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)

Time approach in the sense that it can reproduce the exact results obtained by sorting stocks in portfolios. In the GCT method, regression of returns is performed at the stock level, and the exogenous variables are the products of the market factor with all the characteristics included in the model. 8 These interaction terms arise because the factor exposure (the beta) is represented as a linear function of the characteristics. In vector form, the decomposition of stock returns in the one-factor model reads: (2.4) Here, R i,t denotes the return at period t of an individual stock i , Z i,t is the vector of stock characteristics (1×3-dimensional vector), and ε i,t denotes the idiosyncratic return, assumed to be centred and uncorrelated fromthe exogenous variables. Overall, regression (2.4) comprises a total of six explanatory variables and eight coefficients which are stored in the vector θ . While the subject characteristics in vector Z i,t may vary across both the time dimension and the cross-section, the market factor varies over time but not across stocks. The estimation of the eight coefficients is done by pooling times series and cross-sectional information in a panel regression: the sum of squared residuals ε i,t ² over all stocks and dates is minimised with respect to the eight parameters. We then compute Driscoll and Kraay (1998) nonparametric covariance matrix estimator, which produces heteroskedasticity and auto correlation consistent standard errors that are robust to general forms of spatial and temporal dependence. Prior to the regression, each attribute is transformed into a z-score according to the following formula:

the beta is linear in the vector of returns: the fundamental beta of the portfolio is the weighted sum of the fundamental betas of the constituents:

where ( Cap p,t ) define the size, value and momentum scores of the portfolio at period t . This method is "holding-based": it requires knowledge of portfolio composition and weights, in addition to the constituents' attributes and the model coefficients. 2.2.2 Estimation Our estimation procedure is the Generalised Calendar Time (GCT) method introduced by Hoechle, Schmidt and Zimmermann (2015). In the traditional Calendar Time approach, stocks are first sorted in portfolios on some attribute and each portfolio is then regressed against a set of factors in order to study the relationship between the abnormal performance and the attribute. In contrast, the GCTmodel represents the alpha and the beta of each stock as a function of one or more attribute(s), which can be discrete or continuous variables. Thus, a continuous attribute, such as those that we use in the computation of the fundamental beta, needs not be discretised. This avoids the loss of information incurred by ignoring differences across stocks that belong to the same group. As shown by Hoechle, Schmidt and Zimmermann (2015), the GCT approach nests the Calendar ), ( Bmk p,t ) and ( Ret p,t

8 - The GCT method can also be employed with several

risk factors, like in the Fama-French model.

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