Are the widely used Fama-French three-factor model (market, size, book-to-market ratio) and the Carhart four-factor model (adding momentum) the best factor models of stock returns? In their September 2014 paper entitled “Digesting Anomalies: An Investment Approach”, Kewei Hou, Chen Xue and Lu Zhang construct the q-factor model comprised of market, size, investment and profitability factors and test its ability to predict stock returns. They also test its ability to account for 80 stock return anomalies (16 momentum-related, 12 value-related, 14 investment-related, 14 profitability-related, 11 related to intangibles and 13 related to trading frictions). Specifically, the q-factor model describes the excess return (relative to the risk-free rate) of a stock via its dependence on:

- The market excess return.
- The difference in returns between small and big stocks.
- The difference in returns between stocks with low and high investment-to-assets ratios (change in total assets divided by lagged total assets).
- The difference in returns between high-return on equity (ROE) stocks and low-ROE stocks.

They estimate the q-factors from a triple 2-by-3-by-3 sort on size, investment-to-assets and ROE. They compare the predictive power of this model with the those of the Fama-French and Carhart models. Using returns, market capitalizations and firm accounting data for a broad sample of U.S. stocks during January 1972 through December 2012, *they find that:*

- Over the entire sample period:
- The q-factor model size, investment-to-assets and ROE factors generate average gross monthly excess returns of 0.31%, 0.45% and 0.58%, respectively.
- The Fama-French-Carhart size, book-to-market and momentum factors have average gross monthly excess returns of 0.19%, 0.40% and 0.72%, respectively.

- The investment-to assets factor has a correlation of 0.69 with the book-to-market factor, and the ROE factor has a correlation of 0.50 with the momentum factor, suggesting related pairs.
- The Carhart four-factor model does not account for the investment-to-assets factor or the ROE factor. However, the q-factor model explains the book-to-market factor and the momentum factor, suggesting that the q-factor model is more fundamental.
- Among the 80 anomalies evaluated:
- The q-factor model fully explains about half, including almost all those related to trading frictions (such as volatility, turnover and illiquidity).
- With a few exceptions, the q-factor model is more effective than the Fama-French and Carhart models in explaining the other half.

In summary, *evidence indicates that investors may be able to identify outperforming U.S. stocks more precisely by screening for investment-to-assets ratio (rather than book-to-market ratio) and for ROE (rather than momentum).*

Cautions regarding findings include:

- Reported factor returns are gross, not net. Accounting for trading frictions involved in period factor portfolio reformation would reduce these returns.
- Accumulated factor snooping bias suggests that new factor models should face a higher hurdle than early ones (see “Stock Return Model Snooping” and “Taming the Factor Zoo?”).
- Linear factor models may miss non-linear features of stock returns (see “Linear Factor Stock Return Models Misleading?”).

For other perspectives, see: “A Survey of the Factor Landscape”, “A Five-Factor Model of Differences in Stock Returns”, “A Different Factor Model for Each Group of Stocks?”, “A Better Three-Factor Model?”, “A Market Volatility Factor Model” and “Profitability as a Fourth Stock Return Forecast Factor”.