al o f Fi B. The Submartingale Model Suppose we assume in(1) that for all t and t E(3t+11)≥pu, or equivalently,E(行t+1{)≥0 This is a statement that the price sequence pjt) for security j follows a sub martingale with respect to the information sequence (,), which is to say nothing more than that the expected value of next periods price, as projected on the basis of the information t is equal to or greater than the current price If (6)holds as an equality(so that expected returns and price changes are ero), then the price sequence follows a martingale a submartingale in prices has one important empirical implication. Consider the set of"one security and cash"mechanical trading rules by which we mea systems that concentrate on individual securities and that define the conditions under which the investor would hold a given security, sell it short, or simply hold cash at any time t. Then the assumption of(6) that expected returns conditional on t are non-negative directly implies that such trading rules based only on the information in t cannot have greater expected profits than a policy of always buying-and-holding the security during the future period in question. Tests of such rules will be an important part of the empirical evidence on the efficient markets model.a C. The Random Walk Model rly treatments of the efficient markets model, the statement that price of a security "fully reflects'"'available information was assumed to imply that successive price changes (or more usually, successive one-period returns) are independent. In addition, it was usually assumed that successive changes (or returns )are identically distributed. Together the two hypotheses constitute the random walk model. Formally, the model says f(r3+1④+)=f(r;t1), which is the usual statement that the conditional and marginal probability distributions of an independent random variable are identical. In addition the density function f must be the same for all t. 3. Note that the expected profitability of "one see and-hold is not ruled out by the general expected since in principle it allows equilibrium expected returns to be negative, holding cash(which has zero actual and thus expected return) may have higher expected return than holding d negative equilibrium expected returns for some securities are quite possible. For example, models of Markowitz [30] and Tobin [43]) the equilibrium expected return on a security depends the extent to which the dispersion in the security's return distribution is related to dis in the returns on al curities. A security whose returns on average move opposite to the general market is particularly valuable in reducing dispersion of portfolio returns, and so its quilibrium expected return may well be negative m only follow a random walk if price changes are If one-period returns are independent, id ill not follow a random walk since the distribution of price changes m啦