Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
CHAPTER TEN
254
A PROJECT IS NOT
A BLACK BOX
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
A BLACK BOX is something that we accept and use but do not understand. For most of us a computer
is a black box. We may know what it is supposed to do, but we do not understand how it works and,
if something breaks, we cannot fix it.
We have been treating capital projects as black boxes. In other words, we have talked as if man-
agers are handed unbiased cash-flow forecasts and their only task is to assess risk, choose the right
discount rate, and crank out net present value. Actual financial managers won’t rest until they un-
derstand what makes the project tick and what could go wrong with it. Remember Murphy’s law, “If
anything can go wrong, it will,” and O’Reilly’s corollary, “at the worst possible time.”
Even if the project’s risk is wholly diversifiable, you still need to understand why the venture could

fail. Once you know that, you can decide whether it is worth trying to resolve the uncertainty. Maybe
further expenditure on market research would clear up those doubts about acceptance by con-
sumers, maybe another drill hole would give you a better idea of the size of the ore body, and maybe
some further work on the test bed would confirm the durability of those welds. If the project really
has a negative NPV, the sooner you can identify it, the better. And even if you decide that it is worth
going ahead on the basis of present information, you do not want to be caught by surprise if things
subsequently go wrong. You want to know the danger signals and the actions you might take.
We will show you how to use sensitivity analysis, break-even analysis, and Monte Carlo simulation
to identify crucial assumptions and to explore what can go wrong. There is no magic in these tech-
niques, just computer-assisted common sense. You don’t need a license to use them.
Discounted-cash-flow analysis commonly assumes that companies hold assets passively, and it ig-
nores the opportunities to expand the project if it is successful or to bail out if it is not. However, wise
managers value these opportunities. They look for ways to capitalize on success and to reduce the costs
of failure, and they are prepared to pay up for projects that give them this flexibility. Opportunities to
modify projects as the future unfolds are known as real options. We describe several important real op-
tions, and we show how to use decision trees to set out these options’ attributes and implications.
255
Uncertainty means that more things can happen than will happen. Whenever
you are confronted with a cash-flow forecast, you should try to discover what
else can happen.
Put yourself in the well-heeled shoes of the treasurer of the Otobai Company in
Osaka, Japan. You are considering the introduction of an electrically powered mo-
tor scooter for city use. Your staff members have prepared the cash-flow forecasts
shown in Table 10.1. Since NPV is positive at the 10 percent opportunity cost of cap-
ital, it appears to be worth going ahead.
Before you decide, you want to delve into these forecasts and identify the key
variables that determine whether the project succeeds or fails. It turns out that the
marketing department has estimated revenue as follows:
ϭ .1 ϫ 1 million ϭ 100,000 scooters
Unit sales ϭ new product’s share of market ϫ size of scooter market

NPV ϭϪ15 ϩ
a
10
tϭ1
3
11.102
t
ϭϩ¥3.43 billion
10.1 SENSITIVITY ANALYSIS
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
The production department has estimated variable costs per unit as ¥300,000. Since
projected volume is 100,000 scooters per year, total variable cost is ¥30 billion. Fixed
costs are ¥3 billion per year. The initial investment can be depreciated on a straight-
line basis over the 10-year period, and profits are taxed at a rate of 50 percent.
These seem to be the important things you need to know, but look out for
unidentified variables. Perhaps there are patent problems, or perhaps you will
need to invest in service stations that will recharge the scooter batteries. The
greatest dangers often lie in these unknown unknowns, or “unk-unks,” as scien-
tists call them.
Having found no unk-unks (no doubt you’ll find them later), you conduct a sen-
sitivity analysis with respect to market size, market share, and so on. To do this,
the marketing and production staffs are asked to give optimistic and pessimistic

estimates for the underlying variables. These are set out in the left-hand columns
of Table 10.2. The right-hand side shows what happens to the project’s net present
value if the variables are set one at a time to their optimistic and pessimistic values.
Your project appears to be by no means a sure thing. The most dangerous variables
appear to be market share and unit variable cost. If market share is only .04 (and
all other variables are as expected), then the project has an NPV of Ϫ¥10.4 billion.
If unit variable cost is ¥360,000 (and all other variables are as expected), then the
project has an NPV of Ϫ¥15 billion.
Value of Information
Now you can check whether an investment of time or money could resolve some
of the uncertainty before your company parts with the ¥15 billion investment. Sup-
pose that the pessimistic value for unit variable cost partly reflects the production
department’s worry that a particular machine will not work as designed and that
the operation will have to be performed by other methods at an extra cost of
¥20,000 per unit. The chance that this will occur is only 1 in 10. But, if it does occur,
the extra ¥20,000 unit cost will reduce after-tax cash flow by
ϭ 100,000 ϫ 20,000 ϫ .50 ϭ ¥1 billion
Unit sales ϫ additional unit cost ϫ 11 Ϫ tax rate2
ϭ 100,000 ϫ 375,000 ϭ ¥37.5 billion
Revenue ϭ unit sales ϫ price per unit
256 PART III
Practical Problems in Capital Budgeting
Year 0 Years 1–10
Investment 15
1. Revenue 37.5
2. Variable cost 30
3. Fixed cost 3
4. Depreciation 1.5
5. Pretax profit (1 Ϫ 2 Ϫ 3 Ϫ 4) 3
6. Tax 1.5

7. Net profit (5 Ϫ 6) 1.5
8. Operating cash flow (4 ϩ 7) 3
Net cash flow Ϫ15 ϩ3
TABLE 10.1
Preliminary cash-flow forecasts for Otobai’s
electric scooter project (figures in ¥ billions).
Assumptions:
1. Investment is depreciated over 10 years
straight-line.
2. Income is taxed at a rate of 50 percent.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
It would reduce the NPV of your project by
putting the NPV of the scooter project underwater at ϩ3.43 Ϫ 6.14 ϭϪ¥2.71 billion.
Suppose further that a ¥10 million pretest of the machine will reveal whether it
will work or not and allow you to clear up the problem. It clearly pays to invest
¥10 million to avoid a 10 percent probability of a ¥6.14 billion fall in NPV. You are
ahead by Ϫ10 ϩ .10 ϫ 6,140 ϭϩ¥604 million.
On the other hand, the value of additional information about market size is small.
Because the project is acceptable even under pessimistic assumptions about market
size, you are unlikely to be in trouble if you have misestimated that variable.
Limits to Sensitivity Analysis
Sensitivity analysis boils down to expressing cash flows in terms of key project vari-

ables and then calculating the consequences of misestimating the variables. It forces
the manager to identify the underlying variables, indicates where additional informa-
tion would be most useful, and helps to expose confused or inappropriate forecasts.
One drawback to sensitivity analysis is that it always gives somewhat ambigu-
ous results. For example, what exactly does optimistic or pessimistic mean? The mar-
keting department may be interpreting the terms in a different way from the pro-
duction department. Ten years from now, after hundreds of projects, hindsight
may show that the marketing department’s pessimistic limit was exceeded twice
as often as the production department’s; but what you may discover 10 years hence
is no help now. One solution is to ask the two departments for a complete descrip-
tion of the various odds. However, it is far from easy to extract a forecaster’s sub-
jective notion of the complete probability distribution of possible outcomes.
1
Another problem with sensitivity analysis is that the underlying variables are
likely to be interrelated. What sense does it make to look at the effect in isolation
of an increase in market size? If market size exceeds expectations, it is likely that
a
10
tϭ1
1
11.102
t
ϭ ¥6.14 billion,
CHAPTER 10
A Project Is Not a Black Box 257
Range NPV, ¥ Billions
Variable Pessimistic Expected Optimistic Pessimistic Expected Optimistic
Market size .9 million 1 million 1.1 million ϩ1.1 ϩ3.4 ϩ5.7
Market share .04 .1 .16 Ϫ10.4 ϩ3.4 ϩ17.3
Unit price ¥350,000 ¥375,000 ¥380,000 Ϫ4.2 ϩ3.4 ϩ5.0

