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• Taking each Negative NPV project decreases ¬rm value.

(These rules do not hold if projects have non-zero externalities, as we shall explain below.)
Project NPVs are additive, because all cash ¬‚ows have been translated into the same units,
today™s dollars, and the project interaction term is zero. Project independence makes decisions
a lot easier: for twenty projects, only twenty independent decisions (accept or reject) have to
be made, not a million.


Important: You can evaluate zero interaction projects independently. In this
case, you can simply add project net present values.
¬le=npvadvice.tex: LP
156 Chapter 7. Capital Budgeting (NPV) Applications and Advice.

Positive Project Interactions
Positive interactions mean that the sum of the parts is worth more than the parts individually.
Positive interactions
exist when taking one If one project has a positive in¬‚uence on the net present value of another project, you cannot
project increases the
consider it without considering this positive in¬‚uence. For example, consider creating a prod-
value of another project.
uct as one project and an advertising campaign as another project. The advertising campaign
Projects with positive
interactions are often
project is of lesser use without a product, and the product is of lesser use without the adver-
considered as “bundles.”
tising campaign. You must consider creating a product and an advertising campaign together.
Indeed, in many cases,
what makes a project a Such positive externalities are even more plentiful in smaller decisions: A secretary with word
project in the ¬rm™s
processing skills is less useful without a word processor, and a word processor is less useful
mind is the indivisibility
without a secretary who can use it. A computer keyboard is less useful without a computer,
of its components.
and a computer is less useful without a keyboard. In fact, some projects or products only
make sense if purchased together. In this case, producers may bundle them together and/or
purchasers may only buy them as bundles.
In the corporate context, investment in infrastructure is another classical example of positive
Infrastructure can
bene¬t many different project interactions. For example, building a road, hiring a security ¬rm, or laying a fast Internet
projects.
connection could enhance the value of many divisions simultaneously. The ¬rm should factor
in the increase in value to all divisions when deciding on how much infrastructure to add.
Don™t take positive externalities too lightly: On a philosophical basis, positive project interac-
Positive externalities is
why ¬rms exist to begin tions are the reason why ¬rms exist in the ¬rst place. If there were no cost savings to having all
with.
resources combined in the ¬rm, we might as well not bother and instead all work as individuals.




Important: When deciding whether to take a project, you must credit all positive
interactions to the project. The overall NPV is higher than the individual project
NPVs alone.


Internal con¬‚ict and cost allocation procedures issues (further discussed as “agency con¬‚icts”
Agency issues often
prevent proper crediting. below) often hinder corporations from taking advantage of many positive externalities. For
example, in real life, our division managers might argue that they should not be charged for
the Internet connection, because they did not request it and therefore do not really need it
(even if it were to increase their divisions™ values). After all, division managers would prefer
getting Internet for free from the company instead of paying for it out of their own division
budgets.
Nowadays, managers who want to acquire other companies usually claim the presence of large
Another phrase for
positive externalities. positive externalities. Synergies, the managerial term for positive externalities between an
acquirer and a potential acquisition target, has become an important managerial buzzword. For
example, in the 2001 acquisition of Compaq by Hewlett-Packard, HP touted synergies of $2.5
billion dollars”most from cutting employees. Of course, whether synergies will be realized is
always another question.


Negative Project Interactions
Negative interactions mean that the sum of the parts is worth less than the parts individually.
Negative interactions
exist when taking one In this case, projects have negative in¬‚uences on one another, and thereby decrease one an-
project decreases the
other™s value. Economists sometimes call negative externalities diseconomies of scale. Here
value of another project.
are a few examples.
Pollution,
cannibalization, and
limited attention span
Pollution and Congestion If there is only one major road to two divisions, and the tra¬c of
are examples thereof.
one division clogs up the tra¬c to the other division, it can cause a loss of cash ¬‚ow
in the other division. A division that wants to expand and thereby clog up more of the
existing infrastructure will not want to pay for the congestion cost that its own expansion
will impose on the other divisions. (Of course, it is the overall ¬rm™s headquarters that
should step in and allow the expansion only if the NPV is positive after taking into account
the negative externalities imposed on other divisions.)
¬le=npvadvice.tex: RP
157
Section 7·1. The Economics of Project Interactions.

Cannibalization If a new Apple computer can produce $100,000 in NPV compared to the older
Windows machine that only produced $70,000 in NPV, how should we credit the Apple
machine? The answer is that the Apple would eliminate the positive cash ¬‚ows produced
by the existing Windows machine, so the cash ¬‚ow of the project “replace Windows with
Apple” is only the $100,000 minus the $70,000 that the now unused Windows machine
had produced.

Bureaucratization and Internal Con¬‚ict If more projects are adopted, project management
may ¬nd it increasingly di¬cult to make good decisions in a reasonable time frame. This
may require more cumbersome bureaucracy and reduce cash ¬‚ows for all other divisions.

Resource Exhaustion Perhaps the most common source of negative externalities”and often
underestimated”is limited attention span. Management can only pay so much attention
to so many di¬erent issues. An extra project distracts from the attention previously
received by existing projects.

Although costs always include opportunity costs, in the case of negative project externalities,
they are more obvious. If your project cannibalizes another project or requires more attention,
it™s clearly an opportunity cost.



Important: When deciding whether to take a project, charge all negative inter-
actions to the project. The overall NPV is lower than the individual project NPVs
alone.


Again, as in the case of positive externalities, agency issues and cost allocation systems of- Agency issues often
prevent proper costing.
ten prevent proper accounting for negative externalities in the real world. Whichever division
created the negative externality will argue that it is not its problem, and that the complaining
division overstates the problem. Clearly, companies that are better at overcoming these issues
will end up being more pro¬table.


7·1.C. One More Project: Marginal Rather Than Average Contribution

Usually, managers do not make the decision for all interacting projects simultaneously. Instead, The Capital Budgeting
Rule for one extra
many projects are already in place. Although existing projects should also constantly be evalu-
project requires taking
ated in an ideal world, the manager often has to make a decision about adding or not adding a all project interactions
single new project (or project complex) only in the real world. For practical purposes, the old into account.
projects are present, given, and unalterable. So the new project may have positive or negative
externalities on other existing projects, and the question is how best to decide whether to take
it or not. This simpli¬es the decision even further: the question is now only whether the new
project adds or subtracts value from the total. In this case, economists use the concept of
decision on the margin”holding the existing projects as is, what is the additional contribution
of the new project?
Return to our aquarium example. The aquarium haunts us.


