Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T21:05:52.287Z Has data issue: false hasContentIssue false
  • English
  • Français

RISK SHARING WITH EXPECTED AND DUAL UTILITIES

Published online by Cambridge University Press:  11 May 2017

Tim J. Boonen
Tim J. Boonen*
Affiliation:
Amsterdam School of Economics, University of Amsterdam, Roetersstraat 11, 1018 WB, Amsterdam, The Netherlands
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper analyzes optimal risk sharing among agents that are endowed with either expected utility preferences or with dual utility preferences. We find that Pareto optimal risk redistributions and the competitive equilibria can be obtained via bargaining with a hypothetical representative agent of expected utility maximizers and a hypothetical representative agent of dual utility maximizers. The representative agent of expected utility maximizers resembles an average risk-averse agent, whereas representative agent of dual utility maximizers resembles an agent that has lowest aversion to mean-preserving spreads. This bargaining leads to an allocation of the aggregate risk to both groups of agents. The optimal contract for the expected utility maximizers is proportional to their allocated risk, and the optimal contract for the dual utility maximizing agents is given by “tranching” of their allocated risk. We show a method to derive equilibrium prices. We identify a condition under which prices are locally independent of the expected utility functions, and given in closed form. Moreover, we characterize uniqueness of the competitive equilibrium.

Keywords

Pareto optimal risk sharingcompetitive equilibriaexpected utilitydual utility
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 2 , May 2017 , pp. 391 - 415
Copyright
Copyright © Astin Bulletin 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aase, K.K. (1993) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin, 23, 185–211.Google Scholar
Aase, K.K. (2010) Existence and uniqueness of equilibrium in a reinsurance syndicate. ASTIN Bulletin, 40, 491–517.Google Scholar
Acerbi, C. and Tasche, D. (2002) On the coherence of expected shortfall. Journal of Banking and Finance, 26, 1487–1503.Google Scholar
Arrow, K.J. (1963) Uncertainty and the welfare economics of medical care. American Economic Review, 53, 941–973.Google Scholar
Arrow, K.J. and Debreu, G. (1954) Existence of an equilibrium for a competitive economy. Econometrica, 22, 265–290.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9, 203–228.Google Scholar
Boonen, T.J. (2015) Competitive equilibria with distortion risk measures. ASTIN Bulletin, 45, 703–728.Google Scholar
Boonen, T.J. (2017) Risk redistribution games with dual utilities. ASTIN Bulletin, 47, 303–329.Google 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
Bühlmann, H. (1984) The general economic premium principle. ASTIN Bulletin, 14, 13–21.CrossRefGoogle Scholar
Bühlmann, H. and Jewell, W.S. (1979) Optimal risk exchanges. ASTIN Bulletin, 10, 243–262.CrossRefGoogle Scholar
Carlier, G., Dana, R.-A. and Galichon, A. (2012) Pareto efficiency for the concave order and multivariate comonotonicity. Journal of Economic Theory, 147, 207–229.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
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
Chi, Y. (2012) Reinsurance arrangements minimizing the risk-adjusted value of an insurer's liability. ASTIN Bulletin, 42, 529–557.Google 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.Google 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.Google Scholar
Embrechts, P., Liu, H. and Wang, R. (2016) Quantile-based risk sharing. Working Paper. Available at http://ssrn.com/abstract=2744142, 2016.Google Scholar
Filipović, D. and Kupper, M. (2008) Equilibrium prices for monetary utility functions. International Journal of Theoretical and Applied Finance, 11, 325–343.Google Scholar
Hey, J. D. and Orme, C. (1994) Investigating generalizations of expected utility theory using experimental data. Econometrica, 62, 1291–1326.Google Scholar
Hommes, C.H. (2006) Heterogeneous agent models in economics and finance. Handbook of Computational Economics, 2, 1109–1186.CrossRefGoogle Scholar
Jouini, E., Schachermayer, W. and Touzi, N. (2008) Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18, 269–292.Google Scholar
Kiesel, S. and Rüschendorf, L. (2007) Characterization of optimal risk allocations for convex risk functionals. Statistics and Decisions, 99, 1000–1016.Google Scholar
Kusuoka, S. (2001) On law invariant coherent risk measures. Advances in Mathematical Economics, 3, 83–95.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.Google Scholar
Ludkovski, M. and Young, V.R. (2009) Optimal risk sharing under distorted probabilities. Mathematics and Financial Economics, 2, 87–105.Google 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. Norwell: Kluwer Academic Publishers.Google Scholar
Raviv, A. (1979) The design of an optimal insurance policy. American Economic Review, 69, 84–96.Google Scholar
Rothschild, M. and Stiglitz, J. E. (1970) Increasing risk: A definition. Journal of Economic Theory, 2, 225–243.Google Scholar
Schmeidler, D. (1989) Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.Google Scholar
Strzalecki, T. and Werner, J. (2011) Efficient allocations under ambiguity. Journal of Economic Theory, 146, 1173–1194.CrossRefGoogle Scholar
Tsanakas, A. and Christofides, N. (2006) Risk exchange with distorted probabilities. ASTIN Bulletin, 36, 219–243.Google Scholar
Von Neumann, J. and Morgenstern, O. (1944) Theory of Games and Economic Behavior, 1st edition. Princeton: Princeton University Press.Google Scholar
Wang, S.S. and Young, V.R. (1998) Ordering risks: Expected utility theory versus Yaari's dual theory of risk. Insurance: Mathematics and Economics, 22, 145–161.Google 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. (1987) Arbitrage and the existence of competitive equilibrium. Econometrica, 55, 1403–1418.CrossRefGoogle Scholar
Wilson, R. (1968) The theory of syndicates. Econometrica, 36, 119–132.Google Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55, 95–115.Google Scholar