Improved Risk Reporting with Factor-Based Diversification Measures

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

3. Empirical Analysis for Equity Indices

In this section, we perform an empirical illustration of the benefits of using the ENB measure of portfolio diversification in the context of an improved risk reporting methodology. We compare the ENB measure to the more naive ENC measure and present an application to the measure of diversification for 14 seemingly well-diversified equity portfolios, with a more detailed analysis of the S&P500. We first present some descriptive statistics for 14 equity indices, and then analyse the link between diversification measures and equity index performance. For these empirical studies, we consider a set of popular indices representing each particular universe, and we estimate their diversification measures. The list of equity universes that we consider is given as follows:

We collect data for the indices listed above from Datastream, except for the S&P500 for which we collect the historical data from CRSP. For each equity index, we first extract the list of equity constituents that have at least been once in the sample period part of the constituent list of the index. Then, for each equity constituent, we upload its historical total return series (with reinvested dividend) at the weekly frequency, as well as historical market values, i.e., share prices multiplied by the number of ordinary shares in issue. Finally, we also extract the historical returns with reinvested dividend for each aggregated index. As a base case, we consider the S&P500 index, and collect return data at a weekly frequency on the period starting on 4 January 1957 until 31 December 2012. We use a one-year period in order to estimate the sample covariance matrix of the index constituents, then we robustify our estimator using Section 2.2, and we roll over this one-year window without overlap to generate annual estimates for the diversification measures. We use the same methodology for the 13 other indices for which we estimate the covariance matrices on the maximum historical data available. Eventually, we calculated the ENC and the ENB using the entropy measure, see Equations (2.3), and (2.6). In particular, we calculated the ENB using three methods: principal component analysis (PCA), minimum linear torsion (MLT) model and the Fama-French-Carhart four factor (FF) model and presented a detailed comparison of these three approaches in Section 2.2.

US universe: •  Large cap stock index: S&P 500

•  Technology stock index: NASDAQ 100 •  Industrial stock index: DOW JONES 30

Global European universe: •  Large cap stock index: STOXX Europe 200 •  Broader stock index: STOXX Europe 600 Global Eurozone universe: •  Very large cap stock index: EURO STOXX 50 •  Broader stock index: EURO STOXX 300

European countries: • UK stock index: FTSE 100 •  France stock index: CAC 40 • Germany stock index: DAX •  Switzerland stock index: SPI

3.1 Time-Series Analysis of Diversification Measures We present below some descriptive statistics

Asia: •  Japan stock index: TOPIX 100 • Hong-Kong stock index: HANG SENG

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An EDHEC-Risk Institute Publication