# Finance:Internal rate of return

Short description: Method of calculating an investment's rate of return

Internal rate of return (IRR) is a method of quantifying the merits of a project or investment opportunity. The calculation is termed internal because it depends only on the cash flows of the investment being analyzed and excludes external factors, such as returns available elsewhere, the risk-free rate, inflation, the cost of capital, or financial risk.[1]

The method may be applied either ex-post 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 rate of return (DCFROR)[2] or yield rate.[3]

## Definition (IRR)

The IRR of an investment or project is the "annualized effective compounded return rate" or rate of return when the net present value (NPV) of all cash flows (both positive and negative) from the investment is assumed to be equal to zero.[3][4] Equivalently, it is the interest rate at which the net present value of the future cash flows is equal to the initial investment,[3][4] and it is also the interest rate at which the total present value of costs (negative cash flows) equals the total present value of the benefits (positive cash flows).

IRR, an acronym for Internal Rate of Return, is a crucial concept in the realm of finance. It represents the return on investment achieved when a project reaches its breakeven point, meaning that the project is only marginally justified as valuable. To gain a comprehensive understanding of IRR, it is essential to grasp another fundamental concept known as NPV, or Net Present Value. When NPV demonstrates a positive value, it indicates that the project is expected to generate value, thereby receiving approval from management to proceed. Conversely, if NPV shows a negative value, management will likely decide against moving forward with the project.

In essence, IRR signifies the rate of return attained when the NPV of the project reaches a neutral state, precisely at the point where NPV breaks even. For example, suppose that a project costs a company $100 today and returns$110 a year from today, as shown in the following table. The internal rate of return of this project is 10% because that is the discount rate (R) that makes net present value (NPV) equal to zero. If the 10% internal rate of return exceeds the company's minimal acceptable rate of return, the project is economically attractive.[5]

Example[5]
NPV = 0 = -100 + [110/(1+R)]
100 = [110/(1+R)]
1+R = 110/100 = 1.1
R = 10%

IRR accounts for the time preference 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 on this deposit is paid to the investor at a specified interest rate 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.

### Profitability of an investment

The IRR is an indicator of the profitability, efficiency, quality, or yield of an investment. This is in contrast with the NPV, which is an indicator of the net value or magnitude added by making an investment.

To maximize the value 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. If the estimated IRR of a project or investment - for example, the construction of a new factory - exceeds the firm's cost of capital 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.[6]

The selection of investments may be subject to budget constraints. There may be mutually exclusive competing projects, or limits on a firm's ability to manage multiple projects. For these reasons, corporations use IRR in capital budgeting to compare the profitability 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, 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.[7]

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.[8]

### Fixed income

IRR is used to evaluate investments in fixed income securities, using metrics such as the yield to maturity and yield to call.

### Liabilities

Both IRR and net present value can be applied to liabilities as well as investments. For a liability, a lower IRR is preferable to a higher one.

### Capital management

Corporations use IRR to evaluate share issues and stock buyback programs. A share repurchase proceeds if returning capital to shareholders has a higher IRR 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 (internal rate of return).

### Private equity

IRR is also used for private equity, from the limited partners' perspective, as a measure of the general partner's performance as investment manager.[9] 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, cash flow) representing a project, the NPV is a function of the rate of return. 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 ($\displaystyle{ n }$, $\displaystyle{ C_n }$) where $\displaystyle{ n }$ is a non-negative integer, the total number of periods $\displaystyle{ N }$, and the $\displaystyle{ \operatorname{NPV} }$, (net present value); the internal rate of return is given by $\displaystyle{ r }$ in:

$\displaystyle{ \operatorname{NPV} = \sum_{n=0}^N \frac{C_n}{(1+r)^n} = 0 }$

## Problems with use

### Practitioner preference for IRR over NPV

Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV.[10] 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.

