CHAPTER 10
CAPITAL BUDGETING FOCUS Our focus in this first capital budgeting chapter begins with the time value concepts behind methods and then moves on to computational and decision making techniques. The problems of cash flow estimation and risk encountered in practice are touched upon here in anticipation of a detailed treatment in a later chapter. PEDAGOGY A brief overview of the cost of capital concept is presented early in the chapter even though it is the subject of Chapter 13. The knowledge is necessary to understand and motivate the capital budgeting models. It relates NPV - IRR procedures to the required rate of return idea, something with which students are already familiar. We explicitly tie NPV and IRR together by emphasizing that the IRR comes from the NPV equation as the interest rate that sets NPV=0. This helps to develop an overall understanding of both procedures. TEACHING OBJECTIVES After this chapter students should: 1. appreciate the discounted cash flow basis of capital budgeting theory, and 2. be able to make the computations associated with the major capital budgeting techniques. They should also be marginally aware of the difficulties associated with estimating cash flows and differences in project risk. Along these lines, care should be taken not to form the impression that capital budgeting is an engineering-like process that always gives exactly the right answer. OUTLINE I. CHARACTERISTICS OF BUSINESS PROJECTS The nature of projects requiring capital budgeting decisions. A. Project Types and Risk Replacement, expansion, and new venture projects and their order of risk. B. Stand-alone and Mutually Exclusive Projects Projects considered by themselves and in competition with one another. C. Project Cash Flows Representing projects as streams of cash for analysis. D. The Cost of Capital A brief introduction to the concept of cost of capital at this point makes the NPV and IRR techniques easier to understand. II. CAPITAL BUDGETING TECHNIQUES A general statement defining the techniques as methods of analysis and decision making. A. Payback Period The payback method explained and illustrated. Its uses and drawbacks discussed. B. Net Present Value (NPV) The NPV concept and the defining equation. The relationship to shareholder wealth. Calculations, decision rules, and applications. C. Internal Rate of Return (IRR)
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D. E. F. G. H. I.
The IRR concept and the relation to a required rate of return. The defining equation and its relationship to NPV. Decision rules, calculations, and examples. Comparing IRR and NPV Which is better and why. Possible conflicts NPV and IRR Solutions Using Financial Calculators and Spreadsheets Instruction on using calculators and spreadsheets in capital budgeting. Projects With A Single Outflow and Regular Inflows Solution techniques when annuity methods are possible. Profitability Index (PI) PI as a variation on the NPV concept. Decision rules, calculations, and examples. Comparing Projects with Unequal Lives Chaining and Equivalent Annual Annuity methods. Capital Rationing Allocating a limited Capital Budget among available projects.
QUESTIONS 1. Define mutual exclusivity and describe ways in which projects can be mutually exclusive. ANSWER: A mutually exclusive decision is one in which the selection of any option precludes the selection of all others. In other words, you can't "do both." Mutual exclusivity can be rooted in either the nature of the project or in the availability of resources. Replacements and many expansion projects tend to be mutually exclusive, because there's just one job to be done. Once the method of getting it accomplished is selected, there are simply no other opportunities. Other expansion projects and most new ventures tend to be mutually exclusive because of resource constraints. The firm usually doesn't have enough money to do everything presented as a viable opportunity. 2. Capital budgeting is based on the idea of identifying incremental cash flows, so overheads aren't generally included. Does this practice create a problem for a firm that over a long period of time takes on a large number of projects that are just barely acceptable under capital budgeting rules? ANSWER: Yes! This is a major problem in incremental thinking. If everything is incrementally just viable, over a long period the firm can wind up with no income to support necessary overhead. 3. Relate the idea of cost of capital to the opportunity cost concept. Is the cost of capital the opportunity cost of project money? ANSWER: The cost of capital is an opportunity cost because project funds could always be alternatively used to pay down debt and/or distributed to shareholders as dividends. 4. The payback technique is criticized for not using discounted cash flows. Under what conditions will this matter most? That is, under what patterns of cash flow will payback and NPV or IRR be likely to give different answers? ANSWER: Recognizing the time value of money will matter most when substantial cash flows are projected in the distant future. The discounting methods reduce the value of those flows a lot relative to payback, which gives them their face value. 5. Explain the rationale behind the NPV method in your own words. Why is a higher NPV conceptually better than a lower one?
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ANSWER: The NPV method adds up the PV of all of a project's cash flows thereby calculating its effect on the wealth of the firm (and its shareholders). The more NPV a project has, the bigger is its wealth contribution. It is this direct relation with wealth that makes NPV a very good measure of a project's worth. 6. Projects A and B have approximately the same NPV. Their initial outlays are similar in size. Project A has early positive cash flows, and little or nothing is expected to come in later on. Project B has much larger positive cash flows than A, but they're farther in the future. Can you make any general statement about which project might be better? ANSWER: The project with the cash flows that come in earlier may be better because of the uncertainty of the future. The large flows predicted far out in time are less likely to come true than modest flows predicted in the short run. In fact, this is a major business problem. People tend to predict marvelous results in the distant future that are often very unrealistic. (We'll have a great deal to say about this in later chapters.) 7. Suppose the present value of cash ins and outs is very close to balanced for a project to build a new $50M factory, so that the NPV is +$25,000. The same company is thinking about buying a new trailer truck for $150,000. The NPV of projected cash flows associated with the truck is also about $25,000. Does this mean that the two projects are comparable? Is one more desirable than the other? How are their IRRs likely to compare? If the cash flows have similar risks are the projects equally risky? (Hint: Think in terms of the size of the investment placed at risk relative to the financial rewards expected.) ANSWER: Projects of grossly different sizes are not readily comparable. In this case, the factory project is marginal, because its NPV is minimally positive relative to the size of the investment required to undertake it. Conversely, the truck is a pretty good deal because its NPV is substantial relative to the investment required to get it. The factory's IRR would be just a hair above the cost of capital while the truck's IRR would exceed k by quite a bit. The factory is really a risky project because a small unfavorable percentage variation in the cash flows planned could result in a big dollar loss relative to the capital budgeting analysis. 8. Think about the cash flows associated with putting $100,000 in the bank for five years, assuming you draw out the interest each year and then close the account. Now think about a set of hypothetical cash flows associated with putting the same money in a business, operating for five years, and then selling out. Write an explanation of why the IRR on the business project is like the bank's interest rate. How are the investments different? ANSWER: Both uses of the money involve receiving a series of cash inflows over the five-year period and a large inflow at the end. The IRR is defined as the interest rate that makes the present value of these payments just equal to the initial investment of $100K (project NPV = 0). But the bank's interest rate does exactly the same thing. If we take the present value of the interest payments and the final withdrawal at the bank's interest rate, we'll get the amount of the initial deposit. (This is also exactly like a bond's yield.) There are two major differences between the bank account and the business. The bank account's periodic cash flows will be constant while the business's are likely to vary. Further, all of the bank's flows are nearly certain while the business's are subject to considerable risk. 9. What is it about the cash flows associated with business projects that makes the NPV profile slope downward to the right? Would the NPV profile of any randomly selected set of positive and negative flows necessarily slope one way or the other? Why?
