DUKE UNIVERSITY Fuqua School of Business. FINANCE CORPORATE FINANCE Problem Set #7 Prof. Simon Gervais Fall 2011 Term 2.

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1 DUKE UNIVERSITY Fuqua School of Business FINANCE CORPORATE FINANCE Problem Set #7 Prof. Simon Gervais Fall 2011 Term 2 Questions 1. Suppose the corporate tax rate is 40%, and investors pay a tax rate of 15% on income from dividends or capital gains and a tax rate of 1 3 (i.e., 33.3%) on interest income. Your firm decides to add debt so it will pay an additional $15 million in interest each year. It will pay this interest expense by cutting its dividend (i.e., the operations will not change, and so shareholders will receive less). (a) How much will the debt-holders receive after paying taxes on the interest they earn each year? (b) By howmuchwill thefirmneedtocutitsdividendeach year topaythisinterest expense? (c) By how much will this cut in the dividend reduce equity-holders annual after-tax income (d) How much less will the government receive in total revenues each year? (e) What is the effective tax advantage of debt t? 2. Suppose that, in an effort to reduce the federal deficit, Congress increases the top personal tax rate on interest and dividends to 44% but retains a 20% tax rate on realized capital gains. The corporate tax rate stays at 35%. Assuming that capital gains are half of equity income, compute the total corporate plus personal taxes paid on debt versus equity income if (a) all capital gains are realized immediately; (b) capital gains are deferred forever. 3. The XYZ Co. is assessing its current capital structure and its implications for the welfare of its security holders. XYZ is currently financed entirely with common stock, of which 1,000 shares are outstanding. Given the risk of the underlying cash flows (EBIT, earnings before interest and taxes) generated by XYZ, investors currently require a 20% return on the XYZ common. The company pays out all expected earnings as dividends to common stockholders, and these expected earnings are based on the expected operating earnings (EBIT) generated by the firm s assets. XYZ estimates that operating earnings may be either 1,000, 2,000 or 4,200 with respective probabilities of 0.1, 0.4 and 0.5 depending on future economic conditions. Further, the firm expects to produce a level stream of EBIT in perpetuity (assume that, every year, Depreciation is equal to CapEx and that NWC = 0). Assume that the corporate and personal tax rates are equal to zero. (a) Given the above facts, compute (i) the value of the firm; 1

2 (ii) the market value of a common share; (iii) the expected earnings per share of common stock; (iv) the firm s weighted average cost of capital. (b) The president of XYZ has come to the conclusion that shareholders would be better off if the company had equal proportions of debt and equity. He therefore proposes to issue $7,500 of debt at an interest rate of 10% and use the proceeds to repurchase equity. Analyze this proposition by computing (i) the new value of the firm; (ii) the value of debt; (iii) the value of equity; (iv) the number of common shares repurchased, and the price of one such share; (v) the required rate of return on equity; (vi) the firm s weighted average cost of capital. (c) Lift the assumption of no corporate taxes and now assume that the corporate income tax rate is 40%. Assuming no risk of bankruptcy for a debt-equity ratio less than one, how does the existence of a corporate income tax affect your answers to part (b)? If bankruptcy risks were a concern, how would this (qualitatively) affect your answers? (d) Now lift the assumption that personal taxes are zero. Suppose that returns to equity holders are effectively not taxed but that interest is taxed at the personal level. How will your answer to parts (c.i) to (c.iii) change if the recipients of interest on the firm s debt are in the (i) 30% personal income tax bracket? (ii) 40% personal income tax bracket? 4. Gladstone Corporation is about to launch a new product. Depending on the success of the new product, Gladstone may have one of four values next year: $150 million, $135 million, $95 million, or $80 million. These outcomes are all equally likely (i.e., they occur with a 25% probability each), and this risk is diversifiable. Suppose that the riskfree interest rate is 5% and assume perfect capital markets (i.e., no market imperfections). (a) What is the initial value of Gladstone s equity without leverage? Now suppose that Gladstone has zero-coupon debt with a $100 million face value due next year. (b) What is the initial value of Gladstone s debt? (c) What is the yield-to-maturity of the debt? What is its expected return? (d) What is the initial value of Gladstone s equity? What is Gladstone s total value with debt? 5. As in the last problem, Gladstone Corporation is about to launch a new product. Depending on the success of the new product, Gladstone may have one of four values next year: $150 million, $135 million, $95 million, or $80 million. These outcomes are all equally likely (i.e., they occur with a 25% probability each), and this risk is diversifiable. Suppose that the 2

