



Debt 151,200 2 = = Equity 226,800 3

 this D/E ratio will be applied in your own portfolio.

Personal levered investment value = 150 *42 = $6300, only the equity portion of proposed cap. Structure. Personal lending should be  6300 *

2 =$ 2520 5

of your levered investment value.

So, you need to sell $2520 worth of stocks and lend it to earn 8% interest.

$ 2520 =60 shares need ¿ be sold . 42 After that, you’ll be left with 150-60=90 shares, each of which will bring $5.1674 EPS. You’ll make the following CF from earnings  90 * $5.1674 = $465.06 You’ll also make an interest income of  $2520 * 8% = $201.60 So, you’ll have a total CF of $465.06 + $201.60 = $666.66 You have successfully replicated the payoff of the investor who invests in the unlevered firm by staying in the levered firm and mimicking the unlevered firm’s capital structure in your personal portfolio.

15. MM Arbitrage Example 1: The Veblen Company and the Knight Company are identical in every respect except that Veblen is not levered. Financial information for the two firms appears in the following table. All earnings streams are perpetuities, and neither firm pays taxes. Both firms distribute all earnings available to common stockholders immediately.

Operating Cash Flow Year-end interest on debt Market Value of stock Market Value of debt

Veblen 400,000 ----3,600,000 -----

Knight 400,000 72,000 2,532,000 1,200,000

a) What will the annual cash flow be to an investor who purchases 5% of Knight's equity? Our investor will switch from Knight’s equity to that of Veblen. How much is she going to need extra? Cash flow from Knight to shareholder = 0.05*($400,000 – 72,000) = $16,400 To purchase 5 percent of Knight’s equity, the investor would need: Knight investment = 0.05*($2,532,000) = $126,600 And to purchase 5 percent of Veblen without borrowing would require: Veblen investment = 0.05*($3,600,000) = $180,000 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:

Amount to borrow = $180,000 – 126,600 = $53,400 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 operating income is $400,000, it must pay $72,000 to debt holders before distributing any of its earnings to stockholders. So, the amount available to this shareholder will be: b) What is the annual net cash flow to the investor if 5 percent of Veblen's equity is purchased instead? Assume that borrowing occurs so that the net initial investment in each company is equal. The interest rate on debt is 6 percent per year. Veblen will distribute all of its earnings to shareholders, so the shareholder will receive: Cash flow from Veblen to shareholder = 0.05*($400,000) = $20,000 However, to have the same initial cost, the investor has borrowed $53,400 to invest in Veblen, so interest must be paid on the borrowings. The net cash flow from the investment in Veblen will be: Net cash flow from Veblen investment = $20,000 – 0.06*($53,400) = $16,796 For the same initial cost, the investment in Veblen produces a higher dollar return. c) Given the two investment strategies in (a), which will investors choose? 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 Veblen’s equity to rise and/or the market value of Knight’s equity to fall. Any differences in the dollar returns to the two strategies will be eliminated, and the process will cease when the total market values of the two firms are equal. 16. MM Arbitrage Example 2 There are two firms, U and L, having the same EBIT of $20,000. They are identical in every respect except firm L has a debt of $100,000 at 7% rate of interest. The cost of equity of firm U is 10% and that of firm L is 11.5%. Assume that you own 10% of shares of the firm L. Find how the arbitrage principle will be applied in this setting. Also assume that all earnings streams are perpetuities. There are no taxes and both firms distribute all earnings available to common stockholders.

EBIT Debt RE

Value of Equity = VU equity =

Firm U(unlevered) $20,000 ---10%

EBIT −Interest Cost of Equity

Since firm U has no debt VU = VU equity = $200,000

=

Firm L(Levered) $20,000 $100,000 11.5%

20,000 −0 =$ 200,000 0.10

for firm U.

