CAPITAL BUDGETING

0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 158 CHAPTER 7 INTRODUCTION TO CAPITAL BUDGETING 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 158 Overview 159 The NPV Rule for Judging Investments and Projects 159 The IRR Rule for Judging Investments 161 NPV or IRR, Which to Use? 162 The “Yes–No” Criterion: When Do IRR and NPV Give the Same Answer? 163 Do NPV and IRR Produce the Same Project Rankings? 164 Capital Budgeting Principle: Ignore Sunk Costs and Consider Only Marginal Cash Flows 168 Capital Budgeting Principle: Don’t Forget the Effects of Taxes—Sally and Dave’s Condo Investment 169 Capital Budgeting and Salvage Values 176 Capital Budgeting Principle: Don’t Forget the Cost of Foregone Opportunities 180 In-House Copying or Outsourcing? A Mini-case Illustrating Foregone Opportunity Costs 181 Accelerated Depreciation 184 Conclusion 185 Exercises 186 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 159 CHAPTER 7 Introduction to Capital Budgeting 159 OVERVIEW Capital budgeting is finance terminology for the process of deciding whether or not to undertake an investment project. There are two standard concepts used in capital budgeting: net present value (NPV) and internal rate of return (IRR). Both of these concepts were introduced in Chapter 5; in this chapter we discuss their application to capital budgeting. Here are some of the topics covered: • • • • • • • Should you undertake a specific project? We call this the “yes–no” decision, and we show how both NPV and IRR answer this question. Ranking projects: If you have several alternative investments, only one of which you can choose, which should you undertake? Should you use IRR or NPV? Sometimes the IRR and NPV decision criteria give different answers to the yes–no and the ranking decisions. We discuss why this happens and which criterion should be used for capital budgeting (if there’s disagreement). Sunk costs. How should you account for costs incurred in the past? The cost of foregone opportunities. Salvage values and terminal values. Incorporating taxes into the valuation decision. This issue is dealt with briefly in Section 7.7. We return to it at greater length in Chapters 8–10. Finance Concepts Discussed • • • • • • • IRR NPV Project ranking using NPV and IRR Terminal value Taxation and calculation of cash flows Cost of foregone opportunities Sunk costs Excel Functions Used • • • NPV IRR Data Tables 7.1 The NPV Rule for Judging Investments and Projects In preceding chapters we introduced the basic NPV and IRR concepts and their application to capital budgeting. We start off this chapter by summarizing each of these rules—the NPV rule in this section and the IRR rule in the following section. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 160 160 PART TWO CAPITAL BUDGETING AND VALUATION Here’s a summary of the decision criteria for investments implied by the net present value: The NPV rule for deciding whether or not a specific project is worthwhile: Suppose you are considering a project that has cash flows CF0 , CF1 , CF2 , . . . , CF N. Suppose that the appropriate discount rate for this project is r. Then the NPV of the project is NPV = CF0 + N
CF1 CFt CF2 CF N + + · · · + = CF + 0 2 N (1 + r) (1 + r) (1 + r) (1 + r)t t=1 Rule: A project is worthwhile by the NPV rule if its NPV ⬎ 0. The NPV rule for deciding between two mutually exclusive projects: Suppose you are trying to decide between two projects A and B, each of which can achieve the same objective. For example, your company needs a new widget machine, and the choice is between widget machine A and machine B. You will buy either A or B (or perhaps neither machine, but you will certainly not buy both machines). In finance jargon, these projects are “mutually exclusive.” Suppose project A has cash flows CFA0 , CFA1 , CFA2 , . . . , CFAN and that project B has cash flows CFB0 , CFB1 , CFB2 , . . . , CFBN . Rule: Project A is preferred to project B if NPV(A) = CFA0 + N N

CFAt CFBt > CFB0 + = NPV(B) t (1 + r) (1 + r)t t=1 t=1 The logic  of both NPV rules presented above is that the present value of a project’s cash N [CFt /(1 + r)t ]—is the economic value today of the project. Thus, if we flows—PV = t=1 have correctly chosen the discount rate r for the project, the PV is what we ought to be able to sell the project for in the market.1 The net present value is the wealth increment produced by the project, so that NPV ⬎ 0 means that a project adds to our wealth: NPV = CF 0 ↑ Initial cash flow required to implement the project. This is usually a negative number. + N
t=1  CFt (1 + r)t   ↑ Market value of future cash flows. An Initial Example To set the stage, let’s assume that you’re trying to decide whether to undertake one of two projects. Project A involves buying expensive machinery that produces a better product at a lower cost. The machines for project A cost $1,000 and, if purchased, you anticipate that the project will produce cash flows of $500 per year for the next five years. Project B’s machines are cheaper, costing $800, but they produce smaller annual cash flows of $420 per year for the next five years. We’ll assume that the correct discount rate is 12%. 1 This assumes that the discount rate is “correctly chosen,” by which we mean that it is appropriate to the riskiness of the project’s cash flows. For the moment, we fudge the question of how to choose discount rates; this topic is discussed in Chapter 9. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 161 CHAPTER 7 Introduction to Capital Budgeting 161 Suppose we apply the NPV criterion to projects A and B: A 1 2 Discount rate 3 Year 4 5 0 6 1 7 2 8 3 9 4 10 5 11 12 NPV B C D TWO PROJECTS 12% Project A -1000 500 500 500 500 500 802.39 Project B -800 420 420 420 420 420 714.01 <-- =NPV($B$2,C6:C10)+C5 Both projects are worthwhile, since each has a positive NPV. If we have to choose between the projects, then project A is preferred to project B because it has the higher NPV. EXCEL NOTE EXCEL’S NPV FUNCTION VERSUS THE FINANCE DEFINITION OF NPV We reiterate our Excel note from Chapter 5 (p. 94): Excel’s NPV function computes the present value of future cash flows; this does not correspond to the finance notion of NPV, which includes the initial cash flow. To calculate the finance NPV concept in the spreadsheet, we have to include the initial cash flow. Hence, in cell B12, the NPV is calculated as ⴝNPV($B$2,B6:B10)ⴙB5 and in cell C12 the calculation is ⴝNPV($B$2,C6:C10)ⴙC5. 7.2 The IRR Rule for Judging Investments An alternative to using the NPV criterion for capital budgeting is to use the internal rate of return (IRR). Recall from Chapter 5 that the IRR is defined as the discount rate for which the NPV equals zero. It is the compound rate of return that you get from a series of cash flows. Here are the two decision rules for using the IRR in capital budgeting. The IRR rule for deciding whether or not a specific investment is worthwhile: Suppose we are considering a project that has cash flows CF0 , CF1 , CF2 , . . . , CF N . IRR is an interest rate such that CF0 + N
CFt CF1 CF2 CF N + + ··· + = CF0 + =0 2 N (1 + IRR) (1 + IRR) (1 + IRR) (1 + k)t t=1 Rule: If the appropriate discount rate for a project is r, you should accept the project if its IRR > r and reject it if its IRR < r. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 162 162 PART TWO CAPITAL BUDGETING AND VALUATION The logic behind the IRR rule is that the IRR is the compound return you get from the project. Since r is the project’s required rate of return, it follows that if the IRR > r, you get more than you require. The IRR rule for deciding between two competing projects: Suppose you are trying to decide between two mutually exclusive projects A and B (meaning: both projects are ways of achieving the same objective, and you will choose at most one of the projects). Suppose project A has cash flows CFA0 , CFA1 , CFA2 , . . . , CFAN and that project B has cash flows CFB0 , CFB1 , CFB2 , . . . , CFBN . Rule: Project A is preferred to project B if IRR(A) > IRR(B). Again the logic is clear: Since the IRR gives a project’s compound rate of return, if we choose between two projects using the IRR rule, we prefer the higher compound rate of return. Applying the IRR rule to our projects A and B, we get: A B 1 2 Discount rate 3 Year 4 5 0 6 1 7 2 8 3 9 4 10 5 11 12 IRR C D TWO PROJECTS 12% Project A Project B -1000 -800 500 420 500 420 500 420 500 420 500 420 41% 44% <-- =IRR(C5:C10) Both project A and project B are worthwhile, since each has an IRR > 12%, which is our relevant discount rate. If we have to choose between the two projects by using the IRR rule, project B is preferred to project A because it has a higher IRR. 7.3 NPV or IRR, Which to Use? We can sum up the NPV and IRR rules as follows: Criterion “Yes or No”: Choosing Whether or Not to Undertake a Single Project “Project Ranking”: Comparing Two Mutually Exclusive Projects NPV criterion The project should be undertaken if its NPV > 0. Project A is preferred to project B if NPV(A) > NPV(B). IRR criterion The project should be undertaken if its IRR > r, where r is the appropriate discount rate. Project A is preferred to project B if IRR(A) > IRR(B). 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 163 CHAPTER 7 Introduction to Capital Budgeting 163 Both the NPV rules and the IRR rules look logical. In many cases your investment decision—to undertake a project or not, or which of two competing projects to choose—will be the same whether you use NPV or IRR. There are some cases, however (such as that of projects A and B illustrated above), where NPV and IRR give different answers. In our present value analysis, project A won out because its NPV is greater than project B’s. In our IRR analysis of the same projects, project B was chosen because it had the higher IRR. In such cases, you should always use the NPV to decide between projects. The logic is that if individuals are interested in maximizing their wealth, they should use NPV, which measures the incremental wealth from undertaking a project. 