Digesting Anomalies: An Investment Approach

Kewi Hou, Chen Sue, Lu Zhang. "Digesting anomalies: an investment approach." The Review of Financial Studies; 28.3 (2015): 650-705. (link)

When it was first introduced, the Fama French three factor model was highly successful at explaining the cross section of expected stock returns. However, since the 90s, this model has not performed as well and many anomalies have been discovered which display abnormal returns within this model. In this paper, the authors present a q-factor model in which expected returns are a factor of market excess returns, the difference in returns of small and big stocks, the difference in returns of low and high investment stocks, and the difference in returns of high and low profitability stocks.

In this paper, Hou, Xue, and Zhang present an asset pricing model to explain the cross-section of stock returns. The q-factor model describes excess returns in terms of the market return (MKT), a size factor ($r_{ME}$), an investment factor ($r_{I/A}$), and a profitability factor ($r_{ROE}$). They compare their model to the Fama-French (1993) three factor model as well as the Carhart (1997) four factor model. The results show that their q-factor model performs well in relation to these classic models and has better explanatory power over broad collection of anomalies.

The authors develop a two period economic model relateing firm profitability and investment. The model assumes that a firm will continue to invest until the marginal cost of that investment is equal to the expected return in the next period, times the discount rate. Equation (4) relates expected returns with profitability, $E_{0}[\Pi_{i1}]$, and investment, $1+a(I_{i0}/A_{i0})$. All else held constant, it predicts that highly profitable firms should earn higher returns and highly invested firms should earn lower returns.

\[ E_{0}[r_{i1}^S] = \frac{E_{0} [ \Pi_{i1} ] }{ 1 + a (I_{i0} A_{i0} ) } \qquad (4) \]

Using thirty years of data, the authors construct two size, three investment, and three profitability portfolios. Table 1 reports how their factors compare to the CAPM, FF3 and Carhart models. Size has an average monthly return of 0.31%, while Investment has monthly returns of 0.45% and profitability has monthly returns of 0.58%. The alphas of CAPM, FF3 and Carhart are also generally large enough to indicate that these models cannot capture the q-factors. The authors’ size factor has a correlation of 0.95 with SMB, the investment factor has a correlation of 0.69 with HML, and the profitability factor has a correlation of 0.5 with UMD, all of which are significant. Finally, the q-factor model captures a significant amount of HML and UMD with alphas of 0.06% and 0.13% respectively. They suggest their evidence points to HML and UMD representing less accurate variations of their q-factors.

The paper next tests the q-factor model against a broad collection of 80 anomalies. Equation (5) gives the regression used to test the model.

\[ r_{t}^i - r_{t}^f = \alpha_{q}^i + \beta_{\text{MKT}}^i \text{MKT}_{t} + \beta_{\text{ME}}^i r_{\text{ME}, t} + \beta_{\text{I/A}}^i r_{\text{I/A}, t} + \beta_{\text{ROE}}^i r_{\text{ROE}, t} + \epsilon^i \hspace{10}(5) \]

Of the 80 anomalies, only 35 are found to be significant in the cross section. Table 6 presents the factor loadings for the q-factor model with relation to these significant anomalies. It shows that most of the anomalies are combinations of the investment and profitability factors. Specifically, the momentum and profitability anomalies are mostly captured by the profitability factor while value / growth and investment anomalies are mostly captured by the investment factor. 

Earnings and price momentum are analyzed under the q-factor model. Five of the earnings momentum deciles have significant alphas under FF3 while three are significant under Carhart. Only one is significant for the q-factor model. For price momentum, four alphas are significant under FF3, three under Carhart, and zero under the q-factor. Momentum is mostly loaded under the ROE (profitability) factor. This makes intuitive sense since firms with good earnings and increasing prices are more likely to have increased profitability.

The q-factor model developed by Hou, Xue, and Zhang, attempts to explain the cross-section of expected returns using the market factor, combined with size, investment, and profitability. It succeeds in explaining most of a variety of anomalies using the investment and profitability factors, and outperforms the Fama-French 3 factor model and Carhart 4 factor model in almost all the studied anomalies.