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COMPETITIVE EQUILIBRIA WITH DISTORTION RISK MEASURES

Published online by Cambridge University Press:  03 July 2015

Tim J. Boonen
Tim J. Boonen*
Affiliation:
University of Amsterdam, Valckenierstraat 65-67, 1098 XE, Amsterdam, the Netherlands
*
Email address: t.j.boonen@uva.nl
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Abstract

This paper studies optimal risk redistribution between firms, such as banks or insurance companies. The introduction of the Basel II regulation and the Swiss Solvency Test has increased the use of risk measures to evaluate financial or insurance risk. We consider the case where firms use a distortion risk measure (also called dual utility) to evaluate risk. The paper first characterizes all Pareto optimal redistributions. Thereafter, it characterizes all competitive equilibria. It presents three conditions that are jointly sufficient for existence of a unique equilibrium redistribution. This equilibrium's redistribution and prices are provided in closed form via a representative agent.

Keywords

Competitive equilibriadistortion risk measurescapital asset pricing model
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 3 , September 2015 , pp. 703 - 728
Copyright
Copyright © Astin Bulletin 2015 

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References

Aase, K. (1993) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin, 23, 185–211.CrossRefGoogle Scholar
Aase, K. (2010) Existence and uniqueness of equilibrium in a reinsurance syndicate. ASTIN Bulletin, 40, 491–517.Google Scholar
Acciaio, B. (2007) Optimal risk sharing with non-monotone monetary functionals. Finance and Stochastics, 11, 267–289.CrossRefGoogle Scholar
Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance, 26, 1487–1503.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9, 203–228.CrossRefGoogle Scholar
Asimit, A., Badescu, A. and Tsanakas, A. (2013) Optimal risk transfers in insurance groups. European Actuarial Journal, 3, 159–190.CrossRefGoogle Scholar
Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30, 424–444.CrossRefGoogle Scholar
Bühlmann, H. (1980) An economic premium principle. ASTIN Bulletin, 11, 52–60.CrossRefGoogle Scholar
Burgert, C. and Rüschendorf, L. (2008) Allocations of risks and equilibrium in markets with finitely many traders. Insurance: Mathematics and Economics, 42, 177–188.Google Scholar
Chateauneuf, A., Dana, R.-A. and Tallon, J.-M. (2000) Optimal risk-sharing rules and equilibria with Choquet-expected-utility. Journal of Mathematical Economics, 34, 191–214.CrossRefGoogle Scholar
Chateauneuf, A., Kast, R. and Lapied, A. (1996) Choquet pricing for financial markets with frictions. Mathematical Finance, 6, 323–330.CrossRefGoogle Scholar
Dana, R.-A. (2011) Comonotonicity, efficient risk-sharing and equilibria in markets with short-selling for concave law-invariant utilities. Journal of Mathematical Economics, 47, 328–335.CrossRefGoogle Scholar
Dana, R.-A. and Le Van, C. (2010) Overlapping sets of priors and the existence of efficient allocations and equilibria for risk measures. Mathematical Finance, 20, 327–339.CrossRefGoogle Scholar
De Castro, L. I. and Chateauneuf, A. (2011) Ambiguity aversion and trade. Economic Theory, 48, 243–273.CrossRefGoogle Scholar
De Giorgi, E. and Post, T. (2008) Second-order stochastic dominance, reward-risk portfolio selection, and the CAPM. Journal of Financial and Quantitative Analysis, 43, 525–546.CrossRefGoogle Scholar
Denneberg, D. (1994) Non-Additive Measure and Integral. Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
Dhaene, J., Vanduffel, S., Goovaerts, M., Kaas, R., Tang, Q. and Vyncke, D. (2006) Risk measures and comonotonicity: A review. Stochastic Models, 22, 573–606.CrossRefGoogle Scholar
