Approximating the time-weighted return: The case of flows at unknown time

We consider the problem of approximating the true time-weighted return when a cash flow occurs at an unknown time during the estimation period, which is usually the case of a traditional portfolio evaluated on a daily basis. We aim to provide the best approximation, in terms of mean square error (MSE), under the following main assumptions: the distribution of the log-returns belongs to a subclass of elliptical distributions; a single flow occurs at a uniformly distributed random time; the amount of the flow and the returns of the period are independent. We derive a closed-form formulation for high evaluation frequencies when the returns satisfy the popular assumption of a Geometric Brownian Motion. Besides, with the further assumption of small flows, the Original Dietz return can be obtained as an approximation of our optimal estimator. This implies that under the above-mentioned conditions the Original Dietz return has a MSE close to the minimum. Although further improvements of the MSE seem to be possible only by increasing the estimation frequency, which in turn is usually infeasible, our model provides a rigorous way to handle large flows, which are especially frequent in applications such as performance attribution.