3 riskfree interest rate is 5% and that, in the event of default, 25% of the value of Gladstone s assets will be lost to bankruptcy costs. Ignore all other market imperfections, such as taxes. (a) What is the initial value of Gladstone s equity without leverage? Now suppose that Gladstone has zero-coupon debt with a $100 million face value due next year. (b) What is the initial value of Gladstone s debt? (c) What is the yield-to-maturity of the debt? What is its expected return? (d) What is the initial value of Gladstone s equity? What is Gladstone s total value with leverage? Suppose that Gladstone has 10 million shares outstanding and no debt at the start of the year. (e) If Gladstone does not issue debt, what is its share price? (f) If Gladstone issues debt of $100 million due next year and uses the proceeds to repurchase shares, what will its share price be? Why does your answer differ from that in part (e)? 6. Nadir Inc., an unlevered firm, has expected earnings before interest and taxes of $2 million per year. Nadir s tax rate is 40%, and its market value is V = E = $12 million. The stock has a beta of 1, the riskfree rate is 9%, and assume that r m r f = 6%. Management is considering the use of debt; debt would be issued and used to buy back stock, and the size of the firm would remain constant. The default-free interest rate on debt is 12%. Because interest expenses are tax deductible, the value of the firm would tend to increase as debt is added to the capital structure, but there would be an offset in the form of the rising cost of bankruptcy. The firm s analysts have estimated that the present value of any bankruptcy cost is $8 million, and that the probability of bankruptcy will increase with leverage according to the following schedule: Value of Probability of Debt Failure (%) 2,500, ,000, ,500, ,000, ,000, ,000, ,500, (a) What is the cost of equity and cost of capital at this time? (b) What is the optimal capital structure when bankruptcy costs are considered? (c) What will the value of the firm be at this optimal capital structure? (Optional) 7. Suppose that every investor in the economy is risk-neutral (i.e., does not care about risk), and that the riskfree rate is 10% (this means that all discounting is done at 10% in this economy). 3

4 Suppose further that the economy in one year will turn out to be good with a probability of 1/4, medium with a probability of 1/2, and bad with a probability of 1/4. Now, Xirdneh Imij (XI) is a firm which has two assets: $10 million in cash which is invested in T-bills (and will generate $11 million for sure in one year), and a one-year project whose end-of-year cash flowdependsonthestate oftheeconomy at that time: it will be$349 million, $99 million and $25 million in the good, medium and bad states respectively. Assume that XI will cease to exist at the end of the year, i.e., it will return all of its assets at that time to the appropriate investors. However, if the firm is not solvent (if its assets do not generate enough to pay back the bondholders), it will cost $10 million in legal fees to sort out who gets what. Also, assume that there are no corporate or personal taxes in this economy. (a) Suppose that the firm is currently all-equity financed. What is the value of the equity today? (b) Now suppose that XI is considering a change in its capital structure. The firm will raise $30 million by issuing one-year bonds which promise an interest rate of 10%. The $30 million will be used to buy back some equity. (i) How much does the firm owe the bondholders in one year (including both interest and principal)? (ii) What will the bondholders get in the good, medium and bad state respectively? (iii) Show that the debt is indeed worth $30 million today (by discounting the expected payoff). (iv) What is the new value of the equity? (v) Are the shareholders better off, worse off, or indifferent? Why? (c) Suppose instead that XI tries to raise $40 million by issuing bonds which still promise an interest rate of 10%. (i) Show that investors would not purchase the bonds. (Hint: Proceed as in parts (b)-(i) to (b)-(iii).) (ii) What is the minimum promised rate (the fair rate) at which investors would start buying the bonds? (iii) Show that, at that fair rate, the shareholders are made worse off. Why is that the case? (d) Suppose that XI decides to go forward with the refinancing in part (b) (i.e., suppose XI issues $30 million worth of debt at 10%). Suppose that XI is considering investing all of its available cash ($10 million) in a second one-year project that will generate an end-of-year cash flow of $23 million, $13 million and $3 million in the good, medium and bad states respectively. (i) What is the net present value of the project if taken by itself (i.e., ignoring the firm s other assets and financial structure)? (ii) Show that rational bondholders would have insisted that a clause preventing the firm to undertake such a project be added to their debt contract (despite the fact that the project has a positive NPV). 4

