FILTER RULES AND STOCK-MARKET TRADING* EUGENE F FAMAT AND MARSHALL E. BLUMEt " N THE recent literature there has been tcy? profits than a buy-and-hold pol- I. INTRODUCTION pected a considerable interest in the theory On the other hand, the statistician has of random walks in stock-market different though prices. The basic hypothesis of the theory tions of what constitutes an important is that successive price changes in indi- violation of the independence assump- vidual securities are independent random tion of the random-walk model. He will variables plies, of course, that the past history of a series of changes implies that the history of a price series can hanges cannot be used to predict future not be used to increase expected gains, the revers changes in any"meaningful"way. proposition does not hold. It is possible to construct What constitutes a"meaningful""pre- models where successive price changes are depend iction depends, of course, on the pur- used to increase expected profits.In fact, Mandelbrot di pose for which the data are being ex- 19 and Samuelson (12 show that, under fairly gen. to know whether the history of prices can tingle"which may or may not have the independ be used to increase expected gains. In a the martingale property mplies only that the random-walk market, with either zero of expecled values of future prices will be independent positive drift, no mechanical trading rule of the values of past prices; the distributions of applied to an individual security would future prices, however may very well depend on consistently outperform a policy of the values of past prices. In a martingale, though simply buying and holding the security. cannot be used by the trader to increase his expected Thus, the investor who must choose be- profits. A random walk is a martingale, but a mar- tween the random-walk model and a more complicated model which assumes the behavior of stock-market prices came about be- the existence of an excessive degree of fore the theoretical importance of the martingale either persistence (positive dependence) ly concerned with the theory of random walks. In or reaction (negative dependence)in suc- practice, this is not serious, since in most cases it is cessi ive price changes, should accept the probably impossible to distinguish a that fol- heory of random walks as the better series that follows a random walk. In most cases the model if the actual degree of dependence degree of dependence shown by a martingale will be cannot be used to produce greater e ors have bene. random-walk model fited from discussions with Professors Lawrence The terminology used in this paper will be that Fisher, Benoit Mandelbrot, Merton Miller, Peter of the more familiar theory of random walks rather Pashigian, and Harry Roberts of the University of than the more general(but perhaps simpler)theor Chicage of martingale processes. The reader will note, how Assistant professor of finance graduate School ever, that the bulk of our discussions remain valid if of Business, University of Chicago. e word“ martingale" is substituted for“rand Lecturer in applied mathematics, Graduate walk"and the words"the martingale property"are School of Business, University of Chicago substituted for"independe