Unit variable cost ¥360,000 ¥300,000 ¥275,000 Ϫ15.0 ϩ3.4 ϩ11.1
Fixed cost ¥4 billion ¥3 billion ¥2 billion ϩ.4 ϩ3.4 ϩ6.5
TABLE 10.2
To undertake a sensitivity analysis of the electric scooter project, we set each variable in turn at its most pessimistic or
optimistic value and recalculate the NPV of the project.
1
If you doubt this, try some simple experiments. Ask the person who repairs your television to state a
numerical probability that your set will work for at least one more year. Or construct your own subjec-
tive probability distribution of the number of telephone calls you will receive next week. That ought to
be easy. Try it.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
demand will be stronger than you anticipated and unit prices will be higher. And
why look in isolation at the effect of an increase in price? If inflation pushes prices
to the upper end of your range, it is quite probable that costs will also be inflated.
Sometimes the analyst can get around these problems by defining underlying
variables so that they are roughly independent. But you cannot push one-at-a-time
sensitivity analysis too far. It is impossible to obtain expected, optimistic, and pes-
simistic values for total project cash flows from the information in Table 10.2.
Scenario Analysis
If the variables are interrelated, it may help to consider some alternative plausible
scenarios. For example, perhaps the company economist is worried about the pos-
sibility of another sharp rise in world oil prices. The direct effect of this would be

to encourage the use of electrically powered transportation. The popularity of com-
pact cars after the oil price increases in the 1970s leads you to estimate that an im-
mediate 20 percent price rise in oil would enable you to capture an extra 3 percent
of the scooter market. On the other hand, the economist also believes that higher
oil prices would prompt a world recession and at the same time stimulate inflation.
In that case, market size might be in the region of .8 million scooters and both prices
and cost might be 15 percent higher than your initial estimates. Table 10.3 shows
that this scenario of higher oil prices and recession would on balance help your
new venture. Its NPV would increase to ¥6.5 billion.
Managers often find scenario analysis helpful. It allows them to look at differ-
ent but consistent combinations of variables. Forecasters generally prefer to give an
258 PART III
Practical Problems in Capital Budgeting
Cash Flows, Years 1–10, ¥ Billions
Base Case High Oil Prices and Recession Case
1. Revenue 37.5 44.9
2. Variable cost 30.0 35.9
3. Fixed cost 3.0 3.5
4. Depreciation 1.5 1.5
5. Pretax profit (1 Ϫ 2 Ϫ 3 Ϫ 4) 3.0 4.0
6. Tax 1.5 2.0
7. Net profit (5 Ϫ 6) 1.5 2.0
8. Net cash flow (4 ϩ 7) 3.0 3.5
PV of cash flows ϩ18.4 ϩ21.5
NPV ϩ3.4 ϩ6.5
Assumptions
Base Case High Oil Prices and Recession Case
Market size 1 million .8 million
Market share .1 .13
Unit price ¥375,000 ¥431,300

Unit variable cost ¥300,000 ¥345,000
Fixed cost ¥3 billion ¥3.5 billion
TABLE 10.3
How the NPV of the electric scooter project would be affected by higher oil prices and a world
recession.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
estimate of revenues or costs under a particular scenario than to give some ab-
solute optimistic or pessimistic value.
Break-Even Analysis
When we undertake a sensitivity analysis of a project or when we look at alterna-
tive scenarios, we are asking how serious it would be if sales or costs turned out to
be worse than we forecasted. Managers sometimes prefer to rephrase this question
and ask how bad sales can get before the project begins to lose money. This exer-
cise is known as break-even analysis.
In the left-hand portion of Table 10.4 we set out the revenues and costs of the
electric scooter project under different assumptions about annual sales.
2
In the
right-hand portion of the table we discount these revenues and costs to give
the present value of the inflows and the present value of the outflows. Net present
value is of course the difference between these numbers.
You can see that NPV is strongly negative if the company does not produce a

single scooter. It is just positive if (as expected) the company sells 100,000 scooters
and is strongly positive if it sells 200,000. Clearly the zero-NPV point occurs at a lit-
tle under 100,000 scooters.
In Figure 10.1 we have plotted the present value of the inflows and outflows un-
der different assumptions about annual sales. The two lines cross when sales are
85,000 scooters. This is the point at which the project has zero NPV. As long as sales
are greater than 85,000, the project has a positive NPV.
3
Managers frequently calculate break-even points in terms of accounting profits
rather than present values. Table 10.5 shows Otobai’s after-tax profits at three lev-
els of scooter sales. Figure 10.2 once again plots revenues and costs against sales.
But the story this time is different. Figure 10.2, which is based on accounting prof-
its, suggests a break-even of 60,000 scooters. Figure 10.1, which is based on present
values, shows a break-even at 85,000 scooters. Why the difference?
When we work in terms of accounting profit, we deduct depreciation of ¥1.5 bil-
lion each year to cover the cost of the initial investment. If Otobai sells 60,000 scoot-
ers a year, revenues will be sufficient both to pay operating costs and to recover the
CHAPTER 10
A Project Is Not a Black Box 259
2
Notice that if the project makes a loss, this loss can be used to reduce the tax bill on the rest of the com-
pany’s business. In this case the project produces a tax saving—the tax outflow is negative.
3
We could also calculate break-even sales by plotting equivalent annual costs and revenues. Of course,
the break-even point would be identical at 85,000 scooters.
Inflows Outflows
Year 0 Years 1–10
Unit Sales, Revenue, Variable Fixed PV PV
Thousands Years 1–10 Investment Costs Costs Taxes Inflows Outflows NPV
00 1503Ϫ2.25 0 19.6 Ϫ19.6

