A Martingale System Theorem for Stock Investments

In tlus paper we consider an investment strategy called dollar-cast-averaging (DCA). It is well known that the argument in favor of DCA is flawed: an investor cannot get something for nothing. Probabilists have made tltis type of flaw precise with theorems like the optional sampling theorem or, more generally, the martingale system theorem (see e.g. [4]). It is interesting that the usual martingale system theorem is not formulated in a manner that is convenient for debunking DCA. The purpose of tltis paper is to formulate a new martingale system theorem wltich directly shows that DCA yields no advantage. The story beltind DCA goes as follows. Financial pla111lers argue (see e.g. [2] or [5]) that investing a fixed small amount periodically in a given stock is superior to a single large investment in that stock. These financial pla111lers, perhaps motivated by a larger percentage commission on small investments, argue that, in buying fixed small amounts frequently, the investor will benefit through buying relatively more when the price is low than when it is high. In tltis way, they claim that the investor can make money even when the share price does not exhibit an upward trend. Tltis investment strategy is called dollar-cost-averaging. We compare a general investment strategy to the mean performance of the stock. By introducing a time dependent scale factor to correct for any drift in the mean performance, we may without loss of generality assume that the mean performance would pay one dollar for every dollar invested no matter when the investment is liquidated. For this reason, we will refer to the mean performance as pocketing the money but the reader should realize that what we have in ntind is a riskless investment that matches the mean performance of the stock investment (at least over some reasonable time frame). Of course in reality, there is no riskless investment that matches the performance of a risky investment. Our comparison to the mean performance merely serves as a benchmark to judge the relative merits of possible investment strategies. In Section 3, we propose a better model and show that, on the average, no strategy has an edge. In Section 4, we show how to correct for trends so that the martingale theorem in Section 3 applies in general. Finally, in Section 5 we study the continuous version of the model introduced in Section 3.