• If you already have the octopus in the tank [with its NPV of $125,000], should you add the
shark? If you do, you pay an additional (“marginal”) $50,000 and get nothing”because
the shark will become octopus food, which generate no additional ticket sales. Thus, the
marginal bene¬t of adding the shark is ’$50, 000. Therefore you should not add the
shark.
• If you already have the shark in the tank with its NPV of $70,000, should you add the
octopus? Your marginal cost to add the octopus is $75,000 for the beast itself. In ticket
sales, you would lose the $120,000 in shark receipts but gain $200,000 in octopus re-
ceipts. Your net bene¬t would therefore be $200, 000 ’ $120, 000 + $75, 000 = +$5, 000.
Consequently, you should add the octopus, even though you know that your shark will
become pet food!
¬le=npvadvice.tex: LP
158 Chapter 7. Capital Budgeting (NPV) Applications and Advice.

Of course, if you can sell the shark or put it into its own aquarium, your calculations
would change”though you would then also have to consider the marginal cost of selling
the shark or getting a new aquarium.




Important:

• The decision on whether to take one additional project should be made based
on the rule

Take New Project if

Total Firm NPV with > Total Firm NPV with-
New Project out New Project

• This means that the single new project should be credited with any value
increase or value decrease that it confers on other projects.

• When considering a project on the margin (i.e., extra), credit/charge to this
project all externalities that this project conveys onto the existing ¬rm.

• Everything else equal, projects with positive externalities on the rest of the
¬rm have higher marginal bene¬ts than projects with negative externalities.



We now discuss some more examples of how to think in terms of marginal costs and bene¬ts.


Working with Economies of Scale
Consider an example in which there are economies of scale”the more product we produce,
An example in which the
cost function creates the lower the average product price. Say, our factory can produce at
economies of scale.
$10
(7.3)
Average Price Per Good = $4 + .
x+1
Thus, producing one good costs $4 + $10/(1 + 1) = $9/good, and one-hundred goods costs
$4 + $10/(100 + 1) = $4.10/good. The company is currently selling 5 goods domestically, each
for a price of $8.00. It earns a

$10
Total Pro¬t @ 5 items = 5·$8 ’ 5· $4 + = $11.67 . (7.4)
(5 + 1)

The company is considering a new foreign sales division that would cost $16 to open, and
that could sell another 5 units at $8. The average price for the company producing 10 units
would be $4+$10/11= $4.91/unit. Therefore, 5 units cost $24.55 to produce. The total cost of
$16 + $24.55 = $40.55 exceeds the total pro¬t of 5 · $8 = $40. If considered by itself, opening
a foreign sales division would not be a positive NPV project.
Now compute the total ¬rm pro¬t if the ¬rm were to open the foreign sales division. Ten units
The foreign sales
division also lowers the would sell for a pro¬t of $80. Subtracting the opening costs of $16 and production costs of
cost of domestic
10 · $4.91 = $49.19 would earn a
production!

$10
Total Pro¬t @ 10 items = 10·$8 ’ 10· $4 + ’ $16 ≈ $14.91 . (7.5)
(10 + 1)

This is more than the $11.67 that the ¬rm earned without the foreign sales division. The reason
is that the foreign o¬ce has an additional marginal bene¬t: it reduces the average production
cost experienced by the domestic o¬ce. This cost improvement is a positive externality that
must be credited to the project”or the ¬rm will make the bad decision of not opening the
foreign division.
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159
Section 7·1. The Economics of Project Interactions.

All this is easier to see when we translate it into terms of marginal costs and bene¬ts. The extra Think of these
economies of scale in
marginal cost of each item changes item by item”it is the di¬erence in total costs of each item:
terms of marginal costs.


Units Average Total Marginal Units Average Total Marginal
1 $9.00 $9.00 $9.000 6 $5.42 $32.57 $4.238
2 $7.33 $14.67 $5.667 7 $5.25 $36.75 $4.179
3 $6.50 $19.50 $4.833 8 $5.11 $40.89 $4.139
4 $6.00 $24.00 $4.500 9 $5.00 $45.00 $4.111
5 $5.66 $28.33 $4.333 10 $4.90 $49.09 $4.091


Going from 5 items to 10 items, production creates extra costs of $4.333 to $4.091 for a
marginal cost of $20.76. There would be an additional marginal cost of $16 to open the
foreign o¬ce. The total marginal cost would thus be $36.76. The marginal bene¬t of 5
extra items would be $40. Therefore, the foreign sales division gives us marginal NPV of
$40 ’ $36.76 ≈ $3.242. This is exactly the di¬erence between $11.67 from Formula 7.4 and
$14.91 from Formula 7.5. So thinking in terms of marginal costs and bene¬ts is just a more
convenient way to compare overall project values.
In Figure 7.1, we show three di¬erent cost functions that are based on di¬erent externalities. Graphical Display of
Economies of Scale
When there are positive externalities, as there are when there are economies of scale, then
each item has a positive e¬ect on the next item produced, so the marginal cost is decreasing
with the number of units. When there are negative externalities, as there are when there are
diseconomies of scale (it may become harder and harder to ¬nd the necessary input materials),
then each item has a negative e¬ect on the next item produced, so the marginal cost is increasing
with the number of units.


Figure 7.1. The E¬ects of Economies of Scale on Marginal Costs


+

+
Increasing Marginal Costs
5.5




+
Diseconomies of Scale +
Marginal Production Cost




+
+
+
5.0




+
+
’ +
o o o o o o o o oooooooo o o o o
+
Constant Costs
+
No Benefit or Cost To Scale
4.5




’ +
+

+
’ Decreasing Marginal Costs
+ ’ ’ ’
’ ’ Economies of Scale ’ ’
’ ’ ’’’’’
4.0




5 10 15 20

Number of Units Produced

Here are three examples to show positive externalities, zero externalities, and negative externalities in the production
of goods. The decreasing marginal cost function represents economies of scale. In this case, it also looks as if we have
diminishing decreasing marginal costs”though the production cost is less for each additional product, it only is a
little less good-by-good if we produce many of them. The increasing marginal cost function represents diseconomies
of scale”the more units we produce, the more we have to pay for each additional unit.
¬le=npvadvice.tex: LP
160 Chapter 7. Capital Budgeting (NPV) Applications and Advice.

In my opinion, decreasing marginal costs are responsible for the biggest corporate success
Economies of scale are
often responsible for the stories. For example, Wal-Mart and Dell have managed not only to use their scale to negotiate
big corporate success
considerable supplier discounts, but have also created inventory and distribution systems that
stories of our time.
allow them to spread their ¬xed costs very e¬ciently over the large quantities of goods they
sell. They have the lowest costs and highest industry inventory turnover rates”two factors
that allow them to bene¬t tremendously from their economies of scale. Similarly, Microsoft
enjoys economies of scale”with a large ¬xed cost and almost zero variable cost, they can
swamp the planet with copies of Windows. No commercial alternative can compete”Microsoft
can always drop its price low enough to drive its competitor out of business. The socially
optimal number of operating systems software companies is very small and may even be just
one”it is what economists call a natural monopoly. If we consider our economy as one big
¬rm, we would not want to incur the same huge ¬xed software development cost twice. The
same applies to utilities: we would not want two types of cable strung to our house, two types
of telephone lines, and two types of power lines. Of course, companies with monopolies will
want to charge higher prices to exploit their monopoly powers. This helps the companies,
but hurts the economy overall. Society has therefore often found it advantageous to regulate
monopolists. Unfortunately, the regulatory agencies are themselves often “captured” by the
companies that they are supposed to regulate, which sometimes can hurt the economy even
more than the monopolies themselves. There are no easy and obvious solutions.


