Abstract
We study optimal risk sharing among n agents endowed with distortion risk measures. Our model includes market frictions that can either represent linear transaction costs or risk premia charged by a clearing house for the agents. Risk sharing under third-party constraints is also considered. We obtain an explicit formula for Pareto optimal allocations. In particular, we find that a stop-loss or deductible risk sharing is optimal in the case of two agents and several common distortion functions. This extends recent result of Jouini et al. (Adv Math Econ 9:49–72, 2006) to the problem with unbounded risks and market frictions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aase K.K.: Perspectives of risk sharing. Scand. Actuar. J. 2, 73–128 (2002)
Acciaio B.: Optimal risk sharing with non-monotone monetary functions. Finance Stoch. 11(2), 267–289 (2007)
Arrow K.J.: Uncertainty and the welfare of medical care. Am. Econ. Rev. 53, 941–973 (1963)
Barrieu P., El Karoui N.: Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 9(2), 269–298 (2005)
Bäuerle N., Müller A.: Stochastic orders and risk measures: consistency and bounds. Insurance: Math. Econ. 38(1), 132–148 (2006)
Bernard, C., Tian, W.: Insurance market effects of risk management metrics. Geneva Risk Insur. Rev. (forthcoming) (2009)
Borch K.: Equilibrium in a reinsurance market. Econometrica 30(3), 424–444 (1962)
Burgert C., Rüschendorf L.: Allocation of risks and equilibrium in markets with finitely many traders. Insurance: Math. Econ. 42(1), 177–188 (2008)
Carlier G., Dana R.-A.: Are generalized call-spreads efficient?. J. Math. Econ. 43(5), 581–596 (2007)
Carlier G., Dana R.-A.: Two-persons efficient risk-sharing and equilibria for concave law-invariant utilities. Econ. Theory 36(2), 189–223 (2008)
Chateauneuf A., Dana R.-A., Tallon J.-M.: Optimal risk-sharing rules and equilibria with Choquet-expected-utility. J. Math. Econ. 34(2), 191–214 (2000)
Dana, R.-A., Meilijson, I.: Modelling agents’ preferences in complete markets by second order stochastic dominance, working paper, Cahier du Ceremade 0238 (2003)
Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R., Laeven, R.J.A.: Risk measurement with equivalent utility principles. Working Paper, K.U. Leuven (2006)
Denneberg D.: Non-Additive Measure and Integral. Kluwer, Dordrecht (1994)
Filipovic D., Kupper M.: Optimal capital and risk transfers for group diversification. Math. Finance 18(1), 55–76 (2008)
Filipovic D., Kupper M.: Equilibrium prices for monetary utility functions. Int. J. Theor. Appl. Finance 11, 325–343 (2008)
Föllmer H., Schied A.: Stochastic Finance. An Introduction in Discrete Time, 2nd edn. Walter de Gruyter, Amsterdam (2004)
Gerber H.U.: Pareto-optimal risk exchanges and related decision problems. ASTIN Bull. 10(1), 25–33 (1978)
Jouini E., Schachermayer W., Touzi N.: Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9, 49–72 (2006)
Jouini E., Schachermayer W., Touzi N.: Optimal risk sharing for law invariant monetary utility functions. Math. Finance 18(2), 269–292 (2008)
Landsberger M., Meilijson I.: Co-monotone allocations, Bickel–Lehmann dispersion and the Arrow–Pratt measure of risk aversion. Ann. Oper. Res. 52, 97–106 (1994)
Ludkovski M., Rüschendorf L.: On comonotonicity of Pareto optimal allocations. Stat. Probab. Lett. 78(10), 1181–1188 (2008)
Ludkovski, M., Young, V.R.: Optimal risk sharing under distorted probabilities. Preprint (2008). arxiv.org/0809.3778
Promislow S.D., Young V.R.: Unifying framework for optimal insurance. Insurance: Math. Econ. 36(3), 347–364 (2005)
Raviv A.: The design of an optimal insurance policy. Am. Econ. Rev. 69(1), 84–96 (1979)
Rothschild M., Stiglitz J.E.: Increasing risk, I: a definition. J. Econ. Theory 2, 225–243 (1970)
Rothschild M., Stiglitz J.E.: Increasing risk, II: its economic consequences. J. Econ. Theory 3, 66–84 (1971)
Rothschild M., Stiglitz J.E.: Equilibrium in competitive insurance markets: an essay on the economics of imperfect information. Q. J. Econ. 90, 629–649 (1976)
Tsanakas A., Desli E.: Risk measures and theories of choice. Br. Actuar. J. 9(4), 959–991 (2003)
Wang S.S., Young V.R.: Ordering risks: expected utility theory versus Yaari’s dual theory of risk. Insurance: Math. Econ. 22(2), 145–161 (1998)
Wang S.S., Young V.R., Panjer H.H.: Axiomatic characterization of insurance prices. Insurance: Math. Econ. 21, 173–183 (1997)
Wilson R.: The theory of syndicates. Econometrica 36(1), 119–132 (1968)
Yaari M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)
Young V.R., Browne M.J.: Equilibrium in competitive insurance markets under adverse selection and Yaari’s dual theory of risk. Geneva Pap. Risk Insurance Theory 25, 141–157 (2000)
Acknowledgments
We thank Carole Bernard and Damir Filipovic for useful discussions. We are also grateful to the editor and the anonymous referees for their many suggestions that have greatly improved our presentation.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Ludkovski, M., Young, V.R. Optimal risk sharing under distorted probabilities. Math Finan Econ 2, 87–105 (2009). https://doi.org/10.1007/s11579-009-0015-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11579-009-0015-0