100 37.5 15 30 3 1.5 230.4 227.0 3.4
200 75.0 15 60 3 5.25 460.8 434.4 26.4
TABLE 10.4
NPV of electric scooter project under different assumptions about unit sales (figures in ¥ billions except as noted).
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
initial outlay of ¥15 billion. But they will not be sufficient to repay the opportunity
cost of capital on that ¥15 billion. If we allow for the fact that the ¥15 billion could
have been invested elsewhere to earn 10 percent, the equivalent annual cost of the
investment is not ¥1.5 billion but ¥2.44 billion.
4
260 PART III Practical Problems in Capital Budgeting
Scooter sales, thousands
Break-even point:
NPV = 0
PV outflows
PV inflows
20085
PV, billions of yen
200
19.6
400
FIGURE 10.1

A break-even chart
showing the present
values of Otobai’s cash
inflows and outflows under
different assumptions
about unit sales. NPV
is zero when sales are
85,000.
Profit
Unit Sales, Variable Fixed Total after
Thousands Revenue Costs Costs Depreciation Taxes Costs Tax
0 0 0 3 1.5 Ϫ2.25 2.25 Ϫ2.25
100 37.5 30 3 1.5 1.5 36.0 1.5
200 75.0 60 3 1.5 5.25 69.75 5.25
TABLE 10.5
The electric scooter project’s accounting profit under different assumptions about unit sales (figures in ¥ billions except
as noted).
4
To calculate the equivalent annual cost of the initial ¥15 billion investment, we divide by the 10-year
annuity factor for a 10 percent discount rate:
See Section 6.3.
The annual revenues at 85,000 scooters per year are about ¥31.9 billion. You can check that this is
sufficient to cover variable costs, fixed costs, and taxes and still leave ¥2.44 billion per year to recover
the ¥15 billion initial investment and a 10 percent return on that investment.
ϭ
15
6.145
ϭ ¥2.44 billion
Equivalent annual cost ϭ
investment

10-year annuity factor
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
Companies that break even on an accounting basis are really making a loss—they
are losing the opportunity cost of capital on their investment. Reinhardt has described
a dramatic example of this mistake.
5
In 1971 Lockheed managers found themselves
having to give evidence to Congress on the viability of the company’s L-1011 TriStar
program. They argued that the program appeared to be commercially attractive and
that TriStar sales would eventually exceed the break-even point of about 200 aircraft.
But in calculating this break-even point, Lockheed appears to have ignored the op-
portunity cost of the huge $1 billion capital investment on this project. Had it allowed
for this cost, the break-even point would probably have been nearer to 500 aircraft.
Operating Leverage and Break-Even Points
Break-even charts like Figure 10.1 help managers appreciate operating leverage, that
is, project exposure to fixed costs. Remember from Section 9.5 that high operating
leverage means high risk, other things equal, of course.
The electric scooter project had low fixed costs, only ¥3 billion against projected
revenues of ¥37.5 billion. But suppose Otobai now considers a different production
technology with lower variable costs of only ¥120,000 per unit (versus ¥300,000 per
unit) but higher fixed costs of ¥19 billion. Total forecasted production costs
are lower (12 ϩ 19 ϭ ¥31 billion versus ¥33 billion), so profitability improves—

compare Table 10.6 to Table 10.1. Project NPV increases to ¥9.6 billion.
Figure 10.3 is the new break-even chart. Break-even sales have increased to 88,000
(that’s bad), even though total production costs have fallen. A new sensitivity analy-
sis would show that project NPV is much more exposed to changes in market size,
CHAPTER 10
A Project Is Not a Black Box 261
Scooter sales,
thousands
Break-even point:
Profit = 0
Costs
(including depreciation
and taxes)
Revenues
20060
Accounting revenues
and costs,
billions of yen
20
60
40
FIGURE 10.2
Sometimes break-even
charts are constructed in
terms of accounting
numbers. After-tax profit is
zero when sales are 60,000.
5
U. E. Reinhardt, “Break-Even Analysis for Lockheed’s TriStar: An Application of Financial Theory,”
Journal of Finance 28 (September 1973), pp. 821–838.

Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
market share, or unit price. All of these differences can be traced to the higher fixed
costs of the alternative production technology.
Is the alternative technology better than the original one? The financial man-
ager would have to consider the alternative technology’s higher business risk,
and perhaps recompute NPV at a higher discount rate, before making a final
decision.
6
262 PART III Practical Problems in Capital Budgeting
Year 0 Years 1–10
Investment 15
1. Revenue 37.5
2. Variable cost 12.0
3. Fixed cost 19.0
4. Depreciation 1.5
5. Pretax profit (1 Ϫ 2 Ϫ 3 Ϫ 4) 5.0
6. Tax 2.5
7. Net profit (5 Ϫ 6) 2.5
8. Operating cash flow (4 ϩ 7) 4.0
Net cash flow Ϫ15 ϩ4.0
NPV ϭϪ15 ϩ
a

10
tϭ1
4.0
11.12
t
ϭϩ¥9.6 billion
TABLE 10.6
Cash-flow forecasts and PV for the electric
scooter project, here assuming a production
technology with high fixed costs but low
total costs (figures in ¥ billions). Compare
Table 10.1.
Scooter sales,
thousands
Break-even point:
NPV = 0
PV outflows
PV inflows
PV, billions of yen
20088
200
400
68.8
FIGURE 10.3
Break-even chart for an
alternative production
technology with higher fixed
costs. Notice that break-even
sales increase to 88,000.
Compare Figure 10.1.

6
He or she could use the procedures outlined in Section 9.5 to recalculate beta and come up with a new
discount rate.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
Sensitivity analysis allows you to consider the effect of changing one variable at a
time. By looking at the project under alternative scenarios, you can consider the ef-
fect of a limited number of plausible combinations of variables. Monte Carlo simu-
lation is a tool for considering all possible combinations. It therefore enables you
to inspect the entire distribution of project outcomes. The use of simulation in cap-
ital budgeting was first advocated by David Hertz
7
and McKinsey and Company,
the management consultants.
Imagine that you are a gambler at Monte Carlo. You know nothing about the
laws of probability (few casual gamblers do), but a friend has suggested to you
a complicated strategy for playing roulette. Your friend has not actually tested
the strategy but is confident that it will on the average give you a 2
1
⁄2 percent re-
turn for every 50 spins of the wheel. Your friend’s optimistic estimate for any
series of 50 spins is a profit of 55 percent; your friend’s pessimistic estimate is a
loss of 50 percent. How can you find out whether these really are the odds? An

easy but possibly expensive way is to start playing and record the outcome at
the end of each series of 50 spins. After, say, 100 series of 50 spins each, plot a
frequency distribution of the outcomes and calculate the average and upper
and lower limits. If things look good, you can then get down to some serious
gambling.
An alternative is to tell a computer to simulate the roulette wheel and the strat-
egy. In other words, you could instruct the computer to draw numbers out of its
hat to determine the outcome of each spin of the wheel and then to calculate how
much you would make or lose from the particular gambling strategy.
That would be an example of Monte Carlo simulation. In capital budgeting we
replace the gambling strategy with a model of the project, and the roulette wheel
with a model of the world in which the project operates. Let’s see how this might
work with our project for an electrically powered scooter.
Simulating the Electric Scooter Project
Step 1: Modeling the Project The first step in any simulation is to give the com-
puter a precise model of the project. For example, the sensitivity analysis of the
scooter project was based on the following implicit model of cash flow:
This model of the project was all that you needed for the simpleminded sensi-
tivity analysis that we described above. But if you wish to simulate the whole proj-
ect, you need to think about how the variables are interrelated.
For example, consider the first variable—market size. The marketing depart-
ment has estimated a market size of 1 million scooters in the first year of the pro-
ject’s life, but of course you do not know how things will work out. Actual market
Costs ϭ 1market size ϫ market share ϫ variable unit cost2ϩ fixed cost
Revenues ϭ market size ϫ market share ϫ unit price
Cash flow ϭ 1revenues Ϫ costs Ϫ depreciation2ϫ 11 Ϫ tax rate2ϩ depreciation
CHAPTER 10
A Project Is Not a Black Box 263
10.2 MONTE CARLO SIMULATION
7