Working with Sunk Costs
Sunk costs are, in a sense, the opposite of marginal costs. A sunk cost is a cost that cannot
Sunk costs are
ubiquitous, and the be altered and that therefore should not enter into your decisions today. It is what it is. Sunk
opposite of costs that
costs are ubiquitous, if only because with the passage of time, everything is past or irrevocably
should enter
decided and thus becomes a sunk cost.
decision-making.

For example, consider circuit board production”a very competitive industry. If you have just
An example of how
capital investments completed a circuit board factory for $1 billion, it is a sunk cost. What matters now is not
become sunk, and then
that you spent $1 billion, but how much the production of each circuit board costs. Having
how production itself
invested $1 billion is irrelevant. What remains relevant is that the presence of the factory
becomes sunk.
makes the marginal cost of production of circuit boards very cheap. It is only this marginal
cost that matters when you decide whether to produce circuit boards or not. If the marginal
board production cost is $100 each, but you can only sell them for $90 each, then you should
not build boards, regardless of how much you spent on the factory. Though tempting, the
logic of “we have spent $1 billion, so we may as well put it to use” is just plain wrong. Now,
presume that the market price for boards is $180, so you go ahead and manufacture 1 million
boards at a cost of $100 each. Alas, your production run has just ¬nished, and the price of
boards”contrary to everyone™s best expectations”has dropped from $180 each to $10 each.
At this point, the board production cost is sunk, too. Whether the boards cost you $100 to
manufacture or $1 to manufacture is irrelevant. The cost of the production run is sunk. If
boards now sell at $10 each, assuming you cannot store them, you should sell them for $10
each. Virtually all supply costs eventually become sunk costs, and all that matters when you
want to sell a completed product is the demand for the product.
One more note”time itself often, but not always, decides on what is sunk or not. Contracts
Time is a good proxy,
but not the deciding may allow you to undo things that happened in the past (thereby converting an ex-post sunk
factor.
cost into a cost about which you still can make decisions), or bind you irrevocably to things
that will happen in the future.



Important: A sunk cost has no cost contribution on the margin. It should
therefore be ignored.
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Section 7·1. The Economics of Project Interactions.

Overhead Allocation and Unused Capacity
A closely related mistake is to forget that “overhead” is often a sunk cost. By de¬nition, over- Allocating already
existing overhead to a
head is not a marginal cost, but something that has been incurred already and is allocated to
project valuation is a
departments. For example, assume your ¬rm has spent $500,000 on a computer that is cur- common example of bad
rently idle half the time. It serves only one division. Assume that another division can take an project decision making.
additional project that produces $60,000 in net present value, but that will consume twenty
percent of the computer™s time. Should your ¬rm take this project? If twenty percent of the
cost of the computer is allocated to this new project (i.e., 20% · $500, 000 = $100, 000), the
net present value of the new project would appear to be a negative ’$40, 000. But the correct
decision process is not to allocate the existing overhead as a cost to divisions. The $500,000
on overhead has already been spent. The computer is a sunk cost”assuming that it really
would sit idle otherwise and ¬nd no better purpose. It may seem unfair to have charged only
the original division for the computer and exempt the opportunistic other division. Yet taking
this additional project will produce $60,000 in pro¬ts without cost”clearly, a good thing. I
personally know of plenty of examples in which overhead allocation has killed very pro¬table
projects.
“Capacity” is a subject that is closely related. For example, a garage may be currently only used If capacity is unused, it
should have a zero price.
for half its space. Adding the project “another car” that could also park in the garage would
reduce this car™s depreciation. The garage would then have a positive externality on project
“corporate cars.” The marginal cost of storing other cars in the garage should be zero.


Real World Dilemmas
But should we really charge zippo for parking corporate cars if we suspect that the unused Often we do not have
easy, smooth margins.
capacity will not be unused forever? What if a new division might come along that wants to
rent the ¬ve currently unused garage space in the future? Do we then kick out all current
parkers? Or, how should we charge this new division if it wanted to rent six spaces? Should
we give it the ¬ve remaining unused parking spots for free? Presuming that garages can only
be built in increments of ten parking spots each, should we build another ten-car garage, and
charge it entirely to this new division that needs only one extra parking spot in the new garage?
Should this new division get a refund if other divisions were to want to use the parking space”
but, as otherwise unused parking space, should we not use the garage appropriately by not
charging for the nine extra spaces that will then be a free resource?
When there are high ¬xed and low variable costs, then capacity is often either incredibly cheap Fixed costs are often
responsible. The old
(or even free) or it is incredibly expensive”at least in the short run. Still, the right way to
method works,
think of capacity is in terms of the relevant marginal costs and marginal bene¬ts. From an though”use “on the
overall corporate perspective, it does not matter how or who you charge”just as long as you margin.”
get the optimal capacity utilization. To the extent that cost allocation distorts optimal marginal
decision-making, it should be avoided. In our case, if optimal capacity utilization requires zero
parking cost for the old garage, then so be it. Of course, when it comes to the decision to build
an entirely new garage, you simply weigh the cost of building the 10-spot garage against the
reduced deterioration for 1 car.
Unfortunately, real life is not always so simple. Return to our example on Page 156 of an It becomes much harder
if we do not know the
Internet connection, that has a positive in¬‚uence on all our divisions. We know that our division
right outcome, so we
managers will not want to pay for it if they can enjoy it for free”so we cannot rely on them have to play games with
telling us the correct marginal bene¬t. So, would it solve our problem to charge only divisions our subordinate
managers.
that are voluntarily signing up for the Internet connection, and to forcibly exclude those that
do not sign up? If we do, then we solve the problem of everyone claiming that they do not need
the Internet connection. However, we are then stuck with the problem that we may have a lot
of unused network capacity that sits around, has zero marginal cost, and could be handed to
the non-requesters at a zero cost. It would not impose a cost on anyone else and create more
pro¬t for the ¬rm. Of course, if we do this, or even if we are suspected to do this, then no
division would claim that they need the Internet to begin with, so that they will get it for free.
In sum, what makes these problems so di¬cult in the real world is that as the boss, you often
do not know the right marginal bene¬ts and marginal costs, and you end up having to “play
¬le=npvadvice.tex: LP
162 Chapter 7. Capital Budgeting (NPV) Applications and Advice.

games” with your division managers to try to make the right decision. Such is real life!
Solve Now!
Q 7.1 A company must decide if it should move division DA to a new location. If division DA
moves, it will be housed in a new building that reduces its operating costs by $10,000 per year
forever. The new building costs $120,000. Moving division DA allows division DB to expand
within the old factory. This enables DB to increase its pro¬tability by $3,000 per year forever. If
the discount rate is 10%, should division DA move?


