10.5Evaluating Real Investments with the Internal Rate of Return

Deriving the NPVwith the discounted cash ow is the most popular method for eval-

uating investment projects. Almost as many corporate managers, however, base their

real investment decisions on the internal rate of return (IRR)of their investments.

For this reason, we analyze this approach in depth.

The internal rate of return (IRR) for a cash ow stream C, C, . . . , Cat dates

01T

0, 1, . . . , T,respectively, is the interest rate ythat makes the net present value of a

project equal zero; that is, the ythat solves

CCC
12T
0 C. . .	(10.3)
0yy)2T
1(1(1y)

Since internal rates of return cannot be less than 100 percent, 1 yis positive.

Intuition for the irrMethod

The internal rate of return methodfor evaluating investment projects compares the

IRRof a project to a hurdle rateto determine whether the project should be taken. In

this chapter, where cash ows are riskless, the hurdle rate is always the risk-free rate

of interest. For risky projects, the hurdle rate, which is the project’s cost of capital (see

Chapter 11), may reect a risk premium.

Projects have several kinds of cash ow patterns. Alatercash ow streamstarts

at date 0 with a negative cash ow and then, at some future date, begins to experience

positive cash ows for the remaining life of the project. It has this name because all

the positive cash ows of the project come later, after a period when the rm has

pumped money into the project. Alater cash ow stream can be thought of as the cash

ow pattern for investing. When investing, high rates of return are good. Thus, a proj-

ect with a later cash ow stream enables the rm to realize arbitrage prots by taking

on the project and shorting the tracking portfolio whenever the project’s IRR(its mul-

tiperiod rate of return) exceeds the appropriate hurdle rate (which is the multiperiod

rate of return for the tracking portfolio). Projects with internal rates of return less than

the hurdle rates have low rates of return and are rejected.

The reverse cash ow pattern, when all positive cash ows occur rst and all neg-

ative cash ows occur only after the last positive cash ow, is known as an early cash

ow stream. This cash ow pattern resembles borrowing; when borrowing, it is better

to have a low rate (IRR) than a high rate. Hence, the IRRrule for this cash ow pat-

tern is to reject projects that have an IRRexceeding the hurdle rate and to accept those

where the hurdle rate exceeds the IRR.

Some cash ow patterns, however, are more problematic because they resemble

both investing and borrowing, including those that start out negative, become positive,

and then become negative again. Such cash ow patterns create problems for the IRR

method, as we will see shortly.

Numerical Iteration of the IRR

The computation of the internal rate of return is usually performed using a technique

known as numerical iteration, which can be viewed as an intelligent “trial-and-error

procedure.” Iterative procedures begin with an initial guess at a solution for the inter-

est rate y.They then compute the discounted value of the cash ow stream given on

the right-hand side of equation (10.3) with that y,and then increase or decrease y

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346Part IIIValuing Real Assets

depending on whether the right-hand side of equation (10.3) is positive or negative.

The procedure generally stops when the discounted value of the present and future cash

ows is close enough to zero to satisfy the analyst. Most electronic spreadsheets and

nancial calculators have built-in programs that compute internal rates of return.

Formulas for computing the internal rate of return directly (that is, without numer-

ical iteration) exist only in rare cases, such as when a project has cash ows at only a

few equally spaced dates. For example, if a project has cash ows only at year 0 and

year 1, we can multiply both sides of equation (10.3) by 1 yand solve for y.The

resulting equation, 0 C(1 y) C, has the solution

01

C

y ¹1

C

0

If there are cash ows exactly zero, one, and two periods from now, multiplying both

sides of equation (10.3) by (1 y)²

results in a quadratic equation. This can be solved

with the quadratic formula. Most of the time, however, projects have cash ows that

occur at more than three dates or are timed more irregularly. In this case, numerical

iteration is used to nd the IRR.

NPVand Examples of IRR

This subsection works through several examples to illustrate the relation between NPV

and IRR.¹⁴

Examples with One IRR.Examples 10.9 and 10.10 are simple cases having only one

IRR.

