The
internal rate of return (
IRR) is a
rate of returnIn finance, rate of return , also known as return on investment , rate of profit or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss, or...
used in
capital budgetingCapital budgeting is the planning process used to determine whether an organization's long term investments such as new machinery, replacement machinery, new plants, new products, and research development projects are worth pursuing...
to measure and compare the
profitabilityIn economics, the term profit has two related but distinct meanings. Normal profit represents the total opportunity costs of a venture to an entrepreneur or investor, whilst economic profit In economics, the term profit has two related but distinct meanings. Normal profit represents the total...
of
investmentInvestment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...
s. It is also called the
discounted cash flowIn finance, discounted cash flow analysis is a method of valuing a project, company, or asset using the concepts of the time value of money...
rate of return (DCFROR) or the rate of return (ROR). In the context of savings and loans the IRR is also called the
effective interest rateThe effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.It is used to compare the...
. The term
internal refers to the fact that its calculation does not incorporate environmental factors (e.g., the
interest rateAn interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
or inflation).
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 valueIn finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
(NPV as NET*1/(1+IRR)^year) of all cash flows (both positive and negative) from a particular investment equal to zero.
In more specific terms, the IRR of an investment is the
discount rateThe discount rate can mean*an interest rate a central bank charges depository institutions that borrow reserves from it, for example for the use of the Federal Reserve's discount window....
at which the
net present valueIn finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
of costs (negative cash flows) of the investment equals the
net present valueIn finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
of the benefits (positive cash flows) of the investment.
Internal rates of return are commonly used to evaluate the desirability of investments or projects. The higher a project's internal rate of return, 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
Because the internal rate of return is a rate quantity, it is an indicator of the efficiency, quality, or
yieldIn finance, the term yield describes the amount in cash that returns to the owners of a security. Normally it does not include the price variations, at the difference of the total return...
of an investment. This is in contrast with the net present value, which is an indicator of the value or
magnitudeThe magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of an investment.
An investment is considered acceptable if its internal rate of return is greater than an established
minimum acceptable rate of returnIn business and engineering, the minimum acceptable rate of return, often abbreviated MARR, or hurdle rate is the minimum rate of return on a project a manager or company is willing to accept before starting a project, given its risk and the opportunity cost of forgoing other projects...
or
cost of capitalThe cost of capital is a term used in the field of financial investment to refer to the cost of a company's funds , or, from an investor's point of view "the shareholder's required return on a portfolio of all the company's existing securities"...
. In a scenario where an investment is considered by a firm that has equity holders, this minimum rate is the
cost of capitalThe cost of capital is a term used in the field of financial investment to refer to the cost of a company's funds , or, from an investor's point of view "the shareholder's required return on a portfolio of all the company's existing securities"...
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
valueAn economic value is the worth of a good or service as determined by the market.The economic value of a good or service has puzzled economists since the beginning of the discipline. First, economists tried to estimate the value of a good to an individual alone, and extend that definition to goods...
for the company (i.e., it is economically profitable).
Calculation
Given a collection of pairs (
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
,
cash flowCash flow is the movement of money into or out of a business, project, or financial product. It is usually measured during a specified, finite period of time. Measurement of cash flow can be used for calculating other parameters that give information on a company's value and situation.Cash flow...
) involved in a project, the internal rate of return follows from the
net present valueIn finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
as a function of the
rate of returnIn finance, rate of return , also known as return on investment , rate of profit or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss, or...
. 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
annuityThe term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...
); the value obtained is zero if and only if the NPV is zero.
In the case that the cash flows are
random variableIn probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s, such as in the case of a
life annuityA life annuity is a financial contract in the form of an insurance product according to which a seller — typically a financial institution such as a life insurance company — makes a series of future payments to a buyer in exchange for the immediate payment of a lump sum or a series...
, the
expected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
s are put into the above formula.
Often, the value of
cannot be found analytically. In this case,
numerical methodsNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
or
graphical methodsA plot is a graphical technique for representing a data set, usually as a graph showing the relationship between two or more variables. The plot can be drawn by hand or by a mechanical or electronic plotter. Graphs are a visual representation of the relationship between variables, very useful for...
must be used.
Example
If an investment may be given by the sequence of cash flows
Year ( ) |
Cash Flow ( ) |
| 0 |
-4000 |
| 1 |
1200 |
| 2 |
1410 |
| 3 |
1875 |
| 4 |
1050 |
37.49
then the IRR
is given by
.
