is the geometric mean of returns, which, over n periods, is: The geometric average return is equivalent to the cumulative return over the whole n periods, converted into a rate of return per period. Where the individual sub-periods are each equal (say, 1 year), and there is reinvestment of returns, the annualized cumulative return is the geometric average rate of return. For example, assuming reinvestment, the cumulative return for four annual returns of 50%, -20%, 30%, and −40% is: The geometric average return is: The annualized cumulative return and geometric return are related thus: Comparisons between various rates of return External flows In the presence of external flows, such as cash or securities moving into or out of the portfolio, the return should be calculated by compensating for these movements. This is achieved using methods such as the time-weighted return. Time-weighted returns compensate for the impact of cash flows. This is useful to assess the performance of a money manager on behalf of his/her clients, where typically the clients control these cash flows. Fees To measure returns net of fees, allow the value of the portfolio to be reduced by the amount of the fees. To calculate returns gross of fees, compensate for them by treating them as an external flow, and exclude accrued fees from valuations. Money-weighted rate of return Like the time-weighted return, the money-weighted rate of return (MWRR) or dollar-weighted rate of return also takes cash flows into consideration. They are useful evaluating and comparing cases where the money manager controls cash flows, for example private equity. (Contrast with the true time-weighted rate of return, which is most applicable to measure the performance of a money manager who does not have control over external flows.) Internal rate of return The internal rate of return (IRR) (which is a variety of money-weighted rate of return) is the rate of return which makes the net present value of cash flows zero. It is a solution satisfying the following equation: where: NPV = net present value and = net cash flow at time , including the initial value and final value , net of any other flows at the beginning and at the end respectively. (The initial value is treated as an inflow, and the final value as an outflow.) When the internal rate of return is greater than the cost of capital, (which is also referred to as the