Implications of Value Additivity When Evaluating Mutually Exclusive Projects.

The value additivity property makes it easy to understand how to properly select the

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best project from among a group of projects that are mutually exclusive. For exam-

ple, a rm may choose to congure a manufacturing plant to produce either tractors

or trucks, but it cannot use the plant for both. Value additivity implies that the best

of these mutually exclusive projects is the one with the largest positive net present

value.

Result 10.3

Given a set of investment projects, each with positive NPV,one should select the projectwith the largest positive NPVif allowed to adopt only one of the projects.

One way to understand this is to recognize that adopting the project with the cash

ows that have the largest NPVgenerates the largest net present value of the rm’s

aggregated cash ows. This is because value additivity implies that the NPVof the rm,

a collection of projects, is the sum of the NPVof the rm without the adoption of any

of the proposed projects plus the NPVof the incremental cash ows of whichever proj-

ect is adopted.

Another way to understand that the project with the largest NPVis the best is that

the cost of adopting one of two mutually exclusive projects, each with positive NPV,

is the forgone cash ows of the other project, not a zero NPVinvestment in the nan-

cial markets. Recall that the concept of incremental cash ows compares the cash ows

of the rm with the project with the cash ows of the rm without the project. With

mutual exclusivity, not adopting a particular project means that the rm adopts its next

best alternative project, provided that the latter has positive NPV.Thus, the true incre-

mental cash ows of a particular project are the difference between the cash ows of

the project and those of the forgone alternative. By value additivity, or more precisely

“subtractivity,” the NPVof this differenced cash ow stream is the difference between

the NPVs of the two projects. Example 10.2 illustrates this point.

Example 10.2:Mutually Exclusive Projects and NPV

The law rm of Jacob and Meyer is small and has the resources to take on only one of four

cases.The cash ows for each of the four legal projects and their net present values dis-

counted at the rate of 10 percent per period are given below.

Cash Flows (in $000s)
at Date
	Net Present Value
012	(in $000s)

Project A

7

11

12.1

13

Project B	12212.1	9

Project C	54424.2	15

Project D

1110

9

Which is the best project?

Answer:These cash ows are calculated as the cash ows of the rm with the project

less the cash ows of the rm without any of the four projects.Clearly, project C has the

highest net present value and should be adopted.

It is possible to calculate the cash ows differently.With mutual exclusivity, the cost of

adopting one of the projects is the loss of the others.Hence, computation of the cash ows

relative to the best alternative should provide an equally valid calculation.The best alterna-

tive to projects A, B, and D is project C.The best alternative to project C is project A.The

appropriate cash ow calculation subtracts the cash ows of the best alternative and is

described with the pairwise project comparisonsbelow:

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338Part IIIValuing Real Assets

Cash Flows (in $000s)
at Date
	Net Present Value
012	(in $000s)

Project A (less C)	2	33	36.3	2
Project B (less C)	4	22	12.1	6
Project C (less A)	2	33	36.3	2
Project D (less C)	4	33	24.2	6

Only project C has a positive net present value.Thus, selecting the project with the largest

positive net present value, as in the rst set of NPVcalculations, is equivalent to picking the

only positive net present value project, when cash ows are computed relative to the best

alternative.

We have assumed here that mutual exclusivity implies that only one project can be

selected. Example 10.3 discusses how we might generalize this.

Example 10.3:Ranking Projects

Suppose that it is possible to adopt any two of the four positive NPVprojects from Exam-

ple 10.2.Which two should be selected?

Answer:Projects A and C have the two largest positive NPVs and thus should be

adopted.

Using NPVwith Capital Constraints

The last subsection considered the possibility of having mutually exclusive projects

because of physical constraints. Among those was the possibility that some important

input was in short supply, perhaps land available for building a manufacturing facility

or managerial time.

This subsection considers the possibility that the amount of capital the rm can

devote to new investments is limited. For truly riskless projects, these capital con-

straintsare unlikely to be important because it is almost always possible to obtain out-

side nancing for protable riskless projects. However, the cash ows of most major

projects are uncertain, and their probability distributions are difcult to verify, so the

availability of outside capital for these projects may be constrained.

Protability Index.We are concerned here with how a corporation, constrained in

its choice owing to a limited supply of capital, should allocate the capital that it can

raise.¹⁰Specically, this subsection introduces an extension of the net present value

rule known as the protability index, which is the present value of the project’s future

cash ows divided by C, the negative of the initial cash ow, which is the initial

0

cost of the project. In the absence of a capital constraint, the value maximizing rule

with the protability index is one that adopts projects with a protability index greater

than 1 if Cis negative. (If Cis positive, a rm should adopt projects with a prof-

00

itability index of less than 1.) This is simply another form of the net present value rule.

For example, with annual cash ows, denote

¹⁰Parts

IVand Vof the text explain why these capital constraints exist.

