EDHEC-Risk Institute October 2016

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

Introduction

growth and global inflation. They show that asset-based risk parity portfolios can often concentrate too much in just one component of risk exposures, particularly equity risk, in contrast to factor-based risk parity which allows a more robust risk diversification. A related argument is made by Carli, Deguest and Martellini (2014), who emphasise the importance of reasoning in terms of uncorrelated factors to assess the degree of diversification of a portfolio. Finally, a recent strand of research has started to look at factor investing as a tool for portfolio construction or asset allocation. In this approach, it is the factors that are regarded as the constituents of a portfolio – see Martellini and Milhau (2015) and Maeso and Martellini (2016). Performance and risk attribution models used by practitioners, such as the Barra models, often include "factors" other than those borrowed from asset pricing theory. Typical examples are sector and country factors. The question therefore arises to assess what exactly are the marginal contributions of the various dimensions to the return and the volatility of a given equity portfolio. A straightforward procedure is to introduce the new factors as additional regressors in the econometric model. Menchero and Poduri (2008) develop a multi- factor model in which the set of pricing factors is extended with “custom factors”. We apply this method to the multi- dimensional analysis of various equity portfolios, and we propose an alternative, more parsimonious, holding-based method (as opposed to a purely return- based) approach when information about portfolio holdings is available. Overall, it must be acknowledged that increasing the number of factors raises concerns about their potential overlap: after all,

all these factors, as suggested by Hou, Xue and Zhang (2015), who show that the book-to-market effect is predicted by a four-factor model with the market, the size factor and the investment and profitability factors. Multi-factor models have thus become standard tools for the analysis of the risk and performance of equity portfolios. On the performance side, they allow us to disentangle abnormal returns (alpha) from the returns explained by exposure to common risk factors. The alpha component is interpreted as “abnormal return” because it should be (statistically not different from) zero if factor exposures were able to explain any difference between expected returns. Thus, a non- zero alpha reveals either misspecification of the factor model, from which relevant factors have been omitted, or genuine skill of the manager who was able to exploit pricing anomalies. 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 benchmark). 2 The performance and risk decomposition of a portfolio across factors is receiving increasing attention from sophisticated investors. Recent research (Ang, Goetzmann and Schaefer, 2009; Ang, 2014) has highlighted that risk and allocation decisions could be best expressed in terms of rewarded risk factors, as opposed to standard asset class decompositions. Bhansali et al. (2012) evaluate the benefits of using a factor-based diversification measure over asset-based measures. They use a principal component analysis to extract two risk factors driven by global

2 - In Section 1 of this paper, we apply these decomposition methods to the analysis of ex-post performance, volatility and tracking error of US equity mutual funds. We also review multi-factor models commonly used by practitioners, such as Barra models, which include sector and country attributes in addition to risk factors.

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An EDHEC-Risk Institute Publication