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value equation for cash flows. In the conventional project, where there is an initial

investment and positive cash flows thereafter, there is only one sign change in the cash

flows, and one root - that is, there is a unique IRR. When there is more than one sign

change in the cash flows, there will be more than one internal rate of return.12 In Figure

12 While the number of internal rates of return will be equal to the number of sign changes, some internal

rates of return may be so far out of the realm of the ordinary (eg. 10,000%) that they may not create the

kinds of problems described here.

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5.6, for example, the cash flow changes sign from negative to positive in year 1, and from

positive to negative in year 4, leading to internal rates of return.

Lest this be viewed as some strange artifact that is unlikely to happen in the real

world, note that many long term projects require substantial reinvestment at intermediate

points in the project and that these reinvestments may cause the cash flows in those years

to become negative. When this happens, the IRR approach may run into trouble.

There are a number of solutions suggested to the multiple IRR problems. One is

to use the hurdle rate to bring the negative cash flows from intermediate periods back to

the present. Another is to construct a NPV profile. In either case, it is probably much

simpler to estimate and use the net present value.

Comparing NPV and IRR

While the net present value and the internal rate of return are viewed as

competing investment decision rules, they generally yield similar conclusions in most

cases. The differences between the two rules are most visible when decision makers are

choosing between mutually exclusive projects.

Differences in Scale

The net present value of a project is stated in dollar terms and does not factor in

the scale of the project. The internal rate of return, by contrast, is a percentage rate of

return, which is standardized for the scale of the project. When choosing between

mutually exclusive projects with very different scales, this can lead to very different

results.

Illustration 5.19: NPV and IRR for projects of different scale

Assume that you are a small bank and that you are comparing two mutually

exclusive projects. The first project, which is hire 4 extra tellers at the branches that you

operate, requires an initial investment of $1 million and produces the cash flow revenues

shown below in Figure 5.7. The second project requires investment of $10 million in an

Automated Teller Machine, and is likely to produce the much higher cash flows shown in

Figure 5.9, as well. The hurdle rate is 15% for both projects.

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Figure 5.9: NPV and IRR - Different Scale Projects

Additional Bank Tellers

$ 350,000 $ 450,000 $ 600,000

Cash Flow $ 750,000

Investment $ 1,000,000

NPV = $467,937

IRR= 33.66%

Automated Teller Machines

$ 5,500,000

Cash Flow $ 3,500,000 $ 4,500,000

$ 3,000,000

Investment $ 10,000,000

NPV = $1,358,664

IRR=20.88%

The two decision rules yield different Capital Rationing: This refers to the

scenario where the firm does not have sufficient

results. The net present value rule suggests

funds - either on hand or in terms of access to

that project B is the better project, while

markets - to take on all of the good projects it

the internal rate of return rule leans might have.

towards project A. This is not surprising,

given the differences in scale.

Which rule yields the better decision? The answer depends on the capital

rationing constraints faced by the business

making the decision. When there are no Profitability Index (PI): The profitability

index is the net present value of a project

capital rationing constraints (i.e., the firm

divided by the initial investment in the

has the capacity to raise as much capital as

project вЂ“ it is a scaled version of NPV.

it needs to take prospective projects), the

net present value rule provides the right answer - Project B should be picked over Project

A. If there are capital rationing constraints, however, then taking Project B may lead to

the rejection of good projects later on. In those cases, the internal rate of return rule may

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provide the better solution. The capital rationing question is dealt with in more detail in

Chapter 6.

Another approach to scaling NPV: The Profitability Index

Another way of scaling the net present value is to divide it by the initial

investment in the project. Doing so provides the profitability index which is another

measure of project return.

Net Present Value

Profitability Index =

Initial Investment

In Illustration 5.17, for instance, the profitability index can be computed as follows for

each project:

Profitability Index for Project A = $467,937/$1,000,000 = 46.79%

Profitability Index for Project B = $ 1,358,664/ $10,000,000 = 13.59%

Based on the profitability index, project A is the better project, after scaling for size.

In most cases, the profitability index and the internal rate of return will yield

similar results. As we will see in the next section, the differences between these

approaches can be traced to differences in reinvestment assumptions.

Differences in Reinvestment Rate Assumption

While the differences between the NPV rule and the IRR rules due to scale are

fairly obvious, there is a subtler, and much more significant difference between the two

rules, relating to the reinvestment of intermediate cash flows. As pointed out earlier, the

net present value rule assumes that intermediate cash flows are reinvested at the discount

rate, whereas the IRR rule assumes that intermediate cash flows are reinvested at the IRR.