In this case, the answer is 14.3%.
Numerical solution
Since the above is a manifestation of the general problem of finding the roots of the equation
, there are many
numerical methodsA root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f = 0, for a given function f. Such an x is called a root of the function f....
that can be used to estimate
. For example, using the
secant methodIn numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method. However, the method was developed...
,
is given by
.
where
is considered the
th approximation of the IRR.
This
can be found to an arbitrary degree of
accuracyIn the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...
.
The convergence behaviour of the sequence is governed by the following:
- If the function
has a single realIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
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 +∞In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
.
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:
< 0,
≥ 0 for
≥ 1.
In this case the NPV of the payment stream is a
convexIn mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...
,
strictly decreasingIn mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
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. There is however a much more accurate estimation formula, given by:
-
where
-
.
In this equation,
and
refer to the NPV's of the
inflows only (that is, set
= 0 and compute NPV). For example, using the stream of payments {-4000, 1200, 1410, 1875, 1050} and initial guesses
and
gives
and
. The accurate formula estimates IRR as 14.35% (0.3% error) as compared to IRR = 14.7% (3% error) from the secant method.
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
-
sum of inflows 
-
.
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
investmentInvestment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...
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.
When the calculated IRR is higher than the true reinvestment rate for interim cash flows, the measure will overestimate — sometimes very significantly — the annual equivalent return from the project. The formula assumes that the company has additional projects, with equally attractive prospects, in which to invest the interim cash flows.
This makes IRR a suitable (and popular) choice for analyzing
venture capitalVenture capital is financial capital provided to early-stage, high-potential, high risk, growth startup companies. The venture capital fund makes money by owning equity in the companies it invests in, which usually have a novel technology or business model in high technology industries, such as...
and other
private equityPrivate equity, in finance, is an asset class consisting of equity securities in operating companies that are not publicly traded on a stock exchange....
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 capitalThe cost of capital is a term used in the field of financial investment to refer to the cost of a company's funds , or, from an investor's point of view "the shareholder's required return on a portfolio of all the company's existing securities"...
, it should not be used to compare projects of different duration.
Modified Internal Rate of ReturnThe modified internal rate of return is a financial measure of an investment's attractiveness. It is used in capital budgeting to rank alternative investments of equal size...
(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 powerNuclear power is the use of sustained nuclear fission to generate heat and electricity. Nuclear power plants provide about 6% of the world's energy and 13–14% of the world's electricity, with the U.S., France, and Japan together accounting for about 50% of nuclear generated electricity...
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 theoremIn mathematics, Sturm's theorem is a symbolic procedure to determine the number of distinct real roots of a polynomial. It was named for Jacques Charles François Sturm...
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 returnIn finance, rate of return , also known as return on investment , rate of profit or sometimes just return, is the ratio of money gained or lost on an investment relative to the amount of money invested. The amount of money gained or lost may be referred to as interest, profit/loss, gain/loss, or...
(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 valueIn finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
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
continuousIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
. 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, consider the total value of the cash flows converted to a time between the negative and the positive ones. The resulting function of the rate of return is continuous and monotonically decreasing from positive infinity to negative 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.
See also
- Accounting rate of return
Accounting rate of return, also known as the Average rate of return, or ARR is a financial ratio used in capital budgeting. The ratio does not take into account the concept of time value of money. ARR calculates the return, generated from net income of the proposed capital investment. The ARR is a...
- Capital budgeting
Capital budgeting is the planning process used to determine whether an organization's long term investments such as new machinery, replacement machinery, new plants, new products, and research development projects are worth pursuing...
- Discounted cash flow
In finance, discounted cash flow analysis is a method of valuing a project, company, or asset using the concepts of the time value of money...
- Modified internal rate of return
The modified internal rate of return is a financial measure of an investment's attractiveness. It is used in capital budgeting to rank alternative investments of equal size...
- Modified Dietz Method
The Modified Dietz Method is a calculation used to determine an approximation of the performance of an investment portfolio based on money-weighted cash flow...
- Net present value
In finance, the net present value or net present worth of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values of the individual cash flows of the same entity...
- Simple Dietz Method
Simple Dietz Method is a means of calculating an approximation of investment portfolio performance during a period of external cash flows into/out of the portfolio...
Further reading
- Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0471268496
External links