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Equilibrium Pricing Transforms: New Results Using Buhlmann’s 1980 Economic Model*

Published online by Cambridge University Press:  17 April 2015

Shaun S. Wang
Shaun S. Wang*
Affiliation:
SCOR Reinsurance Company, One Pierce Place, Suite 600, P.O. Box 4049, Itasca, IL 60143-4049 USA, swang@scor.com
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Abstract

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In this paper we revisit an economic model of Buhlmann (ASTIN Bulletin, 1980) and derive equilibrium pricing transforms. We obtain the Esscher Transform and the Wang Transform under different sets of assumptions on the aggregate economic environment. We show that the Esscher Transform and the Wang Transform exhibit very different behaviors when used in pricing insurance risks.

Keywords

Equilibrium PricingEsscher TransformWang TransformDistorted ProbabilityExponential TiltingOptimal Exchange

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 33 , Issue 1 , May 2003 , pp. 57 - 73
Copyright
Copyright © ASTIN Bulletin 2003

Footnotes

*

This paper is dedicated to Dr. Hans Buhlmann for his tremendous contributions to the actuarial profession and the international actuarial community.

References

Aase, K.K. (1993) Equilibrium in reinsurance syndicate; existence, uniqueness and characterization, ASTIN Bulletin 23, 185–211.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J-M. and Heath, D. (1999) Coherent measures of risk, Math. Finance 9/3, 203–228.CrossRefGoogle Scholar
Borch, K. (1962) Equilibrium in a reinsurance market, Econometrica 30, 424–444.CrossRefGoogle Scholar
Buhlmann, H. (1980) An economic premium principle, ASTIN Bulletin 11, 52–60.CrossRefGoogle Scholar
Buhlmann, H. (1984) The general economic premium principle. ASTIN Bulletin 14, 13–21.CrossRefGoogle Scholar
Buhlmann, H., Delbaen, F., Embrechts, P. and Shiryaev, A. (1998) On Esscher transforms in discrete finance model. ASTIN Bulletin 28/2, 171–186.CrossRefGoogle Scholar
Embrechts, P., Mcneil, A. and Straumann, D. (2002) Correlation and dependency in risk management: properties and pitfalls. In Risk Management: Value at Risk and Beyond, ed. Dempster, M.A.H., Cambridge University Press, Cambridge, pp. 176–233.CrossRefGoogle Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas, North American Actuarial Journal, 2/1, 1–25.CrossRefGoogle Scholar
Gerber, H.U. (1981) The Esscher premium principle: a criticism Comment, ASTIN Bulletin 12, 139–140.CrossRefGoogle Scholar
Gerber, H.U. and Pafumi, G. (1998) Utility functions: from risk theory to finance, North American Actuarial Journal, 2/3, 74–91.CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S.W. (1994) Option pricing by Esscher transforms, Transactions of the Society of Actuaries, XLVI, 99–140.Google Scholar
Heilmann, W.R. (1989) Decision theoretic foundations of credibility theory. Insurance: Mathematics and Economics 8, 77–95.Google Scholar
Kamps, U. (1998) On a class of premium principles including the Esscher principle. Scand. Actuarial Journal, 1, 75–80.CrossRefGoogle Scholar
Kaas, R., Van Heerwaarden, A.E. and Goovaerts, M.J. (1994) Ordering of Actuarial Risks, Caire, Brussels.Google Scholar
Meyers, G. (1996) The competitive market equilibrium risk load formula for catastrophe rate-making, Proceedings of the Casualty Actuarial Society, LXXXIII, available from website www.casact.org/pubs/pubs.htm.Google Scholar
Mildenhall, S. (2000) Discussion of Michael Wacek’s 1997 PCAS Paper – Application of the option market paradigm to the solution of insurance problems, Proceedings of the Casualty Actuarial Society, LXXXVII, available from website www.casact.org/pubs/pubs.htm.Google Scholar
Pratt, J.W. (1964) Risk aversion in the small and in the large, Econometrica, 32, 122–136.CrossRefGoogle Scholar
Taylor, G.C. (1992) Risk exchange I: a unification of some existing results, Scandinavian Actuarial Journal, 15–39.CrossRefGoogle Scholar
Van Heerwaarden, A.E., Kaas, R. and Goovaerts, M.J. (1989) Properties of the Esscher premium calculation principle, Insurance: Mathematics and Economics 8, 261–267.Google Scholar
Venter, G.G. (1991) Premium implications of reinsurance without arbitrage, ASTIN Bulletin, 21/2, 223–230.CrossRefGoogle Scholar
Wang, S.S. (1995) Insurance pricing and increased limits ratemaking by proportional hazards transforms, Insurance: Mathematics and Economics, 17, 43–54.Google Scholar
Wang, S.S. (1996) Premium calculation by transforming the layer premium density, ASTIN Bulletin, 26/1, 71–92.Google Scholar
Wang, S.S. (1998) Aggregation of correlated risks: models and algorithms, Proceedings of the Casualty Actuarial Society, LXXXV, 848–939. It is available from website www.casact.org/ pubs/pubs.htm.Google Scholar
Wang, S.S. (2000) A class of distortion operators for pricing financial and insurance risks1, Journal of Risk and Insurance, 67/1, 15–36.CrossRefGoogle Scholar
Wang, S.S. (2002) A universal framework for pricing financial and insurance risks, ASTIN Bulletin, November.CrossRefGoogle Scholar
Wang, S.S. and Dhane, J. (1998) Comonotonicity, correlation order and premium principles, Insurance Mathematics And Economics, 22/3, 235–242.CrossRefGoogle Scholar
Wang, S.S., Young, V.R. and Panjer, H.H. (1997) Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics, 21, 173–183.Google Scholar
Wirch, J.L. and Hardy, M.R. (1999) A Synthesis of risk measures for capital adequacy, Insurance: Mathematics and Economics, 25/3, 337–347.Google Scholar
Zehnwirth, B. (1981) The Esscher premium principle: a criticism, ASTIN Bulletin 12, 77–78.CrossRefGoogle Scholar