DUKE UNIVERSITY Fuqua School of Business FINANCE 351 - CORPORATE FINANCE Problem Set #5
Prof. Simon Gervais
Fall 2011 – Term 2 Questions
1. Digita Digitall Org Organi anics cs (D (DO) O) has the oppor opportuni tunity ty to in inve vest st $1 mil millio lion n no now w (t = 0) and expects after-tax returns of $600,000 in t = 1 and $700,000 in t = 2. The project project will last last for two years ye ars only. only. The appropriate appropriate cost of capital is 12% with all-e all-equit quity y financ financing, ing, the borro borrowing wing ratee is 8%, and DO will borr rat borrow ow $300,0 $300,000 00 aga agains instt the project. project. Thi Thiss debt must be rep repaid aid in two equal installments. Assume that debt tax shields have a net value of $0.30 per dollar of interest paid. Calculate the project’s APV. 2. Environme Environmenta ntally lly Correct Inc. is considering considering a project to make solar water heaters. heaters. The water water heater project requires an investment of $25 million and offers a steady after-tax cash flow of $4. $4.55 mill million ion per ye year ar for 10 ye years ars.. The required required return return on unl unlev evere ered d cas cash h flo flows ws is 12% 12%.. Environmentally Correct is going to finance this project with a $12.5 million debt issue that is privately privately placed. placed. This debt issue has an interest interest rate of 8%. The remaining remaining $12.5 million willl be rai wil raised sed from equit equity y cap capita ital. l. The debt princip principal al of $12 $12.5 .5 mill million ion is to be b e paid bac back k in ten equal installments. Assume the marginal tax rate for corporations is 34%. Calculate the APV of the project. 3. Assuming Assuming all equity equity financing, a project has a net present present value value (NPV) of $1.5 million. To finance the project, debt is issued with associated floatation costs of $60,000. The floatation costs can be amortized over the project’s 5 year life. The debt of $10 million is issued at 10% inter in terest est,, with princip principal al repaid repaid in a lum lump p sum at the end of the fift fifth h year. year. If the firm firm’s ’s tax rate is 34%, calculate the project’s adjusted present value (APV). 4. The Sretaw Sretaw Regor (SR) Corporation Corporation is considering considering a new 5-year 5-year project. Since this project project is very different from SR’s current operations, the adjusted present value will be used to value the project. The project requires an initial investment of $750,000 in new assets, which will be depreciated straightstrai ght-line line to 0 ov over er the project’s 5-year 5-year life. These assets assets will be wor worthles thlesss in five years, years, i.e., i.e ., the they y wil willl not be res resold old.. Eac Each h ye year ar for fiv fivee ye years ars,, the project project is expected expected to gen genera erate te pre-tax revenues of $600,000 and to require pre-tax costs of $240,000. The entire project will be fina finance nced d through through a 5-y 5-year ear bank loan with an annual annual rat ratee of 10%. The princip principal al on the loan will be repaid in equal installments of $150,000 each (i.e., each year, the company pays $150,0 $15 0,000 00 in pri princi ncipal pal,, and pays the in inter terest est on the outstand outstanding ing loan). loan). It is est estima imated ted that the pre-tax costs (payable at time zero) of negotiating the loan will be 4% of the amount borrowed. The project’s risk is very similar to the risk of Ruomlig Divad (RD) Inc.’s (unlevered) assets. This firm is currently financed by 100,000 shares worth $12.50 each, and $750,000 worth of debt. The beta of RD’s stock is 1.5, and the company borrows at a rate of 11%. 1
The riskfree rate in the economy is 8%, and the expected return on the market is 18%. The current corporate tax rate is 45% (assume that it applies to both SR and RD). Ignore personal taxes, and assume that the debt of all the firms is permanent (i.e., not rebalanced). (a) What is the appropriate discount rate for the unlevered project? (b) What is the adjusted present value of the project? (Optional)
