Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T12:26:27.773Z Has data issue: false hasContentIssue false
  • English
  • Français

PRICING IN REINSURANCE BARGAINING WITH COMONOTONIC ADDITIVE UTILITY FUNCTIONS

Published online by Cambridge University Press:  08 April 2016

Tim J. Boonen ,
Ken Seng Tan  and
Sheng Chao Zhuang
Tim J. Boonen
Affiliation:
Amsterdam School of Economics, University of Amsterdam, Roetersstraat 11, 1018 WB, Amsterdam, the Netherlands, E-Mail: t.j.boonen@uva.nl
Ken Seng Tan
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada, E-Mail: kstan@uwaterloo.ca
Sheng Chao Zhuang*
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
*
E-Mail: s2zhuang@uwaterloo.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Optimal reinsurance indemnities have widely been studied in the literature, yet the bargaining for optimal prices has remained relatively unexplored. Therefore, the key objective of this paper is to analyze the price of reinsurance contracts. We use a novel way to model the bargaining powers of the insurer and reinsurer, which allows us to generalize the contracts according to the Nash bargaining solution, indifference pricing and the equilibrium contracts. We illustrate these pricing functions by means of inverse-S shaped distortion functions for the insurer and the Value-at-Risk for the reinsurer.

Keywords

Reinsurance bargainingcomonotonic additivityinverse-S shaped distortion functionValue-at-risk
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 46 , Issue 2 , May 2016 , pp. 507 - 530
Copyright
Copyright © Astin Bulletin 2016 

