RISK, RETURN, AND EQUILIBRIUM 6I If the errors in the Ba are substantially less than perfectly positively cor- lated,the bs of portfolios can be much more precise estima β s than theβ '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 B high observed B, tend to be above the corresponding true Be and low observed Bi tend to be below the true B Forming portfolios on the bas 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 Bp, 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 B2 computed from data for one time period but then using a subsequent period to obtain the B for these portfolios that are used to test the two-parameter model. With fresh data, within a portfolio errors in the individual securit ty 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 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 steger 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 2 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)+ 1 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 o initial portfolio Bpt for the risk-return tests. The subscript t is added to indicate that each month t of the following four years(1935-38)these are recomputed as simple averages of individual security B, thus ad justing the portfolio month by month to allow for delisting of securi ties. The component B2 for securities are themselves updated yearly--that The errors-in-ti problem and the by Blume (1970).T by Friend and blur and Black, Jensen, ar menon that then by black ind Scholes (1972), who offer a solution to