This paper investigates the optimal reinsurance-investment strategy for an insurer whose premium is subject to extrapolative bias. In other words, the insurance premium is dynamically updated by a weighted average of prior claims and the initial estimation of claims. The insurer's surplus follows a diffusion approximation process. He purchases proportional reinsurance or acquires new business to manage insurance risk, and invests his surplus in the financial market, containing a risk-free asset and a risky asset (stock). The price of the risky asset is described by a constant elasticity of variance (CEV) model. The insurer is uncertain about the models of claims and risky asset. In order to derive robust optimal reinsurance-investment strategies, we establish an optimal control problem by maximizing the insurer's expected exponential utility of terminal wealth and solve the optimization problem explicitly. Finally, we present several numerical examples to illustrate our theoretical results.

References

• Azcue, P., & Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Mathematical Finance, 15(open in a new window)(2(open in a new window)), 261–308. https://doi.org/10.1111/mafi.2005.15.issue-2 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Barberis, N., Greenwood, R., Jin, L., & Shleifer, A. (2015). X-capm: An extrapolative capital asset pricing model. Journal of Financial Economics, 115(open in a new window)(1(open in a new window)), 1–24. https://doi.org/10.1016/j.jfineco.2014.08.007 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Bensoussan, A., Siu, C. C., Yam, S., Phillip, C., & Yang, H. (2014). A class of non-zero-sum stochastic differential investment and reinsurance games. Automatica, 50(open in a new window)(8(open in a new window)), 2025–2037. https://doi.org/10.1016/j.automatica.2014.05.033 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Black, F., & Scholes, M. S. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(open in a new window)(3(open in a new window)), 637–654. https://doi.org/10.1086/260062 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Borch, K. (1960). The safety loading of reinsurance premiums. Scandinavian Actuarial Journal, 1960(open in a new window)(3-4(open in a new window)), 163–184. https://doi.org/10.1080/03461238.1960.10410587 
  • View
   (Open in a new window)Google Scholar
• Browne, S. (1995). Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Mathematics of Operations Research, 20(open in a new window)(4(open in a new window)), 937–958. https://doi.org/10.1287/moor.20.4.937 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Chen, Z., & Epstein, L. (2002). Ambiguity, risk, and asset returns in continuous time. Econometrica, 70(open in a new window)(4(open in a new window)), 1403–1443. https://doi.org/10.1111/ecta.2002.70.issue-4 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Chen, S., Hu, D., & Wang, H. (2017). Optimal reinsurance problems with extrapolative claim expectation. Optimal Control Applications and Methods, 39(1), 79–94. 

   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar

• Chen, Z., & Yang, P. (2020). Robust optimal reinsurance-investment strategy with price jumps and correlated claims. Insurance Mathematics and Economics, 92(open in a new window)(5(open in a new window)), 27–46. https://doi.org/10.1016/j.insmatheco.2020.03.001 
  • View
   (Open in a new window)Google Scholar
• Cox, J., & Ross, S. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(open in a new window)(1-2(open in a new window)), 145–166. https://doi.org/10.1016/0304-405X(76)90023-4 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Davydov, D., & Linetsky, V. (2001). The valuation and hedging of barrier and lookback options under the CEV process. Management Science, 47(open in a new window), 949–965. https://doi.org/10.1287/mnsc.47.7.949.9804 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Deng, C., Zeng, X., & Zhu, H. (2018). Non-zero-sum stochastic differential reinsurance and investment games with default risk. European Journal of Operational Research, 264(open in a new window)(3(open in a new window)), 1144–1158. https://doi.org/10.1016/j.ejor.2017.06.065 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Fleming, W., & Soner, M. (2006). Controlled Markov processes and viscosity solutions. Springer-Verlag. 