Q 7.2 A ¬rm can purchase a new punch press for $10,000. The new press will allow the ¬rm
to enter the widget industry, and thereby earn $2,000 in pro¬ts per year forever. However, the
punch press will displace several screw machines that produce $1,500 in pro¬ts per year. If the
interest rate is 10%, should the new punch press be purchased?


Q 7.3 A company rents 40,000 square feet of space and is using 30,000 square feet for its present
operations. It wishes to add a new division that will use the remaining 10,000 square feet. If it
adds the division, equipment will cost $210,000 once, and the operations will generate $50,000
in pro¬ts every year. Presently, the o¬ce sta¬ costs $160,000 per year. However, the expansion
requires a larger sta¬, bringing costs up to $180,000 per year. If the cost of capital r = 10%,
should the ¬rm expand?




7·2. Comparing Projects With Different Lives and Rental Equiv-
alents

Let me switch gears, and cover another interesting and not immediately obvious application
Comparing contracts
with unequal lives. of NPV. A customer who currently pays $350 each year (at year end) is o¬ering you a 5-year
deal, in which she pays $1,000 upfront, followed by $100 each year end, for total payments of
$1,500. The appropriate cost of capital is 10%. Should you accept this?
At ¬rst glance, you only get $1, 500/5 = $300 per year, less than the $350 you are getting now.
You get less per
year”but more present The present value of this new contract is even less:
value if you remember
that you currently are
$100 $100 $100 $100 $100
paid only at year™s end, PV = +$1, 000 + + + + +
(1 + 10%) (1 + 10%) (1 + 10%) (1 + 10%) (1 + 10%)5
2 3 4
too.. (7.6)
= $1, 379.08 ,

which comes to only $275.82 per year. Looks like a bad idea, does it not? Wrong! What you
need to realize is that the customer currently pays only $350 per year in the future, too, so the
current arrangement of $350 per year is an annuity that is worth only

$350 $350 $350 $350 $350
PV = + + + +
(1 + 10%) (1 + 10%) (1 + 10%) (1 + 10%) (1 + 10%)5
2 3 4
(7.7)
= $1, 326.78 .

The new contract is better than the current annual arrangement.
You can also think of this problem in terms of an equivalent “rental per annum value” of the
Rental Value Equivalents.
contract. (It is sometimes abbreviated as EAC for equivalent annual cost.) What kind of annuity
would be equal to our contract? The answer is
x x x x x
+ + + +
(1 + 10%) (1 + 10%) (1 + 10%) (1 + 10%) (1 + 10%)5
2 3 4
(7.8)
x 1
= Annuity( x, 5 years, 10% ) = · 1’ = $1, 379.08 .
(1 + 10%)5
10%
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163
Section 7·2. Comparing Projects With Di¬erent Lives and Rental Equivalents.

The solution is an annual rental value of x = $363.80. So, the new contract would be worth the
same as annual contracted payments of $363.80”more than the $350 that you are currently
receiving.
Rental values are a good way to think of contracts with di¬erent lengths. For example, would Comparing multi-period
contracts.
you prefer our contract to another one in which the customer pays $800 upfront and $200 each
year for two years? The NPV of this contract is only

+$200 +$200
NPV = $800 + + = $1, 147.11 , (7.9)
1 + 10% (1 + 10%)2

which is lower than the $1,379.08 that the earlier contract o¬ered”but this is like comparing
apples and oranges. Again, if you think about it, with the newer contract, you would still have
another three years of product to sell. In this case, the NPV of the new contract of $1,147.11
for two years worth of production is equivalent to rental payments of

x 1
NPV = · 1’ x = $660.95 . (7.10)

(1 + 10%)2
10%

The annual ¬‚ow of $660.95 is much higher than the annual ¬‚ow of $363.80 that the older
contract o¬ered, so you might prefer the newer contract.
However, as appealing as comparing rental equivalents may be, this technique depends on the Rental equivalents work
only if you believe they
very strong assumption that you can repeat contracts when they expire and for quite a number
are repeatable.
of times (at least until both contracts expire in the same year). To see this, ask yourself which
contract you would prefer if you believed that your customer would jump to the competition
after the two-year contract is over. If your project would be otherwise useless, you might now
prefer the ¬ve-year contract. Similarly, you have assumed that you do not value ¬‚exibility”for
example, the shorter contract may allow you to attract an even better customer in 3 years. Or,
it may be that you value the ¬‚exibility of not repeating the contract, if you fear that your input
costs may be much higher in 3 years (so your net cash ¬‚ows would not be what you input
above). On the other hand, you may not like giving your customer the ¬‚exibility of jumping
ship. In this case, ¬‚exibility would be a cost to you, and you would prefer the 5-year contract.
In Section 7·4, we will be talking more about the value of such ¬‚exibility.
Similar rental equivalent value problems also often arise when you compare di¬erent technologies” Bases
Cost
for example, you can purchase a machine that is likely to last for 18 years, and you must com-
pare it against another machine that is likely to last for 22 years. The method for solving these
problems is exactly the same, so try it in the next question.
Solve Now!
Q 7.4 Machine A costs $10,000 upfront, and lasts for 18 years. It has annual maintenance
costs of $1,000 per year. Machine B costs $15,000 upfront, lasts for 22 years, and has annual
maintenance costs of $800 per year. Both machines produce the same product. The interest rate
is 12% per annum.

(a) What is the PV of the costs of each machine?

(b) What is the rental equivalent of either machine?

(c) Which machine is the better purchase if you assume neither value to ¬‚exibility, nor expect
di¬erent machine costs or contracting conditions in the future?
¬le=npvadvice.tex: LP
164 Chapter 7. Capital Budgeting (NPV) Applications and Advice.

7·3. Expected, Typical, and Most Likely Scenarios

Let us move on to a di¬erent, but also common error when managers apply NPV. This error is
The NPV formula
requires expected cash primarily conceptual. Under uncertainty, the NPV formula requires the expected cash ¬‚ows in
¬‚ows, not typical cash
the numerator. The mistake is to think of the typical cash ¬‚ow (in statistical terminology, the
¬‚ows. The difference is
median) or the most likely cash ¬‚ow (the mode), instead. If you do this, you will fail to consider
low-probability events.
low-probability events: a plane crash, a legal suit, an especially severe recession, or a terri¬c
new client.
For example, your business may have the following payo¬s:
An Example.