See D. B. Hertz, “Investment Policies that Pay Off,” Harvard Business Review 46 (January–February
1968), pp. 96–108.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
size will exceed or fall short of expectations by the amount of the department’s
forecast error:
You expect the forecast error to be zero, but it could turn out to be positive or neg-
ative. Suppose, for example, that the actual market size turns out to be 1.1 million.
That means a forecast error of 10 percent, or ϩ.1:
You can write the market size in the second year in exactly the same way:
But at this point you must consider how the expected market size in year 2 is af-
fected by what happens in year 1. If scooter sales are below expectations in year 1,
it is likely that they will continue to be below in subsequent years. Suppose that a
shortfall in sales in year 1 would lead you to revise down your forecast of sales in
year 2 by a like amount. Then
Now you can rewrite the market size in year 2 in terms of the actual market size in
the previous year plus a forecast error:
In the same way you can describe the expected market size in year 3 in terms of
market size in year 2 and so on.
This set of equations illustrates how you can describe interdependence between
different periods. But you also need to allow for interdependence between different
variables. For example, the price of electrically powered scooters is likely to increase
with market size. Suppose that this is the only uncertainty and that a 10 percent

shortfall in market size would lead you to predict a 3 percent reduction in price.
Then you could model the first year’s price as follows:
Then, if variations in market size exert a permanent effect on price, you can define
the second year’s price as
ϭ actual price, year 1 ϫ °1 ϩ
.3 ϫ error in
market size
forecast,
year 2
¢
Price, year 2 ϭ expected price, year 2 ϫ °1 ϩ
.3 ϫ error in
market size
forecast,
year 2
¢
Price, year 1 ϭ expected price, year 1 ϫ °1 ϩ
.3 ϫ error in
market size
forecast,
year 1
¢
Market size, year 2 ϭ market size, year 1 ϫ a1 ϩ
forecast error,
year 2
b
Expected market size, year 2 ϭ actual market size, year 1
Market size, year 2 ϭ expected market size, year 2 ϫ a1 ϩ
forecast error,
year 2

b
Market size, year 1 ϭ 1 ϫ 11 ϩ .12ϭ 1.1 million
Market size, year 1 ϭ expected market size, year 1 ϫ a1 ϩ
forecast error,
year 1
b
264 PART III
Practical Problems in Capital Budgeting
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
Notice how we have linked each period’s selling price to the actual selling prices
(including forecast error) in all previous periods. We used the same type of linkage
for market size. These linkages mean that forecast errors accumulate; they do not
cancel out over time. Thus, uncertainty increases with time: The farther out you
look into the future, the more the actual price or market size may depart from your
original forecast.
The complete model of your project would include a set of equations for each of
the variables: market size, price, market share, unit variable cost, and fixed cost.
Even if you allowed for only a few interdependencies between variables and across
time, the result would be quite a complex list of equations.
8
Perhaps that is not a
bad thing if it forces you to understand what the project is all about. Model build-

ing is like spinach: You may not like the taste, but it is good for you.
Step 2: Specifying Probabilities Remember the procedure for simulating the
gambling strategy? The first step was to specify the strategy, the second was to
specify the numbers on the roulette wheel, and the third was to tell the computer
to select these numbers at random and calculate the results of the strategy:
CHAPTER 10
A Project Is Not a Black Box 265
8
Specifying the interdependencies is the hardest and most important part of a simulation. If all compo-
nents of project cash flows were unrelated, simulation would rarely be necessary.
9
Suppose “near certainty” means “99 percent of the time.” If forecast errors are normally distributed,
this degree of certainty requires a range of plus or minus three standard deviations.
Other distributions could, of course, be used. For example, the marketing department may view any
market size between .85 and 1.15 million scooters as equally likely. In that case the simulation would re-
quire a uniform (rectangular) distribution of forecast errors.
Step 1
Model the strategy
Step 2
Specify numbers on
roulette wheel
Step 3
Select numbers and
calculate results
of strategy
The steps are just the same for your scooter project:
Step 1
Model the project
Step 2
Specify probabilities

for forecast errors
Step 3
Select numbers for
forecast errors and
calculate cash flows
Think about how you might go about specifying your possible errors in fore-
casting market size. You expect market size to be 1 million scooters. You obviously
don’t think that you are underestimating or overestimating, so the expected fore-
cast error is zero. On the other hand, the marketing department has given you a
range of possible estimates. Market size could be as low as .85 million scooters or
as high as 1.15 million scooters. Thus the forecast error has an expected value of 0
and a range of plus or minus 15 percent. If the marketing department has in fact
given you the lowest and highest possible outcomes, actual market size should fall
somewhere within this range with near certainty.
9
That takes care of market size; now you need to draw up similar estimates of the
possible forecast errors for each of the other variables that are in your model.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
Step 3: Simulate the Cash Flows The computer now samples from the distribution
of the forecast errors, calculates the resulting cash flows for each period, and
records them. After many iterations you begin to get accurate estimates of the prob-
ability distributions of the project cash flows—accurate, that is, only to the extent

that your model and the probability distributions of the forecast errors are accu-
rate. Remember the GIGO principle: “Garbage in, garbage out.”
Figure 10.4 shows part of the output from an actual simulation of the electric
scooter project.
10
Note the positive skewness of the outcomes—very large out-
comes are more likely than very small ones. This is common and realistic when
forecast errors accumulate over time. Because of the skewness the average cash
flow is somewhat higher than the most likely outcome; in other words, a bit to the
right of the peak of the distribution.
11
Step 4: Calculate Present Value The distributions of project cash flows should al-
low you to calculate the expected cash flows more accurately. In the final step you
need to discount these expected cash flows to find present value.
266 PART III
Practical Problems in Capital Budgeting
10
These are actual outputs from Crystal Ball™ software used with an EXCEL spreadsheet program. The
simulation assumed annual forecast errors were normally distributed and ran through 10,000 trials. We
thank Christopher Howe for running the simulation.
11
When you are working with cash-flow forecasts, bear in mind the distinction between the expected
value and the most likely (or modal) value. Present values are based on expected cash flows—that is, the
probability-weighted average of the possible future cash flows. If the distribution of possible outcomes is
skewed to the right as in Figure 10.4, the expected cash flow will be greater than the most likely cash flow.
Cash flow,
billions of
yen
Year 10: 10,000 Trials
8.5 9.08.07.57.06.56.05.55.04.54.03.53.02.52.01.51.0.50

0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
Frequency
FIGURE 10.4
Simulation of cash flows for year 10 of the electric scooter project.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
Simulation of Pharmaceutical Research and Development
Simulation, though sometimes costly and complicated, has the obvious merit of
compelling the forecaster to face up to uncertainty and to interdependencies. By
constructing a detailed Monte Carlo simulation, you will gain a better under-
standing of how the project works and what could go wrong with it. You will have
confirmed, or improved, your forecasts of future cash flows, and your calculations
of project NPV will be more confident.

