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)

and the corrected t-Stats obtained with the Driscoll and Kraay methodology. Except for the coefficient θ α ,Ret , all the coefficients are significant. But t-Stats are reduced with the Driscoll and Kraay methodology and some coefficients, in particular alpha components as θ α ,Bmk , appear non-significant. But the beta decomposition is still meaningful because the beta components are still significant with the corrected t-Stats. 2.2.3 Beta as a Function of Attributes Figures 12 and 13 present the results of Table 2 on fundamental beta, first as a function of the two microeconomic characteristics on which the Fama-French size and value factors are based and then as a function of the microeconomic characteristic on which the momentum factor is based. Under the assumption of a standard normal distribution for the z-score variable, almost all the observations (99.73%) lie between -3 and 3, so we restrict the range in each axis to [–3;3]. In this range, fundamental beta lies between 0.4 and 1.4 which is a right scale when considering market exposures. We also see that the fundamental beta is decreasing in market cap and past 1-year return and increasing in book-to-market ratio. These observations can be related to the signs of the size, value and momentum premia in the period under study. Suppose that the conditional CAPM with the fundamental beta holds, that is the conditional expected return on a stock is

where z-score { attribute } i,t denotes the normalised score of asset i for the given attribute, attribute i,t denotes the attribute value of asset i , mean t [ attribute ] denotes the mean value of the attribute across the whole universe of stocks at fixed period and std t [attribute] denotes the standard deviation of the attribute across the whole universe of stocks at fixed period. The result is that each attribute has a null mean and a standard deviation equal to 1 at each period. The pooled model is estimated over the S&P 500 universe with 500 stocks and the period 2002-2015 (51 quarterly returns), hence 51×500=25 500 observations. Because z-scores are centred, the fundamental alpha and beta of an equally-weighted portfolio of all stocks are constant and equal to θ α , 0 and θ β , 0 .

In Table 1, we verify that these estimates are close to the in-sample alpha and beta of the broad EW portfolio. Table 2 displays the complete set of parameter estimates, together with the standard errors and p-values, estimated via the standard OLS formulas or via the Driscoll Kraay covariance matrix. We observe large differences between traditional t-Stats

Table 1: Comparison of Fundamental and Historical Betas for the S&P 500 Equally-Weighted Portfolio The in-sample alpha and beta of the S&P 500 equally-weighted portfolio are estimated by regressing quarterly index returns on market returns from Ken French's library over the period 2002-2015. The coefficients θ α ,0 and θ β ,0 are obtained by a pooled regression of the 500 stock returns. Abnormal Return Market Exposure Coefficient θ α , 0 In-sample α Coefficient θ β , 0 In-sample β 0.051 0.05 0.919 0.92

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