The Journal of finance 十R Moreover, for any portfolio p, defined by the proportions x,p of portfolio funds invested in individual securities j E(1+B+1)=E( (1+氮t+) tR. .+RM 1+ R R E 1+Rut From(2)and (3), it follows directly that E(1+1+ +R E 0.(4) 1+ In words, in a market dominated by growth-optimizers, the expected value of the ratio of one plus the percentage return on any security or portfolio to one plus the return on the market portfolio is equal to 1.0. Assuming the equired correspondence between ex ante assessments of return distribution and ex post return distributions, we can test this condition by testing for differences of average values of the return ratios for securities or portfolios from the theoretical value of 1.0. We work with portfolios and in particular with the time series of returns on twenty portfolios available from our earlier study [71 IV. HOTELLING T TESTS Let +rpt 1+RM/P=1,2, 20;t=1/35,2/35,,,6/68, where the Rpt are monthly percentage returns on the twenty portfolios for the period January 1935 to June 1968, and where the proxy for Rit is "Fishers Arithmetic Index"[9], a simple average of the returns from month t-1 to month t on all common stocks on the New York Stock Exchange. The twenty folios to be discussed in more detail later, are likewise formed from stock on this Exchange. For present purposes, suffice it to say that securities are allocated in approximately equal numbers to each of the twenty portfolios, and the allocation is according to ranked values of estimates of P, where COV (R1,R3 2(Rx) is the risk of security i in the market portfolio M, and where risk is measured 8. The theory, of course, calls for a value weighted index rather than an index where each stock,s return equally, In tests similar to some of those to be carried out here how- er, Roll [2 that his conclusions are insensitive to whether a value weighted or an equally weighted returns is used 9. The data at he Center for Research in Security Prices of the University of chicago