5 Solutions 1. (a) The debt holders will pay t D (r D D) = 1 3 (15) = 5 in taxes, and so they expect to receive 10 after tax every year. (b) Before the debt is issued, the firm pays an expected dividend of EBIT(1 t c ) = 0.60EBIT. After the debt issue, the firm s expected dividend is (EBIT r D D)(1 t c ) = 0.60(EBIT 15). The difference is = 9. That is, because the interest payment is tax-deductible, the firm only needs to lower annual dividends by 15(1 t c ) to afford an annual interest payment of 15. (c) The equity-holders used to receive Now they receive EBIT(1 t c )(1 t E ) = EBIT(1 0.40)(1 0.15) = 0.51EBIT. (EBIT r D D)(1 t c )(1 t E ) = (EBIT 15)(1 0.40)(1 0.15) = 0.51(EBIT 15). The difference is = That is, their after-tax dividend is reduced by 15(1 t c )(1 t E ). (d) Before the debt issue, the government used to receive the firm s earnings that did not go to equity-holders, namely EBIT EBIT(1 t c )(1 t E ) = EBIT [ 1 (1 0.40)(1 0.15) ] = 0.49EBIT. After the debt issue, the government receives the earnings that do not end up in the hands of debt-holders and equity-holders, that is, debt-holders equity-holders { }} { { }} { EBIT (r D D)(1 t D ) (EBIT r D D)(1 t c )(1 t E ) ( = EBIT ) (EBIT 15)(1 0.40)(1 0.15) 3 = EBIT 2 (15) 0.51(EBIT 15) 3 ( ) 2 = 0.49EBIT (15) = 0.49EBIT So the government receives 2.35 less per year. (e) The effective tax advantage of debt is t = [ 1 (1 t ] c)(1 t E ) = 1 t D [ 1 (1 0.40)(1 0.15) ] =

6 2. Because debt is tax deductible at the corporate level, the effective tax rate is 44% in both cases. 1 (a) If all equity income (including capital gains) is realized immediately, the effective personal tax rate on equity income is t E = 1 2 (0.44) (0.20) = 32%. Since equity income is also taxed at the corporate level, a dollar in revenues paid via the equity channel becomes (1 t c )(1 t E ) = (1 0.35)(1 0.32) = in the hands of equity-holders. The effective tax rate is therefore = 55.8%. (b) If only dividends are taxed when paid and capital gains can be deferred forever, the effective personal tax rate on equity income is t E = 1 2 (0.44)+ 1 2 (0) = 22%. Since equity income is also taxed at the corporate level, a dollar in revenues paid via the equity channel becomes (1 t c )(1 t E ) = (1 0.35)(1 0.22) = in the hands of equity-holders. The effective tax rate is therefore = 49.3%. 3. (a) Suppose at first that D = 0 and t c = t E = t D = 0. Since the firm is all-equity financed, we have r A = r E = The expected earnings before interest and taxes are Therefore: EBIT = (0.1)(1,000) +(0.4)(2,000) +(0.5)(4,200) = 3,000. (i) V U = E U = EBIT r A = 3, = 15,000; (ii) P = E U /N U = 15,000/1,000 = 15; (iii) EPS = EBIT N U = 3,000 1,000 = 3; (iv) r A = r E = (b) Now assume that the firm issues $7,500 of debt at r D = 0.10 and uses the proceeds to repurchase shares. (i) Since we are assuming no taxes, we have V L = V U = 15,000; (ii) D L = 7,500; (iii) E L = V L D L = 7,500; (iv) The total wealth of the shareholders is 15,000, as they still have E L and they receive $7,500 in cash from the debt issue. On a per-share basis, this amounts to P = 15,000 1,000 = 15. The number of shares repurchased is therefore n = 7, = 500, and there are now 1, = 500 shares outstanding. 2 (v) The required (expected) return on equity can be calculated using what the equityholders receive every year (earnings after interest payment) and the value of their equity (E L ): r E = EBIT r DD L = 3, = 2,250 E L 7,500 7,500 = Note that this is because the debt is assumed to be perpetual, and so it is paid in the form of interest forever. 2 Note thatwe couldhavesolved this more analytically. We have(1,000 n)p = E L = 7,500 andnp = D L = 7,500. We can then solve for n = 500 and P = 15. 6