Example 10.9:An IRRCalculation and a Comparison with NPV

Joe’s Bowling Alley is considering whether to relacquer its lanes, which will generate addi-

tional business.The project’s cash ows, which occur at dates 0, 1, 2, and 3, are assumed

to be riskless and are described by the following table.

Cash Flows (in $000s) at Date

0123

9453

If the yields of riskless zero-coupon bonds maturing at years 1, 2, and 3 are 8 percent, 5per-

cent, and 6 percent per period, respectively, nd the net present value and the internal rate

of return of the cash ows.

Answer:The NPV,obtained by discounting the cash ows at the zero-coupon yields, is

$4,000$5,000$3,000

$9,000 $1,758

1.081.05²³

1.06

The IRRis approximately 16.6 percent per period;that is

$4,000$5,000$3,000

$9,000 0

1.1661.166²³

1.166

¹⁴As

we discuss later, there are more problematic examples of cash ow streams with multiple IRRs.

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347

In Example 10.9, the internal rate of return of the project exceeds the rate of return

of all of the bonds used to track the project’s cash ow. In this case, it is not surpris-

ing that the NPVis positive. In Example 10.10, the IRRis negative.

Example 10.10:NPVand IRRwith IrregularPeriods

A contractor is considering whether to take on a renovation project that will take two and one-

half years to complete.Under the proposed deal, the contractor has to initially spend more

money to pay workers and acquire material than he receives from the customer to start the

project.After the initial phase is completed, there is a partial payment for the renovation.A

second phase then begins.When the company completes the second phase, the customer

makes the nal payment.The cash ows of the project are described by the following table:

Years from Now:	0	.5	1.25	2.5
Cash ows (in $000s):	10	5	15	18

The respective annualized zero-coupon rates are

Years to Maturity:	.5	1.25	2.5
Yield (%y ear):	6%	6%	8%

Find the net present value and internal rate of return of the project.

Answer:The NPVof the project is:

$5,000$15,000$18,000

$10,000 $4,240

1.06^.51.252.5

1.061.08

This is a negative number.The IRRis approximately 6.10 percent per year.

In Example 10.10, the negative IRRwas below the yield on each of the zero-coupon

bonds used in the project’s tracking portfolio, and the NPVwas negative.

An Example with Multiple IRRs.One reason it was easy to compare the

conclusions derived from the NPVand IRRin Examples 10.9 and 10.10 is that both

examples had only one IRR.Whenever the cash ows have only one IRR,the internal

rate of return method and the net present value method usually will result in

the same decision. However, the project in Example 10.11, for instance, has multi-

ple internal rates of return, making comparisons between the two methods rather

difcult.

Example 10.11:Multiple Internal Rates of Return

Strip Mine, Inc., is considering a project with cash ows described in the following table.

Cash Flows (in $millions) at Date

0123

1041301

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Compute the IRRof this project.

Answer:This project has two internal rates of return, 213.19 percent and 0 percent, deter-

mined by making different initial guesses in the IRRprogram of a nancial calculator or

spreadsheet.

Exhibit 10.1 outlines the net present value of the project in Example 10.11 as a

function of various hypothetical discount rates. The exhibit shows two discount rates

at which the NPVis zero.

Multiple internal rates of return can (but do not have to, as Example 10.10 illus-

trates) arise when there is more than one sign reversal in the cash ow pattern. To bet-

ter understand this concept, let us denote the plus sign () if the cash ow is positive

and minus sign () if a cash ow is negative. In Example 10.11, the sign pattern is

, which has two sign reversals: One sign reversal occurs between dates 0

and 1, when the cash ow sign switches from negative to positive, the other between

dates 1 and 2, when the cash ow sign switches from positive to negative.

