Ross公司理财(第七版)答案 CH7

Ross公司理财(第七版)答案 CH7 Chapter 7: Net Present Value and Capital Budgeting 7.1 a. Yes, the reduction in the sales of the company’s other products, referred to as erosion, should be treated as an incremental cash flow. These lost sales are included because they are a cost (a revenue reduction) that the firm must bear if it chooses to produce the new product. b. Yes, expenditures on plant and equipment should be treated as incremental cash flows. These are costs of the new product line. However, if these expenditures have already occurred, they are sunk costs and are not included as incremental cash flows. c. No, the research and development costs should not be treated as incremental cash flows. The costs of research and development undertaken on the product during the past 3 years are sunk costs and should not be included in the evaluation of the project. Decisions made and costs incurred in the past cannot be changed. They should not affect the decision to accept or reject the project. d. Yes, the annual depreciation expense should be treated as an incremental cash flow. Depreciation expense must be taken into account when calculating the cash flows related to a given project. While depreciation is not a cash expense that directly affects cash flow, it decreases a firm’s net income and hence, lowers its tax bill for the year. Because of this depreciation tax shield, the firm has more cash on hand at the end of the year than it would have had without expensing depreciation. e. No, dividend payments should not be treated as incremental cash flows. A firm’s decision to pay or not pay dividends is independent of the decision to accept or reject any given investment project. For this reason, it is not an incremental cash flow to a given project. Dividend policy is discussed in more detail in later chapters. f. Yes, the resale value of plant and equipment at the end of a project’s life should be treated as an incremental cash flow. The price at which the firm sells the equipment is a cash inflow, and any difference between the book value of the equipment and its sale price will create gains or losses that result in either a tax credit or liability. g. Yes, salary and medical costs for production employees hired for a project should be treated as incremental cash flows. The salaries of all personnel connected to the project must be included as costs of that project. 7.2 Item I is a relevant cost because the opportunity to sell the land is lost if the new golf club is produced. Item II is also relevant because the firm must take into account the erosion of sales of existing products when a new product is introduced. If the firm produces the new club, the earnings from the existing clubs will decrease, effectively creating a cost that must be included in the decision. Item III is not relevant because the costs of Research and Development are sunk costs. Decisions made in the past cannot be changed. They are not relevant to the production of the new clubs. Choice C is the correct answer. B-116 7.3 Cash Flow Chart: Year 0 Year 1 Year 2 Year 3 Year 4 1. Sales revenue - $7,000 $7,000 $7,000 $7,000 2. Operating costs - 2,000 2,000 2,000 2,000 3. Depreciation - 2,500 2,500 2,500 2,500 4. Income before tax - 2,500 2,500 2,500 2,500 [1-(2+3)] 5. Taxes at 34% - 850 850 850 850 6. Net income 0 1,650 1,650 1,650 1,650 [4-5] 7. Cash flow from 0 4,150 4,150 4,150 4,150 operation [1-2-5] 8. Initial Investment -$10,000 - - - - 9. Changes in net working -200 -50 -50 100 200 capital 10. Total cash flow from -10,200 -50 -50 100 200 investment [9+10] 11. Total cash flow -$10,200 $4,100 $4,100 $4,250 $4,350 [7+10] a. Incremental Net Income [from 6]: Year 0 Year 1 Year 2 Year 3 Year 4 0 $1,650 $1,650 $1,650 $1,650 b. Incremental cash flow [from 11]: Year 0 Year 1 Year 2 Year 3 Year 4 -$10,200 $4,100 $4,100 $4,250 $4,350 c. The present value of each cash flow is simply the amount of that cash flow discounted back from the date of payment to the present. For example, discount the cash flow in Year 1 by 1 period (1.12), and 2. Note that since the Year 0 cash flow discount the cash flow that occurs in Year 2 by 2 periods (1.12) occurs today, its present value does not need to be adjusted. PV(C) = -$10,200 0PV(C) = $4,100 / (1.12) = $3,661 12PV(C) = $4,100 / (1.12) = $3,268 23PV(C) = $4,250 / (1.12) = $3,025 34PV(C) = $4,350 / (1.12) = $2,765 4 NPV = PV(C) + PV(C) + PV(C) + PV(C) + PV(C) = $2,519 01234 These calculations could also have been performed in a single step: 234 NPV = -$10,200 + $4,100 / (1.12) + $4,100 / (1.12) + $4,250 / (1.12) + $4,350 / (1.12) = $2,519 The NPV of the project is $2,519. B-117 7.4 The initial payment, which occurs today (year 0), does not need to be discounted: PV = $1,400,000 The expected value of his bonus payment is: (Probability of Occurrence) + C(Probability of Nonoccurrence) Expected Value = C0 1 = $750,000 (0.60) + $0 (0.40) = $450,000 The expected value of his salary, including the expected bonus payment, is $2,950,000 (=$2,500,000 + $450,000). The present value of his three-year salary with bonuses is: TPV Annuity = CA 1 r3 = $2,950,000 A0.1236 = $7,041,799 Remember that the annuity formula yields the present value of a stream of cash flows one period prior to the initial payment. Therefore, applying the annuity formula to a stream of cash flows that begins four years from today will generate the present value of that annuity as of the end of year three. Discount that result by three years to find the present value. TT-1 PV Delayed Annuity = (A) / (1+r)r103 = ($1,250,000 A) / (1.1236) 0.1236 = $4,906,457 Thus, the total PV of his three-year contract is: 3103 PV = $1,400,000 + $2,950,000 A+ ($1,250,000 A) / (1.1236)0.1236 0.1236 = $1,400,000 + $7,041,799 + $4,906,457 = $13,348,256 The present value of the contract is $13,348,256. 7.5 Compute the NPV of both alternatives. If either of the projects has a positive NPV, that project is more favorable to Benson than simply continuing to rent the building. If both of the projects have positive net present values, recommend the one with the higher NPV. If neither of the projects has a positive NPV, the correct recommendation is to reject both projects and continue renting the building to the current occupants. Note that the remaining fraction of the value of the building and depreciation are not incremental and should not be included in the analysis of the two alternatives. The $225,000 purchase price of the building is a sunk cost and should be ignored. B-118 Product A: t = 0 t = 1 - 14 t = 15 Revenues $105,000 $105,000 -Foregone rent 12,000 12,000 -Expenditures 60,000 63,750 ** -Depreciation* 12,000 12,000 Earnings before taxes $21,000 $17,250 -Taxes (34%) 7,140 5,865 Net income $13,860 $11,385 +Depreciation 12,000 12,000 Capital investment -$180,000 A/T-NCF -$180,000 $25,860 $23,385 *Since the two assets, equipment and building modifications, are depreciated on a straight-line basis, the depreciation expense will be the same in each year. To compute the annual depreciation expense, determine the total initial cost of the two assets ($144,000 + $36,000 = $180,000) and divide this amount by 15, the economic life of each of the 2 assets. Annual depreciation expense for building modifications and equipment equals $12,000 (= $180,000 / 15). **Cash expenditures ($60,000) + Restoration costs ($3,750) - The cash flows in years 1 - 14 (CC) could have been computed using the following simplification: 114 After-Tax NCF = Revenue (1 – T) - Expenses (1 - T) + Depreciation (T) CCC = $105,000 (0.66) - $72,000 (0.66) + $12,000 (0.34) = $25,860 The cash flows for year 15 could have been computed by adjusting the annual after-tax net cash flows of the project (computed above) for the after-tax value of the restoration costs. After-Tax value of restoration costs = Restoration Costs (1 - T) C = -$3,750 (0.66) = -$2,475 After-Tax NCF = $25,860 - $2,475 = $23,385 The present value of the initial outlay is simply the cost of the outlay since it occurs today (year 0). PV(C) = -$180,000 0 Since the cash flows in years 1-14 are identical, their present value can be found by determining the value of a 14-year annuity with payments of $25,860, discounted at 12 percent. 14 PV(C) = $25,860 A = $171,404 1-140.12 Because the last cash flow occurs 15 years from today, discount the amount of the cash flow back 15 years at 12 percent to determine its present value. 15 PV(C) = $23,385 / (1.12) 15 = $4,272 NPV = PV(C) + PV(C) + PV(C) A01-1415 = -$4,324 B-119 These calculations could also have been performed in a single step: 1415 = -$180,000 + $25,860 A + $23,385 / (1.12) NPVA0.12 = -$180,000 + $171,404 + $4,272 = -$4,324 Since the net present value of Project A is negative, Benson would rather rent the building to its current occupants than implement Project A. Product B t = 0 t = 1 - 14 t = 15 Revenues $127,500 $127,500 -Foregone rent 12,000 12,000 -Expenditures 75,000 103,125 ** -Depreciation* 14,400 14,400 Earnings before taxes $26,100 -$2,025 -Taxes (34%) 8,874 -689 Net income $17,226 -$1,336 +Depreciation 14,400 14,400 Capital investment -$216,000 A/T-NCF -$216,000 $31,626 $13,064 * Since the two assets, equipment and building modifications, are depreciated on a straight-line basis, the depreciation expense will be the same in each year. To compute the annual depreciation expense, determine the total initial cost of the two assets ($162,000 + $54,000 = $216,000) and divide this amount by 15, the economic life of each of the two assets. Annual depreciation expense for building modifications and equipment is $14,400 (= $216,000/ 15). **Cash expenditures ($75,000) + Restoration costs ($28,125) The cash flows in years 1 - 14 (C - C) could have been computed using the following simplification: 114 After-Tax NCF = Revenue (1 - T) - Expenses (1 - T) + Depreciation (T) = $127,500 (0.66) - $87,000 (0.66) + $14,400 (0.34) = $31,626 The cash flows for year 15 could have been computed by adjusting the annual after-tax net cash flows of the project (computed above) for the after-tax value of the restoration costs. After-tax value of restoration costs = Restoration Costs (1 - T) C = - $28,125(0.66) = -$18,562 After-Tax NCF = $31,626 - $18,562 = $13,064 The present value of the initial outlay is simply the cost of the outlay since it occurs today (year 0). PV(C) = -$216,000 0 Because the cash flows in years 1-14 are identical, their present value can be found by determining the value of a 14-year annuity with payments of $31,626, discounted at 12 percent. 