INTERNAL RATE OF RETURN: A PROJECT PARAMETER, NOT A MEASURE FOR
INVESTMENT RETURN
Antonio Carlos Teixeira Álvares
José Carlos Barbieri
Claude Machline
Abstract: The value of the interest rate at which the net cash inflow of an investment project is
equal to zero has always been called internal rate of return (IRR). The objective of the present article
is to show that the IRR is a project parameter and does not represent, at least in the great majority of
cases, an exact measure of investment return. Only in the case of conventional cash flows, where
there is a single initial payment and a final income (typical case of certain financial investments), the
IRR represents the exact return over the invested capital. The existence of intermediate cash flows,
which happens frequently in projects for the production and operational fields, takes away from the
IRR the condition of measure of investment return. The present paper aims to provide a more
adequate interpretation for the phenomenon of multiple return rates, which may be found in nonconventional cash flows, and to validate the modified internal rate of return (MIRR) as a more
acceptable indicator for estimating the rate of return of a conventional investment project.
Keywords: Engineering Economics, Cash Flow Analysis, Internal Rate of Return, Project Balance,
Modified Internal Rate of Return.
1. Introduction
A conventional investment project presents payments (cash outflows) in its initial phase and
incomes (cash inflows) in future periods. Obviously, in order to have return on investment, it is
necessary that the total cash inflow be greater than the outflows. However, since the inflows take
place after the outflows, it is necessary to take into account the cost of money over time. Thus, being
defined a minimum interest rate that is accepted by the investor, named Minimum Acceptable Rate
of Return (MARR), the investment project may be accepted if the Net Present Value (NPV) of the
cash flow is not negative.
Considering as negative the values relative to the cash outflows and as positive those relative
to the inflows, the condition of non-negative Net Present Value (NPV  0) indicates that the sum of
the discounted inflows (at a MARR) to the initial date of the project is greater than or equals the
outflows discounted in the same manner (at the same rate), meaning the project is acceptable to the
investor.
The specialized literature defines the Internal Rate of Return – IRR as being the interest rate
at which the net present value (NPV) of the cash flow of an investment is equal to zero. The IRR is
2
obtained solving equation 1, in which the unknown value is the rate i, and CFn is the cash flow in
date n.
NPV  CF0 
CFn 1
CFn
CF1
CF2

 .................................................. 

 0 (1)
2
n 1
(1  i) (1  i)
(1  i)
(1  i) n
Multiplying (1) by (1+i)n, the equivalent equation of the net future value (NFV) is obtained:
NFV  CFo  (1  i) n  CF1  (1  i) n1  ................................  CFn1  (1  i)  CFn  0 (2)
That is, the IRR is obtained by solving a polynomial equation of degree n, which, in the
limit, may admit up to n positive roots.
We intend to show in this article that the denomination Internal Rate of Return is mistaken,
but the fact is that, existing only one solution to the IRR, it is a meaningful parameter associated to
the cash flow of the project, even though it does not necessarily mean an exact measure of return on
investment. Thus, first considerations will be made about the IRR in different cash flows, as well as
its interpretation. Cases will be analyzed involving multiple results and their corresponding
treatments. As it will be observed, there is a substantial controversy related to this aspect of the IRR.
Finally, the article provides a suggestion for adjusting the name of this project metric to the
observations made herein.
2. Conventional Cash Flows
In most cases, investment projects have a typical cash flow, as in Figure 1, which the
literature calls “conventional cash flow”, characterized by the following conditions:
1. the payments (net cash outflows) take place in the first years, and incomes (net cash
inflows) in the following years, with only one inversion in the sign of the cash flow; and
2. the sum of incomes is greater than the sum of payments.
Figure 1 - Conventional Cash Flow
CFn-2
0
1
CF2
CF0
CFn
n - 1
n
2
n - 2
CF1
CFn-1
3
The existence of these conditions is a common fact in industrial investments, where the
payments due to equipments and physical installations of the productive facility precede the incomes
with products sales. It is a fundamental condition for accepting the project that the future incomes
are greater than the initial payments. Being observed the condition 1, the sign rule of the Descartes
theorem indicates that there is only one positive root (x* = 1 + i* ) for equation 2. On the other hand,
being observed condition 2, it may be demonstrated, as shown by De Faro (1979, p.61), that this
root (x*) is greater than the unit, that is, i* > 0. In other words, in the cases of conventional cash
flows, it is mathematically proven the existence and singularity of the IRR.
