American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2020103

Optimal reinsurance with default risk: A reinsurer's perspective

Tao Chen 1, , Wei Liu 1, , Tao Tan 1, , Lijun Wu 1,, and  Yijun Hu 2,

1. 

School of Mathematics and System Sciences, Xinjiang University, Urumqi Xinjiang, 830046, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan Hubei, 430072, China

* Corresponding author: Lijun Wu

Received  May 2019 Revised  January 2020 Published  June 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (Nos: 11601463, 11861064, 11771343)

In this paper, we study the optimal reinsurance design with default risk by minimizing the VaR (value at risk) of the reinsurer's total risk exposure. The optimal reinsurance treaty is provided. When the reinsurance premium principle is specified to the expected value and exponential premium principles, the explicit expressions for the optimal reinsurance treaties are given, respectively.

Keywords: Optimal reinsurance, default risk, value at risk, expected value premium principle, exponential premium principle.
Mathematics Subject Classification: 91B30.
Citation: Tao Chen, Wei Liu, Tao Tan, Lijun Wu, Yijun Hu. Optimal reinsurance with default risk: A reinsurer's perspective. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020103
References:
[1]

A. V. Asimit, A. M. Badescu and T. Verdonck, Optimal risk transfer under quantile-based risk measures, Insurance: Mathematics and Economics, 53 (2013), 252-265.  doi: 10.1016/j.insmatheco.2013.05.005.  Google Scholar

[2]

A. V. Asimit, A. M. Badescu and K. C. Cheung, Optimal reinsurance in the presence of counterparty default risk, Insurance: Mathematics and Economics, 53 (2013), 690-697.  doi: 10.1016/j.insmatheco.2013.09.012.  Google Scholar

[3]

H. Assa, On optimal reinsurance policy with distortion risk measures and premiums, Insurance Mathematics and Economics, 61 (2015), 70-75.  doi: 10.1016/j.insmatheco.2014.11.007.  Google Scholar

[4]

K. J. Arrow, Uncertainty and the welfare economic of medical care, Uncertainty in Economics, (1978), 347–375. doi: 10.1016/B978-0-12-214850-7.50028-0.  Google Scholar

[5]

C. Bernard and M. Ludkovski, Impact of counterparty risk on the reinsurance market, North American Actuarial Journal, 16 (2012), 87-111.  doi: 10.1080/10920277.2012.10590634.  Google Scholar

[6]

K. Borch, An attempt to determine the optimal amount of stop loss reinsurance, Transactions of the 16th International Congress of Actuaries, 1 (1960), 597-610.   Google Scholar

[7]

J. Cai, C. Lemieux and F. Liu, Optimal reinsurance with regulatory initial capital and default risk, Insurance: Mathematics and Economics, 57 (2014), 13-24.  doi: 10.1016/j.insmatheco.2014.04.006.  Google Scholar

[8]

J. Cai and K. S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, Astin Bulletin, 37 (2007), 93-112.  doi: 10.1017/S0515036100014756.  Google Scholar

[9]

J. Cai, K. S. Tan, C. Weng and Y. Zhang, Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43 (2008), 185-196.  doi: 10.1016/j.insmatheco.2008.05.011.  Google Scholar

[10]

J. Cai, C. Lemieux and F. Liu, Optimal reinsurance from the perspectives of both an insurer and a reinsurer, Astin Bulletin, 46 (2016), 815-849.  doi: 10.1017/asb.2015.23.  Google Scholar

[11]

J. Cai and C. Weng, Optimal reinsurance with expectile,, Scandinavian Actuarial Journal, (2016), 624–645. doi: 10.1080/03461238.2014.994025.  Google Scholar

[12]

K. C. Cheung, Optimal reinsurance revisited - a geometric approach, Astin Bulletin, 40 (2010), 221-239.  doi: 10.2143/AST.40.1.2049226.  Google Scholar

[13]

K. C. Cheung and A. Lo, Characterizations of optiaml reinsurance treaties: A cost-benefit approach, Scandinavian Actuarial Journal, 2017 (2017), 1-28.  doi: 10.1080/03461238.2015.1054303.  Google Scholar

[14]

K. C. Cheung and W. Wang, Optimal Reinsurance from the perspectives of both insurers and reinsurers under general distortion risk measures,, SSRN Electronic Journa, (2017), 31pp. doi: 10.2139/ssrn.3048626.  Google Scholar

