The internal rate of return or economic rate of return 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. 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.
Definition 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^year) of all cash flows 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 of the investment equals the
net present value of the benefits of the investment. 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 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. Uses of IRR
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. 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. 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.
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 than any other use of the funds for other capital projects,
or than any acquisition candidate at current market prices.
Calculation Given a collection of pairs 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 pairs where is a positive integer,
the total number of periods , and the net present value , 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 and converted to
a yearly period thereafter. Any fixed time can be used in place of the
present; 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
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%. Numerical solution
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 . For example,
using the secant 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 the sequence
is governed 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 when or when may speed up convergence of to .
Numerical solution for single outflow and multiple inflows
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 with always produces an improved
estimate . This is sometimes referred to as the Hit and Trial 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% as compared to IRR = 13.2% from the secant method. Other improved formulas may
be found in 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. Decision criterion
If the IRR is greater than the cost of capital, accept the project.
If the IRR is less than the cost of capital, reject the project.
Problems with using internal rate of return 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.
In cases where one project has a higher initial investment than a second mutually exclusive
project, the first project may have a lower IRR, but a higher NPV and should thus be accepted
over the second project. Since IRR does not consider cost of capital,
it should not be used to compare projects of different duration. Modified Internal Rate
of Return does consider cost of capital and provides a better indication of a project's
efficiency in contributing to the firm's discounted cash flow.
In the case of positive cash flows followed by negative ones and then by positive ones
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, 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 are strip 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. Accordingly, MIRR is used, which has an assumed reinvestment
rate, usually equal to the project's cost of capital.
It has been shown 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 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. 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
Mathematically, the value of the investment is assumed to undergo exponential growth or
decay according to some rate of return, 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. 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. 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 to the value of the first cash flow, so there is a unique
rate of return for which it is zero. Hence, the IRR is also unique. 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.
The reinvestment misconception It is often stated that IRR assumes reinvestment
of all cash flows until the very end of the project. This is a misconception. There is
no hidden reinvestment assumption associated with the calculation of IRR. IRR is the solution
to the equation in the example shown above - nothing more, nothing less. The cash flows
are static. The NPV is set at zero. There is only one unknown variable in the equation
- r. This misconception likely stems from the MIRR
concept, which allows for inclusion of a second, subsequent investment. If the reinvestment
rate is set at IRR, the MIRR equals the IRR. This is hardly a surprise - compounding cash
flows and then discounting them using the same discount factor is obviously a zero-sum
game. There are many, highly reputable sources arguing
that there is a hidden reinvestment assumption in the IRR calculation. There are, however,
also many sources disputing the so-called reinvestment assumption.
See also Accounting rate of return
Capital budgeting Discounted cash flow
Modified internal rate of return Modified Dietz method
Net present value Rate of return
Simple Dietz method References
Further reading Bruce J. Feibel. Investment Performance Measurement.
New York: Wiley, 2003. ISBN 0-471-26849-6 External links
Economics Interactive Lecture from University of South Carolina