### Multiple IRRs

In certain circumstances, there can be more than one internal rate of return that makes net present value equal to zero. When this happens, there is no definitive answer to the question "What is the rate of return?" This may occur when the cash flows of the investment are unconventional, i.e. 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 (+ + − − − +). 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.[11]

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.[12]

See [13] 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 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."[14]

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 [a California public pension fund] provide 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."[15]

### Modified internal rate of return (MIRR)

Modified Internal Rate of Return (MIRR) considers cost of capital, 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.[16] 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.[17] 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. [18]

## 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.[16][19] Other sources have argued that there is no IRR reinvestment assumption.[20][21][22][23][24][25]

## 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,[26] 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.[26] 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.[27][28] 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.[27][28] The resulting rate is called the fixed rate equivalent (FREQ).[26]

## Unannualized internal rate of return

In the context of investment performance measurement, there is sometimes ambiguity in terminology between the periodic rate of return, such as the IRR as defined above, and a holding period return. The term internal rate of return (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.[29]

## References

1. Ross, Stephen A.: Westerfield, Randolph W.; Jordan, Brandon (24 February 2009). Fundamentals of Corporate Finance (ninth; alternate ed.). Boston: McGraw-Hill Irwin. p. 273. ISBN 978007724612-9.
2. Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269
3. Kellison, Stephen G. (2009). The theory of interest (Third ed.). Boston: McGraw-Hill Irwin. pp. 251–252. ISBN 978-0-07-338244-9. OCLC 182552985.
4. Broverman, Samuel A. (2010). Mathematics of investment and credit (5th ed.). Winsted, CT: ACTEX Publications, Inc. pp. 264–265. ISBN 978-1-56698-767-7. OCLC 651487023.
5. Ross et al p. 273-274.
6. Ehsan, Nikbakht, Ehsan and Groppelli, A.A. (2012). Finance (sixth ed.). Hauppagge, NY: Barron's Educational Series. p. 201. ISBN 978-0-7641-4759-3.
7. Pogue, M.(2004). Investment Appraisal: A New Approach. Managerial Auditing Journal.Vol. 19 No. 4, 2004. pp. 565–570
8. Ross et al p. 277-278.
9. Hazen, G. B., "A new perspective on multiple internal rates of return," The Engineering Economist 48(1), 2003, 31–51.
10. Hartman, J. C., and Schafrick, I. C., "The relevant internal rate of return," The Engineering Economist 49(2), 2004, 139–158.
11. Phalippou, Ludovic (June 10, 2020). "Professor Financial Economics Said Business School Oxford University". SSRN Paper: 4.
12. Phalippou, Ludovic (June 10, 2020). "Professor Financial Economics Said Business School Oxford University". SSRN Paper: 15, 16.
13. Internal Rate of Return: A Cautionary Tale
14. Magni, C.A. (2010) "Average Internal Rate of Return and investment decisions: a new perspective". The Engineering Economist, 55(2), 150‒181.
15. Magni, C.A. (2013) "The Internal-Rate-of-Return approach and the AIRR paradigm: A refutation and a corroboration" The Engineering Economist, 58(2), 73‒111.
16. [1] Measuring Investment Returns
17. Dudley, C.L., "A note on reinvestment assumptions in choosing between net present value and internal rate of return." Journal of Finance 27(4), 1972, 907–15.
18. Keane, S.M., "The internal rate of return and the reinvestment fallacy." Abacus 15(1), 1979, 48–55.
19. Lohmann, J.R., "The IRR, NPV and the fallacy of the reinvestment rate assumptions". The Engineering Economist 33(4), 1988, 303–30.
20. Keef, S.P., and M.L. Roush, "Discounted cash flow methods and the fallacious reinvestment assumptions: a review of recent texts." Accounting Education 10(1), 2001, 105-116.
21. Rich, S.P., and J.T. Rose, "Re-examining an Old Question: Does the IRR Method Implicitly Assume a Reinvestment Rate?" Journal of Financial Education 10(1), 2014, 105-116.
22. Dudley, Magni, Carlo Alberto and Martin, John D., "The Reinvestment Rate Assumption Fallacy for IRR and NPV: A Pedagogical Note" 'https://mpra.ub.uni-muenchen.de/83889/', 2017
23. The Mathematics of the Fixed Rate Equivalent, a GreaterThanZero White Paper.
24. 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.
25. 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.
26. [2] Global Investment Performance Standards