RISK,RETURN,AND EQUILIBRIUM 615 If the errors in the B are substantially less than perfectly positively cor- related,the B's of portfolios can be much more precise estimates of true B's than the B's for individual securities. To reduce the loss of information in the risk-return tests caused by using portfolios rather than individual securities,a wide range of values of portfolio B's is obtained by forming portfolios on the basis of ranked values of B:for individual securities.But such a procedure,naively exe- cuted could result in a serious regression phenomenon.In a cross section of Bi,high observed B.tend to be above the corresponding true B:and low observed B:tend to be below the true B.Forming portfolios on the basis of ranked B:thus causes bunching of positive and negative sampling errors within portfolios.The result is that a large portfolio B would tend to over- state the true B,while a low B would tend to be an underestimate. The regression phenomenon can be avoided to a large extent by forming portfolios from ranked B:computed from data for one time period but then using a subsequent period to obtain the Bp for these portfolios that are used to test the two-parameter model.With fresh data,within a portfolio errors in the individual security B:are to a large extent random across securities,so that in a portfolio Bp the effects of the regression phenomenon are,it is hoped,minimized.6 B.Details The specifics of the approach are as follows.Let N be the total number of securities to be allocated to portfolios and let int(N/20)be the largest integer equal to or less than N/20.Using the first 4 years (1926-29)of monthly return data,20 portfolios are formed on the basis of ranked B for individual securities.The middle 18 portfolios each has int(N/20) securities.If N is even,the first and last portfolios each has int(N/20) IN-20 int(N/20)]securities.The last (highest B)portfolio gets an additional security if N is odd. The following 5 years (1930-34)of data are then used to recompute the B,and these are averaged across securities within portfolios to obtain 20 initial portfolio for the risk-return tests.The subscript t is added to indicate that each month t of the following four years (1935-38)these Bare recomputed as simple averages of individual security Bi,thus ad- justing the portfolio Bot month by month to allow for delisting of securi- ties.The component B:for securities are themselves updated yearly-that 6The errors-in-the-variables problem and the technique of using portfolios to solve it were first pointed out by Blume (1970).The portfolio approach is also used by Friend and Blume (1970)and Black,Jensen,and Scholes (1972).The regression phenomenon that arises in risk-return tests was first recognized by Blume (1970) and then by Black,Jensen,and Scholes (1972),who offer a solution to the problem that is similar in spirit to ours