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July  2012, 8(3): 531-547. doi: 10.3934/jimo.2012.8.531

## Optimal investment with a value-at-risk constraint

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong, China 2 Department of Mathematics, Nankai University, Tianjin 300071, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  January 2011 Revised  March 2012 Published  June 2012

We consider constrained investment problem with the objective of minimizing the ruin probability. In this paper, we formulate the cash reserve and investment model for the insurance company and analyze the value-at-risk ($VaR$) in a short time horizon. For risk regulation, we impose it as a risk constraint dynamically. Then the problem becomes minimizing the probability of ruin together with the imposed risk constraint. By solving the corresponding Hamilton-Jacobi-Bellman equations, we derive analytic expressions for the optimal value function and the corresponding optimal strategies. Looking at the value-at-risk alone, we show that it is possible to reduce the overall risk by an increased exposure to the risky asset. This is aligned with the risk diversification effect for negative correlated or uncorrelated risky asset with the stochastic of the fundamental insurance business. Moreover, studying the optimal strategies, we find that a different investment strategy will be in place depending on the Sharpe ratio of the risky asset.
Citation: Jingzhen Liu, Lihua Bai, Ka-Fai Cedric Yiu. Optimal investment with a value-at-risk constraint. Journal of Industrial and Management Optimization, 2012, 8 (3) : 531-547. doi: 10.3934/jimo.2012.8.531
##### References:
 [1] S. Asmussen, "Ruin Probabilities," Advanced Series on Statistical Science & Applied Probability, 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [2] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975. doi: 10.1016/j.insmatheco.2007.11.002. [3] R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk, in "Proceedings of the Casualty Actuarial Society Casualty Actuarial Society" (Arlington, Virginia, 2000), LXXXVII, 31-78. [4] S. Bouma, "Risk Management in the Insurance Industry and Solvency II," European Survey, Capgemini, November 7, 2006. [5] 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, 20 (1995), 937-958. [6] J. Cai and K. S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measure, Astin Bulletin, 37 (2007), 93-112. doi: 10.2143/AST.37.1.2020800. [7] J. Cai, K. S. Tan, C. G. Weng and Y. Zhang, Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43 (2008), 185-196. doi: 10.1016/j.insmatheco.2008.05.011. [8] S. Cheng, H. U. Gerber and E. S. W. Shiu, Discounted probabilities and ruin theory in the compound binomial model, Insurance: Mathematics and Economics, 26 (2000), 239-250. doi: 10.1016/S0167-6687(99)00053-0. [9] Y. H. Dong and G. J. Wang, Ruin probability for renewal risk model with negative risk sums, Journal of Industrial and Management Optimization, 2 (2006), 229-236. [10] W. H. Fleming, and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993. [11] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78. [12] H. Gerber and H. Yang, Absolute ruin probabilities in a jump diffusion risk model with investment, North American Actuarial Journal, 11 (2007), 159-169. [13] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228. [14] R. Kostadinova, Optimal investment for insurers when the stock price follows an exponential Lévy process, Insurance: Mathematics and Economics, 41 (2007), 250-263. doi: 10.1016/j.insmatheco.2006.10.018. [15] C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31. [16] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance Mathematics Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002. [17] H. M. Markowitz, "Portfolio Selection: Efficient Diversification of Investments," 2$^n^d$ edition, Blackwell, USA, 1991. [18] J. Paulsen, Ruin models with investment income, Probability Surveys, 5 (2008), 416-434. [19] D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128. [20] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907. doi: 10.1214/aoap/1031863173. [21] J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in "Contributions to Nonlinear Functional Analysis" (eds. E. H. Zarantonello and Author 2) (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), Academic Press, New York, (1971), 565-601. [22] W. F. Sharpe, The Sharpe ratio, Journal of Portfolio Management, Fall issue, (1994), 49-58. [23] T. K. Siu, J. W. Lau and H. Yang, Ruin theory under a generalized jump-diffusion model with regime switching, Applied Mathematical Sciences, 2 (2008), 1415-1430. [24] Q. Tang and G. Tsitsiashvili, Finite-and infinite-time ruin probabilities in the presence of stochastic returns on investments, Advances in Applied Probability, 36 (2004), 1278-1299. doi: 10.1239/aap/1103662967. [25] L. Xu and R. M. Wang, Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate, Journal of Industrial and Management Optimization, 2 (2006), 165-175. [26] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 35 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009. [27] K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint, Journal of Economic Dynamics and Control, 28 (2004), 1317-1334. [28] K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94. [29] K. C. Yuen, G. J. Wang and K. W. Ng, Ruin probabilities for a risk process with stochastic return on investments, Stochastic Processes and their Applications, 110 (2004), 259-274.

