Internal rate of return

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The internal rate of return (IRR) is a capital budgeting method used by firms to decide whether they should make long-term investments.

The IRR is the 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 interest that could be earned by alternative investments (investing in other projects, buying bonds, even putting the money in a bank account). The IRR should include an appropriate risk premium.

Mathematically the IRR is defined as any discount rate that results in a net present value of zero of a series of cashflows.

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.

1 Methodology
2 Problems with using IRR
3 See also
4 External links

[edit] Methodology

To find the internal rate of return, find the IRR that satisfies the following equation:

$\mbox{Initial Investment} = \sum_{t=1}^N \frac{C_t}{(1+IRR)^t}$

 Example:
 Year     Cash flow
  0        -100
  1        +120

 Calculation of NPV:
 i = interest rate in percent
 NPV = -100 +120/[(1+i/100)^1]
 (This calculation is condensed, see net present value.)

 Calculation of IRR:
 NPV = 0
 -100 +120/[(1+IRR/100)^1] = 0
 IRR = 20%

[edit] Problems with using 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. A method called marginal IRR can be used to adapt the IRR methodology to this case.

The IRR method should not be used in the usual manner for projects that start with an initial positive cash inflow, for example where a customer makes a deposit before a specific machine is built, resulting in a single positive cash flow followed by a series of negative cash flows (+ - - - -). In this case the usual IRR decision rule needs to be reversed.

If there are multiple sign changes in the series of cash flows, e.g. (- + - + -), there may be multiple IRRs for a single project, so that the IRR decision rule may be impossible to implement. 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. Sturm's Theorem can be used to determine if that polynomial has a unique real solution. Importantly, the IRR equation cannot be solved analytically (i.e. in its general form) but only via iterations.

A critical shortcoming of the IRR method is that it is commonly misunderstood to convey the actual annual profitability of an investment. However, this is not the case because intermediate cash flows are almost never reinvested at the project's IRR; and, therefore, the actual rate of return (akin to the one that would have been yielded by stocks or bank deposits) is almost certainly going to be lower. Accordingly, a measure called Modified Internal Rate of Return (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 intuitively more appealing to evaluate investments in terms of percentage rates of return than dollars of NPV. However, NPV remains the "more accurate" reflection of value to the business.