Internal rate of return
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Internal rate of return (IRR) is a method of calculating an
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing i ...
’s
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
. The term ''internal'' refers to the fact that the calculation excludes external factors, such as the
risk-free rate The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ra ...
,
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
, the
cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate new ...
, or
financial risk Financial risk is any of various types of risk associated with financing, including financial transactions that include company loans in risk of default. Often it is understood to include only downside risk, meaning the potential for financial ...
. The method may be applied either
ex-post References Notes References Further reading

* * {{Latin phrases Lists of Latin phrases, E ...
or ex-ante. Applied ex-ante, the IRR is an estimate of a future annual rate of return. Applied ex-post, it measures the actual achieved investment return of a historical investment. It is also called the
discounted cash flow The discounted cash flow (DCF) analysis is a method in finance of valuing a security, project, company, or asset using the concepts of the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate devel ...
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
(DCFROR)Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269 or yield rate.


Definition (IRR)

The internal rate of return on an investment or project is the "annualized effective compounded return rate" or
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
that sets the
net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...
of all cash flows (both positive and negative) from the investment equal to zero. Equivalently, it is the
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
at which the net present value of the future cash flows is equal to the initial investment, and it is also the interest rate at which the total
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has inte ...
of costs (negative cash flows) equals the total present value of the benefits (positive cash flows). IRR accounts for the
time preference In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good or some cash at an earlier date compared with receiving it at a later ...
of money and investments. A given return on investment received at a given time is worth more than the same return received at a later time, so the latter would yield a lower IRR than the former, if all other factors are equal. A fixed income investment in which money is deposited once,
interest In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinct ...
on this deposit is paid to the investor at a specified
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
every time period, and the original deposit neither increases nor decreases, would have an IRR equal to the specified interest rate. An investment which has the same total returns as the preceding investment, but delays returns for one or more time periods, would have a lower IRR.


Uses


Savings and loans

In the context of savings and loans, the IRR is also called the
effective interest rate The effective interest rate (EIR), effective annual interest rate, annual equivalent rate (AER) or simply effective rate is the percentage of interest on a loan or financial product if compound interest accumulates over a year during which no pa ...
.


Profitability of an investment

The internal rate of return is an indicator of the
profitability In economics, profit is the difference between the revenue that an economic entity has received from its outputs and the total cost of its inputs. It is equal to total revenue minus total cost, including both explicit and implicit costs. It i ...
, efficiency, quality, or yield of an investment. This is in contrast with the
net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...
, which is an indicator of the net
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
or
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
added by making an investment. To maximize the
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
of a business, an investment should be made only if its profitability, as measured by the internal rate of return, is greater than a
minimum acceptable rate of return In business and for engineering economics in both industrial engineering and civil engineering practice, 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 ...
. If the estimated IRR of a project or investment - for example, the construction of a new factory - exceeds the firm's
cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate new ...
invested in that project, the investment is profitable. If the estimated IRR is less than the cost of capital, the proposed project should not be undertaken. The selection of investments may be subject to budget constraints. There may be
mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
competing projects, or limits on a firm's ability to manage multiple projects. For these reasons, corporations use IRR in
capital budgeting Capital budgeting in corporate finance is the planning process used to determine whether an organization's long term capital investments such as new machinery, replacement of machinery, new plants, new products, and research development project ...
to compare the
profitability In economics, profit is the difference between the revenue that an economic entity has received from its outputs and the total cost of its inputs. It is equal to total revenue minus total cost, including both explicit and implicit costs. It i ...
of a set of alternative capital projects. For example, a corporation will compare an investment in a new plant versus an extension of an existing plant based on the IRR of each project. To maximize
returns Return may refer to: In business, economics, and finance * Return on investment (ROI), the financial gain after an expense. * Rate of return, the financial term for the profit or loss derived from an investment * Tax return, a blank document or t ...
, the higher a project's IRR, the more desirable it is to undertake the project. There are at least two different ways to measure the IRR for an investment: the project IRR and the equity IRR. The project IRR assumes that the cash flows directly benefit the project, whereas equity IRR considers the returns for the shareholders of the company after the debt has been serviced. Even though IRR is one of the most popular metrics used to test the viability of an investment and compare returns of alternative projects, looking at the IRR in isolation might not be the best approach for an investment decision. Certain assumptions made during IRR calculations are not always applicable to the investment. In particular, IRR assumes that the project will have either no interim cash flows or the interim cash flows are reinvested into the project which is not always the case. This discrepancy leads to overestimation of the rate of return which might be an incorrect representation of the value of the project.


