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CHAPTER 12

RISK TOPICS AND REAL OPTIONS IN CAPITAL BUDGETING

FOCUS

Traditional capital budgeting techniques compute point estimates of NPV and IRR with nomeasure of variability. Hence they dont give managers the information necessary to include atradeoff between risk and expected return in their decisions. This chapter is concerned with modernapproaches to incorporating risk into capital budgeting. The techniques considered includeprobabilistic cash flows, risk adjusted discount rates and the idea of real options.

PEDAGOGY

We begin with the idea that cash flows are random variables, which implies that project NPVsand IRRs are also random variables with associated probability distributions. We then explore theimplications of choosing a high risk project over one with less variability, and conclude thatmanagements would often trade higher return for lower risk if they had the necessary information.With that background we explore the currently available methods for incorporating risk into capital

budgeting calculations including a detailed explanation of real options thinking.

TEACHING OBJECTIVES

Students should gain an appreciation of risk in the capital budgeting context, and be relativelywell-versed in the approaches scholars have taken to incorporating it into the decision making process.At the same time they should understand that putting risk into capital budgeting is difficult, and thatthe methods currently available are often less than completely satisfactory.

OUTLINE

I. RISK IN CAPITAL BUDGETING GENERAL CONSIDERATIONSA. Cash Flows as Random Variables

Incorporating risk by viewing individual cash flows as random variables with probabilitydistributions.NPV and IRR are then also random variables.

B. The Importance of Risk in Capital BudgetingWhy risk matters, making mistakes on individual projects and changing the risk character ofthe company.

C. Incorporating Risk Into Capital Budgeting Numerical and Computer MethodsScenario/Sensitivity Analysis and SimulationDecision Tree Analysis

D. Real OptionsThe concept of an option. Real options thinking applied to capital budgeting.Valuing real options the NPV and risk effects.Designing real options into projects, types of real options.

E. Incorporating Risk Into Capital Budgeting The Theoretical Approach - Risk Adjusted Ratesof ReturnThe concept of adjusting for risk by raising the threshold (hurdle) rate thus making riskyprojects less acceptable.

F. Estimating Risk Adjusted Rates Using CAPMUsing the SML to estimate appropriate risk adjusted rates for project analysis.The kind of risk CAPM estimates and whether it is appropriate for capital budgetingdecisions.

G. Problems with the Theoretical Approach

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Finding the right beta, concerns about the appropriate risk definition.QUESTIONS

1. In 1983 the Bell Telephone System, which operated as AT&T, was broken up, resulting in thecreation of seven regional telephone companies. AT&T stockholders received shares of the newcompanies and the continuing AT&T, which handled long distance services. Prior to the breakup,

telephone service was a regulated public utility. That meant AT&T had a monopoly on the sale of itsservice, but couldnt charge excessive prices due to government regulation. Regulated utilities areclassic examples of low risk modest return companies. After the breakup, the "Baby Bells," as theywere called, were freed from many of the regulatory constraints under which the Bell Systemoperated, and at the same time had a great deal of money. The managements of these young giantswere determined to be more than the staid old-line telephone companies they'd been in the past. Theywere quite vocal in declaring their intentions to undertake ventures in any number of new fields,despite the fact that virtually all of their experience was in the regulated environment of the oldtelephone system. Many stockholders were alarmed and concerned by these statements. Comment onwhat their concerns may have been.

ANSWER: The stockholders probably invested in phone company stocks at least partially because ofthe stability and low risk of regulated public utilities. If the Baby Bells expanded into riskyunregulated businesses in which they had little experience, the nature of the companies could shifttoward higher risk. This would clearly be upsetting to people who had invested for stability in the firstplace.

2. A random variable is defined as the outcome of one or more chance processes. Imagine thatyou're forecasting the cash flows associated with a new business venture. List some of the things thatcome together to produce cash flows in future periods. Describe how they might be considered to beoutcomes of chance processes and therefore random variables. Cash flow forecasts for a project areput together by using Equations (10-1) and (10-2) to calculate the project's NPV and IRR. That makesNPV and IRR random variables as well. Is their variability likely to be greater or less than thevariability of the individual cash flows making them up?

ANSWER:

Revenues depend on The effectiveness of advertising and promotion

Customer response to product and promotion activity Competitive responses to the venture

The effectiveness of distribution channels Product quality

Costs depend on The price and availability of labor and material

Overhead costs The effectiveness of production operations

Each of these items is subject to the influence of a variety of physical, administrative and economicforces that can each be positive or negative. The result of such a confluence of different influencestends to vary randomly around some central value. This kind of pattern represents the essence of arandom variable.

When a random variable is the result of a combination of other random variables, its variabilitytends to be less than that of the constituent pieces. This is because there is usually some offsetting ofthe pluses and minuses among the pieces coming together in a typical observation.

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Risk Topics and Real Options in Capital Budgeting

3. One of the problems of using simulation to incorporate risk into capital budgeting is related tothe idea that the probability distributions of successive cash flows usually are not independent. If thefirst period's cash flow is at the high end of its range, for example, flows in subsequent periods aremore likely to be high than low. Why do you think this is generally the case? Describe an approachthrough which the computer might adjust for this phenomenon to portray risk better.

ANSWER: Suppose a project's cash flows depend, among other things, on customer acceptance of aproduct. Before any marketing is done, we project acceptance in each period as a random variable thatdefines the volume of product sold.

However, acceptance really isn't independent from one period to the next. For most products it'seither good or bad throughout the project's entire life. That means choosing an independentobservation in successive periods isn't a very realistic way to model the process. For example, it'sunlikely that a very good response in one period will be followed by a very poor response in the next.If acceptance is good in the first period, it's likely to be pretty good in the second as well. But treatingacceptance as independent from period to period is likely to result in simulation runs in which it variesback and fourth from great to terrible any number of times. That's not realistic.

