Improved Risk Reporting with Factor-Based Diversification Measures

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

3. Empirical Analysis for Equity Indices

constituents of the index were missing in the database. (The number of missing constituents is illustrated in Figure 3.) However, we can see in Figure 2 that the ENB computed on the equally-weighted portfolio is very close to the ENB computed on the cap-weighted portfolio all along the period. This result is consistent as the equally-weighted scheme of diversification is in fact a naive way to diversify a portfolio in term of risk. From these results, we see that ENC does not provide good measures for risk diversification, but is only good to reflect the level of concentration in the constituents of a given portfolio. It does not account for the presence of differences in volatility and correlation levels amongst constituents, as does the ENB measure. Overall, we find that the equally-weighted version of the S&P 500 index is indeed better diversified in terms of effective number of constituents, but they are not better diversified in terms of effective number of uncorrelated bets. While this finding may be perhaps somewhat surprising, it can intuitively be explained by the fact that equal-weighting is a naive form of portfolio diversification which does not utilise the information in the covariance matrix. Finally, we try to relate the evolutions of the ENC and ENB (based on MLT approach) diversification measures to changes in economic conditions. In order to do so, we support our analysis by computing the correlations of the diversification measures and several economic indicators: the returns of the cap-weighted S&P500 (with reinvested dividends), the variations of the fitted GARCH Volatility, the term spread, the credit spread and the interest rate, and the returns of the four Fama-French factors. We compute in

Table 3 the correlation of each measure of diversification with each economic indicator on the period ranging from 1959 until the end of 2012. The table exhibits very low levels of correlations between the diversification measures and the economic factors. Indeed, all correlations are less than 10% in absolute values, except for the one between the ENC measure and the S&P500 volatility. The ENC measure appears to be negatively correlated (-10.97%) with the index volatility, which means that when the volatility increases then the ENC decreases, leading to increasing levels of concentration in the index. On the other hand, the ENB seems to be uncorrelated to the volatility of the index (-0.72%). Overall, diversification measures seems to be very partially explained by standard state variables, which suggests that they contain some original meaningful information. Finally, we move beyond a contemporaneous analysis and test for the predictive power of diversification measures for the S&P500 index over the sample period, that is between 4 January 1957 and 31 December 2012. We actually compute six linear regressions, each linear regression testing the relationship between the diversification measure at a given week t and the annualised performance on a given period starting at date week t + 1. We compute six different time-series of annualised performances for six different lengths of the predictive period: the following quarter, the following semester, the following year, the following two years, the following five years and the following ten years. We compute the diversification measures and the six time-series of annualised performances at a weekly frequency.

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An EDHEC-Risk Institute Publication