EDHEC-Risk Institute October 2016

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

Executive Summary

Fundamental Betas as Functions of Attributes The traditional approach to measuring the market exposure of a stock or a portfolio is to run a time-series regression of the stock (excess) returns on a market factor over a rolling window. If the joint distribution of stock and market returns were constant over time, the sample beta at date t – 1 would be a consistent estimator of the conditional beta on this date, and the variation in rolling-window estimates would be due to sampling errors only. Factor exposures, however, are not constant over time and the key challenge is therefore to estimate the beta for each stock conditional on the information available to date: where R i,t denotes the return on stock i in period [ t – 1 , t ] in excess of risk-free rate, R m,t is the excess return on the market portfolio and Φ t -1 is the information set available at date t – 1. The traditional measure of conditional market exposure is the beta estimated over a sample period, but if the distributions of stock and market returns change over time, the sample estimates are not good estimators of the true conditional moments. By shifting the sample period (rolling-window estimation), one does generate time dependency in the beta, but the "historical beta" changes relatively slowly due to the overlap between estimation windows. We introduce an alternative estimator for the conditional beta, which we name "fundamental beta" because it is defined as a function of the stock’s characteristics. More specifically, we first consider the following one-factor model for stock returns, in which the alpha and the beta are functions of the three observable attributes that define

firm’s characteristics, so that the attributes can remain attributes in the context of a parsimonious factor model, as opposed to being artificially treated as additional factors. Our approach is somewhat related to the literature on conditional asset pricing models (Jagannathan and Wang, 1996), who also allow the factor exposure to be a function of some state variables. One key difference is that standard conditional versions of the CAPM (e.g. Ferson and Schadt, 1996), stipulate that betas (and possibly alphas and risk premia) are functions of underlying macroeconomic factors such as the T-Bill rate, dividend yield, slope of the term structure, credit spread, etc. In contrast, we take betas to be functions of the time-varying micro attributes or characteristics of the underlying firms that are typically used to define additional factors, including in particular market capitalisation, the book-to-market ratio and past one-year performance. Of course, one could in principle regard the factor exposures as a function of stock specific attributes and pervasive state variables. In what follows, we introduce a formal framework for estimating these so-called fundamental betas , as opposed to historical betas, and we provide evidence of the usefulness of these fundamental betas for (i) parsimoniously embedding the sector dimension in multi-factor portfolio risk and performance analysis, (ii) building equity portfolios with controlled target factor exposure, and also (iii) explaining the cross-section of expected returns.

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