Skip to main content
SpringerLink
Log in

Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

The present paper studies time-consistent solutions to an investment-reinsurance problem under a mean-variance framework. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer jointly. The claim process of the insurer is governed by a Brownian motion with a drift. A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle. Both the insurer and the reinsurer are assumed to invest in a risky asset, which is distinct for each other and driven by a constant elasticity of variance model. The optimal decision is formulated on a weighted sum of the insurer’s and the reinsurer’s surplus processes. Upon a verification theorem, which is established with a formal proof for a more general problem, explicit solutions are obtained for the proposed investment-reinsurance model. Moreover, numerous mathematical analysis and numerical examples are provided to demonstrate those derived results as well as the economic implications behind.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bai L H, Guo J Y. Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance Math Econom, 2008, 42: 968–975

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai L H, Guo J Y. Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection. Sci China Math, 2010, 53: 1787–1804

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai L H, Zhang H Y. Dynamic mean-variance problem with constrained risk control for the insurers. Math Methods Oper Res, 2008, 68: 181–205

    Article  MathSciNet  MATH  Google Scholar 

  4. Bäuerle N. Benchmark and mean-variance problems for insurers. Math Methods Oper Res, 2005, 62: 159–165

    Article  MathSciNet  MATH  Google Scholar 

  5. Björk T, Murgoci A. A general theory of Markovian time-inconsistent stochastic control problems. Working paper, http://papers.ssrn.com/sol3/papers.cfm?abstract id=1694759, 2010

    MATH  Google Scholar 

  6. Björk T, Murgoci A, Zhou X Y. Mean variance portfolio optimization with state dependent risk aversion. Math Finance, 2014, 24: 1–24

    Article  MathSciNet  MATH  Google Scholar 

  7. Black F, Scholes M. The pricing of options and corporate liabilities. J Polit Econ, 1973, 81: 637–654

    Article  MathSciNet  MATH  Google Scholar 

  8. Cai J, Fang Y, Li Z, et al. Optimal reciprocal reinsurance treaties under the joint survival probability and the joint profitable probability. J Risk Insur, 2013, 80: 145–168

    Article  Google Scholar 

  9. Cai J, Tan K S, Weng C, et al. Optimal reinsurance under VaR and CTE risk measures. Insurance Math Econom, 2008, 43: 185–196

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheung K C, Sung C J, Yam S C P. Risk-minimizing reinsurance protection for multivariate risks. J Risk Insur, 2014, 81: 219–236

    Article  Google Scholar 

  11. Cox J. Notes on Option Pricing I: Constant Elasticity of Diffusions. Stanford: Stanford University, 1975

    Google Scholar 

  12. Davydov D, Linetsky V. The valuation and hedging of barrier and lookback option under the CEV process. Manag Sci, 2001, 47: 949–965

    Article  MATH  Google Scholar 

  13. Delong L, Gerrard R. Mean-variance portfolio selection for a nonlife insurance company. Math Methods Oper Res, 2007, 66: 339–367

    Article  MathSciNet  MATH  Google Scholar 

  14. Dias J C, Nunes J P. Pricing real options under the constant elasticity of variance diffusion. J Futur Mark, 2011, 3: 230–250

    Article  Google Scholar 

  15. Dimitrova D S, Kaishev V K. Optimal joint survival reinsurance: An efficient frontier approach. Insurance Math Econom, 2010, 47: 27–35

    Article  MathSciNet  MATH  Google Scholar 

  16. Dufresne D. The integrated square root process. Working Paper, Melbourne: University of Melbourne, 2001

    MATH  Google Scholar 

  17. Ekeland I, Lazrak A. Being serious about non-commitmetn: Subgame perfect equilibrium in continuous time. Working paper, http://arxiv.org/abs/math/0604264, 2006

    Google Scholar 

  18. Gao J W. Optimal investment strategy for annuity contracts under the constant clasticity of variance (CEV) model. Insurance Math Econom, 2009, 45: 9–18

    Article  MathSciNet  MATH  Google Scholar 

  19. Geman H, Shih Y F. Modeling commodity prices under the CEV model. J Altern Invest, 2009, 11: 65–84

    Article  Google Scholar 

  20. Grandell J. Aspects of Risk Theory. New York: Springer, 1991

    Book  MATH  Google Scholar 

  21. Gu A L, Guo X P, Li Z F, et al. Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model. Insurance Math Econom, 2012, 51: 674–684

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo Z J, Duan B X. Dynamic mean-variance portfolio selection in market with jump-diffusion models. Optimization, 2015, 64: 663–674

    MathSciNet  MATH  Google Scholar 

  23. Hald M, Schmidli H. On the maximisation of the adjustment coefficient under proportional reinsurance. Astin Bull, 2004, 34: 75–83

    Article  MathSciNet  MATH  Google Scholar 

  24. Hu Y, Jin H Q, Zhou X Y. Time inconsistent stochastic linear-quadratic control. SIAM J Control Optim, 2012, 50: 1548–1572.