In Example 10.11, it is obvious that 0 percent is an internal rate of return because

the sum of the cash ows is zero. As the interest rate increases to slightly above zero,

the future cash ows have a present value that exceeds $10 million because a small

positive discount rate has a larger effect on the negative cash ows at t 2 and t 3

than it does at t 1. With very high interest rates, however, even the cash ow at

t 1 is greatly diminished after discounting; and subsequently higher discount rates

lower the present value of the future cash ows. Thus, as the interest rate increases,

eventually there will be a discount rate that results in an NPVof zero. This rate is

213.19 percent.

EXHIBIT10.1

Net Present Value forCash Flows in Example 10.11 as a Function of Discount Rates

Net present value

(in $millions)

$3.00

$2.00

$1.00
	Discount
	rate
–100	213.19(%)

–$1.00

–$2.00

–$3.00

–$4.00

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349

An Example with No IRR.It is also possible to have no IRR,as Example 10.12

illustrates.

Example 10.12:An Example Where No IRRExists

A group of statisticians is thinking of taking a 2-period leave from their academic positions

to form a consulting rm for analyzing survey data from polls in political campaigns.The proj-

ect has cash ows described by the following table.

Cash Flows (in $000s) at Date

012

103035

Compute its IRR.

Answer:This project has no internal rate of return.As Exhibit 10.2 illustrates, all posi-

tive and negative discount rates (above 100 percent) make the discounted cash ow stream

from the project positive.

Because the NPVis positive at every discount rate, the project in Example 10.12

is a very good project, indeed! However, this does not mean that the absence of an IRR

implies that one has a positive NPVproject. If the signs on the cash ows of Example

10.12’s project were reversed, all positive and negative discount rates would make the

net present value of this project negative. However, there is still no internal rate of

return. Thus, the lack of an IRRdoes not indicate whether a project creates or destroys

rm value.

EXHIBIT10.2

Net Present Value forCash Flows in Example 10.12 as a Function of Discount Rates

Net present value

(in $000)

25

20

15

10

5

	Discount
	rate (%)
–100

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350Part IIIValuing Real Assets

Term Structure Issues

The previous material illustrated problems that arise with the internal rate of return

method because of multiple internal rates of return in some cases and none in others.

Additional problems exist when the term structure of riskless interest rates is not at.

In this case, there is ambiguity about what the appropriate hurdle rate should be. In

general, long-maturity riskless bonds have different yields to maturity than short-

maturity riskless bonds, implying that the appropriate hurdle rate should be different

for evaluating riskless cash ows occurring at different time periods. Although this cre-

ates no problem for the net present value method, it does create difculties when the

internal rate of return method is applied.

Most businesses, confused about what the appropriate IRRhurdle rate should be,

take a coupon bond of similar maturity to the project and use its yield to maturity as

the hurdle rate. This kind of bond has a yield that is a weighted average of the zero-

coupon yields attached to each of the bond’s cash ows. While this may provide a rea-

sonable approximation of the appropriate hurdle rate in a few instances, using the yield

of a single-coupon bond as the hurdle rate for the IRRcomparison generally will not

provide the same decision criteria as the net present value method. Result 10.5 outlines

a technique for computing a hurdle rate that makes the net present value method and

the internal rate of return method equivalent in cases where the IRRis applicable, even

when the term structure of interest rates is not at.

Result 10.5

The appropriate hurdle rate for comparison with the IRRis that which makes the sum ofthe discounted futurecash ows of the project equal to the selling price of the tracking port-folio of the futurecash ows.

Although the hurdle rate in Result 10.5 generates the same IRR-based decision as

the net present value method, it requires additional computations. Indeed, an analyst

has to compute the PVof the project to compute the appropriate hurdle rate. Because

of the additional work—as well as other pitfalls in its use that will shortly be

discussed—we generally do not recommend the IRRas an approach for evaluating an

investment project. However, in some circumstances, the internal rate of return method

may be useful for communicating the value created by taking on an investment proj-

ect. An assistant treasurer, communicating his or her analysis to the CFO, may have

more luck communicating the value of taking on a project by comparing the project’s

internal rate of return to the rate of return of a comparable portfolio of bonds, rather

than by presenting the project’s NPV.

Example 10.13 illustrates a case that makes this comparison.