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ANSWER: The bulk of negative project cash flows are generally in the early years, while most of the positive flows are in the more distant future. This means that the discounting factors make a bigger impact on the distant positives than on the near term negatives. This in turn means larger interest rates shrink the positives more than the negatives because they tend to be further into the future. Hence total NPV becomes less positive (declines) as the interest rate increases. This produces a downsloping curve when a project's NPV is graphed versus the interest rate. A randomly selected series of flows would not tend to slope one way or the other. 10. The following set of cash flows changes sign twice and has two IRR solutions. Identify the sign changes. Demonstrate mathematically that 25% and 400% are both solutions to the IRR equation. C0 C1 C2 ($320) $2,000 ($2,000) On the basis of this example, why would you expect multiple solutions to be an unusual problem in practice? ANSWER: The sign changes from minus to plus between C0 and C1 and from plus to minus from C1 to C2. k = 25% NPV = −$320 +
$2,000 − $2,000 + = −$320 + $1,600 − $1,280 = $0 (1.25 ) (1.25 ) 2
NPV = −$320 +
$2,000 − $2,000 + = −$320 + $400 − $80 = $0 (5.00 ) (5.00 ) 2
k = 400%
It's not likely that anyone would mistake the 400% solution as real. Further, multiple sign changes with substantial negative flows in the future are rare. 11. Under what conditions will the IRR and NPV methods give conflicting results for mutually exclusive decisions? Will they ever give conflicting results for stand-alone decisions? Why? ANSWER: Results can conflict when the project's NPV profiles cross in the first quadrant of the (k, NPV) plane. The methods will never give conflicting results for standalone decisions. Examination of a graph shows that for a single down-sloping NPV profile, IRR>k always implies a positive NPV and IRR
NPV>0
IRR
IRR>k
NPV<0 IRR
12. Why is the profitability index more appropriately described as a variation on the NPV technique than on the IRR technique?
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ANSWER: The PI separates a project's NPV at a particular cost of capital into positive and negative parts, respectively representing benefits and costs. It then forms a ratio of these pieces. Hence it is closely associated with the NPV idea which is actually the difference between the same two pieces. The IRR finds the point at which NPV is zero, because the present value of positives and negatives just offset one another. If the PI were calculated at that point it would always be 1.0, and wouldn't mean anything. 13. Show that the profitability index (PI), the initial outlay (C0), and the net present value (NPV) of a project are related by the following equation: NPV = C0 (1 − PI) (Hint: State both the NPV and the PI in terms of C0 and the sum of all other cash flows.) ANSWER: Let S be the sum of the present values of all cash flows after C0. Then NPV = C0 + S and PI =
S − C0
The second expression gives S = − C0 (PI) which is substituted into the first to give NPV = C0 − C0(PI) = C0(1 − PI) BUSINESS ANALYSIS 1. You are a financial analyst for the Ajax Company, which uses about $1M of inventory per month. The purchasing manager has come to you for help with a buying decision. He can get a big discount on $15M of inventory by buying it all at once. However, there is some risk of obsolescence when buying that far in advance. He understands that large purchases are frequently analyzed by means of capital budgeting techniques, and asks for your help in deciding whether or not to buy the specially priced inventory. How would you advise him? Is capital budgeting appropriate? ANSWER: Inventory problems are not generally capital budgeting issues, because they don't stretch over long periods of time. In this case, however, capital budgeting thinking may help by putting time value concepts into the purchasing manager's thinking. He or she should realize that if the inventory is acquired, its value to the firm won't be $15M but the present value of a fifteen period annuity of $1M at the firm's cost of capital. The inventory's "NPV" is then the difference between that value and the cost at which it is offered. An estimate of an increased loss for obsolescence and shrinkage should also be subtracted from this figure. Such losses might be larger than usual because of the long time the inventory will held before use. 2. Risk in capital projects is the probability that a project will earn less than expected. Make up and describe one hypothetical project in each of the replacement, expansion and new venture categories, and list a few ways that each might go wrong and cause the cash flows to be less favorable than expected. Can you think of situations in which projects could result in losses? Could the losses exceed the initial investment (C0)? ANSWER: Most replacement projects deal with worn out or obsolete equipment like production machinery, trucks, or computers. The choice is generally between replacing with the same product, something different, or keeping the old unit. The old machine is usually costing money in terms of
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maintenance or poor product quality. Hence the benefit of replacement is that it saves money. The risk is that the replacement unit won't work as expected and therefore will cost more than keeping the old machine. For example, an old computer might be replaced with a state of the art model with "buggy" hardware or software. Such a situation is relatively unusual if the replacement is a proven product. Expansion projects gear operations up to produce more of something that's already successful. For example, suppose a small brewery that's been distributing beer locally expands to distribute nationally. The risk comes from two sources. The demand for the product may not exist nationally, and the new production facilities may not work right. The latter risk is generally small, but the former is substantial, because it includes customer acceptance as well as sales and distribution problems. New Ventures involve doing something the firm has never done before. Suppose, for example, a radio manufacturer decides to make and sell TVs. Here everything is new and anything can go wrong. Production costs can be high, product quality can be low, customers may reject the product for any number of reasons, and sales or distribution problems can arise. Losses can occur in any type of project if cash inflows are less than total outflows including the initial investment. Losses that exceed the initial investment are not unusual in new ventures. The situation occurs when a new business does worse than expected but shows some promise. The firm then continues to pour in operating funds hoping the venture will turn around and produce positive cash flows. The more that's sunk into the business, the more reluctant people are to give it up. This isn't good thinking, but it happens all the time. 3. Charlie Brown is thinking about starting Wing-It Airlines to fly a commuter route in and out of a major city. Four planes are on the market that will do the job, but each has different flight, load, and operating characteristics. Charlie is unsure of the demand for his service, and feels that it may depend to some extent on the type of plane chosen. Whether or not the business is feasible may depend on which airplane is used in conjunction with the demand estimate assumed. Are capital budgeting techniques appropriate for analysis of this problem? If so, is the issue a stand-alone or mutually exclusive decision? ANSWER: This situation is not unusual in new ventures. Essentially we're saying that the level of sales projected for the project depends on the initial investment. The situation is a capital budgeting problem calling for a mutually exclusive decision. However, the cash inflows aren't the same for each option. Generally the larger initial investment results in larger sales. However, success depends on the difference between inflows and outflows, so it isn't certain whether a big investment will be more profitable than a smaller one. 4. The Budwell & Son Oil Company is looking at two drilling proposals. One project lasts for three years, costs $20M to start, pays back quickly, and has an NPV of $15M. The other project also costs about $20M to start, but has an expected life of seven years, takes much longer to pay back, and has an NPV of $17M. Mr. Budwell, the company's founder, favors the shorter project because of the quick investment recovery. His son Billy, however, has taken finance in college and insists that the only way to judge projects is on NPV. He therefore favors the longer project. They've engaged you as their financial advisor to settle the issue. How would you advise them? ANSWER: In theory the higher NPV project is better. However, cash flows in the distant future are more risky than those in the near term, and that risk is hard to quantify. As a practical matter, Dad's judgement may be better, especially considering the high-risk nature of drilling speculative oil wells. 5. Webley Corp. has a capital budget limited to $20M. Five relatively high IRR projects are available that have initial investments totaling $15M. They are all roughly the same size. A sixth project has an IRR only slightly lower than those of the first five, but requires a $8M investment. Several other smaller projects are available with IRRs quite a bit lower than the sixth. The president has stated that
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it's too bad the firm has to pass up the sixth project, but it just doesn't fit into the budget. How would you advise him? ANSWER: If the first five projects are done, doing the sixth would require more capital than is available. However, if the sixth is skipped, the last $5M of budgeted money will be used on lower return projects starting with the seventh. A higher total NPV might be achieved by skipping the fifth project and including the sixth. That might use up the entire budget on relatively high return projects while eliminating the need to do any of those with substantially lower returns. The idea behind capital rationing is to maximize the return on the entire capital program, not necessarily to take on projects in strictly decreasing order of IRR. PROBLEMS
Payback Period: Example 10.1 (page 459) 1.