The discipline Engineering Economics was developed since the decade of 1920 in the USA
by various authors such as Fish, Grant, Thuesen. In that time economists were concerned with the
macroeconomic models, such as the studies of the economic cycles and the theory of the currency
and credit. Engineers, on the other hand, desired to solve practical problems such as, for example, to
build a new plant or not, to buy or rent a machine or to decide between building a bridge using
concrete or steel. Thus, engineers demanded a microeconomic model for investment analysis and the
discipline was given birth with the instigating name of Engineering Economics. The Engineering
Economics dedicated itself, in the early stages, to studying typical investment projects for the
production and operational fields, especially to the construction and expansion of industrial
facilities, mostly involving conventional cash flows, that is, those complying with the two
conditions mentioned above.
An investment project was accepted when the present value of the discounted cash flow of
the incomes was equal or greater than the present value of the payments. In other words, it was
accepted if, considering payments as negative values and incomes as positive values, the net present
value (NPV) of the entire flow was equal to or greater than zero (NPV ≥ 0). The discount rate
should be equal to a minimum interest rate accepted by the investor, named Minimum Acceptable
Rate of Return (MARR).
In the case of a feasibility study of a project with a conventional cash flow, it would always
be mathematically possible to obtain, by solving equation 2, one and only one positive interest rate
at which the present value of the cash flow is equal to zero. For any positive interest rate lower than
that, the net present value of the project would always be positive. Such a rate, which is a parameter
associated to the cash flow of the project, was then named Internal Rate of Return (IRR),
inappropriately, as we intend to show. The condition of NPV ≥ 0 (non-negative net present value)
for accepting the project could be replaced by the equivalent IRR ≥ MARR (internal rate of return
not lower than the minimum acceptable rate of return).
2.1. Interpretation of the IRR
The internal rate of return, as indicated by its name, has been, from its conception,
interpreted as the rate that returns the investment made on the project. Many authors commonly
refer to the IRR as the measure of payback of the investment. For example, Hartman and Schafrick
(2004), say that, when singular, the IRR defines the return of an investment (p. 139).
This interpretation is inaccurate in most cases. The IRR calculation model, such as
conceived, assumes that the intermediate cash flows, if positive (cash inflows), are returned by an
interest rate equal to the IRR, as well as the negative cash flows (cash outflows) are also financed by
the same rate. According to Kassai et alii (1999, p.68), because of this, when the calculated IRR
4
differs by a substantial amount from the rates of the market, its interpretation as rate of return on the
investment is false. If the IRR is exceedingly different from the rates of the market, it may be very
far from indicating the real return of the investment project.
A few investment projects will be analyzed below. In all analyses it will be considered that
all values are expressed in constant currency (without inflation) and that the real interest rate (also
exempt from the effects of inflation) will be constant during the life of the project. In order to
illustrate the distortion caused by the hypothesis of reinvesting the intermediate positive cash flows
(incomes) at the IRR, a project of very high return will be analyzed, first.
Example 1: An investor decided to bet in recovering an oil well, which had been declared
exhausted. He obtained the exploration rights and invested in recovering the pumps. The
initial investment was $1,000 thousand. The well revealed the existence of unexplored areas
and generated, in the first year, a positive cash flow of $1,000 thousand, paying for the
entire investment. In the second year the net incomes were $1,500 thousand; in the third
$2,000 thousand, decreasing to $1,000 thousand in the fourth. The oil reserves were
exhausted in the fifth year, still generating net incomes of $ 500 thousand in that year. By the
end of the fifth year the well was abandoned. The Minimum Acceptable Rate of Return
(MARR) adopted by the investor is 12% /year.
The cash flow diagram of the investment is indicated below:
2.000
1.500
1.000
1.000
500
0
1
2
3
4
5
1.000
The net present value, as a function of the interest rate i, will be:
Graph 1: Function NPV =f(i) of Example 1
5000
4000
NPV  1,000 
NPV
3000
1,000 1,500
2,000 1,000
500




2
3
4
1  i  1  i  1  i  1  i  1  i 5
2000
1000
0
0%
20%
40%
60%
80%
100%
-1000
Interest Rate
120%
140%
160%
5
Graph 1 presents the NPV function of the example. As can be seen, the NPV function has
negative slope and has only one internal rate of return at the point at which it intercepts the x axis
(NPV = 0), yielding: i = IRR = 120.56%
In this example, in which the IRR is more than 10 times larger than the MARR, the distortion
caused by the reinvestment of the intermediate incomes at the IRR is amplified (the conception of
the IRR implies, by the principle of capital equivalency, that all flows run along time at the IRR).
