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A Unified Approach to Generate Risk Measures

Published online by Cambridge University Press:  17 April 2015

Marc J. Goovaerts ,
Rob Kaas ,
Jan Dhaene  and
Qihe Tang
Marc J. Goovaerts
Affiliation:
Center for Risk and Insurance Studies (CRIS), Katholieke Universiteit Leuven, B-3000 Leuven, Belgium, E-mail: [email protected]
Rob Kaas
Affiliation:
Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands, E-mail: [email protected]
Jan Dhaene
Affiliation:
Center for Risk and Insurance Studies (CRIS), Katholieke Universiteit Leuven, B-3000 Leuven, Belgium, E-mail: [email protected]
Qihe Tang
Affiliation:
Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands, E-mail: [email protected]
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Abstract

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The paper derives many existing risk measures and premium principles by minimizing a Markov bound for the tail probability. Our approach involves two exogenous functions v(S) and φ(S, π) and another exogenous parameter α ≤ 1. Minimizing a general Markov bound leads to the following unifying equation:

E [φ (S, π)] = αE [v (S)].

For any random variable, the risk measure π is the solution to the unifying equation. By varying the functions φ and v, the paper derives the mean value principle, the zero-utility premium principle, the Swiss premium principle, Tail VaR, Yaari's dual theory of risk, mixture of Esscher principles and more. The paper also discusses combining two risks with super-additive properties and sub-additive properties. In addition, we recall some of the important characterization theorems of these risk measures.

Keywords

Insurance premium principleRisk measureMarkov inequality
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 33 , Issue 2 , November 2003 , pp. 173 - 191
Copyright
Copyright © ASTIN Bulletin 2003

Footnotes

1

CRIS, Catholic University of Leuven

2

Department of Quantitative Economics, University of Amsterdam.

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