Formula and method
IRR is the periodic rate that makes net present value zero: sum cash flow at period t divided by (1+r)^t. The initial investment is at time zero. This calculator restricts the series to one initial outflow and later nonnegative receipts so the rate is unique.
Worked example
An initial outlay of $100 followed by a $110 receipt one year later has a 10% IRR. If the receipt is $50 instead, the one-year IRR is −50%.
Average return: arithmetic and geometric
Choose “Average return” and enter one return per period. The arithmetic average is (r₁ + r₂ + … + rₙ) ÷ n. The geometric average is ((1 + r₁)(1 + r₂)…(1 + rₙ))^(1/n) − 1: the constant rate that would turn the starting value into the actual ending value.
Example: returns of 10%, −5% and 20% have an arithmetic average of 8.33%, but 1.10 × 0.95 × 1.20 = 1.254 and 1.254^(1/3) − 1 = 7.84%. The geometric figure is never higher than the arithmetic one; the gap grows with volatility. Both are time-weighted: every period counts the same, whatever amount was invested.
Money-weighted return from dated cash flows
Choose “Money-weighted return” when you added or withdrew money on specific dates. Enter each flow as a date and an amount from your side: deposits negative, withdrawals positive, and the ending account value as a positive amount on the last date. The calculator finds the annual rate r that makes Σ CFᵢ ÷ (1 + r)^tᵢ = 0, where tᵢ is actual days since the first date ÷ 365 (the XIRR convention), by bisection.
Microsoft's XIRR example is −10,000 on 1 January 2008, then 2,750, 4,250, 3,250 and 2,750 on 1 March 2008, 30 October 2008, 15 February 2009 and 1 April 2009. It returns 37.34% a year. A money-weighted return rewards good timing of large deposits, so it can differ sharply from the time-weighted average.