Gander, Inc. is considering two projects with the following cash flows. Year Project X Project Y 0 ($100,000) ($100,000) 1 40,000 50,000 2 40.000 0 3 40,000 0 4 40,000 0 5 40,000 250,000 Gander uses the payback period method of capital budgeting and accepts only projects with payback periods of 3 years or less. a. If the projects are presented as standalone opportunities which one(s) would Gander accept? If they were mutually exclusive and Gander disregarded its three year rule, which project would be chosen? b. Is there a flaw in the thinking behind the correct answers to part a? SOLUTION a. Project X has a payback period of 2.5 years. $100,000 / $40,000 = 2.5 Project Y has a payback period of 4.2 years. $50,000 + 0 + 0 +0 + .2(250,000) = $100,000 In a standalone context, payback will accept Project X and reject Project Y. In a mutually exclusive context, payback will choose Project X. b. Yes, project Y is preferable in the longer run, yet the payback method chooses X. That’s because the method ignores cash flows that occur after the shortest payback period. This makes Payback Period a weak method. It’s generally used just to eliminate really bad projects, and shouldn’t be used blindly on Project Y.
Net Present Value (NPV): Example 10.3 (page 462) Profitability Index: Example 10.7 (page 476) 2.
A project has the following cash flows C0 C1 ($700) $200 a. What is the project’s payback period? b. Calculate the projects NPV at 12%.
C2 $500
C3 $244
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SOLUTION: a. The cumulative cash flow is Year 0 1 2 3 Cash Flow ($700) $200 $500 $244 Cumulative ($700) ($500) 0 $244 Cumulative cash flow is zero after two years, hence the payback period is two years. b.
Year 0 1 2 3
Ci ($700) $200 $500 $244
PVFk,i 1.0 .8929 .7972 .7118
c.
PI = ($179 + $399 + $174) / $700 $752 / $700 1.07
Ck,i × PVFk,i ($700) $179 $399 $174 NPV = $ 52
Internal Rate of Return (IRR) – Iterative Procedure: Example 10.5 (page 468) 3. Calculate an IRR for the project in the previous problem using an iterative technique. (Hint: Start by guessing 15%.) SOLUTION: At 15% Year 0 1 2 3
Ci ($700) $200 $500 $244
PVFk,i 1.0 .8696 .7561 .6575
Ck,i × PVFk,i ($700) $174 $378 $160 NPV = $ 12
Since NPV is positive we need to try a higher interest rate At 16% Year 0 1 2 3
Ci ($700) $200 $500 $244
PVFk,i 1.0 .8621 .7432 .6407
Ck,i × PVFk,i ($700) $172 $372 $156 NPV = $ 0
Hence the project’s IRR is 16%. 4.
Clancy Inc. is considering a project with the following cash flows. C0 C1 C2 C3 ($7,800) $2,300 $3,500 $4,153
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a. Clancy has a policy of rejecting all projects that don’t pay back within three years outright, and analyzing those that do more carefully with time value based methods. Does this project warrant further consideration? b. Should Clancy accept the project based on its NPV if the company’s cost of capital is 8%? Is the recommendation definite or marginal? c. What conclusion will the firm reach Based on PI and an 8% cost of capital? Is the recommendation definite or marginal? SOLUTION: a. The cumulative cash flow is Year 0 Cash Flow ($7,800) Cumulative ($7,800)
1 $2,300 ($5,500)
2 $3,500 ($2,000)
3 . $4,153 $2,153
Cumulative cash flow is negative after two years and positive after three, hence the payback period is between two and three years (a little less than 2½ years). Hence Clancy’s policy would require further evaluation using time value based methods. b.
The project’s NPV at 8% is calculated as follows: Year Ci PVF8,i Ci × PVF8,i 0 ($7,800) 1.0 ($7,800) 1 $2,300 .9259 $2,130 2 $3,500 .8573 $3,001 3 $4,153 .7938 $3,297 NPV = $ 628 The project has a positive NPV which implies that it should be accepted. But notice that the magnitude of the NPV is relatively small compared to the size of the investment required. Hence a cautious management might not accept the project even though NPV is positive. c. The PI at 8% is calculated as follows: PI = ($2,130 + $3,001 + $3,297) / $7,800 = $8,428 / $7,800 = 1.08 The conclusion is the same using PI. Acceptance is indicated but not strongly. Notice that the NPV and PI methods both indicate that the present value of inflows exceeds that of outflows by only about 8% over a three-year period. (It is just coincidental that the cost of capital is also 8%.) 5. Should the project being considered in the previous problem be accepted or rejected based on IRR? (Hint: Start by guessing 11% for IRR) Does the IRR method seem to give a more definite result? If so would your recommendation , after considering all four methods be strong or cautious? SOLUTION: To solve for IRR we first guess 11% and compute NPV at that rate. Year 0 1 2 3
Ci ($7,800) $2,300 $3,500 $4,153
PVF11,i Ci × PVF11,i 1.0 ($7,800) .9009 $2,072 .8116 $2,841 .7312 $3,037
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$ 150
Since NPV is positive, we have to try a larger interest rate, say 12%. Year 0 1 2 3
Ci ($7,800) $2,300 $3,500 $4,153
PVF12,i 1.0 .8929 .7972 .7118
Ci × PVF12,i ($7,800) $2,054 $2,790 $2,956 NPV = $ 0
NPV is zero at 12%, so that is the project’s IRR. Since this substantially exceeds Clancy’s 8% cost of capital (by 50%) the IRR method gives a more positive indication that the project should be accepted than did the NPV or PI methods explored in the last problem. However, after considering all four methods, an accept recommendation should be cautious because of the result of the NPV analysis which is the best method. 6.
Hamstring Inc. is considering a project with the following cash flows: C0 ($25,000)
C1 $10,000
C2 $12,000
C3 $5,000
C4 $8,000
The company is reluctant to consider projects with paybacks of more than three years. If projects pass the payback screen, they are considered further by means of the NPV and IRR methods. The firm's cost of capital is 9%. a. What is the project's payback period? Should the project be considered further? b. What is the project's NPV? Does NPV indicate acceptance on a stand-alone basis? c. Calculate the project's IRR using an iterative approach. Start with the cost of capital and the NPV calculation from part (b). Does IRR indicate acceptance on a stand-alone basis? d. What is the project's PI? Does PI indicate acceptance on a stand-alone basis? SOLUTION: a. C0 Cash flows: ($25,000)
C1
C2
$10,000
$12,000
$5,000
$8,000
($3,000)
$2,000
$10,000
Cumulative cash flows: ($25,000) ($15,000)
C3
C4__
Payback period = 2.6 yrs. Yes, since payback < 3yrs b. NPV = −$25,000 + $10,000[PVF9,1] + $12,000[PVF9,2] + $5,000[PVF9,3] + $8,000[PVF9,4] NPV = −$25,000 + $10,000(.9174) + $12,000(.8417) + $5,000(.7722) + $8,000(.7084) NPV = $3,803 Yes, NPV does indicate acceptance on a stand-alone basis. c.
k 9%
NPV $3,803
Capital Budgeting 12% 16%
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$2,138 $ 161
Hence the IRR is slightly more than 16% (a calculator gives 16.3%.) which implies acceptance of the project since 16% > 9%. d.
