﻿ 边际赔偿函数和违约风险下的最优再保险 The Optimal Reinsurance under the Marginal Indemnification Function and the Default Risk

Statistics and Application
Vol.06 No.02(2017), Article ID:21022,10 pages
10.12677/SA.2017.62017

The Optimal Reinsurance under the Marginal Indemnification Function and the Default Risk

Junhong Du*, Lijun Wu#

College of Mathematics and System Sciences, Xinjiang University, Urumqi Xinjiang

*第一作者。

#通讯作者。

Received: May 18th, 2017; accepted: Jun. 15th, 2017; published: Jun. 19th, 2017

ABSTRACT

In this paper, we consider the default risk of reinsurer. Firstly, we use the distortion risk measure and distortion premium principle to establish the total risk model with default risk. Secondly, by the relationship between the Marginal Indemnification Function (MIF) and the ceded loss function, we build MIF reinsurance optimization model equivalent to total risk model. Then, the optimal MIF function is obtained by solving the MIF reinsurance optimization model. Furthermore, the optimal ceded loss function is obtained. Finally, we apply this method to study the optimal loss function by the VaR risk measure and Wang’s premium principle.

Keywords:Optimal Reinsurance, Default Risk, Marginal Indemnification Function, Distortion Risk Measure, Reinsurance Premium, VaR Risk Measure, Wang’s Premium Principle

1. 引言

2. 模型的建立

2.1. 失真风险度量和失真保费原理

(1) 同单调可加性：，对任意两个同单调的随机变量

(2) 平移不变性：，对任意的和随机变量

(3) 单调性：，当随机变量

(1)

2.2. 初始资本和违约风险的再保险模型

2.3. 分出损失函数集的建立

(i)是非减函数。

(ii)

2.4. 最优再保险模型

(2)

3. 失真风险度量下的最优再保险

(3)

。 (4)

。 (5)

。 (6)

(7)

(8)

。 (9)

4. 风险度量VaR和Wang’s保费原理的实例分析

(a) 若；则

(i) 如果

(ii) 如果，复合函数表示为：

(b) 若；记

；当

(i) 如果，则

(ii) 如果，则

(iii) 如果，则

5. 结论

The Optimal Reinsurance under the Marginal Indemnification Function and the Default Risk[J]. 统计学与应用, 2017, 06(02): 146-155. http://dx.doi.org/10.12677/SA.2017.62017

1. 1. Borch, K. (1960) An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance. Transactions of the 16th International Congress of Actuaries, 1, 597-610.

2. 2. Arrow, K.J. (1963) Uncertainty and the Welfare Economics of Medical Care. American Economic Review, 53, 941-973.

3. 3. Young, V.R. (1999) Optimal Insurance under Wang’s Premium Principle. Insurance Mathe-matics and Economics, 25, 109-122. https://doi.org/10.1016/S0167-6687(99)00012-8

4. 4. Kaluszka, M. (2001) Optimal Rein-surance under Mean-Variance Premium Principles. Insurance Mathematics and Economics, 28, 61-67. https://doi.org/10.1016/S0167-6687(00)00066-4

5. 5. Cai, J. and Tan, K.S. (2007) Optimal Retention for a Stop-Loss Reinsur-ance under the VaR and CTE Risk Measures. Astin Bulletin, 37, 93-112. https://doi.org/10.2143/AST.37.1.2020800

6. 6. Cai, J., Tan, K.S., Weng, C., et al. (2008) Optimal Reinsurance under VaR and CTE Risk Measures. Insurance: Mathematics and Economics, 43, 185-196. https://doi.org/10.1016/j.insmatheco.2008.05.011

7. 7. Cheung, K.C. (2010) Optimal Reinsurance Revisited—A Geometric Approach. ASTIN Bulletin: The Journal of the IAA, 40, 221-239. https://doi.org/10.2143/AST.40.1.2049226

8. 8. Chi, Y. and Tan, K.S. (2011) Optimal Reinsurance under VaR and CVaR Risk Measures: A Simplified Approach. Astin Bulletin, 41, 487-509.

9. 9. Chi, Y. and Tan, K.S. (2013) Optimal Reinsurance with General Premium Principles. Insurance: Mathematics and Economics, 52, 180-189. https://doi.org/10.1016/j.insmatheco.2012.12.001

10. 10. Cui, W., Yang, J. and Wu, L. (2013) Optimal Reinsurance Minimizing the Distortion Risk Measure under General Reinsurance Premium Principles. Insurance Mathematics and Economics, 53, 74-85. https://doi.org/10.1016/j.insmatheco.2013.03.007

11. 11. Zheng, Y., Cui, W. and Yang, J. (2015) Optimal Reinsurance under Distortion Risk Measures and Expected Value Premium Principle for Reinsurer. Journal of Systems Science and Complexity, 28, 122-143. https://doi.org/10.1007/s11424-014-2095-z

12. 12. Cai, J., Lemieux, C., Liu, F., et al. (2014) Optimal Reinsurance with Regulatory Initial Capital and Default Risk. Insurance Mathematics and Economics, 57, 13-24. https://doi.org/10.1016/j.insmatheco.2014.04.006

13. 13. Assa, H. (2014) On Optimal Reinsurance Policy with Distortion Risk Measures and Premiums. SSRN Electronic Journal, 61, 70-75. https://doi.org/10.2139/ssrn.2448678

14. 14. Zhuang, S.C., Weng, C., Tan, K.S., et al. (2016) Marginal Indemnification Function Formulation for Optimal Reinsurance. Insurance Mathematics and Economics, 67, 65-76. https://doi.org/10.1016/j.insmatheco.2015.12.003

15. 15. Balas, A., Balbs, B., Balbs, R., et al. (2015) Op-timal Reinsurance under Risk and Uncertainty. Insurance Mathematics and Economics, 60, 61-74. https://doi.org/10.1016/j.insmatheco.2014.11.001