Improved Risk Reporting with Factor-Based Diversification Measures

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

3. Empirical Analysis for Equity Indices

for equity indices and their diversification measures. We first focus the analysis on the S&P500 universe, which we use as a base case since a long sample of returns is available for this index before turning in the next section to results on cross-sectional tests among 14 regional equity indices. We study several diversification measures for the S&P500: the ENC and three versions of the ENB (PCA, MLT and FF), as explained before. First, we observe in Figure 1 that the ENC measure is about 2.5 times smaller than the nominal number of constituents, which shows a substantial level of concentration since the portfolio is on average effectively invested in roughly 200 stocks instead of 500. This finding is consistent with the results in Haugen and Baker (1991), who among others have highlighted the poor efficiency score of capitalisation-weighted indices. Intuitively, the mediocre risk-adjusted performance of cap-weighted portfolios may not be surprising, given that the weighting method automatically gives very high weights to some very large cap stocks and therefore leads to concentrated portfolios. Usually, the 10-30 largest stocks make up the majority of the weighting in the index. Put differently, even if an index has more than 500 components, 90% of the components make up an almost negligible part of the index weights. Even though the ENC is a natural measure to estimate the level of concentration of a given portfolio, it does not provide insights into the level of diversification in terms of uncorrelated sources of risk. In order to do so, we plot on Figure 1 the ENB measure using three different approaches, PCA, MLT and FF. We observe that the ENB of the S&P500 index computed using a PCA approach has very small values (between

1 and 2). This means that the S&P500 is exposed to at most a couple of uncorrelated sources of risk. Remember that when the ENB is equal to one, it means that the index is exposed to a single risk factor, which is the market factor since it is commonly acknowledged that the first factor of the PCA is close to an equally weighted portfolio of the market constituents. As outlined in the previous discussion on the shortcomings of the PCA approach, the methodology is not well-suited for estimating the ENB since by construction it allocates to uncorrelated factors with decreasing powers of explanation. Consequently, the first risk factor is the one that has the highest explanatory power while the last factor has the smallest one. Therefore, the first factor found when computing a PCA should be overweighted when calculating ENB, which would explain why we have low ENB results. We test this assumption and compute the ENB using a PCA approach on the equally- weighted S&P500. We see on Figure 17 that from January 1959 to December 2012, the values of the ENB are approximately equal to 1 (up to a maximum of 2) meaning that a single factor essentially explains the whole exposure to risk of the S&P500 index when the index is equally-weighted. This supports our previous assumption that the first risk factor determined from the PCA approach explains almost all the S&P500 index risk exposure. The ENB calculated using the four Fama-French orthogonal factors is also very low (between 1 and 1.5 during the whole period). Note that in this case, since we only consider four factors and not N factors, where N is the nominal number of constituents, the maximum value that the ENB can achieve is equal to 4. Observing

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An EDHEC-Risk Institute Publication