Event Probability Value
Good Business 50% $1,200,000
Normal Business 45% $1,000,000
’$10, 000, 000
Lawyers Sue For Punitive Damages 5%

The most likely payo¬ is $1,200,000. The typical payo¬ is $1,000,000. The expected payo¬,
however, is only

E (Payo¬) = 50% · $1, 200, 000 + 45% · $1, 000, 000 + 5% · (’$10, 000, 000)
(7.11)
= .
$550, 000

It is the latter that is required in an NPV analysis. If you run this business 100 times, you
would receive $1.2 million 50 times, $1 million forty-¬ve times, and lose $10 million 5 times.
Fortunately, if the statistical distribution is symmetric, e.g., as it is in the case of the normal
distribution, then the center of the distribution is the mean, median, and mode. Unfortunately,
few businesses are immune to low-probability shocks, so the distinction between mean, median,
and mode can rarely be taken lightly.
¬le=npvadvice.tex: RP
165
Section 7·4. Future Contingencies and Real Options.

7·4. Future Contingencies and Real Options

Now we move on to an important complication in estimating future expected cash ¬‚ows. When A real option is the value
of ¬‚exibility to change
the future is uncertain, then the ability to change course in the future”depending on the eco-
course in the future.
nomic environment at the time”creates value. A business may expand its size, accelerate its
production, and venture out into related or spin-o¬ businesses, if the demand for its products
increases, or if the costs of its inputs fall. Similarly, the ¬rm may reduce, delay, or stop produc-
tion if its economic environment deteriorates. This is called a real option (or strategic option).
Conceptually, these options are just a variant of the problem of assessing the expected cash
¬‚ows (and their cost of capital) correctly. Practically, the resulting complications can be so
di¬cult that one could write a whole book on the subject”and some have done so.


7·4.A. A Basic Introduction

Let us illustrate a real option with a simple example. A factory may cost $1 million to build in An Example.
year 1. In year 2, it can produce $2 million worth of inputs into $3 million worth of outputs.
The expected pro¬ts next year of $1 million are no better than the actual cost today. From an
NPV perspective, this factory does not appear worthwhile. Or does it?


Table 7.1. State Contingent Factory Payo¬ Table

Ignore Real Option Recognize Real Option
Always Run Factory Shut down if Optimal
Prob Dumb NPV Smart NPV
1/2
Demand is low $0
$1,000,000
“ $2,000,000
= “$1,000,000 $0
1/2
Demand is high $5,000,000 $5,000,000
“ $2,000,000 “ $2,000,000
$3,000,000 $3,000,000
Expected Value $1,000,000 $1,500,000




Ignore Real Option Recognize Real Option


¨ NPV
B B
¨ NPV
¨¨ = ’$1 million ¨¨ = ’$1 million
eak eak
¨ ¨
W¨ W¨ = $0 million
s¨ s¨
di i
an ¨ 1/2
d¨ 1/2
an
¨ lity ¨¨
D e ¨ abi e m ability
m
¨ ob ¨
D
¨ Pr ¨ Prob
s¨ ¨
s
¨ ¨
rr rr
r Pro r Pro
r r
De r babil De r babil
ma rrity 1 ma rrity 1
nd r /2 nd r /2
is r is r
Str r Str r
on r on r
gr gr
r NPV r NPV
j j
= +$3 million = +$3 million



Expected Value: $1 million Expected Value: $1.5 million
¬le=npvadvice.tex: LP
166 Chapter 7. Capital Budgeting (NPV) Applications and Advice.

If the product can be either in high demand (yielding $5 million) or in low demand (yielding
It is not the expected
value that matters. $1 million)”both equally likely”the expected value of output can indeed be $3 million, as
stated. This is what the ¬rm expects to earn if managers always operate the factory, regardless
of demand. But if managers can shut down the factory when demand is low, then owning the
factory is similar to owning a contingent equity claim with limited liability (Chapter 5): you can
get the upside with less downside. Take a look at the payo¬ table in Table 7.1. It shows that if
the factory always runs, its expected net cash ¬‚ows are less than if managers shut down the
factory when demand is low. It is the managerial ¬‚exibility that increases the expected factory
cash ¬‚ow from $1 million to $1.5 million, which means that the factory is well worth building
under reasonable cost-of-capital scenarios.


7·4.B. More Complex Option Valuation in a Risk-Neutral World

In order to gain some more intuition on how to think about and value real options, let us
We are ignoring
different costs of capital now work with some two-period examples (well, two periods beyond today). By assuming the
for real options.
world is risk-neutral, we are also ignoring that discount rates can be higher when there are
more strategic options. (Depending on the context, not worrying too much about the correct
discount rate can be forgiveable or deadly.) Trust me: it will get tough enough without this
extra complication.


7·4.C. Decision Trees: One Set of Parameters

Assume that you own a ¬rm that can produce 150,000 units of a good at $100/u. The retail
Our more involved
example. price of your good was $500/u recently, but you now expect it to go up or down either $100/u
this year, i.e., either to $400/u or $600/u. The year thereafter, you expect it to go up or down
by $200/u. For example, if the retail price were to become $600/u, you expect it to be either
$400/u or $800/u the following year. All price changes are equally likely. The ¬xed costs of
running the plant are $50 million, and rent (regardless of whether you run the plant or not) is
$10 million.
The world is risk-neutral and the prevailing interest is 10% per year, which applies to this year™s
The cost of capital and
timing assumptions. coming cash ¬‚ows, and twice compounded to the following year™s cash ¬‚ows. Moreover, we
presume that you know at the beginning of each year what the price over the whole year will
be, because you receive customer orders at this point. (To model intra-year uncertainty more
realistically, you would have to deal with more periods, not any more di¬cult in concept but
much more tedious.)
As an example, let us compute the ¬rm value if you know that the price will go to $600/u and
Here is how our
assumptions work then to $400/u, and if you know that you will operate the plant this year but not the following
together.
year. The ¬rst year, you would earn revenues of 150, 000u · ($600/u ’ $100/u) = $75M, pay
¬xed costs of $50M, and rent of $10M. Your net pro¬ts would be $15M, which discounts to
$13.64M. The second year, you would earn no revenues and pay no ¬xed costs, but still pay
rent of $10M. This discounts to $8.3M. In sum, under this price path and with this operating
policy, your ¬rm would have an NPV of $5.4M.
We want to consider the value of this project in a number of scenarios, which di¬er in the
Our goal
assumption of your ability to know and respond to the prevailing environment.

