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September  2021, 17(5): 2971-2987. 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  September 2021 Early access  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, 2021, 17 (5) : 2971-2987. 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

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