14 PV(C) = $31,626 A 1-140.12 = $209,622 B-120 Since the last cash flow occurs 15 years from today, discount the amount of the cash flow back 15 years at 12 percent to determine its present value. 15) = $13,064 / (1.12) PV(C15 = $2,387 NPV = PV(C) + PV(C) + PV(C) B01-1415 = -$216,000 + $209,622 + $2,387 = -$3,991 These calculations could also have been performed in a single step: 1415 NPV = -$216,000 + $31,626 A + $13,064 / (1.12) B0.12 = -$216,000 + $209,622 + $2,387 = -$3,991 Since the net present value of Project B is negative, Benson would rather rent the building to its current occupants than implement Project B. Since the net present values of both Project A and Project B are negative, Benson should continue to rent the building to its current occupants. 7.6 Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 1. Keyboards Produced 10,000 10,000 10,000 10,000 10,000 2 3 4 2. Price per Keyboard 40 40(1.05) 40(1.05)40(1.05)40(1.05) 3. Sales revenue [1*2] 400,000 420,000 441,000 463,050 486,203 2344. Cost per Keyboard 20 20(1.10) 20(1.10) 20(1.10) 20(1.10) 5. Operating costs[1*4] 200,000 220,000 242,000 266,200 292,820 6. Gross Margin [3-5] 200,000 200,000 199,000 196,850 193,383 7. Depreciation 80,000 80,000 80,000 80,000 80,000 8. Pretax Income [6-7] 120,000 120,000 119,000 116,850 113,383 9. Taxes at 34% 40,800 40,800 40,460 39,729 38,549 10. Net income [8-9] 79,200 79,200 78,540 77,121 74,834 11. Cash flow from 159,200 159,200 158,540 157,121 154,834 operations [10+7] 12. Investment -400,000 13. Total Cash Flow -$400,000 $159,200 $159,200 $158,540 $157,121 $154,834 Since the initial investment occurs today (year 0), its present value does not need to be adjusted. PV(C) = -$400,000 0 PV(C) = $159,200 / (1.15) = $138,435 12PV(C) = $159,200 / (1.15) = $120,378 23PV(C) = $158,540 / (1.15) = $104,243 34PV(C) = $157,121 / (1.15) = $89,834 45PV(C) = $154,834 / (1.15) = $76,980 5 NPV = PV(C) + PV(C) + PV(C) + PV(C) + PV(C) + PV(C) = $129,870 012345 B-121 These calculations could also have been performed in a single step: 23NPV = -$400,000+ $159,200 / (1.15) + $159,200 / (1.15) + $158,540 / (1.15) 45 + $157,121 / (1.15) + $154,834 / (1.15) = $129,870 The NPV of the investment is $129,870. 7.7 Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 1. Annual Salary Savings $120,000 $120,000 $120,000 $120,000 $120,000 2. Depreciation 100,000 100,000 100,000 100,000 100,000 3. Taxable Income [1- 2] 20,000 20,000 20,000 20,000 20,000 4. Taxes 6,800 6,800 6,800 6,800 6,800 5. Operating Cash Flow [1- 113,200 113,200 113,200 113,200 113,200 4] 6. $100,000 -100,000 , Net working capital 7. Investment -$500,000 66,000* 8. Total Cash Flow -$400,000 $113,200 $113,200 $113,200 $113,200 $79,200 * When calculating the salvage value, remember that tax liabilities or credits are generated on the difference between the resale value and the book value of the asset. In this case, the computer has a book value of $0 and a resale value of $100,000 at the end of year 5. The total amount received in salvage value is the resale value minus the taxes paid on the difference between the resale value and the book value: $66,000 = $100,000 - 0.34 ($100,000 - $0). PV(C) = -$400,000 0PV(C) = $113,200 / (1.12) = $101,071 12PV(C) = $113,200 / (1.12) = $90,242 23PV(C) = $113,200 / (1.12) = $80,574 34PV(C) = $113,200 / (1.12) = $71,941 45PV(C) = $79,200 / (1.12) = $44,940 5 NPV = PV(C) + PV(C) + PV(C) + PV(C) + PV(C) + PV(C) = -$11,232 012345 These calculations could also have been performed in a single step: 23NPV = -$400,000 + $113,200 / (1.12) + $113,200 / (1.12) + $113,200 / (1.12) + 45 $113,200 / (1.12) + $79,200 / (1.12) = -$11,232 Since the NPV of the computer is negative, it is not a worthwhile investment. B-122 7.8 t = 0 t = 1- 2 t = 3 1. Revenues $600,000 $600,000 2. Expenses 150,000 150,000 3. Depreciation 150,000 150,000 4. Pretax Income $300,000 $300,000 [1-2-3] 5. Taxes (35%) 105,000 105,000 6. Net Income [4-5] $195,000 $195,000 7. Net Working Capital - 25,000 $25,000 8. CF from Operations - 25,000 $345,000 $370,000 [6+3+7] 9. Capital Investment - $750,000 $40,000 10. Tax benefit from $91,000 Capital Loss* 11. A/T-NCF - $775,000 $345,000 $501,000 * The capital loss arises because the resale value ($40,000) is less than the net book value ($300,000). The tax benefit from the capital loss is computed by multiplying the amount of the capital loss by the tax rate ($91,000 = 0.35 * $260,000). This represents the tax shield, i.e. the reduction in taxes from the capital loss. The cash flows in years 1 and 2 could also have been computed using the following simplification: ) - Expenses (1 – T) + Depreciation (T) After-Tax NCF = Revenue (1 – Tccc = $600,000 (0.65) - $150,000 (0.65) + $150,000(0.35) = $345,000 PV(C) = -$775,000 0PV(C) = $345,000/ (1.17) = $294,872 12PV(C) = $345,000/ (1.17) = $252,027 23 PV(C) = $501,000/(1.17)= $312,810 3 NPV = PV(C) + PV(C) + PV(C) + PV(C) = $84,709 0123 These calculations could also have been performed in a single step: 23 NPV = -$775,000 + $345,000/ (1.17) + $345,000/ (1.17) + $501,000/(1.17) = -$775,000 + $294,872 + $252,027 + $312,810 = $84,709 The NPV of the new software is $84,709. 7.9 The least amount of money that the firm should ask for the first-year lease payment is the amount that will make the net present value of the purchase of the building equal to zero. In other words, the least that the firm will charge for its initial lease payment is the amount that makes the present value of future cash flows just enough to compensate it for its $4,000,000 purchase. In order to determine this amount, set the net present value of the project equal to zero. Solve for the amount of the initial lease payment. Since the purchase of the building will occur today (year 0), its present value does not need to be adjusted. PV(Purchase of Building) = -$4,000,000 B-123 Since the initial lease payment also occurs today (year 0), its present value also does not need to be adjusted. However, since it will be recorded as revenue for the firm and will be taxed, the inflow must be adjusted to the corporate tax rate. (1- 0.34) PV(Initial Lease Payment) = C0 Note that in this problem we are solving for C, which is not yet known. 0 The second lease payment represents the first cash flow of a growing annuity. Since lease payments increase by three percent each year, the amount of the second payment is the amount of the first payment multiplied by 1.03, adjusted for taxes, or C(1- 0.66)(1.03). Recall that the appropriate discount rate is 12 0percent, the growth rate is three percent, and that the annuity consists of only 19 payments, since the first of the 20 payments was made at t=0. 19 PV(Remainder of Lease Payments) = C(1- 0.34)(1.03)(GA)* 00.12, 0.03 T* The notation GA represents a growing annuity consisting of T payments growing at a rate of g per r, g payment, discounted at r. Annual depreciation, calculated by the straight-line method (Initial Investment / Economic Life of Investment), is $200,000 (= $4,000,000 / 20). Since net income will be lower by $200,000 per year due to this expense, the firm’s tax bill will also be lower. The annual depreciation tax shield is found by multiplying the annual depreciation expense by the tax rate. The annual tax shield is $68,000 (= $200,000 * 0.34). Apply the standard annuity formula to calculate the present value of the annual depreciation tax shield. 20 PV(Depreciation Tax Shield) = $68,000A0.12 Recall that the least that the firm will charge for its initial lease payment is the amount that makes the present value of future cash flows just enough to compensate it for its $4,000,000 purchase. This is represented in the equation below: PV(Purchase) = PV(Lease Payments) + PV(Depreciation Tax Shield) 1920 $4,000,000 = C(1- 0.34) + C(1- 0.34)(1.03)( GA) + $68,000A 000.12, 0.030.12 C = $523,117 0 Therefore, the least that the firm should charge for its initial lease payment is $523,117. B-124 7.10 The decision to accept or reject the project depends on whether the NPV of the project is positive or negative. (in thousands) Year 0 Year 1 Year 2 Year 3 Year 4 1. Sales revenue - $1,200 $1,200 $1,200 $1,200 2. Operating costs - 300 300 300 300 3. Depreciation - 400 400 400 400 4. Income before tax - 500 500 500 500 [1-2-3] 5. Taxes at 35% - 175 175 175 175 6. Net income 0 325 325 325 325 [4-5] 7. Cash flow from 0 725 725 725 725 operation [1-2-5] 8. Initial Investment -2000 - - - 237.5* 9. Changes in net working -100 - - - 100 capital 10. Total cash flow from -2,100 - - - 337.5 investment [8+9] 11. Total cash flow -2,100 725 725 725 1,062.5 [7+10] * Remember that, when calculating the salvage value, tax liabilities or credits are generated on the difference between the resale value and the book value of the asset. Since the capital asset is depreciated over five years, yet sold in the year 4, the book value at the time of sale is $400,000 (= $2,000,000 – $1,600,000). Since the salvage value of $150,000 is below book value, the resulting capital loss creates a tax credit. After-Tax Resale Value = $150,000 - 0.35 ($150,000 – 400,000) = $237,500 Note that an increase in required net working capital is a negative cash flow whereas a decrease in required net working capital is a positive cash flow. Thus, in year 0, the firm realizes a $100,000 cash outflow while in year 4 the firm realizes a $100,000 cash inflow. Since year 0 is today, year 0 cash flows do not need to be discounted. ) = -$2,100,000 PV(C0PV(C) = $725,000 / (1.1655) = $622,051 12 PV(C) = $725,000 / (1.1655)= $533,720 23PV(C) = $725,000 / (1.1655) = $457,932 34PV(C) = $1,062,500 / (1.1655) = $575,811 4 NPV = PV(C) + PV(C) + PV(C) + PV(C) + PV(C) = $89,514 01234 These calculations could also have been performed in a single step: 2NPV = -$2,100,000 + $725,000 / (1.1655) + $725,000 / (1.1655) + $725,000 / 34(1.1655) + $1,062,500 / (1.1655) = $89,514 Since the NPV of the project is positive, Royal Dutch should proceed with the project. B-125 7.11 To determine the maximum price that MMC should be willing to pay for the equipment, calculate how high the price for the new equipment must be for the project to have an NPV of zero. Determine the cash flows pertaining to the sale of the existing equipment, the purchase of the new equipment, the future incremental benefits that the new equipment will provide to the firm, and the sale of the new equipment in eight years. Sale of existing equipment To find the after-tax resale value of the equipment, take into consideration the current market value and the accumulated depreciation. The difference is the amount subject to capital gains taxes. Purchase Price = $40,000 Depreciation per year = $40,000 / 10 years = $4,000 per year Accumulated Depreciation = 5 years * $4,000 per year = $20,000 Net Book Value of existing equipment = Purchase Price – Accumulated Depreciation = $40,000 - $20,000 = $20,000 (Sale Price – Net Book Value) PV(After-Tax Net Resale Value) = Sale Price – Tc = $20,000 - 0.34 ($20,000 – $20,000) = $20,000 Purchase of new equipment Let I equal the maximum price that MMC should be willing to pay for the equipment. PV(New Equipment) = -$I Lower operating costs Before-tax operating costs are lower by $10,000 per year for eight years if the firm purchases the new equipment. Lower operating costs raise net income, implying a larger tax bill. Increased annual taxes due to higher net income = $10,000 * 0.34 = $3,400 If the firm purchases the new equipment, its net income will be $10,000 higher but it will also pay $3,400 more in taxes. Therefore, lower operating costs increase the firm’s annual cash flow by $6,600. 