How useful are mean-variance considerations in stock trading via feedback control?

In classical finance, when a stochastic investment outcome is characterized in terms of its mean and variance, it is implicitly understood that the underlying probability distribution is not heavily skewed. For example, in the “perfect” case when outcomes are normally distributed, mean-variance considerations tell the entire story. The main point of this paper is that mean-variance based measures of performance may be entirely inappropriate when a feedback control law is used instead of buy-and-hold to modulate one's stock position as a function of time. For example, when using a feedback gain K to increment or decrement one's stock position, we see that the resulting skewness measure S(K) for the trading gains or losses can easily become dangerously large. Hence, we argue in this paper that the selection of this gain K based on a classical mean-variance based utility function can lead to a distorted picture of the prospects for success. To this end, our analysis begins in a so-called idealized market with prices generated by Geometric Brownian Motion (GBM). In addition to the “red flag” associated with skewness, a controller efficiency analysis is also brought to bear. While all feedback gains K lead to efficient (non-dominated, Pareto optimal) controllers, in the mean-variance sense, we show that the same does not hold true when we use a return-risk pair which incorporates more information about the probability distribution for gains and losses. To study the efficiency issue in an application context, the paper also includes a simulation for Pepsico Inc. using the last five years of historical data.