EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

0.64% for the average low book-to- market portfolios.

In the remainder of the Section 1, we will use the Fama-French-Carhart model, which is a standard choice in practice. In this model the returns on a given equity portfolio can be decomposed in five components: market factor, value factor, size factor, momentum factor, and a residual component. 5 The betas can be estimated by performing the following regression model for R i,t , the return on stock i at period t in excess of risk-free rate:

1.1.4 Carhart’s Four-Factor Model Using a four-factor model, Daniel et al. (1997) studied fund performance and concluded that performance persistence in funds is due to the use of momentum strategies by the fund managers, rather than the managers being particularly skilful at picking winning stocks. This momentum anomaly stays unexplained by the Fama and French 3-factor model. Carhart (1997) and Chan, Chen and Lakonishok (2002) report that the four-factor model shows significant improvement over the single market factor CAPM model in explaining equity portfolio performance. This empirical model is an extension of Fama and French’s three-factor model that includes a momentum factor: ) denotes the expected return for asset i • B i,k denotes the k -th factor loadings • E ( R m ) denotes the expected return of the market portfolio • SMB (small minus big) denotes the difference between returns on two portfolios: a small-capitalisation portfolio and a large-capitalisation portfolio • HML (high minus low) denotes the difference between returns on two portfolios: a portfolio with a high book-to- market ratio and a portfolio with a low book-to-market ratio • WML (winner minus losers) denotes the difference between the average of the highest returns and the average of the lowest returns from the previous year. Where • E ( R i

(1.4)

1.2 Portfolio Risk and Performance Analysis with Factors It is widely agreed that factor allocation accounts for a large part of the variability in the return on an investor's portfolio. Indeed recent research (Ang, Goetzmann and Schaefer, 2009) has highlighted that risk and allocation decisions could be best expressed in terms of rewarded risk factors, as opposed to standard asset class decompositions. Risk reporting is increasingly regarded by sophisticated investors as an important ingredient in their decision making process. On the performance side, factor models disentangle the abnormal return (alpha) and normal return (beta). On the risk side, factor models allow us to distinguish between specific risk and systematic risk, from either an absolute or relative risk perspectives. Because of the non-linearity of portfolio risk decomposition with the volatility as risk measure, we explore some methods to handle this issue and give attention to the multi-collinearity issues raised by the simultaneous inclusion of several factors. We illustrate these decompositions with four US equity mutual funds by performing their risk and performance analysis with the Carhart four-factor model.

5 - When assessing the relative importance of the market factor versus long-short factors, one typically obtains for well-diversified portfolios a strong domination of the market factor. In the limit, if the equity portfolio to be analysed is the market index used as a proxy for the market factor, then the contribution of the market factor will be trivially 100%, while the contribution of other factors will be measured as zero. In case one would like to make statements about factor exposures for a cap-weighted market index, it is possible to take equally-weighted factors (for the market factor as well as other factors) to avoid such trivial statements.

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