Several large pharmaceutical companies have used Monte Carlo simulation to an-
alyze investments in research and development (R&D) of new drugs. Figure 10.5
sketches the progression of a new drug from its infancy, when it is identified as a prom-
ising chemical compound, all the way through the R&D required for approval for sale
by the Food and Drug Administration (FDA). At each phase of R&D, the company
must decide whether to press on to the next phase or halt. The R&D effort lasts 10 to
12 years from preclinical testing to FDA approval and can cost $300 million or more.
12
The pharmaceutical companies face two kinds of uncertainty:
1. Will the compound work? Will it have harmful side effects? Will it ultimately
gain FDA approval? (Most drugs do not: Of 10,000 promising compounds,
CHAPTER 10
A Project Is Not a Black Box 267
Phase III clinical
trials (large-scale
testing)
FDA application
FDA approves:
Invest in marketing
and production
Basic research;
identification of
drug candidate
Preclinical
testing
Phase I clinical
trials (first tests
on humans for
safety)
Phase II clinical

trials (small-scale
tests for efficacy
and safety)
STOPSTOPSTOP
STOPSTOPSTOP
Proceed
to Phase III
FIGURE 10.5
Research and testing of a potential new drug from discovery to initial sales. This figure concentrates on the odds that
the drug will pass all required clinical tests and be approved by the Food and Drug Administration (FDA). Only a small
fraction of drug candidates identified in basic research prove safe and effective and achieve profitable production. The
“Stop” signs indicate failure and abandonment.
12
Myers and Howe estimated the average cost of bringing one new drug to market as about $300 mil-
lion after tax. The estimate was based on R&D costs and success rates from the 1970s and 1980s, but ad-
justed for inflation through 1994. See S. C. Myers and C. Howe, “A Life-Cycle Model of Pharmaceuti-
cal R&D,” MIT Program on the Pharmaceutical Industry, April 1997.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
only 1 or 2 may ever get to market. The 1 or 2 that are marketed have to
generate enough cash flow to make up for the 9,999 or 9,998 that fail.)
2. Market success. FDA approval does not guarantee that a drug will sell. A
competitor may be there first with a similar (or better) drug. The company

may or may not be able to sell the drug worldwide. Selling prices and
marketing costs are unknown.
Imagine that you are standing at the top left of Figure 10.5. A proposed research
program will investigate a promising class of compounds. Could you write down
the expected cash inflows and outflows of the program up to 25 or 30 years in the
future? We suggest that no mortal could do so without a model to help; simulation
may provide the answer.
13
Simulation may sound like a panacea for the world’s ills, but, as usual, you pay
for what you get. Sometimes you pay for more than you get. It is not just a matter
of the time and money spent in building the model. It is extremely difficult to esti-
mate interrelationships between variables and the underlying probability distri-
butions, even when you are trying to be honest.
14
But in capital budgeting, fore-
casters are seldom completely impartial and the probability distributions on which
simulations are based can be highly biased.
In practice, a simulation that attempts to be realistic will also be complex. There-
fore the decision maker may delegate the task of constructing the model to man-
agement scientists or consultants. The danger here is that, even if the builders un-
derstand their creation, the decision maker cannot and therefore does not rely on
it. This is a common but ironic experience: The model that was intended to open
up black boxes ends up creating another one.
268 PART III
Practical Problems in Capital Budgeting
13
N. A. Nichols, “Scientific Management at Merck: An Interview with CFO Judy Lewent,” Harvard Busi-
ness Review 72 (January–February 1994), p. 91.
14
These difficulties are less severe for the pharmaceutical industry than for most other industries.

Pharmaceutical companies have accumulated a great deal of information on the probabilities of scien-
tific and clinical success and on the time and money required for clinical testing and FDA approval.
15
Some simulation models do recognize the possibility of changing policy. For example, when a phar-
maceutical company uses simulation to analyze its R&D decisions, it allows for the possibility that the
company can abandon the development at each phase.
10.3 REAL OPTIONS AND DECISION TREES
If financial managers treat projects as black boxes, they may be tempted to think only
of the first accept–reject decision and to ignore the subsequent investment decisions
that may be tied to it. But if subsequent investment decisions depend on those made
today, then today’s decision may depend on what you plan to do tomorrow.
When you use discounted cash flow (DCF) to value a project, you implicitly as-
sume that the firm will hold the assets passively. But managers are not paid to be
dummies. After they have invested in a new project, they do not simply sit back and
watch the future unfold. If things go well, the project may be expanded; if they go
badly, the project may be cut back or abandoned altogether. Projects that can easily
be modified in these ways are more valuable than those that don’t provide such flex-
ibility. The more uncertain the outlook, the more valuable this flexibility becomes.
That sounds obvious, but notice that sensitivity analysis and Monte Carlo
simulation do not recognize the opportunity to modify projects.
15
For example,
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Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
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10. A Project is Not a Black
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© The McGraw−Hill

Companies, 2003
think back to the Otobai electric scooter project. In real life, if things go wrong
with the project, Otobai would abandon to cut its losses. If so, the worst out-
comes would not be as devastating as our sensitivity analysis and simulation
suggested.
Options to modify projects are known as real options. Managers may not al-
ways use the term real option to describe these opportunities; for example, they
may refer to “intangible advantages” of easy-to-modify projects. But when they
review major investment proposals, these option intangibles are often the key to
their decisions.
The Option to Expand
In 2000 FedEx placed an order for 10 Airbus A380 superjumbo transport planes for
delivery in the years 2008–2011. Each flight of an A380 freighter will be capable of
making a 200,000 pound dent in the massive volume of goods that FedEx carries
each day, so the decision could have a huge impact on FedEx’s worldwide busi-
ness. If FedEx’s long-haul airfreight business continues to expand and the super-
jumbo is efficient and reliable, the company will need more superjumbos. But it
cannot be sure they will be needed.
Rather than placing further firm orders in 2000, FedEx has secured a place in the
Airbus production line by acquiring options to buy a “substantial number” of ad-
ditional aircraft at a predetermined price. These options do not commit the com-
pany to expand but give it the flexibility to do so.
Figure 10.6 displays FedEx’s expansion option as a simple decision tree. You
can think of it as a game between FedEx and fate. Each square represents an action
or decision by the company. Each circle represents an outcome revealed by fate. In
this case there is only one outcome in 2007,
16
when fate reveals the airfreight de-
mand and FedEx’s capacity needs. FedEx then decides whether to exercise its op-
tions and buy additional A380s. Here the future decision is easy: Buy the airplanes

only if demand is high and the company can operate them profitably. If demand is
low, FedEx walks away and leaves Airbus with the problem of selling the planes
that were reserved for FedEx to some other customer.
CHAPTER 10
A Project Is Not a Black Box 269
High
demand
2007: Observe
demand for
airfreight
2000: Acquire
delivery option
in 2008–2011
Exercise
delivery
option
Don't take
delivery
Low
demand
FIGURE 10.6
FedEx’s expansion option
expressed as a simple
decision tree.
16
We assume that FedEx can wait until 2007 to decide whether to acquire the additional planes.
Brealey−Meyers:
Principles of Corporate
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III. Practical Problems in

Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
You can probably think of many other investments that take on added value be-
cause of the further options they provide. For example
• When launching a new product, companies often start with a pilot program to
iron out possible design problems and to test the market. The company can
evaluate the pilot and then decide whether to expand to full-scale production.
• When designing a factory, it can make sense to provide extra land or floor
space to reduce the future cost of a second production line.
• When building a four-lane highway, it may pay to build six-lane bridges so
that the road can be converted later to six lanes if traffic volumes turn out to
be higher than expected.
Such options to expand do not show up in the assets that the company lists in its
balance sheet, but investors are very aware of their existence. If a company has
valuable real options that can allow it to invest in new profitable projects, its mar-
ket value will be higher than the value of its physical assets now in place.
In Chapter 4 we showed how the present value of growth opportunities (PVGO)
contributes to the value of a company’s common stock. PVGO equals the fore-
casted total NPV of future investments. But it’s better to think of PVGO as the
value of the firm’s options to invest and expand. The firm is not obliged to grow. It
can invest more if the number of positive-NPV projects turns out high or slow
down if that number turns out low. The flexibility to adapt investment to future op-
portunities is one of the factors that makes PVGO so valuable.
The Option to Abandon
If the option to expand has value, what about the decision to bail out? Projects
don’t just go on until assets expire of old age. The decision to terminate a project
is usually taken by management, not by nature. Once the project is no longer

profitable, the company will cut its losses and exercise its option to abandon the
project.
17
Some assets are easier to bail out of than others. Tangible assets are usually eas-
ier to sell than intangible ones. It helps to have active secondhand markets, which
really exist only for standardized items. Real estate, airplanes, trucks, and certain
machine tools are likely to be relatively easy to sell. On the other hand, the knowl-
edge accumulated by a software company’s research and development program is
a specialized intangible asset and probably would not have significant abandon-
ment value. (Some assets, such as old mattresses, even have negative abandonment
value; you have to pay to get rid of them. It is costly to decommission nuclear
power plants or to reclaim land that has been strip-mined.)
Example. Managers should recognize the option to abandon when they make
the initial investment in a new project or venture. For example, suppose you
must choose between two technologies for production of a Wankel-engine out-
board motor.
1. Technology A uses computer-controlled machinery custom-designed to
produce the complex shapes required for Wankel engines in high volumes
and at low cost. But if the Wankel outboard doesn’t sell, this equipment will
be worthless.
270 PART III
Practical Problems in Capital Budgeting
17
The abandonment option was first analyzed by A. A. Robichek and J. C. Van Horne, “Abandonment
Value in Capital Budgeting,” Journal of Finance 22 (December 1967), pp. 577–590.
Brealey−Meyers:
Principles of Corporate
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III. Practical Problems in
Capital Budgeting

10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
2. Technology B uses standard machine tools. Labour costs are much higher,
but the machinery can be sold for $10 million if the engine doesn’t sell.
Technology A looks better in a DCF analysis of the new product because it was de-
signed to have the lowest possible cost at the planned production volume. Yet you
can sense the advantage of technology B’s flexibility if you are unsure about
whether the new outboard will sink or swim in the marketplace.
We can make the value of this flexibility concrete by expressing it as a real op-
tion. Just for simplicity, assume that the initial capital outlays for technologies A
and B are the same. Technology A, with its low-cost customized machinery, will
provide a payoff of $18.5 million if the outboard is popular with boat owners
and $8.5 million if it is not. Think of these payoffs as the project’s cash flow in
its first year of production plus the present value of all subsequent cash flows.
The corresponding payoffs to technology B are $18 million and $8 million.
CHAPTER 10
A Project Is Not a Black Box 271
Payoffs from Producing
Outboard ($ millions)
Technology A Technology B
Buoyant demand $18.5 $18
Sluggish demand 8.5 8
If you are obliged to continue in production regardless of how unprofitable the project
turns out to be, then technology A is clearly the superior choice. But remember
that at year-end you can bail out of technology B for $10 million. If the outboard
is not a success in the market, you are better off selling the plant and equipment
for $10 million than continuing with a project that has a present value of only
$8 million.

Figure 10.7 summarizes this example as a decision tree. The abandonment op-
tion occurs at the right-hand boxes for Technology B. The decisions are obvious:
continue if demand is buoyant, abandon otherwise. Thus the payoffs to Technol-
ogy B are:
Buoyant continue own business
demand production worth $18 million
Sluggish exercise option receive
demand to sell assets $10 million
Technology B provides an insurance policy: If the outboard’s sales are disap-
pointing, you can abandon the project and recover $10 million. You can think of
this abandonment option as an option to sell the assets for $10 million. The total
value of the project using technology B is its DCF value, assuming that the com-
pany does not abandon, plus the value of the abandonment option. When you
value this option, you are placing a value on flexibility.
Two Other Real Options
These are not the only real options. For example, companies with positive-NPV
projects are not obliged to undertake them right away. If the outlook is uncertain,
you may be able to avoid a costly mistake by waiting a bit. Such options to post-
pone investment are called timing options.
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Principles of Corporate
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III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
When companies undertake new investments, they generally think about the
possibility that at a later stage they may wish to modify the project. After all, to-

day everybody may be demanding round pegs, but, who knows, tomorrow
square ones could be all the rage. In that case you need a plant that provides the
flexibility to produce a variety of peg shapes. In just the same way, it may be
worth paying up front for the flexibility to vary the inputs. For example, in Chap-
ter 22 we will describe how electric utilities often build in the option to switch be-
272 PART III
Practical Problems in Capital Budgeting
Technology A
Demand
revealed
Demand
revealed
Technology B
Buoyant
Sluggish
$18.5 million
$8.5 million
$18 million
$10 million
$8 million
$10 million
Abandon
Continue
Continue
Abandon
Buoyant
Sluggish
FIGURE 10.7
Decision tree for the
Wankel outboard motor

project. Technology B
allows the firm to
abandon the project and
recover $10 million if
demand is sluggish.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
tween burning oil to burning natural gas. We refer to these opportunities as pro-
duction options.
More on Decision Trees
We will return to all these real options in Chapter 22, after we have covered the the-
ory of option valuation in Chapters 20 and 21. But we will close this chapter with
a closer look at decision trees.
Decision trees are commonly used to describe the real options imbedded in cap-
ital investment projects. But decision trees were used in the analysis of projects
years before real options were first explicitly identified.
18
Decision trees can help
to understand project risk and how future decisions will affect project cash flows.
Even if you never learn or use option valuation theory, decision trees belong in
your financial toolkit.
The best way to appreciate how decision trees can be used in project analysis is
to work through a detailed example.