Value of Equity = VL equity =

EBIT −Interest Cost of Equity

=

20,000−(100,000∗0.07 ) =$ 113,043 for firm 0.115

L. VL = VD + VL equity =100,000 + 113,043 = $213,043 Assume that you have 10% of shares in firm L  so your net worth of investment in firm L 10%* 113,043 = $11,304.30 Assume that you sell your shares in firm L and receive $11,304.30 You borrow the same percentage (10% as equity) of debt from L’s debt, at a cost of 7%  10%*100,000 = $10,000 Now, you have $11,304.30+$10,000= $21,304.30 to invest in firm U. If you purchase the same 10% of firm U, you need to pay $20,000 and your interest to be paid is 10,000*0.07= $700 Here you invest $20,000 and receive a 10% return in firm U and pay your interest expenses. So, the net proceeds from this switch from firm L to firm U and adding as much equity as debt you have in firm L can be calculated as (20,000 * 0.1) – 700 = $1300 There also exists a surplus of $21,304.30-$20,000 = $1,304.30 In short, you have reached to a better position by moving from firm L to firm U and introducing as much leverage as equity to your portfolio. That is, you not only kept the same amount of ownership, 10%, this time in firm U, but also maintained a surplus of $1,304.30 17. MM Arbitrage Example 3 Company U and company L are identical in every respect except company U is unlevered and company L has $2,000,000 perpetual debt with an interest rate of 6%. Both companies are expecting to have an EBIT of $500,000 in perpetuity and all earnings will be immediately distributed to common shareholders. The cost of equity Company U is 12% and the cost of equity for company L is 15%. Assume that all Modigliani and Miller assumptions are satisfied (individuals can borrow or lend at the same rate as the corporations). a) Calculate the value of firm U.

V U=

EBIT 500,000 =$ 4,166,667 = 0.12 RA

b) Calculate the total value of firm L and its equity portion.

V L=

EBIT −∫ Expense 500,000−(2,000,000∗6 % ) + debt= +2,000,000 =$ 4,533,333 RE 0.15

Equity portion is the first part of the equation 

500,000−(2,000,000∗6 %) =$ 2,533,333 0.15

c) Assuming an investor who owns 15% of firm L’s equity; show that if she will be able to increase her wealth using the MM arbitrage argument. Also, show that future expected cash flows cash flows are not affected. She’ll switch her investment position from L to U: Step I: If she sells her 15% of equity in firm L, she’ll receive  15%*

$ 2,533,333=$ 380,000 Step II: She’ll borrow 15% of L’s debt  15%*2,000,000 = $300,000 at a cost of 6%. Step III: She’ll purchase 15% of firm U for  15 %∗$ 4,166,667=$ 625,000 Step IV: Profit from this switch = ($380,000 + $300,000) - $625,000 = $55,000, or Step I: If she sells her 15% of equity in firm L, she’ll receive  15%*

$ 2,533,333=$ 380,000 Step II: She’ll borrow as much (in %) as of L’s debt 

D 2,000,000 = *380,000 = $300,000 E 2,533,333

at a cost of 6%. Step III: She’ll purchase 15% of firm U for  15 %∗$ 4,166,667=$ 625,000 Step IV: Surplus (profit) from this switch = ($380,000 + $300,000) - $625,000 = $55,000 CF before  15% (500,000 – (2,000,000*6%)) = $57,000 CF after  15% (500,000) – (300,000*6%) = $57,000 18.MM Arbitrage Example 4 Company U and company L are identical in every respect except company U is unlevered and company L has $3,000,000 perpetual debt with an interest rate of 4%. Both companies are expecting to have an EBIT of $500,000 in perpetuity and all earnings will be immediately distributed to common shareholders. The Company U has 400,000 shares outstanding and the current shares price is $20. Company L has 600,000 shares outstanding and the current share price is $10. Assume that all Modigliani and Miller assumptions are satisfied (individuals can borrow or lend at the same rate as the corporations). a) Calculate the cost of equity for each firm.

EBIT = 500,000 → R E=6.25 % RE RE EBIT −interest expense 500,000−(3,000,000∗0.04) 380,000 = = V L =600,000∗10= 6,000,000= → RE RE 6,000,000 V U =400,000∗20 =8,000,000...