However, it is not possible to reinvest the intermediate incomes at the IRR of the project. The IRR is
the result of a mathematical calculation that, in the given example, is far larger than the reality of the
market. Assuming that it would be possible to reinvest the intermediate incomes until the end of the
project at a 12% /year, by the end of the fifth year the following amount would be obtained:
FV  1,000  1.12  1,500  1.12  2,0001.12  1,000  1.12  500  7,809.71
4
3
2
and the solution:
 7,809.71 
i

 1,000 
1
5
 1  0.5084
If the hypothesis of investing the intermediate incomes of the project at the 12% rate is really
feasible, the 50.84% represents a much more adequate approximation of the return of the
investment. In case the market rate for investing the incomes were only 6% /year, the value of the
added incomes at date 5 would be 6,856.20, resulting in a rate of 46.97% /year. Assuming that the
incomes, due to a contract signed with the governmental agency responsible for the exploration
rights, must be held and could only be withdrawn by the end of the fifth year, upon the expiration of
the exploration rights, with no interest, we would have, by the end of year 5, an income of 6,000 and
a resulting rate of 43.10%. Finally, assuming that the incomes generated by the project could be
reinvested at the IRR (120.56% /year), by the end of the fifth year the amount would be 52,191.27,
resulting in a 120.56% rate of return.
As can be seen, the IRR, in this example, since it is too different from the market rate for
reinvesting the incomes generated during the project, is far from measuring the real return of the
investment. Much more suitable metrics for measuring the return are those that take into account the
market rate for investing the intermediate incomes. Table 1 resumes the values obtained.
6
Table 1: Comparative analysis of investment return
Annual rate for investment of
the project incomes
120,56%
12,00%
6,00%
0
Investment value at the end of
the 5th year
52.191,97
7.809,71
6.856,20
6.000,00
Annual rate of return of the
investment
120,56%
50,84%
46,97%
43,10%
The conception of the IRR leads to the fact that the intermediate cash flows of the project run
along time at the IRR itself. If the IRR is too different from the market rate, it may significantly
differ from the real return rate of the investment. The IRR will only represent the exact measure of
return on investment in a flow of only two points, typical of certain financial applications, in which
all the investment (PV) takes place on date zero and all the income (FV) is concentrated at date n. In
this case, admitting FV > │PV│, the equation 3 below yields a positive interest rate that represents
exactly the return of the investment.
 FV
i  
 | PV


| 
1
n
1
(3)
3. The case of Multiple Rates of Return
As seen, calculation of the IRR results from solving a polynomial equation of degree n. Back
to equation 2:
VFL  CFo  (1  i) n  CF1  (1  i) n1  ................................  CFn1  (1  i)  CFn  0
According to the Descartes theorem, a polynomial equation of this type may admit up to n
positive real roots, the maximum number thereof being equal to the number of times that there are
changes in sign of the coefficients (CFj). In other words, the maximum number of positive real roots
will be equal to the number of times that the sequence of the cash flow changes sign during the life
of the project. Thus, an interesting question arises about the concept of the Internal Rate of Return
when a non-conventional cash flow is presented, in which the cash outflows and inflows alternate
during the life of the project. This possibility was presented by Lorie and Savage (1955). Solomon
(1956) analyzed in detail this situation, making use of the following example.
Example 2: The proposal being considered is the installation of a larger oil pump that
would get a fixed quantity of oil out of the ground more rapidly than the pump is already in
use. By operating the existing pump, the investor can expect $10,000 a year hence and
$10,000 two years hence. By installing the larger pump at a net cost of $1,600 he can expect
$ 20,000 a year hence and nothing the second year. The installation of the larger pump can
be viewed as a project having the following cash flow:
CF0 = -1,600, CF1 = 10,000 and CF2 = -10,000.
7
Equation 2 applied to this cash flow, becomes:
 1,600  (1  i) 2  10,000  (1  i)  10,000  0
which, being an equation of the second degree, has two roots:
(1  i)1  1.25  i  0.25
(1  i) 2  5.00  i  4.00
meaning that there are two distinct values for the rate that makes the net future (or present) value of
the cash flow equal to zero: 25% and 400%. The author questions which of these represents the
correct measure of return of the investment, and finally concludes that neither represents the return
of the project. As a solution for finding a single and significant return rate, he suggests that the
intermediate income be reinvested at the market rate until the project ends.
