Optimal Investment and Proportional Reinsurance with Risk Constraint

Affiliation(s)

School of Insurance, Central University of Finance and Economics, Beijing; Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China.

Department of Mathematics and Statistics, Curtin University, Perth, Australia.

School of Insurance, Central University of Finance and Economics, Beijing; Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China.

Department of Mathematics and Statistics, Curtin University, Perth, Australia.

ABSTRACT

In this paper, we investigate the problem of maximizing the expected exponential utility for an insurer. In the problem setting, the insurer can invest his/her wealth into the market and he/she can also purchase the proportional reinsurance. To control the risk exposure, we impose a value-at-risk constraint on the portfolio, which results in a constrained stochastic optimal control problem. It is difficult to solve a constrained stochastic optimal control problem by using traditional dynamic programming or Martingale approach. However, for the frequently used exponential utility function, we show that the problem can be simplified significantly using a decomposition approach. The problem is reduced to a deterministic constrained optimal control problem, and then to a finite dimensional optimization problem. To show the effectiveness of the approach proposed, we consider both complete and incomplete markets; the latter arises when the number of risky assets are fewer than the dimension of uncertainty. We also conduct numerical experiments to demonstrate the effect of the risk constraint on the optimal strategy.

Cite this paper

J. Liu, K. Yiu, R. Loxton and K. Teo, "Optimal Investment and Proportional Reinsurance with Risk Constraint,"*Journal of Mathematical Finance*, Vol. 3 No. 4, 2013, pp. 437-447. doi: 10.4236/jmf.2013.34046.

J. Liu, K. Yiu, R. Loxton and K. Teo, "Optimal Investment and Proportional Reinsurance with Risk Constraint,"

References

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http://dx.doi.org/10.1287/moor.20.4.937

[2] R. Kostadinova, “Optimal Investment for Insurers When the Stock Price Follows Anexponential Levy Process,” Insurance: Mathematics and Economics, Vol. 41, No. 2, 2007, pp. 250-263.

http://dx.doi.org/10.1016/j.insmatheco.2006.10.018

[3] J. Ma and X. Sun, “Ruin Probabilities for Insurance Models Involving Investments,” Scandinavian Actuarial Journal, Vol. 2003, No. 3, 2003, pp. 217-237.

http://dx.doi.org/10.1080/03461230110106381

[4] D. S. Promislow and V. R. Young, “Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift,” North American Actuarial Journal, Vol. 9, No. 3, 2005, pp. 109-128.

[5] D. S. Promislow and V. R. Young, “Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift,” North American Actuarial Journal, Vol. 9, No. 3, 2005, pp. 109-128.

[6] H. Schmidli, “Optimal Proportional Reinsurance Policies in a Dynamic Setting,” Scandinavian Actuarial Journal, Vol. 2001, No. 1, 2001, pp. 55-68.

http://dx.doi.org/10.1080/034612301750077338

[7] H. Schmidli, “On Minimizing the Ruin Probability by Investment and Reinsurance,” The Annals of Applied Probability, Vol. 12, No. 3, 2002, pp. 890-907.

http://dx.doi.org/10.1214/aoap/1031863173

[8] M. Taksar and C. Markussen, “Optimal Dynamic Reinsurance Policies for Larg Insurance Portfolios,” Finance and Stochastics, Vol. 7, No. 1, 2003, pp. 97-121.

http://dx.doi.org/10.1007/s007800200073

[9] H. L. Yang and L. H. Zhang, “optimal Investment for Insurer with Jump Diffusion Risk Process,” Insurance: Mathematics and Economics, Vol. 37, No. 3, 2005, pp. 615-634. http://dx.doi.org/10.1016/j.insmatheco.2005.06.009

[10] J. Z. Liu, L. Bai and K. F. C. Yiu, “Optimal Investment with a Value-at-Risk Constraint,” Journal of Industrial and Management Optimization, Vol. 8, No. 3, 2012, pp. 531-547. http://dx.doi.org/10.3934/jimo.2012.8.531

[11] J. Z. Liu and K. F. C. Yiu, “Optimal Stochastic Differential Games with Var Constraints,” Discrete and Continuous Dynamical Systems-Series B, Vol. 18, No. 7, 2013, pp. 1889-1907. http://dx.doi.org/10.3934/dcdsb.2013.18.1889

[12] J. Z. Liu, K. F. C. Yiu and T. K. Siu, “Optimal Investment-Reinsurance with Dynamic Risk Constraint and Regime Switching,” Scandinavian Actuarial Journal, Vol. 2013, No. 4, 2013, pp. 263-285.

http://dx.doi.org/10.1080/03461238.2011.602477

[13] J. Z. Liu, K. F. C. Yiu and K. L. Teo, “Optimal Portfolios with Stress Analysis and the in a Effect of a CVaR Constraint,” Pacific Journal of Optimization, Vol. 7, No. 1, 2010, pp. 83-95.

