Bringing Order to the Factor Zoo?

From a purely statistical perspective, how many factors are optimal for explaining both time series and cross-sectional variations in stock anomaly/stock returns, and how do these statistical factors relate to stock/firm characteristics? In their July 2018 paper entitled “Factors That Fit the Time Series and Cross-Section of Stock Returns”, Martin Lettau and Markus Pelger search for the optimal set of equity factors via a generalized Principal Component Analysis (PCA) that includes a penalty on return prediction errors returns. They apply this approach to three datasets:

1. Monthly returns during July 1963 through December 2017 for two sets of 25 portfolios formed by double sorting into fifths (quintiles) first on size and then on either accruals or short-term reversal.
2. Monthly returns during July 1963 through December 2017 for 370 portfolios formed by sorting into tenths (deciles) for each of 37 stock/firm characteristics.
3. Monthly excess returns for 270 individual stocks that are at some time components of the S&P 500 Index during January 1972 through December 2014.

They compare performance of their generalized PCA to that of conventional PCA. Using the specified datasets, they find that:

• Generalized PCA identifies five factors that together explain well both the time series and cross sections of test asset returns (indicating that many conventional factors are redundant). They relate to the following conventional factors:
  1. Market
  2. Value and its interactions with other factors
  3. Momentum and its interactions with other factors
  4. Profitability
  5. A “high-Sharpe ratio” related to short-term reversal
• Generalized PCA works better than conventional PCA. Specifically, both in-sample and out-of-sample, generalized PCA:
  • Has smaller pricing errors than conventional PCA.
  • Has maximum gross 5-factor Sharpe ratio more than twice that of conventional PCA.
  • Finds high-Sharpe ratio factors that affect only a subset of the test assets, while conventional PCA does not.
• Most of the return predictive power is from extreme deciles.
• Linear factor models of returns are stable over time for test portfolios (optimal factor weights do not vary much), but not for individual stocks (optimal factor weights vary considerably).

In summary, evidence from pure statistical analysis indicates that five factors are sufficient to construct a model of stock/stock portfolio returns, corresponding roughly to conventional market, value, momentum, profitability and short-term reversal factors.

Cautions regarding findings include:

• Calculations are gross, not net. Trading frictions for monthly portfolio reformation and shorting costs would reduce returns. Shorting may not always be feasible as assumed. Since these costs may vary by factor, net findings may differ from gross findings.
• The approach used is beyond the reach of most investors, who would bear fees for delegating to a fund manager.

See also “Perfect Factor Model of U.S. Stock Returns?”.