### Digesting Anomalies: An Investment Approach

Ohio State University (OSU) – Department of Finance

University of Cincinnati

Ohio State University – Fisher College of Business; National Bureau of Economic Research (NBER)

**Abstract: **

An empirical q-factor model consisting of the market factor, a size factor, an investment factor, and a profitability factor largely summarizes the cross section of average stock returns. A comprehensive examination of nearly 80 anomalies reveals that about one-half of the anomalies are insignificant in the broad cross section. More importantly, with a few exceptions, the q-factor model’s performance is at least comparable to, and in many cases better than that of the Fama-French (1993) 3-factor model and the Carhart (1997) 4-factor model in capturing the remaining significant anomalies.

### Digesting Anomalies: An Investment Approach – Introduction

In a highly influential article, Fama and French (1996) show that, except for momentum, their 3-factor model, which consists of the market factor, a factor based on market equity (small-minus-big, SMB), and a factor based on book-to-market equity (high-minus-low, HML), summarizes the cross section of average stock returns as of the mid-1990s. Over the past 2 decades, however, it has become clear that the Fama-French model fails to account for a wide array of asset pricing anomalies.

Our contribution is to construct a new empirical model that does a good job in summarizing the cross section of average stock returns. In particular, many (but not all) of the anomalies that prove challenging for the Fama-French model can be captured.2 Our model is in part inspired by investment-based asset pricing, which is in turn built on the neoclassical q-theory of investment. In our model (dubbed the q-factor model), the expected return of an asset in excess of the risk-free rate, denoted E[ri] ? rf , is described by the sensitivities of its returns to 4 factors: the market excess return (MKT), the difference between the return on a portfolio of small size stocks and the return on a portfolio of big size stocks (rME), the difference between the return on a portfolio of low investment stocks and the return on a portfolio of high investment stocks (rI/A), and the difference between the return on a portfolio of high profitability (return on equity, ROE) stocks and the return on a portfolio of low profitability stocks (rROE). Formally,

in which E[MKT], E[rME], E[rI/A], and E[rROE] are expected factor premiums, and ?iMKT, ?iME, ?iI/A, and ?iROE, are the factor loadings on MKT, rME, rI/A, and rROE, respectively.

We construct the q-factors from a triple 2-by-3-by-3 sort on size, investment-to-assets, and ROE. From January 1972 to December 2012, the size factor earns an average return of 0.31% per month (t = 2.12); the investment factor 0.45% (t = 4.95); and the ROE factor 0.58% (t = 4.81). The investment factor has a high correlation of 0.69 with HML, and the ROE factor has a high correlation of 0.50 with the Carhart (1997) momentum factor (up-minus-down, UMD). The alphas of HML and UMD in the q-factor model are small and insignificant, but the alphas of the investment and the ROE factors in the Carhart model that augments the Fama-French model with UMD) are large and significant. As such,

HML and UMD might be noisy versions of the q-factors.

To evaluate the relative performance of the q-factor model, we start with a wide array of nearly 80 variables that cover all major categories of anomalies. Following Fama and French (1996), we construct testing deciles based on the breakpoints from the New York Stock Exchange (NYSE), and calculate value-weighted decile returns. Surprisingly, the high-minus-low deciles formed on about one-half of the anomaly variables, including the vast majority of variables related to trading frictions, have average returns that are insignificant at the 5% level. As such, echoing Schwert (2003) and Harvey, Liu, and Zhu (2013), we suggest that many claims in the anomalies literature seem exaggerated.

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