PI = $28,803 / $25,000 = 1.15
Implies acceptance since PI > 1.0. 7. Project Alpha requires an initial outlay of $35,000 and results in a single cash inflow of $56,367.50 after five years. a. If the cost of capital is 8% what are Alpha's NPV and PI? Is the project acceptable under each of these techniques? b. What is project Alpha's IRR? Is it acceptable under IRR? c. What are Alpha's NPV and PI if the cost of capital is 12%? Is the project acceptable under that condition. d. What is Alpha's payback period? Does payback make much sense for a project like Alpha? Why? SOLUTION: a. NPV = −$35,000 + $56,367.50 [PVF8,5] = −$35,000 + $56,367.50 (.6806) = −$35,000 + $38,363.72 = $3,363.72 Acceptable since NPV > 0. PI = $38,363.72 / $35,000 = 1.10 Acceptable since PI > 1.0. b. Because there’s only one table factor in the NPV computation we can substitute into equation 10.2 and treat it as a time value problem in which the interest rate is the unknown. 0 = −$35,000 + $56,367.50 [PVFIRR,5] PVFIRR,5 = .6209 IRR = 10% Acceptable since IRR > k. = −$35,000 + $56,367.50 [PVF12,5] = −$35,000 + $56,367.50 (.5674) = −$35,000 + $31,982.92 = −$3,017.08 Unacceptable since NPV < 0. c.
NPV
PI = $31,982.92 / $35,000 = .91 Unacceptable since PI < 1.0. d. Five years. Yes, because it takes five years to recover the investment regardless of the fact that the entire return comes in a single sum.
Finding NPV and IRR When Inflows are Regular: Example 10.6 (page 474)
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8. The Sampson Company is considering a project that requires an initial outlay of $75,000 and produces cash inflows of $20,806 each year for five years. Sampson's cost of capital is 10%. a. Calculate the project's payback period by making a single division rather than accumulating cash inflows. Why is this possible in this case? b. Calculate the project's IRR recognizing the fact that the cash inflows are an annuity. Is the project acceptable? Did your calculation in this part result in any number(s) that were also calculated in part a? What is it about this problem that creates this similarity? Will this always happen in such cases? c. What is the project's NPV? Is it acceptable according to NPV rules? SOLUTION: a. $75,000 / $20,806 = 3.6 years The calculation is possible because the returns are constant in amount and regular in time (an annuity). 0 = −$75,000 + $20,806 [PVFAIRR,5] PVFAIRR,5 = 3.6047 IRR = 12% The project is acceptable since IRR > k = 10%. The PVF factor is the same calculation as the payback period. This will always occur when there is only one outflow preceding inflows which are in the form of an annuity. b.
c. NPV = −$75,000 + $20,806 [PVFA10,5] = −$75,000 + $20,806 (3.7908) = −$75,000 + $78,871 = $3,871 The project is acceptable since NPV > 0. 9. Calculate the IRR, NPV, and PI for projects with the following cash flows. Do each NPV and PI calculation at costs of capital of 8% and 12%. Calculate IRRs to the nearest whole percent. a. An initial outlay of $5,000 and inflows of $1,050 for seven years. b. An initial outlay of $43,500 and inflows of 14,100 for four years c. An investment of $78,000 followed by 12 years of income of $11,500. d. An outlay of $36,423 followed by receipts $8,900 of for six years. SOLUTION: IRR:
0 = C0 + C [PVFAIRR,n] [PVFAIRR,n] = − C0 / C
a.
[PVFAIRR,7] = $5,000/$1,050 = 4.7619 IRR = 11%
b.
[PVFAIRR,4] = $43,500/$14,100 = 3.0851 IRR = 11%
c.
[PVFAIRR,12] = $78,000/$11,500 = 6.7826 IRR = 10%
d.
[PVFAIRR,6] = $36,423/$8,900 = 4.0925 IRR = 12%
Capital Budgeting
NPV:
NPV = C0 + C [PVFAk,n]
a. 8%
NPV
= −$5,000+$1,050[PVFA8,7] = −$5,000+$1,050(5.2064) = −$5,000+$5,467 = $467
12%
NPV
= −$5,000+$1,050[PVFA12,7] = −$5,000+$1,050(4.5638) = −$5,000+$4,792 = −$208
b. 8%
NPV
= −$43,500+$14,100[PVFA8,4] = −$43,500+$14,100(3.3121) = −$43,500+$46,701 = $3,201
12%
NPV
= −$43,500+$14,100[PVFA12,4] = −$43,500+$14,100(3.0373) = −$43,500+$42,826 = −$674
c. 8%
NPV
= −$78,000+$11,500[PVFA8,12] = −$78,000+$11,500(7.5361) = −$78,000+$86,665 = $8,665
12%
NPV
= −$78,000+$11,500[PVFA12,12] = −$78,000+$11,500(6.1944) = −$78,000+$71,236 = −$6,764
d. 8%
NPV
= −$36,423+$8,900[PVFA8,6] = −$36,423+$8,900(4.6229) = −$36,423+$41,144 = $4,721
12%
NPV
= −$36,423+$8,900[PVFA12,6] = −$36,423+$8,900(4.1114) = −$36,423+$36,591 = $168
PI:
PI = C [PVFAk,n] / −C0 Figures from the NPV calculation
a. 8% 12%
PI = $5,467 / $5,000 = 1.09 PI = $4,792 / $5,000 = .96
b. 8% 12%
PI = $46,701 / $43,500 = 1.07 PI = $42,826 / $43,500 = .98
c. 8% 12%
NPV = $86,665 / $78,000 = 1.11 NPV = $71,236 / $78,000 = .91
d. 8% 12%
NPV = $41,144 / $36,423 = 1.13 NPV = $36,591 / $36,423 = 1.00
241
Mutually Exclusive Decisions and Judgment Issues: Example 10.4 (page 463) 10. Island Airlines, Inc. needs to replace a short haul computer plane on one of its busier routes. Two aircraft that satisfy the general requirements of the route are on the market. One is more expensive than the other but has better fuel efficiency and load-bearing characteristics that result in better long-
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term profitability. The useful life of both planes is expected to be about seven years, after which time both are assumed to have no value. Cash flow projections for the two aircraft follow:
Initial Cost
Low Cost $775,000
Cash Inflows, years 1 through 7 $154,000
High Cost $950,000 $176,275
a. Calculate the payback period for each plane and select the best choice. b. Calculate the IRR for each plane and select the best option. Use the fact that all the inflows can be represented by an annuity. c. Compare the results of parts a. and b. Both should select the same option, but does one method result in a clearer choice than the other based on the relative sizes of the two payback periods versus the relative sizes of the two IRRs? d. Calculate the NPV and PI of each project assuming a cost of capital of 6%. Use annuity methods. Which plane is selected by NPV? By PI? e. Calculate the NPV and PI of each project assuming the following costs of capital: 2%, 4%, 6%, 8% and 10%. Use annuity methods. Is the same plane selected by NPV and PI at every level of cost of capital? Investigate the relative attractiveness of the two planes under each method. f. Use the results of parts b. and e. to sketch the NPV profiles of the two proposed planes on the same set of axes. Show the IRRs on the graph. Would NPV and IRR ever give conflicting results? Why? SOLUTION: a. P/BH = $950,000 / $176,275 = 5.4 yrs P/BL = $775,000 / $154,000 = 5.0 yrs The best choice appears to be the low cost plane, but the difference in paybacks is marginal (8%). b.
High Cost 0 = C0 + C [PVFAIRR,n] 0 = −$950,000 + $176,275 [PVFAIRR,7] [PVFAIRR,7] = $950,000 / $176,275 = 5.3893 IRR = 7% Low Cost 0 = C0 + C [PVFAIRR,n] 0 = −$775,000 + $154,000 [PVFAIRR,7] [PVFAIRR,7] = $775,000 / $154,000 = 5.0325 IRR = 9% IRR also selects the low cost plane.
c. Both methods choose the low cost plane, but beyond that their message is very different. Payback indicates the decision is marginal in that it chooses the low cost option by a narrow 8% margin. IRR, on the other hand, unambiguously favors the same choice by a 29% margin. d.
