Finance 402: Problem Set 5 Solutions

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1 Finance 402: Problem Set 5 Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. 1. The first step is to determine the cash flows and their timing, which are outlined in Table 1 Table 1: Year (a) EBITDA 0 11,000 11, ,000 (b) Depreciation 0-6,000-6, ,000 (c) EBIT (a + b) 0 5,000 5, ,000 (d) Tax ( c 0.34) 0-1,700-1, ,700 (e) Salvage Value ,000 (f) Cost -60, Total (a + d + e + f) -60,000 9,300 9, ,300 Depreciation is computed using a straight-line (interpolated) method, which amounts to computing the average value loss per year ((60, , 000)/8 = 6, 000). The tax figures are computed assuming a 34% marginal corporate tax rate (5, = 1, 700). The final cash flows are obtained by subtracting the tax from the EBITDA (Earnings Before Interest, Taxes, Depreciation, and Amortization); remember: depreciation is not a cash outflow. 1

2 The next step is compute the present value of these cash flows: $60, $9, ( ) $12, 000 = $8, (1) (1.12) 8 We reject the project since it has a negative NPV. 2. The maximum amount that we would be willing to pay for the machine is the price such that the NPV is equal to zero (at which point we would be indifferent between purchasing and not purchasing the machine). Since the purchase price determines the depreciation, salvage value and tax expense, we cannot simply adjust the $60,000 figure from equation (1) until the NPV is driven to 0. There are several ways to solve this problem. The first is simply a trial and error method (that can be automated by using GOALSEEK in Excel). The other, more elegant and useful way, is to derive an expression for the NPV of the machine in terms of the purchase price (P rice) of the machine and solve for the price that sets NPV = 0. The NPV of the machine may be expressed as: NP V = P rice + 8 t=1 1 (1 + r) Cash Flow Salvage Value t t + (2) (1 + r) 8 The cash flow in any period t are: Cash Flow t = EBIT DA t 0.34(EBIT DA t Depreciation) (3) Substituting 11,000 for the value of EBIT DA t yields Cash Flow t = 7, Depreciation. The depreciation in any period may be expressed as: Depreciation = (P rice Salvage Value)/Life Span = (P rice 12, 000)/8 (4) 2

3 Substituting equations (3), (4) and known values for r and Salvage Value into equation (2) yields: NP V = P rice t=1 { 1 7, ( ) t } (P rice 12, 000) 8 12, 000 ( ) 8 (5) Arithmetic allows us to rewrite equation (5) as: NP V = P rice + 8 t=1 7, (1.12) + P rice t t= (1.12) t t=1 510 (1.12) t + 4, (6) Using our annuity formula, we can evaluate the summation terms as: 8 t=1 8 t=1 8 t=1 reducing equation (6) to 7, 260 = 7, (1.12) 8 (1.12) t = (1.12) 8 (1.12) t = (1.12) 8 (1.12) t 0.12 = 36, = = 2, (7) NP V = ( )P rice + 38, (8) Setting NP V = 0 and solving for P rice in equation (8) yields $48, (your answer may differ by up to several dollars due to rounding) a The cash flows are detailed in Table 2. The depreciation figure follows from the straight-line method (25,000/8 = 3,125) and the tax follows from a 34% marginal tax rate (1, = ). 3

4 Table 2: Year (a) Cost Savings 0 4,500 4, ,500 (b) Depreciation 0-3,125-3, ,125 (c) Increase in Taxable Income (a + b) 0 1,375 1, ,375 (d) Tax (34%) ( c 0.34) (e) Investment -25, Total (a + d + e) -25,000 4, , , Given the cash flows, we now need to find the NPV: NP V = 4, ( ) , 000 = $4, Since this figure is less than zero, we reject the project. 4

5 3.b The cash flows are under the leasing scenario are shown in Table 3. Table 3: Year (a) Cost Savings 0 4,500 4, ,500 (b) Lease Payments 0-4,000-4, ,000 (c) Pre-tax Revenue (a + b) (d) Tax (34%) ( c 0.34) After-tax Cash Flows (c + d) The NPV of these cash flows is NP V = ( ) = $1, c The cash flows are shown in Table 4. Table 4: Year (a) Cost Savings 0 4,500 4, ,500 (b) Depreciation 0-1,250-1, ,250 (c) Increase in Taxable Income (a + b) 0 3,250 3, ,250 (d) Tax (34%) ( c 0.34) 0-1,105-1, ,105 (e) Investment -10, Total (a + d + e) -10,000 3,395 3, ,395 The NPV of these cash flows is NP V = 3, ( ) , 000 = 6, The maximum lease payment we would be willing to make is one that sets the NPV to 6, The yearly cash flows from the lease strategy are 5

