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Open AccessOpen Access JUSTC Mathematics 30 June 2023

Bowley reinsurance with asymmetric information under reinsurer’s default risk

  • Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUSTC-2022-0111
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  • Author Bio:

    Zhenfeng Zou is a Ph.D. candidate at the Department of Statistics and Finance, University of Science and Technology of China (USTC). His research mainly focuses on quantitative risk management and optimal (re)insurance

  • Corresponding author: E-mail: newzzf@mail.ustc.edu.cn
  • Received Date: 08 August 2022
  • Accepted Date: 30 October 2022
    • Available Online: 30 June 2023
    Abstract

  • Abstract

    Bowley reinsurance with asymmetric information means that the insurer and reinsurer are both presented with distortion risk measures but there is asymmetric information on the distortion risk measure of the insurer. Motivated by predecessors research, we study Bowley reinsurance with asymmetric information under the reinsurer’s default risk. We call this solution the Bowley solution under default risk. We provide Bowley solutions under default risk in a closed form under general assumptions. Finally, some numerical examples are provided to illustrate our main conclusions.

    Graphical abstract

    The method, theorem, and simulation study for Bowley reinsurance under default risk.

    Abstract

    Bowley reinsurance with asymmetric information means that the insurer and reinsurer are both presented with distortion risk measures but there is asymmetric information on the distortion risk measure of the insurer. Motivated by predecessors research, we study Bowley reinsurance with asymmetric information under the reinsurer’s default risk. We call this solution the Bowley solution under default risk. We provide Bowley solutions under default risk in a closed form under general assumptions. Finally, some numerical examples are provided to illustrate our main conclusions.

    • Public Summary

    • We study the problem of Bowley reinsurance with asymmetric information under the reinsurer’s default risk.
    • We adopt the path-wise optimization to solve the problem of Bowley reinsurance under default risk.
    • We give two numerical examples to illustrate Theorem 1.