8 PV(Operating Cost Savings) = $6,600 A 0.08= $37,928 Incremental depreciation tax shield The firm will realize depreciation tax benefits on the new equipment. However, the firm also foregoes the depreciation tax shield on the old equipment. Incremental depreciation per year due to new equipment = Annual Depreciation on new equipment – Annual Depreciation on old equipment if it had been retained Annual Depreciation on New Equipment = (Purchase Price/ Economic Life) = ($I/5) Annual Depreciation on Old Equipment = $4,000 B-126 Incremental Depreciation per year due to new equipment = ($I/5) - $4,000 Incremental Depreciation tax shield per year = Incremental Depreciation per year * TC =[($I/5) - $4,000] * 0.34 5PV(Incremental Depreciation Tax Shield) = 0.34[($I/5) - $4,000] A 0.08 Note that since both old and new equipment will be fully depreciated after 5 years, no depreciation tax shield is applicable in years 6-8. Sale of New Equipment The new equipment will be sold at the end of year 8. Since it will have been fully depreciated by year 5, capital gains taxes must be paid on the entire resale price. Sale Price of new equipment = $5,000 Net Book Value of new equipment = $0 (It had been fully depreciated as of year 5.) After-Tax Net Cash Flow = Sale Price – T (Sale Price – Net Book Value) c = $5,000 - 0.34 ($5,000 – 0) = $3,300 8PV(Resale Value) = $3,300 / (1.08) = $1,783 The maximum price that MMC should be willing to pay for the new equipment is the price that makes the NPV of the investment equal to zero. In order to solve for the price, set the net present value of all incremental after-tax cash flows related to the new equipment equal to zero and solve for I. 8580 = ($20,000 – $I) + $6,600 A + [0.34][($I/5) - $4,000] A + $3,300/ (1.08) 0.080.08 I = $74,510 Therefore, the maximum price that MMC should be willing to pay for the equipment is $74,510. 7.12 Purchase of New Equipment = -$28,000,000 Since the old equipment is sold at a price that is greater than its book value, the firm will record a capital gain on the sale, and this sale will be subject to the corporate tax rate. After-Tax Salvage Value = Sale Price – T(Sale Price – Net Book Value) CAfter-Tax Value of Sale of Old Equipment = $20,000,000 - 0.40($20,000,000-$12,000,000) = $16,800,000 After-Tax Operating Cost Savings due to New Equipment Year 1 = (1-0.40)($17,500,000) = $10,500,000 Year 2 = (1-0.40)($17,500,000)(1.12) = $11,760,000 2Year 3 = (1-0.40)($17,500,000)(1.12) = $13,171,200 3Year 4 = (1-0.40)($17,500,000)(1.12) = $14,751,744 Depreciation of Old Equipment Year 1 = ($12,000,000/4) = $3,000,000 Year 2 = ($12,000,000/4) = $3,000,000 Year 3 = ($12,000,000/4) = $3,000,000 Year 4 = ($12,000,000/4) = $3,000,000 B-127 Depreciation of New Equipment Year 1 = ($28,000,000 * 0.333) = $9,324,000 Year 2 = ($28,000,000*0.399) = $11,172,000 Year 3 = ($28,000,000*0.148) = $4,144,000 Year 4 = ($28,000,000*0.120) = $3,360,000 Incremental Depreciation due to New Equipment Year 1 = $9,324,000 - $3,000,000 = $6,324,000 Year 2 = $11,172,000- $3,000,000 = $8,172,000 Year 3 = $4,144,000- $3,000,000 = $1,144,000 Year 4 = $3,360,000- $3,000,000 = $360,000 Incremental Depreciation Tax Shield due to New Equipment Year 1 = $6,324,000 * 0.40 = $2,529,600 Year 2 = $8,172,000 * 0.40 = $3,268,800 Year 3 = $1,144,000 * 0.40 = $457,600 Year 4 = $360,000 * 0.40 = $144,000 a. Net Investment = - Purchase of New Equipment + After-Tax Proceeds from Sale of Old Equipment + Increase in Net Working Capital = -$28,000,000 + $16,800,000 - $5,000,000 = -$16,200,000 Therefore, the cash outflow at the end of year 0 is $16,200,000. b. Year 0 Year 1 Year 2 Year 3 Year 4 Purchase of New Equipment -28,000,000 After-Tax Sale of Old Equipment 16,800,000 -5,000,000 5,000,000 , Net Working Capital After-Tax Operating Cost Savings 10,500,000 11,760,000 13,171,200 14,751,744 Incremental Depreciation Tax Shield 2,529,600 3,268,800 457,600 144,000 After-Tax Incremental Cash Flow -16,200,000 13,029,600 15,028,800 13,628,800 19,895,744 c. IRR Calculation: In order to determine the internal rate return (IRR) of the investment in new equipment, determine the discount rate that makes the NPV of the project equal to zero. 23 + $13,628,800/(1+IRR) + 0 =-$16,200,000 + $13,029,600/(1+IRR) + $15,028,800/(1+IRR)4 $19,895,744/(1+IRR) IRR = 0.7948 = 79.48% The internal rate of return of the investment in new equipment is 79.48%. d. NPV Calculation: 23 NPV =-$16,200,000 + $13,029,600/(1.14) + $15,028,800/(1.14) + $13,628,800/(1.14) + 4 $19,895,744/(1.14) = $27,772,577 The net present value of the investment in new equipment is $27,772,577. B-128 7.13 $12 million = New market share – Current Market Erosion $20 million – $4 million $100*20% – $80*10%*.5 Note: The $10 million for running shoe R & D is a sunk cost and not considered. The $1 million for walking shoe R & D is a capital expenditure, but does not affect revenues. The $1 million for the warehouse extension is a capital expenditure, but does not affect revenues. B-129 7.14 Amount in thousands Period 0 1 2 3 4 5 NPV Signing bonus $5,000 Salary $2,000 $2,000 $2,000 $2,000 $2,000 Annual bonus* $1,000 $1,000 $1,000 $1,000 $1,000 Total Cash Flow $5,000 $3,000 $3,000 $3,000 $3,000 $3,000 2345PV 12% (CF) = CF = CF/1.12 = CF/(1.12) = CF/(1.12) = CF/(1.12) = CF/(1.12) 5,000 2,679 2,392 2,135 1,907 1,702 $15,814,329 *annual bonus = $1 mil ($2 mil x 50%) NPV = $15,814,329. 7.15 Year Year Year Year Year Year 0 1 2 3 4 5 NPV Investments: B-130 Capital equipment (60,000) Change in NWC (28,000) (13,500) (15,525) (17,854) (20,532) Total cash flow from investments (60,000) (28,000) (13,500) (15,525) (17,854) (20,532) Income: Units 5,000 5,750 6,613 7,604 8,745 Sales 225,000 258,750 297,563 342,197 393,526 Fixed Costs 75,000 75,000 75,000 75,000 75,000 Variable Costs 100,000 115,000 132,250 152,088 174,901 Depreciation (12,000) (12,000) (12,000) (12,000) (12,000) Income before depreciation 62,000 80,750 102,313 127,109 155,626 Taxes at 34% 15,500 20,188 25,578 31,777 38,906 Net Income 46,500 60,563 76,734 95,332 116,719 Cash flow from operations: Net Income 46,500 60,563 76,734 95,332 116,719 Depreciation 12,000 12,000 12,000 12,000 12,000 Operating cash flow 58,500 72,563 88,734 107,332 128,719 Total cash flow from project (60,000) 30,500 59,063 73,209 89,478 108,188 PV 25% (CF) (60,000) 24,400 37,800 37,483 36,650 35,451 $111,784 The project’s NPV is $111,784. Note: We are assuming that initial net working capital (NWC) is needed for year 1 production and starts in year 1. 7.16 Year Year Year Year Year Year 0 1 2 3 4 5 NPV Investments: Touch screen system $ (150,000) Annual maintenance $ (5,000) $ (5,000) $ (5,000) $ (5,000) $ (5,000) Change in NWC $ (5,000) $ 5,000 Wages saved $30,000 $30,000 $30,000 $30,000 $30,000 Total cash flow from investments $ (150,000) $20,000 $25,000 $25,000 $25,000 $30,000 Income: Revenue $15,000 $15,000 $15,000 $15,000 $15,000 COGS $ (3,750) $ (3,750) $ (3,750) $ (3,750) $ (3,750) Cash flow from operations $11,250 $11,250 $11,250 $11,250 $11,250 Total cash flow from project $ (150,000) $31,250 $36,250 $36,250 $36,250 $41,250 2345 PV 15% (CF) = CF = CF/1.15 = CF/(1.15) = CF/(1.15) = CF/(1.15) = CF/(1.15) (150,000) 27,174 27,410 23,835 20,726 20,509 ($30,346) B-131 No, the big burrito should not make this investment because it has a negative NPV. 7.17 Year 0 1 2 3 4 Signing $500,000 Salary $800,000 800,000 800,000 800,000 8Deferred 523,919 = 100000/0.105*(1–(1/1.105) Bonus 82,500 82,500 82,500 82,500 = 150000*0.55 SUM $500,000 882,500 882,500 882,500 1,406,419 PV $500,000 798,643 722,753 654,075 943,334 SUM (PV) $3,618,806 7.18 Year Year Year Year Year Year 0 1 2 3 4 5 NPV Investments: Capital equipment (1,500,000) 400,000 Capital gains taxes (106,420) Change in NWC 500,000 (500,000) Salary savings 300,000 300,000 300,000 300,000 300,000 After-tax salary savings 198,000 198,000 198,000 198,000 198,000 Depreciation tax shield 102,000 163,200 97,920 58,650 58,650 Total cash flow from project (1,500,000) 800,000 361,200 295,920 256,650 50,230 PV 15% (CF) (1,500,000) 695,652 273,119 194,572 146,740 24,973 $(164,943) No, it is not worthwhile to purchase the automation machine since it has a negative NPV. Note: We are assuming the reduction in net working capital (NWC) in one-time only. B-132 7.19 Year Year Year Year Year Year 0 1 2 3 4 5 NPV Investments: Capital equipment (10,000) Change in NWC (1,000) 1,000 Profits 6,000 6,000 6,000 6,000 6,000 After-tax profits 3,960 3,960 3,960 3,960 3,960 Depreciation tax shield 680 680 680 680 680 Total cash flow from project (10,000) 3,640 4,640 4,640 4,640 5,640 PV 17% (CF) (10,000) 3,111 3,390 2,897 2,476 2,572 $4,446 Yes, it is worthwhile to purchase the machine since it has a NPV of $4,446. Note: We are assuming the increase in net working capital (NWC) is one-time only and it is associated with the use of the machine in Year 1. 