Upfront Choice First, let us compute the value under in¬‚exible behavior. This is one extreme
benchmark. What is the value if you have to make your decision today of whether to
operate or not in all future scenarios? That is, the ¬rm would either have to operate or
not operate in both future periods.

• If you do not start the plant, you would simply value the ¬rm at $0.
• If you do start the plant, then you must make the calculations that the tree in Fig-
ure 7.2 shows. If the price increases, you earn $75M ’ $50M ’ $10M = $15M. If
it decreases, you earn ’$45M ’ $1M ’ $5M = ’$15M. Therefore, your expected
¬le=npvadvice.tex: RP
167
Section 7·4. Future Contingencies and Real Options.



Figure 7.2. Value Under No Flexibility ” Always Operate The Plant


Retail P=$800, Cost C=$100
Decision: None (Plant Runs)
Revenues: $105,000,000
Fixed Costs: $50,000,000
Rent: $10,000,000
Retail P=$600, Cost C=$100
Net: $45,000,000
 

Decision: None (Plant Runs)
 
Revenues: $75,000,000
  Retail P=$400, Cost C=$100
Fixed Costs: $50,000,000
d Decision: None (Plant Runs)
Rent: $10,000,000
d
d
‚ Revenues: $45,000,000
Net: $15,000,000
Fixed Costs: $50,000,000
¡
! Rent: $10,000,000
¡ Net: ’$15, 000, 000
¡
Retail P=$500 (known)
¡
Flexibility: Plant (or not)
e
e Retail P=$600, Cost C=$100
e Decision: None (Plant Runs)
e
… Revenues: $75,000,000
Retail P=$400, Cost C=$100 Fixed Costs: $50,000,000
 

Decision: None (Plant Runs)
  Rent: $10,000,000
Revenues: $45,000,000
  Net: $15,000,000
Fixed Costs: $50,000,000
d
Rent: $10,000,000
d Retail P=$200, Cost C=$100

d
Net: ’$15, 000, 000 Decision: None (Plant Runs)
Revenues: $15,000,000
Fixed Costs: $50,000,000
Rent: $10,000,000
Net: ’$45, 000, 000
“ “
NPV = $0M ⇐ PV(E (C1 )) = $0M PV(E (C2 )) = $0M




revenues are $0. The following year, you earn either +$45M, ’$15M, +$15M, or
’$45M. This again comes to an expected $0M.

In this example, it really does not matter whether you start or not start the plant”your
¬rm value is always $0.
Importantly, this $0 is also the value if you work with expected outcomes instead of the The example was
rigged”the ¬rm value is
tree. The expected price in both future years is $500/u. At the expected price, your
$0 if you have no
$500/u production cost translates into expected revenues of $60M. You would still have in¬‚exibility.
to pay for rent and ¬xed costs, at $60M per year. Indeed, working with expected values
is the same as assuming that you do not have the ability to make strategic choices in
the future (discussed next)”and a common source of underestimated project values in
practice.

All Real Options ” The Fully Flexible Choice Now assume the opposite extreme benchmark”
you know each year what the price is and you have perfect ¬‚exibility to shut down and
reopen the plant in response to market conditions. (This is the “Timing Option.”) Here,
if the retail price is above $500/u, you would operate. For example, if the price is retail
$600/u, your marginal revenues are $150, 000 · ($600/u ’ $100/u) ’ $50M = $25M. Sub-
tract $10M in sunk rent cost, and you end up with revenues of $15M. If the retail price is
$400/u, you earn $45M, which is not enough to cover the $50M marginal ¬xed cost, so
you are better o¬ not operating and just paying the rent of $10M.
¬le=npvadvice.tex: LP
168 Chapter 7. Capital Budgeting (NPV) Applications and Advice.



Figure 7.3. Value Under Perfect Flexibility ” Full Knowledge and Choice

Retail P=$800, Cost C=$100
Flexibility: Plant (or not)
Decision: Run Plant
Revenues: $105,000,000
Fixed Costs: $50,000,000
Retail P=$600, Cost C=$100
Rent: $10,000,000
Flexibility: Plant (or not)
 

  Net: $45,000,000
Decision: Run Plant
 
Revenues: $75,000,000
Retail P=$400, Cost C=$100
d
Fixed Costs: $50,000,000
Flexibility: (Plant or) Not
d
Rent: $10,000,000
d
‚ Decision: Do Not Run Plant
Net: $15,000,000
Revenues: $0
¡
! Rent: $10,000,000
¡ Net: ’$10, 000, 000
¡
¡
Retail P=$500 (known)
e Retail P=$600, Cost C=$100
e Flexibility: Plant (or not)
ee
… Decision: Run Plant
Revenues: $75,000,000
Retail P=$400, Cost C=$100
Fixed Costs: $50,000,000
 

Flexibility: (Plant or) Not
  Rent: $10,000,000
Decision: Do Not Run Plant
  Net: $15,000,000
Revenues: $0
d
Rent: $10,000,000
d Retail P=$200, Cost C=$100
d

Net: ’$10, 000, 000 Flexibility: (Plant or) Not
Decision: Do Not Run Plant
Revenues: $0
Rent: $10,000,000
Net: ’$10, 000, 000
“ “
NPV = $10.537M ⇐ PV(E (C1 )) = $2.272M PV(E (C2 )) = $8.264M




Figure 7.3 shows your valuation and optimal decision tree now. Again, the ¬gure high-
If you have perfect
¬‚exibility, you get “the lights important ¬‚exibility-related choices in blue as “Flexibility.” The fat boxes indicate
max.”
that you operate the plant, thin boxes that you do not. You earn +$15M or ’$10M in
the ¬rst year. The expected value is $2.5M, which discounts to $2.3M (indicated at the
bottom of the ¬gure). The ¬nal year, you earn +$45M, ’$10M, +$15M, or ’$10M, which
is an expected value of $10M and a discounted value of $8.3M. Therefore, this ¬rm is
worth +$10.5M.
The value to having knowledge and the ¬‚exibility to act on it”knowledge without ¬‚exi-
bility is useless!”has transformed this ¬rm from a nothing into a gem. It is this value-
through-¬‚exibility that your “strategic options to respond” has created. Put di¬erently,
the value of your real option is +$10.5M.

The Option to Delay Choice Often, you do not have full ¬‚exibility. Instead, you have some
strategic options, but not perfect ¬‚exibility. For example, what would happen if you had
the option to delay your decision by one year, more speci¬cally, to run the plant only if
the price appreciates to $600/u, but not if it depreciates to $400/u? If you run the plant
next year, you have to run it the following year. If you do not run the plant next year,
you cannot run it the following year, either. Figure 7.4 shows your revised decision tree.
The value of this strategic option is still a respectable $4.4M. The reason why it does not
reach +$10.5M is that you would still operate the plant in the ¬nal period if the price is
$400/u (which you would rather not do), and you would fail to run the plant in the ¬nal
period if the price is $600/u (which you would rather do).
¬le=npvadvice.tex: RP
169
Section 7·4. Future Contingencies and Real Options.



