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The internal rate of return can be viewed mathematically as a root to the present
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 “ 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 “good” project and why it is that some
firms have a more ready supply of “good” projects than others.
Implicit in the definition of a good project ““ one that earns a return that is greater
than that earned on investments of equivalent risk ““ 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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