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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 no
measure of variability. Hence they don’t give managers the information necessary to include a
tradeoff between risk and expected return in their decisions. This chapter is concerned with modern
approaches to incorporating risk into capital budgeting. The techniques considered include
probabilistic 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 NPVs
and IRRs are also random variables with associated probability distributions. We then explore the
implications of choosing a high risk project over one with less variability, and conclude that
managements 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 relatively
well-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 that
the methods currently available are often less than completely satisfactory.

OUTLINE

I.   RISK IN CAPITAL BUDGETING – GENERAL CONSIDERATIONS
A. Cash Flows as Random Variables
Incorporating risk by viewing individual cash flows as random variables with probability
distributions.
NPV and IRR are then also random variables.
B. The Importance of Risk in Capital Budgeting
Why risk matters, making mistakes on individual projects and changing the risk character of
the company.
C. Incorporating Risk Into Capital Budgeting – Numerical and Computer Methods
Scenario/Sensitivity Analysis and Simulation
Decision Tree Analysis
D. Real Options
The 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 Rates
of Return
The concept of adjusting for risk by raising the threshold (hurdle) rate thus making risky
projects less acceptable.
F. Estimating Risk Adjusted Rates Using CAPM
Using the SML to estimate appropriate risk adjusted rates for project analysis.
The kind of risk CAPM estimates and whether it is appropriate for capital budgeting
decisions.
G. Problems with the Theoretical Approach
Finding the right beta, concerns about the appropriate risk definition.

285
286                                            Chapter 12

QUESTIONS

1.       In 1983 the Bell Telephone System, which operated as AT&T, was broken up, resulting in the
creation of seven regional telephone companies. AT&T stockholders received shares of the new
companies 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 its
service, but couldn’t charge excessive prices due to government regulation. Regulated utilities are
classic examples of low risk – modest return companies. After the breakup, the "Baby Bells," as they
were called, were freed from many of the regulatory constraints under which the Bell System operated,
and at the same time had a great deal of money. The managements of these young giants were
determined to be more than the staid old-line telephone companies they'd been in the past. They were
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 old telephone system.
Many stockholders were alarmed and concerned by these statements. Comment on what their
concerns may have been.

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

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

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
    The effectiveness of production operations

Each of these items is subject to the influence of a variety of physical, administrative and economic
forces that can each be positive or negative. The result of such a confluence of different influences
tends to vary randomly around some central value. This kind of pattern represents the essence of a
random variable.
When a random variable is the result of a combination of other random variables, its variability
tends to be less than that of the constituent pieces. This is because there is usually some offsetting of
the pluses and minuses among the pieces coming together in a typical observation.

3.       One of the problems of using simulation to incorporate risk into capital budgeting is related to
the idea that the probability distributions of successive cash flows usually are not independent. If the
Risk Topics and Real Options in Capital Budgeting                            287

first period's cash flow is at the high end of its range, for example, flows in subsequent periods are
more likely to be high than low. Why do you think this is generally the case? Describe an approach
through 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 a
product. Before any marketing is done, we project acceptance in each period as a random variable that
defines the volume of product sold.
However, acceptance really isn't independent from one period to the next. For most products it's
either good or bad throughout the project's entire life. That means choosing an independent
observation in successive periods isn't a very realistic way to model the process. For example, it's
unlikely 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 treating
acceptance as independent from period to period is likely to result in simulation runs in which it varies
back 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 that
sets the level of the mean values of the periodic acceptance random variables throughout the project's
life. This r.v. would be drawn only once per simulation run and would define acceptance as good or
bad for the whole run. The program would raise the means of all the acceptance r.v.s if a good
observation 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 less
acceptable 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 the
firm and the consequences of a miss in predicting the project's outcome. This makes failure or loss
less likely.

