Internal rate of return
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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 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 (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.
Mathematically the IRR is defined as any discount rate that results in a net present value of zero of a series of cash flows.
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.
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[edit] Method
To find the internal rate of return, find the value(s) of r that satisfies the following equation:
(See net present value for details on this formula.)
[edit] 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:
| Year | Cash Flow |
|---|---|
| 0 | -100 |
| 1 | 39 |
| 2 | 59 |
| 3 | 55 |
| 4 | 20 |
Solution:
We use an iterative solver to determine the value of r that solves the following equation:
The result from the numerical iteration is 
.
[edit] Graph of NPV as a function of r for the example
[edit] 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 makes no assumptions about the reinvestment of the positive cash flow from a project. As a result, IRR should not be used to compare projects of different duration and with a different overall pattern of cash flows. Modified Internal Rate of Return (MIRR) provides a better indication of a project's efficiency in contributing to the firm's discounted cash flow.
The IRR method should not be used in the usual manner for projects that start with an initial positive cash inflow (or in some projects with large negative cash flows at the end), 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 equation. Sturm's Theorem can be used to determine if that equation 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 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.
In addition if the NPV of one project is higher than another and the other project has a higher IRR, then the cross over point method can be used to solve this dispute.
Cross Over Point > IRR = Accept project with higher NPV and if the Cross Over Point < IRR = Accept project with higher IRR
[edit] See also
[edit] External links
- Capital Budgeting, audio lecture with slide show
- Misleading IRR values , tied-up and free capital
- Calculate IRR
- Calculating IRR using Excel
- An example of IRR using a savings account
- IRR online calculator
- IRR web calculator
- International Federation of Accountants International Good Practice Guidance on Project Appraisal Using Discounted Cash Flow
[edit] Further reading
Bruce J. Feibel. Investment Performance Measurement. New York: Wiley, 2003. ISBN 0471268496