High Cost: NPV = C0 + C [PVFAk,n] = −$950,000 + $176,275 [PVFA6,7] = −$950,000 + $176,275 (5.5824) = −$950,000 + $984,038 = $34,038
Capital Budgeting
PI
243
= $984,038 / $950,000 = 1.04
Low Cost: NPV = C0 + C [PVFAk,n] = −$775,000 + $154,000 [PVFA6,7] = −$775,000 + $154,000 (5.5824) = −$775,000 + $859,690 = $84,690 PI = $859,690 / $775,000 = 1.11 Both methods choose the low cost plane. e.
k 2%
NPV PI
High Cost $190,852 1.20
Low Cost $221,688 1.29
4%
NPV PI
$108,020 1.11
$149,323 1.19
6%
NPV PI
$ 34,038 1.04
$ 84,690 1.11
8%
NPV PI
–$ 32,242 .97
$ 26,786 1.03
10%
NPV PI
–$91,823 .90
–$25,263 .97
NPV and PI rank these projects fairly consistently relative to one another (mutually exclusive decision) and in absolute terms (stand-alone decision). f. $200K Low Cost Plane
$100K High Cost Plane
–$100K 2%
4%
6%
8%
10%
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NPV and IRR would not give conflicting results in this case because the NPV profiles do not cross in the first quadrant.
11. Calculate the NPV for the following projects. a. An outflow of $7,000 followed by inflows of $3,000, $2,500 and $3,500 at one-year intervals at a cost of capital of 7% b. An initial outlay of $35,400 followed by inflows of $6,500 for three years and then a single inflow in the fourth year of $18,000 at a cost of capital of 9%. (Recognize the first three inflows as an annuity in your calculations.) c. An initial outlay of $27,500 followed by an inflow of $3,000 followed by five years of inflows of $5,500 at a cost of capital of 10%. (Recognize the last five inflows as an annuity, but notice that it requires a treatment different from the annuity in part b. See Imbedded Annuities, Chapter 6, Page 286.) SOLUTION: a. NPV = −$7,000 + $3,000[PVF7,1] + $2,500[PVF7,2] + $3,500[PVF7,3] NPV = −$7,000 + $3,000(.9346) + $2,500(.8734) + $3,500(.8163) NPV = $844.35 b.
NPV NPV NPV
= −$35,400 + $6,500[PVFA9,3] + $18,000[PVF9,4] = −$35,400 + $6,500(2.5313) + $18,000(.7084) = −$6,195.35
c.
NPV NPV NPV
= −$27,500 + $3,000[PVF10,1] + [PVF10,1]{$5,500[PVFA10,5]} = −$27,500 + $3,000(.9091) + (.9091){$5,500(3.7908)} = −$5,818.51
12. Calculate the IRR for the following projects. a. An initial outflow of $15,220 followed by inflows of $5,000, $6,000, and $6,500. b. An initial outflow of $47,104 followed by inflows of $16,000, $17,000, and 18,000. SOLUTION: NPV = 0 = C0 + C1[PVFIRR,1] + C2[PVFIRR,2] + C3[PVFIRR,3] a.
IRR is approximately 7%: NPV = −$15,220 + $5,000(.9346) + $6,000(.8734) + $6,500(.8163) = $0.65
b.
IRR is approximately 4%: NPV = −$47,104 + $16,000(.9615) + $17,000(.9246) + $18,000(.8890) = $0.20
13. Calculate the NPV at 9% and the IRR for the following projects: a. An initial outlay of $69,724 and an inflow of 15,000 followed by four consecutive inflows of $17,000. b. An initial outlay of $25,424 followed by two zero cash years and then four years of inflows at $10,500. c. An outlay of $10,672 followed by another outlay of $5,000 followed by five inflows of $5,000.
Capital Budgeting
SOLUTION: a. NPV = −$69,724 + $15,000[PVF9,1] + [PVF9,1]{$17,000[PVFA9,4]} = −$69,724 + $15,000(.9174) + (.9174){$17,000(3.2397)} = −$5,437.29 IRR is approximately 6%: NPV = −$69,724 + $15,000(.9434) + (.9434){$17,000(3.4651)} = −$0.42 b. NPV = −$25,424 + [PVF9,2]{$10,500[PVFA9,4]} = −$25,424 + (.8417){$10,500(3.2397)} = $3,207.98 IRR is approximately 12%: NPV = −$25,424 + (.7972){$10,500(3.0373)} = $0.02 c.
NPV
= −$10,672 − $5,000[PVF9,1] + [PVF9,1]{$5,000[PVFA9,5]} = −$10,672 − $5,000(.9174) + (.9174){$5,000(3.8897)} = $2,583.05
IRR is approximately 14%: NPV = −$10,672 − $5,000(.8772) + (.8772){$5,000(3.4331)} = −$0.42 14. Calculate the NPV at 12% and the IRR for the following projects. Find IRR's to the nearest whole percent. a. An initial outflow of $5,000 followed by three inflows of $2,000. b. An initial outflow of $5,000 followed by inflows of $1,000, $2,000, and $3,000. c. An initial outflow of $5,000 followed by inflows of $3,000, $2,000, and $1,000. d. Notice that in parts a, b, and c, a total of $6,000 is received over three years. Compare the NPVs and IRRs to see the impact of shifting $1,000 between years. SOLUTION: NPV = C0 + C [PVFAk,n] a.
NPV
0 PVFAIRR,3 IRR b.
NPV
= −$5,000 + $2,000 [PVFA12,3] = −$5,000 + $2,000 (2.4018) = −$5,000 + $4,803.60 = −$196.40 = −$5,000 + $2,000 [PVFAIRR,3] = 2.5000 = 10% = −$5,000 + $1,000[PVF12,1] + $2,000[PVF12,2] + $3,000[PVFA12,3] = −$5,000 + $1,000(.8929) + $2,000(.7972) + $3,000(.7118) = −$5,000 + $4,622.70 = −$377.30
IRR is approximately 8% (NPV = $22). c.
NPV
= −$5,000 + $3,000[PVF12,1] + $2,000[PVF12,2] + $1,000[PVFA12,3]
245
246
Chapter 10 = −$5,000 + $3,000(.8929) + $2,000(.7972) + $1,000(.7118) = −$5,000 + $4,984.90 = −$15.10 Hence, IRR is approximately 12%. The impact is significant, especially in IRR.
d.
15. Grand Banks Mining Inc. plans a project to strip mine a wilderness area. Setting up operations and initial digging will cost $5 million. The first year's operations are expected to be slow and net a positive cash flow of only $500,000. Then there will be four years of $2 million cash flows after which the ore will run out. Closing the mine and restoring the environment in the sixth year will cost $1 million. a. Calculate the project's NPV at a cost of capital of 12%. b. Calculate the project's IRR to the nearest whole percent. SOLUTION: ($000) = −$5,000 + $500[PVF12,1] + [PVF12,1]{$2,000[PVFA12,4]}− $1,000[PVFA12,6] = −$5,000 + $500(.8929) + (.8929){$2,000(3.0373)} − $1,000(.5066) = $364
a.
NPV
b.
IRR is approximately 15% (NPV = −$32).