show all references

##### References:
 [1] S. Asmussen, "Ruin Probabilities," Advanced Series on Statistical Science & Applied Probability, 2, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [2] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975. doi: 10.1016/j.insmatheco.2007.11.002. [3] R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk, in "Proceedings of the Casualty Actuarial Society Casualty Actuarial Society" (Arlington, Virginia, 2000), LXXXVII, 31-78. [4] S. Bouma, "Risk Management in the Insurance Industry and Solvency II," European Survey, Capgemini, November 7, 2006. [5] 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, 20 (1995), 937-958. [6] J. Cai and K. S. Tan, Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measure, Astin Bulletin, 37 (2007), 93-112. doi: 10.2143/AST.37.1.2020800. [7] J. Cai, K. S. Tan, C. G. Weng and Y. Zhang, Optimal reinsurance under VaR and CTE risk measures, Insurance: Mathematics and Economics, 43 (2008), 185-196. doi: 10.1016/j.insmatheco.2008.05.011. [8] S. Cheng, H. U. Gerber and E. S. W. Shiu, Discounted probabilities and ruin theory in the compound binomial model, Insurance: Mathematics and Economics, 26 (2000), 239-250. doi: 10.1016/S0167-6687(99)00053-0. [9] Y. H. Dong and G. J. Wang, Ruin probability for renewal risk model with negative risk sums, Journal of Industrial and Management Optimization, 2 (2006), 229-236. [10] W. H. Fleming, and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics (New York), 25, Springer-Verlag, New York, 1993. [11] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78. [12] H. Gerber and H. Yang, Absolute ruin probabilities in a jump diffusion risk model with investment, North American Actuarial Journal, 11 (2007), 159-169. [13] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228. [14] R. Kostadinova, Optimal investment for insurers when the stock price follows an exponential Lévy process, Insurance: Mathematics and Economics, 41 (2007), 250-263. doi: 10.1016/j.insmatheco.2006.10.018. [15] C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31. [16] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance Mathematics Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002. [17] H. M. Markowitz, "Portfolio Selection: Efficient Diversification of Investments," 2$^n^d$ edition, Blackwell, USA, 1991. [18] J. Paulsen, Ruin models with investment income, Probability Surveys, 5 (2008), 416-434. [19] D. S. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128. [20] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907. doi: 10.1214/aoap/1031863173. [21] J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in "Contributions to Nonlinear Functional Analysis" (eds. E. H. Zarantonello and Author 2) (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), Academic Press, New York, (1971), 565-601. [22] W. F. Sharpe, The Sharpe ratio, Journal of Portfolio Management, Fall issue, (1994), 49-58. [23] T. K. Siu, J. W. Lau and H. Yang, Ruin theory under a generalized jump-diffusion model with regime switching, Applied Mathematical Sciences, 2 (2008), 1415-1430. [24] Q. Tang and G. Tsitsiashvili, Finite-and infinite-time ruin probabilities in the presence of stochastic returns on investments, Advances in Applied Probability, 36 (2004), 1278-1299. doi: 10.1239/aap/1103662967. [25] L. Xu and R. M. Wang, Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate, Journal of Industrial and Management Optimization, 2 (2006), 165-175. [26] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 35 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009. [27] K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint, Journal of Economic Dynamics and Control, 28 (2004), 1317-1334. [28] K. F. C. Yiu, S. Y. Wang and K. L. Mak, Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains, Journal of Industrial and Management Optimization, 4 (2008), 81-94. [29] K. C. Yuen, G. J. Wang and K. W. Ng, Ruin probabilities for a risk process with stochastic return on investments, Stochastic Processes and their Applications, 110 (2004), 259-274.
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