As a consequence, the two rules can yield different conclusions, even for projects with

the same scale, as illustrated in Figure 5.10.

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Figure 5.10: NPV and IRR - Reinvestment Assumption

Automated Teller Machine 1

$ 5,000,000 $ 4,000,000 $ 3,200,000

Cash Flow $ 3,000,000

Investment $ 10,000,000

NPV = $1,191,712

IRR=21.41%

Automated Teller Machine 2

$ 5,500,000

Cash Flow $ 3,500,000 $ 4,500,000

$ 3,000,000

Investment $ 10,000,000

NPV = $1,358,664

IRR=20.88%

In this case, the net present value rule ranks the second investment higher, while the IRR

rule ranks first investment as the better project. The differences arise because the NPV

rule assumes that intermediate cash flows get invested at the hurdle rate, which is 15%.

The IRR rule assumes that intermediate cash flows get reinvested at the IRR of that

project. While both projects are impacted by this assumption, it has a much greater effect

for project A, which has higher cash flows earlier on. The reinvestment assumption is

made clearer if the expected end balance is estimated under each rule.

End Balance for ATM1 with IRR of 21.41% = $10,000,000*1.21414 = $21,730,887

End Balance for ATM2 with IRR of 20.88% = $10,000,000*1.20884 = $21,353,673

To arrive at these end balances, however, the cash flows in years 1, 2, and 3 will have to

be reinvested at the IRR. If they are reinvested at a lower rate, the end balance on these

projects will be lower than the values stated above, and the actual return earned will be

lower than the IRR even though the cash flows

Modified Internal Rate of Return (MIRR):

on the project came in as anticipated.

This is the internal rate of return, computed

The reinvestment rate assumption made

on the assumption that intermediate

by the IRR rule creates more serious cashflows are reinvested at the hurdle rate.

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consequences the longer the term of the project and the higher the IRR, since it implicitly

assumes that the firm has and will continue to have, a fountain of projects yielding

returns similar to that earned by the project under consideration.

A Solution to the Reinvestment Rate Problem: The Modified Internal Rate of Return

One solution that has been suggested for the reinvestment rate assumption is to

assume that intermediate cash flows get reinvested at the hurdle rate - the cost of equity if

the cash flows are to equity investors and the cost of capital if they are to the firm - and to

calculate the internal rate of return from the initial investment and the terminal value.

This approach yields what is called the modified internal rate of return (MIRR), as

illustrated in Figure 5.11.

Figure 5.11: IRR versus Modified Internal Rate of Return

$ 400 $ 500 $ 600

Cash Flow $ 300

Investment <$ 1000>

$600

$500(1.15)

$575

$400(1.15) 2 $529

$300(1.15) 3 $456

Terminal Value = $2160

Internal Rate of Return = 24.89%

Modified Internal Rate of Return = 21.23%

Modified Internal Rate of Return = ($2160/$1000)1/4 -1 = 21.23%

The modified internal rate of return is lower than the internal rate of return because the

intermediate cash flows are invested at the hurdle rate of 15% instead of the IRR of

24.89%.

There are many who believe that the MIRR is neither fish nor fowl, since it is a

mix of the NPV rule and the IRR rule. From a practical standpoint, the MIRR becomes a

weighted average of the returns on individual projects and the hurdle rates the firm uses,

with the weights on each depending on the magnitude and timing of the cash flows - the

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larger and earlier the cash flows on the project, the greater the weight attached to the

hurdle rate. Furthermore, the MIRR approach will yield the same choices as the NPV

approach for projects of the same scale and lives.

Where Do Good Projects Come From?

In the process of analyzing new investments in the preceding chapters, we have

contended that good projects have a positive net present value and earn an internal rate of

return greater than the hurdle rate. While these criteria are certainly valid from a

measurement standpoint, they do not address the deeper questions about good projects

including the economic conditions that make for a вЂњgoodвЂќ project and why it is that some

firms have a more ready supply of вЂњgoodвЂќ projects than others.

Implicit in the definition of a good project вЂ“вЂ“ one that earns a return that is greater

than that earned on investments of equivalent risk вЂ“вЂ“ is the existence of super-normal

returns to the business considering the project. In a competitive market for real

investments, the existence of these excess returns should act as a magnet, attracting

competitors to take on similar investments. In the process, the excess returns should

dissipate over time; how quickly they dissipate will depend on the ease with which

competition can enter the market and provide close substitutes and on the magnitude of

any differential advantages that the business with the good projects might possess. Take

an extreme scenario, whereby the business with the good projects has no differential

advantage in cost or product quality over its competitors, and new competitors can enter

the market easily and at low cost to provide substitutes. In this case the super-normal

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