References

Aase, K.K. (1993a) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin, 23 (2), 185–211.CrossRefGoogle Scholar
Aase, K.K. (1993b) Premiums in a dynamic model of a reinsurance market. Scandinavian Actuarial Journal, 1993 (2), 134–160.Google Scholar
Aase, K.K. (2002) Perspectives of risk sharing. Scandinavian Actuarial Journal, 2002 (2), 73–128.Google Scholar
Aase, K.K. (2009) The Nash bargaining solution vs. equilibrium in a reinsurance syndicate. Scandinavian Actuarial Journal, 2009 (3), 219–238.Google Scholar
Abdellaoui, M. (2000) Parameter-free elicitation of utility and probability weighting functions. Management Science, 46 (11), 1497–1512.Google Scholar
Artzner, P., Delbaen, F., Eber, J. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9 (3), 203–228.Google Scholar
Asimit, A. V., Badescu, A.M. and Verdonck, T. (2013) Optimal risk transfer under quantile-based risk measurers. Insurance: Mathematics and Economics, 53 (1), 252–265.Google Scholar
Assa, H. (2015) On optimal reinsurance policy with distortion risk measures and premiums. Insurance: Mathematics and Economics, 61 (1), 70–75.Google Scholar
Barrieu, P. and El Karoui, N. (2005) Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics, 9 (2), 269–298.Google Scholar
Bernard, C., He, X., Yan, J.-A. and Zhou, X.Y. (2015) Optimal insurance design under rank-dependent expected utility. Mathematical Finance, 25 (1), 154–186.Google Scholar
Bertrand, J. (1883) Review of ‘Théorie mathématique de la richesse sociale’ and ‘Recherches sur les principles mathématiques de la théorie des richesses’. Journal des Savants, 48, 499–508.Google Scholar
Binmore, K.G. (1998) Game Theory and the Social Contract: Just Playing, Volume 2. Cambridge, MA: MIT Press.Google Scholar
Boonen, T.J. (2015) Competitive equilibria with distortion risk measures. ASTIN Bulletin, 45 (3), 703–728.CrossRefGoogle Scholar
Boonen, T.J. (2016) Nash equilibria of over-the-counter bargaining for insurance risk redistributions: The role of a regulator. European Journal of Operational Research, 250 (3), 955–965.Google Scholar
Boonen, T.J., De Waegenaere, A. and Norde, H. (2012) Bargaining for Over-The-Counter risk redistributions: The case of longevity risk. CentER Discussion Paper Series No. 2012-090.Google Scholar
Boonen, T.J., Tan, K.S. and Zhuang, S.C. (2015) Optimal reinsurance with one insurer and multiple reinsurers, Working Paper, available at SSRN: http://ssrn.com/abstract=2628950.Google Scholar
Borch, K.H. (1960) Reciprocal reinsurance threaties. ASTIN Bulletin, 1 (4), 170–191.Google Scholar
Borch, K.H. (1962) Equilibrium in a reinsurance market. Econometrica, 30 (3), 424–444.Google Scholar
Britz, V., Herings, P. and Predtetchinski, A. (2010) Non-cooperative support for the asymmetric Nash bargaining solution. Journal of Economic Theory, 145 (5), 1951–1967.Google Scholar
Bühlmann, H. and Jewell, W.S. (1979) Optimal risk exchanges. ASTIN Bulletin, 10 (3), 243–262.CrossRefGoogle Scholar
Burgert, C. and Rüschendorf, L. (2006) On the optimal risk allocation problem. Statistics & Decisions, 24, 153–171.CrossRefGoogle Scholar
Burgert, C. and Rüschendorf, L. (2008) Allocations of risks and equilibrium in markets with finitely many traders. Insurance: Mathematics and Economics, 42 (1), 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 (2), 191–214.Google Scholar
Cheung, K.C. and Lo, A. (2015) Characterizations of optimal reinsurance treaties: A cost-benefit approach. Forthcoming in Scandinavian Actuarial Journal.Google Scholar
Chi, Y. (2012) Reinsurance arrangements minimizing the risk-adjusted value of an insurer's liability. ASTIN Bulletin, 42 (2), 529–557.Google Scholar
Chi, Y. and Meng, H. (2014) Optimal reinsurance arrangements in the presence of two reinsurers. Scandinavian Actuarial Journal, 2014 (5), 424–438.Google Scholar
Chi, Y. and Tan, K.S. (2011) Optimal reinsurance under VaR and CVaR risk measures: A simplified approach. ASTIN Bulletin, 41 (2), 487–509.Google Scholar
Chi, Y. and Weng, C. (2013) Optimal reinsurance subject to Vajda condition. Insurance: Mathematics and Economics, 53 (1), 179–189.Google Scholar
Cui, W., Yang, J. and Wu, L. (2013) Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance: Mathematics and Economics, 53 (1), 74–85.Google Scholar
De Castro, L.I. and Chateauneuf, A. (2011) Ambiguity aversion and trade. Economic Theory, 48 (2), 243–273.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 (2), 525–546.Google Scholar
De Waegenaere, A., Kast, R. and Lapied, A. (2003) Choquet pricing and equilibrium. Insurance: Mathematics and Economics, 32 (3), 359–370.Google Scholar
Denuit, M. and Vermandele, C. (1998) Optimal reinsurance and stop-loss order. Insurance: Mathematics and Economics, 22 (3), 229–233.Google Scholar
Filipović, D., Kremslehner, R. and Muermann, A. (2015) Optimal investment and premium policies under risk shifting and solvency regulation. Journal of Risk and Insurance, 82 (2), 261–288.Google Scholar
Filipović, D. and Kupper, M. (2008) Equilibrium prices for monetary utility functions. International Journal of Theoretical and Applied Finance, 11 (3), 325–343.Google Scholar
Föllmer, H. and Schied, A. (2002) Convex measures of risk and trading constraints. Finance and Stochastics, 6 (4), 429–447.CrossRefGoogle Scholar
Frittelli, M. and Rosazza-Gianin, E. (2002) Putting order in risk measures. Journal of Banking and Finance, 26 (7), 1473–1486.Google Scholar
