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WEIGHTED COMONOTONIC RISK SHARING UNDER HETEROGENEOUS BELIEFS

Published online by Cambridge University Press:  12 March 2020

Haiyan Liu
Haiyan Liu*
Affiliation:
Department of Mathematics and Department of Statistics and Probability, Michigan State University, East Lansing, MI48824, USA, E-mail: [email protected]
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Abstract

We study a weighted comonotonic risk-sharing problem among multiple agents with distortion risk measures under heterogeneous beliefs. The explicit forms of optimal allocations are obtained, which are Pareto-optimal. A necessary and sufficient condition is given to ensure the uniqueness of the optimal allocation, and sufficient conditions are given to obtain an optimal allocation of the form of excess of loss or full insurance. The optimal allocation may satisfy individual rationality depending on the choice of the weight. When the distortion risk measure is value at risk or tail value at risk, an optimal allocation is generally of the excess-of-loss form. The numerical examples suggest that a risk is more likely to be shared among agents with heterogeneous beliefs, and the introduction of the weight enables us to prioritize some agents as part of a group sharing a risk.

Keywords

Distortion risk measureheterogeneous beliefsindividual rationalityvalue at risktail value at riskC60G22
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 2 , May 2020 , pp. 647 - 673
Copyright
© Astin Bulletin 2020

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References

Acciaio, B. (2007) Optimal risk sharing with non-monotone monetary functionals. Finance and Stochastics, 11, 267–289.CrossRefGoogle Scholar
Acciaio, B. and Svindland, G. (2009) Optimal risk sharing with different reference probabilities. Insurance: Mathematics and Economics, 44(3), 426–433.Google Scholar
Anthropelos, M. and Kardaras, C. (2017) Equilibrium in risk-sharing games. Finance and Stochastics, 21(3), 815–865.CrossRefGoogle Scholar
Asimit, V. and Boonen, T. (2018) Insurance with multiple insurers: A game-theoretic approach. European Journal of Operational Research, 267, 778–790.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9, 203–228.CrossRefGoogle Scholar
Barrieu, P. and El Karoui, N. (2005) Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics, 9, 269–298.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
Boonen, T. (2016). Optimal reinsurance with heterogeneous reference probabilities. Risks, 4, 26.CrossRefGoogle Scholar
Boonen, T. and Ghossoub, M. (2019) On the existence of a representative reinsurer under heterogeneous beliefs. Insurance: Mathematics and Economics, 88, 209–225.Google Scholar
Boonen, T., Liu, F. and Wang, R. (2018) Competitive equilibria in a comonotone market. SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3091424.Google Scholar
Boonen, T., Tan, K. and Zhuang, S. (2016) The role of a representative reinsurer in optimal reinsurance. Insurance: Mathematics and Economics, 70, 196–204.Google Scholar
Borch, K. (1960) Reciprocal reinsurance treaties seen as a two-person co-operative game. Scandinavian Actuarial Journal, 1960(1–2), 29–58.CrossRefGoogle Scholar
Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30, 424–444.CrossRefGoogle Scholar
Cai, J., Lemieux, C. and Liu, F. (2016) Optimal reinsurance from the perspective of both an insurer and a reinsurer. ASTIN Bulletin, 46(2), 815–849.CrossRefGoogle Scholar
Cai, J., Liu, H. and Wang, R. (2017) Pareto-optimal reinsurance arrangements under general model settings. Insurance: Mathematics and Economics, 77, 24–37.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
Cheung, K.C., Sung, K.C. J., Yam, S.C.P. and Yung, S.P. (2014) Optimal reinsurance under general law-invariant risk measures. Scandinavian Actuarial Journal, 2014, 72–91.CrossRefGoogle Scholar
Chi, Y. (2019) On the optimality of a straight deductible under belief heterogeneity. ASTIN Bulletin, 49(1), 243–262.CrossRefGoogle Scholar
Dhaene, J., Kukush, A., Linders, D. and Tang, Q. (2012) Remarks on quantiles and distortion risk measures. European Actuarial Journal, 2, 319–328.CrossRefGoogle Scholar
Embrechts, P., Liu, H. and Wang, R. (2018a) Quantile-based risk sharing. Operations Research, 66(4), 936–949.CrossRefGoogle Scholar
Embrechts, P., Liu, H., Mao, T. and Wang, R. (2018b) Quantile-based risk sharing with heterogeneous beliefs. Mathematical Programming, forthcoming. https://link.springer.com/article/10.1007/s10107-018-1313-1CrossRefGoogle Scholar
Filipovic, D. and Svindland, G. (2008) Optimal capital and risk allocations for law- and cash-invariant convex functions. Finance and Stochastics, 12, 423–439.CrossRefGoogle Scholar
Frittelli, M. and Rosazza Gianin, E. (2002) Putting order in risk measures. Journal of Banking and Finance, 26(7), 1473–1486.CrossRefGoogle Scholar
Frittelli, M. and Rosazza Gianin, E. (2005) Law-invariant convex risk measures. Advances in Mathematical Economics, 7, 33–46.CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2002) Convex measures of risk and trading constraints. Finance and Stochastics, 6(4), 429–447.CrossRefGoogle Scholar
Gerber, H.U. (1978) Pareto-optimal risk exchanges and related decision problems. ASTIN Bulletin, 10(1), 25–33.CrossRefGoogle Scholar
Ghossoub, M. (2017) Arrow's Theorem of the deductible with heterogeneous beliefs. North American Actuarial Journal, 21(1), 15–35.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
Ludkovski, M. and Rüschendorf, L. (2008) On comonotonicity of Pareto optimal risk sharing. Statistics and Probaility Letters, 78(10), 1181–1188.CrossRefGoogle Scholar
Miettinen, K. (1999) Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston, Massachusetts.Google Scholar
Rockafellar, R.T. (1970) Convex Analysis. Princeton, New Jersey: Princeton University Press.CrossRefGoogle Scholar
Shen, J., Shen, Y., Wang, B. and Wang, R. (2019) Distributional compatibility for change of measures. Finance and Stochastics, 23, 761–794.CrossRefGoogle Scholar
Tsanakas, A. (2009) To split or not to split: Capital allocation with convex risk measures. Insurance: Mathematics and Economics, 44(2), 268–277.Google Scholar
Yaari, M. (1987) The dual theory of choice under risk. Econometrica, 55(1), 95–115.CrossRefGoogle Scholar
Zhuang, S.C., Weng, C., Tan, K.S. and Assa, H. (2016) Marginal indemnification function formulation for optimal reinsurance. Insurance: Mathematics and Economics, 67, 65–76.Google Scholar