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BILATERAL RISK SHARING WITH HETEROGENEOUS BELIEFS AND EXPOSURE CONSTRAINTS

Published online by Cambridge University Press:  07 January 2020

Tim J. Boonen Open the ORCID record for Tim J. Boonen [Opens in a new window]  and
Mario Ghossoub Open the ORCID record for Mario Ghossoub [Opens in a new window]
Tim J. Boonen*
Affiliation:
Amsterdam School of Economics University of AmsterdamRoetersstraat 11, 1018 WB, AmsterdamThe Netherlands E-Mail: [email protected]
Mario Ghossoub
Affiliation:
Department of Statistics and Actuarial Science University of Waterloo200 University Ave. W., Waterloo N2L 3G1, Canada E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

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This paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank-dependent utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level of belief heterogeneity. We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue’s Decomposition Theorem.

Keywords

Risk sharingPareto optimalityheterogeneous beliefsprobability distortionexposure constraintsC02D86G22
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 1 , January 2020 , pp. 293 - 323
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Astin Bulletin 2020

References

Aase, K.K. (1993) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin, 23(2), 185–211.CrossRefGoogle Scholar
Aase, K.K. (2010) Existence and uniqueness of equilibrium in a reinsurance syndicate. ASTIN Bulletin, 40(2), 491–517.Google Scholar
Acciaio, B. and Svindland, G. (2009) Optimal risk sharing with different reference probabilities. Insurance: Mathematics and Economics, 44(3), 426–433.Google Scholar
Aliprantis, C.D. and Border, K.C. (2006) Infinite Dimensional Analysis, Third Edition. New York: Springer-Verlag.Google Scholar
Amarante, M., Ghossoub, M. and Phelps, E.S. (2015) Ambiguity on the insurer’s side: The demand for insurance. Journal of Mathematical Economics, 58, 61–78.CrossRefGoogle Scholar
Arrow, K.J. and Debreu, G. (1954) Existence of an equilibrium for a competitive economy. Econometrica, 22(3), 265–290.CrossRefGoogle 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
Billot, A., Chateauneuf, A., Gilboa, I. and Tallon, J.M. (2002) Sharing beliefs and the absence of betting in the Choquet expected utility model. Statistical Papers, 43(1), 127–136.CrossRefGoogle Scholar
Boonen, T.J. (2016) Optimal reinsurance with heterogeneous reference probabilities. Risks, 4(3), 26.CrossRefGoogle Scholar
Boonen, T.J. (2017) Risk sharing with expected and dual utilities. ASTIN Bulletin, 47(2), 391–415.CrossRefGoogle Scholar
Boonen, T.J., De Waegenaere, A. and Norde, H. (2017) Redistribution of longevity risk: The effect of heterogeneous mortality beliefs. Insurance Mathematics and Economics, 72, 175–188.CrossRefGoogle Scholar
Boonen, T.J., Liu, F. and Wang, R. (2018) Competitive Equilibria in a Comonotone Market. Available at SSRN: https://ssrn.com/abstract=3091424.Google Scholar
Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30(3), 424–44.CrossRefGoogle Scholar
Bühlmann, H. (1980) An economic premium principle. ASTIN Bulletin, 11(1), 52–60.CrossRefGoogle Scholar
Bühlmann, H. and Jewell, W.S. (1979) Optimal risk exchanges. ASTIN Bulletin, 10(3), 243–262.CrossRefGoogle Scholar
Cambanis, S., Simons, G. and Stout, W. (1976) Inequalities for Ek(X,Y) when the marginals are fixed. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 36(4), 285–294.CrossRefGoogle Scholar
Cass, D. and Shell, K. (1983) Do sunspots matter? Journal of Political Economy, 91(2), 193–227.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(2), 191–214.CrossRefGoogle Scholar
Chen, H., Joslin, S. and Tran, N. (2012) Rare disasters and risk sharing with heterogeneous beliefs. Review of Financial Studies, 25(7), 2189–2224.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(2), 370–381.Google Scholar
Chi, Y. (2019) On the optimality of a straight deductible under belief heterogeneity. ASTIN Bulletin, 42(1), 243–262.CrossRefGoogle Scholar
Cohon, J.L. (1978) Multiobjective Programming and Planning, Vol. 140. Mathematics in Science and Engineering. New York: Academic Press.Google Scholar
