EDHEC-Risk Institute October 2016

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

Introduction

errors as the less parsimonious four- factor model of Carhart (1997). We also show that the magnitude of the alphas is further reduced by introducing a time-varying market premium, as in Jagannathan and Wang (1996). The rest of the paper is organised as follows. Section 1 contains a reminder on factor models in asset pricing theory and on the empirical models developed by Fama and French (1993) by Carhart (1997). We also discuss the question of performance and risk attribution, first with respect to the Fama-French and Carhart factors, and then to the same set of factors extended with sectors. We also illustrate these methods on various portfolios of US stocks. In Section 2, we define the fundamental betas and we discuss the estimation procedure in detail. Section 3 presents three applications of the fundamental beta approach to embed the sector dimension in a multi- dimensional performance and risk analysis, the construction of the market- neutral portfolios and the pricing of portfolios sorted on size, book-to-market and past short-term return, respectively. Section 4 concludes.

the fundamental beta approach to include sector effects along with observable attributes that define the Fama-French and Carhart factors in the analysis of expected return and volatility of a portfolio. This provides a parsimonious alternative to the decomposition methods that introduce dedicated factors for additional attributes such as sector and country classifications. The second application of the fundamental beta approach is the construction of portfolios with a target factor exposure. We compare the out-of-sample beta of a portfolio constructed by the fundamental method with that of a portfolio constructed through the rolling-window approach. For both portfolios, the out-of-sample beta is estimated by performing a full- period regression on the market, in order to have a consistent comparison criterion. This protocol is similar to that employed to compare competing estimators of the covariance matrix, when minimum variance portfolios are constructed with various estimators and their out-of-sample variances are computed. We find that out-of-sample, the portfolio constructed on the basis of fundamental betas is indeed closer to neutrality (defined as a target value of 1) compared to the portfolio constructed on the basis of the rolling-window betas. The last application that we consider is a “fundamental CAPM”, which is a form of conditional CAPM where the conditional beta of a stock depends on its characteristics. We compute the alphas of portfolios sorted on size, book-to-market and past one-year return by performing cross-sectional regressions of the Fama and MacBeth (1973) type. We confirm that the unconditional CAPM yields large pricing errors for these portfolios, and we find that the conditional version of the model has roughly the same pricing

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