The Not-So-Well-Known Three-and-One-Half-Factor Model

In the Fama–French three-factor model, the market return is not the return to market beta. By including a separate beta factor, the market portfolio without a coefficient can be described as only “half” a factor. Documenting the returns to a pure beta factor in the US equity market, the authors show that the distinction between the market return and the return to the cross-sectional variation in security betas also applies to portfolio performance measurement. The realized alphas of low-beta (high-beta) portfolios are reduced (increased) when a separate beta factor is included.Equity analysts conceptualize the Fama–French framework as a tool for studying the size and value characteristics of equity portfolios, along with traditional market beta. But the market portfolio return is not the return to market beta. In fact, commercial providers of equity risk models typically include both a market factor and a separate beta factor, along with size and value factors. In other words, in equity risk-modeling practice, the basic Fama–French framework includes four factors, not just three. Unlike the other three factors, the intercept term (i.e., market factor) in the original Fama–French regression equation does not have a coefficient that varies across securities, and so it can be described as just “half” a factor. Similarly, applied versions of the Fama–French plus Carhart momentum model include five factors—or at least four and a half.In our study, we use cross-sectional Fama–MacBeth regressions, with several econometric enhancements now used in industry, on essentially all US common stocks for the last half century (1963–2012). The econometric enhancements include market-capitalization weighting of multivariate regressions and scaling of stock characteristics to unit standard deviations (i.e., z-scores). The regressions produce returns to five factors: the Market intercept term and four standardized factors (z-Beta, z-Small, z-Value, and z-Mom). The returns to the “pure” z-Small, z-Value, and z-Mom portfolios are similar to the returns to the Fama–French SMB, HML, and UMD portfolios but are less correlated with each other over time. The long-term return to the z-Beta factor is negative, in stark contrast to the prediction of the traditional CAPM. The empirical security market line (SML) has been not only “too flat” but actually downward sloping. Direct comparison of the returns to standardized factors indicates that the market beta anomaly was more significant than either the size anomaly or the value anomaly in the US equity market from 1963 to 2012.In the absence of a separate beta factor, commonly used performance measurement (i.e., time-series-based regression) specifications can lead to misconceptions about the source and magnitude of portfolio alpha. We illustrate an alternative performance measurement specification by using returns on the SSgA Sector ETFs over the most recent decade (2003–2012). The alphas of low-beta (high-beta) industrial sector portfolio returns decrease (increase) toward zero once the beta anomaly is properly acknowledged. Similarly, the large positive alpha of the MSCI Minimum Volatility Index is substantially reduced. Understanding the distinction between the market and beta factors in the not-so-well-known three-and-one-half-factor model can help avoid misperceptions about the sources of portfolio performance.