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)

Hence, the coefficients can be estimated separately for each stock, by running a time-series regression. We regress for each stock its excess return on the market return and the market return crossed with the stock’s attributes. For a stock i , the regression equation takes the form:

These quantities are still called a “fundamental alpha” and a “fundamental beta”. For N stocks, the model has 8N coefficients to estimate instead of 8 with the previous model. The increase in the number of parameters has two effects. On the one hand, misspecification risk is reduced because restrictions on parameters are relaxed; on the other hand, we lose degrees of freedom, which may cause loss of robustness. As in the constrained case, the model is estimated by minimising the sum of squared residuals ε i,t over all dates and stocks. But because the coefficients are independent from one stock to the other, the pooled regression is equivalent to N time-series regressions: minimising is equivalent to

(2.5)

The model is estimated over the S&P 500 universe with N = 500 stocks and the period 2002-2015 (which corresponds to 51 quarterly returns). Hence we obtain 500 coefficients of each type (intercept, capitalisation sensitivity, book-to-market sensitivity and past return sensitivity) for the alpha, and 500 others for the beta.

for each i

minimising

Figure 15: Distributions of Coefficients in the More Flexible One-Factor Model The coefficients are estimated through time-series regressions for each of the 500 stocks from the S&P 500 universe with quarterly stock returns, z-score attributes and market returns from Ken French's library over the period 2002-2015. Attributes come from the ERI Scientific Beta US database and are updated quarterly.

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