An Example: Magna Charter
Magna Charter is a new corporation formed by Agnes Magna to provide an executive
flying service for the southeastern United States. The founder thinks there will be a
ready demand from businesses that cannot justify a full-time company plane but nev-
ertheless need one from time to time. However, the venture is not a sure thing. There
is a 40 percent chance that demand in the first year will be low. If it is low, there is a 60
percent chance that it will remain low in subsequent years. On the other hand, if the
initial demand is high, there is an 80 percent chance that it will stay high.
The immediate problem is to decide what kind of plane to buy. A turboprop
costs $550,000. A piston-engine plane costs only $250,000 but has less capacity and
customer appeal. Moreover, the piston-engine plane is an old design and likely to
depreciate rapidly. Ms. Magna thinks that next year secondhand piston aircraft
will be available for only $150,000.
That gives Ms. Magna an idea: Why not start out with one piston plane and buy
another if demand is still high? It will cost only $150,000 to expand. If demand is
low, Magna Charter can sit tight with one small, relatively inexpensive aircraft.
Figure 10.8 displays these choices. The square on the left marks the company’s
initial decision to purchase a turboprop for $550,000 or a piston aircraft for
$250,000. After the company has made its decision, fate decides on the first year’s
demand. You can see in parentheses the probability that demand will be high or
low, and you can see the expected cash flow for each combination of aircraft and
demand level. At the end of the year the company has a second decision to make
if it has a piston-engine aircraft: It can either expand or sit tight. This decision point
is marked by the second square. Finally fate takes over again and selects the level
of demand for year 2. Again you can see in parentheses the probability of high or
low demand. Notice that the probabilities for the second year depend on the first-
period outcomes. For example, if demand is high in the first period, then there is
an 80 percent chance that it will also be high in the second. The chance of high
CHAPTER 10
A Project Is Not a Black Box 273

18
The use of decision trees was first advocated by J. Magee in “How to Use Decision Trees in Capital In-
vestment,” Harvard Business Review 42(September–October 1964), pp. 79–96. Real options were first
identified in S. C. Myers, “Determinants of Corporate Borrowing,” Journal of Financial Economics 5
(November 1977), pp. 146–175.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
274 PART III Practical Problems in Capital Budgeting
Turboprop
–$550
Piston
–$250
Low demand (.2)
High demand (.8)
High demand (.6)
$150
Low demand (.4)
$30
$220
$960
Low demand (.6)
High demand (.4)
$140

$930
Low demand (.2)
High demand (.8)
High demand (.6)
$100
Expand
–$150
Do not
expand
Low demand (.4)
$50
$100
$800
Low demand (.6)
High demand (.4)
$100
$220
Low demand (.2)
High demand (.8)
$180
$410
FIGURE 10.8
Decision tree for Magna Charter. Should it buy a turboprop or a smaller piston-engine plane? A second piston plane can
be purchased in year 1 if demand turns out to be high. (All figures are in thousands. Probabilities are in parentheses.)
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black

Box
© The McGraw−Hill
Companies, 2003
demand in both the first and second periods is .6 ϫ .8 ϭ .48. After the parentheses
we again show the profitability of the project for each combination of aircraft and
demand level. You can interpret each of these figures as the present value at the end
of year 2 of the cash flows for that and all subsequent years.
The problem for Ms. Magna is to decide what to do today. We solve that prob-
lem by thinking first what she would do next year. This means that we start at the
right side of the tree and work backward to the beginning on the left.
The only decision that Ms. Magna needs to make next year is whether to expand
if purchase of a piston-engine plane is succeeded by high demand. If she expands,
she invests $150,000 and receives a payoff of $800,000 if demand continues to be
high and $100,000 if demand falls. So her expected payoff is
low demand)
If the opportunity cost of capital for this venture is 10 percent,
19
then the net pres-
ent value of expanding, computed as of year 1, is
If Ms. Magna does not expand, the expected payoff is
low demand)
The net present value of not expanding, computed as of year 1, is
Expansion obviously pays if market demand is high.
Now that we know what Magna Charter ought to do if faced with the expansion
decision, we can roll back to today’s decision. If the first piston-engine plane is
bought, Magna can expect to receive cash worth $550,000 in year 1 if demand is
high and cash worth $185,000 if it is low:
NPV ϭ 0 ϩ
364
1.10

ϭϩ331, or $331,000
ϭ 1.8 ϫ 4102ϩ 1.2 ϫ 1802ϭϩ364, or $364,000
ϩ 1probability low demand ϫ payoff with
1Probability high demand ϫ payoff with high demand2
NPV ϭϪ150 ϩ
660
1.10
ϭϩ450, or $450,000
ϭ 1.8 ϫ 8002ϩ 1.2 ϫ 1002ϭϩ660, or $660,000
ϩ 1probability low demand ϫ payoff with
1Probability high demand ϫ payoff with high demand2
CHAPTER 10 A Project Is Not a Black Box 275
19
We are guilty here of assuming away one of the most difficult questions. Just as in the Vegetron mop
case in Chapter 9, the most risky part of Ms. Magna’s venture is likely to be the initial prototype proj-
ect. Perhaps we should use a lower discount rate for the second piston-engine plane than for the first.
High demand (.6)
$550,000
Invest
$250,000
Low demand (.4)
$185,000
$100,000 cash flow
plus $450,000 net
present value









$50,000 cash flow
plus net present value of
= $135,000
(.4
×
220)
+
(.6
×
100)
1.10
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Principles of Corporate
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III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
The net present value of the investment in the piston-engine plane is therefore
$117,000:
If Magna buys the turboprop, there are no future decisions to analyze, and
so there is no need to roll back. We just calculate expected cash flows and
discount:
Thus the investment in the piston-engine plane has an NPV of $117,000; the in-
vestment in the turboprop has an NPV of $96,000. The piston-engine plane is the

better bet. Note, however, that the choice would be different if we forgot to take ac-
count of the option to expand. In that case the NPV of the piston-engine plane
would drop from $117,000 to $52,000:
The value of the option to expand is, therefore,
The decision tree in Figure 10.8 recognizes that, if Ms. Magna buys one piston-
engine plane, she is not stuck with that decision. She has the option to expand by
buying an additional plane if demand turns out to be unexpectedly high. But Fig-
ure 10.8 also assumes that, if Ms. Magna goes for the big time by buying a turbo-
prop, there is nothing that she can do if demand turns out to be unexpectedly low.
That is unrealistic. If business in the first year is poor, it may pay for Ms. Magna to
sell the turboprop and abandon the venture entirely. In Figure 10.8 we could rep-
resent this option to bail out by adding an extra decision point (a further square)
if the company buys the turboprop and first-year demand is low. If that happens,
Ms. Magna could decide either to sell the plane or to hold on and hope demand re-
covers. If the abandonment option is sufficiently valuable, it may make sense to
take the turboprop and shoot for the big payoff.
Pro and Con Decision Trees
Any cash-flow forecast rests on some assumption about the firm’s future invest-
ment and operating strategy. Often that assumption is implicit. Decision trees force
the underlying strategy into the open. By displaying the links between today’s and
117 Ϫ 52 ϭϩ65, or $65,000
ϭϩ52, or $52,000
ϩ
.63.814102ϩ .2118024 ϩ .43.412202ϩ .6110024
11.102
2
NPV ϭϪ250 ϩ
.611002ϩ .41502
1.10
ϭϪ550 ϩ

