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# irr

Internal rate of return

## Syntax

`Return = irr(CashFlow)[Return, AllRates] = irr(CashFlow)`

## Description

`Return = irr(CashFlow)` calculates the internal rate of return for a series of periodic cash flows.

`[Return, AllRates] = irr(CashFlow)` calculates the internal rate of return and a vector of all internal rates for a series of periodic cash flows.

## Input Arguments

 `CashFlow` A vector containing a stream of periodic cash flows. The first entry in `CashFlow` is the initial investment. If `CashFlow` is a matrix, `irr` handles each column of `CashFlow` as a separate cash-flow stream.

## Output Arguments

 `Return` An internal rate of return associated to `CashFlow`. If `CashFlow` is a matrix, then `Return` is a vector whose entry `j` is an internal rate of return for column `j` in `CashFlow`. `AllRates` A vector containing all the internal rates of return associated with `CashFlow`. If `CashFlow` is a matrix, then `AllRates` is also a matrix, with the same number of columns as `CashFlow` and one less row. Also, column `j` in `AllRates` contains all the rates of return associated to column `j` in `CashFlow` (including complex-valued rates).

## Definitions

`irr` uses the following conventions:

• If one or more internal rates of returns (warning if multiple) are strictly positive rates, `Return` sets to the minimum.

• If there is no strictly positive rate of returns, but one or multiple (warning if multiple) returns are nonpositive rates, `Return` sets to the maximum.

• If no real-valued rates exist, `Return` sets to `NaN` (no warnings).

## Examples

Find the internal rate of return for a simple investment with a unique positive rate of return. The initial investment is \$100,000 and the following cash flows represent the yearly income from the investment.

• Year 1 — \$10,000

• Year 2 — \$20,000

• Year 3 — \$30,000

• Year 4 — \$40,000

• Year 5 — \$50,000

Calculate the internal rate of return on the investment:

```Return = irr([-100000 10000 20000 30000 40000 50000]) ```

This returns:

```Return = 0.1201```

If the cash flow payments were monthly, then the resulting rate of return is multiplied by 12 for the annual rate of return.

Find the internal rate of return for multiple rates of return. The project has the following cash flows and a market rate of 10%.

` CashFlow = [-1000 6000 -10900 5800]`

Use `irr` with a single output argument:

`Return = irr(CashFlow)`

A warning appears and `irr` returns a 100% rate of return. The 100% rate on the project looks attractive:

```Warning: Multiple rates of return > In irr at 166 Return = 1.0000```

Use `irr` with two output arguments:

```[Return, AllRates] = irr(CashFlow) ```

This returns:

```>> [Return, AllRates] = irr(CashFlow) Return = 1.0000 AllRates = -0.0488 1.0000 2.0488```

The rates of return in `AllRates` are -4.88%, 100%, and 204.88%. Though some rates are lower and some higher than the market rate, based on the work of Hazen, any rate gives a consistent recommendation on the project. However, you can use a present value analysis in these kinds of situations. To check the present value of the project, use `pvvar`:

`PV = pvvar(CashFlow,0.10)`

This returns:

```PV = -196.0932```

The second argument is the 10% market rate. The present value is -196.0932, negative, so the project is undesirable.

## References

Brealey and Myers. Principles of Corporate Finance. McGraw-Hill Higher Education, Chapter 5, 2003.

Hazen G. "A New Perspective on Multiple Internal Rates of Return." The Engineering Economist. Vol. 48-1, 2003, pp. 31–51.