Improved Risk Reporting with Factor-Based Diversification Measures

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

Executive Summary

constituents and achieve the lowest average tracking error with the initial constituents. By construction, the obtained factors are the closest uncorrelated representations of the original constituents, which alleviates the concern over interpretation, and the explanatory power of the factors are not biased in favour of some particular factors. Analysing the Relationship between Portfolio Diversification and Portfolio Performance in Various Market Conditions Our main objective is to analyse the diversification of a portfolio, measured either in terms in effective number of constituents (ENC) or (more appropriately) in terms of effective number of bets (ENB), and its relationship with subsequent portfolio performance. We provide an empirical application of this measure for intra-class and inter-class diversification. For intra-class diversification, we cast the empirical analysis in the context of various popular equity indices, with a particular emphasis on the S&P 500 index. For inter- class diversification, we analyse policy portfolios for the 1,000 largest US pension funds. 3 We first compute the ENC and ENB measures for the S&P 500 index, and test for their predictive power using weekly total return data over a sample period extending from 4 January 1957 to 31 December 2012. For both diversification measures, we actually perform six linear regression analyses, with each linear regression testing the relationship between the diversification measure at a given week and the annualised Diversification Measures for International Equity Indices

performance for each of the following 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 10 years. In Table 1, we show the results obtained from the six linear regressions for the S&P500 index. The predictive power of the diversification measure is statistically significant for both ENC and ENB measures as we obtain a p-value indistinguishable from 0 (at two decimal points) for every recording period chosen for the measure of subsequent performance. These results suggest that there is a positive relationship between the level of diversification as measured via the ENC or ENB indicator and the subsequent performance of the S&P 500 index whatever its period of analysis. It should be noted, however, that the coefficients of proportionality remain relatively low (between 0.21 and 0.42). In addition, we find for both diversification measures that the R-squared and the t-stats of the linear regressions increase with the length of the period of annualised performance computation, which shows that the diversification measures have better forecasting power over long horizons. Lastly, if we only focus on the quarterly and the semi-annual performance computation, we notice that t-statistics are higher for the ENB compared to the ENC, suggesting a stronger relationship between diversification and subsequent performance for the former measure compared to the latter. This result confirms that the entropy of the distribution of risk contributions to the portfolio from uncorrelated factors is a

3 - We also analyse a sample of the world’s 10 largest pension funds.

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An EDHEC-Risk Institute Publication