7.20 Year Year Year Year Year Year 0 1 2 3 4 5 NPV Investments: Capital equipment (1,000,000) Change in NWC (10,000) 10,000 Costs 329,220 329,220 329,220 329,220 329,220 After-tax Costs 217,285 217,285 217,285 217,285 217,285 Depreciation tax shield 68,000 68,000 68,000 68,000 68,000 Total cash flow from project (1,000,000) 275,285 285,285 285,285 285,285 295,285 PV 13% (CF) (1,000,000) 243,615 223,420 197,717 174,971 160,269 0 a. At 26 cents per bottle (rounding down from 26.46 cents), the NPV of the machine will be 0. b. No, it is not worthwhile to make the investment at 25 cents per bottle. Note: We are assuming the increase in net working capital (NWC) is one-time only and it is associated with the use of the machine in Year 1. B133 7.21 Nominal cash flows should be discounted at the nominal discount rate. Real cash flows should be discounted at the real discount rate. Project A’s cash flows are presented in real terms. Therefore, one must compute the real discount rate before calculating the NPV of Project A. Since the cash flows of Project B are given in nominal terms, discount its cash flows by the nominal rate in order to calculate its NPV. Nominal Discount Rate = 0.15 Inflation Rate = 0.04 1 + Real Discount Rate = (1+ Nominal Discount Rate) / (1+ Inflation Rate) Real Discount Rate = 0.1058 =10.58% Project A’s cash flows are expressed in real terms and therefore should be discounted at the real discount rate of 10.58%. Project A: ) = -$40,000 PV(C0 PV(C) = $20,000 / (1.1058) = $18,086 12PV(C) = $15,000/ (1.1058) = $12,267 23PV(C) = $15,000 / (1.1058) = $11,093 3 NPV = PV(C) + PV(C) + PV(C) + PV(C) A0123 = $1,446 These calculations could also have been performed in a single step: 23NPV = -$40,000+ $20,000 / (1.1058) + $15,000 / (1.1058) + $15,000 / (1.1058) A = $1,446 Project B’s cash flows are expressed in nominal terms and therefore should be discounted at the nominal discount rate of 15%. Project B: PV(C) = -$50,000 0PV(C) = $10,000 / (1.15) = $8,696 12PV(C) = $20,000/ (1.15) = $15,123 23PV(C) = $40,000 / (1.15) = $26,301 3 NPV = PV(C) + PV(C) + PV(C) + PV(C) B0123 = $120 These calculations could also have been performed in a single step: 23NPV = -$50,000+ $10,000 / (1.15) + $20,000 / (1.15) + $40,000 / (1.15) B = $120 Since the NPV of Project A is greater than the NPV of Project B, choose Project A. 7.22 Notice that the problem provides the nominal values at the end of the first year, so to find the values for revenue and expenses at the end of year 5, compound the values by four years of inflation, e.g. 4$200,000*(1.03) = $225,102. Since the resale value is given in nominal terms as of the end of year 5, it does not need to be adjusted for inflation. Also, no inflation adjustment is needed for either the depreciation charge or the recovery of net working capital since these items are already expressed in nominal terms. Note that an increase in required net working capital is a negative cash flow whereas a decrease in required net working capital is a positive cash flow. B-134 Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 1. Revenue $200,000 $206,000 $212,180 $218,545 $225,102 2. Expenses 50,000 51,500 53,045 54,636 56,275 3. Depreciation 50,000 50,000 50,000 50,000 50,000 4. Taxable Income 100,000 104,500 109,135 113,909 118,827 [1 –2 –3] 5. Taxes 34,000 35,530 37,106 38,729 40,401 6. Operating Cash Flow 116,000 118,970 122,029 125,180 128,426 [1 – 2 – 5] 7. -10,000 10,000 , Net working capital 8. Investment -250,000 19,800* 9. Total Cash Flow -$260,000 $116,000 $118,970 $122,029 $125,180 $158,226 * When calculating the salvage value of the asset, remember that only the gain on the sale of the asset is taxed. This gain is calculated as the difference between the resale value and the net book value of the asset (Resale Value – Net Book Value). at the time of sale. It follows that the tax associated with the sale is TC Therefore, the after-tax salvage value of the asset is $19,800 [= $30,000 – 0.34($30,000 – 0)]. The nominal cash flow at year 5 is $158,226. 7.23 Since the problem lists nominal cash flows and a real discount rate, one must determine the nominal discount rate before computing the net present value of the project. 1 + Real Discount Rate = (1 + Nominal Discount Rate) / (1 + Inflation Rate) 1.14 = (1+ Nominal Discount Rate) / (1.05) Nominal Discount Rate = 0.197 Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 1. Sales revenue - $50,000 $52,500 $55,125 $57,881 $60,775 $63,814 $67,005 2. Operating costs - 20,000 21,400 22,898 24,501 26,216 28,051 30,015 3. Depreciation - 17,143 17,143 17,143 17,143 17,143 17,143 17,143 4. Income before - 12,857 13,957 15,084 16,237 17,416 18,620 19,847 tax [1-2-3] 5. Taxes at 34% - 4,371 4,745 5,129 5,521 5,921 6,331 6,748 6. Net income - 8,486 9,212 9,955 10,716 11,495 12,289 13,099 [4-5] 7. Cash flow from - 25,629 26,355 27,098 27,859 28,638 29,432 30,242 operation [1-2-5] 8. Initial -120,000 - - - - - - - Investment 10. Total cash flow -120,000 - - - - - - - from investment [9+10] 11. Total cash flow -120,000 25,629 26,355 27,098 27,859 28,638 29,432 30,242 [7+10] B-135 ) = -$120,000 PV(C0 PV(C) = $25,629 / (1.197) = $21,411 12PV(C) = $26,355 / (1.197) = $18,394 23PV(C) = $27,098 / (1.197) = $15,800 34PV(C) = $27,859 / (1.197) = $13,570 45PV(C) = $28,638 / (1.197) = $11,654 56PV(C) = $29,432 / (1.197) = $10,006 67PV(C) = $30,242 / (1.197) = $8,589 7 NPV = PV(C) + PV(C) + PV(C) + PV(C) + PV(C) + PV(C) + PV(C) + PV(C) 01234567 = -$20,576 These calculations could also have been performed in a single step: 23NPV = -$120,000 + $25,629 / (1.197) + $26,025 / (1.197) + $27,098 / (1.197) 456 + $27,859 / (1.197) + $28,638 / (1.197) + $29,432 / (1.197) 7+ $30,242 / (1.197) = -$20,576 To solve the problem using a string of annuities, find the present value of each cash flow. The investment occurs today and therefore is not discounted: PV(Investment) = -$120,000 The PV of the revenues is found by using the growing annuity formula. Note that nominal cash flows must be discounted by nominal rates. The following solution treats revenues as a growing annuity discounted at 19.7 percent and growing at five percent annually over seven years: T PV(Revenues) = C (1 – T) GA 1cr, g7 PV(Revenues) = $50,000 GA (1 - 0.34)* 0.197, 0.05 = $134,775 T* The notation GA represents a growing annuity consisting of T payments growing at a rate of g per r, g payment, discounted at r. The PV of the expenses is found using the same method that was used in finding the PV of the revenues. Again, the expenses are treated as a nominal growing annuity, discounted at 19.7 percent and growing at seven percent annually over seven years: T PV Expenses = C GA (1 – T) 1r, gc7 PV Expenses = $20,000 GA (1 - 0.34) 0.197, 0.07 = $56,534 Since the firm has positive net income, the firm will benefit from the depreciation tax shield. Apply the annuity formula to the string of annual tax shields to find the present value of the taxes saved. T PV(Depreciation Tax Shield) = T (Annual Depreciation) Acr 7PV(Depreciation Tax Shield) = 0.34 ($120,000 / 7) A 0.197 = $21,183 B-136 The present value of the project is the sum of the previous annuities: PV Project = -Investment + Revenue - Expenses + Depreciation Tax Shield PV Project = -$120,000 + $134,775 - $56,534 + $21,183 PV Project = -$20,576 Since the project has a negative NPV, -$20,576, it should be rejected. The nominal cash flow during year 5 is $157,926. 7.24 Apply the growing perpetuity formula to the payments that are declining at a constant rate. Because the payments are declining, they have a negative growth rate. The initial cash flow of the perpetuity occurs one year from today and is expressed in real terms. = $120,000 C1 The real discount rate is 11%. r = 0.11 The real growth rate is -6%. g = -0.06 PV = C/ (r-g) , where r > g 1 = $120,000 / [ 0.11 – (-0.06)] = $120,000 / (0.11 + 0.06) = $120,000 / 0.17 = $705,882 The present value of Phillip’s net cash flows is $705,882. 7.25 Notice that the discount rate is expressed in real terms and the cash flows are expressed in nominal terms. In order to solve the problem, convert all nominal cash flows to real cash flows and discount them using the real discount rate. Year 1 Revenue in Real Terms = $150,000 / 1.06 = $141,509 Year 1 Labor Costs in Real Terms = $80,000 / 1.06 = $75,472 Year 1 Other Costs in Real Terms = $40,000 / 1.06 = $37,736 Year 1 Lease Payment in Real Terms = $20,000 / 1.06 = $18,868 Revenues and labor costs form growing perpetuities and other costs form a declining perpetuity. PV (Revenue) = ($141,509.43) / (0.10 - 0.05) = $2,830,189 PV (Labor Costs) = ($75,471.70) / (0.10 - 0.03) = $1,078,167 PV (Other Costs) = ($37,735.85) / [0.10 - (-0.01)] = $343,053 Since the lease payments are constant in nominal terms, they are declining in real terms by the inflation rate. Therefore, the lease payments form a declining perpetuity. PV(Lease Payments) = ($18,868 / [0.10 – (-0.06)] = $117,925 NPV = PV(Revenue) – PV(Labor Costs) – PV(Other Costs) – PV(Lease Payments) = $2,830,189 - $1,078,167 - $343,053 - $117,925 = $1,291,044 B-137 The NPV of the proposed toad ranch is $1,291,044. Alternatively, one could solve this problem by expressing everything in nominal terms. This approach yields the same answer as given above. However, in this case, the computation would have been much more difficult. When faced with two alternative approaches, where both are equally correct, always choose the simplest one. 7.26 Year 1 Year 2 Year 3 Year 4 Revenues $40,000,000 $80,000,000 $80,000,000 $60,000,000 Labor Costs 30,600,000 31,212,000 31,836,240 32,472,965 Energy Costs 1,030,000 1,060,900 1,092,727 1,125,509 Revenues-Costs 8,370,000 47,727,100 47,071,033 26,401,526 After-tax Revenues-Costs 5,524,200 31,499,886 31,066,882 17,425,007 Since revenues and costs are expressed in real terms, after-tax income will be discounted at the real discount rate of 8%. Remember that the depreciation tax shield also affects a firm’s after-tax cash flows. The present value of the depreciation tax shield must be added to the present value of a firm’s revenues and expenses to find the present value of the cash flows related to the project. The depreciation the firm will recognize each year is: Depreciation = Investment / Economic Life = $32,000,000 / 4 = $8,000,000 Next, find the annual depreciation tax shield. Remember that this reduction in taxes is equal to the tax rate times the depreciation expense for the year. (Annual Depreciation Expense) Annual Depreciation Tax Shield = Tc = 0.34 ($8,000,000) = $2,720,000 Remember that depreciation is a nominal quantity, and thus must be discounted at the nominal rate. To find the nominal rate, use the following equation: 1+ Real Discount Rate = (1+Nominal Discount Rate) / (1+Inflation Rate) 1.08 = (1+Nominal Discount Rate) / (1.05) Nominal Discount Rate = 0.134 To find the present value of the depreciation tax shield, apply the four-year annuity formula to the annual tax savings: 4 PV(Tax Shield) = C A 10.1344 = $2,720,000 A0.134 = $8,023,779 PV(C) = -$32,000,000 = -$32,000,000 0PV(C) = $5,524,200 / (1.08) = $5,115,000 12 PV(C) = $31,499,886 / (1.08)= $27,006,075 23PV(C) = $31,066,882 / (1.08) = $24,661,893 34PV(C) = $17,425,007 / (1.08) = $12,807,900 4PV(Depreciation Tax Shield) = $8,023,779 B-138 NPV = PV(C) + PV(C) + PV(C) + PV(C) + PV(C) + PV(Depreciation Tax Shield) 01234 = $45,614,647 These calculations also could have been performed in a single step: 23 NPV = -$32,000,000+ $5,524,200 / (1.08) + $31,499,886 / (1.08) + $31,066,882 / (1.08) 44 + $17,425,007 / (1.08) + (0.34) ($8,000,000) A0.134 = $45,614,647 The NPV of the project is $45,614,647. 