# Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation

## Author Info

• Bjarne Højgaard

()
(Department of Mathematics, Aalborg University, Fr. Bajersv. 7E, DK-9220 Aalborg Ø, Denmark)

• Søren Asmussen

()
(Department of Mathematical Statistics, University of Lund, Box 118, S-221 00 Lund, Sweden)

• Michael Taksar

()
(Department of AMS SUNY - Stony Brook, NY, USA Manuscript)

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## Abstract

We consider a model of a financial corporation which has to find an optimal policy balancing its risk and expected profits. The example treated in this paper is related to an insurance company with the risk control method known in the industry as excess-of-loss reinsurance. Under this scheme the insurance company divert part of its premium stream to another company in exchange of an obligation to pick up that amount of each claim which exceeds a certain level a. This reduces the risk but it also reduces the potential profit. The objective is to make a dynamic choice of a and find the dividend distribution policy, which maximizes the cumulative expected discounted dividend pay-outs. We use diffusion approximation for this optimal control problem, where two situations are considered: (a) The rate of dividend pay-out are unrestricted and in this case mathematically the problem becomes a mixed singular-regular control problem for diffusion processes. Its analytical part is related to a free boundary (Stephan) problem for a linear second order differential equation. The optimal policy prescribes to reinsure using a certain retention level (depending on the reserve) and pay no dividends when the reserve is below some critical level $x_1$ and to pay out everything that exceeds $x_1$. Reinsurance will stop at a level $x_0\leq x_1$ depending on the claim size distribution. (b) The rate of dividend pay-out is bounded by some positive constant \$M

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## Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 4 (2000)
Issue (Month): 3 ()
Pages: 299-324

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Handle: RePEc:spr:finsto:v:4:y:2000:i:3:p:299-324

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## References

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## Citations

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Cited by:
1. He, Lin & Liang, Zongxia, 2008. "Optimal financing and dividend control of the insurance company with proportional reinsurance policy," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 976-983, June.
2. Jin, Zhuo & Yin, G. & Wu, Fuke, 2013. "Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 733-746.
3. Løkka, Arne & Zervos, Mihail, 2008. "Optimal dividend and issuance of equity policies in the presence of proportional costs," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 954-961, June.
4. Zhuo Jin & George Yin & Chao Zhu, 2011. "Numerical Solutions of Optimal Risk Control and Dividend Optimization Policies under A Generalized Singular Control Formulation," Papers 1111.2584, arXiv.org.
5. Abel Elizalde, 2007. "From Basel I To Basel Ii: An Analysis Of The Three Pillars," Working Papers, CEMFI wp2007_0704, CEMFI.
6. Jean-Paul Décamps & Stéphane Villeneuve, 2007. "Optimal dividend policy and growth option," Finance and Stochastics, Springer, Springer, vol. 11(1), pages 3-27, January.
7. Zeng, Yan & Li, Zhongfei & Lai, Yongzeng, 2013. "Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 498-507.
8. Chen, Mi & Peng, Xiaofan & Guo, Junyi, 2013. "Optimal dividend problem with a nonlinear regular-singular stochastic control," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 448-456.
9. Bai, Lihua & Cai, Jun & Zhou, Ming, 2013. "Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 664-670.
10. Irmina Czarna & Zbigniew Palmowski, 2010. "Dividend problem with Parisian delay for a spectrally negative L\'evy risk process," Papers 1004.3310, arXiv.org, revised Oct 2011.
11. Guan, Huiqi & Liang, Zongxia, 2014. "Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 109-122.
12. Peng, Xiaofan & Chen, Mi & Guo, Junyi, 2012. "Optimal dividend and equity issuance problem with proportional and fixed transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 576-585.
13. Guo, Xin & Liu, Jun & Zhou, Xun Yu, 2004. "A constrained non-linear regular-singular stochastic control problem, with applications," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 109(2), pages 167-187, February.
14. Zhou, Ming & Yuen, Kam C., 2012. "Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle," Economic Modelling, Elsevier, Elsevier, vol. 29(2), pages 198-207.
15. Liang, Zhibin & Young, Virginia R., 2012. "Dividends and reinsurance under a penalty for ruin," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 437-445.
16. Sotomayor, Luz R. & Cadenillas, Abel, 2011. "Classical and singular stochastic control for the optimal dividend policy when there is regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 344-354, May.
17. Gu, Ailing & Guo, Xianping & Li, Zhongfei & Zeng, Yan, 2012. "Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 674-684.
18. Zhu, Jinxia & Chen, Feng, 2013. "Dividend optimization for regime-switching general diffusions," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 439-456.
19. Liu, Wei & Hu, Yijun, 2014. "Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy," Statistics & Probability Letters, Elsevier, Elsevier, vol. 84(C), pages 121-130.
20. Yao, Dingjun & Yang, Hailiang & Wang, Rongming, 2011. "Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs," European Journal of Operational Research, Elsevier, Elsevier, vol. 211(3), pages 568-576, June.
21. Irgens, Christian & Paulsen, Jostein, 2004. "Optimal control of risk exposure, reinsurance and investments for insurance portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 21-51, August.
22. Zhao, Hui & Rong, Ximin & Zhao, Yonggan, 2013. "Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 504-514.
23. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2011. "On Gerber-Shiu functions and optimal dividend distribution for a L\'{e}vy risk-process in the presence of a penalty function," Papers 1110.4965, arXiv.org, revised Mar 2014.

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