Internal Rate of Return

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Internal rate of return
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Not to be confused with Implied repo rate.
This article needs additional citations for verification. Please help improve this
article by adding citations to reliable sources. Unsourced material may be challenged and
removed. (March 2009)
The internal rate of return (IRR) or economic rate of return (ERR) is a rate of return used
in capital budgeting to measure and compare the profitability of investments. It is also called
the discounted cash flow rate of return (DCFROR).[1] In the context of savings and loans, the IRR is
also called the effective interest rate. The term internal refers to the fact that its calculation does not
incorporate environmental factors (e.g., the interest rate or inflation).
Contents
[hide]



1 Definition



2 Uses of IRR



3 Calculation
o

o

3.1 Example


3.1.1 Numerical solution



3.1.2 Numerical solution for single outflow and multiple inflows

3.2 Decision criterion



4 Problems with using internal rate of return



5 Mathematics



6 Controversy



7 The internal rate of return in personal finance



8 Unannualized internal rate of return



9 See also



10 References



11 Further reading



12 External links

Definition[edit]
The internal rate of return on an investment or project is the "annualized effective compounded
return rate" or rate of return that makes the net present value (NPV as NET*1/(1+IRR)^year) of all
cash flows (both positive and negative) from a particular investment equal to zero. It can also be
defined as the discount rate at which the present value of all future cash flow is equal to the initial
investment or in other words the rate at which an investment breaks even.
In more specific terms, the IRR of an investment is the discount rate at which the net present
value of costs (negative cash flows) of the investment equals the net present valueof the benefits
(positive cash flows) of the investment.

Uses of IRR[edit]
IRR calculations are commonly used to evaluate the desirability of investments or projects. The
higher a project's IRR, the more desirable it is to undertake the project. Assuming all projects require
the same amount of up-front investment, the project with the highest IRR would be considered the
best and undertaken first.
A firm (or individual) should, in theory, undertake all projects or investments available with IRRs that
exceed the cost of capital. Investment may be limited by availability of funds to the firm and/or by the
firm's capacity or ability to manage numerous projects.
Because the internal rate of return is a rate quantity, it is an indicator of the efficiency, quality,
or yield of an investment. This is in contrast with the net present value, which is an indicator of the
value or magnitude of an investment.
An investment is considered acceptable if its internal rate of return is greater than an
established minimum acceptable rate of return or cost of capital. In a scenario where an investment
is considered by a firm that has shareholders, this minimum rate is the cost of capital of the
investment (which may be determined by the risk-adjusted cost of capital of alternative investments).
This ensures that the investment is supported by equity holders since, in general, an investment
whose IRR exceeds its cost of capital adds value for the company (i.e., it is economically profitable).
One of the uses of IRR is by corporations that wish to compare capital projects. For example, a
corporation will evaluate an investment in a new plant versus an extension of an existing plant based
on the IRR of each project. In such a case, each new capital project must produce an IRR that is
higher than the company's cost of capital. Once this hurdle is surpassed, the project with the highest
IRR would be the wiser investment, all other things being equal (including risk).
IRR is also useful for corporations in evaluating stock buyback programs. Clearly, if a company
allocates a substantial amount to a stock buyback, the analysis must show that the company's own
stock is a better investment (has a higher IRR) than any other use of the funds for other capital
projects, or than any acquisition candidate at current market prices.

Calculation[edit]

Given a collection of pairs (time, cash flow) involved in a project, the internal rate of return follows
from the net present value as a function of the rate of return. A rate of return for which this function is
zero is an internal rate of return.
Given the (period, cash flow) pairs ( ,
periods
, and the net present value

) where is a positive integer, the total number of
, the internal rate of return is given by in:

The period is usually given in years, but the calculation may be
made simpler if is calculated using the period in which the
majority of the problem is defined (e.g., using months if most of the
cash flows occur at monthly intervals) and converted to a yearly
period thereafter.
Any fixed time can be used in place of the present (e.g., the end of
one interval of an annuity); the value obtained is zero if and only if
the NPV is zero.
In the case that the cash flows are random variables, such as in the
case of a life annuity, the expected values are put into the above
formula.
Often, the value of cannot be found analytically. In this
case, numerical methods or graphical methods must be used.

