Apart from the explicit statistical analyses performed by these researchers, one of the significant unifying points was their recognition of the potential problems in testing the CAPM. Although none of the studies expressed these problems as succinctly as in later analyses, their observations were the basis of the revisions to the basic CAPM that followed. The major hypotheses of the CAPM were tested, using the methods and computing power available at the time, under the assumption that these were testable.

## Testing different portfolio formations

With reference to their earlier (1970) results, that the standard CAPM does not fully explain excess returns, Blume & Friend [1973] set out to determine why this was so. The reasons given previously included the inability to borrow at the risk-free rate, and deficiencies in the models being tested. To alleviate the measurement error problems they grouped the equities into twelve portfolios of around 80 equities each, based on their estimated *β* relative to a Fisher Index from 1950-1954. Two methods of investment - by equal investment in each equity and by investment proportional to the market value of each equity - were used. The portfolios were rebalanced monthly to preserve the weighting scheme. The returns for 1955-1959 were averaged for each portfolio and regressed on the Fisher Index. The returns were then regressed on the estimated *β*s. The same procedure was followed for 1960-1964 and 1965-1968.

Blume & Friend found no evidence of non-linearity between risk and return, but the equal-weighted adjustments were found to provide a better fit. Using techniques similar to those of Black, Jensen & Scholes [1972], they found that the expected return on the zero-*β* portfolio differed significantly from the average risk-free rate. Although their empirical results agreed with those of Black, Jensen & Scholes, and Fama & MacBeth [1973], Blume & Friend offered an alternative explanation. They discussed the effect of the assumption of a perfect short sales mechanism. However, they also proposed that the short sales assumption may be less restrictive than the risk-free borrowing assumption if there are no net short positions in the investor's optimal portfolio. In particular, they concluded that the empirical results support the zero-*β* model for equities listed on the New York Stock Exchange ("NYSE") but may not hold for other financial assets. Blume & Friend suggested that there was a segmentation between equities and bonds, a result that could be extended to include real estate.

## Estimating the return on the zero-*β* portfolio

Although Black's zero-*β* version of the CAPM was widely accepted as a better representation of the market, the estimation of a zero-*β* portfolio is not a simple procedure. Morgan [1975] evaluated different estimates of the minimum-variance zero-*β* portfolio and compared predictions using these estimates with the results from cross-sectional regressions of return on risk. Morgan formed 89 portfolios of six equities each from 48 industry groups. The market portfolio was the equal dollar-weighted portfolio of the 89 portfolios. Five-day returns were calculated for each portfolio and were divided into odd and even numbered period groups. The *β*s estimated from the odd groups were used as instrumental variables in testing the returns in even periods, and vice-versa. The zero-*β* portfolio was estimated using Long's [1971] procedure, to solve the following problem:

subject to

where* x* is a portfolio vector, *R* is the vector of market returns, and **m** is the known market portfolio vector. The regression of returns for the traditional Fama & MacBeth model and the direct route for the zero-*β* model was calculated as follows:

Using the Wilcoxon matched pairs test, Morgan could not reject the null hypothesis that the two models predict returns equally well. The estimates of returns on the zero-*β* portfolio had smaller variances than those of the cross-sectional regressions.

**Rankings based on** *β* **and** *σ*

Foster [1978] tested the CAPM using monthly returns on NYSE equities from 1931 to 1974 and a value-weighted average of the NYSE index. Foster emphasised that his results were a test of consistency with the asset pricing model, rather than a validation of the descriptive authenticity of the model. Since the residual risk in an equity's returns is not rewarded under the CAPM, Foster set out to test whether this holds. Foster followed the basic methodology of Fama & MacBeth [1973]. However, in forming his 16 portfolios, equities were first assigned to one of four portfolios based on their estimated *β*s. Then each of these portfolios was subdivided into four portfolios onthe basis of estimated residual variance. Portfolios were rebalanced annually. As a check on the segmentation of the equities, the excess returns were regressed on the excess returns on the market. The* ß*s and variances declined as expected. The Fama regression was calculated for each month from 1931-1938. For each month the return was also regressed on the estimate of residual risk, in an attempt to separate the effects of *β* estimates from *σ *estimates on the portfolio return. Two types of risk-adjusted returns were calculated:

and

where *R*^{s}_{p} is the risk-adjusted return using the standard CAPM, and *R*^{z}_{p} is the risk- adjusted return of the portfolio using the zero-*β* CAPM. Treasury bills with 30 days to maturity were used as a proxy for the risk-free rate, and the *R*_{z} series used was the time series estimates of the intercepts in equation (5.22). The risk-adjusted returns were then regressed on the residual variances of the portfolio. For both the standard and zero-*β* version of CAPM, the coefficient on the residual risk was not significantly different from zero. The entire process was repeated, grouping portfolios first by residual variance and then by *β*; similar results were obtained. Foster also examined the effect of using a value-weighted versus equal-weighted index for *R*_{m}.The value-weighted index was found to have a slight advantage over the equal-weighted index in explaining returns.