Fixed income

The internal rate of return is used to evaluate investments in fixed income securities, using metrics such as the
yield to maturity The yield to maturity (YTM), book yield or redemption yield of a bond or other fixed-interest security, such as gilts, is an estimate of the total rate of return anticipated to be earned by an investor who buys a bond at a given market price, ho ...
and yield to call.


Liabilities

Both the internal rate of return and the net present value can be applied to liabilities as well as investments. For a liability, a lower internal rate of return is preferable to a higher one.


Capital management

Corporations use internal rate of return to evaluate share issues and stock buyback programs. A share repurchase proceeds if returning capital to shareholders has a higher internal rate of return than candidate capital investment projects or acquisition projects at current market prices. Funding new projects by raising new debt may also involve measuring the cost of the new debt in terms of the
yield to maturity The yield to maturity (YTM), book yield or redemption yield of a bond or other fixed-interest security, such as gilts, is an estimate of the total rate of return anticipated to be earned by an investor who buys a bond at a given market price, ho ...
(internal rate of return).


Private equity

IRR is also used for
private equity In the field of finance, the term private equity (PE) refers to investment funds, usually limited partnerships (LP), which buy and restructure financially weak companies that produce goods and provide services. A private-equity fund is both a ty ...
, from the limited partners' perspective, as a measure of the general partner's performance as investment manager. This is because it is the general partner who controls the cash flows, including the limited partners' draw-downs of committed capital.


Calculation

Given a collection of pairs (
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
,
cash flow A cash flow is a real or virtual movement of money: *a cash flow in its narrow sense is a payment (in a currency), especially from one central bank account to another; the term 'cash flow' is mostly used to describe payments that are expected ...
) representing a project, the
net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...
is a function of the
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
. The internal rate of return is a rate for which this function is zero, i.e. the internal rate of return is a solution to the equation NPV = 0 (assuming no arbitrage conditions exist). Given the (period, cash flow) pairs (n, C_n) where n is a non-negative integer, the total number of periods N, and the \operatorname, (
net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...
); the internal rate of return is given by r in: : \operatorname = \sum_^N \frac = 0 This rational polynomial can be converted to an ordinary polynomial having the same roots by substituting ''g'' (gain) for 1+r and multiplying by g^N to yield the equivalent but simpler condition : \sum_^N C_ng^ = 0 The possible IRR's are the real values of ''r'' satisfying the first condition, and 1 less than the real roots of the second condition (that is, r = g-1 for each root ''g''). Note that in both formulas, C_0 is the negation of the initial investment at the start of the project while C_N is the cash value of the project at the end, equivalently the cash withdrawn if the project were to be liquidated and paid out so as to reduce the value of the project to zero. In the second condition C_0 is the leading coefficient of the ordinary polynomial in ''g'' while C_N is the constant term. The period n is usually given in years, but the calculation may be made simpler if r 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 In investment, an annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account, ...
); the value obtained is zero if and only if the NPV is zero. In the case that the cash flows are
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s, such as in the case of a life annuity, the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
s are put into the above formula. Often, the value of r that satisfies the above equation cannot be found analytically. In this case,
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
or
graphical methods A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent ta ...
must be used.


Example

If an investment may be given by the sequence of cash flows then the IRR r is given by : \operatorname = -123400+\frac + \frac + \frac = 0. 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 A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of the equation \operatorname(r)=0, there are many
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
that can be used to estimate r. For example, using the
secant method In 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 o ...
, r is given by : r_ = r_n-\operatorname_n \cdot \left(\frac\right). where r_n is considered the nth approximation of the IRR. This r can be found to an arbitrary degree of
accuracy Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each other ...
. Different accounting packages may provide functions for different accuracy levels. For example,
Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
and
Google Sheets Google Sheets is a spreadsheet program included as part of the free, web-based Google Docs Editors suite offered by Google. The service also includes: Google Docs, Google Slides, Google Drawings, Google Forms, Google Sites and Google Keep. Google ...
have built-in functions to calculate IRR for both fixed and variable time-intervals; "=IRR(...)" and "=XIRR(...)". The convergence behaviour of by the following: *If the function \operatorname(i) has a single
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
root r, then the sequence converges reproducibly towards r. *If the function \operatorname(i) has n real roots \scriptstyle r_1,r_2,\dots,r_n, 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 \operatorname(i) has no real roots, then the sequence tends towards +∞. Having \scriptstyle when \operatorname_0>0 or \scriptstyle when \operatorname_0<0 may speed up convergence of r_n to r.


