An Anatomy of Trading Strategies 1.The Profitability of Trading Strategies We consider a set of trading strategies that either explicitly mimic or capture the essence of previously implemented strategies.Specifically,consider buying or selling stocks at time t-I based on their performance from time t-2 to t-1,where the period (t-1,t]spans any finite time interval. Also,assume that the "performance"of a stock is determined relative to the average performance of all stocks that are used in the trading strategy. Consequently,if the entire universe of assets is included in the strategy,then each stock's performance is measured relative to the return on the equal- weighted"market portfolio,"Rm.Finally,let wir-I denote the fraction of the trading strategy portfolio devoted to security i [see Lehmann(1990)and Lo and MacKinlay (1990)],that is, w1-1k）=±R-1k-Rm-1k)】 (1) where Ri-(k)is the return on security i at time t-1,i 1,....N, R-1(k)is the return on equal-weighted portfolio of all securities,and k is the length of the time-interval {t-1,t). Since the weights are based entirely on information at time t-1,wi- has a subscript of t-1.The expression for the weights in Equation(1)suc- cinctly captures the philosophy of all return-based trading strategies.First, the positive or negative sign preceding the expression on the right-hand side reflects an investor's (institution's)beliefs;that is,whether the investor be- lieves in price continuations or reversals(and therefore recommends and/or follows a momentum or a contrarian strategy).Second,it is important to note that,regardless of whether a strategy is contrarian or momentum,the premise is that its success is based on the time-series behavior of asset prices.Specifically,a security's past performance relative to some bench- mark (e.g.,the average return of the portfolio of all securities)is supposed to be informative about future innovations in the security's prices.This is quite contrary,for example,to the random walk model of stock prices which implies that changes in stock prices are completely unpredictable (see Section 2 for further details).Third,the dollar weights in Equation(1) [i.e.,w-(k),...,wN-1(k)]lead to an arbitrage (zero cost)portfolio by construction Wit-1(k)=0 Yk, (2a) and the dollar investment long(or short)is given by w-1(k) (2b) 493