EDHEC-Risk Institute October 2016

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

Executive Summary

Attributes Should Remain Attributes Factor models, supported by equilibrium arguments (Merton, 1973) or arbitrage arguments (Ross, 1976), are not the only key cornerstones of asset pricing theory (APT). In investment practice, multi-factor models have also become standard tools for the analysis of the risk and performance of equity portfolios. On the performance side, they allow investors and asset managers to disentangle abnormal return (or alpha) from the return explained by exposure to common rewarded risk factors. On the risk side, factor models allow us to distinguish between specific risk and systematic risk, and this decomposition can be applied to both absolute risk (volatility) and relative risk (tracking error with respect to a given benchmark). In addition to analysing the impact of common factors , equity portfolio managers are also interested in analysing the role of stock-specific attributes in explaining differences in risk and performance across assets and portfolios. For example, it has been documented that small stocks tend to outperform large stocks (Banz, 1981) and that value stocks earn higher average returns than growth stocks (Fama and French, 1992). Moreover, stocks that have best performed over the past three to twelve months tend to outperform the past losers over the next three to twelve months (Jegadeesh and Titman, 1993). A common explanation for these effects, which cannot be explained by Sharpe's (1964) single-factor capital asset pricing model or CAPM (Fama and French, 1993, 2006), is that the size and the value premia are rewards for exposure to systematic sources of risk that are not captured by the market factor. This is the motivation for the introduction of the size and value factors by

Fama and French (1993) as proxies for some unobservable underlying economic factors, perhaps related to a distressed factor. In this process, market capitalisation and the book-to-market ratio are used as criteria to sort stocks and to form long-short portfolios with positive long-term performance. In other words, what is intrinsically an attribute is turned into a factor. A similar approach is also used by Carhart (1997), who introduces a “winners minus losers” factor, also known as the momentum factor. More recently, investment and profitability factors have been introduced, so as to capture the investment and profitability effects: again Fama and French (2015) turn attributes into factors by sorting stocks on operating profit or the growth on total assets, while Hou, Xue and Zhang (2015) replace the former measure by the return on equity when constructing their profitability factor. Overall, the standard practice of treating attributes as factors severely, and somewhat artificially, increases the number of factors to consider, especially in the case of discrete attributes. This raises a serious challenge with respect to how to perform a consistent risk and performance analysis for equity portfolios across multiple dimensions that incorporate both macro factors and micro attributes. In this paper, we explore a novel approach to address this challenge. As opposed to artificially adding new factors to account for differences in expected returns for stocks with different attributes, we seek to maintain a parsimonious factor model and treat attributes as auxiliary variables to estimate the betas with respect to true underlying risk factors. In other words, our goal is to decompose market exposure (beta) and risk-adjusted performance (alpha) in a forward-looking way as a function of the

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