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: : C_0<0,\quad C_n\ge 0\textn\ge 1. \, In this case the NPV of the payment stream is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
, strictly decreasing function of interest rate. There is always a single unique solution for IRR. Given two estimates r_1 and r_2 for IRR, the secant method equation (see above) with n=2 always produces an improved estimate r_3. 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 : r_ = r_n-\operatorname_n\left(\frac\right)\left(1 - 1.4 \frac \right), (which is most accurate when 0 > \operatorname_n > \operatorname_ ) 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 and initial guesses r_1 = 0.25 and r_2 = 0.2 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. 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: ::r_1 = \left( A / , C_0, \right) ^ - 1 \, ::r_2 = (1 + r_)^p - 1 \, where :: A = \text = C_1 + \cdots + C_N \, ::p = \frac. Here, \operatorname_ refers to the NPV of the inflows only (that is, set \mathrm_0 = 0 and compute NPV).


Exact dates of cash flows

A cash flow C_n may occur at any time t_n years after the beginning of the project. t_n may not be a whole number. The cash flow should still be discounted by a factor \frac. And the formula is : \operatorname = C_0 + \sum_^N \frac = 0 For numerical solution we can use
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
: r_ = r_k - \frac where \operatorname' is the derivative of \operatorname and given by : \operatorname' = -\sum_^N \frac An initial value r_1 can be given by : r_1 = \frac \sum_^N C_n - 1


Problems with use


Comparison with NPV investment selection criterion

As a tool applied to making an
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing i ...
decision on whether a project adds value or not, comparing the IRR of a single project with the required rate of return, in isolation from any other projects, is equivalent to the NPV method. If the appropriate IRR (if such can be found correctly) is greater than the required rate of return, using the required rate of return to discount cash flows to their present value, the NPV of that project will be positive, and vice versa. However, using IRR to sort projects in order of preference does not result in the same order as using NPV.


Maximizing net present value

One possible investment objective is to maximize the total net present value of projects. When the objective is to maximize total value, the calculated IRR should not be used to choose between mutually exclusive projects. 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). When the objective is to maximize total value, 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.


Practitioner preference for IRR over NPV

Despite a strong academic preference for net present value, surveys indicate that executives prefer IRR over NPV. Apparently, managers prefer to compare investments of different sizes in terms of forecast investment performance, using IRR, rather than maximize value to the firm, in terms of NPV. This preference makes a difference when comparing mutually exclusive projects.


Maximizing long-term return

Maximizing total value is not the only conceivable possible investment objective. An alternative objective would for example be to maximize long-term return. Such an objective would rationally lead to accepting first those new projects within the capital budget which have the highest IRR, because adding such projects would tend to maximize overall long-term return.


Example

To see this, consider two investors, Max Value and Max Return. Max Value wishes her net worth to grow as large as possible, and will invest every last cent available to achieve this, whereas Max Return wants to maximize his rate of return over the long term, and would prefer to choose projects with smaller capital outlay but higher returns. Max Value and Max Return can each raise ''up to'' 100,000 US dollars from their bank at an annual interest rate of 10 percent paid at the end of the year. Investors Max Value and Max Return are presented with two possible projects to invest in, called Big-Is-Best and Small-Is-Beautiful. Big-Is-Best requires a capital investment of 100,000 US dollars today, and the lucky investor will be repaid 132,000 US dollars in a year's time. Small-Is-Beautiful only requires 10,000 US dollars capital to be invested today, and will repay the investor 13,750 US dollars in a year's time.