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Chapter 10

Investing in Risk-Free Projects

339

CCC

12T

PV . . .

rr)^2T

1(1(1r)

12T

The net present value rule says that one should select projects for which

CPV0

0

or equivalently

PVC

0

or

PVPV

1 if C 0 and 1 if C 0

C⁰C⁰

00

The protability index can be particularly useful if Cis negative for all projects

0

under consideration and if there is a capital constraint in the initial period. If the projects

under consideration can be scaled up or down to any degree, the project with the largest

protability index exceeding 1 is the best. This point is demonstrated in Example 10.4.

Example 10.4:The Protability Index

There is a capital constraint of $10,000 in the initial period.The two scalable projects avail-

able for investment are projects B and C from Example 10.2.The cash ows, NPVs at 10per-

cent, and protability indexes are given below.

Cash Flow (in $000s)

at DateNet Present

ValuesProtability

012(in $000s)Index

Project B	12.1	9	10
	122

Project C	24.2	15	4
	544

Which is the better project?

Answer:Project B is the best because it gives the biggest “bang per buck.”For the max-

imum initial expenditure of $10,000, it is possible to run 10 B projects, but only 2 C projects.

The old net present value rule calculation is deceiving here because it reects the benet

of adopting only one project.But adopting 10 B projects would yield an NPVof $90,000,

which exceeds $30,000, the NPVfrom two C projects.The protability index reects this

because it represents the present value of future cash ows per dollar invested.

Net Protability Rate.Perhaps a better representation of the relative protability of

two investments is offered by the net protability rate, where the

Net protability rate (1 Risk-free rate) (Protability index) 1

If Cis negative, the net protability rate represents the additional value next period

0

of all future cash ows per dollar invested in the initial period. It is, in essence, the

net present value translated into a rate of return.¹¹

¹¹The net protability rate amortizes the project’s net present value over the rst period of the

project’s life. It can be used to make decisions about the economic protability of both riskless and risky

projects.

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Strategy, Second Edition

340Part IIIValuing Real Assets

Using NPVto Evaluate Projects That Can Be Repeated over Time

Capital allocation takes place in the presence of many different kinds of constraints, so

it is often inappropriate to look at projects in isolation. One type of constraint is a space

constraint, which would exist, for example, if there was space for only one piece of

equipment on the shop oor. Here, the mere fact that projects have different lives may

affect the choice of project. Longer-lived projects use space, the scarce input, to a

greater degree and should be penalized in some fashion. Example 10.5 demonstrates

how to solve a capital allocation problem with a space constraint.

Example 10.5:Evaluating Projects with Different Lives

Two types of canning machines, denoted A and B, can be placed only in the same corner

of a factory.Machine A, the old technology, has a life of two periods.Machine B costs more

to purchase but cans products faster.However, machine B wears out more quickly and has

a life of only one period.The delivery and setup of each machine takes place one period

after initially paying for it.Immediately upon the setup of either machine, positive cash ows

begin to be produced.The cash ows from each machine and the net present values of their

cash ows at a discount rate of 10 percent are as follows:

Cash Flow (in $millions)

at Date

Net Present Value

012(in $millions)

Machine A	.8	1.1	1.21	1.2
Machine B	1.9	3.30		1.1

It appears that machine A is a better choice.Is this true?

Answer:A second picture tells a different story.

Cash Flow (in $millions)

at Date

Net Present Value

012(in $millions)

Machine A	.8	1.1	1.21	1.2
First machine B	1.9	3.3		1.1
Second machine B		1.9	3.30	1.0

Sum of the two
B machines	1.9	1.4	3.30	2.1

Over the life of machine A, one could have adopted machine B, used it to the end of its

useful life, purchased a second machine B, and let it live out its useful life as well.In this

case, machine B seems to dominate.

Example 10.5 points out the value of being able to repeat a project over time. One

makes an appropriate comparison between repeatable mutually exclusive projects only

after nding a common date where, after repeating in time, both projects have lived

out their full lives. This date is the least common multiple of the terminal dates of the

projects under consideration.

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Markets and Corporate		Projects	Companies, 2002
Strategy, Second Edition

Chapter 10

Investing in Risk-Free Projects

341

Two other approaches also can help in the selection between repeatable, mutually

exclusive projects with different lives. One of these approaches repeats the project in

time innitely. It values the project’s cash ows as the sum of the present values of per-

petuities. The other approach computes the project’s equivalent annual benet; that is,

the periodic payment for an annuity that ends at the same date and has the same NPV

as the project. The project with the largest equivalent annual benet is the best project.

At a 10 percent discount rate, the cash ows at dates 0, 1, and 2 of machine Afrom

Example 10.5 (respectively, $0.8 million, $1.1 million, and $1.21 million) have the same

present value (as cash ows of $0.63 million at date 1 and $0.63 million at date 2). The

equivalent annual benet of machine Ais thus $0.63 million. Similarly, the cash ows

at dates 0 and 1 of machine B from Example 10.5 (respectively, $1.9 million and $3.3

million) have the same present value as a cash ow of $1.21 million at date 1. The

equivalent annual benet of machine B is thus $1.21 million. Since the equivalent annual

benet of machine B exceeds the equivalent annual benet of machine A, machine B is

the better project given the factory space constraint described in Example 10.5.