5. You are asked to value the equity of Roberts Corporation, a company that started to manufacture melamine particle boards. The capital cost of the project is $10 million in year 0. The melamine project will generate the following operating cash flows (in million dollars) before interest and taxes: (−4, 4, 6, 6, 6) for years 1 through 5. At the end of five years, the melamine technology will be outdated and the plant will be sold for $1 million. Ignore depreciation and tax on salvage in this problem. To finance this cost, Roberts has been offered a $10 million loan at 10%. The loan has to be repaid in three equal installments at the end of years 3, 4 and 5, i.e., the repayment schedule is (3.33, 3.33, 3.34) million dollars at the end of years 3, 4, 5. The tax rate is 30%. The equity beta of Dr. K Melamine, a melamine particle board company is 1.2. Dr. K Melamine has a debt/equity ratio of 0.7, and its debt is practically riskfree. The market risk premium is 7% and the riskfree rate is 8%. Make sure you allow for loss carry-forwards in the cash flows. (a) Calculate the unlevered value (base case NPV) of Roberts. (b) Calculate the present value of the tax shields due to the loan. Hint :
In this problem, we assume that the firm does not have income other than from this project. Thus, no corporate taxes are paid in years when the earnings after interest and before tax from this project are nonpositive. (Optional)
6. Honda and GM are competing to sell a fleet of cars to Hertz. Hertz’s policies on its rental cars include use of straight-line depreciation and disposing of the cars after five years. Hertz expects that the autos will have no salvage value. The firm expects a fleet of 25 cars to generate $100,000 per year in pretax income (i.e., extra revenues minus extra expenses). Hertz is in the 34% tax bracket, and the firm’s overall (all-equity) required return is 10%. The addition of the new fleet will not add to the risk of the firm. Treasury bills are priced to yield 6%. (a) What is the maximum price that Hertz should be willing to pay for the fleet of cars, assuming that Hertz stay unlevered (and it is optimal to do so)? (b) Suppose the price of the fleet (in U.S. dollars) is $325,000; both suppliers are charging this price. Hertz is able to issue $200,000 in debt to finance the project. The five-year coupon bonds can be issued at par and will carry an 8% interest rate (i.e., only the interest on the $200,000 is paid back at the end of every year for five years, and the principal is paid back at the end of five years). Hertz will incur no costs to issue the debt and no costs of financial distress. What is the APV of this project if Hertz uses debt to finance the auto purchase? 2
(c) To entice Hertz to buy the cars from Honda, the Japanese government is willing to lend Hertz $200,000 at 5% (also a five-year coupon bond). Now what is the maximum price that Hertz is willing to pay Honda for the fleet of cars?
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Solutions 1. We have rA = 12%, rD = 8%, and tc = 30%. The net present value of the project for an all-equity firm would be: 600,000 700,000 + = 93,750. 1.12 (1.12)2
NPV = −1,000,000 +
If $300,000 of the project is financed with debt to be repaid in two equal installment of I , we must have: I I 300,000 = + I = 168,230.77. ⇒ 1.08 (1.08)2 In the first year’s installment, $300,000×0.08 = $24,000 consist of interest; the rest (144,230.77) is principal repayment, reducing the debt outstanding in the second year to $155,769.23. This also implies that the interest payment in the second year is $155, 769.23 × 0.08 = $12,461.54. In sum, the tax shields at the end of each of the next two years will be calculated as follows: Year 1 2 Total
Debt outstanding at start of year 300,000.00 155,769.23
Interest
Interest tax shield
PV (@8%) of tax shield
24,000.00 12,461.54
7,200.00 3,738.46
6,666.67 3,205.13 9,871.79
The adjusted present value of the project is therefore APV = NPV + PV (tax shields) = 93,750 + 9,871.79 = 103,621.79. 2. The unlevered net present value of the project is NPV U = −25 + 4.5a 1012%
4.5 1 = −25 + 1− = 0.426. 0.12 (1.12)10
The annual tax shield is calculated based on the debt repayment schedule: End of Year 1 2
0 Debt outstanding Interest payment Principal payment Total payment on the debt Interest tax shield
12.50
11.250 1.000 1.250 2.250 0.340
10.000 0.900 1.250 2.150 0.306
···
9
10
···
1.250 0.200 1.250 1.450 0.068
0.000 0.100 1.250 1.350 0.034
··· ··· ··· ···
The present value of these tax shields is PV (tax shields) =
0.340 0.306 0.068 0.034 + + ··· + + = 1.398. 2 9 1.08 (1.08) (1.08) (1.08)10
Thus the adjusted present value of the project is APV = 0.426 + 1.398 = 1.824. 4
3. We know that APV = NPV U (project) + PV (financing). Here, NPV U (project) = 1,500,000, and PV (financing) = PV (floatation costs) + PV (debt tax shields). Now, the present value of the floatation costs is 60,000 (0.34)a 510% 5 12,000(0.34) 1 = −60,000 + 1− 0.10 (1.10)5 = −44,533.59,
PV (floatation costs) =
60,000 +
−
and the present value of the debt tax shields is PV (debt tax shields) = (0.34 × 10,000,000 × 10%)a 510% 340,000 1 = 1− 0.10 (1.10)5 = 1,288,867.50.
