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Risk Exchange with Distorted Probabilities

Published online by Cambridge University Press:  17 April 2015

Andreas Tsanakas  and
Nicos Christofides
Andreas Tsanakas
Affiliation:
Market Risk and Reserving Unit, Lloyd’s of London, One Lime Street, London EC3M 7HA, United Kingdom, E-mail: andreas.tsanakas@lloyds.com
Nicos Christofides
Affiliation:
Centre for Quantitative Finance, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom, E-mail: n.christofides@imperial.ac.uk
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Abstract

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An exchange economy is considered, where agents (insurers/banks) trade risks. Decision making takes place under distorted probabilities, which are used to represent either rank-dependence of preferences or ambiguity with respect to real-world probabilities. Pricing formulas and risk allocations, generalising the results of Bühlmann (1980, 1984) are obtained via the construction of aggregate preferences from heterogeneous agents’ utility and distortion functions. This involves the introduction of a novel ‘collective ambiguity aversion’ coefficient. It is shown that probability distortion changes insurers’ behaviour, who trade not only to share the aggregate market risk, but are also found to bet against each other. Moreover, probability distortion tends to increase the price of insurance (increase asset returns). While the cases of rank-dependence and ambiguity are formally similar, an important distinction emerges as for rank-dependent preferences equilibria are determinate, while for ambiguity they are generally indeterminate.

Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 36 , Issue 1 , May 2006 , pp. 219 - 243
Copyright
Copyright © ASTIN Bulletin 2006

Footnotes

*

The views expressed in this paper are solely those of the authors and do not necessarily reflect the views of Lloyd’s of London.

1

The authors are grateful to an anonymous referee for insightful suggestions that significantly improved the paper.

References

Aase, K. (1993) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterisation. ASTIN Bulletin 23(2), 185–211.CrossRefGoogle Scholar
Aase, K. (2002) Perspectives of Risk Sharing. Scandinavian Actuarial Journal 2, 73–128.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (1999) Coherent measures of risk. Mathematical Finance 9(3), 203–228.CrossRefGoogle Scholar
Aubin, J-P. (1981) Cooperative Fuzzy Games. Mathematics of Operations Research 6(1), 1–13.CrossRefGoogle Scholar
Aumann, R.J. and Shapley, L.S. (1974) Values of Non-Atomic Games. Princeton University Press, Princeton.Google Scholar
Basle Committee On Banking Supervision (2003) The New Basel Capital Accord.Google Scholar
Billot, A., Chateauneuf, A., Gilboa, I. and Tallon, J.-M. (2000) Sharing Beliefs: Between Agreeing and Disagreeing. Econometrica 68(3), 685–694.CrossRefGoogle Scholar
Borch, K. (1962) Equilibrium in a Reinsurance Market. Econometrica 30(3), 424–444.CrossRefGoogle Scholar
Borch, K. (1985) A Theory of Insurance Premiums. The Geneva Papers of Risk and Insurance, 192–208.CrossRefGoogle Scholar
Bühlmann, H. (1980) An Economic Premium Principle. ASTIN Bulletin 11(1), 52–60.CrossRefGoogle Scholar
Bühlmann, H. (1984) The General Economic Premium Principle. ASTIN Bulletin 14(1), 13–21.CrossRefGoogle Scholar
Carlier, G. and Dana, R.A. (2002) Core of convex distortions of a probability. Journal of Economic Theory 113(2), 199–222.CrossRefGoogle 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
Chen, Z. and Epstein, L. (2002) Ambiguity, risk, and asset returns in continuous time. Econometrica 70(4), 1403–1443.CrossRefGoogle Scholar
Chew, S.H., Karni, E. and Safra, Z. (1987) Risk Aversion in the Theory of Expected Utility with Rank Dependent Probabilities. Journal of Economic Theory 42, 370–381.Google Scholar
Choquet, G. (1953) Theory of capacities. Annales de l’Institut Fourier 5, 131–295.CrossRefGoogle Scholar
Dana, R-A. (2002) On Equilibria when Agents have Multiple Priors. Annals of Operations Research 114, 105–115.CrossRefGoogle Scholar
Denneberg, D. (1990) Distorted Probabilities and Insurance Premiums. Methods of Operations Research 63, 3–5.Google Scholar
Denneberg, D. (1994) Non-additive Measure and Integral. Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002) The Concept of Comonotonicity in Actuarial Science and Finance: Theory. Insurance: Mathematics and Economics 31(1), 3–33.Google Scholar
Dubey, P., Neyman, A. and Weber, R.J. (1981) Value Theory Without Efficiency. Mathematics of Operations Research 6, 122–128.CrossRefGoogle Scholar
Duffie, D. (1996) Dynamic Asset Pricing Theory. Princeton University Press, Princeton.Google Scholar
Epstein, L.G. and Wang, T. (1994) Intertemporal asset pricing under Knightian uncertainty. Econometrica 62(3), 283–322.CrossRefGoogle Scholar
Gilboa, I. (1985) Subjective Distortions of Probabilities and Non-additive Probabilities. Preprint, The Foerder Institute of Economic Research, Tel Aviv University.Google Scholar
Gilboa, I. and Schmeidler, D. (1989) Maxmin Expected Utility with a Non-unique Prior. Journal of Mathematical Economics 18, 141–153.CrossRefGoogle Scholar
Knight, F.H. (1921) Risk, Uncertainty and Profit. Houghton Mifflin, Boston.Google Scholar
Landsberger, M. and Meilijson, I. (1994) Co-monotone Allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion. Annals of Operations Research 52, 97–106.CrossRefGoogle Scholar
Mehra, R. and Prescott, E. (1985) The Equity Premium: A Puzzle. Journal of Monetary Economics 15, 145–161.CrossRefGoogle Scholar
Quiggin, J. (1982) A Theory of Anticipated Utility. Journal of Economic Behavior and Organization 3, 323–343.CrossRefGoogle Scholar
Quiggin, J. (1993) Generalized Expected Utility Theory: The Rank-dependent Model. Kluwer Academic Publishers, Boston.CrossRefGoogle Scholar
Rothschild, M. and Stiglitz, J. (1970) Increasing Risk: I. A Definition. Journal of Economic Theory 2, 225–243 CrossRefGoogle Scholar
Rubinstein, M. (1974) An Aggregation Theorem for Securities Markets. Journal of Financial Economics 1, 225–244.CrossRefGoogle Scholar
Samet, D., and Tauman, Y. (1982) The Determination of Marginal Cost Prices under a Set of Axioms. Econometrica 50(4), 895–909.CrossRefGoogle Scholar
Schmeidler, D. (1989) Subjective Probability and Expected Utility without Additivity. Econometrica 57(3), 571–587.CrossRefGoogle Scholar
Tasche, D. (2000) Conditional Expectation as Quantile Derivative. Preprint, TU Munich.Google Scholar
Tsanakas, A. and Desli, E. (2003) Risk measures and theories of choice. British Actuarial Journal 9(4), 959–991.CrossRefGoogle Scholar
Von Neumann, J. and Morgenstern, O. (1944) Theory of Games and Economic Behavior. Princeton University Press, Princeton.Google Scholar
Wang, S.S. (1996) Premium Calculation by Transforming the Premium Layer Density. ASTIN Bulletin, 26(1), 71–92.CrossRefGoogle Scholar
Wilson, R. (1968) The Theory of Syndicates. Econometrica 36(1), 119–132.CrossRefGoogle Scholar
Yaari, M. (1987) The Dual Theory of Choice under Risk. Econometrica 55(1), 95–115.CrossRefGoogle Scholar