Replacement Chain and EAA – Examples 10.8 and 10.9 (pages 478 and 479) 16. Bagel Pantry Inc. is considering two mutually exclusive projects with widely differing lives. The company's cost of capital is 12%. The project cash flows are summarized as follows: C0 C1 C2 C3 C4 C5 C6 C7 C8 C9
Project A ($25,000) $14,742 $14,742 $14,742
Project B ($23,000) $ 6,641 $ 6,641 $ 6,641 $ 6,641 $ 6,641 $ 6,641 $ 6,641 $ 6,641 $ 6,641
a. Compare the projects by using Payback. b. Compare the projects by using NPV. c. Compare the projects by using IRR. d. Compare the projects by using the replacement chain approach. e. Compare the projects by using the EAA method. f. Chose a project and justify your choice. SOLUTION: a. Payback Period A: P/B = $25,000 / $14,742 = 1.7 years B: P/B = $23,000 / $6,641 = 3.5 years Project A is preferred.
Capital Budgeting
b.
247
NPV = C0 + C [PVFAk,n] A: NPV = −$25,000 + $14,742 [PVFA12,3] = −$25,000 + $14,742 (2.4018) = $10,407 B: NPV = −$23,000 + $6,641 [PVFA12,9] = −$23,000 + $6,641 (5.3282) = $12,385 Project B is preferred.
c.
0 = C0 + C [PVFAIRR,n] A: $25,000 PVFAIRR,3 IRR
= $14,742 [PVFAIRR,3] = $25,000 / $14,742 = 1.6958 = 35%
B: $23,000 PVFAIRR,9 IRR
= $6,641 [PVFAIRR,9] = $23,000 / $6,641 = 3.4633 = 25%
Project A is preferred. d. Year 0 1 2 3 4 5 6 7 8 9
Chain project A to nine years. 1 ($25,000) $14,742 $14,742 $14,742
Link 2
($25,000) $14,742 $14,742 $14,742
3
($25,000) $14,742 $14,742 $14,742
Yearly Cash Flow ($25,000) $14,742 $14,742 ($10,258) $14,742 $14,742 ($10,258) $14,742 $14,742 $14,742
PV PVF12,n 1.0000 .8929 .7972 .7118 .6355 .5674 .5066 .4523 .4039 .3606
A chained to B's time horizon is the preferred option based on NPV. e.
A: NPV EAA
= $10,407 = EAA [PVFA12,3] = $10,407 / 2.4018 = $4,333
B: NPV EAA
= $12,385 = EAA [PVFA12,9] = $12,385 / 5.3282 = $2,324
Project A is preferred, because it has the larger EAA.
PV ($25,000) $13,163 $11,752 ($ 7,302) $ 9,369 $ 8,365 ($ 5,197) $ 6,668 $ 5,954 $ 5,316 NPV = $23,088
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f. Project A is preferred on all counts except the original NPV calculation, and that disparity is due to the time horizon problem. Hence A is the best choice.
Calculator Problems The problems in this section should be solved using a financial calculator. See pages 472-474. 17. Callaway Associates, Inc. is considering the following mutually exclusive projects. Callaway’s Cost of capital is 12%. Year Project A Project B 0 ($80,000) ($80,000) 1 $44,000 $65,000 2 $34,000 $30,000 3 $14,000 $ 0 4 $14,000 $ 5,000 a. b.
Calculate each project’s NPV and IRR. Which project should be undertaken? Why?
Solution: a. Set the CF mode on your calculator, enter each project’s cash flows, and solve for NPV and IRR as follows Project A Project B CFo (80000) (80000) C01 44000 65000 F01 1.0 1.0 C02 34000 30000 F02 1.0 1.0 C03 14000 0 F03 2.0 1.0 C04 5000 F04 1.0 Press NPV I 12 12 Results NPV 5,252 5,129 IRR 15.9% 17.1% b. First note that the results of the NPV and IRR methods conflict. Project A should be selected because NPV is the preferred method in the case of such conflicts. 18. Tutak Industries is considering a project requiring an initial investment of $200,000 followed by annual cash inflows of $45,000 for the next six years. A second six-year project has an initial outlay of $325,000. a. How much would the second project have to generate in annual cash flows to have the same IRR as the first? b. If Tutak’s cost of capital is 8%, how much would the second project have to generate in annual cash flows to have the same NPV as the first project. SOLUTION:
Capital Budgeting a.
249
PV = 200,000 N=6 PMT = (45,000) FV = 0 I/Y = ? = 9.3124%, so the IRR of the first project is 9.3124% Use the information from the second project and solve for the PMT using 9.3124% as the IRR. PV = 325,000 N=6 FV = 0 I/Y = 9.3124 PMT = ? = $73,119.84
b.
First calculate the present value of the payment stream in part a at 8%: PMT = 45,000 N=6 I/Y = 8 FV = 0 PV = ? = 208,029.58. Subtract the initial outlay to determine that the project’s NPV is $8,029.58. To have the same NPV the second project’s cash flows would have to have a present value of $325,000 + $8,029.58 = $333,029.58 Use that to solve for the required payment: PV = 333,029.58 N=6 I/Y = 8 FV = 0 PMT = ? = $72,039.42
19.
Provide the missing information for the following projects using the present value of an annuity function [time value of money (TVM) keys rather than the cash flow (CF) function keys]. (Hint: The present value of the annuity of the annual cash flows minus the initial outlay must equal the NPV. For example, for project A calculate the present value of five $35,000 cash inflows and subtract the initial outlay (C0) to get the project’s NPV.) Project
Initial Outlay (C0)
Length (in years)
A B C D E
($100,000) ($200,000) ($300,000) ($400,000) ?
5 4 7 ? 6
Annual Cash Flow $35,000 ? $50,000 $56,098 $75,000
SOLUTION: A. PMT = 35,000 N=5 I/Y = 8 FV = 0 PV = ? = 139,744.85 which make the NPV = $39,744.85
Cost of Capital 8% 13% ? 9% 10%
NPV ? $35,000 $15,000 $20,000 $25,000
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Chapter 10 B.
C.
PV = (235,000) N=4 I/Y = 13 FV = 0 PMT = ? = $79,005.64 PV = (315,000) PMT = 50,000 N=7 FV = 0 I/Y = ? = 2.71%
D.
PV = (420,000) I/Y = 9 PMT = 56,098 FV = 0 N = ? = 13
E.
PMT = 75,000 N=6 I/Y = 10 FV = 0 PV = ? = 326,644.55 which means the Initial outlay was $25,000 less or $301,644.55
20.
Calculate IRRs for the projects in the previous problem.
SOLUTION: At the IRR the present value of the cash inflows is equal to the initial outlay so the TVM keys can be used as follows. Alternately just use the CF mode on the results of the previous problem. A B C D E PV (100,000) (200,000) (300,000) (400,000) (301,644.55) PMT 35,000 79,005.64 50,000 56,098 75,000 N 5 4 7 13 6 FV 0 0 0 0 0 I/Y 22.11% 21.19% 4.01% 9.93% 12.78% 21.
Huron Valley Homes is considering a project requiring a $1 million initial investment. Expected cash inflows will be $25,000 in the first year, $100,000 in the second year, and $200,000 per year for the next six years.
a. b.
Calculate the project’s IRR and the NPV assuming an 8% cost of capital. How much would each of the last six payments have to be to make the project’s NPV $100,000?
SOLUTION: Use the NPV/IRR feature of the calculator rather than the TVM keys. a. CFo = (1,000,000) CF1 = 25,000 CF2 = 100,000
Capital Budgeting
b.