[15]

Y. Chi and K. S. Tan, Optimal reinsurance with general premium principles, Insurance: Mathematics and Economics, 52 (2013), 180-189.  doi: 10.1016/j.insmatheco.2012.12.001.  Google Scholar

[16]

Y. Chi and K. S. Tan, Optimal reinsurance under VaR and CVaR risk measures: A simplified approach, Astin Bulletin, 41 (2011), 487-509.   Google Scholar

[17]

Y. Chi, Reinsurance arrangements minimizing the risk-adjusted value of an insurer's liability, Astin Bulletin, 42 (2012), 529-557.   Google Scholar

[18]

Y. Chi, Optimal reinsurance under variance related premium principles, Insurance: Mathematics and Economics, 51 (2012), 310-321.  doi: 10.1016/j.insmatheco.2012.05.005.  Google Scholar

[19]

Y. Chi and C. Weng, Optimal reinsurance subject to Vajda condition, Insurance: Mathematics and Economics, 53 (2013), 179-189.  doi: 10.1016/j.insmatheco.2013.05.002.  Google Scholar

[20]

J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Theory, Insurance: Mathematics and Economics, 31 (2002), 3-33.  doi: 10.1016/S0167-6687(02)00134-8.  Google Scholar

[21]

G. Huberman, D. Mayers and C. W. Smith, Optimal insurance policy indemnity schedules, Bell Journal of Economics, 14 (1983), 415-426.  doi: 10.2307/3003643.  Google Scholar

[22]

W. Jiang, H. Hong and J. Ren, On Pareto-optimal reinsurance with constraints under distortion risk measures, European Actuarial Journal, 8 (2018), 215-243.  doi: 10.1007/s13385-017-0163-1.  Google Scholar

[23]

Z. Y. Lu, L. P. Liu, Q. J. Shen and L. L. Meng, Optimal reinsurance under VaR and CTE risk measures when ceded loss function is concave, Communications in Statistics Theory and Methods, 43 (2014), 3223-3247.  doi: 10.1080/03610926.2012.716136.  Google Scholar

[24]

Z. Y. Lu, L. L. Meng, Y. Wang and Q. Shen, Optimal reinsurance under VaR and TVaR risk measures in the presence of reinsurer's risk limit, Insurance: Mathematics and Economics, 68 (2016), 92-100.  doi: 10.1016/j.insmatheco.2016.03.001.  Google Scholar

[25]

A. Lo and Z. Tang, Pareto-optimal reinsurance policies in the presence of individual risk constraints, Annals of Operations Research, 274 (2019), 395-423.  doi: 10.1007/s10479-018-2820-4.  Google Scholar

[26] E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Space, Princeton University Press, Princeton, 2005.   Google Scholar
[27]

K. S. Tan, C. Weng and Y. Zhang, VaR and CTE criteria for optimal quota-share and stop-loss reinsurance, North American Actuarial Journal, 13 (2009), 459-482.  doi: 10.1080/10920277.2009.10597569.  Google Scholar

[28]

S. Vajda, Minimum variance reinsurance, Astin Bulletin, 2 (1962), 257-260.  doi: 10.1017/S0515036100009995.  Google Scholar

[29]

W. Wang and X. Peng, Reinsurer's optimal reinsurance strategy with upper and lower premium constraint under distortion risk measures, Journal of Computational and Applied Mathematics, 315 (2017), 142-160.  doi: 10.1016/j.cam.2016.10.017.  Google Scholar

[30]

V. R. Young, Optimal insurance under Wang's premium principle, Insurance: Mathematics and Economics, 25 (1999), 109-122.  doi: 10.1016/S0167-6687(99)00012-8.  Google Scholar

[31]

Y. Zheng and W. Cui, Optimal reinsurance with premium constraint under distortion risk measures, Insurance: Mathematics and Economics, 59 (2014), 109-120.  doi: 10.1016/j.insmatheco.2014.08.010.  Google Scholar

[32]

Y. Zheng, W. Cui and J. Yang, Optimal reinsurance under distortion risk measures and expected value premium principle for reinsurer, Journal of Systems Science and Complexity, 28 (2015), 122-143.  doi: 10.1007/s11424-014-2095-z.  Google Scholar

show all references

References:
[1]

