ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Management 30 June 2023

Bowley reinsurance with asymmetric information under reinsurer’s default risk

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

    Zhenfeng Zou is a Ph.D. candidate in 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
  • The problem of Bowley reinsurance with asymmetric information was recently introduced by Boonen et al. [Scandinavian Actuarial Journal 2021, 623-644] and Boonen and Zhang [Scandinavian Actuarial Journal 2022, 532-551]. 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 these two papers, we study Bowley reinsurance with asymmetric information under the reinsurer's default risk in this paper. We call this solution the Bowley solution under default risk. We provide Bowley solutions under default risk in closed form under general assumptions. Finally, some numerical examples are presented to illustrate our main conclusions.
    The method, theorem and simulation study for Bowley reinsurance under default risk.
    The problem of Bowley reinsurance with asymmetric information was recently introduced by Boonen et al. [Scandinavian Actuarial Journal 2021, 623-644] and Boonen and Zhang [Scandinavian Actuarial Journal 2022, 532-551]. 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 these two papers, we study Bowley reinsurance with asymmetric information under the reinsurer's default risk in this paper. We call this solution the Bowley solution under default risk. We provide Bowley solutions under default risk in closed form under general assumptions. Finally, some numerical examples are presented to illustrate our main conclusions.
    • 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 the main Theorem 1.

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  • [1]
    Borch, K. (1960). An attempt to determine the optimum amount of stop loss reinsurance. In: Transactions of the 16th International Congress of Actuaries, 1, 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. and 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. and Zhang, Y. 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. and 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., Sung, K., Yam, S. and Yung, S. 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. and 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. and Hu, J. 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. and 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, 3: 219–238.
    [14]
    Boonen, T., Tan, K. S. and 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. and 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. and Zhang, Y. Risk-adjusted Bowley reinsurance under distorted probabilities. Insurance: Mathematics and Economics, 2019b, 86: 64–72. doi: 10.1016/j.insmatheco.2019.02.006
    [17]
    Gavagan, J., Hu, L., Lee, G., Liu, H. and Weixel, A. 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. and 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., and Yuen, F. Reinsurance contract design with adverse selection. Scandinavian Actuarial Journal, 2019a, 2019 (9): 784–798. doi: 10.1080/03461238.2019.1616323
    [20]
    Chan, F. and 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. and 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. and 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. and 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. and Cheung, K. C. Optimal reinsurance in the presence of counterparty default risk. Insurance: Mathematics and Economics, 2013a, 53 (3): 690–697. doi: 10.1016/j.insmatheco.2013.09.012
    [25]
    Asimit, A. V., Badescu, A. M. and Verdonck, T. Optimal risk transfer under quantile-based risk measurers. Insurance: Mathematics and Economics, 2013b, 53 (1): 252–265. doi: 10.1016/j.insmatheco.2013.05.005
    [26]
    Cai, J., Lemieux, C. and 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. (2016). How does reinsurance create value to an insurer? A cost-benefit analysis incorporating default risk. Risks, 4(4), Artical number 48(16 pages).
    [28]
    Huberman, G., Mayers, D. and 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. and Assa, H. 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 Deviation1. ASTIN Bulletin, 1990, 20 (2): 181–190. doi: 10.2143/AST.20.2.2005441
    [32]
    Laffont, J.J. and Martimort, D. (2009). The theory of incentives: the principal-agent model. Princeton university press.
    [33]
    Boonen, T. J. and 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
  • 加载中

Catalog

    Figure  1.  Plot of the function $ \phi(t) $ on $ t \in [0, t_1] $ for three different values of $ p $. Corresponding to Example 4.1. (a) $ p = 0.1 $, (b) $ p = 0.4 $ and (c) $ p = 0.9 $

    [1]
    Borch, K. (1960). An attempt to determine the optimum amount of stop loss reinsurance. In: Transactions of the 16th International Congress of Actuaries, 1, 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. and 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. and Zhang, Y. 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. and 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., Sung, K., Yam, S. and Yung, S. 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. and 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. and Hu, J. 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. and 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, 3: 219–238.
    [14]
    Boonen, T., Tan, K. S. and 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. and 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. and Zhang, Y. Risk-adjusted Bowley reinsurance under distorted probabilities. Insurance: Mathematics and Economics, 2019b, 86: 64–72. doi: 10.1016/j.insmatheco.2019.02.006
    [17]
    Gavagan, J., Hu, L., Lee, G., Liu, H. and Weixel, A. 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. and 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., and Yuen, F. Reinsurance contract design with adverse selection. Scandinavian Actuarial Journal, 2019a, 2019 (9): 784–798. doi: 10.1080/03461238.2019.1616323
    [20]
    Chan, F. and 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. and 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. and 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. and 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. and Cheung, K. C. Optimal reinsurance in the presence of counterparty default risk. Insurance: Mathematics and Economics, 2013a, 53 (3): 690–697. doi: 10.1016/j.insmatheco.2013.09.012
    [25]
    Asimit, A. V., Badescu, A. M. and Verdonck, T. Optimal risk transfer under quantile-based risk measurers. Insurance: Mathematics and Economics, 2013b, 53 (1): 252–265. doi: 10.1016/j.insmatheco.2013.05.005
    [26]
    Cai, J., Lemieux, C. and 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. (2016). How does reinsurance create value to an insurer? A cost-benefit analysis incorporating default risk. Risks, 4(4), Artical number 48(16 pages).
    [28]
    Huberman, G., Mayers, D. and 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. and Assa, H. 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 Deviation1. ASTIN Bulletin, 1990, 20 (2): 181–190. doi: 10.2143/AST.20.2.2005441
    [32]
    Laffont, J.J. and Martimort, D. (2009). The theory of incentives: the principal-agent model. Princeton university press.
    [33]
    Boonen, T. J. and 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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