7.4 The “Yes–No” Criterion: When Do IRR and NPV Give the Same Answer? Consider the following project. The initial cash flow of ⫺$1,000 represents the cost of the project today, and the remaining cash flows for years 1–6 are projected future cash flows. The discount rate is 15%. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 B C SIMPLE CAPITAL BUDGETING EXAMPLE Discount rate 15% Year -1,000 1 100 2 200 3 300 4 400 5 500 6 600 PV of future cash flows NPV IRR Cash flow 0 1,172.13 <-- =NPV(B2,B6:B11) 172.13 <-- =B5+NPV(B2,B6:B11) 19.71% <-- =IRR(B5:B11) The NPV of the project is $172.13, meaning that the present value of the project’s future cash flows ($1,172.13) is greater than the project’s cost of $1,000.00. Thus, the project is worthwhile. If we graph the project’s NPV we can see that the IRR—the point where the NPV curve crosses the x-axis—is very close to 20%. As you can see in cell B15, the actual IRR is 19.71%. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 164 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 PART TWO A Discount rate 0% 3% 6% 9% 12% 15% 18% 21% 24% 27% 30% CAPITAL BUDGETING AND VALUATION B C D E F G NPV 1,100.00 <-- =$B$5+NPV(A19,$B$6:$B$11) 849.34 <-- =$B$5+NPV(A20,$B$6:$B$11) 637.67 457.83 NPV of Cash Flows 304.16 172.13 1,200 58.10 1,000 -40.86 800 -127.14 -202.71 600 -269.16 400 NPV 164 200 0 -200 0% 3% 5% 8% 10% 13% 15% 18% 20% 23% 25% 28% 30% -400 Discount rate Accept or Reject? Should We Undertake the Project? It is clear that the above project is worthwhile: • Its NPV ⬎ 0, so that by the NPV criterion the project should be accepted. • Its IRR of 19.71% is greater than the project discount rate of 15%, so that by the IRR criterion the project should be accepted. A General Principle We can derive a general principle from this example: For conventional projects, projects with an initial negative cash flow and subsequent nonnegative cash flows (CF0 < 0, CF1 ≥ 0, CF2 ≥ 0, . . . , CF N ≥ 0), the NPV and IRR criteria lead to the same “Yes–No” decision: If the NPV criterion indicates a “Yes” decision, then so will the IRR criterion (and vice versa). 7.5 Do NPV and IRR Produce the Same Project Rankings? In the previous section we saw that, for conventional projects, NPV and IRR give the same “Yes–No” answer about whether to invest in a project. In this section we see that NPV and IRR do not necessarily rank projects the same, even if the projects are both conventional. Suppose we have two projects and can choose to invest in only one. The projects are mutually exclusive: They are both ways to achieve the same end, and thus we would choose only one. In this section we discuss the use of NPV and IRR to rank the projects. To sum up our results before we start: • Ranking projects by NPV and IRR can lead to possibly contradictory results. Using the NPV criterion may lead us to prefer one project whereas using the IRR criterion may lead us to prefer the other project. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 165 CHAPTER 7 • Introduction to Capital Budgeting 165 Where a conflict exists between NPV and IRR, the project with the larger NPV is preferred. That is, the NPV criterion is the correct criterion to use for capital budgeting. This is not to impugn the IRR criterion, which is often very useful. However, NPV is preferred over IRR because it indicates the increase in wealth that the project produces. An Example Below we show the cash flows for project A and project B. Both projects have the same initial cost of $500 but have different cash flow patterns. The relevant discount rate is 15%. A 1 2 3 4 5 6 7 8 9 10 11 12 13 B C D RANKING PROJECTS WITH NPV AND IRR Discount rate Year 15% Project A 0 1 2 3 4 5 -500 100 100 150 200 400 NPV IRR 74.42 19.77% Project B -500 250 250 200 100 50 119.96 <-- =C5+NPV(B2,C6:C10) 27.38% <-- =IRR(C5:C10) Comparing the Projects Using IRR: If we use the IRR rule to choose between the projects, then B is preferred to A, since the IRR of project B is higher than that of project A. Comparing the Projects Using NPV: Here the choice is more complicated. When the discount rate is 15% (as illustrated above), the NPV of project B is higher than that of project A. In this case the IRR and the NPV agree: Both indicate that project B should be chosen. Now suppose that the discount rate is 8%; in this case the NPV and IRR rankings conflict: A 1 2 3 4 5 6 7 8 9 10 11 12 13 B C D RANKING PROJECTS WITH NPV AND IRR Discount rate Year Project A 0 1 2 3 4 5 NPV IRR 8% -500 100 100 150 200 400 216.64 19.77% Project B -500 250 250 200 100 50 212.11 <-- =C5+NPV(B2,C6:C10) 27.38% <-- =IRR(C5:C10) In this case we have to resolve the conflict between the ranking on the basis of NPV (project A is preferred) and the ranking on the basis of IRR (project B is preferred). As we stated in the introduction to this section, the solution to this conflict is that you should choose on the basis of NPV. We explore the reasons for this later on, but first we discuss a technical question. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 166 166 PART TWO CAPITAL BUDGETING AND VALUATION Why Do NPV and IRR Give Different Rankings? Below we build a table and graph that show the NPV for each project as a function of the discount rate: A B D E F 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% 30% Project A NPV 450.00 382.57 321.69 266.60 216.64 171.22 129.85 92.08 57.53 25.86 -3.22 -29.96 -54.61 -77.36 -98.39 -117.87 Project B NPV 350.00 <-- =$C$5+NPV(A17,$C$6:$C$10) 311.53 <-- =$C$5+NPV(A18,$C$6:$C$10) 275.90 500 242.84 400 212.11 183.49 300 156.79 200 131.84 108.47 100 86.57 66.00 0 46.66 0% 5% 10% 15% -100 28.45 Discount rate 11.28 -200 -4.93 -20.25 G H Project A NPV Project B NPV NPV 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 C TABLE OF NPVs AND DISCOUNT RATES 15 20% 25% 30% From the graph you can see why contradictory rankings occur: • Project B has a higher IRR (27.38%) than project A (19.77%). (Remember that the IRR is the point at which the NPV curve crosses the x-axis.) • When the discount rate is low, project A has a higher NPV than project B, but when the discount rate is high, project B has a higher NPV. There is a crossover point (in the next subsection you will see that this point is 8.51%) that marks the disagreement/agreement range. • Project A’s NPV is more sensitive to changes in the discount rate than project B’s NPV. The reason for this is that project A’s cash flows are more spread out over time than those of project B; another way of saying this is that project A has substantially more of its cash flows at later dates than project B. To summarize: Criterion Discount Rate < 8.51% Discount Rate = 8.51% Discount Rate > 8.51% NPV criterion Project A preferred: NPV(A) > NPV(B) Indifferent between projects A and B: NPV(A) ⫽ NPV(B) Project B preferred: NPV(B) > NPV(A) IRR criterion Project B is always preferred to project A, since IRR(B) > IRR(A) Calculating the Crossover Point The crossover point—which we claimed earlier was 8.51%—is the discount rate at which the NPVs of the two projects are equal. A bit of formula manipulation will show you that the crossover point is the IRR of the differential cash flows. To see what this means, consider the 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 167 CHAPTER 7 167 Introduction to Capital Budgeting following example: A B C 34 Calculating the crossover point 35 36 37 38 39 40 41 42 43 IRR Year 0 1 2 3 4 5 D Project A Project B -500 -500 100 250 100 250 150 200 200 100 400 50 E Differential cash flows: cash flow(A) - cash flow(B) 0 -150 -150 -50 100 350 8.51% <-- =B36-C36 <-- =B37-C37 <-- =IRR(D36:D41) Column D in the above example contains the differential cash flows—the difference between the cash flows of project A and project B. In cell D43 we use the Excel IRR function to compute the crossover point. A bit of theory (can be skipped): To see why the crossover point is the IRR of the differential cash flows, suppose that for some rate r, NPV(A) ⫽ NPV(B): CFA1 CFA2 CFAN + + ··· + 2 (1 + r) (1 + r) (1 + r) N CFB1 CFB2 CFBN = CFB0 + + + ··· + = NPV(B) 2 (1 + r) (1 + r) (1 + r) N NPV(A) = CFA0 + Subtracting and rearranging shows that r must be the IRR of the differential cash flows: CFA0 − CFB0 + CFA1 − CFB1 CFAN − CFBN CFA2 − CFB2 =0 + ··· + + 2 (1 + r) (1 + r) N (1 + r) What to Use? NPV or IRR? Let’s go back to the initial example and suppose that the discount rate is 8%: A 1 2 3 4 5 6 7 8 9 10 11 12 13 B C D RANKING PROJECTS WITH NPV AND IRR Discount rate Year Project A 0 1 2 3 4 5 NPV IRR 8% -500 100 100 150 200 400 216.64 19.77% Project B -500 250 250 200 100 50 212.11 <-- =C5+NPV(B2,C6:C10) 27.38% <-- =IRR(C5:C10) In this case, we know there is disagreement between the NPV (which would lead us to choose project A) and the IRR (by which we choose project B). Which is correct? The answer to this question—for the case where the discount rate is 8%—is that we should choose based on the NPV (that is, choose project A). This is just one example of the general principle discussed in Section 7.3: Using the NPV is always preferred, since the NPV is the additional wealth that you get, whereas IRR is the compound rate of return. The economic assumption is that consumers maximize their wealth, not their rate of return. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 168 168 PART TWO CAPITAL BUDGETING AND VALUATION WHERE IS THIS CHAPTER GOING? Until this point in the chapter, we’ve discussed general principles of project choice using the NPV and IRR criteria. The following sections discuss some specifics: • • • • Ignoring sunk costs and using marginal cash flows (Section 7.6) Incorporating taxes and tax shields into capital budgeting calculations (Section 7.7) Incorporating the cost of foregone opportunities (Section 7.9) Incorporating salvage values and terminal values (Section 7.11) 7.6 Capital Budgeting Principle: Ignore Sunk Costs and Consider Only Marginal Cash Flows This is an important principle of capital budgeting and project evaluation: Ignore the cash flows you can’t control and look only at the marginal cash flows—the outcomes of financial decisions you can still make. In the jargon of finance: Ignore sunk costs, costs that have already been incurred and thus are not affected by future capital budgeting decisions. Here’s an example: You recently bought a plot of land and built a house on it. Your intention was to sell the house immediately, but it turns out that the house is really badly built and cannot be sold in its current state. The house and land cost you $100,000, and a friendly local contractor has offered to make the necessary repairs, which will cost $20,000. Your real estate broker estimates that even with these repairs you’ll never sell the house for more than $90,000. What should you do? There are two approaches to answering this question: • • “My father always said ‘Don’t throw good money after bad.’ ” If this is your approach, you won’t do anything. This attitude is typified in column B below, which shows that if you make the repairs you will have lost 25% on your money. “My mother was a finance professor, and she said, ‘Don’t cry over spilt milk. Look only at the marginal cash flows.’” These turn out to be pretty good. In column C below you see that making the repairs will give you a 350% return on your $20,000. A 1 2 House cost 3 Fix up cost 4 5 6 7 8 IRR Year 0 1 B C D IGNORE SUNK COSTS 100,000 20,000 Cash flow Cash flow wrong! right! -120,000 -20,000 90,000 90,000 -25% 350% <-- =IRR(C6:C7) Of course, your father was wrong and your mother right (this often happens): Even though you made some disastrous mistakes (you never should have built the house in the first place), you should—at this point—ignore the sunk cost of $100,000 and make the necessary repairs. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 169 CHAPTER 7 Introduction to Capital Budgeting 169 7.7 Capital Budgeting Principle: Don’t Forget the Effects of Taxes—Sally and Dave’s Condo Investment In this section we discuss the capital budgeting problem faced by Sally and Dave, two business school grads who are considering buying a condominium apartment and renting it out for the income. We use Sally and Dave and their condo to emphasize the place of taxes in the capital budgeting process. No one needs to be told that taxes are very important.2 In the capital budgeting process, the cash flows that are to be discounted are after-tax cash flows. We postpone a fuller discussion of this topic to Chapters 9 and 10, where we define the concept of free cash flow. For the moment, we concentrate on a few obvious principles, which we illustrate with the example of Sally and Dave’s condo investment. Sally and Dave—fresh out of business school with a little cash to spare—are considering buying a nifty condo as a rental property. The condo will cost $100,000, and (in this example at least) they’re planning to buy it with all cash. Here are some additional facts: • Sally and Dave figure they can rent out the condo for $24,000 per year. They’ll have to pay property taxes of $1,500 annually and they’re figuring on additional miscellaneous expenses of $1,000 per year. • All the income from the condo has to be reported on their annual tax return. Currently, Sally and Dave have a tax rate of 30%, and they think this rate will continue for the foreseeable future. • Their accountant has explained to them that they can depreciate the full cost of the condo over ten years—each year they can charge $10,000 depreciation (= (condo cost)/ (10-year depreciable life)) against the income from the condo.3 This means that they can expect to pay $3,450 in income taxes per year if they buy the condo and rent it out and have a net income from the condo of $8,050: A 1 2 3 4 5 6 7 8 9 10 11 12 13 B C SALLY & DAVE'S CONDO Cost of condo Sally & Dave's tax rate Annual reportable income calculation Rent Expenses Property taxes Miscellaneous expenses Depreciation Reportable income Taxes (rate = 30%) Net income 100,000 30% 24,000 -1,500 -1,000 -10,000 11,500 <-- =SUM(B6:B10) -3,450 <-- =-B3*B11 8,050 <-- =B11+B12 2 Will Rogers said, “The difference between death and taxes is death doesn’t get worse every time Congress meets.” 3 You may want to read the box on depreciation on the next page before going on. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 170 170 PART TWO CAPITAL BUDGETING AND VALUATION WHAT IS DEPRECIATION? In computing the taxes they owe, Sally and Dave get to subtract expenses from their income. Taxes are computed on the basis of the income before taxes (= income − expenses − depreciation − interest). When Sally and Dave get the rent from their condo, this is income— money earned from their asset. When Sally and Dave pay to fix the faucet in their condo, this is an expense—a cost of doing business. The cost of the condo is neither income nor an expense. It’s a capital investment—money paid for an asset that will be used over many years. Tax rules specify that each year part of the capital investments can be taken off the income (“expensed,” in accounting jargon). This reduces the taxes paid by the owners of the asset and takes account of the fact that the asset has a limited life. There are many depreciation methods in use. The simplest method is straight-line depreciation. In this method the asset’s annual depreciation is a percentage of its initial cost. In the case of Sally and Dave, for example, we’ve specified that the asset is depreciated over ten years. This results in annual depreciation charges of straight-line depreciation = initial asset cost $100,000 = = $10,000 annually depreciable life span 10 In some cases depreciation is taken on the asset cost minus its salvage value: If you think that the asset will be worth $20,000 at the end of its life (this is the salvage value), then the annual straight-line depreciation might be $8,000: straight-line depreciation = initial asset cost − salvage value with salvage value depreciable life span $100,000 − $20,000 = = $8,000 annually 10 ACCELERATED DEPRECIATION Although historically depreciation charges are related to the life span of the asset, in many cases this connection has been lost. Under United States tax rules, for example, an asset classified as having a five-year depreciable life (trucks, cars, and some computer equipment are in this category) will be depreciated over six years (yes six) at 20%, 32%, 19.2%, 11.52%, 11.52%, and 5.76% in each of the years 1, 2, . . . , 6. Notice that this method accelerates the depreciation charges—more than one-sixth of the depreciation is taken annually in years 1–3 and less in later years. Since, as we show in the text, depreciation ultimately saves taxes, this benefits the asset’s owner, who now gets to take more of the depreciation in the early years of the asset’s life. Two Ways to Calculate the Cash Flow In the previous spreadsheet you saw that Sally and Dave’s net income was $8,050. In this section you’ll see that the cash flow produced by the condo is much more than this amount. It all has to do with depreciation: Because the depreciation is an expense for tax purposes but not a cash expense, the cash flow from the condo rental is different. So even though the net income from 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 171 CHAPTER 7 Introduction to Capital Budgeting 171 the condo is $8,050, the annual cash flow is $18,050—you have to add back the depreciation to the net income to get the cash flow generated by the property. A 16 17 18 19 Cash flow, method 1: Add back depreciation Net income Add back depreciation Cash flow B C 8,050 <-- =B13 10,000 <-- =-B10 18,050 <-- =B18+B17 In the above calculation, we’ve added back the depreciation to the net income to get the cash flow. An asset’s cash flow (the amount of cash produced by an asset during a particular period) is computed by taking the asset’s net income (also called profit after taxes or sometimes just “income”) and adding back noncash expenses like depreciation.4 Tax Shields There’s another way of calculating the cash flow, which involves a discussion of tax shields. A tax shield is a tax saving that results from being able to report an expense for tax purposes. In general, a tax shield just reduces the cash cost of an expense: In the above example, since Sally and Dave’s property taxes of $1,500 are an expense for tax purposes, the after-tax cost of the property taxes is (1 − 30%) ∗ $1,500 = $1,500 − 30% ∗ 1,500 = $1,050    ↑ This $450 is the tax shield The tax shield of $450 (= 30% ∗ $1,500) has reduced the cost of the property taxes. Depreciation is a special case of a noncash expense that generates a tax shield. A little thought will show you that the $10,000 depreciation on the condo generates $3,000 of cash. Because depreciation reduces Sally and Dave’s reported income, each dollar of depreciation saves them $0.30 of taxes, without actually costing them anything in out-of-pocket expenses (the $0.30 comes from the fact that Sally and Dave’s tax rate is 30%). Thus, $10,000 of depreciation is worth $3,000 of cash. This $3,000 depreciation tax shield is a cash flow for Sally and Dave. In the spreadsheet below we calculate the cash flow in two stages: • • We first calculate Sally and Dave’s net income ignoring depreciation (cell B29). If depreciation were not an expense for tax purposes, Sally and Dave’s net income would be $15,050. We then add to this figure the depreciation tax shield of $3,000. The result (cell B32) gives the cash flow for the condo. 