7.27 In order to determine how much Sparkling Water, Inc. is worth today, find the present value of its cash flows. Sparkling will receive $2.50 per bottle in revenues in real terms at the end of year 1. After-Tax Revenue in Year 1 in real terms = (2,000,000 * $2.50)(1-0.34) = $3,300,000 Sparkling’s revenues will grow at seven percent per year in real terms forever. Apply the growing perpetuity formula. PV(Revenues) = C/ (r-g) , where r > g 1 = $3,300,000 / (0.10 – 0.07) = $110,000,000 Per bottle costs will be $0.70 in real terms at the end of year 1. After-Tax Costs in Year 1 in real terms = (2,000,000 * $0.70)(1-0.34) = $924,000 Sparkling’s costs will grow at 5% per year in real terms forever. This string of payments forms a growing perpetuity. PV(Costs) = C/ (r-g) , where r > g 1 = $924,000 / (0.10 – 0.05) = $18,480,000 Value of the firm = PV(Revenues) – PV(Costs) = $110,000,000 - $18,480,000 = $91,520,000 Sparkling Water, Inc., is worth $91,520,000 today. 7.28 Since all cash flows are stated in nominal terms and the growth rates of both the sales price and the variable cost are stated in real terms, these rates must be restated in nominal terms in order to solve the problem. Since the discount rate is expressed in nominal terms, it does not need to be adjusted. Alternatively, one could solve this problem by expressing everything in real terms. This approach yields the same answer. Find the nominal growth rates: 1 + Real Rate = (1 + Nominal Rate) / (1 + Inflation Rate) 1.05 = (1 + Nominal Selling Price Growth Rate) / (1.05) 0.1025 = Nominal Selling Price Growth Rate B-139 1.02 = (1 + Nominal Variable Cost Growth Rate) / (1.05) 0.071 = Nominal Variable Cost Growth Rate The revenue stream is a five-year growing annuity. Pre-tax revenue in year 1 is found by multiplying the selling price ($3.15) by the number of units produced (1,000,000). The cash flows are growing at the nominal rate of 0.1025 and are discounted at 0.20. In order to find the after-tax present value, multiply ). revenues by (1-TC T PV (Revenues) = (1 – T) (Year 1 Selling Price) (Year 1 Production) GA * cr,g5 PV (Revenues) = (1 - 0.34) ($3.15) (1,000,000) GA 0.20, 0.1025 = $7,364,645 T* The notation GA represents a growing annuity consisting of T payments growing at a rate of g per r, gpayment, discounted at r. The PV of the variable costs is also calculated using the five-year growing annuity formula. Pre-tax variable costs in year 1 are found by multiplying the variable cost ($0.2625) by the number of units (1,000,000). The cash flows are growing at the nominal rate of 0.071 and are discounted at 0.20. In order to find the after-tax present value, multiply variable costs by (1-T). C T PV (Variable Costs) = (1 – T) (Year 1 Variable Costs) (Year 1 Production) GAcr,g 5 PV (Variable Costs) = (1 - 0.34) ($0.2625) (1,000,000) GA 0.20, 0.071 = $582,479 Since the firm is subject to corporate taxes, it will benefit from the depreciation tax shield. First, find the annual depreciation tax shield, which is the tax rate multiplied by the annual depreciation expense. Next, find the PV of all annual tax shields via the annuity formula, using the nominal discount rate of 0.20. Depreciation is a nominal quantity, and therefore must be discounted at the nominal rate. Annual Depreciation Expense = (Investment) / (Economic Life) = $6,000,000 / 5 = $1,200,000 To find the annual depreciation tax shield, perform the following calculation: Annual Depreciation Tax Shield = T (Annual Depreciation Expense) c = 0.34 ($1,200,000) = $408,000 Next, apply the annuity formula to calculate the PV of the annual depreciation tax shields. 5PV(Depreciation Tax Shield) = $408,000 A 0.20 = $1,220,170 The last relevant cash flow is the salvage value of the factory. Since the resale value ($638,140.78) is higher than the book value ($0), the firm must pay capital gains taxes on the difference. Once the after-tax value is calculated, the value must be discounted back five years to the present (year 0). Remember that the salvage value is expressed in nominal terms, and thus must be discounted by the nominal discount rate, 0.20. After-Tax Salvage Value = Salvage Value – T (Salvage Value – Book Value) c = $638,140.78 - 0.34 ($638,140.78 - $0) = $421,173 B-140 5 PV(After-Tax Salvage Value) = C / (1+r) 55 = $421,173 / (1.20) = $169,260 To compute the NPV of the project, consider the PVs of all the relevant after-tax cash flows. NPV = -Investment + PV(Revenues) - PV(Costs) + PV(Depreciation Tax Shield) + PV(Salvage Value) = -$6,000,000 + $7,364,645 - $582,479 + $1,220,170 + $169,260 = $2,171,596 These calculations could also have been performed in a single step: 5 NPV = -$6,000,000 + (1 - 0.34) ($3.15) (1,000,000) A – (1 - 0.34) ($.2625) 0.20, 0.102555 (1,000,000) A + 0.34 ($6,000,000 / 5) A + 0.20, 0.0710.205 [$638,140.78 - 0.34 ($638,140.78 - $0)] / (1.20) = $2,171,596 The NPV of the project is $2,171,596. 7.29 Since the problem asks which medicine the company should produce, solve for the NPV of both medicines and select the one with the higher NPV. Headache-only medicine: First, find the PV of the initial investment. Since the cash outlay occurs today, no discounting is necessary. PV(Initial Investment) = -$10,200,000 Find the PV of the revenues if the headache-only medicine were produced. The problem states that the selling price will be $4 in real terms. Since the discount rate, 0.13, is also given in real terms, no adjustment is necessary and inflation can be ignored. The problem also indicates that 5 million packages will be sold in each of the next three years. The PV will be expressed as a three-year annuity discounted at 0.13. Remember to find the after-tax revenues by multiplying pre-tax revenues by (1 - T). c Annual Revenues Headache-only = $4 * 5,000,000 = $20,000,000 TPV(Headache-only revenues) = (1 - T) C A c1r,3 = (1 - 0.34) $20,000,000 A0.13 = $31,167,214 Annual costs per unit will be $1.50 in real terms. The PV will be expressed as a three-year annuity discounted at the real discount rate of 0.13. Remember to find the after-tax costs by multiplying pre-tax costs by (1 - T). c Annual Costs Headache-only = -$1.50 * 5,000,000 = -$7,500,000 TPV(Headache-only costs) = (1 - T) C A c1r,3 = (1 - 0.34)( -$7,500,000 A) 0.13 = -$11,687,705 B-141 Since Pill, Inc. has positive pre-tax income, the firm will benefit from a depreciation tax shield. Remember, depreciation is a nominal quantity and therefore must be discounted at the nominal rate. 1 + Real Rate = (1 + Nominal Rate) / (1 + Inflation Rate) 1.13 = (1 + Nominal Rate) / (1.05) Nominal Rate = 0.1865 Annual depreciation, calculated by the straight-line method (Initial Investment / Economic Life of Investment), is $3,400,000 (= $10,200,000 / 3 Years). The string of annual tax shields forms an annuity. The present value of this annuity is: T (Annual Depreciation Expense) A PV(Depreciation Tax Shield) = Tcr 3PV(Depreciation Tax Shield) = 0.34 ($3,400,000) A 0.1865 = $2,487,521 Since the resale value of the headache-only equipment is $0, it has no effect the NPV of the project. To find the NPV of the project, find the sum of the present values of the initial investment, after-tax revenues, after-tax costs, and the depreciation tax shield. NPV = - Initial Investment + PV(Revenues) – PV(Costs) + Depreciation Tax Shield = -$10,200,000 + $31,167,214 – $11,687,705 + $2,487,521 = $11,767,030 These calculations could also have been performed in a single step: 33NPV = -$10,200,000 + (1 - 0.34) $20,000,000 A – (1 - 0.34) $7,500,000 A+ 0.130.13 30.34 ($3,400,000) A 0.1865 = $11,767,030 Headache and Arthritis medicine: First, find the PV of the initial investment. Since the cash outlay occurs today, no discounting is necessary. PV(Initial Investment) = -$12,000,000 Find the PV of the revenues if the headache and arthritis medicine were produced. The problem states that the selling price will be $4 in real terms. Since the discount rate, 0.13, is also given in real terms, no adjustment is necessary, and inflation can be ignored. The problem also indicates that 10 million packages will be sold in each of the next 3 years. The PV will be expressed as a three-year annuity discounted at 0.13. Remember to find the after-tax revenues by multiplying pre-tax revenues by (1 - T). c Annual Revenues Headache and Arthritis = $4 * 10,000,000 = $40,000,000 TPV(Headache and Arthritis revenues) = (1 - T) C A c1r,3 = (1 - 0.34) $40,000,000 A0.13 = $62,334,429 The annual costs will be calculated using the same method. The problem states that annual costs per unit will be $1.70. Again, since the costs and discount rate are given in real terms, inflation can be ignored. The PV will be expressed as a three-year annuity discounted at 0.13. Remember to find the after-tax costs by multiplying pre-tax costs by (1 - T). c Annual Costs Headache and Arthritis = -$1.70 * 10,000,000 = -$17,000,000 B-142 T PV(Headache and arthritis costs) = (1 - T) C A c1r,3 = (1 - 0.34) -$17,000,000 A 0.13 = -$26,492,132 Since Pill, Inc. has positive income, it will benefit from a depreciation tax shield. Remember, depreciation is a nominal quantity and therefore must be discounted using the nominal rate. 1 + Real Discount Rate = (1 + Nominal Discount Rate) / (1 + Inflation Rate) 1.13 = (1 + Nominal Discount Rate) / (1.05) Nominal Discount Rate = 0.1865 Annual depreciation, calculated by the straight-line method (Initial Investment / Economic Life of Investment), is $4,000,000 (= $12,000,000 / 3 Years). The string of annual tax shields forms an annuity. The present value of this annuity is: T PV(Depreciation Tax Shield) = T (Annual Depreciation) Acr 3PV(Depreciation Tax Shield) = 0.34 ($4,000,000) A 0.1865 = $2,926,496 Unlike the Headache-only medicine equipment, the Headache and Arthritis medicine equipment has a resale value of $1 million at the end of three years. Since the net book value of the equipment is $0, Pill, Inc. must