An Anatony of Trading Strategies the role of o2u(k)]since it has a small effect on profits to trading strategies that use weekly returns (see also Tables 2 and 4).We,on the other hand, define P(k)to emphasize that total expected profits to return-based trading strategies do not result entirely from time-series predictability in returns. 2.1 The random walk model Although Equation(4)provides a convenient decomposition of expected profits,we need a benchmark model for the return-generating process of financial assets to interpret the two different potential sources of profits to trading strategies.Let us assume that all security prices follow random walks,so that returns can be depicted as Rir(k)=ui(k)+nir(k) i=1,..,N (5) where E[nir(k)]=ovi,k and E[nir(k)nit1(k)]=0i,j,k.6 The usefulness of the random walk model in Equation(5)as a benchmark, particularly for this study,becomes obvious since trading strategies that rely on time-series predictability in returns cannot be profitable by construction because Cov[Rir(k),Rit-1(k)]=0vi,j,k.Equivalently,Equation (5) implies that there is no return predictability in either individual securities or across different securities,and hence the very basis of return-based trad- ing strategies is ruled out.The model in Equation(5)also has economic appeal as a benchmark because changes in stock prices will(generally)be unpredictable in a risk-neutral world with an informationally efficient stock market [see,e.g.,Samuelson (1965)]. The most important property of the model in Equation(5),when com- bined with the decomposition of total expected profits in Equation (4), however,lies in the fact that it helps demonstrate that momentum(con- trarian)strategies will be profitable (unprofitable)even if asset returns are completely unpredictable.More specifically,from Equations (4)and(5)it follows that E[π，(k)]=σ2[u(k)] (6) Equation(6)implies that as long as there are any cross-sectional differ- ences in mean returns of individual securities,momentum strategies will generate profits equal to o-[u(k)].Conversely,contrarian strategies will generate losses of an equal amount.Under the assumption that the mean returns of individual securities are stationary,these profits (losses)have no relation to any time-series predictability in returns.The"profits"in Equa- Technically,all we need in our benchmark model is that the's are uncorrelated:but forease of exposition. we assume a random walk model for stock prices. 7 Of course.although predictability in asset returns is a necessary condition for the success of trading strategies considered in this article,it is not a sufficient condition for "abnormal"gains to be reaped from these strategies.As others have pointed out,time variation in expected returns could also lead to predictability in stock returns [see.e.g.,Fama(1970,1991)]. 499