How Superadditive Can a Risk Measure Be?

Forthcoming in SIAM Journal on Financial Mathematics (SIFIN)

32 Pages Posted: 1 Jan 2014 Last revised: 4 Jun 2015

See all articles by Ruodu Wang

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Valeria Bignozzi

Università di Milano Bicocca - Dipartimento di Statistica e Metodi Quantitativi

Andreas Tsanakas

City University London - Cass Business School

Date Written: March 13, 2015

Abstract

In this paper, we study the extent to which any risk measure can lead to superadditive risk assessments, implying the potential for penalizing portfolio diversification. For this purpose we introduce the notion of extreme-aggregation risk measures. The extreme-aggregation measure characterizes the most superadditive behavior of a risk measure, by yielding the worst-possible diversification ratio across dependence structures. One of the main contributions is demonstrating that, for a wide range of risk measures, the extreme-aggregation measure corresponds to the smallest dominating coherent risk measure. In our main result, it is shown that the extreme- aggregation measure induced by a distortion risk measure is a coherent distortion risk measure. In the case of convex risk measures, a general robust representation of coherent extreme-aggregation measures is provided. In particular, the extreme-aggregation measure induced by a convex short- fall risk measure is a coherent expectile. These results show that, in the presence of dependence uncertainty, quantification of a coherent risk measure is often necessary, an observation that lends further support to the use of coherent risk measures in portfolio risk management.

Keywords: distortion risk measures; shortfall risk measures; expectiles; model uncertainty; risk aggregation; superadditivity; coherence

Suggested Citation

Wang, Ruodu and Bignozzi, Valeria and Tsanakas, Andreas, How Superadditive Can a Risk Measure Be? (March 13, 2015). Forthcoming in SIAM Journal on Financial Mathematics (SIFIN), Available at SSRN: https://ssrn.com/abstract=2373149 or http://dx.doi.org/10.2139/ssrn.2373149

Ruodu Wang (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

Valeria Bignozzi

Università di Milano Bicocca - Dipartimento di Statistica e Metodi Quantitativi ( email )

Via Bicocca degli Arcimboldi, 8
Milano, 20126
Italy

Andreas Tsanakas

City University London - Cass Business School ( email )

106 Bunhill Row
London, EC1Y 8TZ
United Kingdom

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