Example 10.13:Term Structure Issues and the IRR

Assume the following for Finalcon’s consulting project:Zero-coupon bonds maturing at date

1 have a 6 percent yield to maturity, those maturing at date 2 have an 8 percent yield, while

cash ows are given in the table below.

Cash Flows (in $000s) at Date

012

804050

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351

The NPVof the project is $602.79, which is positive, indicating a good project.The IRRof

the project is 7.92 percent.Does the internal rate of return also indicate that it is a good

project?

Answer:There is ambiguity about which number the internal rate of return must be com-

pared to.If one compares the IRRwith 6 percent, it appears to be a good project, but it is

a bad project if compared with a hurdle rate of 8 percent.

The appropriate comparison is with some weighted average of 6 percent and 8 percent.

That number is another internal rate of return:the yield to maturity of a bond portfolio with

payments of $40,000 at date 1 and $50,000 at date 2.In a market with no arbitrage, this

bond should sell for its present value, $80,602.79 because

$40,000$50,000

$80,602.79

1.06(1.08)²

Its yield to maturity is the number ythat makes

$40,000$50,000

$0 $80,602.79

yy)²

1(1

which is 7.39 percent.Since the project’s internal rate of return, 7.92 percent, exceeds the

7.39 percent IRR(or yield to maturity) of the portfolio of zero-coupon bonds that track the

future cash ows of the project, it should be accepted.

Cash Flow Sign Patterns and the Number of Internal Rates of Return

Adopting a project only when the internal rate of return exceeds the hurdle rate is con-

sistent with the net present value rule as long as the pattern of cash ows has one sign

reversaland cash outows precede cash inows.One example of this is the later cash ow

sign pattern (of pluses and minuses) seen below for cash ows at six consecutive dates.

Date 0-5 Time Line

Cash Flow Sign at Date

012345

LaterCash Flow Streams.While the illustration above has two negative cash ows

followed by four positive ones, a later cash ow stream can have an arbitrary number

of both negative and positive cash ows. Having only onesign reversal, however,

means that a later cash ow stream can have only one IRR.

The uniqueness of the IRRin a later cash ow stream is best demonstrated by look-

ing at the value of the cash ow stream at the date when the sign of the cash ow has

just changed. If, for some discount rate, the value of the entire cash ow stream dis-

counted to that date is zero, then that discount rate corresponds to an IRR.Let’s look

at the date 2 value of the cash ow stream in the blue-shaded time line above. Date 2

is where the cash ow sign has changed from negative to positive. If the date 2 value

of the entire stream of the six cash ows is zero for discount rate r,then the value of

the stream at date 0 is simply the date 2 value divided by (1 r)²

. Thus, if the date

2 value of the six cash ows in the date 05 time line is zero, their value at date 0 is

also zero, which makes r,by denition, an IRR.

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Without assigning actual numbers to these cash ows, it is possible to analyze the

date 2 values of the positive and negative cash ows separately as the discount rate

changes. The date 2 value of the four positive cash ows in the time line always dimin-

ishes as the discount rate increases. Moreover, for a very large discount rate, their date

2 value is close to zero. For a very small discount rate (that is, close to 100 percent),

their date 2 value becomes large.

Now, consider the value at date 2 of the two negative cash ows at dates 0 and

1. Their negative date 2 value is becoming more negative as the discount rate

increases. For a very small discount rate (close to 100percent), the value is close

to zero. For a very large discount rate, the date 2 value of the cash ows at dates 0

and 1 is extremely negative. Hence, summing the date 2 values of the positive and

negative cash ows, one nds that at low discount rates the sum is large and posi-

tive and close to the date 2 value of the four positive cash ows, while at high inter-

est rates it is negative and close to the date 2 value of the two negative cash ows.

Moreover, as one increases the discount rate, the sum of the date 2 values of the pos-

itive and negative cash ow streams decreases. Thus, because the IRRcoincides with

the discount rate where the date 2 values of the stream’s positive and negative cash

ows just offset each other.