=Solution

= The cost of capital for both investors is 10 percent. Both Big-Is-Best and Small-Is-Beautiful have positive NPVs: :\operatorname(\text) = \frac - 100,000 = 20,000 :\operatorname(\text) = \frac - 10,000 = 2,500 and the IRR of each is (of course) greater than the cost of capital: :\mathit(\text) = \frac - 100,000 = 0 so the IRR of Big-Is-Best is 32 percent, and :\operatorname(\text) = \frac - 10,000 = 0 so the IRR of Small-Is-Beautiful is 37.5 percent. Both investments would be acceptable to both investors, but the twist in the tale is that these are mutually exclusive projects for both investors, because their capital budget is limited to 100,000 US dollars. How will the investors choose rationally between the two? The happy outcome is that Max Value chooses Big-Is-Best, which has the higher NPV of 20,000 US dollars, over Small-Is-Beautiful, which only has a modest NPV of 2,500, whereas Max Return chooses Small-Is-Beautiful, for its superior 37.5 percent return, over the attractive (but not as attractive) return of 32 percent offered on Big-Is-Best. So there is no squabbling over who gets which project, they are each happy to choose different projects. How can this be rational for both investors? The answer lies in the fact that the investors do not have to invest the full 100,000 US dollars. Max Return is content to invest only 10,000 US dollars for now. After all, Max Return may rationalize the outcome by thinking that maybe tomorrow there will be new opportunities available to invest the remaining 90,000 US dollars the bank is willing to lend Max Return, at even higher IRRs. Even if only seven more projects come along which are identical to Small-Is-Beautiful, Max Return would be able to match the NPV of Big-Is-Best, on a total investment of only 80,000 US dollars, with 20,000 US dollars left in the budget to spare for truly unmissable opportunities. Max Value is also happy, because she has filled her capital budget straight away, and decides she can take the rest of the year off investing.


Multiple IRRs

When the sign of the cash flows changes more than once, for example when positive cash flows are followed by negative ones and then by positive ones (+ + − − − +), the IRR may have multiple real values. 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 real IRRs for a single project, like in the example 0% as well as 10%. Examples of this type of project are
strip mine Surface mining, including strip mining, open-pit mining and mountaintop removal mining, is a broad category of mining in which soil and rock overlying the mineral deposit (the overburden) are removed, in contrast to underground mining, in whic ...
s and
nuclear power Nuclear power is the use of nuclear reactions to produce electricity. Nuclear power can be obtained from nuclear fission, nuclear decay and nuclear fusion reactions. Presently, the vast majority of electricity from nuclear power is produced b ...
plants, where there is usually a large cash outflow at the end of the project. The IRR satisfies 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 by iteration. With multiple internal rates of return, the IRR approach can still be interpreted in a way that is consistent with the present value approach if the underlying investment stream is correctly identified as net investment or net borrowing. See for a way of identifying the relevant IRR from a set of multiple IRR solutions.


Limitations in the context of private equity

In the context of
survivorship bias Survivorship bias or survival bias is the logical error of concentrating on entities that passed a selection process while overlooking those that did not. This can lead to incorrect conclusions because of incomplete data. Survivorship bias is ...
which makes the high IRR of large private equity firms a poor representation of the average, according to Ludovic Phalippou, "...a headline figure that is often shown prominently as a rate of return in presentations and documents is, in fact, an IRR. IRRs are not rates of return. Something large PE firms have in common is that their early investments did well. These early winners have set up those firms' since-inception IRR at an artificially sticky and high level. The mathematics of IRR means that their IRRs will stay at this level forever, as long as the firms avoid major disasters. In passing, this generates some stark injustice because it is easier to game IRRs on LBOs in Western countries than in any other PE investments. That means that the rest of the PE industry (e.g. emerging market growth capital) is sentenced to look relatively bad forever, for no reason other than the use of a game-able performance metric." Also, "Another problem with the presentation of pension fund performance is that for PE, time-weighted returns...are not the most pertinent measure of performance. Asking how much pension funds gave and got back in dollar terms from PE, i.e. MoM, would be more pertinent. I went through the largest 15 funds websites to collect information on their performance. Few of them post their PE fund returns online. In most cases, they post information on their past performance in PE, but nothing that enables any meaningful benchmarking. ''E.g.'', CalSTRS California public pension fundprovide only the net IRR for each fund they invest in. As IRR is often misleading and can never be aggregated or compared to stock-market returns, such information is basically useless for gauging performance."