Example[edit]
Year ( )

Cash flow (

0

-123400

1

36200

2

54800

3

48100

)

If an investment may be given by the sequence of cash flows
then the IRR

is given by

In this case, the answer is 5.96% (in the calculation, that is, r = .
0596).
Numerical solution[edit]
Since the above is a manifestation of the general problem of
finding the roots of the equation
, there are
many numerical methods that can be used to estimate . ant
method]], is given by

where

is considered the

th

approximation of the IRR.

This can be found to an arbitrary degree of accuracy. An
accuracy of 0.00001% is provided by Microsoft Excel
The convergence behaviour of by the following:


If the function
has a single real root , then
the sequence converges reproducibly towards .



If the function
has real roots
,
then the sequence converges to one of the roots, and
changing the values of the initial pairs may change the
root to which it converges.



If function
has no real roots, then the
sequence tends towards +∞.

Having
n

when
or
whe
may speed up convergence of

to .

Numerical solution for single outflow and multiple
inflows[edit]
Of particular interest is the case where the stream of
payments consists of a single outflow, followed by multiple
inflows occurring at equal periods. In the above notation,
this corresponds to:
In this case the NPV of the payment stream is
a convex, strictly decreasing function of interest rate.
There is always a single unique solution for IRR.
Given two estimates
and
for IRR, the secant
method equation (see above) with
always
produces an improved estimate . This is sometimes
referred to as the Hit and Trial (or Trial and Error)
method. More accurate interpolation formulas can also
be obtained: for instance the secant formula with
correction

,
(which is most accurate
when
) has been
shown to be almost 10 times more accurate than

the secant formula for a wide range of interest rates
and initial guesses. For example, using the stream
of payments {−4000, 1200, 1410, 1875, 1050} and
initial guesses
and
the
secant formula with correction gives an IRR
estimate of 14.2% (0.7% error) as compared to IRR
= 13.2% (7% error) from the secant method. Other
improved formulas may be found in [2]
If applied iteratively, either the secant method or
the improved formula always converges to the
correct solution.
Both the secant method and the improved formula
rely on initial guesses for IRR. The following initial
guesses may be used:

where

Here,
refers to the NPV
of the inflows only (that is,
set
and compute NPV).

Decision criterion[edit]
If the IRR is greater than the cost
of capital, accept the projects.
If the IRR is less than the cost of
capital, reject the projects.

Problems with using
internal rate of
return[edit]
As an investment decision tool, the
calculated IRR should not be used
to rate mutually exclusive projects,
but only to decide whether a single
project is worth investing in.

NPV vs discount rate comparison
for two mutually exclusive
projects. Project 'A' has a higher
NPV (for certain discount rates),
even though its IRR (= x-axis
intercept) is lower than for project
'B' (click to enlarge)

In cases where one project has a
higher initial investment than a
second mutually exclusive project,
the first project may have a lower
IRR (expected return), but a higher
NPV (increase in shareholders'
wealth) and should thus be
accepted over the second project
(assuming no capital constraints).
IRR should not be used to
compare projects of different
duration. For example, the net
present value added by a project
with longer duration but lower IRR
could be greater than that of a
project of similar size, in terms of
total net cash flows, but with
shorter duration and higher IRR.
Modified Internal Rate of
Return (MIRR) considers cost of
capital, and is intended to provide
a better indication of a project's
probable return.
In the case of positive cash flows
followed by negative ones and
then by positive ones (for example,
+ + − − − +) the IRR may have
multiple values. In this case a
discount rate may be used for the
borrowing cash flow and the IRR