251
CF3 = 200,000 F03 = 6 I=8 NPV = ? = (98,443.13) IRR = 5.74% First, compute the NPV of the initial outlay and the first two unequal cash flows and add the $100,000 required NPV. CFo = (1,000,000) CF1 = 25,000 CF2 = 100,000 I=8 NPV = ? = (891,117.97) + (100,000) = (991,117.97) Now switch to the TVM mode and calculate the future value of that amount two periods out: PV = (991,117.97) N=2 PMT = 0 I/Y = 8 FV = ? = 1,156,040 This amount becomes the PV of the six year annuity which begins in the third year. PV = 1,156,040 N=6 I/Y = 8 FV = 0 PMT = ? = $250,069.24 Alternatively use an iterative approach in the NPV/IRR mode by trying new values for C03 until one produces an NPV that’s close to $100,000. Press CF to reenter the cash flow mode where the inputs from part a are still in place. Press ↓ repeatedly until C03 appears and input a new value. Press enter and then NPV. I=8 will reappear. Press enter, ↓, CPT, and read the new NPV. To try another value press CF again and repeat the procedure. Keep adjusting C03 until NPV = $100,000.
22.
Consider two mutually exclusive projects, A and B. Project A requires an initial cash outlay of $100,000 followed by five years of $30,000 cash inflows. Project B requires an initial cash outlay of $240,000 with cash inflows of $40,000 in the first two years, $80,000 in the next two years and $100,000 in the fifth year.
a. b.
Compute the IRR for each project. Compute the NPV for each project for each of the following costs of capital: 0%, 4%, 8%, 12% and 16%, and record your results in a table. For which costs of capital do the IRR and NPV methods select the same project? Examine the table created in part b and determine the costs of capital between which the methods begin to select different projects? Is your answer consistent with the result of part a. Explain your answer in terms of NPV profiles.
c. d.
SOLUTION: a.
A CFo = (100,0000) CF1 = 30,000 F01 = 5
B CFo = (240,000) CF1 = 40,000 F01 = 2
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IRR = 15.24%
CF2 = 80,000 F02 = 2 CF3 = 100,000 IRR = 10.86%
b.
NPV @ 0% = 50,000 NPV @ 4% = 33,555 NPV @ 8% = 19,781 NPV @ 12% = 8,143 NPV @ 16% = (1,771)
NPV @ 0% = 100,000 NPV @ 4% = 57,141 NPV @ 8% = 21,698 NPV @ 12% = (7,871) NPV @ 16% = (32,744)
c.
The IRR method selects A. The NPV method selects A at 12%. At 16%, both projects have negative NPVs, but A’s is less negative.
d.
The NPV selection shifts when the cost of capital is between 8% and 12%. This implies that the NPV profiles cross in the first quadrant between 8% and 12%. That’s consistent with part a because the crossover must be to the left of both IRRs.
Mutually Exclusive Decisions and Judgment Issues: Example 10.4 (page 463) 23. Kneelson and Botes Inc. (K&B) is a construction company that does road and bridge work for the state highway authority. The state government solicits bids on construction projects from private contractors. The winning contractor is chosen based on its bid price as well as its perceived ability to do the work. Sophisticated contractors develop bids using capital budgeting techniques because most projects require cash outlays for hiring, equipment and materials before getting started (C0). After that the state makes progress payments to cover costs and profits until the job is finished (C1…..Cn). Contractors know that even after they’ve won a bid, realizing the planned profits and cash flows isn’t assured in part because government budgets can change while construction progresses. If funding is up, officials tend to add to the work originally ordered leading to increased profits and cash flows. But if funding is down, officials start to nit pick the contract looking for cost savings, which generally leads to lower cash inflows. State budget projections are fairly good for a year or two, but tend to be inaccurate over longer periods. K&B has been offered two, four year contracts, but doesn’t have enough cash or management depth to take on both (mutually exclusive because of resource limitations). One project involves road repair, most of which will be done and paid quickly. The other requires working on a new bridge. The bulk of the cash inflows on bridge projects generally occur near completion. K&B’s estimating department has put together the following projections of the two projects’ cash flows: ($000) Road Repair Bridge Work C0 ($3,000) ($4,500) C1 3,000 100 C2 2,000 2,000 C3 1,000 3,000 C4 100 4,500 K&B doesn’t know its exact cost of capital, but feels it’s between 10% and 15%. This is not uncommon in smaller companies. (In Chapter 13 we’ll learn that estimating the cost of capital can be difficult and less than precise for firms of any size.) The company has hired you as a financial consultant to make a recommendation as to which project to accept.
Capital Budgeting
253
a. Calculate the payback period for both projects. Which does payback choose? b. Calculate the IRR for both projects. Which does the IRR method choose. Is the choice clear or is it a close decision? Is the choice consistent with the result of the payback method. c. Calculate NPVs for both projects for costs of capital from 10% to 15% in 1% increments. Then plot both projects’ NPV profiles on a graph similar to that shown in Figure 10-2 on page 471. Does the NPV method give a meaningful result. If so is it consistent with the results of the payback and IRR methods. Which method is theoretically the best? Does that help in this situation? d. You must make a recommendation to K&B’s management regardless of any technical difficulties you’ve encountered. Provide another, less quantitative argument that tends to support one project over the other. (Hint: See question 6 on page 483 and Business Analysis 4 on page 484.) e. What is your recommendation and why? SOLUTION: ($000) a. The Road project pays back the initial investment of $3,000K in exactly one year. The Bridge project recovers $2,100K of the initial outlay in the first two years leaving $2,400, which is recovered in ($2,400/$3,000 = .8) 8/10 of the third year. Hence the payback period is 2.8 years The payback method chooses the Road project by a wide margin. b. Enter the project’s cash flows in your calculator in accordance with the instructions in the chapter as follows: CFo C01 F01 C02 F02 C03 F03 C04 F04
Road (3000) 3000 1.0 2000 1.0 1000 1.0 100 1.0
Bridge (4500) 100 1.0 2000 1.0 3000 1.0 4500 1.0
Then for each project press IRR and then compute: IRR
56.9%
27.1%
The Road project has the highest IRR by a wide margin, hence the IRR method clearly chooses the Road project which is consistent with the payback method. c. With a project’s cash flows in your calculator, press NPV. The machine prompts for the cost of capital which it calls I (we’ve called it k). Type in a value and press enter. Then press the ↓ and then compute. The calculator displays the NPV at that value of the cost of capital. Continue with all the values of the cost of capital for both projects. The results are as follows: k 10% 11% 12% 13% 14% 15%
Road $2,200 2,123 2,048 1,976 1905* 1,836*
Bridge $2,571* 2,371* 2,179* 1,994* 1,816 1,645
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* Highest NPV implies NPV choice at that k. NPV Profiles
The NPV Method does not give a meaningful result in this case, because we can’t clearly place the cost of capital (k) on either side of the crossover of the NPV profile. Even if we could, the result wouldn’t be meaningful because the projects’ NPVs are very close in the neighborhood of the crossover which is between 13% and 14%. That means the NPV method is more or less indifferent between the projects at that cost of capital. The NPV result doesn’t support the IRR result but there isn’t really a conflict either, because NPV is essentially neutral in this case. NPV is theoretically the best method, but that doesn’t help us much here since it doesn’t give us a clear answer. d. Another less quantitative argument involves the timing of the two projects’ cash flows. The Bridge is inherently more risky than the Road because its positive cash flows are further in the future and therefore are less likely to be realized. In this case we were told that’s especially problematic because of the state government’s tendency to run out of money. The fact that most managements are risk averse (just like the investors we studied in chapter 9) argues in favor of choosing the Road project. The Road project also requires placing a smaller investment at risk which is generally positive to managements. Along the same lines, the payback method is unusually telling in this case, because it shows that the Road project recovers its outlay almost immediately, so the risk of loss is minimal. e. Recommend the road project because 1. Its IRR and payback are better without conflicting with NPV, 2. It’s less risky, and 3. It gets K&B’s money back faster.
