Certain attributes or characteristics, such as the size (SMB) and value (HML) factors popularized by Fama and French, have been proposed extensively in previous literature as being true, or systemic, risk factors. The authors argue that true risk factors must be economically related to security returns, regardless of how the portfolios have been grouped or sorted. In other words, true risk factors should be able to fit the regression of any set of securities, not just the set used to create the factors. But size and value factors do not explain returns in arbitrary sets of portfolios that are not sorted on size and value. The authors conclude that these factors do not represent true risk factors and only work for portfolios sorted on size and value.

The authors demonstrate that factors created by sorting return data based on characteristics create a good mechanical fit in the regression of portfolio returns. Such factors are approximately linear functions of the sorted portfolios. The effects of including these factors in running regressions on portfolio returns are to decrease the significance of intercepts and increase the significance of the slope coefficients in time-series regressions (TSRs), to produce significant t-statistics in cross-sectional regressions (CSRs), and to increase R² in both TSRs and CSRs.

According to the authors, the logic behind the creation of artificial factors that mechanically fit regression models applies not only to specific samples but also to any random sample, although the factor-creation procedure to achieve the best fit for a particular dataset might require a small adjustment in some cases. They further argue that for such randomly generated factors to fit regressions, it is not necessary for them to have a correlation with the true risk factors. The linear restrictions created by the very structure of such factor proxies ensure that they will fit the regression well, whether or not they have a correlation with the true risk factor.

The authors use two artificial factors, Factor 1 (F1) and Factor 2 (F2), to test their proposed theories and successfully demonstrate how well these factors explain returns in regressions of the dataset from which they are created. F1 and F2 explain security returns even better than do such so-called true risk factors as size and the book-to-market ratio. The authors compare the performance of F1 and F2 with the performance of the factors SMB (a proxy for size) and HML (a proxy for value) in 25 sample portfolios.

They report that SMB and HML seem to explain returns only for those datasets or portfolios to which they are mechanically related—that is, the ones used to create the factors. Therefore, their relationship to returns is most likely a mechanical one, not a true economic one. For portfolios or datasets not sorted on SMB and HML, inclusion of these factors in regressions actually deteriorates the performance of the model and decreases R². In almost all the sample portfolios, the size and value factors fail to explain returns better than the two artificial factors, F1 and F2.

The authors use both realized and simulated returns with constant-value and flexible-value weights to test their hypotheses. For realized returns, they use the universe of NYSE, AMEX, and NASDAQ monthly stock returns from July 1963 to December 1991. The portfolios are sorted on the basis of capital asset pricing model (CAPM) alphas. Two artificial factors are generated: F1 is the difference between the averages of the first 13 and the last 12 portfolio returns, and F2 is the difference between the first 12 and the last 5 portfolio returns. This methodology remains the same throughout the different simulations and regressions.

Next, the authors regress portfolio returns on (1) the market factor only; (2) the market factor and F1; and (3) the market factor, F1, and F2. In all scenarios, the market factor alone does a poor job of explaining the returns. When F1 and F2 are added to the regression, together they explain the returns much better than does the market factor alone, resulting in decreased significance of the intercepts and improved R², although the respective contributions of the two artificial factors vary from case to case.

For simulations, the authors use 89 securities with continuous monthly returns over the period of July 1963–December 1991 and generate a random sample of returns from a multivariate normal distribution. They again regress portfolio returns on the market factor only; on the market factor and F1; and on the market factor, F1, and F2. They repeat the entire procedure 500 times. Once again, the market factor alone fails to explain returns. F1 proves to be a good fit and, according to the authors, shows robustness to differences among samples. F2, on the other hand, fails to improve the regression’s performance and demonstrates that some factors, even when they are based on characteristics that explain returns, may perform well in some samples but not in others. The authors suggest that with a little adjustment in the grouping of the portfolios, the factors can work better.

The authors base their conclusions on robust statistical analysis using both actual and simulated data. Their assertion that size and book value might not be true risk factors in the CAPM needs to be explored further in more independent studies because it could have important ramifications for security valuation and portfolio construction. The authors also highlight how grouping assets may affect the way portfolio returns appear to be explained by various factors or factor proxies, something investors should be aware of when evaluating portfolio performance.