An error formula for Monte Carlo based portfolio optimization

One–period utility based portfolio optimization is a classic method in financial economics [13]. The theory is extremely well developed, and extends in many directions, for example to dynamic portfolio optimization in continuous time [12]. Nonetheless, the solutions to these problems, characterized theoretically, often remain dismayingly difficult to compute in practise. Very often, finance practitioners retreat from the economically sound methods of utility based optimization to a computationally simple but economically unjustified mean– variance method. This paper will examine a straightforward Monte Carlo method for computing utilityoptimal portfolios for problems of investing inM assets over a single period. After reviewing the basics of portfolio theory in sections 2 and 3, in section 4 we propose a formula for the error between the true solution and the Monte Carlo estimate and give a theoretical justification for it (but not a proof). As is typical in Monte Carlo methods, the error term is difficult to estimate accurately. In this paper we give rough estimates of the theoretical error in two ways: first by using the Monte Carlo simulation itself, and second by using an approximate formula derived under a Gaussian assumption. We find these two approaches to be mutually consistent, and consistent with the observed convergence of the Monte Carlo estimates. ∗Research supported by the Natural Sciences and Engineering Research Council of Canada and Mathematics of Information Technology and Complex Systems, Canada