Figure 7.4. Value to One-Year-Ahead Information (or Ability to Delay Choice Until Year 1)


Retail P=$800, Cost C=$100
Decision: None (Plant Runs)
Revenues: $105,000,000
Fixed Costs: $50,000,000
Retail P=$600, Cost C=$100 Rent: $10,000,000
Flexibility: Commit Fully Net: $45,000,000

 
 
Decision: Run Plant
 
Revenues: $75,000,000
Retail P=$400, Cost C=$100
d
Fixed Costs: $50,000,000
Decision: None (Plant Runs)
d
Rent: $10,000,000
d
‚ Revenues: $45,000,000
Net: $15,000,000
Fixed Costs: $50,000,000
¡
! Rent: $10,000,000
¡ Net: ’$15, 000, 000
¡
¡
Retail P=$500 (known)
e
e Retail P=$600, Cost C=$100
ee
… Decision: None (Plant Closed)
Revenues: $0
Retail P=$400, Cost C=$100
Rent: $10,000,000
 

Flexibility: Abandon Fully
  Net: ’$10, 000, 000
Decision Do Not Run Plant
 
Revenues: $0
d
Rent: $10,000,000
d

d Retail P=$200, Cost C=$100
Net: ’$10, 000, 000
Decision: None (Plant Closed)
Revenues: $0
Rent: $10,000,000
Net: ’$10, 000, 000

“ “
NPV = $4.338M ⇐ PV(E (C1 )) = $2.272M PV(E (C2 )) = $2.066M




The Option to Starting Later An alternative scenario would allow you to start the plant any-
time you wish, but once you start the plant, you cannot stop it. Figure 7.5 shows the
tree for this scenario”the plant value now comes to +$9.5M. This is more than you get
from the option to delay for this scenario, because there is one node (where the price
hits $600/u) where you now could make money where previously you had to have already
committed yourself not to operate. But this is less than what you get under perfect ¬‚exi-
bility, because you are still robbed of the option to shut down if the retail price is $400/u
in the ¬nal period.

The Option to Stopping Later Yet another alternative scenario would force you to keep a once-
closed plant stopped. That is, you cannot restart a plant once you have shut the burners
o¬ and allowed your skilled workers to leave. This is called the “abandonment option.”
This case also illustrates that decisions trees can become complex. If the price falls to What should you do if
the price falls to $400/u
$400/u at ¬rst, should you run the plant or not? If you do not run the plant, you save
at time 1?
money but you lose the strategic option to operate if the price then appreciates to $600.
Actually, you have no choice but to compute the best value both ways. Figure 7.6 and
Figure 7.7 show the two decision trees. If you close the plant, your ¬rm would be worth
$5.4M (Figure 7.6). If you keep the plant open”eating a loss of $15M rather than just
$10M”your ¬rm would be worth $8.3M, because you keep the strategic option to operate
if the retail price were to increase again to $600. Therefore, keeping the plant open is the
better strategy.
¬le=npvadvice.tex: LP
170 Chapter 7. Capital Budgeting (NPV) Applications and Advice.



Figure 7.5. Value to Flexible Plant Starting (But Not Stopping)


Retail P=$800, Cost C=$100
Decision: None (Plant Runs)
Revenues: $105,000,000
Fixed Costs: $50,000,000
Retail P=$600, Cost C=$100 Rent: $10,000,000
Flexibility: Commit Fully Net: $45,000,000
 

 
Decision: Run Plant
 
Revenues: $75,000,000
Retail P=$400, Cost C=$100
d
Fixed Costs: $50,000,000
Decision: None (Plant Runs)
d
Rent: $10,000,000
d
‚ Revenues: $45,000,000
Net: $15,000,000
Fixed Costs: $50,000,000
¡
! Rent: $10,000,000
¡ Net: ’$15, 000, 000
¡
¡
Retail P=$500 (known)
e Retail P=$600, Cost C=$100
e Flexibility: Commit Plant
ee
… Decision: Run Plant
Revenues: $75,000,000
Retail P=$400, Cost C=$100
Fixed Costs: $50,000,000

 
Flexibility: Wait
  Rent: $10,000,000
Decision Do Not Run Plant
  Net: $15,000,000
Revenues: $0
d
Rent: $10,000,000
d Retail P=$200, Cost C=$100

d
Net: ’$10, 000, 000 Flexibility: (Plant or) Not
Decision: Do Not Run Plant
Revenues: $0
Rent: $10,000,000
Net: ’$10, 000, 000
“ “
NPV = $9.504M ⇐ PV(E (C1 )) = $2.272M PV(E (C2 )) = $7.231M




Solving such trees is a di¬cult problem, because your optimal strategy next year does not
You really need to
consider all possible just depend on that year, but on future years. In fact, in our previous examples, I have
future strategies in
cheated in making it too easy for you: I had told you the strategy at each node. Real option
response to all possible
problems are di¬cult to value, precisely because your optimal strategy at any node can
future price paths.
depend both on the current state of your ¬rm and on all future possible scenarios.
The web chapter on real options explains how you can solve such problems more sys-
You can ¬nd more
sophisticated tematically. Decisions are often worked out “backwards”: you start with the ¬nal year
instructions in advanced
and work your way towards today. Another important tool explained in the web chapter
books or chapters.
is a form of automated scenario analysis called Monte-Carlo simulation, in which you
can specify a whole range of possible future scenarios. The spreadsheet itself can then
compute the expected outcome under many di¬erent scenarios under di¬erent decision
strategies that you would specify.
Real option valuation methods are so di¬cult that they are not used as often as NPV, but
Here is what the world is
already using they are also not obscure. Recall the survey of CFOs from Chapter 1. About 50% of their
respondents stated that they use sensitivity analysis, which is the minimum tool to do at
least some real options analysis. About 27% even try to explicitly value real options, which
is a very di¬cult task; and 13% do simulation analyses. Again, these methods are di¬cult
and beyond the scope of this chapter. They are covered in the web chapter, which”you
should be warned”is also the most di¬cult chapter.
¬le=npvadvice.tex: RP
171
Section 7·4. Future Contingencies and Real Options.