5.      Rationalize the appropriateness of using the cost of capital to analyze normally risky projects
and 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 desirable
to discriminate against such projects in the capital budgeting process. Using a risk-adjusted rate
instead of the cost of capital when project risk is high is a mechanical method of making risky projects
less acceptable. The method simply requires that a risky project have a higher expected return to
qualify than a lower risk project.
The cost of capital reflects the current state of the business, and therefore can be associated with
normally 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 risk
CAPM (Chapter 9) and the knowledge you're now developing of capital budgeting. The issue is
concisely summarized by Figure 12.7. Is the special concept of risk developed in portfolio theory
applicable here? Don't be intimidated into thinking that because the idea is presented in textbooks, it's
necessarily 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.
288                                           Chapter 12

1.       Ed Draycutt is the engineering manager of Airway Technologies, a firm that makes computer
systems 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 a
recent proposal for a new procedural approach to handling in flight calls from planes experiencing
emergencies. 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 respond
within another year with one of two possible changes in their technology. These can simply be called
A and B. The A response is far more likely, also having a probability of about 98%. Ed’s device
works with the A system and is a stroke of engineering genius. If the A system becomes the industry
standard and Airway has Ed’s product, it will make a fortune before anyone else can market a similar
device. On the other hand if the A system isn’t adopted, Airway will lose whatever it’s put into the
new device’s development.
Developing Ed’s device will cost about \$20 million, which is a very substantial investment for a small
company 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 \$20
million investment and a huge NPV and IRR reflecting the adoption of the A system. Everyone on the
committee is very excited. You’re the CFO and are a lot less excited. You asked Ed how he reflected
the admittedly remote possibility that the A system would never be put in place. Ed, obviously proud
of his business sophistication, said he’d taken care of that with a statistical calculation. He said
adoption 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 inflow
estimates by 4%. He maintains this correctly accounts for risk in the project.
Does Ed have the right expected NPV? What’s wrong with his analysis? Suggest an
approach that will give a more insightful result. Why might the firm consider passing on the proposal
in spite of the tremendous NPV and IRR Ed has calculated?

Ed’s calculation has given him the correct expected NPV but it’s masked the real risk in the
proposal. The problem that isn’t shown in his presentation is that there’s a 4% probability that Airway
will lose \$20M on the project and probably be put out of business. That the project involves a
substantial risk of ruin should be made explicit in any presentation to management.

2.      Might Ed’s case in the preceding problem be helped by a real option? If so What kind? How
would it help?

Yes, an abandonment option would help if a substantial portion of the development cost could
be avoided or recovered if, during the first year, it starts to look like the A system isn’t going to be
installed. This could reduce the magnitude of the loss but wouldn’t affect its probability.

3.       Charlie Henderson, a senior manager in the Bartok Company, is known for taking risks. He
recently proposed that the company expand its operations into a new and untried field. He put
together a set of cash flow projections and calculated an IRR of 25% for the project. The firm's cost of
capital is about 10%. Charlie maintains that the favorability of the calculated IRR relative to the cost
of capital makes the project an easy choice for acceptance, and urges management to move forward
immediately.
Several knowledgeable people have looked at the proposal and feel Charlie's projections represent
an optimistic scenario that has about one chance in three of happening. They think the project also has
about one chance in three of failing miserably. An important consideration is that the project is large
enough to bankrupt the company if it fails really badly.
Risk Topics and Real Options in Capital Budgeting                            289

Charlie doesn't want to talk about these issues, claiming the others are being "negative" and that he
has a history of success with risky ventures like this. When challenged, he falls back on the 25% IRR
versus the 10% cost of capital as justification for his idea.
issue of the 25% IRR versus the 10% cost of capital. Should this project be evaluated by using
different standards? How does the possibility of bankruptcy as a result of the project affect the
analysis? Are capital budgeting rules still appropriate? How should Charlie's successful record be
factored into the president's thinking?

ANSWER: This project is indeed high risk since it involves as much as a one third chance of ruining
the firm. For that reason measuring the IRR against the cost of capital is meaningless. It should be
measured against a risk-adjusted rate that is much higher. Intuitively, it's likely that such a rate would
be 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 the
significant probability of bankruptcy involved. When the risk of ruin is substantial, risk-averse people
often walk away from projects regardless of high expected returns. Stated another way, the project is a
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 describe
that 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 in
the 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 the
acceptability of Charlie's proposal. What practical concerns would you overlay on top of the theory
you've just described? Do they make the project more or less acceptable? Does the fact that Bartok
has never done this kind of business before matter? How would you adjust for that inexperience? Is
the risk of bankruptcy still important? What would you advise doing about that? All things
considered, 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 rate
with 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 subjective
estimates 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 probably
too 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 the
rate 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, and
advise against the project regardless of the IRR.
290                                            Chapter 12

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 \$2.5M
are more likely. Develop a range of NPVs for the project if Glendale’s 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 Glendale’s management in the last problem attaches a probability of .7 to the better
outcome, what is the project’s most likely (expected) NPV?