4 In Chapter 3 we introduced the concept of free cash flow, which is an extension of the cash flow concept discussed here. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 172 172 PART TWO CAPITAL BUDGETING AND VALUATION A B Cash flow, method 2: Compute after-tax income without depreciation, then add depreciation tax shield Rent Expenses Property taxes Miscellaneous expenses Depreciation Reportable income Taxes (rate = 30%) Net income without depreciation 21 22 23 24 25 26 27 28 29 30 31 Depreciation tax shield 32 Cash flow 33 C D 24,000 -1,500 -1,000 0 21,500 <-- =SUM(B22:B26) -6,450 <-- =-B3*B27 15,050 <-- =B27+B28 3,000 <-- =B3*10000 18,050 <-- =B31+B29 This is what the net income would have been if depreciation were not an expense for tax purposes. The effect of depreciation is to add a $3,000 tax shield. Is Sally and Dave’s Condo Investment Profitable?—A Preliminary Calculation At this point Sally and Dave can make a preliminary calculation of the net present value and internal rate of return on their condo investment. Assuming a discount rate of 12% and assuming that they hold the condo for only ten years, the NPV of the condo investment is $1,987 and its IRR is 12.48%: A B C SALLY & DAVE'S CONDO--PRELIMINARY VALUATION 1 12% 2 Discount rate 3 4 Year Cash flow 0 -100,000 5 1 18,050 6 2 18,050 7 3 18,050 8 4 18,050 9 5 18,050 10 6 18,050 11 7 18,050 12 8 18,050 13 9 18,050 14 10 18,050 15 16 1,987 <-- =B5+NPV(B2,B6:B15) 17 Net present value, NPV 12.48% <-- =IRR(B5:B15) 18 Internal rate of return, IRR Is Sally and Dave’s Condo Investment Profitable?—Incorporating Terminal Value into the Calculations A little thought about the previous spreadsheet reveals that we’ve left out an important factor: the value of the condo at the end of the ten-year horizon. In finance an asset’s value at the end of the investment horizon is called the asset’s salvage value or terminal value. In the above spreadsheet, we’ve assumed that the terminal value of the condo is zero, but this assumption is implausible. To make a better calculation about their investment, Sally and Dave will have to make an assumption about the condo’s terminal value. Suppose they assume that at the end of the 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 173 CHAPTER 7 173 Introduction to Capital Budgeting ten years they’ll be able to sell the condo for $80,000. The taxable gain relating to the sale of the condo is the difference between the condo’s sale price and its book value at the time of sale—the initial price minus the sum of all the depreciation since Sally and Dave bought it. Since Sally and Dave have been depreciating the condo by $10,000 per year over a ten-year period, its book value at the end of ten years will be zero. In cell E10 below, you can see that the sale of the condo for $80,000 will generate a cash flow of $56,000: A 1 2 3 4 5 Cost of condo Sally & Dave's tax rate C D E F 100,000 30% Annual reportable income calculation 6 7 8 9 Rent Expenses Property taxes Miscellaneous expenses 10 11 12 13 14 Depreciation Reportable income Taxes (rate = 30%) Net income 15 16 17 18 B SALLY & DAVE'S CONDO: PROFITABILITY AND TERMINAL VALUE Cash flow, method 1 Add back depreciation Net income Add back depreciation Cash flow 24,000 -1,500 -1,000 -10,000 11,500 <-- =SUM(B6:B10) -3,450 <-- =-B3*B11 8,050 <-- =B11+B12 Terminal value Estimated resale value, year 10 Book value Taxable gain Taxes Net after-tax cash flow from terminal value 80,000 0 80,000 <-- =E6-E7 24,000 <-- =B3*E8 56,000 <-- =E8-E9 8,050 <-- =B13 10,000 <-- =-B10 18,050 <-- =B17+B16 To compute the rate of return on Sally and Dave’s condo investment, we put all the numbers together: A 20 Discount rate 21 Year 22 23 0 24 1 25 2 26 3 27 4 28 5 29 6 30 7 31 8 32 9 33 10 34 35 NPV of condo investment 36 IRR of investment B C D 12% Cash flow -100,000 18,050 <-- =B18, Annual cash flow from rental 18,050 18,050 18,050 18,050 18,050 18,050 18,050 18,050 74,050 <-- =B32+E10 20,017 <-- =B23+NPV(B20,B24:B33) 15.98% <-- =IRR(B23:B33) Assuming that the 12% discount rate is the correct rate, the condo investment is worthwhile: Its NPV is positive and its IRR exceeds the discount rate.5 5 When we say that a discount rate is “correct,” we usually mean that it is appropriate to the riskiness of the cash flows being discounted. In Chapter 9 we have our first discussion in this book on how to determine a correct discount rate. For the moment, let’s assume that the discount rate is appropriate to the riskiness of the condo’s cash flows. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 174 174 PART TWO CAPITAL BUDGETING AND VALUATION BOOK VALUE VERSUS TERMINAL VALUE The book value of an asset is its initial purchase price minus the accumulated depreciation. The terminal value of an asset is its assumed market value at the time you “stop writing down the asset’s cash flows.” This sounds like a weird definition of terminal value, but often when we do present value calculations for a long-lived asset (like Sally and Dave’s condo, or like the company valuations we discuss in Chapters 9 and 10), we write down only a limited number of cash flows. Sally and Dave are reluctant to make predictions about condo rents and expenses beyond a ten-year horizon. Past this point, they’re worried about the accuracy of their guesses. So they write down ten years of cash flows; the terminal value is their best guess of the condo’s value at the end of year 10. Their thinking is, “Let’s examine the profitability of the condo if we hold on to it for ten years and sell it.” This is what we mean when we say that “the terminal value is what the asset is worth when we stop writing down the cash flows.” Taxes: If Sally and Dave are right in their terminal value assumption, they will have to take account of taxes. The tax rules for selling an asset specify that the tax bill is computed on the gain over the book value. So, in the example of Sally and Dave, terminal value − taxes on gain over book = terminal value − tax rate ∗ (terminal value − book value) = 80,000 − 30% ∗ (80,000 − 0) = 56,000 Doing Some Sensitivity Analysis (Advanced Topic) A sensitivity analysis can show how the IRR of the condo investment varies as a function of the annual rent and the terminal value. Using Excel’s Data Table (see Chapter 30), we build a sensitivity table: A 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 B C D E F G H 26,000 16.47% 16.84% 17.19% 17.54% 17.87% 18.19% 18.50% 18.80% 19.09% 19.37% 19.65% 19.91% 28,000 18.10% 18.44% 18.76% 19.08% 19.38% 19.68% 19.96% 20.24% 20.51% 20.78% 21.03% 21.28% Data table--Condo IRR as function of annual rent and terminal value Rent Terminal value --> =B36 15.98% 50,000 60,000 70,000 80,000 90,000 100,000 110,000 120,000 130,000 140,000 150,000 160,000 18,000 9.72% 10.26% 10.77% 11.25% 11.71% 12.15% 12.58% 12.98% 13.37% 13.75% 14.11% 14.46% 20,000 11.45% 11.93% 12.40% 12.84% 13.27% 13.67% 14.06% 14.44% 14.80% 15.15% 15.49% 15.82% 22,000 13.15% 13.59% 14.01% 14.42% 14.81% 15.19% 15.55% 15.90% 16.23% 16.56% 16.87% 17.18% 24,000 14.82% 15.22% 15.61% 15.98% 16.34% 16.69% 17.02% 17.35% 17.66% 17.96% 18.26% 18.55% Note: The data table above computes the IRR of the condo investment for combinations of rent (from $18,000 to $26,000 per year) and terminal value (from $50,000 to $160,000). Data tables are very useful though not trivial to compute. See Chapter 30 for more information. The calculations in the data table aren’t that surprising: For a given rent, the IRR is higher when the terminal value is higher, and for a given terminal value, the IRR is higher given a higher rent. Building the Data Table6 Here’s how the data table was set up: • 6 We build a table with terminal values in the left-hand column and rent in the top row. This subsection doesn’t replace Chapter 30, but it may help reinforce what we say there. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 175 CHAPTER 7 • 175 Introduction to Capital Budgeting In the top left-hand corner of the table (cell B40), we refer to the IRR calculation in the spreadsheet example (this calculation occurs in cell B36). At this point the table looks like this: A B C 38 Data table--Condo IRR as function of annual rent and terminal value Rent 39 18,000 15.98% 40 Terminal value --> 50,000 41 60,000 42 70,000 43 80,000 44 =B36 90,000 45 100,000 46 110,000 47 120,000 48 130,000 49 140,000 50 150,000 51 160,000 52 D E 20,000 F 22,000 G 24,000 H 26,000 28,000 Using the mouse, we now mark the whole table. We use the Data|Table command and fill in the cell references from the original example: 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 176 176 PART TWO CAPITAL BUDGETING AND VALUATION The dialog box tells Excel to repeat the calculation in cell B36, varying the rent number in cell B6 and varying the terminal value number in cell E6. Pressing OK does the rest. MINI-CASE A mini-case for this chapter looks at Sally and Dave’s condo once more—this time under the assumption that they take out a mortgage to buy the condo. Highly recommended! 