EDHEC-Risk Institute October 2016

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

3. Applications of Fundamental Beta

no more evidence against it than against the four-factor model. Hence, while these tests lead to reject the static CAPM, they do not allow us to reject the fundamental CAPM and the four-factor model, and the two models have comparable success in explaining the returns to the test assets. 3.3.2 Fundamental CAPM with Time-Varying Market Risk Premium The expression of the conditional CAPM in Equation (3.4) naturally leads to the introduction of the conditional market premium. In this subsection, we relax the assumption that the conditional premium

Across the three models studied here, it is the fundamental CAPM that delivers the market premium estimate (7.14%) that is the closest to the sample average (6.88%). The other two models fall short of the historical mean, with estimates of about 4.5%. For the Carhart model, the premia estimated by Fama-MacBeth regressions have reasonable values, though they do not align well with the realised premia of Table 5. In closing, these tests raise less evidence against the fundamental CAPM than against the static CAPM, and they raise Table 5: Annualised Ex-Post Risk Premia Over 1973-2014 Market Size

Value

Momentum

6.88% 8.20% We compute average of factor annualised returns with quarterly data from Ken French's library over the period 1973-2014. 2.80% 4.48%

Table 6: Average Alphas and Estimated Factor Risk Premia From Fama-MacBeth Regressions

Average Alpha Market Premium Size Premium Value Premium Momentum Premium

Static CAPM

5.04% (1.66) 2.87% (0.89) 2.86% (0.79)

4.54% (0.76) 4.07% (0.74) 7.14% (0.90)

Carhart

5.85% (8.36)

8.01% (9.65)

6.48% (4.91)

Fundamental CAPM

For each of the three pricing models, we perform Fama-MacBeth regressions of the returns to 30 portfolios sorted on size, book-to- market or past one-year return on the market factor (for the static and the fundamental CAPM) or the market augmented with the size, value and momentum factors (for the Carhart model). The difference between the static and the fundamental CAPM is that in the former model, a single beta is estimated for each test portfolio, while in the latter, each portfolio has a time-varying beta that is a function of the attributes of the constituents. The test portfolios are equally weighted and their constituents are the 500 stocks from the S&P 500 universe. Regressions are done on the period 1973-2014. The first column contains the average alpha across the 30 portfolios, and the next columns display the estimated risk premium of each factor included in the model. The numbers in brackets in the first column are the average t-statistics estimated by the Fama-MacBeth method. In the other columns, they are t-statistics for the risk premia estimates.

Table 7: Distribution of Estimated Alphas over the Cross-Section of Sorted Portfolios Mean Standard Deviation First Quartile

Median

Third Quartile

Static CAPM

5.04%

2.74%

3.35%

4.95%

6.35%

Carhart

2.87%

1.06%

2.35%

2.66%

3.57%

Fundamental CAPM 4.19% For each of the three pricing models, we perform Fama-MacBeth regressions of the returns to 30 portfolios sorted on size, book-to- market or past one-year return on the market factor (for the static and the fundamental CAPM) or the market augmented with the size, value and momentum factors (for the Carhart model). The difference between the static and the fundamental CAPM is that in the former model, a single beta is estimated for each test portfolio, while in the latter, each portfolio has a time-varying beta that is a function of the attributes of the constituents. The test portfolios are equally weighted and their constituents are the 500 stocks from the S&P 500 universe. Regressions are done on the period 1973-2014. This table shows the mean, the standard deviation and the 25th, the 50th and the 75th percentiles of the distribution of alphas across the 30 test portfolios. 2.86% 2.72% 1.08% 2.76%

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