SOLUTION:
Scenario             Outcome                 Probability              Outcome x Prob.
Good               \$ 867,500                   .7                      \$ 607,250
Bad             (\$ 2,114,100)                  .3                     (\$ 643,230)
Expected NPV =            (\$ 26,980)

3.      Keener Clothiers Inc. is considering investing \$2 million in an automatic sewing machine to
produce 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, some
uncertainty about production costs associated with the new machine. The production
department has estimated operating costs at 70% of revenues, but senior management realizes
that this figure could turn out to be as low as 65% or as high as 75%. The new machine will
be depreciated at a rate of \$200,000 per year for six years (straight line, zero salvage).
Keener’s cost of capital is 14% and its marginal tax rate is 35%. Calculate a point estimate
along with best and worst case scenarios for the project’s NPV.
Risk Topics and Real Options in Capital Budgeting                        291

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              1,800
Depreciation                      200                  200                200
EBT Impact                        640                  520                400
Tax                               224                  182                140
EAT Impact                        416                  338                260
Add back depreciation.            200                  200                200
Cash 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               460
F01                                6                     6                 6
I                                 14                    14                14
NPV                         395.419                92.103          (211.213)

4.      Assume Keener Clothiers of the last problem assigns the following probabilities to production
cost as a percent of revenue
% of Revenue                      Probability
65%                            .30
70%                            .50
75%                            .20
Sketch a probability distribution (histogram) for the project’s NPV, and compute its expected
NPV.

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

Notice that because the probabilities of the 65% and 75% scenarios are not equal, the expected NPV is
not that of the 70% scenario.
292                                             Chapter 12

5.       The Blazingame Corporation is considering a three-year project that has an initial cash
outflow (C0) of \$175,000 and three cash inflows that are defined by the independent probability
distributions 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 its
probability?
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 (333) each of which is obtained by
selecting one cash flow from each column and combining with the initial outflow. The probability of
any pattern is the product of the probabilities of its three uncertain cash flows. For example, a
particular pattern might be as follows.

C0     C1        C2       C3
CI              (\$175) \$50       \$120     \$80
Probability     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 the
best, 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

175    45      99       56      25                  .25.25.25 = .015625
60      29                  .25.25.50 = .031250
64      33                  .25.25.25 = .015625
Risk Topics and Real Options in Capital Budgeting                         293

175    55       33      56      31             .50.25.25 = .031250
60      27             .50.25.50 = .062500
64      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 = .015625
60      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 = .015625
1.000000

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

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               .015625
1.000000
294                                           Chapter 12

Prob(NPV)

.25

.20

.15

.10

.05
NPV
<40 35        25     15    5       5    15      25           35     45      >50
Mid Points of NPV Ranges (\$000)

6.      Sanville Quarries is considering acquiring a new drilling machine which is expected to be
more efficient than their current machine. The project is to be evaluated over four years. The initial
outlay required to get the new machine operating is \$675,000. Incremental cash flows associated with
the machine are uncertain, so management developed the following probabilistic forecast of cash
flows by year (\$000). Sanville’s 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 project’s 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             205
C02                      250                 200             214.5
C03                      400                 350             377.5
C04                      375                 300             351
I                         10                  10              10
NPV                      360.995              94.517         211.995

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

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

7.       Using the information from the previous problem, randomly select four NPV outcomes from
the data. (Select one cash flow from each year and compute the project NPV and the probability of
that NPV implied by those selections.) Do your selections give a sense of where NPV outcomes are
likely to cluster?
Risk Topics and Real Options in Capital Budgeting                           295

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 one
building. Business has been good lately and management is considering a project that will add five
screens at an estimated cost of \$3 million. The success of the expansion depends on whether local
demand over the next two years will support the additional capacity. Demand is believed to depend on
the local economy. An economist at a nearby university has predicted a 90% probability of continued
prosperity 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 in
the second. A moderate downturn would produce contributions of \$1.5 and \$2 million. Northwest’s
cost 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 project’s NPV.
d. Calculate the project’s 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 project’s paths is:
Path 1
NPV = -3000 + 2000(PVF12,1) + 3000(PVF12,2)
= -3000 + 2000(.8929) + 3000(.7972)
= -3000 + 1785.8 + 2391.6
= 1177.4
Path 2
NPV = -3000 + 1000(PVF12,1) + 1500(PVF12,2)
= -3000 + 1000(.8929) + 1500(.7972)
= -3000 + 892.9 + 1195.8
= -911.3