Figure 7.6. Value to Flexible Plant Stopping (But Not Starting) ” Strategy 1: Close at $400


Retail P=$800, Cost C=$100
Decision: Run Plant
Revenues: $105,000,000
Fixed Costs: $50,000,000
Retail P=$600, Cost C=$100 Rent: $10,000,000
Flexibility: Try Plant Net: $45,000,000
 

 
Decision: Run Plant
 
Revenues: $75,000,000
Retail P=$400, Cost C=$100
d
Fixed Costs: $50,000,000
Flexibility: Abandon Plant
d
Rent: $10,000,000
d
‚ Decision: Do Not Run Plant
Net: $15,000,000
Revenues: $0
¡
! Rent: $10,000,000
¡ Net: ’$10, 000, 000
¡
¡
Retail P=$500 (known)
e
e Retail P=$600, Cost C=$100
ee
… Decision: None (Plant is dead)
Revenues: $0
Retail P=$400, Cost C=$100
Rent: $10,000,000

 
Flexibility: Abandon Plant
  Net: ’$10, 000, 000
Decision: Do Not Run Plant
 
Revenues: $0
d
Rent: $10,000,000
d

d Retail P=$200, Cost C=$100
Net: ’$10, 000, 000
Decision: None (Plant is dead)
Revenues: $0
Rent: $10,000,000
Net: ’$10, 000, 000

“ “
NPV = $5.371M ⇐ PV(E (C1 )) = $2.272M PV(E (C2 )) = $3.099M




In sum, real options are the value to ¬‚exible choice. The value of the ¬rm depends on how Real options could be
called the “value of
you can respond to situations in the future”and the more optimally you (can) respond, the
future ¬‚exibility.”
higher is the value today. You cannot simply work with the expected retail prices or expected
outcomes. Instead, you have to consider the whole range of future scenarios, and take into
account how you can and will respond.


7·4.D. Decision Trees: One Set of Parameters

This example was a little arti¬cial, because it kept the same parameters throughout. This More common examples.
symmetry made it easy to explain and compare options. More commonly, the parameters
themselves will change and determine the extent of your ¬‚exibility (and thus the value of your
strategic option). This is best to explain by example.

Fixed vs. Flexible Technology Choice Reconsider the fully ¬‚exible choice in Figure 7.3. Now
assume that you have an alternative technology available, which eliminates your ¬xed
operating costs, but requires an upfront $80M investment. (You are installing robots that
will replace expensive manpower.) At ¬rst blush, this seems like a great idea”you no
longer have to spend $100M, which discounts to $86.8M. Alas, this ignores the strategic
option that human workers have over robots: they can be hired and ¬red. Figure 7.8
shows that despite the low cost of the robots and despite the seeming savings of $6.8M
in present value, you would allow the value of the ¬rm to decline to $6.8M. This is lower
than even the value of the ¬rm if you can not restart the plant”once your workers are
gone, they are gone”as in Figure 7.7. Robots, therefore, are not a great idea. Incidentally,
it is often suggested that the value of smart employees is not their initial or even expected
¬le=npvadvice.tex: LP
172 Chapter 7. Capital Budgeting (NPV) Applications and Advice.



Figure 7.7. Value to Flexible Plant Stopping (But Not Starting) ” Strategy 2: Run at $400


Retail P=$800, Cost C=$100
Decision: Run Plant
Revenues: $105,000,000
Fixed Costs: $50,000,000
Retail P=$600, Cost C=$100 Rent: $10,000,000
Flexibility: Try Plant Net: $45,000,000
 

 
Decision: Run Plant
 
Revenues: $75,000,000
Retail P=$400, Cost C=$100
d
Fixed Costs: $50,000,000
Flexibility: Abandon Plant
d
Rent: $10,000,000

d Decision: Do Not Run Plant
Net: $15,000,000
Revenues: $0
¡
! Rent: $10,000,000
¡ Net: ’$10, 000, 000
¡
¡
Retail P=$500 (known)
e
e Retail P=$600, Cost C=$100
e Decision: Run Plant
e
… Revenues: $75,000,000
Retail P=$400, Cost C=$100
Fixed Costs: $50,000,000
Flexibility: Keep Plant Alive

 
  Rent: $10,000,000
Decision: Run Plant
  Net: $15,000,000
Revenues: $45,000,000
d
Fixed Costs: $50,000,000
d Retail P=$200, Cost C=$100
Rent: $10,000,000
d
‚ Flexibility: Abandon Plant
Net: ’$15, 000, 000
Decision: Do Not Run Plant
Revenues: $0
Rent: $10,000,000
Net: ’$10, 000, 000
“ “
NPV = $8.264M ⇐ PV(E (C1 )) = $0M PV(E (C2 )) = $8.264M




value, but the fact that smart people have the ¬‚exibility to attack novel problems for which
they are not initially hired. Your value may be primarily that of a real option!

Adding Plant Capacity Another interesting real option is the option to expand. You can view
this as the choice to build currently unused capacity.
For example, say you can choose between

• your current fully ¬‚exible production technology that allows you to produce 150,000
units at $100/u (as in Figure 7.3);
• and another production technology that builds the following extra capacity: you can
still produce 150,000 units at $100/u, but you can also produce 300,000 units at a
cost of $200/u, though with higher machine costs of $100,000.

Note that doubling increases the cost of all goods, not just the cost of the extra 150,000
units. It would cost you $60M in variable production costs rather than just $15M, and
$100M in ¬xed costs rather than just $50M”that is, almost $95M more if you ever wanted
to use such extra capacity! Would you be willing to pay $3M to upgrade your plant to
such a technology?
Figure 7.9 shows you the ¬rm value with the option to expand. If the retail price hits its all
time high of $800/u, the unusued capacity is worth a tremendous amount. Therefore, the
value of the ¬rm increases to $15.7M from your earlier optimal value of $10.5M, easily
enough to justify a $3M expenditure.
¬le=npvadvice.tex: RP
173
Section 7·4. Future Contingencies and Real Options.



Figure 7.8. Value of a Fixed Cost Technology With Di¬erent Parameters


Retail P=$800, Cost C=$100
Decision: Run Plant
Revenues: $105,000,000
Rent: $10,000,000
Net: $95,000,000
Retail P=$600, Cost C=$100
 

 
Decision: Run Plant
 
Revenues: $75,000,000
d Retail P=$400, Cost C=$100
Rent: $10,000,000
d Decision: Run Plant
Net: $65,000,000

d Revenues: $45,000,000
Rent: $10,000,000
¡
!
¡ Net: $35,000,000
¡
Retail P=$500 (known)
¡
Flexibility: Technology?
e
Fixed Costs: $80,000,000
e Retail P=$600, Cost C=$100
ee
… Decision: Run Plant
Revenues: $75,000,000
Retail P=$400, Cost C=$100 Rent: $10,000,000

 
 
Decision: Run Plant Net: $65,000,000
 
Revenues: $45,000,000
d
Rent: $10,000,000
d
Net: $35,000,000
d
‚ Retail P=$200, Cost C=$100
Decision: Run Plant
Revenues: $15,000,000
Rent: $10,000,000
Net: $5,000,000

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