Replacement Chain and EAA: Examples 10.8 and 10.9 (pages 478-479) 24. Haley Motors is considering a maintenance contract for its heavy equipment. One firm has offered Haley a four-year contract for $100,000 to be paid in advance. Another firm has offered an
Capital Budgeting
255
eight year contract for $165,000 also to be paid in advance. Both contracts require full payment in advance. Haley will be able to save $34,000 per year under either contract because its employees will no longer have to do the work themselves. a. If Haley’s cost of capital is 10%, which project should be selected? Use both the replacement chain and the equivalent annual annuity (EAA) method to justify your answer. b. If Haley’s cost of capital is 12%, does it change the decision? What about 14%? SOLUTION: 4-year contract
4-year contract w/ R/C
8-year contract
(100,000) 34,000 4
(100,000) 34,000 3 (66,000) 1 34,000 4
(165,000) 34,000 8
NPV @10% NPV @12% NPV @14%
7,775.42 3,269.87 (933.78)
13,086.15 5,347.94 (1,486.65)
16,387.49 3,899.75 (7,278.62)
EAA @10% PV N I/Y FV PMT (EAA)
7,775.42 4 10 0 2,452.92
CFo CF1 F01 CF2 F02 CF3 F03
16,387.49 8 10 0 3,071.74
a.
At 10%, both the EAA and replacement chain favor the 8-year contract
b.
At 12%, the NPV of the replacement chain is higher on the 4-year contract. At 14%, both of the projects have a negative NPV, so neither should be selected.
25. Cassidy and Sons is reviewing a project with an initial cash outflow of $250,000. An additional $100,000 will have to be invested after the first year, followed by an additional investment of $50,000 at the end of the second year. Beginning at the end of year three, the project is expected to generate cash flows of $90,000 per year for the next eight years. a. Calculate the project’s Payback Period, and IRR, and its NPV and PI at a cost of capital of 8%. b. What concerns might Cassidy have regarding this project beyond the financial calculations in part a? SOLUTION: a. CF Cum
0 (250) (250)
1 (100) (350)
2 (50) (400)
3 90 (310)
Payback Period = 6 + 40/90 = 6.44 years
4 90 (220)
5 90 (130)
6 90 (40)
7 90 50
8 90
9 90
10 90
256
Chapter 10 IRR (based on cash flows shown above) = 10.75% NPV = $57,954
The present value of the three outflows is (385,460). Hence the present value of the positive cash flows is 385,460 + 57,954 so the PI is calculated as follows. PI = (385,460 + 57,954)/(385,460) = 1.15 b. The length of the payback period is an important concern. Projects that don’t pay back for that long a time are very risky simply because of the uncertainty of the future. 26. Zuker Distributors handles the warehousing of perishable foods and is considering replacing one of its primary cold storage units. One supplier has offered a unit for $250,000 with an expected life of 10 years. The unit is projected to reduce electricity costs by $50,000 per year. However, it requires a $20,000 refurbishing every two years, beginning two years after purchase. Another supplier has offered a cold storage unit with similar capabilities for $300,000. It will produce the same savings in electricity costs, but requires refurbishing every five years at a cost of $40,000. Zuker’s cost of capital is 8.5%. Use NPV to determine which cold storage unit Zuker should select. SOLUTION: 2-year refurb.
5-year refurb.
CF0 CF1 CF2 CF3 CF4 CF5 CF6 CF7 CF8 CF9 CF10
(250,000) 50,000 30,000 50,000 30,000 50,000 30,000 50,000 30,000 50,000 30,000
(300,000) 50,000 50,000 50,000 50,000 10,000 50,000 50,000 50,000 50,000 10,000
I = 8.5% NPV
15,129
(16,226)
NPV tells us to select the $250,000 storage unit.
Capital Rationing: (page 481) 27.
Griffin-Kornberg is reviewing the following projects for next year’s capital program. Initial Length Annual Project Investment in years Cash Flow A $3.0 million 6 $ 719,374 B $3.5 million 5 $ 970,934 C $4.0 million 7 $ 904,443 D $5.0 million 4 $1,716,024 E $6.0 million 6 $1,500,919 F $7.0 million 5 $1,941,868
Capital Budgeting G
$8.0 million
7
257
$1,725,240
Projects A and B are mutually exclusive and so are Projects D and E. Griffin-Kornberg has a 9% cost of capital and a maximum of $14 million to spend on capital projects next year. Use capital rationing to determine which projects should be included in Griffin-Kornberg’s capital program. SOLUTION: (000) At the IRR the present values of the annuities of cash inflows are just equal to the initial outlays so we can find the projects’ IRRs by solving time value of annuity problems for the interest rates as follows. A B C D E PV (3,000) (3,500) (4,000) (5,000) (6,000) PMT 719.374 970.934 904.443 1716.024 1500.919 N 6 5 7 4 6 FV 0 0 0 0 0 I/Y 11.5 12.0 13.0 14.0 13.0 X X A and B are mutually exclusive, so B should be selected. D and E are mutually exclusive, so D should be selected. X below A and E indicate they are eliminated. In order of IRR the projects are listed as follows Project
IRR
C0
D C F B G
14 13 12 12 11.5
$5000 4000 7000 3500 8000
F (7,000) 1941.868 5 0 12.0
G (8,000) 1725.240 7 0 11.5
D should be first; C should be next, which would leave $5 million to invest. Both B and F have a 12% return, but F is too large to be funded, so projects D, C and B should be included in the capital program leaving $1.5 million unused. COMPUTER PROBLEMS: DEVELOPING SOFTWARE 28. Write a spreadsheet program to calculate the NPV of a project with an irregular pattern of cash flows for up to 10 periods without using the spreadsheet software's NPV function. Essentially, the task is to program Equation 10.1a with n = 10. First, input the interest rate (k) in a single cell. Next, set up three horizontal rows of 11 cells (including C0). The top row will receive the cash flows as inputs. Program the present value factor for each period into the second row of cells using the interest rate you input earlier as follows. Period
0
1
2. . .
. . .10
258 Factor
Chapter 10 1
1 1 +k
1 (1 + k )
2
...
...
1 (1 + k )10
Note that we're calling the interest rate k, but it will appear as a cell name in your program. Next, form the third row by multiplying the top two cells in each column together. This makes the third row the present value of each cash flow. Finally, sum the values along the third row in another cell to form the project's NPV. Notice that your program will handle a project of less than ten periods if you simply input zero (or leave blank) the cash flow cells from n+1 to ten. Also notice that you can easily extend your program to any reasonable number of periods by extending the horizontal rows and the programming logic. Test your program on the data in Example 10.4 to make sure it works correctly. SOLUTION: A sample solution is in Instructor's Resources. 29. The Tallahassee Motor Company is thinking of automating one of its production facilities. The equipment required will cost a total of $10.0 million and is expected to last 10 years. The company's cost of capital is 9%. The project's benefits include labor savings, and a quality improvement that will lower warranty costs. Savings are estimated as follows. Year 1 2 3 4 5 6 7 8 9 10
Cost Savings ($000) $ 574 864 1,246 2,748 3,367 2,437 2,276 1,839 1,264 623
a. Use the program developed in the previous problem to find the project's NPV. Is the project acceptable? b. Use the program to develop the data for an NPV profile. Evaluate the NPV for interest rates (costs of capital) from 6 to 14 percent. c. Use the program to iteratively find the project's IRR to one tenth of a percent. SOLUTION: A sample solution is on the Instructor's Resource CD-ROM.