c. The project’s NPV probability distribution is
NPV                  Probability
(\$911,300)                 .1
\$1,177,400                  .9

d. The expected NPV is the sum of the products of each path’s NPV and probability.
Expected NPV = \$1,177,400 (.9) - \$911,300 (.1)
= \$1,059,660 – \$91,130
= \$968,0530
296                                            Chapter 12

e. Recommendation: The project has a positive expected NPV, which is substantial relative to the size
of 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 project
unless 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 ―ergonomic‖
products that are designed to be easier on the bodies of office workers’ who suffer from aliments such
as back and neck pain due to sitting for long periods. Unfortunately customer acceptance of
ergonomic 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 with
the 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 lost
more than \$1.5 million. The firm’s cost of capital is 14%.

a. Develop a probability distribution for NPV based on the forecast. I.e., calculate the
project’s NPV along each path of the decision tree and the associated probability.
b. Calculate the project’s expected NPV.
considering both expected NPV and risk

SOLUTION:
a. The NPV along each of the project’s four paths and the probability of each of those outcomes is
calculated as follows:

Path 1
NPV = -6000 + 4000(PVF14,1) + 7000(PVF14,2)
= -6000 + 4000(.8772) + 7000(.7695)
= -6000 + 3508.8 + 5386.5
= 2895.3                            Probability = .6  .3 = .18
Risk Topics and Real Options in Capital Budgeting                           297

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 = .08
1.00

b. The expected NPV for the entire project is the sum of the products of each path’s NPV and
probability.

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 size
of 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 means
accepting 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 the
first year under which it will recover \$5 million of the initial investment in year 2. What is the value
of 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

.4
\$2,000                            \$5,000      3

Paths 3 and 4 condense into a new path 3 with the following NPV and probability.
298                                           Chapter 12

Path 3
NPV = 6,000 + 2000(.8772) + 5000(.7695)
= 6000 + 1754.4 + 3847.5
= 398.1                             Probability = .40

The project’s 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. The
expected 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 the
project.
Perhaps more important, however, is the fact that the project’s risk characteristics have
changed dramatically. The ability to abandon and recover most of the initial investment has
eliminated the potentially ruinous outcomes that were major concerns before. These outcomes have
been 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 is
concerned 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 next
year. The state of the economy probably won’t 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 the
second year. Incremental cash flows are as follows:

Year 1:
Viola opens a store - \$700,000
Viola doesn’t open a store - \$900,000
Year 2:
Viola opens a store, strong economy - \$850,000
Viola opens a store, weak economy - \$700,000
Viola doesn’t open a store, strong economy - \$1,500,000
Viola doesn’t open a store, weak economy - \$1,200,000

Perform a decision tree analysis of the refurbishment project. Draw the decision tree diagram
and calculate the probabilities and NPVs along each of its four paths. Then calculate the
overall expected NPV. Assume Vaughn’s cost of capital is 10%.
Risk Topics and Real Options in Capital Budgeting                         299

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 3
NPV = -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 path’s 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 Vaughn’s is refurbished. Carlson
300                                            Chapter 12

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

SOLUTION:
Vaughn would exercise Carlson’s option if Viola opened a new store and thus avoid
completing paths 1 or 2 in the previous problem. The recovery would be 70% of the initial
outlay 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 = .24
1.00

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
Risk Topics and Real Options in Capital Budgeting                         301

The value of the real option is this figure less the project’s 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 the
project’s NPV. The decision is marginal, however, because the incremental value added is small.

13.      Spitfire Aviation Inc. manufactures small, private aircraft. Management is evaluating a
proposal to introduce a new high performance plane. High performance aviation is an expensive sport
undertaken largely by people who are both young and wealthy. Spitfire sees its target market as
affluent 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 very
handsome, but lately there are serious concerns about a market downturn. If the market remains
strong, Spitfire estimates it will sell 50 of the new planes per year for five years, each of which will
result in a net cash flow contribution of \$200,000. If the market turns down, however, only about 20
units a year will be sold. Economists think there’s about a 40% chance the market will turn down in
the near future.
There are also some concerns about the design of the new plane. Not everyone is convinced it
will perform as well as the engineering department thinks. Indeed, the engineers have sometimes been
too optimistic about their projects in the past. If performance is below the engineering estimate, word
of mouth communication among fliers will erode the product’s reputation, and unit sales after the first
year, will be 50% of the forecasts above. Management thinks there’s a 30% chance the plane won’t
perform as well as the engineers think it will. The cost to bring the plane through design and into
production is estimated at \$15M. Spitfire’s cost of capital is 12%.
a. Draw and fully label the decision tree diagram for the project
b. 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
302                                                        Chapter 12