A conceptual way around the problem is to define an initial random variable for acceptance thatsets the level of the mean values of the periodic acceptance random variables throughout the project'slife. This r.v. would be drawn only once per simulation run and would define acceptance as good orbad for the whole run. The program would raise the means of all the acceptance r.v.s if a goodobservation is drawn, and lower them if a poor one comes up.

4. Why is it desirable to construct capital budgeting rules so that higher risk projects become lessacceptable than lower risk projects?

ANSWER: A high risk project has a significant probability of turning out worse than expected.Requiring a higher expected outcome for acceptance for such a project puts a cushion between thefirm and the consequences of a miss in predicting the project's outcome. This makes failure or lossless likely.

5. Rationalize the appropriateness of using the cost of capital to analyze normally risky projectsand higher rates for those with more risk.

ANSWER: Risky projects are less desirable than low risk projects with the same expected return,because there is more chance that a significantly unfavorable outcome will occur. Hence, it's desirableto discriminate against such projects in the capital budgeting process. Using a risk-adjusted rateinstead of the cost of capital when project risk is high is a mechanical method of making risky projectsless acceptable. The method simply requires that a risky project have a higher expected return toqualify than a lower risk project.

The cost of capital reflects the current state of the business, and therefore can be associated withnormally risky projects. Higher rates are logically appropriate for higher risk projects.

6. Evaluate the conceptual merits of applying CAPM theory to the problem of determining riskadjusted interest rates for capital budgeting purposes. Form your own opinion based on your study ofCAPM (Chapter 9) and the knowledge you're now developing of capital budgeting. The issue isconcisely summarized by Figure 12.7. Is the special concept of risk developed in portfolio theoryapplicable here? Don't be intimidated into thinking that because the idea is presented in textbooks, it'snecessarily correct. Many scholars and practitioners feel this application stretches theory too far.On the other hand, others feel it has a great deal of merit. What do you think and why?

ANSWER: This is an opinion question for which there isn't a right or wrong answer.

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BUSINESS ANALYSIS

1. Ed Draycutt is the engineering manager of Airway Technologies, a firm that makes computersystems for air traffic control installations at airports. He has proposed a new device the success of

which depends on two separate events. First the Federal Aviation Administration (FAA) must adopt arecent proposal for a new procedural approach to handling in flight calls from planes experiencingemergencies. Everyone thinks the probability of the FAA accepting the new method is at least 98%,but it will take a year to happen. If the new approach is adopted, radio makers will have to respondwithin another year with one of two possible changes in their technology. These can simply be calledA and B. The A response is far more likely, also having a probability of about 98%. Eds deviceworks with the A system and is a stroke of engineering genius. If the A system becomes the industrystandard and Airway has Eds product, it will make a fortune before anyone else can market a similardevice. On the other hand if the A system isnt adopted, Airway will lose whatever its put into thenew devices development.Developing Eds device will cost about $20 million, which is a very substantial investment for a smallcompany like Airway. In fact, a loss of $20 million would put the firm in danger of failing.

Ed just presented his idea to the executive committee as a capital budgeting project with a $20million investment and a huge NPV and IRR reflecting the adoption of the A system. Everyone on thecommittee is very excited. Youre the CFO and are a lot less excited. You asked Ed how he reflectedthe admittedly remote possibility that the A system would never be put in place. Ed, obviously proudof his business sophistication, said hed taken care of that with a statistical calculation. He saidadoption of the A system required the occurrence of two events each of which has a 98% probability.The probability of both happening is (.98x.98=.96) 96%. He therefore reduced all of his cash inflowestimates by 4%. He maintains this correctly accounts for risk in the project.

Does Ed have the right expected NPV? Whats wrong with his analysis? Suggest anapproach that will give a more insightful result. Why might the firm consider passing on the proposalin spite of the tremendous NPV and IRR Ed has calculated?

ANSWER:

Eds calculation has given him the correct expected NPV but its masked the real risk in theproposal. The problem that isnt shown in his presentation is that theres a 4% probability that Airwaywill lose $20M on the project and probably be put out of business. That the project involves asubstantial risk of ruin should be made explicit in any presentation to management.

2. Might Eds case in the preceding problem be helped by a real option? If so What kind? Howwould it help?

ANSWER:

Yes, an abandonment option would help if a substantial portion of the development cost couldbe avoided or recovered if, during the first year, it starts to look like the A system isnt going to beinstalled. This could reduce the magnitude of the loss but wouldnt affect its probability.

3. Charlie Henderson, a senior manager in the Bartok Company, is known for taking risks. Herecently proposed that the company expand its operations into a new and untried field. He puttogether a set of cash flow projections and calculated an IRR of 25% for the project. The firm's cost ofcapital is about 10%. Charlie maintains that the favorability of the calculated IRR relative to the costof capital makes the project an easy choice for acceptance, and urges management to move forwardimmediately.

Several knowledgeable people have looked at the proposal and feel Charlie's projections representan optimistic scenario that has about one chance in three of happening. They think the project also has

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about one chance in three of failing miserably. An important consideration is that the project is largeenough to bankrupt the company if it fails really badly.

Charlie doesn't want to talk about these issues, claiming the others are being "negative" and that hehas a history of success with risky ventures like this. When challenged, he falls back on the 25% IRRversus the 10% cost of capital as justification for his idea.

The company president has asked you for your comments on the situation. Specifically address the

issue of the 25% IRR versus the 10% cost of capital. Should this project be evaluated by usingdifferent standards? How does the possibility of bankruptcy as a result of the project affect theanalysis? Are capital budgeting rules still appropriate? How should Charlie's successful record befactored into the president's thinking?