There always is a unique internal rate of return for a later cash ow stream. More-

over, if discounting all cash ows(present and future) with the hurdle rate yields a pos-

itive NPV, the IRRis larger than the hurdle rate. In other words, only a larger discount

rate achieves a zero discounted value. Conversely, a negative NPVimplies that the IRR

lies to the left of the hurdle rate.

Early Cash Flow Streams.Examples 10.14 and 10.15 illustrate the intuition devel-

oped earlier about how to apply the internal rate of return method, depending on

whether the positive portion of the cash ow stream is later or earlier.

Example 10.14:Implementing the IRRRule forInvesting

Refer to the Joe’s Bowling Alley project in Example 10.9, where the stream of cash ows

(in $000s) at dates 0, 1, 2, and 3 was C 9, C 4, C 5, and C 3,respectively,

0123

with associated zero-coupon yields of 8 percent, 5 percent, and 6 percent at maturity

dates 1, 2, and 3, respectively.How does the IRRcompare with the hurdle rate for the

project?

Answer:The yield to maturity (or IRR), y,of the bond portfolio that tracks the future cash

ows of the project satises the equation

$4,000$5,000$3,000$4,000$5,000$3,000

1.05²³²(1y)³

1.081.06(1y)(1y)

The left-hand side, which equals $10,758, is the present value of the futurecash ows

from the project (which is why the $9,000 initial cash ow does not appear).It is the

project’s NPVless its initial cash ow.It also represents the price of the portfolio of bonds

that tracks the future cash ows of the project.The IRR,or y,on the portfolio of bonds is

the constant discount rate that makes the portfolio a zero-NPVinvestment.This number,

about 5.92 percent, represents the hurdle rate for the project.Since the project’s internal

rate of return, the 16.6 percent found in Example 10.9, exceeds the hurdle rate, one should

adopt the project.

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Example 10.15:Implementing the IRR Rule forBorrowing Needs

Consider the cancellation of the project in Example 10.14, immediately after Joe’s

Bowling Alley made the adoption decision.In other words, the decision was made not to

relacquer the lanes following the earlier decision to relacquer them.What are the cash

ows from canceling the project? How does the IRRcompare with the hurdle rate in this

case?

Answer:The relevant incremental cash ows for the cancellation decision (a project

in its own right) are found by subtracting the cash ow position of the bowling alley when

it adopted the lane relacquering project from its cash ow position without the project.

These cash ows are simply the negatives of the cash ows in Example 10.3:$9,000 at

t 0, $4,000 at t 1, $5,000 at t 2, and $3,000 at t 3.The net present value

calculation is

$4,000$5,000$3,000

$9,000$1,758

1.081.05²³

1.06

The internal rate of return and the hurdle rate are still the same, approximately 16.6 percent

and 5.92 percent, respectively.Reversing signs does not change the IRRcalculation! Thus,

as in Example 10.14, the IRRexceeds the hurdle rate.

In Example 10.15, the net present value rule says that canceling the project, once

adopted, is a bad decision. This makes sense because Example 10.14 determined that

adopting the project was a good decision. From the standpoint of the IRR,it also makes

sense. The pattern of cash ows from the cancellation of the project has signs that can

be described by the early cash ow pattern . Like the later cash ows, this

pattern has only one sign change and thus has only one IRR.Because this cash ow

sign pattern is more like borrowing than investing, the appropriate IRR-based invest-

ment evaluation rule is to accepta project when the IRRis lowerthan the hurdle rate

and reject it otherwise (as in this case).

The Correct Hurdle Rate with Multiple Hurdle Rates.The internal rate of return

is unique when a cash ow sign pattern exhibits the pattern of an early cash ow stream

or a later cash ow stream. Even in these cases, however, there may still be multiple

hurdle rates if the term structure of interest rates is not at. The point of this subsec-

tion is that no one should worry about these cases.

