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Entropy, Diversification and the Inefficient Frontier


The essential

  • Diversification measures can be divided in three main groups: (i) metrics in portfolio weights; (ii) approaches in risk contributions; (iii) measures that are based on fundamental factors.
  • Diversification metrics in asset risk contribution depends on asset volatility and correlation, the latter being particularly relevant because correlation instability is shown to produce unstable asset allocation.
  • Diversification and risk are not strictly related: more diversification does not always imply less risk.
  • Mean-variance portfolios are poorly diversified: in our case study the efficient portfolios exhibit only half of the available diversification.
  • A small penalty in terms of risk-return can allow a significant improvement in portfolio diversification.
  • Leverage-aversion is very costly in terms of portfolio diversification especially for high risk profiles.


In “The Battle for Investment Survival” [2], G. M. Loeb claimed that “diversification might be necessary where no intelligent supervision is likely”. Hence diversification is not an issue for the intelligent investor who is able to perfectly forecast market returns. Conversely portfolio diversification is desirable if there is a lack of information or uncertainty in the financial markets. Investors should only accept to lower portfolio diversification in presence of strong market convictions.

The bear equity market over the last decade once again calls into question the predictability of asset risk premia, and the effect of incorrect assumptions in the portfolio selection process. In the eighties and nineties, many researchers ([3, 4, 5]) demonstrated that the sub-optimality due to estimation risk can be dramatic.

Entropy is a concept that originates from Physics, but nowadays applied in many disciplines, such as Computer Science, Sociology, Economics, Medicine, Mathematics and Finance. In Information Theory, entropy is related to the degree of predictability of a dynamical system: the higher the entropy, the less predictable the system; on the other hand entropy decreases when additional information is available.

Entropy and diversification are closely related: in this focus we investigate their relationship. Even though diversification is one of the most popular ideas in finance, there is no agreement on a common measure. This reflects the fact that a portfolio can be seen from different angles. A single measure is probably not able to fully describe portfolio diversification.

In [1] we will address the role of estimation risk in portfolio diversification showing clearly that the main problems come from instability in correlation. Our study demonstrates that in an out-of-sample exercise unstable correlation produces “bad” portfolio turnover that responds only to noise in time-series; portfolio turnover is only acceptable (“good” portfolio turnover) if it corresponds to significant changes in asset risk-return characteristics.

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[1] For a more complete and technical analysis, refer to “Is your portfolio effectively diversified? A critical assessment of diversification measures for portfolio construction.“ G. Pola – AMUNDI Working paper in preparation 2014. We are particularly grateful to Sylvie de Laguiche (Head of Quant Research in Paris) for very useful comments that improved the quality of the analysis and of the manuscript. Ali Zerrad was a trainee in Amundi Quantitative Research in 2013.
[2] G. M. Loeb. “The Battle for Investment Survival”. John Wiley & Sons, 2007.
[3] J. D. Jobson and B. Korkie. “Estimation for Markowitz efficient portfolios”. Journal of the American Statistical Association, 75:544–554, 1980.
[4] M. J. Best and R. R. Grauer. “On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results”. Review of Financial Studies, 4:315–342, 1991.
[5] V. Chopra and W. T. Ziemba. “The effects of errors in means, variances, and covariances on optimal portfolio choice”. The Journal of Portfolio Management, pages 6–11, 1993.







Investors should only accept to lower portfolio diversification in presence of strong market convictions

Gianni POLA, Quantitative research at Amundi
Ali ZERRAD, Quantitative Research at Amundi

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Gianni POLA

Quantitative research at Amundi