EDHEC-Risk Institute October 2016

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

Introduction

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). None of these effects can be explained by the traditional CAPM, as the return spreads cannot be justified by differences in market exposures (Fama and French, 1993, 2006). 1 A common explanation for these effects 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). The size factor is defined as the excess return of a portfolio of small stocks over a portfolio of large stocks, and the value factor is defined as the excess return of value over growth stocks. 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. This approach has been extended to the momentum factor by Carhart (1997), who shows that the continuation of past short-term performance is not accounted for by the Fama-French three- factor model (Fama and French, 1996), but it can be somewhat tautologically explained by introducing a “winners minus losers” portfolio as a fourth factor in an augmented version of the Fama-French model. More recently, the investment and profitability factors have been introduced, so as to capture the investment and profitability effects: Fama and French (2015) sort stocks on operating profit or the growth on total assets, and Hou, Xue and Zhang (2015) replace the former measure by the return on equity. There is some overlap between

Factor models seek to explain the differences in expected returns across assets by their exposure to a set of common factors, which represents the risk factors that are of concern to investors given that they require compensation in the form of higher expected returns for bearing exposure to these factors. Historically, the Capital Asset Pricing Model (CAPM) of Treynor (1961) and Sharpe (1963, 1964) was the first of these factor models. It explains differences in expected returns across securities by their respective sensitivities to a single factor, which is the return on the market portfolio. In 1976, Steve Ross introduced the Arbitrage Pricing Theory for the purpose of valuing assets under the assumptions that there was no arbitrage and that asset returns could be decomposed into a systematic part and an idiosyncratic part. An independent theory of multi-factor asset pricing models has been developed by Merton (1973), with the Intertemporal CAPM. In the ICAPM, expected returns are determined by the exposures to those factors that drive conditional expected returns and volatilities. According to most empirical studies, the CAPM in its original form has very limited success in capturing differences in expected returns. The positive relationship between the expected return and the market beta is seriously challenged by the existence of a low beta anomaly (Frazzini and Pedersen, 2014), and it has long been documented that market exposure is not the only determinant of expected returns. For instance, small stocks tend to outperform large stocks (Banz, 1981; Van Dijk, 2011) and value stocks – stocks with a high book-to-market ratio – earn higher average returns than growth stocks (Stattman, 1980; Fama and

1 - In fact, this statement appears to depend on the period under study. For instance, Fama and French (2006) show that the CAPM fails to explain the value premium between 1963 and 2004, since value stocks have lower betas than growth stocks. However, in the period from 1926 to 1963, the CAPM accounts for the value premium. Rejection of the CAPM over the whole period thus seems to be due to the second half of the sample.

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