ANSWER: This project is indeed high risk since it involves as much as a one third chance of ruiningthe firm. For that reason measuring the IRR against the cost of capital is meaningless. It should bemeasured against a risk-adjusted rate that is much higher. Intuitively, it's likely that such a rate wouldbe a good deal more than 25%, and would therefore show the project to be unacceptable.

Charlie is committing a rather serious management theory error. He's using part of a theory(traditional capital budgeting) to support a position that the full theory (including risk adjustments)would probably reject. That kind of thinking is extremely dangerous.

Many people would argue that capital budgeting isn't appropriate in a case like this because of thesignificant probability of bankruptcy involved. When the risk of ruin is substantial, risk-averse peopleoften walk away from projects regardless of high expected returns. Stated another way, the project isa little like playing Russian Roulette, and not many people will do that.

Charlie's track record might induce the president to take his advice, but it would be hard to describethat action as anything but a high stakes gamble.

4. In evaluating the situation presented in the last problem, you've found a pure play company inthe proposed industry whose beta is 2.5. The rate of return on short-term treasury bills is currently 8%and a typical stock investment returns 14%. Explain how this information might affect theacceptability of Charlie's proposal. What practical concerns would you overlay on top of the theoryyou've just described? Do they make the project more or less acceptable? Does the fact that Bartokhas never done this kind of business before matter? How would you adjust for that inexperience? Isthe risk of bankruptcy still important? What would you advise doing about that? All thingsconsidered, would you advise the president to take on the project or not?

ANSWER: The information lets us use the CAPM to calculate a more appropriate risk adjusted ratewith which to evaluate the project. The calculation uses the SML as follows.

k = kRF + (kM kRF)b

= 8% + (14% 8%) 2.5

= 23%

The 25% IRR compares favorably to this risk adjusted rate, but not by a great deal. The result is

especially marginal considering that the cash flows in Charlie's analysis are probably subjectiveestimates and likely to be upwardly biased by his enthusiasm.The fact that Bartok has never done this kind of business before argues that the new venture will

probably be even riskier than the pure play company. That says the 23% risk adjusted rate is probablytoo low and should be raised.

It's also important to note that the CAPM approach adjusts only for market risk. In this situation,total risk is probably more appropriate. Intuitively, it seems that considering total risk would raise therate considerably and be likely to make the project unacceptable.

The high risk of ruin is still important. Many, if not most, people would find it a show stopper, andadvise against the project regardless of the IRR.

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PROBLEMS

Scenario/Sensitivity Analysis: Example 12-1 (page 528)1. The Glendale Corp. is considering a real estate development project that will cost $5M to

undertake and is expected to produce annual inflows between $1M and $4M for two years.Management feels that if the project turns out really well the inflows will be $3M in the first year and$4M in the second. If things go very poorly, on the other hand, inflows of $1M followed by $2.5Mare more likely. Develop a range of NPVs for the project if Glendales cost of capital is 12%.

SOLUTION:

The NPV for either scenario is

NPV = C0 + C1[PVF12,1] + C2[PVF12,2] = -$5M + C1(.8929) + C2(.7972)

Then for the good scenario we have

NPV =

$5,000,000 + $3M(.8929) + $4M(.7972)= $5,000,000 + $2,678,700 + $3,188,800

= $867,500

While for the unfavorable scenario we have

NPV = $5,000,000 + $1,000,000(.8929) + $2,500,000(.7972)

= $5,000,000 + $892,900 + $1,993,000

= $2,114,100

Hence the likely NPV range for the project is between $0.9M and $2.1M.

2. If Glendales management in the last problem attaches a probability of .7 to the betteroutcome, what is the projects most likely (expected) NPV?

SOLUTION:

Scenario Outcome Probability Outcome x Prob.Good $ 867,500 .7 $ 607,250Bad ($ 2,114,100) .3 ($ 643,230)

Expected NPV = ($ 26,980)

3. Keener Clothiers Inc. is considering investing $2 million in an automatic sewing machine toproduce a newly designed line of dresses. The dresses will be priced at $200, and

management expects to sell 12,000 per year for six years. There is, however, someuncertainty about production costs associated with the new machine. The productiondepartment has estimated operating costs at 70% of revenues, but senior management realizesthat this figure could turn out to be as low as 65% or as high as 75%. The new machine willbe depreciated at a rate of $200,000 per year for six years (straight line, zero salvage).Keeners cost of capital is 14% and its marginal tax rate is 35%. Calculate a point estimatealong with best and worst case scenarios for the projects NPV.

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SOLUTION: ($000)Cost as a % of Revenue

Cash Flow in Years 1-5 @65% @70% @75%Revenues $2,400 $2,400 $2,400

Operating Costs 1,560 1,680 1,800Depreciation 200 200 200EBT Impact 640 520 400Tax 224 182 140EAT Impact 416 338 260Add back depreciation. 200 200 200Cash Flow 616 538 460

Calculating NPVs using a financial calculator yields:Best Point Est Worst

CFo (2,000) (2,000) (2,000)C01 616 538 460F01 6 6 6I 14 14 14NPV 395.419 92.103 (211.213)

4. Assume Keener Clothiers of the last problem assigns the following probabilities to productioncost as a percent of revenue

% of Revenue Probability65% .3070% .5075% .20

Sketch a probability distribution (histogram) for the projects NPV, and compute its expectedNPV.

SOLUTION:

Prob

.5

.3

.2

NPV($211) 0 $92 $395

The most likely, or expected, NPV is calculated as follows:Expected NPV = $395,419(.30) + $92,103(.5) - $211,213(.2) = $

= $118,625.70 + $46,051.50 - $42,242.60= $122,434.60

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Notice that because the probabilities of the 65% and 75% scenarios are not equal, the expected NPV isnot that of the 70% scenario.