Path 2
NPV     = 15 +10[PVF12,1] + 5[PVFA12,4][PVF12,1]
= 15 + 10(.8929) + 5(3.0373)(.8929) = 7.5
Probability = .6  .3 = .18

Path 3
NPV = 15 + 4[PVFA12,5] = 15 + 4(3.6048) = 15 + 14.4 = -.6
Probability = .4  .7 = .28

Path 4
NPV     = 15 +4[PVF12,1] + 2[PVFA12,4][PVF12,1]
= 15 + 4(.8929) + 2(3.0373)(.8929) = -6.0
Probability = .4  .3 = .12

c.           Path                   NPV               Probability       Product
1                   \$21.0M                 .42             \$8.82
2                     7.5                  .18              1.35
3                     .6                  .28              .17
4                    6.0                  .12              .72
1.00             \$9.28 = Expected Value

d.
.50

.40
Probabilities

.30

.20

.10

\$6.0   \$0.6 0         \$7.5    \$21.0
NPV (in \$M)

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 that
there’s a significant chance (12%) of a substantial loss (\$6M) associated with the project. If
management is risk averse, they’re likely to reject a project with that big a loss potential even though
the 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 if
Spitfire can recover \$8M of the initial investment into other uses at that time? If the recovery is
\$13M?
Risk Topics and Real Options in Capital Budgeting                         303

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

10 — 10 — 10 — 10                  1
.7
10
.3
.6                                5 —     5 —      5 —     5        2

15

.4
8
+4 ——————— 0 — 0 — 0 — 0                                  3
12

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              Product
1             \$21.0M             .42                    \$8.82M
2               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 = \$10.25
Problem 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 that’s been in
the owner’s family since colonial times. Surprisingly, the firm doesn’t own its own brewing facilities,
but rents time on the equipment of large brewers who have excess capacity. Other small brewers have
been doing the same thing lately, so capacity has become difficult to find, and must be contracted for
New England’s sales have been increasing steadily, and marketing consultants think there’s a
possibility that demand will really take off soon. Last year’s sales generated net cash flows after all
costs and taxes of \$5M. The consultants predict that sales will probably be at a level that will produce
304                                             Chapter 12

net cash flows of \$6M per year for the next three years, but they also see a 20% probability that sales
could 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 contracting
requirements for brewing capacity. Unless New England arranges for extra facilities now, there’s a
70% chance that brewing capacity won’t 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 New
England until the last minute for an immediate, nonrefundable payment of \$1M. New England’s cost
of capital is 9%.
a. Draw a decision tree reflecting New England’s cash flows for the next three years without
the option and calculate the expected NPV of operating cash flows. (Note that there’s no need
to include an initial outlay because we’re 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 England’s 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 option’s value is the increase in expected NPV it generates before consideration
of it’s own cost. In this case that’s
\$4.94M  \$4.73M = \$.21M
Since that’s substantially less than the \$1M cost of the option, New England would be better off not

c. Real options generally have favorable risk effects when they reduce the probability or magnitude of
losses. In this case the option doesn’t do that. Rather it guarantees the ability to take advantage of a
potential opportunity. That generally wouldn’t be viewed as reducing risk.
Risk Topics and Real Options in Capital Budgeting                           305

Risk Adjusted Rates – SML: Example 12-6 (page 546)
16.     Hudson Furniture specializes in office furniture for self-employed individuals who work at
home. Hudson’s furniture emphasizes style rather than utility, and has been quite successful.
The firm is now considering entering the more competitive industrial furniture market where
volumes 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 annually
for eight years if Hudson enters the field. An average stock currently earns 8%, and the return
on treasury bills is 4%. Hudson’s beta is .5 while that of an important competitorthat
operates solely in the industrial marketis 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 competitor’s
beta.
kx = kRF + (kM - kRF)bX
= 4 + (8 - 4) 1.5
= 10%
Use a calculator to arrive at the project’s NPV
CFo             (10)
C01              1.7
F01              8
I                10
NPV             (.930625)
Since the NPV is negative, Hudson should not enter the new market.