7 Alternatively, we could have calculated r E using the levering formula: 3 ( ) DL r E = r A + (1 t c )(r A r D ) E L ( ) 7,500 = (1 0)( ) = ,500 (vi) Without taxes, WACC should be equal to r A and r E. Let us verify this: ( ) ( ) DL EL WACC L = r D + r E = 0.5(0.1) +0.5(0.3) = V L V L Notice that the value of the firm is unaffected by the increase in debt. This is because the reshuffling of claims to the cash flows generated by the firm s assets has not changed (a) the cash flows generated by the firm s assets (still equal to $3,000 now split up as dividends of $2,250 to shareholders and interest of $750 to debt-holders) nor (b) the riskiness of those cash flows. Hence, the discount rate applicable has not changed and the firm s weighted average cost of capital remains unchanged. The required return on common stock is now greater than the expected return on assets (i.e., r E > r A ): because the bondholders are paid ahead of the stockholders, the stockholders face more risk and so demand compensation for bearing that risk. (c) (i) The value of the unlevered firm will change when we introduce corporate taxes, as the government now receives a fraction of the cash flows generated by the firm s assets. The after-tax cash flows generated by the assets of the firm will now be Thus, the value of the unlevered firm is (1 t c )EBIT = (1 0.4)3,000 = 1,800. V U = (1 t c)ebit r A The value of the levered firm is then = (1 t c)ebit r E = 1, = 9,000. V L = V U +t c D L = 9, (7,500) = 12,000. (ii) The value of debt remains at D L = 7,500: the debt-holders pay $7,500 and receive a claim worth $7,500. (iii) The value of the equity is now E L = V L D L = 12,000 7,500 = 4,500. (iv) Upon the announcement of the debt issue (but before the debt is actually issued), the value of the firm goes up to V L = V U +t c D L = 9, (7,500) = 12,000. Since there are still 1,000 shares outstanding at this point, each share must be trading at P = 12,000 1,000 = 12. The proceeds from the debt issue can therefore be used to buy back n = 7, = 625 shares. 4 3 We know that r A is unaffected by the capital structure, as the firm s projects have not changed. 4 Again, we could have solved this in a more analytical manner. We have (1,000 n)p = E L = 4,500 and np = D L = 7,500. We can then solve for n = 625 and P = 12. 7

8 (v) The required rate of return on equity is again r E = (1 t c)(ebit r D D L ) (1 0.4)(3, ) = = E L 4,500 (vi) However, the weighted average cost of capital for the firm is now WACC L = (1 t c)ebit V L = (1 0.4)3,000 12,000 = Notice that alternatively ( ) ( ) DL EL WACC L = (1 t c )r D +r E V L V L ( ) ( ) 7,500 4,500 = (1 0.4)(0.10) +(0.3) 12,000 12,000 = Bankruptcy risks would reduce the amount by which the firm s value would increase due to leverage and would increase the weighted average cost of capital. (d) If t c > 0, t D > 0 and t E = 0, then V L = V U +D L t [{ }} { 1 1 t ] c, 1 t D where we still have V U = 9,000 since r E = r A and so V U = (1 t E)(1 t c )EBIT (1 t E )r E = (1 0)(1 0.4)3,000 (1 0)0.20 = 9,000. Also, D L = 7,500 in all three cases (the bondholders get what they pay for). Therefore, if (i) t D = 0.3, then [ V L = 9,000+7, ] = 10,071.43, and E L = V L D L = 10, ,500 = 2,571.43; (ii) t D = 0.4, then [ V L = 9,000+7, ] = 9,000, and E L = V L D L = 9,000 7,500 = 1,500; 4. (a) The initial value (in $million) of Gladstone s equity is E = (0.25)(150) +(0.25)(135) +(0.25)(95) +(0.25)(80) =