It is important to remark that, in the past, the calculation of the IRR without the help of a
financial calculator was not an easy task. The usual procedure was then to calculate a solution
(generally close to the MARR) and “stop there as if the rate found was the only one”, as
recommended by Mattos (1978, p. 25). According to Hirschfeld (2000, p. 293), when researching
the IRR by trial, it was usual to consider the first value found, disregarding others, also due to the
fact that “rate factor is not found in the usual tables”. Some authors even described as being
“common sense” to consider as the IRR the rate closest to the market rate. In the case of the problem
illustrated by Solomon, it would mean to consider only the 25% value, disregarding the second
solution of 400%. However, the existence of more than one IRR close to the market rate is perfectly
possible, as will be seen in the next example.
Example 3: An event company has invested $1,000 thousand in a large temporary
convention center to celebrate the new millennium. The net income, one year later, is $2,400
thousand. However, at the end of the following year, it is condemned to pay the city $ 430
thousand concerning the disassembly of the facility, and a $1,000 thousand penalty. The
Minimum Acceptable Rate of Return defined by the company is 20% /year.
The calculation of the net present value of the investment as a function of the interest rate is
obtained by the equation:
NPV  1,000 
2,400 1,430

(1  i) (1  i) 2
Assuming i = 20%, one obtains a NPV = 6.94 > 0, that is, in spite of the problems, the
investment met the policy of the company. However, the calculation of the rate that makes the NPV
= 0 indicates two distinct values for the IRR: 10% and 30%, which are equidistant to the MARR.
8
Now what? How does the IRR method stand? Choosing IRR = 30% makes the project acceptable
(IRR > MARR), but choosing IRR = 10% makes the same unacceptable (IRR < MARR).
In order to better understand this situation, we will analyze Graph 2 of the function NPV =
f(i). The NPV is negative for interest rates (MARR) lower than 10% or higher than 30%. This means
the investment should be rejected in this situation. On the other hand, the investment would be
acceptable if the MARR was between 10% and 30%.
The situation sounds strange at least: if instead of 20%, situation in which the project is
accepted, the MARR was altered to 9%, the project should be refused. As a principle, a lower
MARR should mean a less conservative policy for investment acceptance. What this case is
showing, definitely, is that the IRR, defined as the interest rate at which the net present value of a
cash flow is zero, does not represent the measure of investment return.
Graph2: Function NPV=f(i) of Example 2
10,00
NPV
0,00
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
-10,00
-20,00
-30,00
Interest Rate
The financial interpretation of the occurrence of multiple IRR may be found in Bierman and
Smidt (1993 p. 95) and is related to the fact that, according to the mathematical model of the IRR,
the intermediate flows are invested or taken at the IRR itself. If a significant sign inversion happens
in the flow, all the investment (cash outflows) may be withdrawn at the IRR at an intermediate date.
In this case, the investor, according to the model conception, would see all the investment returned
at the IRR and would borrow, from the same project, the exceeding resources at a rate equal to the
IRR. Well, from the company’s standpoint, the decision to borrow resources has an opposite logic to
the decision of investing. While, in the case of a loan, it is desirable to have the lowest possible
interest rate, the exact opposite takes place in the case of an investment.
3.1. The popularity of the IRR and the approaches for treating multiple solutions
The internal rate of return method has become very popular among executives, exactly due to
the false interpretation that the IRR represents an exact measure of investment return. According to
Bierman and Smidt (1993, p. 102), administrators like the method because it is important to know
the difference between the IRR in a proposed investment and the MARR. Weston and Brigham
(2004, p. 538) say the IRR method is not only familiar, but widely spread in the industry. According
to the same authors, north-american executives prefer the IRR method to the NPV by a margin of 3
to 1 (p. 545). However, the question of the non-applicability of the IRR method upon nonconventional cash flows remained as a ghost haunting more enlightened financial executives. After
9
the seminal article of Lorie and Savage (1955) many authors struggled over the mathematical
paradox of the multiple internal rates of return. The analysis of the literature first indicates (until the
1970s) two positions for the problem:
1. to recommend not using the IRR method in any case of non-conventional cash flow;
2. to establish, by means of complex mathematical approach, the conditions to verify the
applicability of the method in non-conventional cash flows.
The authors that have dedicated their attention to the mathematical approach sought to make
apparent the conditions for the polynomial equation, determined from the net present (or future)
value function, to admit a single non-negative real root. Complex analyses, some very elegant, were
realized by Kaplan (1965), Norstrom (1972), Bernhard (1979), De Faro (1975, 1976, 1983) and
many others. The applicability of the model in financial terms was not very questioned at first. The
paradox was analyzed much more as a mathematical problem. The studies provided, however, a
very important clue for later financial approaches of the IRR model, as they made clear that it was
not every non-conventional flow that admitted multiple internal rates of return. The condition for
existing multiple rates demanded significant inversions of sign in the cash flow of the project.
The concept of project balance (PB) at the end of a period k (future value of the cash flow until
period k), presented by Teichroew, Robichek and Montalbano (1965, p. 165), and used by Galesne,
Fenterseifer and Lamb (1999 p. 84-90), is useful for determining the existence of multiple IRR in a
non-conventional cash flow. The inversion in sign (cash inflow) must be so significant that it makes
the project balance positive, that is, returns all the investment made until that period, so the surplus
was then lent to the investor, since the project would need new investments in the future. A simple
flow allows visualizing the concept of project balance.
Example 4: CF0 = -1,000; CF1 = 3,000; CF2 = -2,000
Considering null the interest rate, at the end of the first year the project balance would be:
PB1 = -1000 + 3000 = 2000
The intermediate positive balance is indicating the payment of the whole investment and the
consequent loan of the project to the investor $ 2,000. By the end of the second year, the net future
value, considering the loan would be returned to the project, without interest, would be:
NFV = 2,000 – 2,000 = 0
That is, the rate of 0% is a solution for the IRR. Considering now a rate of 100%, the project
balance at the end of year 1 would indicate:
PB1 = -1000x(1+1) + 3000 = -2000 + 3000 = 1000
What means that all the investment would be paid at the end of the first year, considering the
annual interest rate of 100%, and with a surplus of $ 1,000. This value, borrowed at the 100% rate
and returned at the end of year 2, would result in the net future value of the project:
10
NFV = 1000x(1+1) – 2000 = 2000 – 2000 = 0
what means that 100% is another possible solution for the IRR.
An investment project begins with a cash outflow, that is, with a negative balance. If, during
the life of the project, there are sign inversions that make the project balance positive, the investor
would start using such positive balance as a loan, which would be returned to the project at a future
date. In such condition, the logic of maximizing the interest rate is inverted, since who borrows
seeks to minimize the interest rate and the IRR model loses its financial meaning. As was wellpointed by Lapponi (1996), in the case of an investment, the rates should be calculated over the
project balance, so the IRR would be an indicator of project return of the non-amortized part of the
investment (p. 94). Thus, after all the investment has been amortized, it is not feasible to use the
concept of return on investment.
4. Modified Internal Rate of Return - MIRR
Possibly due to the popularity of the IRR, besides the mathematical approach, new trends
were started, suggesting the modification of the cash flow in order to eliminate the paradox of the
multiple solutions. Solomon (1956), in his pioneer article, suggests the reinvestment of the
intermediate income at a market rate for capital investment. Other authors, such as Oliveira (1979, p.
89), proposed that the intermediate investments corresponding to the negative cash flows were
discounted to the initial date of the project, considering a market rate for borrowing money. In both
cases the non-conventional flow would be transformed into a conventional one, eliminating the
multiple solutions for the IRR. However, the flow modification, either carrying the intermediate
incomes to the final date of the project or carrying the intermediate payments to the initial date, does
not necessarily eliminate all the intermediate flows and, therefore, according to what was shown in
the previous section, the IRR found would still not represent an adequate measure of investment
return.
A solution, proposed by Lin (1976), was to simultaneously adopt the procedures for carrying
to the final date of the project the intermediate incomes (positive cash flows) at a market rate for
capital reinvestment and bringing to the initial date the intermediate payments (negative cash flows)
at a market rate for financing. These combined procedures would transform any cash flow into a
two-point flow, which IRR would be given by equation 3, as seen in Example 1. This new parameter
of the cash flow of the project is associated to a market rate for reinvestment ir and to a market rate
for financing if, and was named Modified Internal Rate of Return (MIRR).
Returning to Example 1, in which the calculation of the IRR has indicated as a result an
annual rate of 120.56%. Admitting the reinvestment of the annual incomes of the project at a rate of
12% /year until its end in date 5, the result of the MIRR is 50.84% /year. In the opinion of Kassai et
alii (1999), the MIRR is an improved version of the IRR and indicates the true internal rate of return
of a project (p. 73). According to Brigham, Gapenski and Ehrard (2001), the MIRR is superior to the
IRR as an indicator of the “true return rate or long-term return rate of a project” (p. 436).
We agree that the MIRR is a better indicator of the long-term return rate of an investment
project, if it is a conventional one, for taking into account the market reality. As for being exact or
true, we would have to admit that the model precisely represents reality, which may not be the case.
For instance, in the case of non-conventional flows with multiple internal rates, there is no sense (at
least from the financial standpoint), to define a return rate, be it modified or not. Volkman (1999)
shows that the MIRR method does not correct the deficiency in the cases where there is multiplicity
11
of the IRR, since the problem of both methods would lie in not distinguishing the investment flows
from the financing flows (p. 82). However, in the cases where there is singularity of the IRR, the
MIRR is a much more adequate approximation of the project return. The difference between the
values obtained by the MIRR and the IRR in such cases is closely connected to the difference
between the calculated value of the IRR and the market rates for investment and financing and also
to the amount and intensity of the presence of intermediate cash flows. Thus, in the limit, happening
a single initial payment and a single final income, the IRR and the MIRR will be the same and will
indicate exactly the return of the investment.
4.1 The consortium problem
Consortiums of customers to buy vehicles have become very popular in Brazil. Through this
instrument, a group of people commit themselves to deposit every month a fraction of the value of a
new automobile, which is bought and drawn among the participants that have not been drawn in
previous draws.
Example 5: Consider that a consortium for buying automobiles have 50 associates. Each
participant pays monthly 2% of the value of the automobile, added by a 0.1% administration
rate. The vehicle bought every month is distributed by means of a draw. It is assumed the
model of the vehicle will not change significantly in the next 50 months. No inflation is
assumed.
Considering the value of the vehicle as being $10,000, the monthly payment of the
consortium would be of $210. A participant drawn in a generic month t would have a cash flow
associated to the operation according to Figure 2.
FIGURE 2: Cash flow associated to the operation described in Example 5
10.000
0 1
49
t
210
Considering t > 0, we have a typical non-conventional cash flow, in which the investor
invests $210 during t periods, has as an income the value of the vehicle ($10,000) at date t and at
this date borrows from the “project consortium” the difference between this income and the future
value (at date t) of the t+1 payments (made from date 0 to date t) of $210, to be paid in 49-t
payments of also $210. This is an interesting and rare case of non-conventional cash flow, which is
of general knowledge, at least in Brazil. It is intuitive that the participant drawn on date 0 is the one
with the greatest benefits, while the one receiving the vehicle in the last month bears the loss of
having made an investment that in date 49 has an amount ($10,000) lower than the simple sum of
the invested values ($210 x 50 = $10,500).
12
We will now analyze a particular situation in which the vehicle is drawn upon the twentieth
payment (since the first payment is done in date 0, this means the draw is done on date t = 19). The
NPV function of the cash flow in this particular case is indicated below:
NPV  PV (i  i; n  50; PMT  210; ; type  1)  PV (i  i; n  19; ; FV  10,000)
In the calculation notation using functions of financial mathematics was used the mixed
convention HP12C-Excel (Álvares, 1998), whose abstract is attached. From the Descartes theorem
we know that, since the flow has two sign changes, at most there will be two real positive solutions.
Graph 3 shows function NPV, obtained with MS-Excel. As can be seen, there are two solutions for
the IRR, their values being 1.13% and 4.47% according to the MS-Excel calculation.
Graph 3: NPV function of the flow of a vehicle drawn on the twentieth month
200,00
NPV
100,00
0,00
0,0%
-100,00
0,5%
1,0%
1,5%
2,0%
2,5%
3,0%
3,5%
4,0%
4,5%
5,0%
-200,00
-300,00
-400,00
Interest Rate
-500,00
Now we calculate the MIRR. For using correctly the method, it is shown, in Figure 3, a new
representation, this time of the net cash flow.
FIGURE 3: new representation of the net cash flow
9.790
0
1 2
18
20
48 49
19
210
Assuming a reinvestment rate of 0.5% /month, the future value of the cash inflows is
calculated:
13
FV  FV (i  0.5%; n  30; ; PV  9,790)  11,370.11
Assuming a financing rate of 2.5% per month, the present value of the cash outflows can be
calculated with the help of function NPV of HP12C, as indicated below:
PV  NPV (CF0  210; CF j  210; n j  18; CF j  0; CF j  210; n j  30; i  2.5%)  5,973.63
That is, the flow has two points (PV and FV). And the calculated MIRR indicates:
MIRR  RATE (n  49; PV  5,973.63; ; FV  11,370.11)  1.32%
The MIRR may also be easily obtained by means of the function MIRR of Excel, as shown
below, and the addressed sequence in the Excel of the cash flows CFj must necessarily register the
cash outflows with a negative sign and the cash inflows with a positive sign.
MIRR  MIIR(sequenceCF j ; i f  2.5%; ir  0.5%)  1.32%
However, what is the meaning of the MIRR in this particular case? Does it mean it is better
to accept the project, since a financial investment would yield lower rates (0.5% / month)? Or
should we accept the project because the cost of obtaining financing would be higher (2.5% /
month)? Maybe because of both?
Actually, it is not appropriate, from the financial standpoint, to define a return for some nonconventional flows. An investment project starts with cash payments in its implementation years,
that is with negative balance. If, during its execution, there are intermediate cash inflows that make
the balance positive for a certain interest rate, this means that all the investment has been paid for
and the investor began to receive the profits of the project. With the investment being nonconventional, such profits (in whole or in part) must be returned to the project in a future date and so
the withdrawal is characterized as a loan of the project to the investor and, with such logic, there is
no sense in calculating a “return rate” for the loan.
Consider again the project consortium. Instead of making the payments, the investor could
have applied the value at 0.5% / month, In date 19 the future value of the monthly investments (total
of 20, considering the payments start at date 0) would be:
FV  FV (i  0.5%; n  20; PMT  210; ; type  1)  4,427.64
On date 19, the investor/consumer ends the investment and borrows the difference of
5,572.36 to buy the vehicle. Considering a rate of 2.5% /month, the value of the monthly payment of
the loan to be paid in the following 30 months would be:
PMT  PMT (i  2.5%; n  30; PV  5,572.36)  266.23
That is, the project consortium is preferable to the investment/financing combined
alternative, in this case (210 < 266.23).
14
Finally, supposing the vehicle was drawn on date 29. With like procedures, the calculation of
the MIRR would indicate 1.21%. And now, how do we stand? We still have a MIRR higher than the
investment rate (0.5%) and lower to the financing rate (2.5%). Calculating again, the future value of
the investments, in date 29:
FV  FV (i  0.5%; n  30; PMT  210; ; type  1)  6,812.70
The value to be financed through the 20 remaining months for buying the vehicle would be $
3,187.30. Considering the rate of 2.5% /month., the monthly payment would be:
PMT  PMT (i  2.5%; n  20; PV  3,187.30)  204.46
That is, it is preferable to combine investment and financing than having the consortium
drawn in month 29 (204.46 < 210.00). Starting in date 29, what was an investment became a loan. In
fact, from this moment on, the investor becomes a consumer that buys in parcels and his logic is
inverted, since it then chooses for the lowest financing cost. The MIRR, as we have seen, was not
capable of indicating the best choice from the economic standpoint. The great lesson of this example
is showing that the MIRR, like the IRR, lacks financial meaning when the cash flow is nonconventional, with multiple solutions for the regular IRR.
5. Conclusions
The fact that the interest rate at which the present value of the cash flow of an investment is
zero has been named, since its conception in the early twentieth century, internal rate of return, has
enabled considerable misleads. Many executives and scholars became believers that this parameter
was a correct and precise indication of the project return.
When the possibility of having multiple solutions for the IRR was realized, it was observed
that, at least in a number of cases, the model was absolutely inadequate. Since then, complex
mathematical analyses, some elegantly built, were dedicated to discover the conditions at which the
rate that zeroes the net cash flow, if the same existed, would be singular and thus the IRR could be
validated as a measure for investment return. Little was mentioned that the IRR, as originally
conceived, assumes the manipulation of the intermediate cash flows at the IRR itself. So, with the
IRR being a project parameter, it does not actually exist as a return and, therefore, even in the socalled conventional cash flows, it may be very far from providing the measure of investment return.
It is worth mentioning that the Net Present Value method also admits manipulation of the
intermediate cash flows at the same Minimum Acceptable Rate of Return (MARR). The difference,
however, lays in the fact the MARR is defined based on a market reality, which avoids large
distortions in the calculation of the NPV. The IRR, on the other hand, is a totally virtual parameter,
associated with a specific cash flow. It will only be an exact measure of investment return when
there are no intermediate cash flows (positive or negative), that is, when there is a single initial
investment and a single final income (two-point flow).
In the 1970s started to take place the strongest criticism against the IRR. Lin (1976) presents
the concept of the MIRR. The calculation method of the MIRR, upon transforming the cash flow of
the project based on market hypothesis for reinvestment and financing rates, turns out to be a much
more adequate metric for indicating the investment return, in the cases where there is a single IRR.