For a later cash ow stream where the present values of the futurecash ows

of the project are negative (which can only happen if the stream starts off with two

or more negative cash ows), two interest rates can be candidates for the hurdle rate

if hurdle rates are computed with the procedure suggested in Result 10.5.¹⁵A

negative PVand a negative initial cash flow imply a negative NPV.Clearly, the

correct hurdle rate is one that results in rejection. Fortunately, both of these hurdle

¹⁵As

we demonstrate shortly, the reason that two hurdle rates may sometimes arise is that the hurdle

rates are themselves internal rates of return of a portfolio of zero-coupon bonds. If this bond portfolio

has a cash ow pattern with more than one sign reversal (implying both long and short positions in zero-

coupon bonds), it is possible for two internal rates of return to exist. In the following sections, the

existence and implications of multiple internal rates of return are described in greater detail.

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rates will exceed the IRR,which thus tells us to reject the project, as Example 10.16

illustrates.¹⁶

Example 10.16:Multiple Hurdle Rates

The following time line describes the cash ows of a bad project developed by a team at

Idiots, Inc.The appropriate discount rates for riskless cash ows of various maturities are

given below the cash ows.

Cash Flows and Discount Rates for Date

0123

$5	$10	$5	$5
	4%	5%	6%

Compute the IRRand the two hurdle rates for the project.

Answer:The unique IRRis 19.81 percent.The hurdle rates are found by rst dis-

counting the cash ows from years 13, which gives a present value of $.882.The cash

ows of the (zero NPV) bond portfolio that tracks the futurecash ows of the project are

therefore

Tracking Portfolio Cash Flows at Date

0123

$.882$10$5$5

Note that, in contrast to the cash ows of the project, the cash ows of the tracking portfo-

lio have two sign changes and thus may have more than one IRR.The two hurdle rates of

the four cash ows of the tracking portfolio are 6.86 percent and 976.04 percent.Both hur-

dle rates exceed the IRR,implying the project should be rejected.

These considerations suggest the following IRRadoption rule for projects with later

and early cash ow streams:

Result 10.6

(The appropriate internal rate of return rule.)In the absence of constraints, a project witha later cash ow stream should be adopted only if its internal rate of return exceeds thehurdle rate(s). Aproject with an early cash ow stream should only be adopted if the hur-dle rate(s) exceed the internal rate of return of the project.

¹⁶A

similar problem arises with an early cash ow stream when the present value of its future cash

ows is positive. In this case, the project has a positive NPV.If the project has two hurdle rates because

of this, both will exceed the IRRand indicate that the project should be adopted. Since a project has a

positive NPVwhen it has an early cash ow stream with future cash ows that have a positive present

value, the internal rate of return rule in this case makes the correct decision, regardless of which of the

two hurdle rates is used.

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355

Sign Reversals and Multiple Internal Rates of Return

The last subsection noted that multiple internal rates of return for the bond portfolio

that tracks the project, and thus determines the hurdle rates, do not prevent us from

using the IRRto determine whether projects with later or early cash ow streams should

be adopted. By contrast, the existence of multiple internal rates of return for the proj-

ect’s cash ows has a major impact on one’s ability to use the IRRfor evaluating invest-

ments.¹⁷

Aproject could have more than one internal rate of return if its cash ows exhibit

more than one sign reversal. These multiple sign reversals can arise for many reasons.

For example, environmental regulations may require the owner of a strip mine to restore

the land to a pristine state after the mine is exhausted. Or the tax authorities may not

require a rm to pay the corporate income tax on the sale of a protable product until

one year after the prot is earned. In this case, the last cash ow for the project is

merely the tax paid on the last year the project earned prots. The next section illus-

trates how mutually exclusive projects can create sign reversals, even when the cash

ows of the projects per se do not exhibit multiple sign reversals.

In the event of multiple sign reversals, many practitioners employ very complicated

and often ad hoc adjustments to the internal rate of return calculation. In principle

(although rarely in practice), these adjustments can ensure that the IRRrule yields the

same decisions as the net present value rule. However, it makes little sense to bother

with these troublesome procedures when the net present value rule gives the correct

answer and is easier to implement.