7.8 Capital Budgeting and Salvage Values In the Sally–Dave condo example, we focused on the effect of noncash expenses on cash flows: Accountants and the tax authorities compute earnings by subtracting certain kinds of expenses from sales, even though these expenses are noncash expenses. In order to compute the cash flow, we add back these noncash expenses to accounting earnings. We showed that these noncash expenses create tax shields—they create cash by saving taxes. In this section, we consider a capital budgeting example in which a firm sells its asset before it is fully depreciated. We show that the asset’s book value at the date of the terminal value creates a tax shield and we look at the effect of this tax shield on the capital budgeting decision. Here’s the example. Your firm is considering buying a new machine. Here are the facts: • • • • • • • The machine costs $800. Over the next eight years (the life of the machine) the machine will generate annual sales of $1,000. The annual cost of the goods sold (COGS) is $400 per year and other costs—selling, general, and administrative expenses (SG&A)—are $300 per year. Depreciation on the machine is straight-line over eight years (that is, $100 per year). At the end of eight years, the machine’s salvage value (or terminal value) is zero. The firm’s tax rate is 40%. The firm’s discount rate for projects of this kind is 15%. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 177 CHAPTER 7 177 Introduction to Capital Budgeting Should the firm buy the machine? Here’s the analysis in Excel: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 B C D E F G BUYING A MACHINE--NPV ANALYSIS Cost of the machine Annual anticipated sales Annual COGS Annual SG&A Annual depreciation 800 1,000 400 300 100 Tax rate Discount rate 40% 15% Annual profit and loss (P&L) Sales Minus COGS Minus SG&A Minus depreciation Profit before taxes Subtract taxes Profit after taxes 1,000 -400 -300 -100 200 <-- =SUM(B12:B15) -80 <-- =-B8*B16 120 <-- =B16+B17 Calculating the annual cash flow Profit after taxes Add back depreciation Cash flow NPV Analysis Year Cash flow 0 -800 <-- =-B2 1 220 <-- =$B$23 2 220 3 220 4 220 5 220 6 220 7 220 8 220 NPV 187 <-- =F7+NPV(B9,F8:F15) 120 100 220 Notice that we first calculate the profit and loss (P&L) statement for the machine (cells B12 to B18) and then turn this P&L into a cash flow calculation (cells B21 to B23). The annual cash flow is $220. Cells F7 to F15 show the table of cash flows, and cell F17 gives the NPV of the project. The NPV is positive, and the firm should therefore buy the machine. Salvage Value—A Variation on the Theme Suppose the firm can sell the machine for $300 at the end of year 8. To compute the cash flow produced by this salvage value, we must make the distinction between book value and market value: Book value An accounting concept: The book value of the machine is its initial cost minus the accumulated depreciation (the sum of the depreciation taken on the machine since its purchase). In our example, the book value of the machine in year 0 is $800, in year 1 it is $700, . . . , and at the end of year 8 it is zero. Market value The market value is the price at which the machine can be sold. In our example, the market value of the machine at the end of year 8 is $300. Taxable gain The taxable gain on the machine at the time of sale is the difference between the market value and the book value. In our case, the taxable gain is positive ($300), but it can also be negative (see an example on p. 180). 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 178 178 PART TWO CAPITAL BUDGETING AND VALUATION Here’s the NPV calculation including the salvage value: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 B C D E F G BUYING A MACHINE--NPV ANALYSIS with salvage value Cost of the machine Annual anticipated sales Annual COGS Annual SG&A Annual depreciation Tax rate Discount rate 800 1,000 400 300 100 40% 15% Annual profit and loss (P&L) Sales Minus COGS Minus SG&A Minus depreciation Profit before taxes Subtract taxes Profit after taxes Calculating the annual cash flow Profit after taxes Add back depreciation Cash flow 1,000 -400 -300 -100 200 <-- =SUM(B12:B15) -80 <-- =-B8*B16 120 <-- =B16+B17 NPV Analysis Year Cash flow 0 -800 <-- =-B2 1 220 <-- =$B$23 2 220 3 220 4 220 5 220 6 220 7 220 8 400 <-- =$B$23+B30 NPV 246 <-- =F7+NPV(B9,F8:F15) 120 100 220 Calculating the cash flow from salvage value Machine market value, year 8 300 Book value, year 8 0 Taxable gain 300 <-- =B26-B27 Taxes paid on gain 120 <-- =B8*B28 Cash flow from salvage value 180 <-- =B26-B29 Note the calculation of the cash flow from the salvage value (cell B30) and the change in the year 8 cash flow (cell F15). One More Example Suppose we change the example slightly: • • The annual sales, SG&A, COGS, and depreciation are still as specified in the original example. The machine will still be depreciated on a straight-line basis over eight years. However, you think you may sell the machine at the end of year 7 for an estimated salvage value of $450. At the end of year 7 the book value of the machine is $100. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 179 CHAPTER 7 179 Introduction to Capital Budgeting Here’s how the calculations look now: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 B C D E F G BUYING A MACHINE--NPV ANALYSIS with salvage value Machine sold at end of year 7 Cost of the machine Annual anticipated sales Annual COGS Annual SG&A Annual depreciation Tax rate Discount rate 800 1,000 400 300 100 40% 15% Annual profit and loss (P&L) Sales Minus COGS Minus SG&A Minus depreciation Profit before taxes Subtract taxes Profit after taxes Calculating the annual cash flow Profit after taxes Add back depreciation Cash flow 1,000 -400 -300 -100 200 <-- =SUM(B12:B15) -80 <-- =-B8*B16 120 <-- =B16+B17 NPV Analysis Year Cash flow 0 -800 <-- =-B2 1 220 <-- =$B$23 2 220 3 220 4 220 5 220 6 220 7 530 <-- =$B$23+B30 NPV 232 <-- =F7+NPV(B9,F8:F15) 120 100 220 Calculating the cash flow from salvage value Machine market value, year 7 450 Book value, year 7 100 Taxable gain 350 <-- =B26-B27 Taxes paid on gain 140 <-- =B8*B28 Cash flow from salvage value 310 <-- =B26-B29 Note the subtle changes from the previous example: • The cash flow from salvage value is salvage value − tax ∗ (salvage value − book value)    ↑ Taxable gain at time of machine sale • In our example this is $310 (cell B30). Another way to write the cash flow from the salvage value is salvage value ∗ (1 − tax) + tax ∗ book  value    ↑ ↑ Tax shield on book After-tax proceeds from machine sale if the whole salvage value is taxed value at time of machine sale 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 180 180 PART TWO CAPITAL BUDGETING AND VALUATION Using this example, you can see the role taxes play even if the machine is sold at a loss. Suppose, for example, that the machine is sold in year 7 for $50, which is less than the book value: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 B C D E F G BUYING A MACHINE--NPV ANALYSIS with salvage value Machine sold at end of year 7 Cost of the machine Annual anticipated sales Annual COGS Annual SG&A Annual depreciation Tax rate Discount rate Annual profit and loss (P&L) Sales Minus COGS Minus SG&A Minus depreciation Profit before taxes Subtract taxes Profit after taxes Calculating the annual cash flow Profit after taxes Add back depreciation Cash flow 800 1,000 400 300 100 40% 15% 1,000 -400 -300 -100 200 <-- =SUM(B12:B15) -80 <-- =-B8*B16 120 <-- =B16+B17 NPV Analysis Year Cash flow 0 -800 <-- =-B2 1 220 <-- =$B$23 2 220 3 220 4 220 5 220 6 220 7 290 <-- =$B$23+B30 NPV 142 <-- =F7+NPV(B9,F8:F15) 120 100 220 Calculating the cash flow from salvage value Machine market value, year 7 50 Book value, year 7 100 Taxable gain -50 <-- =B26-B27 Taxes paid on gain -20 <-- =B8*B28 Cash flow from salvage value 70 <-- =B26-B29 In this case, the negative taxable gain (cell B28, the jargon often heard is “loss over book”) produces a tax shield—the negative taxes of ⫺$20 in cell B29. This tax shield is added to the market value to produce a salvage value cash flow of $70 (cell B30). Thus, even selling an asset at a loss can produce a positive cash flow. 7.9 Capital Budgeting Principle: Don’t Forget the Cost of Foregone Opportunities This is another important principle of capital budgeting. An example: You’ve been offered the project below, which involves buying a widget-making machine for $300 to make a new product. The cash flows in years 1–5 have been calculated by your financial analysts: 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 181 CHAPTER 7 A 1 2 Discount rate 3 Year 4 5 0 6 1 7 2 8 3 9 4 10 5 11 12 NPV 13 IRR Introduction to Capital Budgeting B 181 C DON'T FORGET THE COST OF FOREGONE OPPORTUNITIES 12% Cash flow -300 185 249 155 135 420 498.12 <-- =NPV(B2,B6:B10)+B5 62.67% <-- =IRR(B5:B10) Looks like a fine project! But now someone remembers that the widget process makes use of some already existing but underused equipment. Should the value of this equipment be somehow taken into account? The answer to this question has to do with whether the equipment has an alternative use. For example, suppose that, if you don’t buy the widget machine, you can sell the equipment for $200. Then the true year 0 cost for the project is $500, and the project has a lower NPV: A 16 Discount rate 17 Year 18 19 20 21 22 23 24 25 26 NPV 27 IRR 0 1 2 3 4 5 B C 12% Cash flow The $300 direct cost + $200 -500 <-- value of the existing machines 185 249 155 135 420 298.12 31.97% While the logic here is clear, the implementation can be murky: What if the machine is to occupy space in a building that is currently unused? Should the cost of this space be taken into account? It all depends on whether there are alternative uses, now or in the future.7 7.10 In-House Copying or Outsourcing? A Mini-case Illustrating Foregone Opportunity Costs Your company is trying to decide whether to outsource its photocopying or continue to do it in-house. The current photocopier won’t do anymore—it either has to be sold or thoroughly 7 There’s a fine Harvard case on this topic: “The Super Project,” Harvard Business School case 9-112-034. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 182 182 PART TWO CAPITAL BUDGETING AND VALUATION fixed up. Here are some details about the two alternatives: • • The company’s tax rate is 40%. Doing the copying in-house requires an investment of $17,000 to fix up the existing photocopy machine. Your accountant estimates that this $17,000 can immediately be booked as an expense, so that its after-tax cost is (1 − 40%) ∗ $17,000 = $10,200. Given this investment, the copier will be good for another five years. Annual copying costs are estimated to be $25,000 on a before-tax basis; after-tax this is (1 − 40%) ∗ $25,000 = $15,000. • The photocopy machine is on your books for $15,000, but its market value is in fact much less—it could be sold today for only $5,000. This means that the sale of the copier will generate a loss for tax purposes of $10,000; at your tax rate of 40%, this loss gives a tax shield of $4,000. Thus, the sale of the copier will generate a cash flow of $9,000. • If you decide to keep doing the photocopying in-house, the remaining book value of the copier will be depreciated over five years at $3,000 per year. Since your tax rate is 40%, this will produce a tax shield of 40% ∗ $3,000 = $1,200 per year. • Outsourcing the copying will cost $33,000 per year—$8,000 more expensive than doing it in-house on the rehabilitated copier. Of course, this $33,000 is an expense for tax purposes, so that the net savings from doing the copying in-house are (1 − tax rate) ∗ outsourcing costs = (1 − 40%) ∗ $33,000 = $19,800 • The relevant discount rate is 12%. We show two ways to analyze this decision. The first method values each of the alternatives separately. The second method looks only at the differential cash flows. We recommend the first method—it’s simpler and leads to fewer mistakes. The second method produces a somewhat “cleaner” set of cash flows that take explicit account of foregone opportunity costs. Method 1: Write Down the Cash Flows of Each Alternative This is often the simplest way to do things; if you do it correctly, this method takes care of all the foregone opportunity costs without your thinking about them. Below we write down the cash flows for each alternative: Year 0 In-House Outsourcing −(1 − tax rate) ∗ machine rehab cost = −(1 − 40%) ∗ 17,000 Sale price of machine + tax rate ∗ loss over book value = $5,000 + 40% ∗ ($15,000 − $5,000) = $9,000 = −$10,200 Years 1–5 Annual Cash Flow −(1 − tax rate) ∗ in-house costs + tax rate ∗ depreciation = −(1 − 40%) ∗ $25,000 + 40% ∗ $3,000 = −$13,800 −(1 − tax rate) ∗ outsourcing costs = −(1 − 40%) ∗ $33,000 = −$19,800 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 183 CHAPTER 7 Introduction to Capital Budgeting 183 Putting these data in a spreadsheet and discounting at the discount rate of 12% shows that it is cheaper to do the in-house copying. The NPV of the in-house cash flows is ⫺$59,946, whereas the NPV of the outsourcing cash flows is ⫺$62,375. Note that both NPVs are negative; but the in-house alternative is less negative (meaning: more positive) than the outsourcing alternative; therefore, the in-house alternative is preferred: A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 B C SELL THE PHOTOCOPIER OR FIX IT UP? Annual cost savings (before tax) after fixing up the machine Book value of machine Market value of machine Rehab cost of machine Tax rate Annual depreciation if machine is retained Annual copying costs In-house Outsourcing Discount rate 8,000 15,000 5,000 17,000 40% 3,000 25,000 33,000 12% Alternative 1: Fix up machine and do copying in-house Year 0 1 2 3 4 5 NPV of fixing up machine and in-house copying Cash flow -10,200 <-- =-B5*(1-B6) -13,800 <-- =-$B$9*(1-$B$6)+$B$6*$B$7 -13,800 -13,800 -13,800 -13,800 -59,946 <-- =B15+NPV(B11,B16:B20) Alternative 2: Sell machine and outsource copying Year 0 1 2 3 4 5 NPV of selling machine and outsourcing Cash flow 9,000 <-- =B4+B6*(B3-B4) -19,800 <-- =-(1-$B$6)*$B$10 -19,800 -19,800 -19,800 -19,800 -62,375 <-- =B25+NPV(B11,B26:B30) Method 2: Discounting the Differential Cash Flows In this method we subtract the cash flows of Alternative 2 from those of Alternative 1: A 34 Subtract Alternative 2 CFs from Alternative 1 CFs Year 35 0 36 1 37 38 2 39 3 40 4 41 5 42 NPV(Alternative 1 - Alternative 2) B C Cash flow -19,200 <-- =B15-B25 6,000 <-- =B16-B26 6,000 6,000 6,000 6,000 2,429 <-- =B36+NPV(B11,B37:B41) 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 184 184 PART TWO CAPITAL BUDGETING AND VALUATION The NPV of the differential cash flows is positive. This means that Alternative 1 (in-house) is better than Alternative 2 (outsourcing): NPV(in-house − outsourcing) = NPV(in-house) − NPV(outsourcing) > 0 This means that NPV(in-house) > NPV(outsourcing) If you look carefully at the differential cash flows, you’ll see that they take into account the cost of the foregone opportunities: Year Differential Cash Flow Explanation Year 0 ⫺$19,200 This is the after-tax cost of rehabilitating the old copier (⫺$10,200) and the foregone opportunity cost of selling the copier (⫺$9,000). In other words: This is the cost in year 0 of deciding to do the copying in-house. Years 1–5 $6,000 This is the after-tax saving of doing the copying in-house: If you do it in-house, you save $8,000 pretax (⫽$4,800 after tax) and you get to take depreciation on the existing copier (⫽tax shield of $1,200). Relative to in-house copying, the outsourcing alternative has a foregone opportunity cost of theloss of the depreciation tax shield. If you examine the convoluted prose in the table above (“the outsourcing alternative has a foregone opportunity cost of the loss of the depreciation tax shield”), you’ll agree that it may just be simpler to list each alternative’s cash flows separately. 7.11 Accelerated Depreciation As you know by now, the salvage value for an asset is its value at the end of its life; another term sometimes used is terminal value. Here’s a capital budgeting example that illustrates the importance of accelerated depreciation in computing the Net present value: • Your company is considering buying a machine for $10,000. • If bought, the machine will produce annual cost savings of $3,000 for the next five years; these cash flows will be taxed at the company’s tax rate of 40%. • The machine will be depreciated over the five-year period using the accelerated depreciation percentages allowable in the United States. At the end of year 6, the machine will be sold; your estimate of its salvage value at this point is $4,000, even though for accounting purposes its book value is $576 (cell B19 below). You have to decide what the NPV of the project is, using a discount rate of 12%. Here are the relevant calculations: 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 185 CHAPTER 7 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 B C 185 Introduction to Capital Budgeting D E F G CAPITAL BUDGETING WITH ACCELERATED DEPRECIATION Machine cost Annual materials savings, before tax Salvage value, end of year 5 Tax rate Discount rate 10,000 3,000 4,000 40% 12% Accelerated depreciation schedule (ACRS) Year 1 2 3 4 5 6 Terminal value Year 6 sale price, estimated Year 6 book value Taxable gain Taxes Net cash flow from terminal value ACRS Depreciation depreciation Actual tax shield percentage depreciation 20.00% 2,000 800 <-- =$B$5*C10 32.00% 3,200 1,280 <-- =$B$5*C11 19.20% 1,920 768 <-- =$B$5*C12 11.52% 1,152 461 <-- =$B$5*C13 11.52% 1,152 461 5.76% 576 230 The book value at the end of year 6 is the initial cost of the machine ($10,000) minus the sum of all the depreciation taken on the 4,000 <-- =B4 machine through year 6 ($9,424). 576 <-- =B2-SUM(C10:C14) 3,424 <-- =B18-B19 1,370 <-- =B5*B20 The net cash flow from the 2,630 <-- =B18-B21 terminal value equals the year 6 sale price minus applicable taxes. Net present value calculation Year 26 0 27 28 1 29 2 30 3 31 4 32 5 33 6 34 35 Net present value 36 IRR After-tax cost Depreciation tax shield savings Cost Terminal value -10,000 1,800 1,800 1,800 1,800 1,800 800 1,280 768 461 461 2,630 Total cash flow -10,000 2,600 <-- =SUM(B28:E28) 3,080 2,568 2,261 2,261 2,630 657 <-- =F27+NPV(B6,F28:F33) 14.36% <-- =IRR(F27:F33) The annual after-tax cost saving is $1,800 = (1 − 40%) ∗ $3,000. The depreciation tax shields are determined by the accelerated depreciation schedule (rows 10–15), When the asset is sold at the end of year 6, its book value is $576. This leads to a taxable gain of $3,424 (cell B20) and to taxes of $1,370 (cell B21). The net cash flow from selling the asset at the end of year 6 is its sale price of $4,000 minus the taxes (cell B22). The NPV of the asset is $657 and the IRR is 14.36% (cell B35 and B36). CONCLUSION In this chapter we’ve discussed the basics of capital budgeting using NPV and IRR. Capital budgeting decisions can be separated crudely into “Yes–No” decisions (“Should we undertake a given project?”) and into “ranking” decisions (“Which of the following list of projects do we prefer?”). We’ve concentrated on two important areas of capital budgeting: • The difference between NPV and IRR in making the capital budgeting decision. In many cases these two criteria give the same answer to the capital budgeting question. However, there are cases—especially when we rank projects—where NPV and IRR give different answers. Where they differ, NPV is the preferable criterion to use because the NPV is the additional wealth derived from a project. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 186 186 PART TWO • CAPITAL BUDGETING AND VALUATION Every capital budgeting decision ultimately involves a set of anticipated cash flows, so when you do capital budgeting, it’s important to get these cash flows right. We’ve illustrated the importance of sunk costs, taxes, foregone opportunities, and salvage values in determining the cash flows. EXERCISES 1. You are considering a project whose cash flows are given below: A 3 Discount rate 4 Year 5 6 0 7 1 8 2 9 3 10 4 11 5 12 6 B 25% Cash flow -1,000 100 200 300 400 500 600 (a) Calculate the present values of the future cash flows of the project. (b) Calculate the project’s net present value. (c) Calculate the internal rate of return. (d) Should you undertake the project? 2. Your firm is considering two projects with the following cash flows: A 5 Year 6 7 8 9 10 11 B C Project A 0 1 2 3 4 5 -500 167 180 160 100 100 Project B -500 200 250 170 25 30 (a) If the appropriate discount rate is 12%, rank the two projects. (b) Which project is preferred if you rank by IRR? (c) Calculate the crossover rate—the discount rate r for which the NPVs of both projects are equal. (d) Should you use NPV or IRR to choose between the two projects? Give a brief discussion. 3. Your uncle is the proud owner of an up-market clothing store. Because business is down, he is considering replacing the languishing tie department with a new sportswear department. In order to examine the profitability of such a move, he hired a financial advisor to estimate the cash flows of the new department. After six months of hard work, the financial advisor came up with the 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 187 CHAPTER 7 Introduction to Capital Budgeting 187 following calculations: Investment (at t = 0) Rearranging the shop Loss of business during renovation Payment for financial advisor Total 40,000 15,000 12,000 67,000 Profits (from t = 1 to infinity) Annual earnings from the sport department Loss of earnings from the tie department Loss of earnings from other departments* Additional worker for the sport department Municipal taxes Total 75,000 ⫺20,000 ⫺15,000 ⫺18,000 ⫺15,000 7,000 * Some of your uncle’s stuck-up clients will not buy in a shop that sells sportswear. The discount rate is 12%, and there are no additional taxes. Thus, the financial advisor calculated the NPV as follows: −67,000 + 7,000 = −8,667 0.12 Your surprised uncle asked you (a promising finance student) to go over the calculation. What are the correct NPV and IRR of the project? 4. You are the owner of a factory that supplies chairs and tables to schools in Denver. You sell each chair for $1.76 and each table for $4.40 based on the following calculation: Number of units Cost of material Cost of labor Fixed cost Total cost Cost per unit Plus 10% profit Chair Department Table Department 100,000 80,000 40,000 40,000 160,000 1.60 1.76 20,000 35,000 20,000 25,000 80,000 4.00 4.40 You have received an offer from a school in Colorado Springs to supply an additional 10,000 chairs and 2,000 tables for the price of $1.50 and $3.50, respectively. Your financial advisor advises you not to take up the offer because the price does not even cover the cost of production. Is the financial advisor correct? 5. A factory’s management is considering the purchase of a new machine for one of its units. The machine costs $100,000. The machine will be depreciated on a straight-line basis over its ten-year life to a salvage value of zero. The machine is expected to save the company $50,000 annually, 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 188 188 PART TWO CAPITAL BUDGETING AND VALUATION but in order to operate it the factory will have to transfer an employee (with a salary of $40,000 a year) from one of its other units. A new employee (with a salary of $20,000 a year) will be required to replace the transferred employee. What is the NPV of the purchase of the new machine if the relevant discount rate is 8% and the corporate tax rate is 35%? 6. You are considering the following investment: Year 0 1 2 3 4 5 6 7 7. 8. 9. 10. EBDT (Earnings Before Depreciation and Taxes) ⫺10,500 3,000 3,000 3,000 2,500 2,500 2,500 2,500 The discount rate is 11% and the corporate tax rate is 34%. (a) Calculate the project NPV using straight-line depreciation. (b) What will be the company’s gain if it uses the ACRS depreciation schedule of Section 7.11? A company is considering buying a new machine for one of its factories. The cost of the machine is $60,000 and its expected life span is five years. The machine will save the cost of a worker estimated at $22,500 annually. The book value of the machine at the end of year 5 is $10,000 but the company estimates that the market value will be only $5,000. Calculate the NPV of the machine if the discount rate is 12% and the tax rate is 30%. Assume straight-line depreciation over the fiveyear life of the machine. The ABD Company is considering buying a new machine for one of its factories. The machine cost is $100,000 and its expected life span is eight years. The machine is expected to reduce the production cost by $15,000 annually. The terminal value of the machine is $20,000 but the company believes that it would only manage to sell it for $10,000. If the appropriate discount rate is 15% and the corporate tax is 40%: (a) Calculate the project NPV. (b) Calculate the project IRR. You are the owner of a factory located in a hot tropical climate. The monthly production of the factory is $100,000 except during June–September when it falls to $80,000 due to the heat in the factory. In January 2003 you get an offer to install an air-conditioning system in your factory. The cost of the air-conditioning system is $150,000 and its expected life span is ten years. If you install the air-conditioning system, the production in the summer months will equal the production in the winter months. However, the cost of operating the system is $9,000 per month (only in the four months that you operate the system). You will also need to pay a maintenance fee of $5,000 annually in October. What is the NPV of the air-conditioning system if the discount rate is 12% and the corporate tax rate is 35% (the depreciation costs are recognized in December of each year)? The Cold and Sweet (C&S) Company manufactures ice-cream bars. The company is considering the purchase of a new machine that will top the bar with high quality chocolate. The cost of the machine is $900,000. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 189 CHAPTER 7 11. 12. 13. 14. Introduction to Capital Budgeting 189 Depreciation and terminal value: The machine will be depreciated over ten years to zero salvage value. However, management intends to use the machine for only five years. Management thinks that the sale price of the machine at the end of five years will be $100,000. The machine can produce up to one million ice-cream bars annually. The marketing director of C&S believes that if the company will spend $30,000 on advertising in the first year and another $10,000 in each of the following years, the company will be able to sell 400,000 bars for $1.30 each. The cost of producing each bar is $0.50; and other costs related to the new products are $40,000 annually. C&S’s cost of capital is 14% and the corporate tax rate is 30%. (a) What is the NPV of the project if the marketing director’s projections are correct? (b) What is the minimum price that the company should charge for each bar if the project is to be profitable? Assume that the price of the bar does not affect sales. (c) The C&S Marketing Vice President suggested canceling the advertising campaign. In her opinion, the company sales will not be reduced significantly due to the cancellation. What is the minimum quantity that the company needs to sell in order to be profitable if the Vice President’s suggestion is accepted. (d) Extra: Use a two-dimensional data table to determine the sensitivity of the profitability to the price and quantity sold. The Less Is More Company manufactures swimsuits. The company is considering expanding into the bathrobe market. The proposed investment plan includes: • Purchase of a new machine: The cost of the machine is $150,000 and its expected life span is five years. The terminal value of the machine is 0, but the chief economist of the company estimates that it can be sold for $10,000. • Advertising campaign: The head of the marketing department estimates that the campaign will cost $80,000 annually. • Fixed cost of the new department will be $40,000 annually. • Variable costs are estimated at $30 per bathrobe but due to the expected rise in labor costs they are expected to rise at 5% per year. • Each of the bathrobes will be sold at a price of $45 at the first year. Management estimates that it can raise the price of the bathrobes by 10% in each of the following years. The Less Is More Company discount rate is 10% and the corporate tax rate is 36%. (a) What is the break-even point of the bathrobe department? (b) Plot a graph in which the NPV is the dependent variable of the annual production. The Car Clean Company operates a car wash business. The company bought a machine two years ago at the price of $60,000. The life span of the machine is six years and the machine has no disposal value; the current market value of the machine is $20,000. The company is considering buying a new machine. The cost of the new machine is $100,000 and its life span is four years. The new machine has a disposal value of $20,000. The new machine is faster than