EDHEC-Risk Institute October 2016

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

3. Applications of Fundamental Beta

momentum effect in this sample, at least between the two extreme sides of the classification. For each model, Table 6 reports the average alpha across the 30 portfolios, as well as the risk premia estimated by the Fama-MacBeth procedure. These premia are to be compared with the historical average returns to the factors, which are displayed in Table 5. In Table 6, the static CAPM gives the largest average alpha, at 5.04%. Table 7 provides more information about the dispersion of alphas across test portfolios by showing the 25th, 50th and 75th percentiles of the distribution: the static CAPM has the largest quantiles, and 25% of the portfolios have alphas greater than 6.35%. Moreover, alphas tend to be more statistically significant for this model than for the other two, as appears from the t-statistics. On the other hand, the Carhart model and the conditional CAPM have comparable average alphas, of 2.87% and 2.86% respectively. From Table 7, the conditional CAPM is the one that can achieve the smallest pricing errors, since it has the lowest first quartile. For both models, alphas are, on average, insignificant, as can be seen from the low t-statistics. These results confirm the results from the literature, saying that the return spreads between portfolios sorted on size, value or short- term past return are not explained by the standard CAPM. But they show that from a statistical standpoint, a model in which the characteristics are used as drivers of factor exposures performs as well as one in which the characteristics are used as sorting criteria to define additional factors. In other words, the differences across the average returns to size, value and momentum portfolios can be as well explained with the fundamental CAPM as with a multi-factor model.

If the model is correctly specified, should not be significantly different from zero, and should be close to the average market return. In order to test these restrictions, we need to estimate the standard deviations of the estimators. Simple estimates are given by: 10

and

.

It is also interesting to compare the fundamental CAPM with a multi-factor model in which stocks’ attributes are used to construct additional factors instead of being taken as instrumental variables for the estimation of the beta. The multi- factor model that corresponds to our choice of characteristics is the Carhart four-factor model, in which expected returns are given by: ( λ and β i denote respectively the market premium and the market beta.) The Fama- MacBeth procedure can be immediately extended to this multi-factor framework, and it yields estimates for the alphas of the test portfolios as well as for the four risk premia. Test Assets and Results Our test assets are the 30 decile portfolios formed by sorting stocks on size, book-to-market or past one-year return. We consider the stocks of the S&P 500 universe, and we construct equally-weighted portfolios for which we measure quarterly returns over the period 1973-2014. Appendix A2 provides descriptive statistics on these portfolios: we verify that there is a value, a size and a

10 - As pointed by Jagannathan and Wang

(1996), these estimates do not take into account the sampling errors in estimated betas.

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