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# Internal Rate of Return

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 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.

## Methodology

### Example:

Year Cash flow
0 -100
1 +120

### Calculation of NPV:

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

### Calculation of IRR:

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

### 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. The discount rate often used in capital budgeting that makes the net present value of all cash flows from a particular project equal to zero. Generally speaking, the higher a project’s internal rate of return, the more desirable it is to undertake the project. As such, IRR can be used to rank several prospective projects a firm is considering. Assuming all other factors are equal among the various projects, the project with the highest IRR would probably be considered the best and undertaken first.

IRR is sometimes referred to as "economic rate of return (ERR)".

You can think of IRR as the rate of growth a project is expected to generate. While the actual rate of return that a given project ends up generating will often differ from its estimated IRR rate, a project with a substantially higher IRR value than other available options would still provide a much better chance of strong growth.

IRR’s can also be compared against prevailing rates of return in the securities market. If a firm can’t find any projects with IRR’s greater than the returns that can be generated in the financial, it may simply choose to invest its retained earnings into the market.

### IRR - True interest yield expected from an investment

The Internal Rate of Return (IRR) is the discount rate that results in a net present value of zero for a series of future cash flows. It is an Discounted Cash Flow < (DCF) approach to valuation and investing just as Net Present Value (NPV). Both IRR and NPV are widely used to decide which investments to undertake and which investments not to make. The major difference is that while Net Present Value is expressed in monetary units (Euro’s or Dollars for example), the IRR is the true interest yield expected from an investment expressed as a percentage.

Internal Rate of Return is the flip side of Net Present Value and is based on the same principles and the same math. NPV shows the value of a stream of future cash flows discounted back to the present by some percentage that represents the minimum desired rate of return, often your company’s cost of capital. IRR, on the other hand, computes a break-even rate of return. It shows the discount rate below which an investment results in a positive NPV (and should be made) and above which an investment results in a negative NPV (and should be avoided). It’s the break-even discount rate, the rate at which the value of cash outflows equals the value of cash inflows.

Many people find the percentages of IRR easier to understand than Net Present Value. Another benefit from IRR is that it can be calculated without having to estimate the cost of capital. When IRR is used, the usual approach is to select the projects whose IRR exceeds the cost of capital (often called hurdle rate when used in the IRR context). This may seem simple and straightforward at first sight. However a major disadvantage of using the Internal Rate of Return instead of Net Present Value is that if managers focus on maximizing IRR and not NPV, there is a significant risk in companies where the return on investment is greater than the Weighted Average Cost of Capital (WACC) that managers will not invest in projects expected to earn greater than the WACC, but less than the return on existing assets.

IRR is a true indication of a project’s annual return of investment only when the project generates no interim cash flows - or when those interim investments can be invested at the actual IRR.

The aim of the value-oriented manager should be to invest in any project that has a positive NPV! If IRR usage is unavoidable, then managers are advised to use so called modified internal rate of return (which, while not perfect, at least allows to set more realistic interim reinvestment rates) and additionally to keep a close look on interim cash-flows, especially if they are biased to the beginning of the project period (the distortion is bigger then).

In other words, the aim should not be to maximize the Internal Rate of Return, but to maximize net present value .

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