the old one; thus, management believes the revenue will increase from $1 million annually to $1.03 million. In addition, the new machine is expected to save the company $10,000 in water and electricity costs. The discount rate of the Car Clean Company is 15% and the corporate tax rate is 40%. What is the NPV of replacing the old machine? A company is considering whether to buy a regular or color photocopier for the office. The cost of the regular machine is $10,000, its life span is five years, and the company has to pay another $1,500 annually in maintenance costs. The color photocopier’s price is $30,000, its life span is also five years, and the annual maintenance costs are $4,500. The color photocopier is expected to increase the revenue of the office by $8,500 annually. Assume that the company is profitable and pays 40% corporate tax; the relevant discount rate is 11%. Which photocopy machine should the firm buy? The Coka Company is a soft drink company. Until today, the company bought empty cans from an outside supplier that charges Coka $0.20 per can. In addition, the transportation cost is $1,000 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 190 190 PART TWO CAPITAL BUDGETING AND VALUATION per truck that transports 10,000 cans. The Coka Company’s management is considering whether to start manufacturing cans in its plant. The cost of a can machine is $1,000,000 and its life span is twelve years. The terminal value of the machine is $160,000. Maintenance and repair costs will be $150,000 for every three-year period. The additional space for the new operation will cost the company $100,000 annually. The cost of producing a can in the factory will be $0.17. The cost of capital for Coka is 11% and the corporate tax rate is 40%. (a) What is the minimum number of cans that the company has to sell annually in order to justify self-production of cans? (b) Advanced: Use data tables to show the NPV and IRR of the project as a function of the number of cans. 15. The ZZZ Company is considering investing in a new machine for one of its factories. The company has two alternatives from which to choose: Considerations Machine A Machine B Cost Annual fixed cost per machine Variable cost per unit Annual production $4,000,000 $300,000 $1.20 400,000 $10,000,000 $210,000 $0.80 550,000 The life span of each machine is five years. ZZZ sells each unit for a price of $6. The company has a cost of capital of 12% and its tax rate is 35%. (a) If the company manufactures 1,000,000 units per year, which machine should it buy? (b) Plot a graph showing the profitability of investment in each machine type depending on the annual production. 16. The Easy Sight Company manufactures sunglasses. The company has two machines, each of which produces 1,000 sunglasses per month. The book value of each of the old machines is $10,000 and their expected life span is five years. The machines are being depreciated on a straight-line basis to zero salvage value. The company assumes it will be able to sell a machine today (January 2006) for the price of $6,000. The price of a new machine is $20,000 and its expected life span is five years. The new machine will save the company $0.85 for every pair of sunglasses produced. Demand for sunglasses is seasonal. During the five months of the summer (May– September) demand is 2,000 sunglasses per month, while during the winter months it falls down to 1,000 per month. Assume that due to insurance and storage costs it is uneconomical to store sunglasses at the factory. How many new machines should Easy Sight buy if the discount rate is 10% and the corporate tax rate is 40%? 17. Poseidon is considering opening a shipping line from Athens to Rhodes. In order to open the shipping line, Poseidon will have to purchase two ships that cost 1,000 gold coins each. The life span of each ship is ten years, and Poseidon estimates that he will earn 300 gold coins in the first year and that the earnings will increase by 5% per year. The annual costs of the shipping line are estimated at 60 gold coins annually, Poseidon’s interest rate is 8%, and Zeus’s tax rate is 50%. (a) Will the shipping line be profitable? (b) Due to Poseidon’s good connections on Olympus, he can get a tax reduction. What is the maximum tax rate at which the project will be profitable? 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 191 CHAPTER 7 Introduction to Capital Budgeting 191 18. At the board meeting on Olympus, Hera tried to convince Zeus to keep the 50% tax rate intact due to the budget deficit. According to Hera’s calculations, the shipping line will be more profitable if Poseidon buys only one ship and sells tickets only to first class passengers. Hera estimated that Poseidon’s annual costs will be 40 gold coins. (a) What are the minimum annual average earnings required for the shipping line to be profitable, assuming that earnings are constant throughout the ten years? (b) Zeus, who is an old fashioned god, believes that “blood is thicker than money.” He agrees to give Poseidon a tax reduction if he buys only one ship. Use data tables to show the profitability of the project, dependent on the annual earnings and the tax rate. 19. Kane Running Shoes is considering the manufacture of a special shoe for race walking, which will indicate if an athlete is running (that is, both legs are not touching the ground). The chief economist of the company presented the following calculation for the Smart Walking Shoe (SWS): • R&D: $200,000 annually in each of the next four years For the manufacturing project: • Expected life span: ten years • Investment in machinery: $250,000 (at t = 4) expected life span of the machine ten years • Expected annual sales: 5,000 pairs of shoes at the expected price of $150 per pair • Fixed cost: $300,000 annually • Variable cost: $50 per pair of shoes Kane’s discount rate is 12%, the corporate tax rate is 40%, and R&D expenses are tax deductible against other profits of the company. Assume that at the end of project (that is, after fourteen years) the new technology will have been superseded by other technologies and therefore will have no value. (a) What is the NPV of the project? (b) The International Olympic Committee (IOC) decided to give Kane a loan without interest for six years in order to encourage the company to take on the project. The loan will have to be paid back in six equal annual payments. What is the minimum loan that the IOC should give in order that the project will be profitable? 20. (Continuation of previous problem) After long negotiations, the IOC decided to lend Kane $600,000 at t = 0. The project went ahead. After the research and development stage was completed (at t = 4) but before the investment was made, the IOC decided to cancel race walking as an Olympic event. As a result, Kane is expecting a large drop in sales of the SWS shoes. What is the minimum number of shoes Kane has to sell annually for the project to be profitable in each of the following two cases: (a) If, in the event of cancellation, the original loan term continues? (b) If, in the event of cancellation, the company has to return the outstanding debt to the IOC immediately? 21. The Aphrodite Company is a manufacturer of perfume. The company is about to launch a new line of products. The marketing department has to decide whether to use an aggressive or regular campaign. Aggressive Campaign Initial cost (production of commercial advertisement using a top model): $400,000 First month profit: $20,000 Monthly growth in profit (months 2–12): 10% After 12 months the company is going to launch a new line of products and it is expected that the monthly profits from the current line would be $20,000 forever. 0195301501_158-192_ch7.qxd 11/3/05 12:47 PM Page 192 192 PART TWO CAPITAL BUDGETING AND VALUATION Regular Campaign Initial cost (using a less famous model): $150,000 First month profit: $10,000 Monthly growth in profits (months 2–12): 6% Monthly profit (months $13–∞ ): $20,000 (a) The cost of capital is 7%. Calculate the NPV of each campaign and decide which campaign the company should undertake. (b) The manager of the company believes that, due to the recession expected next year, the profit figures for the aggressive campaign (both first month profit and monthly growth in profits for months 2–12) are too optimistic. Use a data table to show the differential NPV as a function of first month payment and growth rate of the aggressive campaign. 22. The Long-Life Company has a ten-year monopoly for selling a new vaccine that is capable of curing all known cancers. The price at which the company can sell the new drug is given by the following equation: P = 10,000 − 0.3 ∗ X 0 ≤ X < 25,000 where P is the price per vaccine and X is the quantity. In order to mass-produce the new drug, the company needs to purchase new machines. Each machine costs $70,000,000 and is capable of producing 150,000 vaccines per year. The expected life span of each machine is five years; over this time it will be depreciated on a straight-line basis to zero salvage value. The R&D cost for the new drug is $1,500,000,000, the variable costs are $1,000 per vaccine, and fixed costs are $120,000,000 annually. If the discount rate is 12% and the tax rate is 30%, how many vaccines will the company produce annually? (Use either Excel’s Goal Seek or its Solver—see Chapter 32.) 23. (Continuation of Exercise 22). The independent senator from Alaska, Michele Carey, has suggested that the government pay Long-Life $2,000,000 in exchange for the company guaranteeing that it will produce under the zero profit policy (that is, produce as long as NPV ⱖ 0). How many vaccines will the company produce annually?