Therefore, APV = 1,500,000 − 44,533.59 + 1,288,867.50 = 2,744,333.91. 4. We are given rf = 0.10, rm = 0.18, and tc = 45%. (a) Let us first calculate the expected return on RD’s stock using the CAPM: rE = rf + (rm − rf )β E = 0.08 + (0.18 − 0.08)(1.5) = 0.23. RD is financed with 100,000 × $12.50 = $1,250,000 of equity and $750,000 of debt, i.e., E = 1,250,000, D = 750,000, V = D + E = 2,000,000, D/V = 0.375, and E/V = 0.625. This means that the present value of its tax shields is tc × D = 0.45 × 750,000 = 337,500, and the value of its unlevered assets is A = V − PV (tax shields) = 2,000,000 − 337,500 = 1,662,500. ,500 In other words, a fraction 12,,662 = 0.83125 of the firm’s value comes from its assets, 000,000 and a fraction 1 − 0.83125 = 0.16875 comes from its tax shields. Since the firm’s tax shields have the same risk as its debt, we have
0.83125rA + 0.16875rD = 0.375rD + 0.625rE ⇔ ⇔
0.83125rA + 0.16875(0.11) = 0.375(0.11) + 0.625(0.23) rA = 0.2002.
Since the project has the same risk as RD’s assets, the appropriate discount rate is 20.02% (when unlevered). 5
Alternatively, we could have figured out this rate by unlevering the beta of RD’s equity, rD −rf 0.03 after observing through CAPM that RD’s debt has β D = rm −rf = 0.10 = 0.3: β A =
β E +
1+
D E D E
(1 − tc )β D (1 − tc )
1.5 + = 1+
Using the CAPM, we find
0.375 0.625 0.375 0.625
(1 − 0.45)0.3 = 1.2022556. (1 − 0.45)
rA = rf + β A (rm − rf ) = 0.08 + (1.2022556)(0.18 − 0.08) = 20.02%. Yet another alternative to get this rate is by unlevering RD’s weighted average cost of capital (WACC). Indeed, RD’s WACC is WACC L = (1 − tc )rD
D E + rE = (1 − 0.45)(0.11)(0.375) + (0.23)(0.625) = 0.1664375. V V
Since WACC L = rA 1 − tc D , we have V rA =
WACC L 0.1664375 = = 0.2002. 1 − (0.45)(0.375) 1 − tc D V
(b) The adjusted present value (APV) of the project is given by APV = −750,000 + NPV (project) + PV (interest tax shields) −
PV (after-tax issuance costs).
In each of the project’s five years, the after-tax cash flows will be CF = (after-tax profits) + (depreciation tax shields) 750,000 = 360,000(1 − tc ) + tc 5 = 360,000(1 − 0.45) + 0.45(150,000) = 265,500. Therefore, using the discount rate rA calculated in part (a), we have NPV (project) = 265,500a 50 .2002
265,500 1 = 1− = 793,613.59. 0.2002 (1.2002)5
The present value of the interest tax shields can be calculated using the following table:
Year 1 2 3 4 5
Debt outstanding at start of year 750,000 600,000 450,000 300,000 150,000 6
Interest 75,000 60,000 45,000 30,000 15,000
Interest tax shield 33,750 27,000 20,250 13,500 6,750
Therefore, we have PV (interest tax shields) =
33,750 6,750 + ··· + = 81,621.89. 1.10 (1.10)5
Finally, the after-tax issuance costs are PV (after-tax issuance costs) = 4% × 750,000 × (1 − 0.45) = 16,500. The adjusted present value of the project is therefore APV = −750,000 + 793,613.59 + 81,621.89 − 16,500 = 108,735.48. 5. (a) In calculating the base case NPV, it is important to note that we are considering a similar but fictitious firm with no leverage. We first find the cost of equity for the unlevered firm (that has the same business risk). Using Dr. K Melamine as the appropriate benchmark firm, the asset beta (unlevered beta) is β A =
1.2 β E = = 0.805. 1 + (0.7)(1 − 0.30) 1+ D (1 − tc ) E
Using the CAPM, we obtain the expected return on the unlevered assets (i.e., the cost of unlevered equity): rA = rf + β A (rm − rf ) = 0.08 + (0.805)(0.07) = 13.6%. The cash flows given in the problem are earnings before interest and taxes. In year 1, we have an EBIT of −$4 million and thus no taxes are paid. This loss is carried forward to year 2 and offsets the EBIT in the second year. Thus taxes are to be considered only on incomes for years 3, 4, and 5, as shown in the following table (which is in 000’s).
EBIT Loss carry-forwards
1 -4,000 0
2 4,000 -4,000
3 6,000 0
4 6,000 0
5 6,000 0
Taxable income Taxes (T A ) After-tax cash flows (EBIT − T A )
-4,000 0 -4,000
0 0 4,000
6,000 1,800 4,200
6,000 1,800 4,200
6,000 1,800 4,200
We can now find the value of the unlevered firm: salvage value
after-tax cash flows
1,000 V U = −10,000 + (1.136)5 = −2,286.
