Drawing on the clean surplus accounting relation and the classical dividend discount model, the authors add profitability and investment factors to their earlier three-factor asset pricing model. They investigate various “left-hand-side” portfolios that caused problems—in terms of unexplained intercept terms—for their earlier model.

For the factor-based equity investor, the authors provide a lucid account of the sorting methodology they use to measure the size, market β, book-to-market, profitability, and investment risk factors. They also provide detailed descriptions of the finer sorts they use to construct left-hand-side portfolios that load in various ways on the “right-hand-side” risk factors.

The most important contribution of the research is the authors’ explanation of the average excess returns from holding certain “anomaly” portfolios. The market β excess return is earned by going long low-market-β stocks and short high-market-β stocks. The net share issues excess return is earned by going long firms that buy back their own equity and short net issuers. The volatility excess return is earned by going long firms with low price volatility and short firms with high price volatility. The authors explain that the average excess returns of these portfolios are positively loaded on the new profitability and investment risk factors; these portfolios behave like the returns of firms that are profitable and invest conservatively.

The five-factor model still fails to explain the accruals and momentum anomalies.

The data sample is from July 1963 to December 2014 and consists of NYSE, AMEX, and NASDAQ stocks. Accounting data are from Compustat.

The first factor of the model is the excess return of the value-weighted portfolio over the one-month US Treasury bill rate. The high-minus-low (HML) risk factor is constructed using independent 2 × 3 sorts on size and book value/market cap. The new robust-minus-weak (RMW) and conservative-minus-aggressive (CMA) risk factors also use 2 × 3 sorts, with profitability defined as operating profit/book value and investment defined as the annual percentage change in total assets. The small-minus-big (SMB) factor is calculated as the average of the SMB factor for the HML, RMW, and CMA portfolios.

The left-hand-side anomaly portfolios use finer sorts (typically 5 × 5) on size and the appropriate anomaly variable.

Positive average excess return portfolios, such as those formed by going long low-volatility stocks and short high-volatility stocks, behave like the returns of profitable firms that invest conservatively. Negative average excess return portfolios, such as those that go long net share issuers and short firms that buy back their own equity, behave like the returns of unprofitable firms that invest aggressively.

Although the momentum anomaly can be explained by including momentum as a sixth risk factor, the authors concede that this hurdle is lower than that set for the other anomaly portfolios because the left-hand-side portfolios are merely finer sorts of the right-hand-side risk factor.

The authors deliberately set themselves the difficult challenge of explaining the average returns of anomaly portfolios rather than the returns of portfolios constructed using finer sorts of the right-hand-side risk factors. It will be interesting to see whether their results carry over into international equity markets.