9 (b) Gladstone will be able to repay its promise (of $100 million) to debt-holders in the first two states, but not in the last two (it will pay whatever is available instead). The value of the debt is therefore D = (0.25)(100) +(0.25)(100) +(0.25)(95) +(0.25)(80) = (c) For a zero-coupon bond, the yield to maturity is the promised interest rate on the debt contract. In this case, it is y = = 12.0%. The expected rate of return is the rate that debt-holders can expect from this debt. By definition, it must beequal to r D, which is equal to r f (i.e., 5%) in this problem. Still, let us verify it. The debt-holders expect to receive (0.25)(100) +(0.25)(100) +(0.25)(95) + (0.25)(80) = Since their debt is currently worth 89.29, their expected rate of return is r D = = 5.0% (d) Gladstone s equity-holders receive the value that is available after the debt-holders are paid. Thus, the initial value of Gladstone s equity is E = The total value of the firm is still 5 (0.25)(50) +(0.25)(35) +(0.25)(0) +(0.25)(0) = V = D+E = = (a) The initial value (in $million) of Gladstone s equity is as before: E = (0.25)(150) +(0.25)(135) +(0.25)(95) +(0.25)(80) = (b) As before, Gladstone will be able to repay its promise (of $100 million) to debt-holders in the first two states, but not in the last two (i.e., in the two bad states). In the two bad states, 25% of the value will be lost, as the firm goes into bankruptcy. Thus, there will only be 95(1 0.25) = and 80(1 0.25) = 60 to repay the debt-holders in these two states. The value of the debt is therefore D = (0.25)(100) +(0.25)(100) +(0.25)(71.25) +(0.25)(60) = (c) For a zero-coupon bond, the yield to maturity is the promised interest rate on the debt contract. In this case, it is y = = 26.8%. 5 In my calculations, kept all the decimals. This is why there is a small rounding error. 9

10 The expected rate of return is the rate that debt-holders can expect from this debt. By definition, it must be equal to r D, which is equal to r f (i.e., 5%) in this problem. Still, let us verify it. The debt-holders expect to receive (0.25)(100) + (0.25)(100) + (0.25)(71.25) + (0.25)(60) = Since their debt is currently worth 78.87, their expected rate of return is r D = = 5.0%. (d) Gladstone s equity-holders receive the value that is available after the debt-holders are paid. Thus, the initial value of Gladstone s equity is E = The total value of the firm is now (0.25)(50) +(0.25)(35) +(0.25)(0) +(0.25)(0) V = D +E = = = This is smaller than in part (a), when the firm was unlevered. The difference is due to the present value of bankruptcy costs which is (0.25)(0) +(0.25)(0) +(0.25)( )+(0.25)( ) PV(bankruptcy costs) = = (e) Using the value of the firm calculated in part (a), the price per share is $ = $ (f) As soon as the Gladstone announces its plan to issue the debt and repurchase shares, the value of the firm will drop to $99.11, as calculated in part (d). This means that the share price will drop to $ = This also means that the debt issued by Gladstone will allow it to raise $78.87 (from part (b)) and to repurchase $78.87 $9.91 = 7.96 million shares. The other 2.04 million shares also trade at $9.91 per share, and so are worth $20.24 (which we found in part (d)). 6. (a) The cost of equity is given by the CAPM: r E = r f +(r m r f )β E = 0.09+(0.06)1 = 15%. Since the firm is all equity financed, the cost of capital is equal to the cost of equity. Notice that this implies that Nadir s earnings are expected to grow at 5% per year as V = EBIT(1 t c) r E g = 2,000,000(1 0.40) 0.15 g g = 5%. (b) We from the lecture notes that V L = V U +PV(tax shield) PV(bankruptcy costs), so that we should try to find the debt level which maximizes PV(tax shield) PV(bankruptcy costs). 10

