# Internal rate of return

fr:taux de rentabilité interne The internal rate of return (IRR) is defined as the interest rate that gives a net present value (NPV) of zero. The NPV is calculated from an annualized cash flow by discounting all future amounts to the present.

``` Example:
Year     Cash flow
0        -100
1        +120
```
``` Calculation of NPV:
i = interest rate in percent
NPV = -100 +120/[(1+i/100)^1]
```
``` Calculation of IRR (in percent):
NPV = 0
-100 +120/[(1+IRR/100)^1] = 0
IRR = 20
```

As an investment decision tool the calculated IRR is used to rate alternative investments. The investment alternative with the highest IRR is preferred. Note that placing the initial investment amount in the bank is always an alternative. Thus, any investments which do not match the bank's going deposit rate will not be realized.

It should also be noted that zeros of NPV as a function of IRR may lack existence or uniqueness if there is some alternation of positive and negative cash flow. The IRR exists and is unique if one or more years of net investment (negative cash flow) are followed by years of net revenues.

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.

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