• Previous Article
    Portfolio optimization and risk measurement based on non-dominated sorting genetic algorithm
  • JIMO Home
  • This Issue
  • Next Article
    An approximation algorithm for the $k$-level facility location problem with submodular penalties
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 & 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 (2000).   Google Scholar

[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.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk,, in, (2000), 31.   Google Scholar

[4]

S. Bouma, "Risk Management in the Insurance Industry and Solvency II,", European Survey, (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.2143/AST.37.1.2020800.  Google Scholar

[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.  doi: 10.1016/j.insmatheco.2008.05.011.  Google Scholar

[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.  doi: 10.1016/S0167-6687(99)00053-0.  Google Scholar

[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.   Google Scholar

[10]

W. H. Fleming, and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Applications of Mathematics (New York), 25 (1993).   Google Scholar

[11]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin,, North American Actuarial Journal, 2 (1998), 48.   Google Scholar

[12]

H. Gerber and H. Yang, Absolute ruin probabilities in a jump diffusion risk model with investment,, North American Actuarial Journal, 11 (2007), 159.   Google Scholar

[13]

C. Hipp and M. Plum, Optimal investment for insurers,, Insurance: Mathematics and Economics, 27 (2000), 215.   Google Scholar

[14]

R. Kostadinova, Optimal investment for insurers when the stock price follows an exponential Lévy process,, Insurance: Mathematics and Economics, 41 (2007), 250.  doi: 10.1016/j.insmatheco.2006.10.018.  Google Scholar

[15]

C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin,, North American Actuarial Journal, 8 (2004), 11.   Google Scholar

[16]

S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance Mathematics Economics, 42 (2008), 434.  doi: 10.1016/j.insmatheco.2007.04.002.  Google Scholar

[17]

H. M. Markowitz, "Portfolio Selection: Efficient Diversification of Investments,", 2$^n^d$ edition, (1991).   Google Scholar

[18]

J. Paulsen, Ruin models with investment income,, Probability Surveys, 5 (2008), 416.   Google Scholar

[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.   Google Scholar

[20]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, The Annals of Applied Probability, 12 (2002), 890.  doi: 10.1214/aoap/1031863173.  Google Scholar

[21]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations,, in, (1971), 565.   Google Scholar

[22]

W. F. Sharpe, The Sharpe ratio,, Journal of Portfolio Management, (1994), 49.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1239/aap/1103662967.  Google Scholar

[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.   Google Scholar

[26]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance: Mathematics and Economics, 35 (2005), 615.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[27]

K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint,, Journal of Economic Dynamics and Control, 28 (2004), 1317.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

show all references

References:
[1]

S. Asmussen, "Ruin Probabilities,", Advanced Series on Statistical Science & Applied Probability, 2 (2000).   Google Scholar

[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.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar

[3]

R. Bingham, Risk and return: Underwriting, investment and leverage probability of surplus drawdown and pricing for underwriting and investment risk,, in, (2000), 31.   Google Scholar

[4]

S. Bouma, "Risk Management in the Insurance Industry and Solvency II,", European Survey, (2006).   Google Scholar

[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.   Google Scholar

[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.  doi: 10.2143/AST.37.1.2020800.  Google Scholar

[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.  doi: 10.1016/j.insmatheco.2008.05.011.  Google Scholar

[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.  doi: 10.1016/S0167-6687(99)00053-0.  Google Scholar

[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.   Google Scholar

[10]

W. H. Fleming, and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,", Applications of Mathematics (New York), 25 (1993).   Google Scholar

[11]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin,, North American Actuarial Journal, 2 (1998), 48.   Google Scholar

[12]

H. Gerber and H. Yang, Absolute ruin probabilities in a jump diffusion risk model with investment,, North American Actuarial Journal, 11 (2007), 159.   Google Scholar

[13]

C. Hipp and M. Plum, Optimal investment for insurers,, Insurance: Mathematics and Economics, 27 (2000), 215.   Google Scholar

[14]

R. Kostadinova, Optimal investment for insurers when the stock price follows an exponential Lévy process,, Insurance: Mathematics and Economics, 41 (2007), 250.  doi: 10.1016/j.insmatheco.2006.10.018.  Google Scholar

[15]

C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin,, North American Actuarial Journal, 8 (2004), 11.   Google Scholar

[16]

S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance Mathematics Economics, 42 (2008), 434.  doi: 10.1016/j.insmatheco.2007.04.002.  Google Scholar

[17]

H. M. Markowitz, "Portfolio Selection: Efficient Diversification of Investments,", 2$^n^d$ edition, (1991).   Google Scholar

[18]

J. Paulsen, Ruin models with investment income,, Probability Surveys, 5 (2008), 416.   Google Scholar

[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.   Google Scholar

[20]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, The Annals of Applied Probability, 12 (2002), 890.  doi: 10.1214/aoap/1031863173.  Google Scholar

[21]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations,, in, (1971), 565.   Google Scholar

[22]

W. F. Sharpe, The Sharpe ratio,, Journal of Portfolio Management, (1994), 49.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1239/aap/1103662967.  Google Scholar

[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.   Google Scholar

[26]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance: Mathematics and Economics, 35 (2005), 615.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[27]

K. F. C. Yiu, Optimal portfolio under a value-at-risk constraint,, Journal of Economic Dynamics and Control, 28 (2004), 1317.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[1]

Jean-Claude Zambrini. On the geometry of the Hamilton-Jacobi-Bellman equation. Journal of Geometric Mechanics, 2009, 1 (3) : 369-387. doi: 10.3934/jgm.2009.1.369

[2]

Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251

[3]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161

[4]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933

[5]

Yiling Chen, Baojun Bian. optimal investment and dividend policy in an insurance company: A varied bound for dividend rates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5083-5105. doi: 10.3934/dcdsb.2019044

[6]

Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019050

[7]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223

[8]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[9]

Rongfei Liu, Dingcheng Wang, Jiangyan Peng. Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial & Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058

[10]

Yinghui Dong, Guojing Wang. Ruin probability for renewal risk model with negative risk sums. Journal of Industrial & Management Optimization, 2006, 2 (2) : 229-236. doi: 10.3934/jimo.2006.2.229

[11]

Federica Masiero. Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 223-263. doi: 10.3934/dcds.2012.32.223

[12]

Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial & Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012

[13]

Helmut Mausser, Oleksandr Romanko. CVaR proxies for minimizing scenario-based Value-at-Risk. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1109-1127. doi: 10.3934/jimo.2014.10.1109

[14]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295

[15]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298

[16]

Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial & Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191

[17]

Qingwu Gao, Zhongquan Huang, Houcai Shen, Juan Zheng. Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims. Journal of Industrial & Management Optimization, 2016, 12 (1) : 31-43. doi: 10.3934/jimo.2016.12.31

[18]

Yuebao Wang, Qingwu Gao, Kaiyong Wang, Xijun Liu. Random time ruin probability for the renewal risk model with heavy-tailed claims. Journal of Industrial & Management Optimization, 2009, 5 (4) : 719-736. doi: 10.3934/jimo.2009.5.719

[19]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461

[20]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]