5. The Blazingame Corporation is considering a three-year project that has an initial cashoutflow (C0) of $175,000 and three cash inflows that are defined by the independent probabilitydistributions shown below. All dollar figures are in thousands. Blazingame's cost of capital is 10%.

C1 C2 C3 Probability$50 $40 $75 .25$60 $80 $80 .50$70 $120 $85 .25

a. Estimate the project's most likely NPV by using a point estimate of each cash flow. What is itsprobability?b. What are the best and worst possible NPVs? What are their probabilities?c. Choose a few outcomes at random, calculate their NPVs and the associated probabilities, and

sketch the probability distribution of the project's NPV.

[Hint:The project has 27 possible cash flow patterns (333) each of which is obtained by

selecting one cash flow from each column and combining with the initial outflow. The probability ofany pattern is the product of the probabilities of its three uncertain cash flows. For example, aparticular pattern might be as follows.

C0 C1 C2 C3CI ($175) $50 $120 $80Probability 1.0 .25 .25 .50

The probability of this pattern would be .25 .25 .50 = .03125.]

SOLUTION: Although the problem asks for only a few outcomes, we'll list them all and identify thebest, worst, and most likely. First restate the matrix of outcomes by multiplying each Ci by PVF10,i and

rounding to the nearest $1,000:

C1 C2 C3 Probability$45 $33 $56 .25$55 $66 $60 .50$64 $99 $64 .25

Next enumerate the possible cash flows and calculate their probabilities.

($000)

C0 C1 C2 C3 NPV Probability

175 45 33 56 41 .25.25.25 = .015625

Worst 60 37 .25.25.50 = .031250

64 33 .25.25.25 = .015625

175 45 66 56 8 .25.50.25 = .031250

60 4 .25.50.50 = .062500

64 0 .25.50.25 = .031250

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175 45 99 56 25 .25.25.25 = .015625

60 29 .25.25.50 = .031250

64 33 .25.25.25 = .015625

175 55 33 56 31 .50.25.25 = .031250

60 27 .50.25.50 = .06250064 23 .50.25.25 = .031250

175 55 66 56 2 .50.50.25 = .062500

Most likely 60 6 .50.50.50 = .125000

64 10 .50.50.25 = .062500

175 55 99 56 35 .50.25.25 = .031250

60 39 .50.25.50 = .062500

64 43 .50.25.25 = .031250

175 64 33 56 22 .25.25.25 = .01562560 18 .25.25.50 = .031250

64 14 .25.25.25 = .015625

175 64 66 56 11 .25.50.25 = .031250

60 15 .25.50.50 = .062500

64 19 .25.50.25 = .031250

175 64 99 56 44 .25.25.25 = .015625

60 48 .25.25.50 = .031250

Best 64 52 .25.25.25 = .0156251.000000

Finally, sorting the outcomes and grouping within NPV ranges yields the following probabilitydistribution.

NPV Range ($000) Probability

NPV < $40 .015625

$40 < NPV < $30 .078125

$30 < NPV < $20 .109375

$20 < NPV < $10 .046875

$10 < NPV < $00 .125000 $00 < NPV < $10 .250000

$10 < NPV < $20 .125000$20 < NPV < $30 .046875$30 < NPV < $40 .109375$40 < NPV < $50 .078125

NPV > $50 .0156251.000000

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Prob(NPV)

.25

.20

.15

.10

.05

50Mid Points of NPV Ranges ($000)

6. Sanville Quarries is considering acquiring a new drilling machine which is expected to bemore efficient than their current machine. The project is to be evaluated over four years. The initialoutlay required to get the new machine operating is $675,000. Incremental cash flows associated withthe machine are uncertain, so management developed the following probabilistic forecast of cashflows by year ($000). Sanvilles cost of capital is 10%.

Year1 Prob Year2 Prob Year3 Prob Year4 Prob$150 .30 $200 .35 $350 .30 $300 .25

$175 .40 $210 .45 $370 .25 $360 .35$300 .30 $250 .20 $400 .45 $375 .40

a. Calculate the projects best and worst NPVs and their probabilities.b. What is the value of the most likely NPV outcome?

SOLUTION: ($000)Best Worst Most Likely

CFo (675) (675) (675)C01 300 150 205C02 250 200 214.5C03 400 350 377.5C04 375 300 351I 10 10 10NPV 360.995 94.517 211.995

Notice that calculating the most likely outcome requires taking the mean of each cash flowdistribution.

Probabilities:Best: (.3)(.2)(.45)(.4) = .01080Worst: (.3)(.35)(.3)(.25) = .00788

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7. Using the information from the previous problem, randomly select four NPV outcomes fromthe data. (Select one cash flow from each year and compute the project NPV and the probability ofthat NPV implied by those selections.) Do your selections give a sense of where NPV outcomes arelikely to cluster?

SOLUTION:The solutions should fall between the best and worst possible outcomes of $360,995 and$94,517, and should tend to cluster around the most likely solution of $211,995.

Decision Trees: Example 12-2 (page 532)8. Northwest Entertainment Inc. operates a multiplex cinema that has nine small theaters in onebuilding. Business has been good lately and management is considering a project that will add fivescreens at an estimated cost of $3 million. The success of the expansion depends on whether localdemand over the next two years will support the additional capacity. Demand is believed to depend onthe local economy. An economist at a nearby university has predicted a 90% probability of continuedprosperity in the area and a 10% chance of a moderate downturn. Management feels that if prosperity

continues the new theaters will generate a profit margin of $2 million in the first year and $3 million inthe second. A moderate downturn would produce contributions of $1.5 and $2 million. Northwestscost of capital is 12%.

a. Draw a decision tree for the project.b. Calculate the NPV along each path.c. Develop the probability distribution of the projects NPV.d. Calculate the projects expected NPV.e. Make a recommendation on the project with an appropriate comment on risk.