A. V. Asimit, A. M. Badescu and T. Verdonck, Optimal risk transfer under quantile-based risk measures, Insurance: Mathematics and Economics, 53 (2013), 252-265.  doi: 10.1016/j.insmatheco.2013.05.005.  Google Scholar

[2]

A. V. Asimit, A. M. Badescu and K. C. Cheung, Optimal reinsurance in the presence of counterparty default risk, Insurance: Mathematics and Economics, 53 (2013), 690-697.  doi: 10.1016/j.insmatheco.2013.09.012.  Google Scholar

[3]

H. Assa, On optimal reinsurance policy with distortion risk measures and premiums, Insurance Mathematics and Economics, 61 (2015), 70-75.  doi: 10.1016/j.insmatheco.2014.11.007.  Google Scholar

[4]

K. J. Arrow, Uncertainty and the welfare economic of medical care, Uncertainty in Economics, (1978), 347–375. doi: 10.1016/B978-0-12-214850-7.50028-0.  Google Scholar

[5]

C. Bernard and M. Ludkovski, Impact of counterparty risk on the reinsurance market, North American Actuarial Journal, 16 (2012), 87-111.  doi: 10.1080/10920277.2012.10590634.  Google Scholar

[6]

K. Borch, An attempt to determine the optimal amount of stop loss reinsurance, Transactions of the 16th International Congress of Actuaries, 1 (1960), 597-610.   Google Scholar

[7]

J. Cai, C. Lemieux and F. Liu, Optimal reinsurance with regulatory initial capital and default risk, Insurance: Mathematics and Economics, 57 (2014), 13-24.  doi: 10.1016/j.insmatheco.2014.04.006.  Google Scholar

[8]

J. Cai and K. S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures, Astin Bulletin, 37 (2007), 93-112.  doi: 10.1017/S0515036100014756.  Google Scholar

[9]

J. Cai, K. S. Tan, C. Weng and Y. Zhang, Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43 (2008), 185-196.  doi: 10.1016/j.insmatheco.2008.05.011.  Google Scholar

[10]

J. Cai, C. Lemieux and F. Liu, Optimal reinsurance from the perspectives of both an insurer and a reinsurer, Astin Bulletin, 46 (2016), 815-849.  doi: 10.1017/asb.2015.23.  Google Scholar

[11]

J. Cai and C. Weng, Optimal reinsurance with expectile,, Scandinavian Actuarial Journal, (2016), 624–645. doi: 10.1080/03461238.2014.994025.  Google Scholar

[12]

K. C. Cheung, Optimal reinsurance revisited - a geometric approach, Astin Bulletin, 40 (2010), 221-239.  doi: 10.2143/AST.40.1.2049226.  Google Scholar

[13]

K. C. Cheung and A. Lo, Characterizations of optiaml reinsurance treaties: A cost-benefit approach, Scandinavian Actuarial Journal, 2017 (2017), 1-28.  doi: 10.1080/03461238.2015.1054303.  Google Scholar

[14]

K. C. Cheung and W. Wang, Optimal Reinsurance from the perspectives of both insurers and reinsurers under general distortion risk measures,, SSRN Electronic Journa, (2017), 31pp. doi: 10.2139/ssrn.3048626.  Google Scholar

[15]

Y. Chi and K. S. Tan, Optimal reinsurance with general premium principles, Insurance: Mathematics and Economics, 52 (2013), 180-189.  doi: 10.1016/j.insmatheco.2012.12.001.  Google Scholar

[16]

Y. Chi and K. S. Tan, Optimal reinsurance under VaR and CVaR risk measures: A simplified approach, Astin Bulletin, 41 (2011), 487-509.   Google Scholar

[17]

Y. Chi, Reinsurance arrangements minimizing the risk-adjusted value of an insurer's liability, Astin Bulletin, 42 (2012), 529-557.   Google Scholar

[18]

Y. Chi, Optimal reinsurance under variance related premium principles, Insurance: Mathematics and Economics, 51 (2012), 310-321.  doi: 10.1016/j.insmatheco.2012.05.005.  Google Scholar

[19]

Y. Chi and C. Weng, Optimal reinsurance subject to Vajda condition, Insurance: Mathematics and Economics, 53 (2013), 179-189.  doi: 10.1016/j.insmatheco.2013.05.002.  Google Scholar