calculated for the investment cash
flow. This applies for example
when a customer makes a deposit
before a specific machine is built.
In a series of cash flows like (−10,
21, −11), one initially invests
money, so a high rate of return is
best, but then receives more than
one possesses, so then one owes
money, so now a low rate of return
is best. In this case it is not even
clear whether a high or a low IRR
is better. There may even be
multiple IRRs for a single project,
like in the example 0% as well as
10%. Examples of this type of
project arestrip mines and nuclear
power plants, where there is
usually a large cash outflow at the
end of the project.
In general, the IRR can be
calculated by solving a polynomial
equation. Sturm's theorem can be
used to determine if that equation
has a unique real solution. In
general the IRR equation cannot
be solved analytically but only
iteratively.
When a project has multiple IRRs
it may be more convenient to
compute the IRR of the project
with the benefits reinvested.
[3]
Accordingly, MIRR is used,
which has an assumed
reinvestment rate, usually equal to
the project's cost of capital.
It has been shown[4] that with
multiple internal rates of return, the
IRR approach can still be
interpreted in a way that is
consistent with the present value
approach provided that the
underlying investment stream is
correctly identified as net
investment or net borrowing.
See also [5] for a way of identifying
the relevant value of the IRR from
a set of multiple IRR solutions.
Despite a strong academic
preference for NPV, surveys

indicate that executives prefer IRR
over NPV.[6] Apparently, managers
find it easier to compare
investments of different sizes in
terms of percentage rates of return
than by dollars of NPV. However,
NPV remains the "more accurate"
reflection of value to the business.
IRR, as a measure of investment
efficiency may give better insights
in capital constrained situations.
However, when comparing
mutually exclusive projects, NPV is
the appropriate measure.

Mathematics[edit]
Mathematically, the value of the
investment is assumed to undergo
exponential growth or decay
according to some rate of
return (any value greater than
−100%), with discontinuities for
cash flows, and the IRR of a series
of cash flows is defined as any
rate of return that results in a net
present value of zero (or
equivalently, a rate of return that
results in the correct value of zero
after the last cash flow).
Thus, internal rate(s) of return
follow from the net present value
as a function of the rate of return.
This function is continuous.
Towards a rate of return of −100%
the net present value approaches
infinity with the sign of the last
cash flow, and towards a rate of
return of positive infinity the net
present value approaches the first
cash flow (the one at the present).
Therefore, if the first and last cash
flow have a different sign there
exists an internal rate of return.
Examples of time series without an
IRR:


Only negative cash flows —
the NPV is negative for every
rate of return.



(−1, 1, −1), rather small
positive cash flow between
two negative cash flows; the
NPV is a quadratic function of
1/(1 + r), where r is the rate of
return, or put differently, a
quadratic function of
the discount rate r/(1 + r); the
highest NPV is −0.75, for r =
100%.

In the case of a series of
exclusively negative cash flows
followed by a series of exclusively
positive ones, the resulting
function of the rate of return is
continuous and monotonically
decreasing from positive infinity
(when the rate of return
approaches -100%) to the value of
the first cash flow (when the rate of
return approaches infinity), so
there is a unique rate of return for
which it is zero. Hence, the IRR is
also unique (and equal). Although
the NPV-function itself is not
necessarily monotonically
decreasing on its whole domain,
it is at the IRR.
Similarly, in the case of a series of
exclusively positive cash flows
followed by a series of exclusively
negative ones the IRR is also
unique.
Finally, by Descartes' rule of signs,
the number of internal rates of
return can never be more than the
number of changes in sign of cash
flow.

Controversy[edit]
There is controversy over whether
or not the internal rate of return
method contains an implicit
assumption. The question is
whether or not the flows to and/or
from the investment during its life
are assumed to be reinvested, to
obtain a return on the whole net
capital over the whole life of the
investment, and the rate of return

reinvested capital flows would
earn.
The argument relates to
the modified internal rate of
return (MIRR) method, in which
flows to and from the investment
during its life return a fixed
reinvestment rate outside the first
investment. If a second investment
exists, such as a bank account,
which earns (or costs) the
reinvestment rate, and if
furthermore, payments can be
made between the first investment
and the second one, timed to
coincide with the flows to or from
the first investment, then the MIRR
can be earned on the entire net
capital over the entire life of the
investment. This does not hold for
the IRR: the IRR is not earned on
the entire net capital over the
entire life of the investment except in the exceptional case
when a reinvestment rate is
available which happens to equal
the IRR. Nor would it hold for any
of the competing methods of
compensating for flows to
calculate ex post returns, such as
the time-weighted return or
the modified Dietz method.
Sources arguing there is such a
hidden assumption include those
cited below.[3][7] Other sources
argue the opposite.[8]

The internal rate of
return in personal
finance[edit]
The IRR can be used to measure
the money-weighted performance
of financial investments such as an
individual investor's brokerage
account. For this scenario, an
equivalent,[9] more intuitive
definition of the IRR is, "The IRR is
the annual interest rate of the fixed
rate account (like a somewhat
idealized savings account) which,

when subjected to the same
deposits and withdrawals as the
actual investment, has the same
ending balance as the actual
investment." This fixed rate
account is also called
thereplicating fixed rate
account for the investment. There
are examples where the replicating
fixed rate account encounters
negative balances despite the fact
that the actual investment did not.
[9]
In those cases, the IRR
calculation assumes that the same
interest rate that is paid on positive
balances is charged on negative
balances. It has been shown that
this way of charging interest is the
root cause of the IRR's multiple
solutions problem.[10][11] If the model
is modified so that, as is the case
in real life, an externally supplied
cost of borrowing (possibly varying
over time) is charged on negative
balances, the multiple solutions
issue disappears.[10][11] The resulting
rate is called the fixed rate
equivalent (FREQ).[9]

Unannualized internal
rate of return[edit]
In the context of investment
performance measurement, there
is sometimes ambiguity in
terminology between the
periodic rate of return, such as the
internal rate of return as defined
above, and a holding period return.
The term internal rate of
return or IRR or Since Inception
Internal Rate of Return (SI-IRR) is
in some contexts used to refer to
the unannualized return over the
period, particularly for periods of
less than a year.[12]

See also[edit]


Accounting rate of return



Capital budgeting



Discounted cash flow



Modified Dietz method



Modified internal rate of return



Net present value



Rate of return



Simple Dietz method

References[edit]
1.

Jump up^ Project
Economics and Decision
Analysis, Volume I:
Deterministic Models,
M.A.Main, Page 269

2.

Jump up^ Moten, J. and
Thron, C., Improvements on
Secant Method for Estimating
Internal Rate of
Return, International Journal
of Applied Mathematics and
Statistics 42:12(2013),http://
www.ceser.in/ceserp/index.p
hp/ijamas/article/view/1929.

3.

^ Jump up to:a b Internal Rate
of Return: A Cautionary Tale

4.

Jump up^ Hazen, G. B., "A
new perspective on multiple
internal rates of return," The
Engineering
Economist 48(2), 2003, 31–
51.

5.

Jump up^ Hartman, J. C.,
and Schafrick, I. C., "The
relevant internal rate of
return," The Engineering
Economist 49(2), 2004, 139–
158.

6.

Jump up^ Pogue, M.(2004).
Investment Appraisal: A New
Approach. Managerial

Auditing Journal.Vol. 19 No.
4, 2004. pp. 565–570
7.

Jump up^ [1] Measuring
Investment Returns

8.

Jump up^ [2], Schmidt, R.,
"What is IRR and How Does
it
Work?" PropertyMetrics.com
June 9, 2014.

9.

^ Jump up to:a b c The
Mathematics of the Fixed
Rate Equivalent, a
GreaterThanZero White
Paper.

10. ^ Jump up to:a b Teichroew,
D., Robicheck, A., and
Montalbano, M.,
Mathematical analysis of
rates of return under
certainty, Management
Science Vol. 11 Nr. 3,
January 1965, 395–403.
11. ^ Jump up to:a b Teichroew,
D., Robicheck, A., and
Montalbano, M., An analysis
of criteria for investment and
financing decisions under
certainty, Management
Science Vol. 12 Nr. 3,
November 1965, 151–179.
12. Jump up^ [3] Global
Investment Performance
Standards

Further reading[edit]
1. Bruce J.
Feibel. Investment
Performance
Measurement. New York:
Wiley, 2003. ISBN 0-47126849-6

External links[edit]


Economics Interactive Lecture
from University of South
Carolina





GIPS Global Investment
Performance Standards 2010,
CFA Institute
Categories:
Corporate finance



Investment

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