Mutually Exclusive Projects and the Internal Rate of Return

When selecting one project from a group of alternatives, the net present value rule indi-

cates that the project with the largest positive net present value is the best project.

Do Not Select the Project with the Largest IRR.It is easy, but foolhardy, to think

that one should extend this idea to the internal rate of return criterion and adopt the

project with the largest IRR.Even if all the projects under consideration are riskless

and have the later cash ow stream pattern, each project might have different hurdle

rates if the term structure of interest rates is not at. If the project with a large IRR

also has a larger hurdle rate than the other projects, how does one decide? Is it then

more appropriate to look at the difference between the internal rate of return and the

cost of capital or the ratio? Fortunately, it is unnecessary to answer this question

because even if all the projects have the same hurdle rate, the largest internal rate of

return project is not the best, as Example 10.17 illustrates.

Example 10.17:Mutually Exclusive Projects and the IRR

Centronics, Inc., has two projects, each with a discount rate of 2 percent per period.The fol-

lowing table describes their cash ows, net present values, and internal rates of return.

¹⁷However,

Cantor and Lippman (1983) have shown that multiple internal rates of return can have

useful interpretations. Specically, when lending is possible at a rate of rbut borrowing is not possible

and projects are repeated in time because cash from the rst run of the project is needed to fund the

second project, and so on, then if discounting at ryields a positive NPVfor the project, the smallest IRR

above rrepresents the growth rate of the project if the project is repeated in perpetuity.

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356Part IIIValuing Real Assets

Cash Flows at Date

NPV at 2%

012(in $millions)IRR

Project A	10	16	30	$3.149	10.79%
Project B	10	2	11	$2.534	15.36%

Given that Centronics must select only one project, should it choose A or B?

Answer:The net present value of project A, about $3.149 million, is higher than the net

present value of B, about $2.534 million.Thus, project A is better than project B, even though

it has a lower internal rate of return.

The appropriate IRR-based procedure for evaluating mutually exclusive projects is

similar to that used for the net present value rule: Subtract the cash ows of the next

best alternative. If one is lucky and the difference in the cash ow streams of the two

projects have an early or later sign pattern, one can choose which of the two projects

is better. In Example 10.17, the cash ows of project Aless those of project B can be

described by the following table:

Cash Flows at Date
	NPVat 2%
012	(in $millions)	IRR

Project A–B:

0

218

19

.615

5.56%

There is only one sign reversal and a later cash ow. The implication is that project A

is better than project B because the 5.56 percent IRRis higher than the 2 percent hur-

dle rate.

How Multiple Internal Rates of Return Arise from Mutually Exclusive Projects.

The last subsection showed how differencing the cash ows of two projects might lead

to a correct IRR-based rule for evaluating mutually exclusive projects. The key con-

cept is that the comparison of the pair leads to a differenced cash ow that has only

one sign reversal. If there are many projects, a large number of comparisons of two

projects like those described in the last subsection must be made. As Example 10.18

demonstrates, it is highly likely that some of the differenced cash ows will have more

than one sign reversal, even if all of the projects have later or early cash ow sign

patterns.

Example 10.18:Multiple IRRs from Mutually Exclusive Projects

Reconsider Example 10.6, where NASA must select one of three commercial projects for

the next space shuttle mission.Each of these projects has industrial spin-offs that will pay

off in the next two years.The cash ows, net present values, and internal rates of return of

the projects are described on the following page.

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Chapter 10

Investing in Risk-Free Projects

357

Cash Flows at Date

NPV at 2%

012(in $millions)IRR

Project A	17.0	12	9	3.415	16.16%
Project B	16.8	10	11	3.577	15.98
Project C	16.9	11	10	3.496	16.07

Which project should NASA select?

Answer:The net present value criterion indicates that project B is the best project, even

though it has the lowestIRR.The internal rate of return criterion fails because the differ-

ences in the cash ows of various pairs of projects have multiple internal rates of return.In

this case, there are three pairs of cash ow differences.