pay capital gains taxes on the total $1 million resale value. Because the resale value is stated in real terms, it is discounted using the real discount rate. After-Tax Salvage Value = Salvage Value – T (Salvage Value – Book Value) c = $1,000,000 – 0.34 ($1,000,000 – $0) = $660,000 The salvage value must then be discounted in order to find the PV. 3PV(Salvage Value) = C / (1 + r) 33 = $660,000 / (1.13) = $457,413 To find the NPV of the project, find the sum of the present values of the initial investment, after-tax revenues, after-tax costs, the depreciation tax shield, and the resale. NPV = - Initial Investment + PV(Revenues) – PV(Costs) + PV(Tax Shield) + PV(Resale Value) = -$12,000,000 + $62,334,429 – $26,492,132 + $2,926,496 + $457,413 = $27,226,206 These calculations could also have been performed in a single step: 33NPV = -$12,000,000 + (1 - 0.34) $40,000,000 A – (1 - 0.34) $17,000,000 A 0.130.1333 + 0.34 ($4,000,000) A+ [$1,000,000 – 0.34 ($1,000,000 – $0)] / (1.13) 0.1865 = $27,226,206 Pill, Inc. should produce the Headache and Arthritis medicine since it has the higher NPV. B-143 7.30 A time line of the costs of operating a series of such machines in perpetuity is shown below: t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 ... $12,000 $6,000 $6,000 $6,000 $4,000 $12,000 $6,000 $6,000 ... The present value of one cycle is: 34 + $4,000 / 1.06 PV = $12,000 + $6,000 A0.064 = $12,000 + $6,000 (2.6730) + $4,000 / 1.06 = $31,206 In order to calculate the equivalent annual cost (EAC) of the machine, set the NPV equal to an annuity with the same economic life as the machine. 4$31,206 = EAC * A0.06 EAC = $9,006 Making a payment of $9,006 for four years is equivalent to making one cycle of payments, in present value terms. Therefore, the present value of the costs of operating a series of such machines in perpetuity is equal to the present value of a perpetuity with yearly payments of $9,006. PV = C / r 1= $9,006 / 0.06 = $150,100 The present value of operating the machines in perpetuity is $150,100. 7.31 In order to find the equivalent annual cost, first find the net present value of all costs related to the investment, net of any benefits the investment may yield. The initial investment is not discounted because it occurs today. PV(Initial Investment) = -$60,000 Each year, the machine requires $5,000 of maintenance. Apply the three-year annuity formula, discounted at 0.14. Remember to adjust the maintenance cost for taxes. T PV(Maintenance) = (1 – T) C Ac1r 3 = (1 – 0.34) (-$5,000 A) 0.14 = -$7,661 Since the firm generates positive income and is subject to corporate taxes, the firm benefits from a depreciation tax shield. Annual Depreciation Expense = $60,000 / 3 = $20,000. Annual Depreciation Tax Shield = (T) (Annual Depreciation Expense) c = (0.34) ($20,000) = $6,800 To find the PV of the annual depreciation tax shields, apply the formula for a three-year annuity. B-144 TPV(Annual Depreciation Tax Shields) = C A 1r3 = $6,800 A0.14 = $15,787 The NPV of the project is the combination of the above cash flows. NPV = -Initial Investment – PV(Maintenance) + PV(Tax Shield) = -$60,000 - $7,661 + $15,787 = -$51,874 In order to calculate the equivalent annual cost (EAC), set the NPV of the equipment equal to an annuity with the same economic life. Since the project has an economic life of three years and is discounted at 14 percent, set the NPV equal to a three-year annuity, discounted at 14 percent. 3-$51,874 = EAC * A 0.14 EAC = -$22,344 The equivalent annual cost for the project is $22,344. 7.32 In order to find equivalent annual cost, first find the net present value of all costs related to the investment, net of any benefits the investment may yield. PV(Initial Investment) = -$60,000 The new system will incur maintenance costs of $2,000 per year for five years. The cost is treated as a five-year annuity, discounted at 0.18. Remember to adjust the maintenance cost for taxes. 5 PV(Maintenance Costs) = (1 – 0.35) (-$2,000) A0.18 = -$4,065 Since the firm generates positive income, it will benefit from a depreciation tax shield. Annual Depreciation Expense = $60,000 / 5 = $12,000 Annual Depreciation Tax Shield = (T) (Annual Depreciation Expense) c = (0.35) ($12,000) = $4,200 To find the PV of the annual depreciation tax shields, use the formula for a five-year annuity. TPV(Annual Depreciation Tax Shields) = C A 1r5 = $4,200 A0.18 = $13,134 The NPV of the equipment is the combination of the above cash flows. NPV = -Initial Investment – PV(Maintenance) + PV(Depreciation Tax Shield) = -$60,000 - $4,065 + $13,134 = -$50,931 B-145 In order to calculate the equivalent annual cost (EAC), set the net present value of the project equal to a five-year annuity. Solve for the payment amount, which is the equivalent annual cost. 5-$50, 931 = EAC * A0.18 EAC = -$16,286 Therefore, the equivalent annual cost of the new admitting system is $16,286. 7.33 In order to find equivalent annual cost, first find the net present value of all costs related to the investment, net of any benefits the investment may yield. PV(Initial Investment) = -$45,000 The project requires annual maintenance of $5,000, beginning a year from now. The cost is treated as a three-year annuity, discounted at 0.12. Remember to adjust the maintenance cost for taxes. TPV(Maintenance) = (1 – T) C Ac1r 3 = (1 – 0.34)(-$5,000) A 0.12 = -$7,926 Since the firm generates positive income, it benefits from the depreciation tax shield. The annual depreciation expense is $15,000 (= $45,000 / 3). The annual depreciation tax shield is the annual depreciation expense multiplied by the tax rate. Annual Depreciation Tax Shield = (T) (Annual Depreciation Expense) c = (0.34) ($15,000) = $5,100 The string of annual depreciation tax shields forms a three-year annuity, discounted at 12%. TPV(Depreciation Tax Shield) = C A 1r3 = $5,100 A0.12 = $12,249 At the end of its life, the equipment will have a $10,000 salvage value. Since the equipment has been fully depreciated, a gain on the sale equal to the salvage value must be recognized. After-Tax Salvage Value = Salvage Value – T (Salvage Value – Book Value) c = $10,000 – 0.34 ($10,000 – $0) = $6,600 The after-tax salvage value must be discounted back three periods to find its present value. 3PV(After-Tax Salvage Value) = $6,600 / (1.12) = $4,698 The NPV of the equipment is the combination of the above cash flows. NPV = -Initial Investment – PV(Maintenance) + PV(Depreciation Tax Shield) + PV(Salvage) = -$45,000 - $7,926 + $12,249 + $4,698 = -$35,979 B-146 In order to calculate the equivalent annual cost, set the NPV of the equipment equal to an annuity with the same economic life. Since the project has an economic life of three years and is discounted at 12 percent, set the NPV equal to a three-year annuity, discounted at 12 percent. 3 -$35,979 = EAC * A0.12 EAC = -$14,980 The equivalent annual cost for the project is $14,980. 7.34 Since the cash flows are given in real terms, they must be discounted at the real discount rate. 1+ Real Discount Rate = (1+ Nominal Discount Rate) / (1+ Inflation Rate) Real Discount Rate = [(1.14) / (1.05)] – 1 = 0.0857 Find the equivalent annual cost (EAC) of each of the copiers. The firm will choose the model with the lower equivalent annual cost. XX40 Find the present value of both the initial cash outlay and the maintenance expenses. Since the initial cash outlay occurs today (year 0), it does not need to be discounted. To find the present value of the maintenance expenses, use the annuity formula. 3PV of cash outflows from XX40 = $700 + $100 A 0.0857 = $955 3$955 = EAC * A 0.0857EAC = $374 The equivalent annual cost of model XX40 is $374. RH45 Find the present value of both the initial cash outlay and the maintenance expenses. Since the initial cash outlay occurs today (year 0), it does not need to be discounted. To find the present value of the maintenance expenses, use the annuity formula. 5PV of cash outflows from RH45 = $900 + $110 A 0.0857 = $1,333 5$1,333 = EAC * A0.0857 EAC = $339 The equivalent annual cost of model RH45 is $339. Since the equivalent annual cost of model RH45 is lower, the firm should choose model RH45. B-147 7.35 Use the equivalent annual cost (EAC) method to determine which facility Plexi Glasses should purchase. Facility 1: The first step is to find the NPV of the project. The initial investment is not discounted because it occurs today (year 0). PV(Initial Investment) = -$2,100,000 Maintenance costs are $60,000 and are incurred at the end of the year. These costs form a seven-year annuity, discounted at 0.10. Remember to adjust the maintenance cost for taxes. T PV(Maintenance Costs) = (1 – T) C Ac1r 7 = (1 – 0.34)(-$60,000) A 0.10 = -$192,789 The annual depreciation expense is $300,000 (= $2,100,000 / 7). The annual depreciation tax shield is the annual depreciation expense multiplied by the tax rate. Annual Depreciation Tax Shield = (T) (Annual Depreciation Expense) c = (0.34) ($300,000) = $102,000 The string of annual depreciation tax shields form a seven-year annuity, discounted at 0.10. TPV(Depreciation Tax Shield) = C A 1r7 = $102,000 A0.10 = $496,579 The NPV of the project is the combination of the above cash flows. NPV = -Initial Investment – PV(Maintenance Costs) + PV(Depreciation Tax Shield) = -$2,100,000 - $192,789 + $496,579 = -$1,796,210 In order to calculate the equivalent annual cost, set the NPV of the equipment equal to an annuity with the same economic life. Since the project has an economic life of seven years and is discounted at 10 percent, the NPV is equal to a seven-year annuity, discounted at 10 percent. T NPV = EAC * Ar 7 -$1,796,210 = EAC * A 0.10 EAC = -$368,951 The equivalent annual cost for the project is $368,951. Facility 2: The first step is to find the NPV of the project. The initial investment is not discounted because it occurs today (year 0). PV(Initial Investment) = -$2,800,000 B-148 Maintenance costs are $100,000 and are incurred at the end of the year. These costs form a 10-year annuity, discounted at 0.10. Remember to adjust the maintenance cost for taxes. T) C A PV(Maintenance Costs) = (1 – Tc1r 10 = (1 – 0.34)(-$100,000) A 0.10 = -$405,541 The annual depreciation expense is $280,000 (= $2,800,000 / 10). The annual depreciation tax shield is the annual depreciation expense multiplied by the tax rate. Annual Depreciation Tax Shield = (T) (Annual Depreciation Expense) c = (0.34) ($280,000) = $95,200 Apply the annuity formula to calculate the PV of the annual depreciation tax shields. TPV(Depreciation Tax Shield) = C A 1r10 = $95,200 A0.10 = $584,963 The NPV of the project is the combination of the above cash flows. NPV = -Initial Investment – PV(Maintenance Costs) + PV(Depreciation Tax Shield) = -$2,800,000 - $405,541.43+ $584,963 = -$2,620,578 In order to calculate the equivalent annual cost, set the NPV of the equipment equal to an annuity with the same economic life. Since the project has an economic life of 10 years and is discounted at 10 percent, the NPV is equal to a 10-year annuity, discounted at 10 percent. 