4,000 4,000 4,200 4,200 4,200 + + + + − 2 3 4 1.136 (1.136) (1.136) (1.136) (1.136)5
In the next part, we need to calculate the present value of interest tax shields resulting from $1 million in interest in years 1, 2, and 3. (There is additional interest in years 4 and 5 that I handle later.) 7
The twist in this problem is that having interest deductions in years 1 and 2 does you no good because you are not paying taxes anyway. (Why? Because you have a loss in year 1, and the loss is carried forward to completely shield the income you make in year 2, so you pay no taxes in year 1 or year 2 even without interest deductions.) The interest that you pay in years 1 and 2 makes your carry forward a loss larger than that above; that is, the interest is “stored up” (i.e., “carried forward”) and taken in year 3, along with the additional $1 million of interest you have in year 3. (The firm is quite profitable in year 3, so it takes all $3 million of interest deduction at that time.) So, having debt adds $3tc million to your cash flow in year 3, but nothing (relative to having no debt) in year 1 or year 2. (b) First, we have to figure out the debt repayment schedule:
Debt outstanding Interest payment (@10%)
0 10,000
1 10,000 1,000
2 10,000 1,000
3 6,667 1,000
4 3,333 667
5 0 333
We can now calculate the tax shields that this debt generates:
EBIT − Interest payment EBT Loss carry-forwards
1 -4,000 1,000 -5,000 0
2 4,000 1,000 3,000 -3,000
3 6,000 1,000 5,000 -2,000
4 6,000 667 5,333 0
5 6,000 333 5,667 0
Taxable income Taxes (T B ) T B − T A
-5,000 0 0
0 0 0
3,000 900 900
5,333 1,600 200
5,667 1,700 100
There are two ways of computing the amount of tax shield due to the loan. One way is by comparing the amount of taxes Roberts pays with the loan vs without the loan (see the two tables). For example, in year 3, Roberts pays $0.9 million in taxes with the loan vs. paying $1.8 million in taxes without using the loan. Thus, it is a saving of $0.9 million due to the loan tax shield. An alternative way of finding out the tax shield amount is to use the tax rate (30%) to multiply the interest payment in years 4 and 5. For year 3, use 30% to multiply 1 + 1 + 1 = 3 million because all the interest payment in years 1, 2, and 3 are used to offset taxable income in year 3. There is no tax shield available in years 1 and 2 because there is no positive taxable income. The present value of these tax shields is PV (tax shields) =
900 200 100 + + = 875. 3 4 (1.10) (1.10) (1.10)5
Thus the adjusted present value is APV = −2,286 + 875 = −1,411. 6. (a) If Hertz is an all-equity firm, we only need to discount the cash flows of the project at rA = 10% to find the net present value of the project. The maximum price P that Hertz 8
should be willing to pay for the fleet of cars is the price that will make that net present value equal to zero: 0 = NPV = −P + 100,000(1 − 0.34) a 510% +
income
P (0.34) a 510% . 5
depreciation tax shield
Since a 510% = 3.7907868, we can solve for P to get a maximum price of P = 337,082.99. (b) We know that the adjusted net present value (APV) of the project can be calculated in two parts. First, let us calculate the net present value of the project, assuming that it is undertaken by an all-equity firm: 325,000 (0.34) a 510% 5 325,000 = −325,000 + 100,000(1 − 0.34)(3.7907868) + (0.34)(3.7907868) 5 = 8,968.31.
NPV U =
325,000 + 100,000(1 − 0.34) a 5 10% +
−
The second part has to do with the tax shields resulting from the interest payments on the debt used to finance the project. The present value of these tax shields is given by PV (interest tax shields) = tc rD ∆D a 58% = (0.34)(8%)(200,000)(3.9927100) = 21,720.34. So, the adjusted present value of the project is APV = NPV U + PV (interest tax shields) = 8,968.31 + 21,720.34 = 30,688.66. (c) If the Japanese government offers Hertz this “cheap” loan, the APV also needs to account for the fact that Hertz is effectively making a profit out of the loan. Indeed, we assumed in class that all transactions in the capital markets are zero NPV transactions. However, since the Japanese government is now effectively willing to give Hertz the loan at a “below-market” rate, this means that Hertz profits from the loan itself, not just the tax shields. So, the APV of the project is now given by
APV =
depreciation tax shield income P (0.34) a 510% −P + 100,000(1 − 0.34) a 510% + 5 + PV (interest tax shields) + NPV (loan),
(1)
where PV (interest tax shields) = tc rD ∆D a 58% = (0.34)(5%)(200,000)(3.9927100) = 13,575.21.
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and NPV (loan) = ∆D − rD ∆D a 58% −
∆D (1.08)5
= 200,000 − (5%)(200,000)(3.9927100) −
200,000 (1.08)5
= 23,956.26. We can solve for P in (1) to obtain the maximum price that Hertz would be willing to pay for the fleet of cars, that is P = 387,649.05.
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