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    • References(33)
  • [1]
    Borch K. An attempt to determine the optimum amount of stop loss reinsurance. In: Transactions of the 16th International Congress of Actuaries. Brussels, Belgium: International Congress of Actuaries, 1960: 597–610.
    [2]
    Arrow K J. Uncertainty and the welfare economics of medical care. The American Economic Review, 1963, 53 (5): 941–973.
    [3]
    Kaluszka M. Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics and Economics, 2001, 28 (1): 61–67. doi: 10.1016/S0167-6687(00)00066-4
    [4]
    Kaluszka M, Krzeszowiec M. Pricing insurance contracts under cumulative prospect theory. Insurance: Mathematics and Economics, 2012, 50 (1): 159–166. doi: 10.1016/j.insmatheco.2011.11.001
    [5]
    Cai J, Tan K S, Weng C, et al. Optimal reinsurance under VaR and CTE risk measures. Insurance: Mathematics and Economics, 2008, 43 (1): 185–196. doi: 10.1016/j.insmatheco.2008.05.011
    [6]
    Cheung K C. Optimal reinsurance revisited: A geometric approach. ASTIN Bulletin, 2010, 40 (1): 221–239. doi: 10.2143/AST.40.1.2049226
    [7]
    Cui W, Yang J, Wu L. Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance: Mathematics and Economics, 2013, 53 (1): 74–85. doi: 10.1016/j.insmatheco.2013.03.007
    [8]
    Cheung K C, Sung K, Yam S, et al. Optimal reinsurance under general law-invariant risk measures. Scandinavian Actuarial Journal, 2014, 2014 (1): 72–91. doi: 10.1080/03461238.2011.636880
    [9]
    Cai J, Liu H, Wang R. Pareto-optimal reinsurance arrangements under general model settings. Insurance: Mathematics and Economics, 2017, 77: 24–37. doi: 10.1016/j.insmatheco.2017.08.004
    [10]
    Asimit A V, Cheung K C, Chong W F, et al. Pareto-optimal insurance contracts with premium budget and minimum charge constraints. Insurance: Mathematics and Economics, 2020, 95: 17–27. doi: 10.1016/j.insmatheco.2020.08.001
    [11]
    Jiang W, Hong H, Ren J. Pareto-optimal reinsurance policies with maximal synergy. Insurance: Mathematics and Economics, 2021, 96: 185–198. doi: 10.1016/j.insmatheco.2020.11.009
    [12]
    Borch K. The optimal reinsurance treaty. ASTIN Bulletin, 1969, 5 (2): 293–297. doi: 10.1017/S051503610000814X
    [13]
    Aase K. The Nash bargaining solution vs. equilibrium in a reinsurance syndicate. Scandinavian Actuarial Journal, 2009, 2009 (3): 219–238. doi: 10.1080/03461230802425834
    [14]
    Boonen T, Tan K S, Zhuang S C. Pricing in reinsurance bargaining with comonotonic additive utility functions. ASTIN Bulletin, 2016, 46 (2): 507–530. doi: 10.1017/asb.2016.8
    [15]
    Chen L, Shen Y. On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game between an insurer and a reinsurer. ASTIN Bulletin, 2018, 48 (2): 905–960. doi: 10.1017/asb.2018.3
    [16]
    Cheung K C, Yam S C P, Zhang Y. Risk-adjusted Bowley reinsurance under distorted probabilities. Insurance: Mathematics and Economics, 2019, 86: 64–72. doi: 10.1016/j.insmatheco.2019.02.006
    [17]
    Gavagan J, Hu L, Lee G, et al. Optimal reinsurance with model uncertainty and Stackelberg game. Scandinavian Actuarial Journal, 2022, 2022 (1): 29–48. doi: 10.1080/03461238.2021.1925735
    [18]
