Skip to main content
SpringerLink
Log in

Minimisation of penalty payments by investments and reinsurance

  • Original Research Paper
  • Published:
European Actuarial Journal Aims and scope Submit manuscript

Abstract

This paper considers an optimal investment and reinsurance problem for an insurance company, where the surplus follows a linear diffusion. Contrary to classical models the insurer can continue doing business even if the surplus becomes negative, but penalty payments occur depending on the level of the current surplus. The insurer can invest in n risky assets and reduce the insurance risk either by excess of loss or by proportional reinsurance. The aim is to find an optimal investment and reinsurance strategy which minimises the penalty payments. We consider various penalty functions and derive closed form solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Albrecher H, Gerber HU, Shiu ESW (2011) The optimal dividend barrier in the Gamma-Omega model. Eur Actuar J 1:43–55

    Article  MathSciNet  MATH  Google Scholar 

  2. Asmussen S, Højgaard B, Taksar MI (2000) Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation. Financ Stoch 4:299–324

    Article  MathSciNet  MATH  Google Scholar 

  3. Azcue P, Muler N (2005) Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Math Financ 15:261–308

    Article  MathSciNet  MATH  Google Scholar 

  4. Azcue P, Muler N (2010) Optimal investment policy and dividend payment strategy in an insurance company. Ann Appl Probab 20:1253–1302

    Article  MathSciNet  MATH  Google Scholar 

  5. Azcue P, Muler N (2014) Stochastic optimization in insurance. A dynamic programming approach. Springer, New York

    MATH  Google Scholar 

  6. Bai L, Guo J (2008) Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insur Math Econ 42:968–975

    Article  MathSciNet  MATH  Google Scholar 

  7. Eisenberg J (2010) On optimal control of capital injections by reinsurance and investments. Blätter der DGVFM 31:329–345

    Article  MathSciNet  MATH  Google Scholar 

  8. Eisenberg J, Schmidli H (2011) Optimal control of capital injections by reinsurance with riskless rate of interest. J Appl Probab 48:733–748

    Article  MathSciNet  MATH  Google Scholar 

  9. Højgaard B, Taksar MI (1997) Optimal proportional reinsurance policies for diffusion models. Scand Actuar J 2:166–180

    MathSciNet  MATH  Google Scholar 

  10. Højgaard B, Taksar MI (1998) Optimal proportional reinsurance policies for diffusion models with transaction costs. Insur Math Econ 2:41–51

    Article  MathSciNet  MATH  Google Scholar 

  11. Højgaard B, Taksar MI (2004) Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quant Financ 4(3):315–327

    Article  MathSciNet  Google Scholar 

  12. Markussen C, Taksar MI (2003) Optimal dynamic reinsurance policies for large insurance portfolios. Financ Stoch 7:97–121

    Article  MathSciNet  MATH  Google Scholar 

  13. Paulsen J, Gjessing HK (1997) Ruin theory with stochastic return on investments. Adv Appl Probab 29(4):965–985

    Article  MathSciNet  MATH  Google Scholar 

  14. Schmidli H (1994) Corrected diffusion approximations for a risk process with the possibility of borrowing and investment. Schweiz Ver der Versicher Mitt 94:71–81

    MathSciNet  MATH  Google Scholar 

  15. Schmidli H (2002) On minimising the ruin probability by investment and reinsurance. Ann Appl Probab 12:890–907

    Article  MathSciNet  MATH  Google Scholar 

  16. Schmidli H (2008) Stochastic control in insurance. Springer, London

    MATH  Google Scholar 

  17. Shreve SE, Lehocsky JP, Gaver DP (1984) Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J Control Optim 22:55–75

    Article  MathSciNet  MATH  Google Scholar 

  18. Taksar MI (2000) Optimal risk and dividend distribution control models for an insurance company. Math Methods Oper Res 51:1–42

    Article  MathSciNet  MATH  Google Scholar 

  19. Taksar MI, Hunderup C (2007) Influence of bankruptcy value on optimal risk control for diffusion models with proportional reinsurance. Insur Math Econ 40:311–321

    Article  MathSciNet  MATH  Google Scholar 

  20. Vierkötter M, Schmidli H (2015) On optimal dividends with linear and exponential penalty payments. University of Cologne (Preprint)

  21. Zhang XL, Zhang KC, Yu XJ (2009) Optimal proportional reinsurance and investment with transaction costs, I: maximizing the terminal wealth. Insur Math Econ 44:473–478

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Matthias Vierkötter

Authors
  1. Matthias Vierkötter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Vierkötter.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vierkötter, M. Minimisation of penalty payments by investments and reinsurance. Eur. Actuar. J. 6, 233–255 (2016). https://doi.org/10.1007/s13385-016-0128-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13385-016-0128-9

Keywords

Mathematics Subject Classification

Search

Navigation