    Article  MathSciNet  MATH  Google Scholar 

  25. Jeanblanc M, Yor M, Chesney M. Mathematical Methods for Financial Markets. London: Springer-Verlag, 2009

    Book  MATH  Google Scholar 

  26. Kronborg M, Steffensen M. Inconsistent investment and consumption problems. Appl Math Optim, 2015, 71: 473–515

    Article  MathSciNet  MATH  Google Scholar 

  27. Kryger E M, Steffensen M. Some solvable portfolio problems with quadratic and collective objectives. Working paper, http://papers.ssrn.com/sol3/papers.cfm?abstractid=1577265, 2010

  28. Li D P, Rong X M, Zhao H. Optimal reinsurance-investment problem for maximizing the product of the insurer’s and the reinsurer’s utilities under a CEV model. J Comput Appl Math, 2014, 255: 671–683

    Article  MathSciNet  MATH  Google Scholar 

  29. Li D P, Rong X M, Zhao H. Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model. J Comput Appl Math, 2015, 283: 142–162

    Article  MathSciNet  MATH  Google Scholar 

  30. Li Y W, Li Z F. Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion. Insurance Math Econom, 2013, 53: 86–97

    Article  MathSciNet  MATH  Google Scholar 

  31. Li Z F, Zeng Y, Lai Y Z. Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model. Insurance Math Econom, 2012, 51: 191–203

    Article  MathSciNet  MATH  Google Scholar 

  32. Lin X, Li Y F. Optimal reinsurance and investment for a jump diffusion risk process under the CEV model. N Am Actuar J, 2011, 15: 417–431

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu Y, Ma J. Optimal reinsurance/investment problems for general insurance models. Ann Appl Probab, 2009, 19: 1273–1685

    Article  MathSciNet  MATH  Google Scholar 

  34. Lo C F, Yuen P H, Hui C H. Constant elasticity of variance option pricing model with time-dependent parameters. Int J Theor Appl Financ, 2000, 3: 661–674

    Article  MathSciNet  MATH  Google Scholar 

  35. Schmidli H. On minimizing the ruin probability by investment and reinsurance. Ann Appl Probab, 2002, 12: 890–907

    Article  MathSciNet  MATH  Google Scholar 

  36. Shen Y, Zeng Y. Optimal investment C reinsurance with delay for mean-variance insurers: A maximum principle approach. Insurance Math Econom, 2014, 57: 1–12

    Article  MathSciNet  MATH  Google Scholar 

  37. Shen Y, Zeng Y. Optimal investment C reinsurance strategy for mean-variance insurers with square-root factor process. Insurance Math Econom, 2015, 62: 118–137

    Article  MathSciNet  MATH  Google Scholar 

  38. Shen Y, Zhang X, Siu T K. Mean-variance portfolio selection under a constant elasticity of variance model. Oper Res Lett, 2014, 42, 337–342

    Article  MathSciNet  Google Scholar 

  39. Widdicks M, Duck P W, Andricopoulos A D, et al. The Black-Scholes equation revisited: Symptotic expansions and singular perturbations. Math Financ, 2005, 15: 373–391

    Article  MATH  Google Scholar 

  40. Xu L, Wang R, Yao D. On maximizing the expected terminal utility by investment and reinsurance. J Ind Manag Optim, 2008, 4: 801–815

    Article  MathSciNet  MATH  Google Scholar 

  41. Yong J M. Time-inconsistent optimal control problems and the equilibrium HJB equation. Math Control Related Fields, 2012, 2: 271–329

    Article  MathSciNet  MATH  Google Scholar 

  42. Yong J M. Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J Control Optim, 2013, 51: 2809–2838

    Article  MathSciNet  MATH  Google Scholar 

  43. Yong J M. Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations: Time-Consistent Solutions. Providence: Amer Math Soc, 2015

    Google Scholar 

  44. Yong J M, Zhou X Y. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York: Springer-Verlag, 1999.

    Book  MATH  Google Scholar 

  45. Zeng Y, Li Z F. Optimal time-consistent investment and reinsurance policies for mean-variance insurers. Insurance Math Econom, 2011, 49: 145–154

    Article  MathSciNet  MATH  Google Scholar 

  46. Zeng Y, Li Z F, Lai Y Z. Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps. Insurance Math Econom, 2013, 52: 498–507

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11301376, 71201173 and 71571195), China Scholarship Council, the Natural Sciences and Engineering Research Council of Canada (NSERC), and Society of Actuaries Centers of Actuarial Excellence Research Grant, Guangdong Natural Science Funds for Distinguished Young Scholar (Grant No. 2015A030306040), Natural Science Foundation of Guangdong Province of China (Grant No. 2014A030310195) and for Ying Tung Eduction Foundation for Young Teachers in the Higher Education Institutions of China (Grant No. 151081).

Author information

Authors and Affiliations

  1. School of Science, Tianjin University, Tianjin, 300072, China

    Hui Zhao

  2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    ChengGuo Weng

  3. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P 3, Canada

    Yang Shen

  4. Lingnan (University) College, Sun Yat-sen University, Guangzhou, 510275, China

    Yan Zeng

Authors
  1. Hui Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. ChengGuo Weng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yan Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yan Zeng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, H., Weng, C., Shen, Y. et al. Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models. Sci. China Math. 60, 317–344 (2017). https://doi.org/10.1007/s11425-015-0542-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-015-0542-7

Keywords

MSC(2010)

Search

Navigation