SOLUTION: ($000)

a. Path.9 $2,000 $3,000 1

$3,000

.4 $1,000 $1,500 2

b. The NPV along each of the projects paths is:Path 1

NPV = -3000 + 2000(PVF12,1) + 3000(PVF12,2)= -3000 + 2000(.8929) + 3000(.7972)= -3000 + 1785.8 + 2391.6

= 1177.4Path 2NPV = -3000 + 1000(PVF12,1) + 1500(PVF12,2)

= -3000 + 1000(.8929) + 1500(.7972)= -3000 + 892.9 + 1195.8= -911.3

c. The projects NPV probability distribution isNPV Probability

($911,300) .1

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$1,177,400 .9

d. The expected NPV is the sum of the products of each paths NPV and probability.Expected NPV = $1,177,400 (.9) - $911,300 (.1)

= $1,059,660 $91,130= $968,0530

e. Recommendation: The project has a positive expected NPV, which is substantial relative to the sizeof the initial investment. This argues strongly for acceptance. There is substantial risk, however,since there is a modest chance of a significant loss. Most managers would probably accept this projectunless the underlying business is so weak that the loss would seriously damage it.

More Complex Decision Trees: Examples 12-2 and 12-3 (pages 532 and 534)9. Work Station Inc. manufactures office furniture. The firm is interested in ergonomicproducts that are designed to be easier on the bodies of office workers who suffer from aliments suchas back and neck pain due to sitting for long periods. Unfortunately customer acceptance ofergonomic furniture tends to unpredictable, so a wide range of market response is possible.

Management has made the following two-year, probabilistic estimate of the cash flows associated withthe project arranged decision tree format ($000).

Path

$7,000 1.3

$4,000.7

.6 $5,000 2

$6,000

.4 $3,000 3.8

$2,000.2

$2,400 4

Work Station is a relatively small company, and would be seriously damaged by any project that lostmore than $1.5 million. The firms cost of capital is 14%.

a. Develop a probability distribution for NPV based on the forecast. I.e., calculate theprojects NPV along each path of the decision tree and the associated probability.

b. Calculate the projects expected NPV.c. Analyze your results and make a recommendation about the projects advisabilityconsidering both expected NPV and risk

SOLUTION:

a. The NPV along each of the projects four paths and the probability of each of those outcomes iscalculated as follows:

Path 1

NPV = -6000 + 4000(PVF14,1) + 7000(PVF14,2)

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= -6000 + 4000(.8772) + 7000(.7695)= -6000 + 3508.8 + 5386.5

= 2895.3 Probability = .6 .3 = .18

Path 2 NPV = -6000 + 4000(.8772) + 5000(.7695)= -6000 + 3508.8 + 3847.5

= 1356.3 Probability = .6 .7 = .42

Path 3

NPV = -6000 + 2000(.8772) + 3000(.7695)= -6000 + 1754.4 + 2308.5

= -1937.1 Probability = .4 .8 = .32

Path 4

NPV = -6000 +1754.4 + 2400(.7695)= -6000 + 1754.4 + 1846.8

= -2398.8 Probability = .4 .2 = .081.00

b. The expected NPV for the entire project is the sum of the products of each paths NPV andprobability.

Expected NPV = 2895.3(.18) + 1356.3(.42) 1937.1(.32) 2398.8(.08)

= 521.2 + 569.6 619.9 191.9= 279.0

c. Recommendation: The project has a positive expected NPV, which is quite small relative to the sizeof the initial investment. This argues weakly for acceptance. However, risk considerations tell

another story. Two paths with probabilities totaling 40% have ruinously negative NPVs. That meansaccepting the project has as much as a 40% probability of seriously damaging or sinking the company.Most rational managers would forego such a project in spite of the positive overall expected NPV.

Abandonment Options: Example 12-5 (page 539)10. Resolve the last problem assuming Work Station Inc has an abandonment option at the end of thefirst year under which it will recover $5 million of the initial investment in year 2. What is the valueof the ability to abandon the project? How does your overall recommendation change?

SOLUTION:

The ability to abandon the project changes the decision tree as follows:

Path

$7,000 1.3

$4,000.7

.6 $5,000 2

$6,000

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.4$2,000 $5,000 3

Paths 3 and 4 condense into a new path 3 with the following NPV and probability.

Path 3 NPV = 6,000 + 2000(.8772) + 5000(.7695)

= 6000 + 1754.4 + 3847.5

= 398.1 Probability = .40

The projects expected NPV is then this outcome combined with those of original paths 1 and 2.(Notice that the probabilities of original paths 1 and 2 and the new path 3 sum to 1.0)

Expected NPV = 2895.3(.18) + 1356.3(.42) 398.1(.4)= 521.2 + 569.6 159.2= 931.6

Compare this probability distribution and expected NPV with those of the last problem. Theexpected NPV is considerably higher here, it has risen from 279.0 to 931.6, an increase of $652.6.This difference is the least the abandonment option would be worth if Work Station undertakes theproject.

Perhaps more important, however, is the fact that the projects risk characteristics havechanged dramatically. The ability to abandon and recover most of the initial investment haseliminated the potentially ruinous outcomes that were major concerns before. These outcomes havebeen changed to a 40% probability of a modest loss, which is likely to be acceptable.

