EDHEC-Risk Institute October 2016

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

Executive Summary

the Fama-French-Carhart factors: market capitalisation ( Cap i,t ), the 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:

Hence, the coefficients can be estimated separately for each stock, by running a time-series regression. More specifically, we regress each stock’s 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:

For N stocks, the model involves 8 N parameters θ which tie the alphas and betas to the underlying stock characteristics. These parameters are estimated by minimising the sum of squared residuals over all dates and stocks in a procedure known as pooled regression . Because the coefficients are independent from one stock to the other, the pooled regression is actually equivalent to N time-series regressions: minimise is equivalent

Prior to the regression, each attribute is transformed into a zero mean and unit standard deviation z-score so as to avoid scale effects. 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. Exhibit 1 displays the distributions of the four coefficients that appear in the

for each i

to minimise

Exhibit 1: 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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