Internal rate of return 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 (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 value of
the benefits (positive cash flows) 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 (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.
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 (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). Calculation
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 (, ) 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 (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 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 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 will converge reproducibly towards. If the function has real roots, then the sequence
will converge 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
will tend 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 (see above) with will always
produce 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 If applied iteratively, either the secant
method or the improved formula will always converge 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 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 (expected return), but a higher NPV (increase
in shareholders' wealth) and should thus be accepted over the second project (assuming
no capital constraints). IRR assumes reinvestment of interim cash flows
in projects with equal rates of return (the reinvestment can be the same project or a
different project). Therefore, IRR overstates the annual equivalent rate of return for a
project whose interim cash flows are reinvested at a rate lower than the calculated IRR. This
presents a problem, especially for high IRR projects, since there is frequently not another
project available in the interim that can earn the same rate of return as the first
project. This makes IRR a suitable (and popular) choice
for analyzing venture capital and other private equity investments, as these strategies usually
require several cash investments throughout the project, but only see one cash outflow
at the end of the project (e.g., via IPO or M&A).
Since IRR does not consider cost of capital, it should not be used to compare projects
of different duration. Modified Internal Rate of Return (MIRR) 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 (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 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 (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.