# Internal rate of return

Not to be confused with Implied repo rate.

The internal rate of return (IRR) or external rate of return (ERR) is a method of calculating rate of return. The term internal refers to the fact that its calculation does not incorporate environmental factors (e.g., the interest rate or inflation).

It is also called the discounted cash flow rate of return (DCFROR).[1]

In the context of savings and loans, the IRR is also called the effective interest rate.

## 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 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.

Equivalently, the IRR of an investment is the discount rate at which the present value of costs (negative cash flows) of the investment equals the present value of the benefits (positive cash flows) of the investment resulting to a zero net present value.

## Uses of IRR

IRR is used in capital budgeting to measure and compare the profitability of investments.

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.

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).

One of the uses of IRR is by corporations that wish to compare capital projects. For example, a corporation will evaluate an investment in a new plant versus an extension of an existing plant based on the IRR of each project. In such a case, each new capital project must produce an IRR that is higher than the company's cost of capital. Once this hurdle is surpassed, the project with the highest IRR would be the wiser investment, all other things being equal (including risk).

IRR is also useful for corporations in evaluating stock buyback programs. Clearly, if a company allocates a substantial amount to a stock buyback, the analysis must show that the company's own stock is a better investment (has a higher IRR) than any other use of the funds for other capital projects, or than any acquisition candidate at current market prices.

IRR is also suitable for private equity, from the limited partners' perspective, as a measure of the general partner's performance as investment manager.[2] 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) 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 (${\displaystyle n}$, ${\displaystyle C_{n}}$) where ${\displaystyle n}$ is a positive integer, the total number of periods ${\displaystyle N}$, and the ${\displaystyle \mathrm {NPV} }$, (net present value); the IRR is given by ${\displaystyle r}$ in:

${\displaystyle \mathrm {NPV} =\sum _{n=0}^{N}{\frac {C_{n}}{(1+r)^{n}}}=0}$

The period ${\displaystyle n}$ is usually given in years, but the calculation may be made simpler if ${\displaystyle r}$ 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); 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 ${\displaystyle r}$ 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

Year (${\displaystyle n}$) Cash flow (${\displaystyle C_{n}}$)
0 -123400
1 36200
2 54800
3 48100

then the IRR ${\displaystyle r}$ is given by

${\displaystyle \mathrm {NPV} =-123400+{\frac {36200}{(1+r)^{1}}}+{\frac {54800}{(1+r)^{2}}}+{\frac {48100}{(1+r)^{3}}}=0.}$

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 of the equation ${\displaystyle \mathrm {NPV} (r)=0}$, there are many numerical methods that can be used to estimate ${\displaystyle r}$. For example, using the secant method, ${\displaystyle r}$ is given by

${\displaystyle r_{n+1}=r_{n}-\mathrm {NPV} _{n}\cdot \left({\frac {r_{n}-r_{n-1}}{\mathrm {NPV} _{n}-\mathrm {NPV} _{n-1}}}\right).}$

where ${\displaystyle r_{n}}$ is considered the ${\displaystyle n}$th approximation of the IRR.

This ${\displaystyle r}$ can be found to an arbitrary degree of accuracy. An accuracy of 0.00001% is provided by Microsoft Excel.

The convergence behaviour of by the following:

• If the function ${\displaystyle \mathrm {NPV} (i)}$ has a single real root ${\displaystyle r}$, then the sequence converges reproducibly towards ${\displaystyle r}$.
• If the function ${\displaystyle \mathrm {NPV} (i)}$ has ${\displaystyle n}$ real roots ${\displaystyle \scriptstyle r_{1},r_{2},\dots ,r_{n}}$, 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 ${\displaystyle \mathrm {NPV} (i)}$ has no real roots, then the sequence tends towards +∞.

Having ${\displaystyle \scriptstyle {r_{1}>r_{0}}}$ when ${\displaystyle \mathrm {NPV} _{0}>0}$ or ${\displaystyle \scriptstyle {r_{1} when ${\displaystyle \mathrm {NPV} _{0}<0}$ may speed up convergence of ${\displaystyle r_{n}}$ to ${\displaystyle r}$.

