Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T20:24:40.984Z Has data issue: false hasContentIssue false
  • English
  • Français

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: Marc.Goovaerts@econ.kuleuven.ac.be
Rob Kaas
Affiliation:
Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands, E-mail: r.kaas@uva.nl
Jan Dhaene
Affiliation:
Center for Risk and Insurance Studies (CRIS), Katholieke Universiteit Leuven, B-3000 Leuven, Belgium, E-mail: Jan.Dhaene@econ.kuleuven.ac.be
Qihe Tang
Affiliation:
Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands, E-mail: q.tang@uva.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Artzner, Ph. (1999) Application of coherent risk measures to capital requirements in insurance. North American Actuarial Journal 3(2), 11–25.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9, 203–228.CrossRefGoogle Scholar
Borch, K. (1968) The economics of uncertainty. Princeton University Press, Princeton.Google Scholar
Borch, K. (1974) The mathematical theory of insurance. Lexington Books, Toronto.Google Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. Springer-Verlag, Berlin.Google Scholar
Denuit, M., Dhaene, J. and Van Wouwe, M. (1999) The economics of insurance: a review and some recent developments. Mitt. Schweiz. Aktuarver. 2, 137–175.Google Scholar
De Vijlder, F. and Goovaerts, M.J. (1979) An invariance property of the Swiss premium calculation principle. Mitt. Verein. Schweiz. Versicherungsmath. 79(2), 105–120.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002a) The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31(1), 3–33.CrossRefGoogle Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002b) The concept of comonotonicity in actuarial science and finance: application. Insurance Math. Econom. 31(2), 133–161.CrossRefGoogle Scholar
Embrechts, P., Mcneil, A.J. and Straumann, D. (2002) Correlation and dependence in risk management: properties and pitfalls. Risk management: value at risk and beyond, 176–223. Cambridge Univ. Press, Cambridge.CrossRefGoogle Scholar
Gerber, H.U. (1974) On additive premium calculation principles. ASTIN Bulletin 7, 215–222.CrossRefGoogle Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8, distributed by Irwin, Richard D., Inc., Homewood, Illinois.Google Scholar
Gerber, H.U. and Goovaerts, M.J. (1981) On the representation of additive principles of premium calculation. Scand. Actuar. J. 4, 221–227.CrossRefGoogle Scholar
Goovaerts, M.J., De Vijlder, F. and Haezendonck, J. (1984) Insurance Premiums. North-Holland Publishing Co., Amsterdam.Google Scholar
Goovaerts, M.J., Kaas, R. and Dhaene, J. (2003) Economic capital allocation derived from risk measures, North American Actuarial Journal 7(2), 44–59.Google Scholar
Haezendonck, J. and Goovaerts, M.J. (1982) A new premium calculation principle based on Orlicz norms. Insurance Math. Econom. 1(1), 41–53.CrossRefGoogle Scholar
Hardy, G.H., Littlewood, J.E. and Polya, G. (1952) Inequalities. 2nd ed. Cambridge University Press.Google Scholar
Jarrow, R. (2002) Put option premiums and coherent risk measures. Math. Finance 12(2), 135–142.CrossRefGoogle Scholar
Kaas, R., Goovaerts, M.J., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory. Dordrecht: Kluwer Acad. Publ.Google Scholar
Runnenburg, J.Th. and Goovaerts, M.J. (1985) Bounds on compound distributions and stop-loss premiums. Insurance Math. Econom. 4(4), 287–293.CrossRefGoogle Scholar
von Neumann, J., and Morgenstern, O. (1944) Theory of Games and Economic Behavior. Princeton University Press, Princeton, New Jersey.Google Scholar
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin 26(1), 71–92.CrossRefGoogle Scholar
Wang, S. and Young, V.R. (1998) Ordering risks: expected utility theory versus Yaari's dual theory of risk. Insurance Math. Econom. 22(2), 145–161.CrossRefGoogle Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica 55(1), 95–115.CrossRefGoogle Scholar