   (Open in a new window)Google Scholar

• Flor, C., & Larsen, L. (2014). Robust portfolio choice with stochastic interest rates. Annals of Finance, 10(open in a new window), 243–265. https://doi.org/10.1007/s10436-013-0234-5 
  • View
   (Open in a new window)Google Scholar
• Gal, S., & Landsberger, M. (1988). On ‘small sample’ properties of experience rating insurance contracts. The Quarterly Journal of Economics, 103(open in a new window)(1(open in a new window)), 233–243. https://doi.org/10.2307/1882652 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Gao, J. (2009a). Optimal portfolios for dc pension plans under a CEV model. Insurance Mathematics and Economics, 44(open in a new window)(3(open in a new window)), 479–490. https://doi.org/10.1016/j.insmatheco.2009.01.005 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Gao, J. (2009a). Optimal portfolios for DC pension plans under a CEV model. Insurance Mathematics and Economics, 44(open in a new window)(3(open in a new window)), 479–490. https://doi.org/10.1016/j.insmatheco.2009.01.005 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Grandell, J. (1991). Aspects of risk theory. World Publishing Co. 
  • View
   (Open in a new window)Google Scholar
• Gu, A., Guo, X., Li, Z., & Zeng, Y. (2012). Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model. Insurance Mathematics and Economics, 51(open in a new window)(3(open in a new window)), 674–684. https://doi.org/10.1016/j.insmatheco.2012.09.003 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Gu, M., Yang, Y., Li, S., & Zhang, J. (2010). Constant elasticity of variance model for proportional reinsurance and investment strategies. Insurance Mathematics and Economics, 46(open in a new window)(3(open in a new window)), 580–587. https://doi.org/10.1016/j.insmatheco.2010.03.001 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Guelman, L., & Guillén, M. (2014). A causal inference approach to measure price elasticity in automobile insurance. Expert Systems with Applications: An International Journal, 41(open in a new window)(2(open in a new window)), 387–396. https://doi.org/10.1016/j.eswa.2013.07.059 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Hu, D., Chen, S., & Wang, H. (2018). Robust reinsurance contracts with uncertainty about jump risk. European Journal of Operational Research, 266(open in a new window)(3(open in a new window)), 1175–1188. https://doi.org/10.1016/j.ejor.2017.10.061 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Hu, D., & Wang, H. (2019). Optimal proportional reinsurance with a loss-dependent premium principle. Scandinavian Actuarial Journal, 2019(9), 752–767. 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Kraft, H., Munk, C., F. T. Seifried, & Wagner, S. (2016). Consumption habits and humps. Economic Theory, 64(open in a new window), 305–330. https://doi.org/10.1007/s00199-016-0984-1 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Li, D., Rong, X., & Zhao, H. (2014). Optimal reinsurance-investment problem for maximizing the product of the insurer's and the reinsurer's utilities under a CEV model. Journal of Computational and Applied Mathematics, 255(open in a new window), 671–683. https://doi.org/10.1016/j.cam.2013.06.033 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Li, D., Rong, X., & Zhao, H. (2016). Equilibrium excess-of-loss reinsurance-investment strategy for a mean-variance insurer under stochastic volatility model. Communications in Statistics – Theory and Methods, 46(open in a new window)(19(open in a new window)), 9459–9475. https://doi.org/10.1080/03610926.2016.1212071 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Liang, Z., Yuen, K., & Cheung, K. (2012). Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model. Applied Stochastic Models in Business and Industry, 28(open in a new window), 585–597. https://doi.org/10.1002/asmb.v28.6 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Maenhout, P. (2004). Robust portfolio rules and asset pricing. Review of Financial Studies, 17(open in a new window)(4(open in a new window)), 951–983. https://doi.org/10.1093/rfs/hhh003 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Øksendal, B. (2013). Stochastic differential equations: An introduction with applications. Springer. 

   (Open in a new window)Google Scholar

• Øksendal, B., & Sulem, A. (2019). Applied stochastic control of jump diffusions. Springer. 
  • View
   (Open in a new window)Google Scholar
• Schmidli, H. (2001). Optimal proportional reinsurance policies in a dynamic setting. Scandinavian Actuarial Journal, 2001(open in a new window)(1(open in a new window)), 55–68. https://doi.org/10.1080/034612301750077338 
  • View
   (Open in a new window)Google Scholar
• Schmidli, H. (2008). Stochastic control in continuous time. Springer. 
  • View
   (Open in a new window)Google Scholar
• Shen, Y., & Zeng, Y. (2014). Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach. Insurance Mathematics and Economics, 57(open in a new window)(7(open in a new window)), 1–12. https://doi.org/10.1016/j.insmatheco.2014.04.004 
  • View
   (Open in a new window)Google Scholar
• Xiao, J., Hong, Z., & Qin, C. (2007). The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts. Insurance Mathematics and Economics, 40(open in a new window)(2(open in a new window)), 302–310. https://doi.org/10.1016/j.insmatheco.2006.04.007 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Yi, B., Li, Z., Viens, F., & Zeng, Y. (2013). Robust optimal control for an insurer with reinsurance and investment under Heston's stochastic volatility model. Insurance Mathematics and Economics, 53(open in a new window)(3(open in a new window)), 601–614. https://doi.org/10.1016/j.insmatheco.2013.08.011 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Zeng, Y., Li, D., & Gu, A. (2016). Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps. Insurance Mathematics and Economics, 66(open in a new window)(1(open in a new window)), 138–152. https://doi.org/10.1016/j.insmatheco.2015.10.012 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Zhao, H., Rong, X., & Zhao, Y. (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, 53(open in a new window)(3(open in a new window)), 504–514. https://doi.org/10.1016/j.insmatheco.2013.08.004 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar

To cite this article: Ailing Gu, Xuanzhen Zhang, Shumin Chen & Ling Zhang (30 Aug 2024): Robust optimal reinsurance-investment strategy with extrapolative bias premiums and ambiguity aversion, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2024.2393062

To link to this article: https://doi.org/10.1080/24754269.2024.2393062