10 -$2,620,578 = EAC * A 0.10 EAC = -$426,487 The equivalent annual cost for the project is $426,487. The firm should choose facility 1 since it has the lower EAC. 7.36 Find the net present value (NPV) of each option. The firm will choose the option with the higher NPV. Remember to take into account both the maintenance costs and depreciation tax shields associated with both the old and new machines. Note that the replacement machine will be bought in five years regardless of the option chosen and therefore is not incremental to this decision. Option 1 Sell old machine and purchase new machine now. To find the cash flow from selling the old machine, consider both the sales price and the net book value of the machine. Since the firm will be selling the old machine ($2,000,000) for more than its net book value ($1,000,000), the resultant capital gain will be subject to corporate taxes. After-Tax Salvage Value = Sale Price – T(Sale Price – Net Book Value) C = $2,000,000 – 0.34($2,000,000 - $1,000,000) = $1,660,000 PV(Salvage Value) = $1,660,000 B-149 The new machine is purchased today (year 0) and does not need to be discounted. PV(New Machine) = -$3,000,000 To find the present value of the new machine’s maintenance costs, use a five-year annuity, discounted at 12 percent. Remember to account for taxes. 5PV(Maintenance Costs) = (1 – 0.34)(-$500,000)A0.12 = -$1,189,576 The firm will also recognize a depreciation tax shield from the new machine. The annual depreciation expense is $600,000 (= $3,000,000 / 5 years). Annual Depreciation Tax Shield = T * Depreciation per year C = 0.34 * $600,000 = $204,000 The present value of the depreciation tax shields can be found by using a five-year annuity, discounted at 12 percent. T PV(Depreciation Tax Shield) = C A 1r5 = $204,000 A0.12 = $735,374 The new machine will be sold at the end of its economic life. Since the resale price ($500,000) is higher than the net book value ($0), the sale of the machine is subject to capital gains taxes. Since the sale occurs at the end of year 5, discount the after-tax salvage value back 5 periods. After-Tax Salvage Value = Sale Price – T(Sale Price – Net Book Value) C = $500,000 – 0.34($500,000 – 0) = $330,000 5PV(Salvage Value) = $330,000 / (1.12) = $187,251 NPV(Option 1) = $1,660,000 - $3,000,000 - $1,189,576 + $735,374 + $187,251 = -$1,606,950 The net present value (NPV) of selling the old machine and purchasing the new machine now is -$1,606,950. Option 2 Sell old machine in five years and purchase new machine in five years. The company will have to make the scheduled maintenance costs for the old machine. Use a five-year annuity, discounted at 12 percent to find the present value of the costs. Remember to account for taxes. 5PV(Maintenance Costs) = (1 – 0.34)(-$400,000)A0.12 = -$951,661 B-150 The firm will continue to recognize depreciation on the old machine. The annual depreciation expense is $200,000 per year, and the firm will recognize a depreciation tax shield. The present value of the tax shield is found by using a five-year annuity, discounted at 12 percent. Annual Depreciation Tax Shield = 0.34 * $200,000 = $68,000 5PV(Depreciation Tax Shield) = $68,000 A0.12 = $245,125 The salvage value at the end of the old machine’s economic life of five years will be $200,000. Since the machine will have been depreciated to $0, the firm must pay capital gains taxes on the sale. To find the present value, discount the after-tax salvage value by five periods. After-Tax Salvage Value = Sale Price – T(Sale Price – Net Book Value) C = $200,000 – 0.34($200,000 – 0) = $132,000 5 PV(Salvage Value) = $132,000 / (1.12) = $74,900 NPV(Option 2) = -$951,661 + $245,125 + $74,900 = -631,636 The net present value (NPV) of selling the old machine and purchasing the new machine in five years is -631,636. Since the NPV of Option 2 is higher than the NPV of Option 1, the firm will choose to sell the old equipment and purchase new equipment in five years. 7.37 SAL 5000 The first step is to find the NPV of the costs associated with the SAL 5000. Find the NPV of one SAL 5000, and later, when finding the equivalent annual cost (EAC) of the decision, multiply the final answer by 10. The initial investment is not discounted because it occurs today (year 0). PV(Initial Investment) = -$3,750 Each year, the computer requires $500 of maintenance. Apply the eight-year annuity formula, discounted at 11 percent, to find the PV of the cost. T PV(Maintenance Costs) = C A1r 8 = -$500 A 0.11 = -$2,573 At the end of the computer’s economic life, it will have a resale value of $500. Since there are no capital gains taxes, the PV is just that cash flow, discounted by eight periods. 8 PV(Salvage Value) = C / (1 + r)8 8 = $500 / (1.11) = $217 The NPV of the computer is the combination of the above cash flows. NPV = -Initial Investment – PV(Maintenance Costs) + PV(Salvage Value) = -$3,750 - $2,573+ $217 = -$6,106 B-151 In order to calculate the equivalent annual cost, set the NPV of the computer equal to an annuity with the same economic life. Since the computer has an economic life of eight years, set the NPV equal to an eight- year annuity, discounted at 11 percent. 8 -$6,106 = EAC * A 0.11 EAC = -$1,187 Since Gold Star Industries would have to buy 10 SAL 5000s, the EAC here would be: Total EAC = (Number of SAL 5000s purchased) (EAC of one SAL 5000) = (10) (-$1,187) = -$11,870 The equivalent annual cost (EAC) for the decision to buy the SAL 5000 is $11,870. DET 1000 The first step is to find the NPV of the costs associated with the DET 1000. Find the NPV of one DET 1000, and later, when finding the equivalent annual cost (EAC) of the decision, multiply the final answer by 8. The initial investment is not discounted because it occurs today (year 0). PV(Initial Investment) = -$5,250 Each year, the computer requires $700 of maintenance. Apply the six-year annuity formula, discounted at 11 percent, to find the PV of the cost. T PV(Maintenance Costs) = C A1r 6 = -$700 A 0.11 = -$2,961 At the end of the computer’s economic life, it will have a resale value of $600. Since there are no capital gains taxes, the PV is just that cash flow, discounted by six periods. 6 PV (Salvage Value) = C / (1 + r)6 6 = $600 / (1.11) = $321 The NPV of the computer is the combination of the above cash flows. NPV = -Initial Investment – PV(Maintenance) + PV(Salvage) = -$5,250 - $2,961+ $321 = -$7,890 In order to calculate the equivalent annual cost, set the NPV of the computer equal to an annuity with the same economic life. Since the computer has an economic life of six years, set the NPV equal to a six-year annuity, discounted at 11 percent. 6 -$7,890 = EAC * A 0.11 EAC = -$1,865 B-152 Since Gold Star Industries would have to buy eight DET 1000s, the EAC here would be: Total EAC = (Number of DET 1000s purchased) (EAC of one DET 1000) = (8) (-$1,865) = -$14,920 The equivalent annual cost for the decision to buy the DET 1000 is $14,920. Gold Star should purchase the SAL 5000 since it has a lower equivalent annual cost (EAC). 7.38 To evaluate the word processors, compute their equivalent annual costs (EAC). EVF Find the net present value of the costs associated with this model of word processor. The present value of purchasing the 10 EVF word processors is: PV(Purchase) = 10 * -$8,000 = -$80,000 The present value of the maintenance costs is found by using a four-year annuity, discounted at 14 percent. 4 PV(Maintenance Costs) = (-$2,000*10) A0.14 = -$58,274 NPV = -$80,000 -$58,274 = -$138,274 In order to calculate the equivalent annual cost, set the NPV of the word processor equal to an annuity with the same economic life. Since the computer has an economic life of four years, set the NPV equal to a four-year annuity, discounted at 14 percent. 4$138,274 = EAC * A0.14 EAC = $47,456 The equivalent annual cost of the EVF word processor is $47,456. AEH Find the net present value of the costs associated with the AEH model. The present value of purchasing the 11 AEH word processors now is: PV(Purchase) = -$5,000 * 11 = -$55,000 The present value of the maintenance costs is found by applying the three-year annuity formula, discounted at 14 percent. 3 PV(Maintenance Costs) = (-$2,500*11) A 0.14= -$63,845 B-153 At the end of the computer’s economic life, it will have a resale value of $500. Since there are no capital gains taxes, the PV is just that cash flow, discounted back three periods. 3 PV(Resale) = (11*500)/(1.14) = $3,712 NPV = -$55,000 - $63,845 + $3,712 = -$115,133 In order to calculate the equivalent annual cost, set the NPV of the word processor equal to an annuity with the same economic life and discount rate. Since the computer has an economic life of three years, set the NPV equal to a three-year annuity, discounted at 14 percent. 3$115,133 = EAC * A0.14 EAC = $49,591 The equivalent annual cost of the AEH word processor is $49,591. Harwell should purchase the EVF word processors since their equivalent annual cost is lower. 7.39 Mixer X The first step is to find the NPV of the savings associated with Mixer X. The initial investment is not discounted because it occurs today (year 0). PV(Initial Investment) = -$400,000 Each year, the mixer generates after-tax cash flow savings of $120,000. Apply the five-year annuity formula, discounted at 11 percent, to find the PV of the cash flow savings. T PV(Savings) = C A1r 5 = $120,000 A 0.11 = $443,507 The NPV of the mixer is the sum of the above cash flows. NPV = -Initial Investment + PV(Savings) = -$400,000 + $443,507 = $43,507 In order to calculate the equivalent annual benefit (EAB) of the mixer, set the NPV of the mixer equal to an annuity with the same economic life. Since the mixer has an economic life of five years, set the NPV equal to a five-year annuity, discounted at 11 percent. 