[14] K. F. C. Yiu, “Optimal Portfolio under a Value-at-Risk Constraint,” Journal of Economic Dynamics and Control, Vol. 28, No. 7, 2004, pp. 1317-1334.

http://dx.doi.org/10.1016/S0165-1889(03)00116-7

[15] L. Bai and J. Guo, “Optimal Proportional Reinsurance and Investment with Multiple Risky Assets and NoShorting Constrain,” Insurance: Mathematics and Economics, Vol. 42, No. 3, 2008, pp. 968-975.

http://dx.doi.org/10.1016/j.insmatheco.2007.11.002

[16] B. Hujgaard and M. Taksar, “Optimal Proportional Reinsurance Policies for Diffusion Models,” Scandinavian Actuarial Journal, Vol. 1998, No. 2, 1998, pp. 166-180.

http://dx.doi.org/10.1080/03461238.1998.10414000

[17] H. Gerber, “An Introduction to Mathematical Risk Theory,” Richard D Irwin, Bloomsbury, 1979.

[18] R. C. Loxton, K. L. Teo and V. Rehbock, “Optimal Control Problems with Multiple Characteristic Time Points in the Objective and Constraints,” Automatica, Vol. 44, No. 11, 2008, pp. 2923-2929.

http://dx.doi.org/10.1016/j.automatica.2008.04.011

[19] R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, “Optimal Control Problems with Continuous Constraints on the State and the Control,” Automatica, Vol. 45, No. 10, 2009, pp. 2250-2257.

http://dx.doi.org/10.1016/j.automatica.2009.05.029

[20] K. L. Teo, “Control Parametrization Enhancing Transform to Optimal Control Problems Nonlinear Analysis,” Vol. 63, No. 5-7, 2005, pp. 2223-2236.

http://dx.doi.org/10.1016/j.na.2005.03.066

[21] J. W. Pratt, “Risk Aversion in the Small and in the Large,” Econometrica, Vol. 32, No. 1/2, 1964, pp. 122-136.

http://dx.doi.org/10.2307/1913738

[22] K. L. Teo, J. Goh, and K. H. Wong, “A Unified Computational Approach to Optimal Control Problems,” Longman Scientific and Technical, Harlow, 1991.

[23] Ekeland and R. Temam, “Convex Analysis and Variational Problems,” Society for Industrial and Applied Mathematics, Philadelphia, 1976.

[1] S. Browne, “Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin,” Mathematics of Operations Research, Vol. 20, No. 4 1995, pp. 937-958.

http://dx.doi.org/10.1287/moor.20.4.937

[2] R. Kostadinova, “Optimal Investment for Insurers When the Stock Price Follows Anexponential Levy Process,” Insurance: Mathematics and Economics, Vol. 41, No. 2, 2007, pp. 250-263.

http://dx.doi.org/10.1016/j.insmatheco.2006.10.018

[3] J. Ma and X. Sun, “Ruin Probabilities for Insurance Models Involving Investments,” Scandinavian Actuarial Journal, Vol. 2003, No. 3, 2003, pp. 217-237.

http://dx.doi.org/10.1080/03461230110106381

[4] D. S. Promislow and V. R. Young, “Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift,” North American Actuarial Journal, Vol. 9, No. 3, 2005, pp. 109-128.

[5] D. S. Promislow and V. R. Young, “Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift,” North American Actuarial Journal, Vol. 9, No. 3, 2005, pp. 109-128.

