In 1993, Fama and French (Journal of Financial Economics 1993) developed a three-factor asset pricing model, which included market risk, size, and value. They later expanded the model (Journal of Financial Economics 2015) by introducing the investment and profitability factors. In this follow-up paper, the authors dive deeper into factor analysis.

The authors undertake two limited tasks, with the maximum squared Sharpe ratio for model factors—Sh²(f)—as the evaluating metric:

• comparing nested versions of the five-factor model augmented with a momentum factor
• illustrating how Sh²(f) can be used to choose among non-nested models

The authors’ tests use monthly US stock returns for July 1963–June 2016. The six factors included are defined as follows:

• The market factor is the return on the value-weighted portfolio of NYSE, AMEX, and NASDAQ stocks in excess of the one-month US Treasury bill rate.
• The value factor uses high minus low book-to-market equity. The size factor is small-cap minus big-cap stocks.
• The profitability factor is defined as robust minus weak profitability.
• The investment factor is based on the change in total assets to sort groups into conservative minus aggressive.
• The momentum factor is defined as stocks that are up for the month minus stocks that are down.
• With respect to the profitability factor, the authors also investigate whether cash profitability (which is unaffected by accruals) delivers superior performance compared with operating profitability.

The non-nested models are 12 versions of this six-factor model.

A problem arises comparing the non-nested models when the inputs are sample estimates because factors whose average returns are high relative to expected returns are weighted too heavily in the estimated tangency portfolio. The authors’ solution is to conduct bootstrap simulations of in-sample and out-of-sample Sh²(f) estimates for competing models. A benefit of this approach is that the effects of parameter nonstationarity are similar in sample and out of sample. A downside is that the sample estimates are half the size of the full sample.

The authors add momentum to the five-factor model because of popular demand but do so reluctantly; they fail to see the theoretical justification. They show that, unsurprisingly, when comparing nested versions of these models, the six-factor model wins.

Illustrating how Sh²(f) can be used to choose among non-nested models, three issues about the factors in the six-factor model are studied.

• Profitability factor: The authors show that models using cash profitability factors dominate operating profitability in terms of increasing Sh²(f).
• The value, investment, profitability, and momentum spread factors: Each is an average of spread portfolio returns for small and big stocks. However, the small-stock components of these factors have larger average returns than their big-stock counterparts. As such, the authors investigate models that use only the small or big components of these factors.
• Whether spread factors produce higher Sh²(f) than factors that are simply excess returns on the long or short ends of spread factors. The authors find that the winning model on Sh²(f) combines the market risk and the size factors with the small-cap spread factors for value, cash profitability, investment, and momentum.

The authors have made great contributions to the academic literature on factors in the past, so this new research should be of significant interest to investment professionals, particularly those operating in the quant space.