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Diversification in heavy-tailed portfolios: properties and pitfalls

Published online by Cambridge University Press:  19 November 2012

Georg Mainik  and
Paul Embrechts
Georg Mainik*
Affiliation:
RiskLab, Department of Mathematics, ETH Zurich
Paul Embrechts
Affiliation:
RiskLab, Department of Mathematics, ETH Zurich
*
*Correspondence to: Georg Mainik, RiskLab, Department of Mathematics, ETH Zurich, Raemistrasse 101, 8092 Zurich, Switzerland. E-mail: [email protected]
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Abstract

We discuss risk diversification in multivariate regularly varying models and provide explicit formulas for Value-at-Risk asymptotics in this case. These results allow us to study the influence of the portfolio weights, the overall loss severity, and the tail dependence structure on large portfolio losses. We outline sufficient conditions for the sub- and superadditivity of the asymptotic portfolio risk in multivariate regularly varying models and discuss the case when these conditions are not satisfied. We provide several examples to illustrate the resulting variety of diversification effects and the crucial impact of the tail dependence structure in infinite mean models. These examples show that infinite means in multivariate regularly varying models do not necessarily imply negative diversification effects. This implication is true if there is no loss-gain compensation in the tails, but not in general. Depending on the loss-gain compensation, asymptotic portfolio risk can be subadditive, superadditive, or neither.

Keywords

DiversificationValue-at-Riskheavy tailstail dependencerisk subadditivityrisk superadditivitymultivariate regular variationinfinite means
Type
Papers
Information
Annals of Actuarial Science , Volume 7 , Issue 1: Special issue on Enterprise Risk Management , March 2013 , pp. 26 - 45
Copyright
Copyright © Institute and Faculty of Actuaries 2012

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Footnotes

†

Senior SFI Professor

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