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Information Decomposition and Synergy

Tail Risk Constraints and Maximum Entropy

Department of Applied Mathematics & Statistics, Johns Hopkins University, Baltimore, MD 21218-2608, USA
Department of Economics, Mathematics and Statistics, Birkbeck, University of London, London WC1E 7HX, UK
Polytechnic School of Engineering, New York University, New York, NY 11201, USA
Author to whom correspondence should be addressed.
Academic Editor: Rick Quax
Entropy 2015, 17(6), 3724-3737;
Received: 3 March 2015 / Revised: 15 May 2015 / Accepted: 22 May 2015 / Published: 5 June 2015
(This article belongs to the Special Issue Information Processing in Complex Systems)
Portfolio selection in the financial literature has essentially been analyzed under two central assumptions: full knowledge of the joint probability distribution of the returns of the securities that will comprise the target portfolio; and investors’ preferences are expressed through a utility function. In the real world, operators build portfolios under risk constraints which are expressed both by their clients and regulators and which bear on the maximal loss that may be generated over a given time period at a given confidence level (the so-called Value at Risk of the position). Interestingly, in the finance literature, a serious discussion of how much or little is known from a probabilistic standpoint about the multi-dimensional density of the assets’ returns seems to be of limited relevance. Our approach in contrast is to highlight these issues and then adopt throughout a framework of entropy maximization to represent the real world ignorance of the “true” probability distributions, both univariate and multivariate, of traded securities’ returns. In this setting, we identify the optimal portfolio under a number of downside risk constraints. Two interesting results are exhibited: (i) the left- tail constraints are sufficiently powerful to override all other considerations in the conventional theory; (ii) the “barbell portfolio” (maximal certainty/ low risk in one set of holdings, maximal uncertainty in another), which is quite familiar to traders, naturally emerges in our construction. View Full-Text
Keywords: risk management; barbell portfolio strategy; maximum entropy risk management; barbell portfolio strategy; maximum entropy
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MDPI and ACS Style

Geman, D.; Geman, H.; Taleb, N.N. Tail Risk Constraints and Maximum Entropy. Entropy 2015, 17, 3724-3737.

AMA Style

Geman D, Geman H, Taleb NN. Tail Risk Constraints and Maximum Entropy. Entropy. 2015; 17(6):3724-3737.

Chicago/Turabian Style

Geman, Donald, Hélyette Geman, and Nassim Nicholas Taleb. 2015. "Tail Risk Constraints and Maximum Entropy" Entropy 17, no. 6: 3724-3737.

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