# Internal rate of return

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Internal rate of return

The internal rate of return (IRR) is a capital budgeting metric used by firms to decide whether they should make investments. It is an indicator of the efficiency or quality of an investment, as opposed to net present value (NPV), which indicates value or magnitude.

The IRR is the annualized effective compounded return rate which can be earned on the invested capital, i.e., the yield on the investment.

A project is a good investment proposition if its IRR is greater than the rate of return that could be earned by alternate investments of equal risk (investing in other projects, buying bonds, even putting the money in a bank account). Thus, the IRR should be compared to any alternate costs of capital including an appropriate risk premium.

In general, if the IRR is greater than the project's cost of capital, or "hurdle rate," the project will add value for the company.

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

Method

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.

Thus, in the case of cash flows at whole numbers of years, to find the internal rate of return, find the value(s) of $r$ that satisfies the following equation:

$mbox\left\{NPV\right\} = sum_\left\{t=0\right\}^\left\{N\right\} frac\left\{C_t\right\}\left\{\left(1+r\right)^\left\{t = 0$

Note that instead of converting to the present we can also convert to any other fixed time; 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 formula.

Example

Calculate the internal rate of return for an investment of 100 value in the first year followed by returns over the following 4 years, as shown below:Solution:

We use an iterative solver to determine the value of $r$ that solves the following equation:

$mbox\left\{NPV\right\} = -100 + frac\left\{40\right\}\left\{\left(1+r\right)^1\right\} + frac\left\{59\right\}\left\{\left(1+r\right)^2\right\} + frac\left\{55\right\}\left\{\left(1+r\right)^3\right\} + frac\left\{20\right\}\left\{\left(1+r\right)^4\right\} = 0$

The result from the numerical iteration is $r approx 28.09%$.

Graph of NPV as a function of $r$ for the example

Problems with using internal rate of return (IRR)

As an investment decision tool, the calculated IRR should not be used to rate mutually exclusive projects, but only to decide whether a single project is worth investing in. In cases where one project has a higher initial investment than a second mutually exclusive project, the first project may have a lower IRR (expected return), but a higher NPV (increase in shareholders' wealth) and should thus be accepted over the second project (assuming no capital constraints).

IRR assumes reinvestment of positive cash flows during the project at the same calculated IRR. When positive cash flows cannot be reinvested back into the project, IRR overstates returns. IRR is best used for projects with singular positive cash flows at the end of the project period.

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. [ [http://www.cfo.com/article.cfm/3304945/1/c_3348836 Internal Rate of Return: A Cautionary Tale] ]

This makes IRR a suitable (and popular) choice for analyzing venture capital and other private equity investments, as these strategies usually require several cash investments throughout the project, but only see one cash outflow at the end of the project (e.g. via IPO or M&A).

Since IRR does not consider cost of capital, it should not be used to compare projects of different duration. Modified Internal Rate of Return (MIRR) does consider cost of capital and provides a better indication of a project's efficiency in contributing to the firm's discounted cash flow.

In the case of positive cash flows followed by negative ones (+ + - - -) the IRR is a rate for lending/owing money, so the lowest IRR is best. 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.

A potential shortcoming of the IRR method is that it does not take into account that the intermediate positive cash flows possibly come at inconvenient moments. Their reinvestment may have a lower yield. In that case it may be more realistic to compute the IRR of the project including the reinvestments until e.g. the end date of the project. [ [http://www.cfo.com/article.cfm/3304945 Internal Rate of Return: A Cautionary Tale] ] Accordingly, MIRR is used, which has an assumed reinvestment rate, usually equal to the project's cost of capital.

Despite a strong academic preference for NPV, surveys indicate that executives prefer IRR over NPV. Apparently, managers find it easier to compare investments of different sizes in terms of percentage rates of return than by dollars of NPV. However, NPV remains the "more accurate" reflection of value to the business. IRR, as a measure of investment efficiency may give better insights in capital constrained situations. However, when comparing mutually exclusive projects, NPV is the appropriate measure.

Mathematics

Mathematically the value of the investment is assumed to undergo exponential growth or decay according to some rate of return (any value greater than -100%), with discontinuities for cash flows, and the IRR of a series of cash flows is defined as any rate of return that results in a net present value of zero (or equivalently, a rate of return that results in the correct value of zero after the last cash flow).

Thus internal rate(s) of return follow from the net present value as a function of the rate of return. This function is continuous. Towards a rate of return of -100% the net present value approaches infinity with the sign of the last cash flow, and towards a rate of return of positive infinity the net present value approaches the first cash flow (the one at the present). Therefore, if the first and last cash flow have a different sign there exists an internal rate of return. Examples of time series without an IRR:
*Only negative cash flows - the NPV is negative for every rate of return.
*(-1, 1, -1), rather small positive cash flow between two negative cash flows; the NPV is a quadratic function of 1/(1+"r"), where "r" is the rate of return, or put differently, a quadratic function of the discount rate "r"/(1+"r"); the highest NPV is -0.75, for "r" = 100%.

In the case of a series of exclusively negative cash flows followed by a series of exclusively positive ones, 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.

*Extended Internal Rate of Return: The Internal rate of return calculates the rate at which the investment made will generate cash flows. This method is convenient if the project has a short duration, but for projects which has an outlay of many years this method is not practical as IRR ignores the Time Value of Money. To take into consideration the Time Value of Money Extended Internal Rate of Return was introduced where all the future cash flows are first discounted at a discount rate and then the IRR is calculated. This method of calculation of IRR is called Extended Internal Rate of Return or XIRR.

ee also

*Accounting
*Capital budgeting
*Cost of capital
*Finance
*Net present value
*Discounted cash flow

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References

* [http://www.ofdict.com/definition/internal-rate-of-return.php Internal rate of return (IRR) in simple details]
* [http://www.advanced-excel.com/internal_rate_of_return.html Using Excel to find IRR]
* [http://sporkforge.com/finance/cash_flow.php Using the web to find IRR]

Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0471268496

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