    Horst U, Moreno-Bromberg S. Risk minimization and optimal derivative design in a principal agent game. Mathematics and Financial Economics, 2008, 2 (1): 1–27. doi: 10.1007/s11579-008-0012-8
    [19]
    Cheung K C, Yam S C P, Yuen F. Reinsurance contract design with adverse selection. Scandinavian Actuarial Journal, 2019, 2019 (9): 784–798. doi: 10.1080/03461238.2019.1616323
    [20]
    Chan F, Gerber H. The reinsurer’s monopoly and the Bowley solution. ASTIN Bulletin, 1985, 15 (2): 141–148. doi: 10.2143/AST.15.2.2015025
    [21]
    Boonen T J, Cheung K C, Zhang Y. Bowley reinsurance with asymmetric information on the insurer’s risk preferences. Scandinavian Actuarial Journal, 2021, 2021 (7): 623–644. doi: 10.1080/03461238.2020.1867631
    [22]
    Boonen T J, Zhang Y. Bowley reinsurance with asymmetric information: a first-best solution. Scandinavian Actuarial Journal, 2022, 2022 (6): 532–551. doi: 10.1080/03461238.2021.1998922
    [23]
    Liang X, Wang R, Young V. Optimal insurance to maximize RDEU under a distortion-deviation premium principle. Insurance: Mathematics and Economics, 2022, 104: 35–59. doi: 10.1016/j.insmatheco.2022.01.007
    [24]
    Asimit A V, Badescu A M, Cheung K C. Optimal reinsurance in the presence of counterparty default risk. Insurance: Mathematics and Economics, 2013, 53 (3): 690–697. doi: 10.1016/j.insmatheco.2013.09.012
    [25]
    Asimit A V, Badescu A M, Verdonck T. Optimal risk transfer under quantile-based risk measurers. Insurance: Mathematics and Economics, 2013, 53 (1): 252–265. doi: 10.1016/j.insmatheco.2013.05.005
    [26]
    Cai J, Lemieux C, Liu F. Optimal reinsurance with regulatory initial capital and default risk. Insurance: Mathematics and Economics, 2014, 57: 13–24. doi: 10.1016/j.insmatheco.2014.04.006
    [27]
    Lo A. How does reinsurance create value to an insurer? A cost-benefit analysis incorporating default risk. Risks, 2016, 4 (4): 48. doi: /10.3390/risks4040048
    [28]
    Huberman G, Mayers D, Smith Jr C W. Optimal insurance policy indemnity schedules. Bell Journal of Economics, 1983, 14 (2): 415–426. doi: 10.2307/3003643
    [29]
    Zhuang S C, Weng C, Tan K S. et al. Marginal indemnification function formulation for optimal reinsurance. Insurance: Mathematics and Economics, 2016, 67: 65–76. doi: 10.1016/j.insmatheco.2015.12.003
    [30]
    Wang S S. Premium calculation by transforming the layer premium density. ASTIN Bulletin, 1996, 26: 71–92. doi: 10.2143/AST.26.1.563234
    [31]
    Denneberg D. Premium calculation: Why standard deviation should be replaced by absolute deviation. ASTIN Bulletin, 1990, 20 (2): 181–190. doi: 10.2143/AST.20.2.2005441
    [32]
    Laffont J J, Martimort D. The Theory of Incentives: The Principal-Agent Model. Princeton, USA: Princeton University Press, 2009.
    [33]
    Boonen T J, Jiang W. Mean-variance insurance design with counterparty risk and incentive compatibility. ASTIN Bulletin, 2022, 52 (2): 645–667. doi: 10.1017/asb.2021.36
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    Figure  1.  Plots of the function $ \phi(t) $ on $t \in [0, 1]$ for three different values of $ p $. Corresponding to Example 1, (a) $ p = 0.1 $, (b) $ p = 0.4 $, and (c) $ p = 0.9. $