11. Vaughn Clothing is considering refurbishing its store at a cost of $1.4 million. Management isconcerned about the economy and whether a competitor, Viola Apparel, will open a store in the

neighborhood. Vaughn estimates that there is a 60% chance that Viola will open a store nearby nextyear. The state of the economy probably wont affect Vaughn until the second year of the plan.Management thinks there is a 40% chance of a strong economy and a 60% chance of a downturn in thesecond year. Incremental cash flows are as follows:

Year 1:Viola opens a store - $700,000Viola doesnt open a store - $900,000Year 2:Viola opens a store, strong economy - $850,000Viola opens a store, weak economy - $700,000Viola doesnt open a store, strong economy - $1,500,000

Viola doesnt open a store, weak economy - $1,200,000

Perform a decision tree analysis of the refurbishment project. Draw the decision tree diagramand calculate the probabilities and NPVs along each of its four paths. Then calculate theoverall expected NPV. Assume Vaughns cost of capital is 10%.

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SOLUTION:

A decision tree for Vaughn refurbishment project is as follows ($000):Path

$850 1.4

$700 .6.6 $700 2

$1,400

.4 $1,500 3.4

$900.6

$1,200 4

Path 1

NPV = -1400 + 700(PVF10,1) + 850(PVF10,2)= -1400 + 700(.9091) + 850(.8264)= -1400 + 636.37 + 702.44

= -61.19 Probability = .6 .4 = .24

Path 2

NPV = -1400 + 636.37 + 700(.8264)= -1400 + 636.37 + 578.48

= -185.15 Probability = .6 .6 = .36

Path 3NPV = -1400 + 900(.9091) + 1500(.8264)

= -1400 + 818.19 + 1239.60

= 657.79 Probability = .4 .4 = .16

Path 4

NPV = -1400 +818.19 + 1200(.8264)= -1400 + 818.19 + 991.68

= 409.87 Probability = .4 .6 = .24

1.00

The expected NPV for the entire project is the sum of the products of each paths NPV and

probability.

Expected NPV = -61.19(.24) - 185.15(.36) + 657.79(.16) + 409.87(.24)= -14.69 - 66.65 + 105.25 + 98.37= 122.58

Real Options: Example 12-4 (page 537)12. Vaughn Clothing of the previous problem has a real option possibility. Carlson Flooring has

expressed an interest in trading buildings with Vaughn after Vaughns is refurbished. Carlson

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has offered to reimburse Vaughn for 70% of its refurbishment costs at the end of the first yearif they make the trade. Vaughn would then forego all incremental cash flows for the secondyear. Carlson is willing to keep the option open for one year in return for a non-refundablepayment of $150,000 now. Should Vaughn pay the $150,000 to keep the option available?.

SOLUTION:

Vaughn would exercise Carlsons option if Viola opened a new store and thus avoidcompleting paths 1 or 2 in the previous problem. The recovery would be 70% of the initialoutlay or ($000)

$1,400 x .70 = $980.The decision tree then becomes:

Path

$1680 $0 1

.6

$1,550

.4 $1,500 2.4

$900.6

$1,200 3

Path 1

NPV = -1,550 + 1,680(PVF10,1)

= -1,550 + 1,680(.9091)= -1,550 + 1,527.29= -22.71 Probability = .6

Path 2

NPV = -1550 + 900(.9091) + 1500(.8264)= -1550 + 818.19 + 1239.60

= 507.79 Probability = .4 .4 = .16

Path 3

NPV = -1550 +818.19 + 1200(.8264)= -1550 + 818.19 + 991.68

= 259.87 Probability = .4

.6 = .241.00

The expected NPV for the project including the real option is:

Expected NPV = -22.71(.6) + 507.79(.16) + 259.87(.24)= -13.63 + 81.25 + 62.37= 129.99

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The value of the real option is this figure less the projects original NPV.Value of Real Option = $129.99 - $122.58 = $7.71

Hence technically Vaughn should spend the money to keep the option open since it adds to theprojects NPV. The decision is marginal, however, because the incremental value added is small.

13. Spitfire Aviation Inc. manufactures small, private aircraft. Management is evaluating aproposal to introduce a new high performance plane. High performance aviation is an expensive sportundertaken largely by people who are both young and wealthy. Spitfire sees its target market asaffluent professionals under 35, who have made a lot of money in the stock market in recent years.

Stock prices have been rising rapidly for some time, so investment profits have been veryhandsome, but lately there are serious concerns about a market downturn. If the market remainsstrong, Spitfire estimates it will sell 50 of the new planes per year for five years, each of which willresult in a net cash flow contribution of $200,000. If the market turns down, however, only about 20units a year will be sold. Economists think theres about a 40% chance the market will turn down inthe near future.

There are also some concerns about the design of the new plane. Not everyone is convinced itwill perform as well as the engineering department thinks. Indeed, the engineers have sometimes beentoo optimistic about their projects in the past. If performance is below the engineering estimate, wordof mouth communication among fliers will erode the products reputation, and unit sales after the firstyear, will be 50% of the forecasts above. Management thinks theres a 30% chance the plane wontperform as well as the engineers think it will. The cost to bring the plane through design and intoproduction is estimated at $15M. Spitfires cost of capital is 12%.

a. Draw and fully label the decision tree diagram for the projectb. Calculate the NPV and probability along each path.c. Calculate the expected NPV of the project.d. Sketch a probability distribution for NPV.e. Describe the risk situation in words compared to a point estimate of NPV.