Filipović, D. and Kupper, M. (2008) Equilibrium prices for monetary utility functions. International Journal of Theoretical and Applied Finance, 11, 325–343.CrossRefGoogle Scholar
Filipović, D. and Svindland, G. (2008) Optimal capital and risk allocations for law- and cash-invariant convex functions. Finance and Stochastics, 12, 423–439.CrossRefGoogle Scholar
Flåm, S. (2011) Exchanges and measures of risks. Mathematics and Financial Economics, 5, 249–267.CrossRefGoogle Scholar
Heath, D. and Ku, H. (2004) Pareto equilibria with coherent measures of risk. Mathematical Finance, 14, 163–172.CrossRefGoogle Scholar
Jouini, E., Schachermayer, W. and Touzi, N. (2008) Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18, 269–292.CrossRefGoogle Scholar
Kaina, M. and Rüschendorf, L. (2009) On convex risk measures on Lp-spaces. Mathematical Methods of Operations Research, 69, 475–495.CrossRefGoogle Scholar
Kiesel, S. and Rüschendorf, L. (2007) Characterization of optimal risk allocations for convex risk functionals. Statistics and Decisions, 99, 1000–1016.Google Scholar
Kiesel, S. and Rüschendorf, L. (2010) On optimal allocation of risk vectors. Insurance: Mathematics and Economics, 47, 167–175.Google Scholar
Landsberger, M. and Meilijson, I. I. (1994) Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion. Annals of Operations Research, 52, 97–106.CrossRefGoogle Scholar
Ludkovski, M. and Rüschendorf, L. (2008) On comonotonicity of Pareto optimal risk sharing. Statistics & Probability Letters, 78, 1181–1188.CrossRefGoogle Scholar
Ludkovski, M. and Young, V. R. (2009) Optimal risk sharing under distorted probabilities. Mathematics and Financial Economics, 2, 87–105.CrossRefGoogle Scholar
Rigotti, L. and Shannon, C. (2012) Sharing risk and ambiguity. Journal of Economic Theory, 147, 2028–2039.CrossRefGoogle Scholar
Schmeidler, D. (1989) Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.CrossRefGoogle Scholar
Schmidt, U. (1999) Efficient risk-sharing and the dual theory of choice under risk. The Journal of Risk and Insurance, 66, 597–608.CrossRefGoogle Scholar
Sharpe, W. F. (1964) Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19, 425–442.Google Scholar
Strzalecki, T. and Werner, J. (2011) Efficient allocations under ambiguity. Journal of Economic Theory, 146, 1173–1194.CrossRefGoogle Scholar
Tsai, J., Wang, J. and Tzeng, L. (2010) On the optimal product mix in life insurance companies using conditional value at risk. Insurance: Mathematics and Economics, 46, 235–241.Google Scholar
Tsanakas, A. (2004) Dynamic capital allocation with distortion risk measures. Insurance: Mathematics and Economics, 35, 223–243.Google Scholar
Tsanakas, A. and Christofides, N. (2006) Risk exchange with distorted probabilities. ASTIN Bulletin, 36, 219–243.CrossRefGoogle Scholar
Wang, J., Huang, H., Yang, S. and Tsai, J. (2010) An optimal prodict mix for hedging longevity risk in life insurance companies: The immunization theory approach. Journal of Risk and Insurance, 77, 473–497.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, 71–92.CrossRefGoogle Scholar
Wang, S. S. (2000) A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance, 67, 15–36.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
Werner, J. (2001) Participation in risk-sharing under ambiguity. Technical report, University of Minnesota.Google Scholar
Wilson, R. (1968) The theory of syndicates. Econometrica, 36, 119–132.CrossRefGoogle Scholar
Wirch, J. and Hardy, M. (2001) Distortion risk measures: Coherence and stochastic dominance. Working Paper, available at: http://www.researchgate.net/publication/228982123. Accessed June 18, 2015.Google Scholar
Yaari, M. E. (1987) The dual theory of choice under risk. Econometrica, 55, 95–115.CrossRefGoogle Scholar