    [1]
    Borch K. An attempt to determine the optimum amount of stop loss reinsurance. In: Transactions of the 16th International Congress of Actuaries. Brussels, Belgium: International Congress of Actuaries, 1960: 597–610.
    [2]
    Arrow K J. Uncertainty and the welfare economics of medical care. The American Economic Review, 1963, 53 (5): 941–973.
    [3]
    Kaluszka M. Optimal reinsurance under mean-variance premium principles. Insurance: Mathematics and Economics, 2001, 28 (1): 61–67. doi: 10.1016/S0167-6687(00)00066-4
    [4]
    Kaluszka M, Krzeszowiec M. Pricing insurance contracts under cumulative prospect theory. Insurance: Mathematics and Economics, 2012, 50 (1): 159–166. doi: 10.1016/j.insmatheco.2011.11.001
    [5]
    Cai J, Tan K S, Weng C, et al. Optimal reinsurance under VaR and CTE risk measures. Insurance: Mathematics and Economics, 2008, 43 (1): 185–196. doi: 10.1016/j.insmatheco.2008.05.011
    [6]
    Cheung K C. Optimal reinsurance revisited: A geometric approach. ASTIN Bulletin, 2010, 40 (1): 221–239. doi: 10.2143/AST.40.1.2049226
    [7]
    Cui W, Yang J, Wu L. Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance: Mathematics and Economics, 2013, 53 (1): 74–85. doi: 10.1016/j.insmatheco.2013.03.007
    [8]
    Cheung K C, Sung K, Yam S, et al. Optimal reinsurance under general law-invariant risk measures. Scandinavian Actuarial Journal, 2014, 2014 (1): 72–91. doi: 10.1080/03461238.2011.636880
    [9]
    Cai J, Liu H, Wang R. Pareto-optimal reinsurance arrangements under general model settings. Insurance: Mathematics and Economics, 2017, 77: 24–37. doi: 10.1016/j.insmatheco.2017.08.004
    [10]
    Asimit A V, Cheung K C, Chong W F, et al. Pareto-optimal insurance contracts with premium budget and minimum charge constraints. Insurance: Mathematics and Economics, 2020, 95: 17–27. doi: 10.1016/j.insmatheco.2020.08.001
    [11]
    Jiang W, Hong H, Ren J. Pareto-optimal reinsurance policies with maximal synergy. Insurance: Mathematics and Economics, 2021, 96: 185–198. doi: 10.1016/j.insmatheco.2020.11.009
    [12]
    Borch K. The optimal reinsurance treaty. ASTIN Bulletin, 1969, 5 (2): 293–297. doi: 10.1017/S051503610000814X
    [13]
    Aase K. The Nash bargaining solution vs. equilibrium in a reinsurance syndicate. Scandinavian Actuarial Journal, 2009, 2009 (3): 219–238. doi: 10.1080/03461230802425834
    [14]
    Boonen T, Tan K S, Zhuang S C. Pricing in reinsurance bargaining with comonotonic additive utility functions. ASTIN Bulletin, 2016, 46 (2): 507–530. doi: 10.1017/asb.2016.8
    [15]
    Chen L, Shen Y. On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game between an insurer and a reinsurer. ASTIN Bulletin, 2018, 48 (2): 905–960. doi: 10.1017/asb.2018.3
    [16]
    Cheung K C, Yam S C P, Zhang Y. Risk-adjusted Bowley reinsurance under distorted probabilities. Insurance: Mathematics and Economics, 2019, 86: 64–72. doi: 10.1016/j.insmatheco.2019.02.006
    [17]
    Gavagan J, Hu L, Lee G, et al. Optimal reinsurance with model uncertainty and Stackelberg game. Scandinavian Actuarial Journal, 2022, 2022 (1): 29–48. doi: 10.1080/03461238.2021.1925735
    [18]
    Horst U, Moreno-Bromberg S. Risk minimization and optimal derivative design in a principal agent game. Mathematics and Financial Economics, 2008, 2 (1): 1–27. doi: 10.1007/s11579-008-0012-8
    [19]
    Cheung K C, Yam S C P, Yuen F. Reinsurance contract design with adverse selection. Scandinavian Actuarial Journal, 2019, 2019 (9): 784–798. doi: 10.1080/03461238.2019.1616323
    [20]
    Chan F, Gerber H. The reinsurer’s monopoly and the Bowley solution. ASTIN Bulletin, 1985, 15 (2): 141–148. doi: 10.2143/AST.15.2.2015025
    [21]
    Boonen T J, Cheung K C, Zhang Y. Bowley reinsurance with asymmetric information on the insurer’s risk preferences. Scandinavian Actuarial Journal, 2021, 2021 (7): 623–644. doi: 10.1080/03461238.2020.1867631
    [22]
    Boonen T J, Zhang Y. Bowley reinsurance with asymmetric information: a first-best solution. Scandinavian Actuarial Journal, 2022, 2022 (6): 532–551. doi: 10.1080/03461238.2021.1998922
    [23]
    Liang X, Wang R, Young V. Optimal insurance to maximize RDEU under a distortion-deviation premium principle. Insurance: Mathematics and Economics, 2022, 104: 35–59. doi: 10.1016/j.insmatheco.2022.01.007
    [24]
    Asimit A V, Badescu A M, Cheung K C. Optimal reinsurance in the presence of counterparty default risk. Insurance: Mathematics and Economics, 2013, 53 (3): 690–697. doi: 10.1016/j.insmatheco.2013.09.012
    [25]
    Asimit A V, Badescu A M, Verdonck T. Optimal risk transfer under quantile-based risk measurers. Insurance: Mathematics and Economics, 2013, 53 (1): 252–265. doi: 10.1016/j.insmatheco.2013.05.005
    [26]
    Cai J, Lemieux C, Liu F. Optimal reinsurance with regulatory initial capital and default risk. Insurance: Mathematics and Economics, 2014, 57: 13–24. doi: 10.1016/j.insmatheco.2014.04.006
    [27]
    Lo A. How does reinsurance create value to an insurer? A cost-benefit analysis incorporating default risk. Risks, 2016, 4 (4): 48. doi: /10.3390/risks4040048
    [28]
    Huberman G, Mayers D, Smith Jr C W. Optimal insurance policy indemnity schedules. Bell Journal of Economics, 1983, 14 (2): 415–426. doi: 10.2307/3003643
    [29]
    Zhuang S C, Weng C, Tan K S. et al. Marginal indemnification function formulation for optimal reinsurance. Insurance: Mathematics and Economics, 2016, 67: 65–76. doi: 10.1016/j.insmatheco.2015.12.003
    [30]
    Wang S S. Premium calculation by transforming the layer premium density. ASTIN Bulletin, 1996, 26: 71–92. doi: 10.2143/AST.26.1.563234
    [31]
    Denneberg D. Premium calculation: Why standard deviation should be replaced by absolute deviation. ASTIN Bulletin, 1990, 20 (2): 181–190. doi: 10.2143/AST.20.2.2005441
    [32]
    Laffont J J, Martimort D. The Theory of Incentives: The Principal-Agent Model. Princeton, USA: Princeton University Press, 2009.
    [33]
    Boonen T J, Jiang W. Mean-variance insurance design with counterparty risk and incentive compatibility. ASTIN Bulletin, 2022, 52 (2): 645–667. doi: 10.1017/asb.2021.36
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