[6] H. Schmidli, “Optimal Proportional Reinsurance Policies in a Dynamic Setting,” Scandinavian Actuarial Journal, Vol. 2001, No. 1, 2001, pp. 55-68.

http://dx.doi.org/10.1080/034612301750077338

[7] H. Schmidli, “On Minimizing the Ruin Probability by Investment and Reinsurance,” The Annals of Applied Probability, Vol. 12, No. 3, 2002, pp. 890-907.

http://dx.doi.org/10.1214/aoap/1031863173

[8] M. Taksar and C. Markussen, “Optimal Dynamic Reinsurance Policies for Larg Insurance Portfolios,” Finance and Stochastics, Vol. 7, No. 1, 2003, pp. 97-121.

http://dx.doi.org/10.1007/s007800200073

[9] H. L. Yang and L. H. Zhang, “optimal Investment for Insurer with Jump Diffusion Risk Process,” Insurance: Mathematics and Economics, Vol. 37, No. 3, 2005, pp. 615-634. http://dx.doi.org/10.1016/j.insmatheco.2005.06.009

[10] J. Z. Liu, L. Bai and K. F. C. Yiu, “Optimal Investment with a Value-at-Risk Constraint,” Journal of Industrial and Management Optimization, Vol. 8, No. 3, 2012, pp. 531-547. http://dx.doi.org/10.3934/jimo.2012.8.531

[11] J. Z. Liu and K. F. C. Yiu, “Optimal Stochastic Differential Games with Var Constraints,” Discrete and Continuous Dynamical Systems-Series B, Vol. 18, No. 7, 2013, pp. 1889-1907. http://dx.doi.org/10.3934/dcdsb.2013.18.1889

[12] J. Z. Liu, K. F. C. Yiu and T. K. Siu, “Optimal Investment-Reinsurance with Dynamic Risk Constraint and Regime Switching,” Scandinavian Actuarial Journal, Vol. 2013, No. 4, 2013, pp. 263-285.

http://dx.doi.org/10.1080/03461238.2011.602477

[13] J. Z. Liu, K. F. C. Yiu and K. L. Teo, “Optimal Portfolios with Stress Analysis and the in a Effect of a CVaR Constraint,” Pacific Journal of Optimization, Vol. 7, No. 1, 2010, pp. 83-95.

[14] K. F. C. Yiu, “Optimal Portfolio under a Value-at-Risk Constraint,” Journal of Economic Dynamics and Control, Vol. 28, No. 7, 2004, pp. 1317-1334.

http://dx.doi.org/10.1016/S0165-1889(03)00116-7

[15] L. Bai and J. Guo, “Optimal Proportional Reinsurance and Investment with Multiple Risky Assets and NoShorting Constrain,” Insurance: Mathematics and Economics, Vol. 42, No. 3, 2008, pp. 968-975.

http://dx.doi.org/10.1016/j.insmatheco.2007.11.002

[16] B. Hujgaard and M. Taksar, “Optimal Proportional Reinsurance Policies for Diffusion Models,” Scandinavian Actuarial Journal, Vol. 1998, No. 2, 1998, pp. 166-180.

http://dx.doi.org/10.1080/03461238.1998.10414000

[17] H. Gerber, “An Introduction to Mathematical Risk Theory,” Richard D Irwin, Bloomsbury, 1979.

[18] R. C. Loxton, K. L. Teo and V. Rehbock, “Optimal Control Problems with Multiple Characteristic Time Points in the Objective and Constraints,” Automatica, Vol. 44, No. 11, 2008, pp. 2923-2929.

http://dx.doi.org/10.1016/j.automatica.2008.04.011

[19] R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, “Optimal Control Problems with Continuous Constraints on the State and the Control,” Automatica, Vol. 45, No. 10, 2009, pp. 2250-2257.

http://dx.doi.org/10.1016/j.automatica.2009.05.029

[20] K. L. Teo, “Control Parametrization Enhancing Transform to Optimal Control Problems Nonlinear Analysis,” Vol. 63, No. 5-7, 2005, pp. 2223-2236.

http://dx.doi.org/10.1016/j.na.2005.03.066

[21] J. W. Pratt, “Risk Aversion in the Small and in the Large,” Econometrica, Vol. 32, No. 1/2, 1964, pp. 122-136.

http://dx.doi.org/10.2307/1913738

[22] K. L. Teo, J. Goh, and K. H. Wong, “A Unified Computational Approach to Optimal Control Problems,” Longman Scientific and Technical, Harlow, 1991.

[23] Ekeland and R. Temam, “Convex Analysis and Variational Problems,” Society for Industrial and Applied Mathematics, Philadelphia, 1976.