Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T17:20:56.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal dividend and reinsurance in the presence of two reinsurers

Part of: Applications Markov processes Special processes

Published online by Cambridge University Press:  21 June 2016

Mi Chen  and
Kam Chuen Yuen
Mi Chen*
Affiliation:
Fujian Normal University
Kam Chuen Yuen*
Affiliation:
The University of Hong Kong
*
* Postal address: School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350108, China.
** Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper the optimal dividend (subject to transaction costs) and reinsurance (with two reinsurers) problem is studied in the limit diffusion setting. It is assumed that transaction costs and taxes are required when dividends occur, and that the premiums charged by two reinsurers are calculated according to the exponential premium principle with different parameters, which makes the stochastic control problem nonlinear. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. The problem is formulated as a mixed classical-impulse stochastic control problem. Explicit expressions for the value function and the corresponding optimal strategy are obtained. Finally, a numerical example is presented to illustrate the impact of the parameters associated with the two reinsurers' premium principle on the optimal reinsurance strategy.

Keywords

Dividendreinsurancetransaction costsexponential premium principleoptimal reinsurance with two reinsurers

MSC classification

Primary: 60J65: Brownian motion 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 554 - 571
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asimit, A. V., Badescu, A. M. and Verdonck, T. (2013).Optimal risk transfer under quantile-based risk measurers.Insurance Math. Econom. 53, 252–265.Google Scholar
Bai, L., Guo, J. and Zhang, H. (2010).Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes.Quant. Finance 10, 1163–1172.Google Scholar
Cadenillas, A., Choulli, T., Taksar, M. and Zhang, L. (2006).Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm.Math. Finance 16, 181–202.Google Scholar
Chen, M., Peng, X. and Guo, J. (2013).Optimal dividend problem with a nonlinear regular-singular stochastic control.Insurance Math. Econom. 52, 448–456.Google Scholar
Chi, Y. and Meng, H. (2014).Optimal reinsurance arrangements in the presence of two reinsurers.Scand. Actuarial J. 2014, 424–438.Google Scholar
Choulli, T., Taksar, M. and Zhou, X. Y. (2001).Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction.Quant. Finance 1, 573–596.Google Scholar
De Finetti, B. (1957).Su un'impostazione alternativa della teoria collettiva del rischio. In Transactions of the XVth International Congress of Actuaries, Vol. 2, pp.433–443.Google Scholar
Fleming, W. H. and Soner, H. M. (2006).Controlled Markov Processes and Viscosity Solutions (Stoch. Modelling Appl. Prob.25), 2nd edn.Springer, New York.Google Scholar
Guan, H. and Liang, Z. (2014).Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs.Insurance Math. Econom. 54, 109–122.Google Scholar
Guo, X., Liu, J. and Zhou, X. Y. (2004).A constrained non-linear regular-singular stochastic control problem, with applications.Stoch. Process. Appl. 109, 167–187.Google Scholar
He, L. and Liang, Z. (2008).Optimal financing and dividend control of the insurance company with proportional reinsurance policy.Insurance Math. Econom. 42, 976–983.Google Scholar
He, L. and Liang, Z. (2009).Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs.Insurance Math. Econom. 44, 88–94.Google Scholar
Kaluszka, M. (2001).Optimal reinsurance under mean-variance premium principles.Insurance Math. Econom. 28, 61–67.CrossRefGoogle Scholar
Kaluszka, M. (2005).Optimal reinsurance under convex principles of premium calculation.Insurance Math. Econom. 36, 375–398.Google Scholar
Løkka, A. and Zervos, M. (2008).Optimal dividend and issuance of equity policies in the presence of proportional costs.Insurance Math. Econom 42, 954–961.CrossRefGoogle Scholar
Meng, H. and Siu, T. K. (2011a).On optimal reinsurance, dividend and reinvestment strategies.Econom. Modelling 28, 211–218.Google Scholar
Meng, H. and Siu, T. K. (2011b).Optimal mixed impulse-equity insurance control problem with reinsurance.SIAM J. Control Optimization 49, 254–279.CrossRefGoogle Scholar
Moore, K. S. and Young, V. R. (2003).Pricing equity-linked pure endowments via the principle of equivalent utility.Insurance Math. Econom. 33, 497–516.Google Scholar
Musiela, M. and Zariphopoulou, T. (2004).An example of indifference prices under exponential preferences.Finance Stoch. 8, 229–239.Google Scholar
Peng, X., Chen, M. and Guo, J. (2012).Optimal dividend and equity issuance problem with proportional and fixed transaction costs.Insurance Math. Econom. 51, 576–585.CrossRefGoogle Scholar
Scheer, N. and Schmidli, H. (2011).Optimal dividend strategies in a Cramer–Lundberg model with capital injections and administration costs.Europ. Actuarial J. 1, 57–92.CrossRefGoogle Scholar
Schmidli, H. (2002).On minimizing the ruin probability by investment and reinsurance.Ann. Appl. Prob. 12, 890–907.Google Scholar
Yao, D., Yang, H. and Wang, R. (2014).Optimal risk and dividend control problem with fixed costs and salvage value: variance premium principle.Econom. Modelling 37, 53–64.Google Scholar
Young, V. R. (1999).Optimal insurance under Wang's premium principle.Insurance Math. Econom. 25, 109–122.Google Scholar
Young, V. R. (2003).Equity-indexed life insurance: pricing and reserving using the principle of equivalent utility.N. Amer. Actuarial J. 7, 68–86.CrossRefGoogle Scholar
Young, V. R. and Zariphopoulou, T. (2002).Pricing dynamic insurance risks using the principle of equivalent utility.Scand. Actuarial J. 2002, 246–279.CrossRefGoogle Scholar
Zhou, M. and Yuen, K. C. (2012).Optimal reinsurance and dividend for a diffusion model with capital injection: variance premium principle.Econom. Modelling 29, 198–207.Google Scholar