It’s important to notice that the opposite result would have been obtained using Hudson’s own
beta to calculate a risk adjusted return as follows:
kx = kRF + (kM - kRF)bX
= 4 + (8 - 4) .5
= 6%
and
CFo              (10)
C01               1.7
F01               8
I                 6
NPV              .5566
So a positive NPV would have indicated going ahead with the new venture, probably an
incorrect 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 and
residential construction work is hard to get. As a result, management is considering a venture into
commercial 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 firm
has done until now.
The local commercial concrete business is dominated by two firms. Readi-Mix Inc., and
Toddy Concrete Inc. Readi-Mix has been in business for 50 years, has a market share of 70%, and a
beta of 1.3. Toddy has been in the area for only five years and has a beta of 2.4. Crest’s own beta is .9
and its cost of capital is 9.3%. Both of these were developed during a long period in which the
housing market was prosperous and growing steadily. The stock market is currently returning 11%
and treasury bills are yielding 4.2%.
306                                            Chapter 12

Crest will have to spend \$950,000 to get started in the commercial field, and expects net cash
inflows 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?

SOLUTION
The new project is relatively risky for Crest, so it should be evaluated using a risk adjusted
rate rather than the firm’s own cost of capital, which reflects a low risk business (developed during a
stable period). We’ll apply the Capital Asset Pricing Model to a pure play company in the field to get
a reasonable risk adjusted rate. It makes sense to use Toddy rather than Readi-Mix because Crest
would initially be a small player and not a market leader like Readi-Mix.
Applying the CAPM we have
k = kRF + (kM-kRF)b Toddy
= 4.2% + (11% - 4.2%)2.4
= 20.52%
Then use a financial calculator as follows:
CFo               (950000)
C01                250000
F01                 1.0
C02                400000
F02                 1.0
C03                700000
F03                 1.0
Press NPV
I                20.52
Result:          NPV              (67,308)
This implies the project should be rejected.
Now compute the project’s NPV using Crest’s 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 the
traditional approach gives a definite signal to accept.

18.       Illinois Fabrics Inc. makes upholstery that’s used in high-quality furniture, largely chairs and
sofas. Illinois has traditionally sold their fabric to manufacturers who use it to cover furniture frames
they 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 the
highest value element they contain is the Illinois fabric. They further found that generally the frames
were shoddily produced.
Illinois’ VP of Manufacturing, Harrison Flatley, has proposed starting a new business called
Illinois Furniture which will produce and market the end product using the fabric the firm already
manufactures. Harrison has put together a proposal to start such a venture which results in a steady
stream of cash income of \$5 million per year after an initial investment of \$25 million to be spent on
manufacturing facilities and the development of a sales relationship with retailers. The analysis comes
up with an NPV for the project assuming the income stream is a perpetuity and taking its present value
at Illinois’ 10% cost of capital.
NPV = -\$25M + \$5M / .10 = -\$25M + \$50M = \$25M
Top 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 always
entering and leaving the industry. The average beta of the publicly traded end product manufacturers
is a relatively high 1.9. By contrast, Illinois’ beta is .9.
Second, management fears that an economic downturn would impact a new business more
seriously than it would the existing competitors. Management fears that there’s a 40% chance of a
downturn in the near future which would reduce Harrison’s income projections by 20%.
Risk Topics and Real Options in Capital Budgeting                          307

Re-analyze Harrison’s proposal and make a recommendation to management. Treasury bills
are yielding 4% and the S&P 500 index is yielding 10%.

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

Then recalculate the project’s 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   NPV
Good   .6        x     \$7.5M =   \$4.5M
Down   .4        x     \$1.0M =   \$0.4M
\$4.9M

Hence the project’s risk adjusted NPV has an expected value of about \$4.9M without too
much chance of being above \$7.5M or below \$0.4M. This passes the NPV decision rule and doesn’t
involve a chance of a crippling loss which argues in favor of acceptance. Nevertheless, the NPV is
modest relative to the size of the investment required. The recommendation should therefore be a less
than enthusiastic approval as long as there aren’t better uses for the \$25M.
It’s theoretically important to notice that our procedure has not double counted risks. The risk
adjusted discount rate deals with market risk which does not include the extra 20% income decline due
to an economic downturn. That response to an economic downturn is an incremental, business
specific risk incurred solely by the venture because it will be a new entrant and therefore more
vulnerable to economic conditions than the existing competitors whose betas average 1.9.

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