The MS-Excel software has a function that calculates the MIRR of a cash flow, given the
15
reinvestment and financing rates (MIRR function). It is probable that the MIRR becomes more
popular among investment analysts.
The IRR represents a notable project parameter. Its main quality is not, however, to indicate
the return of an investment. Since the IRR does not depend of external factors to the project (it is a
parameter defined only by the cash flow), when applicable, it allows a fast and precise analysis of
sensibility to variations in the MARR. In the cases in which the IRR exists and is singular, the
condition IRR ≥ MARR clearly classifies the project as acceptable.
The present work shows that the IRR exists and is singular in conventional cash flows, as
defined in section 2. Certain non-conventional cash flows may also admit a single positive IRR. For
decades this theme was subject of complex mathematical analyses. A quick method for checking the
existence and singularity was presented by Norstrom (1972) demonstrated that the non-conventional
cash flow will admit a single positive solution for the IRR if the first flow is negative, the last one is
positive, the algebraic sum of the flows is positive and that there is not more than one variation in
sign in the algebraic sum of the accumulated cash flows in each period (project balances at a null
interest rate). This condition, although sufficient, is not, however, necessary, but allows a safe
increase in the use of the internal rate of the project.
As can be seen, this paper is not about abandoning the IRR, but recognizing its serious
interpretative problems, to the point it is mistakenly understood as an exact measure of return, as can
be seen in many textbooks and even in articles, some mentioned herein. Anyway, it is registered in
the present paper the proposal for modifying the name of the interest rate at which the cash flow of a
project is zero from Internal Rate of Return (IRR) to Internal Rate of the Project (IRP), as a
helpful contribution to eliminate the confusion of this rate with an exact measure of investment
return.
Bibliography
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uniqueness of the internal rate of return. The Journal of Industrial Engineering, 18.ago.
Brigham, E. F.; Gapenski, L. C.; Ehrhardt, M. C. , 2001, Administração financeira: teoria e prática,
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17
Attachment – Conventions for Financial Functions
The handling of financial calculators was widely spread in Brazil, not only for its simplicity, but
mainly due to the extreme necessity of using compound interest algebra during more than three
decades of high annual inflation rates. The textbook authors started to incorporate the use of
calculators in place of the compound interest tables and were then confronted with the lack of an
appropriate notation. The textbooks then started to indicate the financial calculations, representing
the buttons of the calculator. With the growth of computing, particularly the software MS-Excel,
investment analysis was made easier, allowing calculations which were before very complex. The
experience in the classroom led us then to replace the “button notation”, matching the notation used
for indicating the financial functions of the Excel with those of the most popular financial calculator,
the HP-12C. The mixed convention HP-Excel was born with the objective of allowing a perfect
observation of the operation made in the HP-12C and simultaneously easily transmitting to the
students the financial functions of the Excel, as can be seen in the table below.
ARGUMENT/FINANCIAL
FUNCTION
HP-12C
Notation
Excel
Notation
Mixed Convention
Interest Rate
i
RATE
Number of Periods
n
NPER
RATE (n; PMT ; PV ; FV ; type )
NPER (i; PMT ; PV ; FV ; type )
Present Value
PV
PV
Future Value
FV
FV
PV (i; n; PMT ; FV ; type )
FV (i; n; PMT ; PV ; type )
PMT
PMT
PMT (i; n; PV ; FV ; type )
HP-Excel
Periodic Payment
The mixed convention indicates the sequence at which the Excel recognizes data. When a certain
data is not available, it should be hidden, but in case there is another, a blank space must be left in
the sequence between the semicolons. Example: inexistence of the PMT, in the FV function the
notation is: FV(i;n;;PV;type).
Type – indicate 1 when the periodic payment takes place by the end of each period (corresponds to
the mode “begin” in HP-12C). Example: to obtain the present value of the series of 12 anticipated
monthly payments of $ 1,000.00, assuming an interest rate of 1.5% / month.
PV  PV (i  1.5%; n  12; PMT  1,000; ; type  1)  11,071.12
The explicit indication of the HP-12C notation allows the easy observation by those using financial
calculators, without confusing Excel users. HP-12C users should pay attention to the fact that the
indication type = 1 means “begin” mode.
Sign convention: both the functions PV, FV and PMT of the HP-12C, and PV, FV and PMT of the
Excel invert the signs of the solution, which is avoided inverting the signs of the cash flows.
Source: Álvares, 1998