6 given by: Cash Flow = (1 0.34)(4, 500 Lease Payment) Therefore, the maximum lease payment solves: 6, = (1 0.34)(4, 500 Lease Payment) 1 (1.12) which implies Lease Payment = $2, Any payment greater than this amount and we would prefer to buy. Any payment less than this amount and we would prefer to lease a The present value of the cash flows (in $thousands) to both machines for one operating cycle are: NP V Machine A = (1.1) (1.1) (1.1) 4 = NP V Machine B = (1.1) (1.1) 3 = In order to compare these investments, we must now compute their equivalent annuity payments. From the annuity formula c A 0 = (1 + i) + c (1 + i) c 2 (1 + i) + c N 1 (1 + i) N we have For machine A, For machine B = c 1 (1 + i) N i c = A 0 i 1 (1 + i) N c = = ( ) 0.10 c = = ( ) Therefore, machine B is the better choice. 6

7 4.b Using the results above, we can consider the value of each machine used in perpetuity. NP V MachineA = NP V MachineB = = = Thus, over the entire lifetime, machine B will provide the company with $ $ = $4.12 additional thousands of dollars. The up front cost of $10,000 for machine B negates this advantage, implying that we should choose machine A under this scenario. 4.c They should replace the existing machine as soon as it s annual cash flows fall below the annual equivalent of its replacement. This occurs in year 3. 4.d Comparing one-year annual equivalent cash flows produces the correct answer regardless of the time-horizon. Thus, without the re-tooling fee of $10,000 on machine B, we should choose machine B. The aggregate NPV of both machines is now, using the perpetuity formula A 0 = c (1 + i) + c (1 + i) + c 2 (1 + i) +... = c 3 i NP V MachineA = ( ) 24 ) 0.1 NP V MachineB = ( ) 24 ) 0.1 = = Therefore, with the re-tooling fee for machine B, we should choose machine A. 7

8 5. The approach for solving this problem is to compute the NPV for each replacement strategy and then compare the annual equivalent for each strategy, choosing the largest amount. One Year Replacement The cash flows are in Table 5 Table 5: Year 0 1 (a) Maintenance Costs 0-1,000 (b) Depreciation 0-9,000 (c) Change in Taxable Income (a + b) 0-10,000 (d) Tax Benefit(34%) ( c 0.34) 0 3,400 (e) Salvage 0 6,000 (f) Cost -15,000 0 Total (a + d + e + f) -15,000 8,400 The NPV of these cash flows is $8,400/ $15,000 = -$7,500. Two Year Replacement The cash flows are in Table 6 Table 6: Year (a) Maintenance Costs 0-1,000-2,000 (b) Depreciation 0-6,000-6,000 (c) Change in Taxable Income (a + b) 0-7,000-8,000 (d) Tax Benefit(34%) ( c 0.34) 0 2,380 2,720 (e) Salvage 0 0 3,000 (f) Cost -15, Total (a + d + e + f) -15,000 1,380 3,720 The NPV of these cash flows is $1, 380/ $3, 720/ $15, 000 = $10, 802. Three Year Replacement The cash flows are in Table 7 8

9 Table 7: Year (a) Maintenance Costs 0-1,000-2,000-3,000 (b) Depreciation 0-5,000-5,000-5,000 (c) Change in Taxable Income (a + b) 0-6,000-7,000-8,000 (d) Tax Benefit(34%) ( c 0.34) 0 2,040 2,380 2,720 (e) Cost -15, Total (a + d + e) -15,000 1, The NPV of these cash flows is $1, 040/ $380/ / $15, 000 = $13, 967. The annual equivalents (AE) for each of these cash flows are: 0.12 AE 1 = 7, 500 = 8, (1.12) AE 2 = 10, 802 = 6, (1.12) AE 3 = 13, 967 = 5, (1.12) 3 Hence, option 3 gives the lowest per-year operating cost and is thus, the best strategy of the three a The IRR is found by solving the following equation: 0 = (1 + IRR) (1 + IRR) 2, which is quadratic in IRR. The solution may be found by applying the quadratic formula to get IRR = 46%. 1 Since the IRR is greater than the 1 Consider the following general quadratic equation: ax 2 + bx + c = 0, 9

