Single-Period Mean–Variance Analysis in a Changing World (corrected)

Ideally, financial analysts would like to be able to optimize a consumption–investment game with many securities, many time periods, transaction costs, and changing probability distributions. We cannot. For a small optimizable version of such a game, we consider in this article how much would be lost by following one or another heuristic that could be easily scaled to handle large games. For the games considered, a particular mean–variance heuristic does almost as well as the optimum strategy. Ideally, financial analysts would like to solve investment problems involving large universes of securities, many time periods, asset illiquidities (such as transaction costs and unrealized taxable capital gains), and changing probability distributions (with perhaps some predictability). Currently, such problems are well beyond state-of-the-art optimization. We consider in this article how much we would lose if we used one or another heuristic (rule of thumb) to solve investment problems. We cannot determine the loss that would come from use of a heuristic rather than an optimum solution for large-scale problems because we cannot compute optimum solutions. For sufficiently small problems, however, we can compute the expected payoffs to an optimum strategy and to various heuristics. In this article, we compare the expected payoffs of the optimum solution and various heuristics for a simple dynamic model for which the optimum solution can be solved. The heuristics considered can be scaled to much larger problems. The simplified model we analyze consists of two assets (stock and cash) and discrete time periods (months). In the case we consider, an investor seeks to maximize total utility, which is defined as the sum of the present values of one-period utilities. The investor has a market prediction model that may predict a very optimistic, optimistic, neutral, pessimistic, or very pessimistic state. We analyze the model for two levels of transaction costs. One of the heuristics tried is a “mean–variance surrogate for the ‘derived’ utility function” (“MV surrogate heuristic” for short). We know from dynamic programming that the expected value of total utility for the game as a whole can be maximized by maximizing the expected values of a sequence of single-period utility functions, using a so-called derived utility function. For complex games, the derived utility function cannot be computed. The MV surrogate heuristic replaces this incomputable function with a simple function of portfolio mean and variance. In addition to the MV surrogate heuristic, we test other heuristics: all stock, all cash, a 60/40 stock/cash mix, a “very active” heuristic that always moves to what would be optimum if there were no transaction costs, and an “inactive” heuristic that does nothing. For the models considered, the MV surrogate heuristic did better than the other heuristics and almost as well as the optimum strategy. We offer suggestions as to how the MV surrogate heuristic could be applied to more complex models (such as the model we tested but with more securities) and to models with other kinds of prediction rules or illiquidities (such as unrealized taxable capital gains).