SOLUTION:

a. The decision tree is as follows ($M)Path

10 10 10 10 1.7

10.3

.6 5 5 5 5 2

15

.4 4 4 4 4 3.7

4.3

2 2 2 2 4

b. Path 1

NPV = 15 + 10[PVFA12,5] = 15 + 10(3.6048) = 15 + 36.048 = 21.0

Probability = .6 .7 = .42

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Path 2

NPV = 15 +10[PVF12,1] + 5[PVFA12,4][PVF12,1]

= 15 + 10(.8929) + 5(3.0373)(.8929) = 7.5


Path 3NPV = 15 + 4[PVFA12,5] = 15 + 4(3.6048) = 15 + 14.4 = -.6


Path 4

NPV = 15 +4[PVF12,1] + 2[PVFA12,4][PVF12,1]

= 15 + 4(.8929) + 2(3.0373)(.8929) = -6.0


c. Path NPV Probability Product1 $21.0M .42 $8.82

2 7.5 .18 1.353 .6 .28 .17

4 6.0 .12 .72

1.00 $9.28 = Expected Value

d.

e. The project has an expected NPV of over $9M, which is a likely value for a point-estimated NPV,and would strongly imply accepting the project. The detailed probability distribution shows thattheres a significant chance (12%) of a substantial loss ($6M) associated with the project. Ifmanagement is risk averse, theyre likely to reject a project with that big a loss potential even thoughthe expected NPV is very positive.

14. If Spitfire elects to do the project, what is an abandonment option at the end of year 1 worth ifSpitfire can recover $8M of the initial investment into other uses at that time? If the recovery is$13M?

46

.50

.40

.30

.20

.10

$6.0 $0.6 0 $7.5 $21.0

Probabilities

NPV (in $M)

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SOLUTION:

With an abandonment option, the decision tree becomes ($M)Path

10 10 10 10 1

.710

.3.6 5 5 5 5 2

15

.48

+4 0 0 0 0 312

The NPVs and probabilities of paths 1 and 2 are the same as in problem 6.For the new third path ($M):

Path 3

NPV = 15 + 12[PVF12,1] = 15 + 12(.8929) = 15 + 10.7 = 4.3

Probability = .4

Expected NPV calculations

Path NPV Probability Product1 $21.0M .42 $8.82M2 7.5 .18 1.35M

3

4.3 .40

1.72M

1.00 $8.45M = Exp Value

Hence the abandonment option is worth nothing since it reduces expected NPV.

If the recovery is $13M we have:

Additional NPV along path 3 $5

Contribution to expected NPV $5 .4 .8929 = $1.8

New expected NPV $8.45 + $1.8 = $10.25Problem 6 expected NPV $9.28

Increase in expected NPV $10.25 $9.28 = $0.97

Hence the abandonment option is worth just under $1M.

15. The New England Brewing Company produces a super premium beer using a recipe thats been inthe owners family since colonial times. Surprisingly, the firm doesnt own its own brewing facilities,but rents time on the equipment of large brewers who have excess capacity. Other small brewers havebeen doing the same thing lately, so capacity has become difficult to find, and must be contracted forseveral years in advance.

New Englands sales have been increasing steadily, and marketing consultants think theres apossibility that demand will really take off soon. Last years sales generated net cash flows after all

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costs and taxes of $5M. The consultants predict that sales will probably be at a level that will producenet cash flows of $6M per year for the next three years, but they also see a 20% probability that salescould be high enough to generate net cash inflows of $8M per year.

Meeting such an increase in demand presents a problem because of the advance contractingrequirements for brewing capacity. Unless New England arranges for extra facilities now, theres a70% chance that brewing capacity wont be available if the increased demand materializes. An option

arrangement is available with one of the large brewers under which it will hold capacity for NewEngland until the last minute for an immediate, nonrefundable payment of $1M. New Englands costof capital is 9%.

a. Draw a decision tree reflecting New Englands cash flows for the next three years withoutthe option and calculate the expected NPV of operating cash flows. (Note that theres no needto include an initial outlay because were dealing with ongoing operations.)b. Redraw the decision tree to include the capacity option as a real option in your calculations.What is its value? Should it be purchased?c. Does the real option reduce New Englands risk in any way?

SOLUTION: a. The decision tree without the option is as follows:

.3 $8 $8 $8.2

.7 $6 $6 $6

.8 $6 $6 $6

NPV = (.2)(.3)($8M)[PVFA9,3] + (.2)(.7)($6M)[PVFA9,3] +(.8)($6M)[PVFA9,3]= $.48M(.7722) + $.84M(.7722) + $4.8M(.7722)= $6.12M(.7722)= $4.73M

b. The capacity option changes the decision tree to the following:

$8 $8 $8.2

($1).8

$6 $6 $6

NPV = $1 + (.2)($8M)[PVFA9,3] + (.8)($6M)[PVFA9,3]

= $1 + $1.6M(.7722) + $4.8M(.7722)

= $1 + $6.4M(.7722)

= $1 + $4.94M

= $3.94M

The capacity options value is the increase in expected NPV it generates before considerationof its own cost. In this case thats

$4.94M $4.73M = $.21M

Since thats substantially less than the $1M cost of the option, New England would be better off notbuying it.

c. Real options generally have favorable risk effects when they reduce the probability or magnitude oflosses. In this case the option doesnt do that. Rather it guarantees the ability to take advantage of apotential opportunity. That generally wouldnt be viewed as reducing risk.

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Risk Adjusted Rates SML: Example 12-6 (page 546)16. Hudson Furniture specializes in office furniture for self-employed individuals who work at

home. Hudsons furniture emphasizes style rather than utility, and has been quite successful.The firm is now considering entering the more competitive industrial furniture market wherevolumes are higher but pricing is more competitive. A $10 million investment is required to

enter the new market. Management anticipates positive cash flows of $1.7 million annuallyfor eight years if Hudson enters the field. An average stock currently earns 8%, and the return

on treasury bills is 4%. Hudsons beta is .5 while that of an important competitorthat

operates solely in the industrial marketis 1.5. Should Hudson consider entering the

industrial furniture market?

