EDHEC-Risk Institute October 2016

Multi-Dimensional Risk and Performance Analysis for Equity Portfolios — October 2016

Introduction

A more recent literature has re-assessed the ability of the CAPM to explain the anomalies, by focusing on a conditional version of the model. In fact, traditional measures of alpha and beta in the CAPM are conducted as if these quantities were constant over time, by performing time-series regressions of stock returns on a market factor. As a result, it is an unconditional version of the CAPM that is tested. The conditional version of the model posits that conditional expected returns are linearly related to conditional market betas, the slope being the conditional market premium. Different specifications have been studied in the literature. Gibbons and Ferson (1985) allow for changing expected returns but assume constant betas, while Harvey (1989) emphasises the need for time-varying conditional covariances between stocks and the market factor. Jagannathan and Wang (1996) introduce both time-varying betas and a time-varying market premium. A crucial point in empirical studies is how the set of conditioning information is specified. Ferson and Schadt (1996) let the conditional betas be a function of lagged macroeconomic variables, namely the T-Bill rate, the dividend yield, the slope of the term structure, the spread of the corporate bond market, plus a dummy variable for the January effect. Jagannathan and Wang (1996) model the conditional market premium as a function of the default spread in the bond market, but do not explicitly model the conditional betas. Lettau and Ludvigson (2001) use the log consumption-wealth ratio. Lewellen and Nagel (2006) do not specify a set of conditioning variables, and they estimate alphas and betas over rolling windows, assuming that conditional alphas and betas are stable over the estimation window (one month

sorts based on sector, country, size or book-to-market are only different ways of segmenting the same universe. Hence, it is desirable to have a decomposition method that keeps the number of factors reasonably low while being flexible enough to handle a wide variety of attributes. Furthermore, the risk-based explanation of the size, value and momentum effects, and the need for the related factors, is debated. First, there is no consensual interpretation of the size, value and momentum factors as risk factors in the sense of asset pricing theory. Indeed, the factors that can explain differences in expected returns are those that affect the marginal expected utility from consumption in consumption-based asset pricing models, and those that determine the comovements between stocks in models based on the APT. However, there is no unique and definitive explanation of why small, value and winner stocks would be more exposed to such systematic risk than large, growth and loser stocks. Second, all these effects can be explained without the help of additional factors. For instance, Daniel and Titman (1997) argue that expected returns depend on the size and the book-to-market ratio rather than the exposure to the Fama French long-short factors. They also reject the interpretation of these factors as “common risk factors”, arguing that the high correlations within small or value stocks simply reveal similarities in the firms’ activities. It has also been documented that behavioural models, in which investors display excessive optimism or reluctance with respect to some stocks, can explain the observed outperformance of small, value or winner stocks (Merton, 1987; Lakonishok, Shleifer and Vishny, 1994; and Hong and Stein, 1999).

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