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Internal rate of return

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

`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.

`irr` uses the following conventions:

If one or more internal rate of returns (warning if multiple) are strictly positive rates,

`Return`sets to the minimum.If 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).

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.

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.

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