SOLUTION: ($M)First calculate the appropriate discount rate for the project using the pure play competitorsbeta.

kx = kRF + (kM - kRF)bX= 4 + (8 - 4) 1.5= 10%

Use a calculator to arrive at the projects NPVCFo (10)C01 1.7F01 8I 10NPV (.930625)

Since the NPV is negative, Hudson should not enter the new market.

Its important to notice that the opposite result would have been obtained using Hudsons ownbeta to calculate a risk adjusted return as follows:

kx = kRF + (kM - kRF)bX= 4 + (8 - 4) .5

= 6%and

CFo (10)C01 1.7F01 8I 6NPV .5566

So a positive NPV would have indicated going ahead with the new venture, probably anincorrect result.

17. Crest Concrete Inc. has been building basements and slab foundations for new homes in La

Crosse, Wisconsin for more than 20 years. However, new home sales have slowed recently andresidential construction work is hard to get. As a result, management is considering a venture intocommercial construction. Although Crest would still be pouring concrete in commercial building,almost everything else about the business differs substantially from homebuilding which is all the firmhas done until now.

The local commercial concrete business is dominated by two firms. Readi-Mix Inc., andToddy Concrete Inc. Readi-Mix has been in business for 50 years, has a market share of 70%, and abeta of 1.3. Toddy has been in the area for only five years and has a beta of 2.4. Crests own beta is .9and its cost of capital is 9.3%. Both of these were developed during a long period in which the

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housing market was prosperous and growing steadily. The stock market is currently returning 11%and treasury bills are yielding 4.2%.

Crest will have to spend $950,000 to get started in the commercial field, and expects net cashinflows of $250,000 in the first year, $400,000 in the second year and $700,000 in the third.

Should Crest give commercial construction a try?

SOLUTIONThe new project is relatively risky for Crest, so it should be evaluated using a risk adjustedrate rather than the firms own cost of capital, which reflects a low risk business (developed during astable period). Well apply the Capital Asset Pricing Model to a pure play company in the field to geta reasonable risk adjusted rate. It makes sense to use Toddy rather than Readi-Mix because Crestwould initially be a small player and not a market leader like Readi-Mix.

Applying the CAPM we havek = kRF + (kM-kRF)b Toddy

= 4.2% + (11% - 4.2%)2.4= 20.52%

Then use a financial calculator as follows:CFo (950000)C01 250000F01 1.0C02 400000F02 1.0C03 700000F03 1.0

Press NPVI 20.52

Result: NPV (67,308)This implies the project should be rejected.Now compute the projects NPV using Crests 9.3% cost of capital to get a value of $149,644.

Notice that a reasonable consideration of risk implies the project should be rejected while thetraditional approach gives a definite signal to accept.

18. Illinois Fabrics Inc. makes upholstery thats used in high-quality furniture, largely chairs andsofas. Illinois has traditionally sold their fabric to manufacturers who use it to cover furniture framesthey produce. These manufacturers then wholesale the finished product to furniture stores.Management has analyzed the finished chairs and sofas of several manufacturers and found that thehighest value element they contain is the Illinois fabric. They further found that generally the frameswere shoddily produced.

Illinois VP of Manufacturing, Harrison Flatley, has proposed starting a new business calledIllinois Furniture which will produce and market the end product using the fabric the firm alreadymanufactures. Harrison has put together a proposal to start such a venture which results in a steadystream of cash income of $5 million per year after an initial investment of $25 million to be spent onmanufacturing facilities and the development of a sales relationship with retailers. The analysis comesup with an NPV for the project assuming the income stream is a perpetuity and taking its present valueat Illinois 10% cost of capital.

NPV = -$25M + $5M / .10 = -$25M + $50M = $25MTop management likes the idea but is concerned about risk in two areas. First, furniture

manufacturing seems to be a riskier business than making fabric as manufacturing firms are alwaysentering and leaving the industry. The average beta of the publicly traded end product manufacturersis a relatively high 1.9. By contrast, Illinois beta is .9.

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Second, management fears that an economic downturn would impact a new business moreseriously than it would the existing competitors. Management fears that theres a 40% chance of adownturn in the near future which would reduce Harrisons income projections by 20%.

Re-analyze Harrisons proposal and make a recommendation to management. Treasury billsare yielding 4% and the S&P 500 index is yielding 10%.

SOLUTION:

First develop the appropriate risk adjusted discount rate for analysis using the SMLkX = kRF + (kM kRF)bX = 4 + (10 4)1.9 = 4 + 11.4 = 15.4%

Then recalculate the projects NPV at the $5M income level

NPV = -$25M + $5M / .154 = -$25M + $32.5M = $7.5M

Next calculate the NPV at a 20% lower income level, i.e., $5M x .8 = $4M

NPV = -$25M + $4M / .154 = -$25M + $26M = $1M

Hence a probability distribution of risk adjusted outcomes is

Economy Probability NPVGood .6 x $7.5M = $4.5MDown .4 x $1.0M = $0.4M

$4.9M

Hence the projects risk adjusted NPV has an expected value of about $4.9M without toomuch chance of being above $7.5M or below $0.4M. This passes the NPV decision rule and doesntinvolve a chance of a crippling loss which argues in favor of acceptance. Nevertheless, the NPV ismodest relative to the size of the investment required. The recommendation should therefore be a lessthan enthusiastic approval as long as there arent better uses for the $25M.

Its theoretically important to notice that our procedure has not double counted risks. The riskadjusted discount rate deals with market risk which does not include the extra 20% income decline dueto an economic downturn. That response to an economic downturn is an incremental, businessspecific risk incurred solely by the venture because it will be a new entrant and therefore morevulnerable to economic conditions than the existing competitors whose betas average 1.9.

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