102
1.10
ϩ
670
11.102
2
ϭϩ96, or $96,000
ϩ
.63.819602ϩ .2122024 ϩ .43.419302ϩ .6114024
11.102
2
NPV ϭϪ550 ϩ
.611502ϩ .41302
1.10
NPV ϭϪ250 ϩ
.615502ϩ .411852
1.10
ϭϩ117, or $117,000
276 PART III Practical Problems in Capital Budgeting
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
tomorrow’s decisions, they help the financial manager to find the strategy with the
highest net present value.

The trouble with decision trees is that they get so _____ complex so _____
quickly (insert your own expletives). What will Magna Charter do if demand is
neither high nor low but just middling? In that event Ms. Magna might sell the tur-
boprop and buy a piston-engine plane, or she might defer expansion and aban-
donment decisions until year 2. Perhaps middling demand requires a decision
about a price cut or an intensified sales campaign.
We could draw a new decision tree covering this expanded set of events and
decisions. Try it if you like: You’ll see how fast the circles, squares, and branches
accumulate.
Life is complex, and there is very little we can do about it. It is therefore unfair
to criticize decision trees because they can become complex. Our criticism is re-
served for analysts who let the complexity become overwhelming. The point of
decision trees is to allow explicit analysis of possible future events and decisions.
They should be judged not on their comprehensiveness but on whether they
show the most important links between today’s and tomorrow’s decisions. Deci-
sion trees used in real life will be more complex than Figure 10.8, but they will
nevertheless display only a small fraction of possible future events and decisions.
Decision trees are like grapevines: They are productive only if they are vigor-
ously pruned.
Decision trees can help identify the future choices available to the manager
and can give a clearer view of the cash flows and risks of a project. However, our
analysis of the Magna Charter project begged an important question. The option
to expand enlarged the spread of possible outcomes and therefore increased the
risk of investing in a piston aircraft. Conversely, the option to bail out would
narrow the spread of possible outcomes, reducing the risk of investment. We
should have used different discount rates to recognize these changes in risk, but
decision trees do not tell us how to do this. But the situation is not hopeless.
Modern techniques of option pricing can value these investment options. We
will describe these techniques in Chapters 20 and 21, and turn again to real op-
tions in Chapter 22.

Decision Trees and Monte Carlo Simulation
We have said that any cash-flow forecast rests on assumptions about future in-
vestment and operating strategy. Think back to the Monte Carlo simulation model
that we constructed for Otobai’s electric scooter project. What strategy was that
based on? We don’t know. Inevitably Otobai will face decisions about pricing, pro-
duction, expansion, and abandonment, but the model builder’s assumptions about
these decisions are buried in the model’s equations. The model builder may have
implicitly identified a future strategy for Otobai, but it is clearly not the optimal
one. There will be some runs of the model when nearly everything goes wrong and
when in real life Otobai would abandon to cut its losses. Yet the model goes on pe-
riod after period, heedless of the drain on Otobai’s cash resources. The most unfa-
vorable outcomes reported by the simulation model would never be encountered
in real life.
On the other hand, the simulation model probably understates the project’s po-
tential value if nearly everything goes right: There is no provision for expanding to
take advantage of good luck.
CHAPTER 10
A Project Is Not a Black Box 277
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
III. Practical Problems in
Capital Budgeting
10. A Project is Not a Black
Box
© The McGraw−Hill
Companies, 2003
278 PART III Practical Problems in Capital Budgeting
Most simulation models incorporate a business-as-usual strategy, which is fine
as long as there are no major surprises. The greater the divergence from expected

levels of market growth, market share, cost, etc., the less realistic is the simulation.
Therefore the extreme high and low simulated values—the “tails” of the simulated
distributions—should be treated with extreme caution. Don’t take the area under
the tails as realistic probabilities of disaster or bonanza.
SUMMARY
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There is more to capital budgeting than grinding out calculations of net present
value. If you can identify the major uncertainties, you may find that it is worth un-
dertaking some additional preliminary research that will confirm whether the proj-
ect is worthwhile. And even if you decide that you have done all you can to resolve
the uncertainties, you still want to be aware of the potential problems. You do not
want to be caught by surprise if things go wrong: You want to be ready to take cor-
rective action.
There are three ways in which companies try to identify the principal threats to
a project’s success. The simplest is sensitivity analysis. In this case the manager con-
siders in turn each of the determinants of the project’s success and recalculates
NPV at very optimistic and very pessimistic levels of that variable. This establishes
a range of possible values. The project is “sensitive to” the variable if the range is
wide, especially on the pessimistic side.
Sensitivity analysis of this kind is easy, but it is not always helpful. Variables do
not usually change one at a time. If costs are higher than you expect, it is a good
bet that prices will be higher also. And if prices are higher, it is a good bet that sales
volume will be lower. If you don’t allow for the dependencies between the swings
and the merry-go-rounds, you may get a false idea of the hazards of the fairground
business. Many companies try to cope with this problem by examining the effect
on the project of alternative plausible combinations of variables. In other words,
they will estimate the net present value of the project under different scenarios and
compare these estimates with the base case.
In a sensitivity analysis you change variables one at a time: When you analyze
scenarios, you look at a limited number of alternative combinations of variables. If

you want to go whole hog and look at all possible combinations of variables, then
you will probably use Monte Carlo simulation to cope with the complexity. In that
case you must construct a complete model of the project and specify the probabil-
ity distribution of each of the determinants of cash flow. You can then ask the com-
puter to select a random number for each of these determinants and work out the
cash flows that would result. After the computer has repeated this process a few
thousand times, you should have a fair idea of the expected cash flow in each year
and the spread of possible cash flows.
Simulation can be a very useful tool. The discipline of building a model of the
project can in itself lead you to a deeper understanding of the project. And once
you have constructed your model, it is a simple matter to see how the outcomes
would be affected by altering the scope of the project or the distribution of any of
the variables.
Elementary treatises on capital budgeting sometimes create the impression that,
once the manager has made an investment decision, there is nothing to do but sit
back and watch the cash flows unfold. In practice, companies are constantly mod-