Minimax Policies for Selling an Asset and Dollar Averaging

The results of this paper fall into two areas. First, it is shown that the conventional wisdom of dollar averaging is related to hedging against, large regrets rather than unfavorable outcomes. Suppose that a given sum of dollars is to be irreversibly converted into stock within a given number of periods. Suppose that the share price of dollars follows an arithmetic random walk. It is shown that dollar averaging is a nonsequential minimax strategy, if the largest possible price increase in each period is equal to the largest possible decrease. On the other hand, it is shown that dollar averaging cannot be a nonsequential expected utility maximizing strategy for any strictly concave utility function and any arithmetic random walk. Second, sequential minimax strategies are examined. Here the analysis may be viewed as an extension of the literature on the optimal time to sell an asset or the optimal stopping problem. The minimax sequential strategy is shown to be of the following form. At any time there exists a critical value which depends only on n, the difference between the maximum price since conversions began and the current price. Funds held in excess of this value are converted into stock, otherwise no funds are converted. The critical values are shown to be decreasing functions of n. When the share price of dollars rises (i.e., stock prices fall) few funds are converted while decreases produce large conversions. These results seem to correspond to the second thoughts of those, embarking on nonsequential policies of the dollar averaging type. Buying and selling minimax policies are shown to be symmetric. Sequential minimaxing will tend to have a reinforcing effect on price movements.