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Managing Diversification
Extended Version1 Attilio Meucci2
attilio_meucci@symmys.com

this version: April 2010 last version available at ssrn.com

Abstract We propose a unified, fully general methodology to analyze and act on diversification in any environment, including long-short trades in highly correlated markets. First, we build the diversification distribution, i.e. the distribution ofthe uncorrelated bets in the portfolio that are consistent with the portfolio constraints. Next, we summarize the wealth of information provided by the diversification distribution into one single diversification index, the effective number of bets, based on the entropy of the diversification distribution. Then, we introduce the mean-diversification efficient frontier, a diversification approach toportfolio optimization. Finally, we describe how to perform mean-diversification optimization in practice in the presence of transaction and market impact costs, by only trading a few optimally chosen securities. Fully documented code illustrating our approach can be downloaded from MATLAB Central File Exchange. JEL Classification: C1, G11 Keywords: entropy, mean-diversification frontier, transaction costs,market impact, selection heuristics, systematic risk, idiosyncratic risk, principal component analysis, principal portfolios, r-square, risk contributions, random matrix theory, relative-value strategy, market-neutral strategy, bond immunization, long-short equity pairs

shorter version appears as A. Meucci, "Managing diversification", Risk, May 2009 author is grateful to Gianluca Fusai, SridharGollamudi, Harvey Stein, Thomas Deuker and two anonymous referees for their helpful feedback
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Electronic copy available at: http://ssrn.com/abstract=1358533

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Introduction

The qualitative definition of diversification is very clear to portfolio managers: a portfolio is well-diversified if it is not heavily exposed to individual shocks. However, oddly enough, there existsno broadly accepted, unique, satisfactory methodology to precisely quantify and manage diversification. In the special case of systematic-plus-idiosyncratic factor models, diversification is measured as the percentage of risk explained by the systematic factors. However, "idiosyncratic" shocks are actually correlated with each other. Furthermore, such measure fails to analyze the degree ofdiversification within the pseudo-idiosyncratic component of the portfolio, but this becomes necessary in market-neutral and relative-value strategies such as equity pairs trading, where the systematic risk is hedged away. In the special case of long-only portfolios differential diversification is defined as the difference between the weighted sum of the volatilities of each position and the total portfoliovolatility. This measure does not cover residual portfolios in such strategies as fixed-income immunization, where the portfolio manager is interested in the diversification net of the parallel shifts of the curve, which are hedged away. More diversification measures have been introduced, including naive measures based only on portfolio weights which do not account for correlations and volatilities,refer to Appendix A.1 for a detailed list and a discussion. However, in addition to not applying in full generality, none of those measures highlights where diversification, or the lack thereof, arises in a given portfolio. The contributions of this article are fourfold. First, instead of focusing on one single number, we introduce the diversification distribution, a tool to analyze the fine structure ofa portfolio’s concentration profile in fully general markets and with fully general long-short positions. Second, we introduce the effective number of uncorrelated bets, an actionable index of diversification based on the entropy of the diversification distribution. Third, we introduce the meandiversification frontier, a quantitative framework to manage the trade-off between the expected return and...