Modified internal rate of return (MIRR)

Modified Internal Rate of Return (MIRR) considers
cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate new ...
, and is intended to provide a better indication of a project's probable return. It applies a discount rate for borrowing cash, and the IRR is calculated for the investment cash flows. This applies in real life for example when a customer makes a deposit before a specific machine is built. When a project has multiple IRRs it may be more convenient to compute the IRR of the project with the benefits reinvested.Internal Rate of Return: A Cautionary Tale
/ref> Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital.


Average internal rate of return (AIRR)

Magni (2010) introduced a new approach, named AIRR approach, based on the intuitive notion of mean, that solves the problems of the IRR. However, the above-mentioned difficulties are only some of the many flaws incurred by the IRR. Magni (2013) provided a detailed list of 18 flaws of the IRR and showed how the AIRR approach does not incur the IRR problems.


Mathematics

Mathematically, the value of the investment is assumed to undergo exponential growth or decay according to some
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
(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 The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...
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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** 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 In mathematics, Descartes' rule of signs, first described by René Descartes in his work ''La Géométrie'', is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots i ...
, the number of internal rates of return can never be more than the number of changes in sign of cash flow.


The reinvestment debate

It is often stated that IRR assumes reinvestment of all cash flows until the very end of the project. This assertion has been a matter of debate in the literature. Sources stating that there is such a hidden assumption have been cited below.
Measuring Investment Returns
Other sources have argued that there is no IRR reinvestment assumption. When comparing investments, making an implicit assumption that cash flows are reinvested at the same IRR would lead to false conclusions. If cash flows received are not reinvested at the same rate as the IRR, a project with a relatively short duration and a high IRR does not necessarily add more value over a longer time span than another project with a longer duration and a lower IRR. This is why IRR should not be used on a stand-alone basis, but in combination with NPV. The Modified Internal Rate of Return (MIRR) addresses this issue by allowing for the inclusion a second investment at a potentially different rate of return, to calculate a portfolio return without external cash flows over the life of the project. However, for capital budgeting, when the objective is to maximize value, finance theory holds that NPV using the firm's cost of capital is the optimal metric.


In personal finance

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,The Mathematics of the Fixed Rate Equivalent
a GreaterThanZero White Paper.
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 the ''replicating 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. 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.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.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. 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. The resulting rate is called the ''fixed rate equivalent'' (''FREQ'').


Unannualized internal rate of return

In the context of investment performance measurement, there is sometimes ambiguity in terminology between the periodic
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
, 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.
Global Investment Performance Standards


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 i ...
*
Capital budgeting Capital budgeting in corporate finance is the planning process used to determine whether an organization's long term capital investments such as new machinery, replacement of machinery, new plants, new products, and research development project ...
*
Discounted cash flow The discounted cash flow (DCF) analysis is a method in finance of valuing a security, project, company, or asset using the concepts of the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate devel ...
*
Modified Dietz method The modified Dietz method is a measure of the ''ex post'' (i.e. historical) performance of an investment portfolio in the presence of external flows. (External flows are movements of value such as transfers of cash, securities or other instruments ...
* Modified internal rate of return *
Net present value The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...
*
Rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
*
Simple Dietz method The simple Dietz method is a means of measuring historical investment portfolio performance, compensating for external flows into/out of the portfolio during the period. The formula for the simple Dietz return is as follows: :R=\frac where : R is ...
*
Marginal efficiency of capital Marginal may refer to: * ''Marginal'' (album), the third album of the Belgian rock band Dead Man Ray, released in 2001 * ''Marginal'' (manga) * '' El Marginal'', Argentine TV series * Marginal seat or marginal constituency or marginal, in polit ...
*
Return on investment Return on investment (ROI) or return on costs (ROC) is a ratio between net income (over a period) and investment (costs resulting from an investment of some resources at a point in time). A high ROI means the investment's gains compare favourably ...


References


Further reading

* Bruce J. Feibel. ''Investment Performance Measurement''. New York: Wiley, 2003. * Ray Martin
INTERNAL RATE OF RETURN REVISITED


External links



{{DEFAULTSORT:Internal Rate Of Return Corporate finance Investment Capital budgeting