5 $43,507 = EAB * A 0.11 EAB = $11,772 The equivalent annual benefit of Mixer X is $11,772. Mixer Y The first step is to find the NPV of the savings associated with Mixer Y. The initial investment is not discounted because it occurs today (year 0). PV(Initial Investment) = -$600,000 B-154 Each year, the mixer generates after-tax cash flow savings of $130,000. Apply the eight-year annuity formula, discounted at 11 percent, to find the PV of the cash flow savings. T A PV(Savings) = C1r 8 = $130,000 A 0.11 = $668,996 The NPV of the mixer is the sum of the above cash flows. NPV = -Initial Investment + PV(Savings) = -$600,000 + $668,996 = $68,996 In order to calculate the equivalent annual benefit of the mixer, set the NPV of the mixer equal to an annuity with the same economic life. Since the mixer has an economic life of eight years, set the NPV equal to an eight-year annuity, discounted at 11 percent. 8$68,996 = EAC * A 0.11 EAC = $13,407 The equivalent annual benefit of Mixer Y is $13,407. DJ Party, Inc. should buy Mixer Y since it yields a higher equivalent annual benefit. 7.40 Tamper A Tamper A is purchased today (year 0) and does not need to be discounted. PV(Purchase) = -$600,000 The present value of the maintenance costs is found by applying the five-year annuity formula, discounted at 12 percent. 5 PV(Maintenance Costs) = -$110,000 A0.12 = -$396,525 NPV = -$600,000 - $396,525 = -$996,525 In order to calculate the equivalent annual cost (EAC) of the tamper, set the NPV equal to an annuity with the same economic life. Since the tamper has an economic life of five years, set the NPV equal to a five- year annuity, discounted at 12 percent. 5 -$996,525 = EAC * A 0.12 EAC = -$276,446 Tamper A has an equivalent annual cost of $276,446. Tamper B Tamper B is purchased today (year 0) and does not need to be discounted. PV(Purchase) = -$750,000 The present value of the maintenance costs is found by applying the seven-year annuity formula, discounted at 12 percent. B-155 7 PV(Maintenance Costs) = -$90,000 A0.12 = -$410,738 NPV = -$750,000 - $410,738 =-$1,160,738 In order to calculate the equivalent annual cost of the tamper, set the NPV equal to an annuity with the same economic life. Since the tamper has an economic life of seven years, set the NPV equal to a seven- year annuity, discounted at 12 percent. 7 -$1,160,738 = EAC * A0.12 EAC = -$254,338 Tamper B has an equivalent annual cost of $254,338. KZD Construction should choose Tamper B since it has a lower equivalent annual cost. 7.41 Klious needs to compare the equivalent annual cost (EAC) of the new machine to the cost incurred by keeping the old autoclave one additional year. First, find the EAC of the new autoclave. Next, find the total one-year cost, including the opportunity cost of not selling the old autoclave at the beginning of that particular year. If the EAC of the new autoclave is higher than the one-year total cost of keeping the existing autoclave, then Klious should not replace the old machine. If the total one-year cost of the existing autoclave is higher than the EAC of the new machine, Klious should replace. The first step of the problem is to calculate the NPV of the new machine. The initial investment is not discounted because it occurs today (year 0). PV(Initial Investment) = -$3,000 Each year, the autoclave generates $20 of maintenance costs. Apply the five-year annuity formula, discounted at 0.10 to calculate the PV of the maintenance costs. T PV(Maintenance Costs) = C A1r 5 = -$20 A 0.10 = -$76 The autoclave has a salvage value of $1,200 at the end of its economic life. Remember that the cash flow occurs at the end of year 5, and therefore must be discounted back five years. 5 PV(Salvage Value) = C / (1 + r) 55 = $1,200 / (1.10) = $745 The NPV of the autoclave is the combination of the above cash flows. NPV = -Initial Investment - PV(Maintenance Costs) + PV(Salvage Value) = -$3,000 - $76 + $745 = -$2,331 In order to calculate the equivalent annual cost of the new autoclave, set the NPV equal to an annuity with the same economic life. Since the autoclave has an economic life of five years, set the NPV equal to a five- year annuity, discounted at 10 percent. 5-$2,331 = EAC * A 0.10 EAC = -$615 The equivalent annual cost of the new autoclave is $615. B-156 To make its decision, Klious must compare the total yearly cost from keeping the old autoclave with the $615 yearly cost of the new autoclave. The matrix below illustrates the relevant costs of keeping the existing autoclave. Replacement Date/Year Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Keep through Year 1 -900 -200 - - - - 850 Keep through Year 2 - - -275 - - - -850 775 Keep through Year 3 - - - -325 - - -775 700 Keep through Year 4 - - - - -450 - -700 600 Keep through Year 5 - - - - - -500 -600 500 Compute the total end-of-year cost of the old autoclave for an additional year. Remember to state the costs in terms of end-of-year dollars. This is necessary because the EAC of the new machine is stated in terms of end-of-year dollars. Keeping the old autoclave through Year 1: The foregone resale value is already stated as of the beginning of the year, and therefore does not need further discounting. PV(Foregone Resale Value) = -$900 Both the maintenance cost and the realizable resale value must be discounted back one year since these cash flows occur at the end of the year. 1 PV(Maintenance Costs) = -$200 / (1.10) = -$182 1 PV(Resale Value) = $850 / (1.10) = $773 The NPV of keeping the old autoclave through the first year is the combination of the above cash flows. NPV = -$900 – $182 + $773 = -$309 Because the EAC of the new machine is expressed in terms of end-of-year dollars, multiply the NPV of the old machine’s costs by the discount rate in order to find its future value as of the end of year 1. FV = (-$309) (1.10) = -$340 The cost of the old autoclave in terms of end-of-year 1 dollars is $340. Since it is cheaper to operate the old autoclave ($340) than to purchase the new one ($615), Klious should continue to operate the old machine in year 1. B-157 Keeping the old autoclave through Year 2: The foregone resale value is already stated as of the beginning of the year, and therefore does not need further discounting. PV(Foregone Resale Value) = -$850 Both the maintenance cost and the realizable resale value must be discounted back one year since they occur at the end of the year. 1 PV(Maintenance Costs) = -$275 / (1.10) = -$250 1 PV(Resale Value) = $775 / (1.10) = $705 The NPV of keeping the old autoclave through the second year is the combination of the above cash flows. NPV = -$850 – $250 + $705 = -$395 Because the EAC of the new machine is expressed in terms of end-of-year dollars, multiply the NPV of the old machine’s costs by the discount rate in order to find its future value as of the end of year 2. FV = (-$395) (1.10) = -$435 The cost of the old autoclave in terms of end-of-year 2 dollars is $435. Since it is cheaper to operate the old autoclave ($435) than to purchase the new one ($615), Klious should continue to operate the old machine in year 2. Keeping the old autoclave through Year 3: The foregone resale value is already stated as of the beginning of the year, and therefore does not need further discounting. PV(Foregone Resale Value) = -$775 Both the maintenance cost and the realizable resale value must be discounted back one year since they occur at the end of the year. 1 PV(Maintenance Costs) = -$325 / (1.10) = -$295 1 PV(Resale Value) = $700 / (1.10) = $636 The NPV of keeping the old autoclave through the third year is the combination of the above cash flows. NPV = -$775 – $295 + $636 = -$434 Because the EAC of the new machine is expressed in terms of end-of-year dollars, multiply the NPV of the old machine’s costs by the discount rate in order to find its future value in terms of end-of-year 3 dollars. FV = (-$434) (1.10) = -$477 B-158 The cost of the old autoclave in terms of end-of-year 3 dollars is $477. Since it is cheaper to operate the old autoclave ($477) than to purchase the new one ($615), Klious should continue to operate the old machine in year 3. Keeping the old autoclave through Year 4: The foregone resale value is already stated as of the beginning of the year, and therefore does not need further discounting. PV(Foregone Resale) = -$700 Both the maintenance cost and the realizable resale value must be discounted back one year since they occur at the end of the year. 1 PV(Maintenance) = -$450 / (1.10) = -$409 1 PV(Resale Value) = $600 / (1.10) = $545 The NPV of keeping the old autoclave through the fourth year is the combination of the above cash flows. NPV = -$700 – $409 + $545 = -$564 Because the EAC of the new machine is expressed in terms of end-of-year dollars, multiply the NPV of the old machine’s costs by the discount rate in order to find its future value as of the end of year 4. FV = (-$564) (1.10) = -$620 The cost of the old autoclave in terms of end-of-year 4 dollars is $620. Since it is more expensive to operate the old autoclave ($620) than to purchase the new one ($615), Klious should purchase the new autoclave at the end of year 3. B-159 Case Study: I.Q. Inc. Year Year Year Year Year Year (Unit: $Million) 0 1 2 3 4 5 NPV Investments: Capital equipment (1.000) 0.300 Capital Gains taxes (0.034) Change in NWC (1.000) 1.000 Revenue 8.600 9.939 11.486 13.274 Costs 0.550 0.636 0.735 0.849 SG&A $ 0.50 0.515 0.530 0.546 Profits 7.550 8.788 10.221 11.878 After-tax Profits 4.983 5.800 6.746 7.840 Depreciation tax shield 0.068 0.068 0.068 0.068 Total Cash flow from project (2.000) 5.051 5.868 6.814 9.174 PV 15% (CF) (2.000) 4.392 4.437 4.480 5.245 16.555 The project should be undertaken since it has a positive NPV. B-160 Case Study: Jimmy’s Hot Dog Stands Year Year Year Year Year Year 0 1 2 3 4 5 NPV Investments: Real Estate (400,000) Capital equipment (65,000) Change in NWC (20,000) Total cash flow from investments (465,000) (20,000) – – – – Income: Revenue 510,000 510,000 510,000 510,000 510,000 Labor Costs 96,000 96,000 96,000 96,000 96,000 Raw Materials 157,080 157,080 157,080 157,080 157,080 Depreciation 13,000 13,000 13,000 13,000 13,000 Income before Taxes 243,920 243,920 243,920 243,920 243,920 Taxes 82,933 82,933 82,933 82,933 82,933 Net Income 160,987 160,987 160,987 160,987 160,987 B-161 Cash flow from operations: Net Income 160,987 160,987 160,987 160,987 160,987 Depreciation 13,000 13,000 13,000 13,000 13,000 Operating cash flow 173,987 173,987 173,987 173,987 173,987 Total cash flow from project (465,000) 153,987 173,987 173,987 173,987 173,987 PV 10% (CF) (465,000) 139,988 143,791 130,719 118,836 108,032 $176,367 The NPV of this investment is $176,367.