Improved Risk Reporting with Factor-Based Diversification Measures

Improved Risk Reporting with Factor-Based Diversification Measures — February 2014

3. Empirical Analysis for Equity Indices

ENB values around 1 or 2 means that, as for the PCA approach, only one factor explains almost all the exposure to risk of the S&P500. In our context, similarly to the PCA method, it is very likely that the dominant market factor that has the highest explanatory power, and in that sense, both the PCA and four Fama-French factor approaches generate downward- biased estimates of the effective number of independent bets. In order to avoid overloading a single risk factor in the orthogonal risks decomposition, a more reliable approach consists in using an MLT of the original constituents in order to extract risk factors that are close to the original constituents, and compute ENB measures from these risk factors regarded as orthogonal versions of the original constituents. In the graph representing the evolution of the ENB of the S&P500 index using an MLT approach on Figure 1, we note that the ENB reaches more reasonable values ranging between 150 and 350. We complement the above analysis by displaying, in Table 1 the correlations between the different diversification measures computed on the period starting in December 1959 and ending in December 2012. We have argued before that the ENB computed using a principal component analysis and the ENB computed using a Fama-French factor model do not show satisfying results in terms of magnitude. We also find that the correlations obtained between the different ENB models are extremely low and sometimes even negative, and the one that is the most positively correlated with the ENC measure, which reflects a reasonable albeit naive diversification measure, is the one computed with the MLT approach (11.82%). On the

other hand, the correlation between the ENB using an MLT model and the ENB using a PCA is equal to 1.58%, the correlation between the ENB using an MLT model and ENB using a FF model is -29.10% and the correlation between ENB using a Fama-French factor model and ENB using a PCA approach is 10.43%. Note that the ENB using a Fama-French factor model is also positively correlated to the ENC (1.64%) but very close to 0. Since the MLT seems to be the most reliable approach to computing the effective number of (uncorrelated) bets, we choose to focus on this approach in the remainder of the paper. We next try and analyse whether an equally-weighted portfolio achieves a higher level of diversification compared to a cap-weighted portfolio (see for example DeMiguel et al. (2009) for evidence that equally-weighted portfolios dominate their cap-weighted counterpart in terms of Sharpe ratio). We can observe the results of the ENC and ENB diversification measures in Figure 2, each figure displaying a diversification measure computed with the equally-weighted S&P500 (in green) and with the cap-weighted S&P500 (in blue). We note that the ENC computed from the equally-weighted S&P500 is approximately equal to 500 during the whole period. This is consistent with the definition of the Effective Number of Constituents, which reaches a maximum equal to the total number of constituents in the index when all constituents have the same weight in the portfolio, which is exactly the definition of an equally-weighted portfolio. If the ENC is not always exactly equal to 500 it is because the S&P500 index does not always exactly contains 500 constituents or because at some dates data on some

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An EDHEC-Risk Institute Publication