10 25% required rate of return, we should accept the project. This is the correct decision rule in this case since there is only one sign change in the cash flow stream. Thus, the IRR rule and the NPV rule coincide. As a check, the NPV of the project is: (1.25) (1.25) 2 = 145, which is greater than 0, implying that we should accept the project. 6.b The IRR for the new cash flow stream is found by solving the following equation: 0 = (1 + IRR) 575 (1 + IRR), (1 + IRR). 3 This equation can be solved in closed form or numerically in Excel. The resulting IRR is 27%, which is indeed higher than 25%. However, in this case the negative cashflows are preceeded by positive cashflows so that a correct use of the IRR rule in this case requires that the IRR be higher than the required rate of return. Indeed, computation of the NPV shows NP V = (1.25) 575 (1.25) (1.25) 3 = 31.2 leading to a rejection of the project.(see Brealey and Myers, 7th Ed., Pages for additional discussion of this issue). The required rate of return is lower than the IRR, but the project should not be taken. where a,b and c are constants. equation is: x = b + b 2 4ac 2a We can algebraically rearrange equation (9) to get The quadratic formula shows that the solution to this 575(1 + IRR) (1 + IRR) = 0 implying that a = 575, b = 500 and c = 500. Substituting these values into equation (9) yields two solutions, 0.46 and Since is not a realistic discount rate, we choose (9)

11 7. There are two approaches to solving this problem. The first compares the NPV for starting the project now versus waiting and compares NPV s. The second subtracts the cash flows for the start now alternative from the cash flows of the start in 3 years alternative and computes the NPV of those incremental cash flows. We take the latter approach. Table 8 shows the incremental cash flows, where Table 9 shows the tax calculations. The NPV of the incremental cash flows is $

12 Table 8: Year T (a) Increase(Decrease) In Revenue (b) Increase(Decrease) In Salvage 3, (c) Increase(Decrease) In Expenses (d) Increase(Decrease) In Maintenance (e) Increase(Decrease) In Lease Expenses 1,000 1,000 1, (f) Increase(Decrease) In Taxes Owed (g) Increase(Decrease) In Net Cash After Taxes 2, (a + b c d e f) (h) Discount Factor (1.016) T (i) Increase(Decrease) In Present Value (g/h) 2,

13 Table 9: Year T (a) Increase (Decrease) in Revenues (b) Increase (Decrease) in Book Gain Salvage (c) Increase (Decrease) in Expenses (d) Increase (Decrease) in Maintenance (150) (150) (150) (e) Increase (Decrease) in Depreciation (600) (600) (600) (f) Increase (Decrease) in Lease Expense 1,000 1,000 1, (g)increase (Decrease) in Net Income before Taxes 350 (250) (250) (a + b c d e f) (h)increase (Decrease) in Taxes Owed (g 0.34) 119 (85) (85)

14 8. 8.a Table 10: New Retail Outlet Year (a) Investment 1,300 (b) Revenue 2,000 2,100 2,205 2, , (c) Expenses 1,100 1,155 1, , , (d) Depreciation (e) Discount Factor (f) Pretax Revenue ( a + b c d) (g) Taxes Owed (f 0.34) (h) Net Cash ( a + b c g) -1, (i) Net Cash PV (h/e) -1, Cash flows are in thousands. The total cash PV is $1,

15 Table 11: Product Enhancement Year (a)investment 1, (b)revenue 1, , , , , (c)expenses (d)depreciation (e)discount Factor (f)pretax rev ( a + b c d) (g)taxes Owed (f 0.34) (h)net Cash ( a + b c g) -1, (i)net Cash PV (h/e) -1, Cash flows are in thousands. The total cash PV is $

8.b As shown above the PV of the new retail outlet is $1,167.18 and of the product enhancement is $761.97. The NPV of investing in the new retail outlet relative to product enhancement is: 1,167.

16 8.b As shown above the PV of the new retail outlet is $1, and of the product enhancement is $ The NPV of investing in the new retail outlet relative to product enhancement is: 1, =$ Hence, investing in the retail outlet is the preferred decision. 8.c 9. If the firm considers the cash flows from the first generation product then it is indifferent between the two projects. Nevertheless, building plant two is the correct decision (see Table 12). Note that both projects have about the same NPV in terms of the expected cash flows from each. The characteristic that distinguishes plant 2 is the fact that this project provides management with an option to enter the market for the second generation product which is not provided with plant 1. This real option intrinsically has some non-zero value, hence plant 2 is the preferred alternative. 16

17 Table 12: Year Plant 1-3,322 1, , Plant 2-4, , , , , Discount PV(Plant 1) -3,322 1, , PV(Plant 2) -4, , NPV(Plant 1) NPV(Plant 2)

CHAPTER 6 MAKING CAPITAL INVESTMENT DECISIONS

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