Gerber, H. and Pafumi, G. (1998) Utility functions: From risk theory to finance. North American Actuarial Journal, 2 (3), 74–91.Google Scholar
He, X.D. and Zhou, X.Y. (2011) Portfolio choice under cumulative prospect theory: An analytical treatment. Management Science, 57 (2), 315–331.Google Scholar
Heath, D. and Ku, H. (2004) Pareto equilibria with coherent measures of risk. Mathematical Finance, 14 (2), 163–172.Google Scholar
Jin, H. and Zhou, X.Y. (2008) Behavioral portfolio selection in continuous time. Mathematical Finance, 18 (3), 385–426.Google Scholar
Jouini, E., Schachermayer, W. and Touzi, N. (2008) Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18 (2), 269–292.Google Scholar
Kalai, E. (1977) Nonsymmetric Nash solutions and replications of 2-person bargaining. International Journal of Game Theory, 6 (3), 129–133.Google Scholar
Kihlstrom, R.E. and Roth, A.E. (1982) Risk aversion and the negotiation of insurance contracts. Journal of Risk and Insurance, 49 (3), 372–387.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 (2), 97–106.Google Scholar
Lemaire, J. (1990) Borch's theorem: A historical survey of applications. In Risk, Information and Insurance. Essays in the memory of Karl H. Borch (ed. Loubergé, H.), pp. 15–37. Boston, Dordrecht, London: Kluwer Academic Publishers.Google Scholar
Ludkovski, M. and Rüschendorf, L. (2008) On comonotonicity of Pareto optimal risk sharing. Statistics & Probability Letters, 78 (10), 1181–1188.Google Scholar
Ludkovski, M. and Young, V.R. (2009) Optimal risk sharing under distorted probabilities. Mathematics and Financial Economics, 2 (2), 87–105.Google Scholar
Markowitz, H. (1952) Portfolio selection. Journal of Finance, 7 (1), 77–91.Google Scholar
Miyakawa, T. (2012) Non-cooperative foundation of Nash bargaining solution under incomplete information, Osaka University of Economics Working Paper Serier No. 2012-2.Google Scholar
Nash, J.F. (1950) The bargaining problem. Econometrica, 18 (2), 155–162.Google Scholar
Osborne, M.J. and Rubinstein, A. (1990) Bargaining and markets, Volume 34. San Diego: Academic press.Google Scholar
Pritsker, M. (1997) Evaluating Value at Risk Methodologies: Accuracy versus Computational Time. Journal of Financial Services Research, 12 (2), 201–242.Google Scholar
Quiggin, J. (1982) A theory of anticipated utility. Journal of Economic Behavior and Organization, 3 (4), 323–343.Google Scholar
Quiggin, J. (1991) Comparative statics for rank-dependent expected utility theory. Journal of Risk and Uncertainty, 4 (4), 339–350.Google Scholar
Quiggin, J. (1992) Generalized expected utility theory: The rank dependent model. Boston: Springer Science & Business Media.Google Scholar
Quiggin, J. and Chambers, R.G. (2009) Bargaining power and efficiency in insurance contracts. Geneva Risk and Insurance Review, 34 (1), 47–73.Google Scholar
Raviv, A. (1979) The design of an optimal insurance policy. American Economic Review, 69 (1), 84–96.Google Scholar
Rieger, M.O. and Wang, M. (2006) Cumulative prospect theory and the St. Petersburg paradox. Economic Theory, 28 (3), 665–679.Google Scholar
Schlesinger, H. (1984) Two-person insurance negotiation. Insurance: Mathematics and Economics, 3 (3), 147–149.Google Scholar
Schmeidler, D. (1986) Integral representation without additivity. Proceedings of the American Mathematical Society, 97 (2), 255–261.Google Scholar
Schmeidler, D. (1989) Subjective probability and expected utility without additivity. Econometrica, 57 (3), 571–587.Google Scholar
Taylor, G.C. (1992a) Risk exchange I: A unification of some existing results. Scandinavian Actuarial Journal, 1992 (1), 15–39.Google Scholar
Taylor, G.C. (1992b) Risk exchange II: Optimal reinsurance contracts. Scandinavian Actuarial Journal, 1992 (1), 40–59.Google Scholar
Tsanakas, A. and Christofides, N. (2006) Risk exchange with distorted probabilities. ASTIN Bulletin, 36 (1), 219–243.Google Scholar
Tversky, A. and Fox, C.R. (1995) Weighing risk and uncertainty. Psychological Review, 102 (2), 269–283.Google Scholar
Tversky, A. and Kahneman, D. (1992) Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5 (4), 297–323.Google Scholar
Wang, S.S., Young, V.R. and Panjer, H.H. (1997) Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21 (2), 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 (1), 119–132.Google Scholar
Wu, G. and Gonzalez, R. (1999) Nonlinear decision weights in choice under uncertainty. Management Science, 45 (1), 74–85.Google Scholar
Xu, Z.Q. and Zhou, X.Y. (2013) Optimal stopping under probability distortion. Annals of Applied Probability, 23 (1), 251–282.Google Scholar
Xu, Z.Q., Zhou, X.Y. and Zhuang, S.C. (2015) Optimal insurance with rank-dependent utility and increasing indemnities, Working Paper, available at SSRN: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2660113.Google Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55 (1), 95–115.Google Scholar
Young, V.R. (1999) Optimal insurance under Wang's premium principle. Insurance: Mathematics and Economics, 25 (2), 109–122.Google Scholar
Zhou, R., Li, J.S.-H. and Tan, K.S. (2015a) Economic pricing of mortality-linked securities: A tâtonnement approach. Journal of Risk and Insurance, 82 (1), 65–96.Google Scholar
Zhou, R., Li, J.S.-H. and Tan, K.S. (2015b) Modeling longevity risk transfers as Nash bargaining problems: Methodology and insights. Economic Modelling, 51, 460–472.Google Scholar