David, A. (2008) Heterogeneous beliefs, speculation, and the equity premium. Journal of Finance, 63(1), 41–83.CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2016) Stochastic Finance: An Introduction in Discrete Time, Fourth Edition. Berlin: Walter de Gruyter.CrossRefGoogle Scholar
Gerber, H.U. (1978) Pareto-optimal risk exchanges and related decision problems. ASTIN Bulletin, 10(1), 25–33.CrossRefGoogle Scholar
Ghirardato, P. and Siniscalchi, M. (2018) Risk sharing in the small and in the large. Journal of Economic Theory, 175, 730–765.CrossRefGoogle Scholar
Ghossoub, M. (2015) Equimeasurable rearrangements with capacities. Mathematics of Operations Research, 40(2), 429–445.CrossRefGoogle Scholar
Ghossoub, M. (2016) Optimal insurance with heterogeneous beliefs and disagreement about zero-probability events. Risks, 4(3), 29.CrossRefGoogle Scholar
Ghossoub, M. (2017) Arrow’s theorem of the deducible with heterogeneous beliefs. North American Actuarial Journal, 21(1), 15–35.CrossRefGoogle Scholar
Ghossoub, M. (2019a) Budget-constrained optimal insurance with belief heterogeneity. Insurance Mathematics and Economics, 89, 79–91.CrossRefGoogle Scholar
Ghossoub, M. (2019b) Optimal insurance under rank-dependent expected utility. Insurance: Mathematics and Economics, 87, 51–66.Google Scholar
Gollier, C. (2007) Whom should we believe? Aggregation of heterogeneous beliefs. Journal of Risk and Uncertainty, 35(2), 107–127.CrossRefGoogle Scholar
He, X., Kouwenberg, R. and Zhou, X.Y. (2017) Rank-dependent utility and risk taking in complete markets. SIAM Journal on Financial Mathematics, 8(1), 214–239.CrossRefGoogle Scholar
Jin, H., Xia, J. and Zhou, X.Y. (2019) Arrow-Debreu equilibria for rank-dependent utilities with heterogeneous probability weighting. Mathematical Finance, 29(3), 898–927.CrossRefGoogle Scholar
Jouini, E., Schachermayer, W. and Touzi, N. (2008) Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18(2), 269–292.CrossRefGoogle Scholar
Kaluszka, M. (2004) An extension of the Gerber-Bühlmann-Jewell conditions for optimal risk sharing. ASTIN Bulletin, 34(1), 27–48.CrossRefGoogle Scholar
Ludkovski, M. and Young, V.R. (2009) Optimal risk sharing under distorted probabilities. Mathematics and Financial Economics, 2(2), 87–105.CrossRefGoogle Scholar
Miettinen, K. (1998) Nonlinear Multiobjective Optimization, Vol. 12. International Series in Operations Research and Management Science. New York: Springer Science+Business Media.CrossRefGoogle Scholar
Moriguti, S. (1953) A modification of Schwarz’s inequality with applications to distributions. Annals of Mathematical Statistics, 24, 107–113.CrossRefGoogle Scholar
Quiggin, J. (1982) A theory of anticipated utility. Journal of Economic Behavior and Organisation, 3(4), 323–343.CrossRefGoogle Scholar
Quiggin, J. (1991) Comparative statics for rank-dependent expected utility theory. Journal of Risk and Uncertainty, 4(4), 339–350.CrossRefGoogle Scholar
Quiggin, J. (1993) Generalized Expected Utility Theory - The Rank-Dependent Model. Dordrecht: Springer Science+Business Media.CrossRefGoogle Scholar
Schmeidler, D. (1989) Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.CrossRefGoogle Scholar
Simsek, A. (2013a) Belief disagreements and collateral constraints. Econometrica, 81(1), 1–53.Google Scholar
Simsek, A. (2013b) Speculation and risk sharing with new financial assets. Quarterly Journal of Economics, 128(3), 1365–1396.CrossRefGoogle Scholar
Tallon, J.M. (1998) Do sunspots matter when agents are Choquet-expected-utility maximizers? Journal of Economic Dynamics and Control, 22(3), 357–368.CrossRefGoogle Scholar
Tsanakas, A. and Christofides, N. (2006) Risk exchange with distorted probabilities. ASTIN Bulletin, 36(1), 219.CrossRefGoogle Scholar
Tversky, A. and Kahneman, D. (1992) Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323.CrossRefGoogle Scholar
Wang, R., Xu, Z.Q. and Zhou, X.Y. (2019) Dual utilities on risk aggregation under dependence uncertainty. Finance and Stochastics, 23(4), 1025–1048.CrossRefGoogle Scholar
Wilson, R. (1968) The theory of syndicates. Econometrica, 36(1), 119–132.CrossRefGoogle Scholar
Xia, J. and Zhou, X.Y. (2016) Arrow-Debreu equilibria for rank-dependent utilities. Mathematical Finance, 26(3), 558–588.CrossRefGoogle Scholar
Xu, Z.Q. (2016) A note on the quantile formulation. Mathematical Finance, 26(3), 558–588.CrossRefGoogle Scholar
Yaari, M. (1987) The dual theory of choice under risk. Econometrica, 55(1), 95–115.CrossRefGoogle Scholar