Improved Risk Reporting with Factor-Based Diversification Measures

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

2. Portfolio Diversification Measures

of constituents in a portfolio. However, the question of which α to use remains. When dealing with longshort portfolios, it seems natural to use the ENC 2 measure since ENC 1 is not defined for negative weights due to the logarithmic function. The presence of negative weights, and the resulting leverage, penalises the concentration measure as we can see on the following simple example. Consider a portfolio with w 1 = , and w 2 = w 3 = w 4 = . This leads to ENC 2 ( ) = 1 which is the highest degree of concentration achieved by a long-only portfolio (corresponding to w 1 = 1, and w 2 = w 3 = w 4 = 0). However, if we increase the size of the short position, and consider the portfolio given by: w 1 = −1, and w 2 = w 3 = w 4 = , then we have ENC 2 ( ) = < 1. This shows that ENC 2 penalises short positions, and values between 0 and 1 can be achieved, but only when portfolio contain large short positions. However, when we consider long-only portfolios, both the ENC 1 and ENC 2 measures are well-defined and can be therefore be used. Note that since we will only deal with long-only portfolios in the following, our empirical analysis will be done using ENC 1 as a measure of the effective number of constituents. A robustness check will be presented in the Appendix to show that our main results remain qualitatively valid when ENC 2 is used instead. In spite of their intuitive appeal, these weight-based measures suffer from a number of major shortcomings. In particular, ENC measures can be deceiving when applied to assets with non homogenous risks. Consider for example a position invested for 50% in a 1% volatility bond, and the other 50% in a 30% volatility stock, and assume for simplicity that the stock

and bond returns are uncorrelated. The weights are perfectly distributed, but the risk is highly concentrated. This is due to differences in the total variance of each constituent, with (50%) 2 ×(30%) 2 being much larger than (50%) 2 × (1%) 2 , thus implying that the equity allocation has a much larger contribution to portfolio risk compared to the bond allocation. On the other hand, ENC measures can be deceiving when applied to assets with correlated risks. For instance, consider a portfolio with equal weights invested in two bonds with similar duration and volatility. Despite the fact that both dollar contributions and risk contributions are homogeneously distributed within the portfolio, risk is still very concentrated because of the high correlation between the two bonds. In other words, the main shortcoming of the ENC measure as a measure of portfolio diversification comes from the fact that it does not use information about differences in volatility and pairs of correlations across assets. To account for information in the covariance matrix, a number of risk-based measures of diversification have also been introduced by various authors. Before introducing them, we want to stress the fact that a low number of observations with a large number of constituents in the portfolios may lead to non-robust sample covariance estimates. Hence, we first robustify the sample covariance matrix Σ smp by identifying implicit factors using principal component analysis (PCA), and proceeding as follows: 1. First, we compute the sample correlation 2.2 Risk-Based Measures of Portfolio Diversification

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An EDHEC-Risk Institute Publication