11 The following table shows the present value of the debt tax shield and the present value of bankruptcy costs and their difference for the different debt levels. Value of Probability of P V(tax shield) P V(bank. costs) Value Added Debt [D] Failure [p D ] [0.4 D] [p D 8M] of Debt 2,500, ,000, ,000,000 5,000, ,000, ,000 1,360,000 7,500, ,000,000 1,640,000 1,360,000 8,000, ,200,000 2,400, ,000 9,000, ,600,000 3,600, ,000, ,000,000 4,200, ,000 12,500, ,000,000 5,600, ,000 Obviously, the optimal debt level is $5,000,000 or $7,500, (c) With this optimal capital structure, the total value of the firm is V L = V U +value added of debt = 12,000,000 +1,360,000 = 13,360, (a) XI s end-of-year assets in the different states of the economy will be: 7 State of the economy Good Medium Bad (1/4) (1/2) (1/4) Cash Project Total Assets If the firm is all-equity financed, the equity is worth E U = V U = 360(1/4) +110(1/2) +36(1/4) = 140. (b) (i) The firm will owe $30 million plus 10% interest, i.e. $33 million. (ii) The firm s assets at the end of the year will be sufficient to pay the $33 million that it owes to the bondholders in all three states of the economy. (iii) The debt is worth D L = (iv) The equity is now worth 33(1/4) +33(1/2) +33(1/4) = 30. E L = (360 33)(1/4) +(110 33)(1/2) +(36 33)(1/4) = More precisely, since we are not given the probabilities of failure for all the debt levels between $5,000,000 and $7,500,000, we can only conclude that the optimal debt level is somewhere between $5,000,000 and $7,500, All the figures in this solution are in $million. 11

12 (v) The shareholders got $30 million in cash (from the share repurchase) and still have $110 million in equity. Therefore, their total wealth is still $140 million, which implies that they should be indifferent about this new capital structure. (c) (i) The bonds now promise $44 million, which implies that the firm will not be solvent in the bad state of the economy (since the firm only has $36 million worth of assets in that state). As a result, bankruptcy costs of $10 million will be incurred in the bad state, and so the bondholders will only receive $36 million $10 million = $26 million in that state. Their debt is then worth D L = 44(1/4) +44(1/2) +26(1/4) = , which is less than what they pay for it. Obviously, investors would simply refuse to buy these bonds. (ii) We would like to find the promised amount A, which will make the debt worth $40 million: A(1/4)+A(1/2) +26(1/4) 40 = A = 50. So the firm has to promise the bondholders = 25% in order for them to expect a return of 10% on average (and therefore accept to buy the bonds). (iii) The shareholders will then receive nothing in the bad state of the economy. Their equity is now worth E L = (360 50)(1/4) +(110 50)(1/2) +(0)(1/4) = Sincethey soldback someequity for $40 million, thismeans that their total wealth is now $ million+$40 million = $ million, which is less than the $140 million that they originally had. The reason why their wealth is now lower is that the bondholders get what they pay for (they are not fooled by the potential bankruptcy costs). As a result, the bankruptcy costs are imposed on the shareholders, who lose in wealth. PV(bankruptcy costs) = 10(1/4) = (d) (i) The present value PV of the expected end-of-year cash flow of the project is PV = (23)(1/4) +(13)(1/2) +(3)(1/4) = Since the project require an initial investment of $10 million, its net present value is therefore NPV = = > 0. 12

13 (ii) If XI undertakes this second project, the firm s end-of-year assets in the three different states will be as follows: State of the economy Good Medium Bad (1/4) (1/2) (1/4) 1st project nd project Total Assets This means that the firmwould nolonger besolvent in thebad state (the $28 million in assets is not sufficient to pay the $33 million that was promised to the bondholders), so that D L = 33(1/4) +33(1/2) +(28 10)(1/4) = <