Cash Flows at Date
	NPV at 2%
012	(in $millions)	IRR

Project A-B	.2	2	2	.162	12.7% and 787.3%
Project A-C	.1	1	1	.081	12.7%and 787.3

Project B-1	1	.081	12.7%and 787.3
C.1

The multiple internal rates of return arise from the two sign reversals in the cash ow dif-

ferences.The pair of IRRs are identical for all three differences and thus cannot provide infor-

mation about the best project.

However, the net present value, based on cash ow differences, selects project B, which

was the project with the highest NPVwithout cash ow differencing.The reason that cash

ow differences alone tell us that project B is the best of the three projects is twofold:

•	The difference between the cash ows of projects A and B has a negative NPV,indicating A is worse than B.
•	The difference between the cash ows of projects B and C has a positive NPV,indicating B is better than C.

The net present value criterion tells us to adopt the project with the largest positive

net present value. The appropriateness of this rule follows from the value additivity

property of the net present value formula. However, the value additivity property does

not apply to the IRR.We can summarize the implications of this as follows:

Result 10.7

The project with the largest IRRis generally not the project with the highest NPVand thusis not the best among a set of mutually exclusive projects.

10.6	Popularbut Incorrect Procedures forEvaluating Real Investments

Other evaluation methods besides NPVand IRRexist, including ratio comparisons,

exemplied by the price to earnings multiple, and the competitive strategies approach.¹⁸

¹⁸See

Chapter 12.

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Strategy, Second Edition

358Part IIIValuing Real Assets

This section briey discusses two popular rules of thumb used by managers to eval-

uate investment projects: the paybackmethod and the accounting rate of return

method.

The Payback Method

The payback methodevaluates projects based on the number of years needed to

recover the initial capital outlay for a project. For example, an investment that costs

$1million and returns $250,000 per year would have a payback of four years. By the

payback criterion, this investment would be preferred to a $1 million investment that

returned $200,000 per year and thus had a payback of ve years.

Amajor problem with the payback method is that it ignores cash ows that occur

after the project is paid off. For example, most of us would prefer the second project

over the rst if the second project generated $200,000 per year for the next 20 years

and the rst project generated $250,000 per year for only 5 years and nothing thereafter.

There is no reason to ignore cash ows after the payback period except that the pay-

back method provides a simple rule of thumb that may help managers make quick deci-

sions on relatively minor projects.¹⁹Experienced managers do not need a nancial cal-

culator to tell them that a routine minor project with a one-year payback has a positive

NPV.They know this is almost always the case.

The Accounting Rate of Return Criterion

The accounting rate of returncriterion evaluates projects by comparing the project’s

return on assets, which is the accounting prot earned on the project divided by the

amount invested to acquire the project’s assets. The accounting rate of return is then

compared with some hurdle rate in the same manner as the internal rate of return deci-

sion rule.

Our principal objection to this criterion is that accounting prots are often very dif-

ferent from the cash ows generated by a project. When making a comparison between

a project and its tracking portfolio, we understand why it would be inappropriate to use

earnings in such a calculation.

The wealth creation associated with the net present value rule is based on the

date 0 arbitrage prots from a long position in the project and a short position in

the project’s tracking portfolio. This combined position is supposed to eliminate all

future cash ows to allow a comparison between the date 0 cost of the investment

strategy in real assets and the date 0 cost of the tracking portfolio’s nancial assets

as the basis for deciding whether value is created or destroyed. However, all future

cash ows are not eliminated in attempting to match the project’s earnings with a

tracking portfolio. Paying the cash ows on the tracking portfolio to some outside

investor requires actual cash, which is not balanced by the reported earnings gener-

ated by the project. Earnings differ from cash ows for a number of reasons, includ-

ing, notably, depreciation.

¹⁹Variations

of the payback method, such as discounted payback, also exist. This method suffers from

a similar deciency of ignoring cash ows after the payback period.

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Markets and Corporate		Projects	Companies, 2002
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Chapter 10

Investing in Risk-Free Projects

359