Previous anomaly research may have misinterpreted corrected, for the market index, mean returns on small firms. Assuming mean-variance preferences, it is shown theoretically that corrected mean returns (i.e., market line deviations) are not indicative of the relative desirability of increasing the proportional investment in small firms. The correct improvement criterion is derived and estimated. Tests indicate that the value-weighted market index is not significantly improved with greater weight on small firms, in the average month. When seasonality is considered, the observed performance improvements due to small or large firms are significant in some months, but the required portfolio position is unclear. If a case exists for a small firm anomaly in January, it probably exists in other months and it also exists for large firms. It is doubtful whether such an anomalous stock market exists.

Bernell Stone's paper extends the single-factor market model to a two-factor model to “better” explain the stochastic process that generates security returns. The inductive search for new models (of which his paper is one) presumably is predicated upon some unsatisfactory results of joint tests of the single-index market model and the capital asset pricing model. It is well known that there are other components of systematic or covariance risk that are not explained by the single-market factor. In the most general sense then, one would conclude that the truth of the return generating process is a multiple factor model, given that the process is indeed linear in the factors. Professor Stone chooses a two-factor (or index) model, in which the known factors are: (1) the return on an equity index, and (2) the return on a bond index. To this extent his interesting work is a special case of the more general work of others.

The purpose of this paper is to provide a link between the various multivariate tests of asset pricing and a performance measure for asset sets. The paper includes a unified summary of various F tests for mean-variance efficiency, intersection, and spanning for sets and subsets of financial assets. Both the risk-free asset and no risk-free asset environments are discussed. These tests are then related to the concept of potential performance for asset sets. The potential performance measure can be viewed as an extension of the Sharpe performance measure for single portfolios. The economic intuition behind the tests is that the multivariate tests of portfolio efficiency, intersection, and spanning are tests of zero potential performance at particular margins between the asset or portfolio subset and the full asset set.

J. D. Jobson, Bob Korkie, Statistical Inference in Two-Parameter Portfolio Theory with Multiple Regression Software, The Journal of Financial and Quantitative Analysis, Vol. 18, No. 2 (Jun., 1983), pp. 189-197

The marginal performance contribution made by new assets in a portfolio is identified. The maximum change in a portfolio's Sharpe performance from the addition of new assets is a simple function of a generalized Jensen index and the unexplained covariances from a multivariate market model. Deviations from a higher dimension market line may be used to rank the desirability of asset additions to an existing portfolio. Statistical tests for the equality of the performance contributions by new assets is possible.

Harry Turtle, Adolf Buse, Bob Korkie, Tests of Conditional Asset Pricing with Time-Varying Moments and Risk Prices, The Journal of Financial and Quantitative Analysis, Vol. 29, No. 1 (Mar., 1994), pp. 15-29

We develop conditional alpha performance measures that are consistent with conditional mean–variance analysis and the magnitude and sign of the implied true conditional time-varying alphas. The sequence of conditional alphas and betas is estimable from surprisingly simple unconditional regressions. Other common performance measures are derivable from the conditional investment opportunity set based on its conditional asset return moments. Our bootstrap analysis of Morningstar mutual fund returns data demonstrates that the differences between existing conditional alpha measures and our proposed alpha are substantive for typical parameterizations.