EDHEC-Risk Institute October 2016

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

3. Applications of Fundamental Beta

3.1 Introducing Sectors in Multi- Dimensional Risk and Performance Analysis Risk and performance analysis for equity portfolios is most often performed according to one single dimension, typically based on sector, country or factor decompositions. In reality, the risk and performance of a portfolio can be explained by a combination of several such dimensions, and the question therefore arises to assess what the marginal contributions are of various sectors, countries and factors to the performance and risk of a given equity portfolio. In this section, we introduce fundamental betas augmented with sector attributes for multi-dimensional risk and performance analysis which intend to allow asset managers to decompose the risk and performance of a given portfolio across multiple dimensions. For simplicity of exposure, we will focus on two dimensions, namely factor and sectors, but the method presented here can easily be extended to include additional dimensions such as regions. In what follows, we first provide a broad overview of the methodology, before applying it to the equally-weighted portfolio of the S&P 500 universe. 3.1.1 One-Factor Model with Beta as Function of Attributes (Fundamentals and Sectors) We consider the following one-factor model for stock returns, in which the alpha and the beta are functions both of the sector attribute and of the three observable attributes that were used in Section 2, namely the market capitalisation, the book-to-market ratio and the past one- year return. Hence, we have the following relations:

In this section, we present three applications of the fundamental beta. We first use this approach to embed the sector dimension in our multi-factor risk and performance analysis. In complement to the previous methods presented in Section 1.3.2 and Section 1.3.3, we show that the fundamental beta approach is more convenient when the multi-factor analysis is extended to additional dimensions (e.g. sector and regions). We then compare the fundamental and the rolling-window betas as estimators of the conditional beta, by constructing market-neutral portfolios based on the two methods. We show that the fundamental method results in more accurate estimates of market exposures, since the portfolios constructed in this way achieve better ex- post market neutrality than those in which the beta was estimated by regressing past stock returns on the market. The third application is a comprehensive asset pricing test to compare the conditional CAPM with two standard alternative factor models: the unconditional CAPM, where the beta and the market premium are constant, and the multi-factor Carhart model. We follow the Fama and MacBeth (1973) cross- sectional method and we compare the alphas of the equally-weighted portfolios sorted on book-to-market ratio, market capitalisation, and past one-year return under the three models. We show that the introduction of fundamental betas is usefully complemented by that of a time- varying market premium, which further reduces the alphas. The conditional model based on fundamental betas proves to be the most effective at explaining the cross- section of expected returns.

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