#### 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:

${\displaystyle C_{0}<0,\quad C_{n}\geq 0{\text{ for }}n\geq 1.\,}$

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 ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ for IRR, the secant method equation (see above) with ${\displaystyle n=2}$ always produces an improved estimate ${\displaystyle r_{3}}$. 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

${\displaystyle r_{n+1}=r_{n}-\mathrm {NPV} _{n}\left({\frac {r_{n}-r_{n-1}}{\mathrm {NPV} _{n}-\mathrm {NPV} _{n-1}}}\right)\left(1-1.4{\frac {\mathrm {NPV} _{n-1}}{\mathrm {NPV} _{n-1}-3\mathrm {NPV} _{n}+2C_{0}}}\right)}$,

(which is most accurate when ${\displaystyle 0>\mathrm {NPV} _{n}>\mathrm {NPV} _{n-1}}$) 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 ${\displaystyle r_{1}=0.25}$ and ${\displaystyle r_{2}=0.2}$ 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. Other improved formulas may be found in [3]

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:

${\displaystyle r_{1}=\left(A/|C_{0}|\right)^{2/(N+1)}-1\,}$
${\displaystyle r_{2}=(1+r_{1})^{p}-1\,}$

where

${\displaystyle A={\text{ sum of inflows }}=C_{1}+\cdots +C_{N}\,}$
${\displaystyle p={\frac {\log(\mathrm {A} /|C_{0}|)}{\log(\mathrm {A} /\mathrm {NPV} _{1,in})}}.}$

Here, ${\displaystyle \mathrm {NPV} _{1,in}}$ refers to the NPV of the inflows only (that is, set ${\displaystyle \mathrm {C_{0}=0} }$ and compute NPV).

## 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.

NPV vs discount rate comparison for two mutually exclusive projects. Project 'A' has a higher NPV (for certain discount rates), even though its IRR (= x-axis intercept) is lower than for project 'B' (click to enlarge)

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 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.

Modified Internal Rate of Return (MIRR) considers cost of capital, and is intended to provide a better indication of a project's probable return.

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.[4] Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital.

It has been shown[5] 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[6] 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.[7] 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.

Magni (2010) introduced a new approach, named AIRR approach, based on the intuitive notion of mean, that solves the problems of the IRR.[8] 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. [9]

## 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.

## The reinvestment misconception

It is often stated that IRR assumes reinvestment of all cash flows until the very end of the project. This is a misconception. There is no hidden reinvestment assumption associated with the calculation of IRR. IRR is simply the solution to the equation in the example shown above. The cash flows are static. The NPV is set at zero. There is only one unknown variable in the equation, namely r.

Sources stating that there is such a hidden assumption include those cited below.[4][10] Sources arguing that there is no IRR reinvestment assumption include[11][12][13][14][15]

When IRR is used in place of NPV in capital budgeting or investment analysis, a user may make an implicit assumption of reinvestment when investments of different duration are compared. This assumption arises from the naive use of IRR, not in its calculation. Clearly a shorter project with a higher IRR is not necessarily better if the proceeds cannot be reinvested at the same rate. The Modified Internal Rate of Return (MIRR) addresses this issue by including a second investment at potentially different rate of return. However for capital budgeting, finance theory holds that NPV using the firm's cost of capital is a superior metric.

## The internal rate of return 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,[16] 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.[16] 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.[17][18] 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.[17][18] The resulting rate is called the fixed rate equivalent (FREQ).[16]

## 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 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.[19]

## References

1. ^ Project Economics and Decision Analysis, Volume I: Deterministic Models, M.A.Main, Page 269
2. ^ "Global Investment Performance Standards". CFA Institute. Retrieved 31 December 2015.
3. ^ Moten, J. and Thron, C., Improvements on Secant Method for Estimating Internal Rate of Return, International Journal of Applied Mathematics and Statistics 42:12(2013), http://www.ceser.in/ceserp/index.php/ijamas/article/view/1929.
4. ^ a b Internal Rate of Return: A Cautionary Tale
5. ^ Hazen, G. B., "A new perspective on multiple internal rates of return," The Engineering Economist 48(2), 2003, 31–51.
6. ^ Hartman, J. C., and Schafrick, I. C., "The relevant internal rate of return," The Engineering Economist 49(2), 2004, 139–158.
7. ^ Pogue, M.(2004). Investment Appraisal: A New Approach. Managerial Auditing Journal.Vol. 19 No. 4, 2004. pp. 565–570
8. ^ Magni, C.A. (2010) "Average Internal Rate of Return and investment decisions: a new perspective". The Engineering Economist, 55(2), 150‒181.
9. ^ 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.
10. ^ [1] Measuring Investment Returns
11. ^ 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.
12. ^ Keane, S.M., “The internal rate of return and the reinvestment fallacy.” Abacus 15(1), 1979, 48–55.
13. ^ Lohmann, J.R., “The IRR, NPV and the fallacy of the reinvestment rate assumptions”. The Engineering Economist 33(4), 1988, 303–30.
14. ^ 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.
15. ^ 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.
16. ^ a b c The Mathematics of the Fixed Rate Equivalent, a GreaterThanZero White Paper.
17. ^ a b 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.
18. ^ a b 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.
19. ^ [2] Global Investment Performance Standards