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)

Other authors have sought to capture the determinants of time variation in the betas. A first idea to generate a time- varying beta is to introduce a dynamic model. Ghysels (1998) examines various parametric models, including those of Ferson (1990), Ferson and Harvey (1991, 1993) and Ferson and Korajczyk (1995), but showed that these models are less accurate and estimate highly volatile betas. Indeed, they tend produce large pricing errors and may have tendency to overstate the beta time variation. 2.1.2 Using Instrumental Variables Ferson and Schadt (1996) take a different approach by summarising conditioning information in the current value of a state vector Z and assuming a linear specification for the beta: , where z t = Z t - E ( Z ) is a vector of the deviations of Z t from the unconditional means. is the unconditional mean of the conditional beta and can be interpreted as an “average beta”: = E [ β i ( Z t )]. The elements of B i are the response coefficients of the conditional beta to the information variables Z t . In general, the average conditional beta does not equal the unconditional beta, obtained by replacing the conditional moments in (2.1) by their unconditional counterparts. We show in Appendix A1 that a sufficient condition for this equality to hold is that the conditional expected return and the conditional variance of the market be constant. Ferson and Schadt use the time-varying beta to construct a conditional CAPM, that is a CAPM in which proportionality between expected returns and the market beta holds period-by-period in terms of

beta otherwise than by regressing stock or portfolio returns on a market proxy.

2.1.1 Alternatives to Regression Technique

Wang and Menchero (2014) argue that the market exposure of a stock is an aggregation of the exposures to multiple factors including country, sector and investment styles. Formally, stock returns are regressed on the K risk factors:

so the beta can be decomposed as:

This can be rewritten as:

where β Fk is the beta of the market factor with respect to factor k and β ε i is the beta of the specific stock return. This beta is called by Wang and Menchero a “predicted beta”, although there is no explicit modelling for the time variation. It is intended as a better estimate for the conditional market exposure in that it takes into account the fundamental sources of risk that determine the market exposure. In the empirical application, the multi-factor model is a version of the Barra model (see Section 1.3.1), where the betas b k,i are fixed according to stock’s characteristics such as nationality, type of activity, size, liquidity, volatility, dividend yield, book-to-price ratio, etc. Hence, the predicted beta is a function of some fundamental characteristics of the stock.

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