EDHEC-Risk Institute October 2016

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

2. From Historical Betas (and Alphas) to Fundamental Betas (and Alphas)

conditional alpha and beta than what would be obtained with rolling-window regressions: indeed, changes in firm’s characteristics are only slowly reflected in rolling-window estimates, while they are instantaneously incorporated in the fundamental parameters. In our tests of the one-factor model, we will also ask whether a conditional CAPM based on the fundamental beta can explain the size, value and momentum effects. An important difference between the conditional beta of Ferson and Schadt (1996) and ours is that their state vector Z t consists of variables that are identical for all stocks while we include microeconomic data. Hence, in Ferson and Schadt’s model, cross-sectional variation in the beta comes only from the differences in the coefficients and B i across stocks, while our model, such variation can be generated even with uniform coefficients. We thus study separately two versions of the conditional beta. The first one, which we present in this subsection, has the same coefficients for all stocks, while the second one, which we develop in Section 2.3, relaxes this restriction. 2.2.1 Model Formulation We 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 the Fama-French-Carhart factors: market capitalisation ( Cap i,t ), book-to-market ratio ( Bmk i,t ) and past 1-year return ( Ret i,t ) for the stock i at date t . Hence we have the following relations: 2.2 First Specification of Fundamental Beta

The conditional beta β i,t of a stock i at period t is ameasure of themarket exposure over the next period conditional on the attributes at date t . In the above model, stocks that have the same attributes on a given date will also have the same market exposure at this instant. But their market exposures can diverge at subsequent dates as their attributes become different. We call β i,t a “fundamental beta” because it is a function of variables measured at the stock level. As is clear from the model equations, there are eight coefficients to estimate. The coefficients θ α , 0 and θ β , 0 are respectively the “alpha intercept” and the “beta intercept“ and the other parameters represent the sensitivities of the alpha and the beta with respect to the characteristics.

The mean of the conditional beta (or “average beta”) is:

This “average beta” is different from the unconditional beta (or “historical beta”) estimated with OLS regression over the entire sample.

To obtain the conditional beta of a portfolio, it suffices to use the fact that

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