[20]

J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Theory, Insurance: Mathematics and Economics, 31 (2002), 3-33.  doi: 10.1016/S0167-6687(02)00134-8.  Google Scholar

[21]

G. Huberman, D. Mayers and C. W. Smith, Optimal insurance policy indemnity schedules, Bell Journal of Economics, 14 (1983), 415-426.  doi: 10.2307/3003643.  Google Scholar

[22]

W. Jiang, H. Hong and J. Ren, On Pareto-optimal reinsurance with constraints under distortion risk measures, European Actuarial Journal, 8 (2018), 215-243.  doi: 10.1007/s13385-017-0163-1.  Google Scholar

[23]

Z. Y. Lu, L. P. Liu, Q. J. Shen and L. L. Meng, Optimal reinsurance under VaR and CTE risk measures when ceded loss function is concave, Communications in Statistics Theory and Methods, 43 (2014), 3223-3247.  doi: 10.1080/03610926.2012.716136.  Google Scholar

[24]

Z. Y. Lu, L. L. Meng, Y. Wang and Q. Shen, Optimal reinsurance under VaR and TVaR risk measures in the presence of reinsurer's risk limit, Insurance: Mathematics and Economics, 68 (2016), 92-100.  doi: 10.1016/j.insmatheco.2016.03.001.  Google Scholar

[25]

A. Lo and Z. Tang, Pareto-optimal reinsurance policies in the presence of individual risk constraints, Annals of Operations Research, 274 (2019), 395-423.  doi: 10.1007/s10479-018-2820-4.  Google Scholar

[26] E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Space, Princeton University Press, Princeton, 2005.   Google Scholar
[27]

K. S. Tan, C. Weng and Y. Zhang, VaR and CTE criteria for optimal quota-share and stop-loss reinsurance, North American Actuarial Journal, 13 (2009), 459-482.  doi: 10.1080/10920277.2009.10597569.  Google Scholar

[28]

S. Vajda, Minimum variance reinsurance, Astin Bulletin, 2 (1962), 257-260.  doi: 10.1017/S0515036100009995.  Google Scholar

[29]

W. Wang and X. Peng, Reinsurer's optimal reinsurance strategy with upper and lower premium constraint under distortion risk measures, Journal of Computational and Applied Mathematics, 315 (2017), 142-160.  doi: 10.1016/j.cam.2016.10.017.  Google Scholar

[30]

V. R. Young, Optimal insurance under Wang's premium principle, Insurance: Mathematics and Economics, 25 (1999), 109-122.  doi: 10.1016/S0167-6687(99)00012-8.  Google Scholar

[31]

Y. Zheng and W. Cui, Optimal reinsurance with premium constraint under distortion risk measures, Insurance: Mathematics and Economics, 59 (2014), 109-120.  doi: 10.1016/j.insmatheco.2014.08.010.  Google Scholar

[32]

Y. Zheng, W. Cui and J. Yang, Optimal reinsurance under distortion risk measures and expected value premium principle for reinsurer, Journal of Systems Science and Complexity, 28 (2015), 122-143.  doi: 10.1007/s11424-014-2095-z.  Google Scholar

[1]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023
[2]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041
[3]

Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021024
[4]

Samira Shahsavari, Saeed Ketabchi. The proximal methods for solving absolute value equation. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 449-460. doi: 10.3934/naco.2020037
[5]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[6]

Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025
[7]

Kuei-Hu Chang. A novel risk ranking method based on the single valued neutrosophic set. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021065
[8]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[9]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151
[10]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021094
[11]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019
[12]

Yingxu Tian, Junyi Guo, Zhongyang Sun. Optimal mean-variance reinsurance in a financial market with stochastic rate of return. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1887-1912. doi: 10.3934/jimo.2020051
[13]

Kai Kang, Taotao Lu, Jing Zhang. Financing strategy selection and coordination considering risk aversion in a capital-constrained supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021042
[14]

Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020179
[15]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398
[16]

Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062
[17]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003
[18]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206
[19]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055
[20]

Mehmet Duran Toksari, Emel Kizilkaya Aydogan, Berrin Atalay, Saziye Sari. Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021044

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (84)
  • HTML views (300)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences