Optimal reinsurance-investment strategies for insurers under mean-CaR criteria

Previous Article
Integrated inventory model with stochastic lead time and controllable variability for milk runs
JIMO Home
This Issue
Next Article
Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria

July 2012, 8(3): 673-690. doi: 10.3934/jimo.2012.8.673

Optimal reinsurance-investment strategies for insurers under mean-CaR criteria

1.	Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, China
2.	Lingnan (University) College/Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou 510275, China

Received June 2011 Revised February 2012 Published June 2012

Abstract
Related Papers

This paper considers an optimal reinsurance-investment problem for an insurer, who aims to minimize the risk measured by Capital-at-Risk (CaR) with the constraint that the expected terminal wealth is not less than a predefined level. The surplus of the insurer is described by a Brownian motion with drift. The insurer can control her/his risk by purchasing proportional reinsurance, acquiring new business, and investing her/his surplus in a financial market consisting of one risk-free asset and multiple risky assets. Three mean-CaR models are constructed. By transforming these models into bilevel optimization problems, we derive the explicit expressions of the optimal deterministic rebalance reinsurance-investment strategies and the mean-CaR efficient frontiers. Sensitivity analysis of the results and a numerical example are provided.

Keywords: Capital-at-Risk, Optimal proportional reinsurance-investment strategy, insurers, Hamilton-Jacobi-Bellman equation..

Mathematics Subject Classification: Primary: 90C26; Secondary: 91B28, 49N1.

Citation: Yan Zeng, Zhongfei Li. Optimal reinsurance-investment strategies for insurers under mean-CaR criteria. Journal of Industrial & Management Optimization, 2012, 8 (3) : 673-690. doi: 10.3934/jimo.2012.8.673

References:

[1]	S. Asmussen, B. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation,, Finance and Stochastics, 4 (2000), 299. Google Scholar
[2]	P. Azcue and N. Muler, Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints,, Insurance: Mathematics and Economics, 44 (2009), 26. doi: 10.1016/j.insmatheco.2008.09.006. Google Scholar
[3]	L. H. Bai and J. Y. 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
[4]	L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers,, Mathematical Methods of Operations Research, 68 (2008), 181. doi: 10.1007/s00186-007-0195-4. Google Scholar
[5]	N. Bäuerle, Benchmark and mean-variance problems for insurers,, Mathematical Methods of Operations Research, 62 (2005), 159. Google Scholar
[6]	F. Black and A. F. Perold, Theory of constant proportion portfolio insurance,, Journal of Economic Dynamics and Control, 16 (1992), 403. doi: 10.1016/0165-1889(92)90043-E. Google Scholar
[7]	S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing probability of ruin,, Mathematics of Operations Research, 20 (1995), 937. doi: 10.1287/moor.20.4.937. Google Scholar
[8]	Y. S. Cao and N. Q. Wan, Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation,, Insurance: Mathematics and Economics, 45 (2009), 157. doi: 10.1016/j.insmatheco.2009.05.006. Google Scholar
[9]	Ł. Delong and R. Gerrard, Mean-variance portfolio selection for a non-life insurance company,, Mathematical Methods of Operations Research, 66 (2007), 339. doi: 10.1007/s00186-007-0152-2. Google Scholar
[10]	S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bound downside risks,, Working paper, (2000). Google Scholar
[11]	S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bounded captial at risk,, Mathematical Finance, 11 (2001), 365. doi: 10.1111/1467-9965.00121. Google Scholar
[12]	P. Gänssler and W. Stute, "Wahrscheinlichkeitstheorie,", Springer-Verlag, (1977). Google Scholar
[13]	J. Grandlle, "Aspects of Risk Theory,", Springer Series in Statistics: Probability and its Applications, (1991). Google Scholar
[14]	C. Hipp and M. Plum, Optimal investment for insurers,, Insurance: Mathematics and Economics, 27 (2000), 215. doi: 10.1016/S0167-6687(00)00049-4. Google Scholar
[15]	D. Iglehart, Diffusion approximations in collective risk theory,, Journal of Applied Probability, 6 (1969), 285. Google Scholar
[16]	C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investment for insurance portfolios,, Insurance: Mathematics and Economics, 35 (2004), 21. doi: 10.1016/j.insmatheco.2004.04.004. Google Scholar
[17]	R. Korn, "Optimal Portfolios,", World Scientific, (1997). Google Scholar
[18]	Z.-F. Li, K. W. Ng and X.-T. Deng, Continuous-time optimal portfolio selection using mean-CaR models,, Nonlinear Dynamics and Systems Theory, 7 (2007), 35. Google Scholar
[19]	Z.-F. Li, Y. Zeng and Y. L. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model,, Insurance: Mathematics and Economics, (2011). doi: 10.1016/j.insmatheco.2011.09.002. Google Scholar
[20]	Z.-B. Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion,, Acta Mathematicae Applicatae Sinica, 23 (2007), 477. Google Scholar
[21]	Z.-B. Liang and J.-Y. Guo, Optimal proportional reinsurance and ruin probability,, Stochastic Models, 23 (2007), 333. doi: 10.1080/15326340701300894. Google Scholar
[22]	Z.-B. Liang and J.-Y. Guo, Upper bound for ruin probabilities under optimal investment and proportional reinsurance,, Applied Stochastic Models in Business and Industry, 24 (2008), 109. doi: 10.1002/asmb.694. Google Scholar
[23]	C. S. Liu and H. L. Yang, Optimal investment for an insurer to minimize its probability of ruin,, North American Actuarial Journal, 8 (2004), 11. Google Scholar
[24]	S.-Z. Luo, Ruin minimization for insurers with borrowing constraints,, North American Actuarial Journal, 12 (2009), 143. Google Scholar
[25]	S.-Z. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance: Mathematics and Economics, 42 (2008), 434. doi: 10.1016/j.insmatheco.2007.04.002. Google Scholar
[26]	R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model,, Review of Economics and Statistics, 51 (1969), 247. doi: 10.2307/1926560. Google Scholar
[27]	R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model,, Journal of Economic Theory, 3 (1971), 373. Google Scholar
[28]	A. F. Perold and W. F. Sharpe, Dynamic strategies for asset allocation,, Financial Analyst Journal, 44 (1988), 16. doi: 10.2469/faj.v44.n1.16. Google Scholar
[29]	S. D. 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
[30]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting,, Scandinavian Actuarial Journal, 1 (2001), 55. Google Scholar
[31]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, The Annals of Applied Probability, 12 (2002), 890. Google Scholar
[32]	M. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios,, Finance and Stochastics, 7 (2003), 97. Google Scholar
[33]	Z. W. Wang, J. M. Xia and L. H. Zhang, Optimal investment for an insurer: The martingale approach,, Insurance: Mathematics and Economics, 40 (2007), 322. doi: 10.1016/j.insmatheco.2006.05.003. Google Scholar
[34]	L. Xu, R. M. Wang and D. J. Yao, On maximizing the expected terminal utility by investment and reinsurance,, Journal of Industrial and Management Optimization, 4 (2008), 801. Google Scholar
[35]	H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance: Mathematics and Economics, 37 (2005), 615. doi: 10.1016/j.insmatheco.2005.06.009. Google Scholar
[36]	Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers,, Insurance: Mathematics and Economics, 49 (2011), 145. doi: 10.1016/j.insmatheco.2011.01.001. Google Scholar
[37]	Y. Zeng, Z. F. Li and J. J. Liu, Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers,, Journal of Industrial and Management Optimization, 6 (2010), 483. Google Scholar

show all references

References:

[1]	S. Asmussen, B. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation,, Finance and Stochastics, 4 (2000), 299. Google Scholar
[2]	P. Azcue and N. Muler, Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints,, Insurance: Mathematics and Economics, 44 (2009), 26. doi: 10.1016/j.insmatheco.2008.09.006. Google Scholar
[3]	L. H. Bai and J. Y. 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
[4]	L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers,, Mathematical Methods of Operations Research, 68 (2008), 181. doi: 10.1007/s00186-007-0195-4. Google Scholar
[5]	N. Bäuerle, Benchmark and mean-variance problems for insurers,, Mathematical Methods of Operations Research, 62 (2005), 159. Google Scholar
[6]	F. Black and A. F. Perold, Theory of constant proportion portfolio insurance,, Journal of Economic Dynamics and Control, 16 (1992), 403. doi: 10.1016/0165-1889(92)90043-E. Google Scholar
[7]	S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing probability of ruin,, Mathematics of Operations Research, 20 (1995), 937. doi: 10.1287/moor.20.4.937. Google Scholar
[8]	Y. S. Cao and N. Q. Wan, Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation,, Insurance: Mathematics and Economics, 45 (2009), 157. doi: 10.1016/j.insmatheco.2009.05.006. Google Scholar
[9]	Ł. Delong and R. Gerrard, Mean-variance portfolio selection for a non-life insurance company,, Mathematical Methods of Operations Research, 66 (2007), 339. doi: 10.1007/s00186-007-0152-2. Google Scholar
[10]	S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bound downside risks,, Working paper, (2000). Google Scholar
[11]	S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bounded captial at risk,, Mathematical Finance, 11 (2001), 365. doi: 10.1111/1467-9965.00121. Google Scholar
[12]	P. Gänssler and W. Stute, "Wahrscheinlichkeitstheorie,", Springer-Verlag, (1977). Google Scholar
[13]	J. Grandlle, "Aspects of Risk Theory,", Springer Series in Statistics: Probability and its Applications, (1991). Google Scholar
[14]	C. Hipp and M. Plum, Optimal investment for insurers,, Insurance: Mathematics and Economics, 27 (2000), 215. doi: 10.1016/S0167-6687(00)00049-4. Google Scholar
[15]	D. Iglehart, Diffusion approximations in collective risk theory,, Journal of Applied Probability, 6 (1969), 285. Google Scholar
[16]	C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investment for insurance portfolios,, Insurance: Mathematics and Economics, 35 (2004), 21. doi: 10.1016/j.insmatheco.2004.04.004. Google Scholar
[17]	R. Korn, "Optimal Portfolios,", World Scientific, (1997). Google Scholar
[18]	Z.-F. Li, K. W. Ng and X.-T. Deng, Continuous-time optimal portfolio selection using mean-CaR models,, Nonlinear Dynamics and Systems Theory, 7 (2007), 35. Google Scholar
[19]	Z.-F. Li, Y. Zeng and Y. L. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model,, Insurance: Mathematics and Economics, (2011). doi: 10.1016/j.insmatheco.2011.09.002. Google Scholar
[20]	Z.-B. Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion,, Acta Mathematicae Applicatae Sinica, 23 (2007), 477. Google Scholar
[21]	Z.-B. Liang and J.-Y. Guo, Optimal proportional reinsurance and ruin probability,, Stochastic Models, 23 (2007), 333. doi: 10.1080/15326340701300894. Google Scholar
[22]	Z.-B. Liang and J.-Y. Guo, Upper bound for ruin probabilities under optimal investment and proportional reinsurance,, Applied Stochastic Models in Business and Industry, 24 (2008), 109. doi: 10.1002/asmb.694. Google Scholar
[23]	C. S. Liu and H. L. Yang, Optimal investment for an insurer to minimize its probability of ruin,, North American Actuarial Journal, 8 (2004), 11. Google Scholar
[24]	S.-Z. Luo, Ruin minimization for insurers with borrowing constraints,, North American Actuarial Journal, 12 (2009), 143. Google Scholar
[25]	S.-Z. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios,, Insurance: Mathematics and Economics, 42 (2008), 434. doi: 10.1016/j.insmatheco.2007.04.002. Google Scholar
[26]	R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model,, Review of Economics and Statistics, 51 (1969), 247. doi: 10.2307/1926560. Google Scholar
[27]	R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model,, Journal of Economic Theory, 3 (1971), 373. Google Scholar
[28]	A. F. Perold and W. F. Sharpe, Dynamic strategies for asset allocation,, Financial Analyst Journal, 44 (1988), 16. doi: 10.2469/faj.v44.n1.16. Google Scholar
[29]	S. D. 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
[30]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting,, Scandinavian Actuarial Journal, 1 (2001), 55. Google Scholar
[31]	H. Schmidli, On minimizing the ruin probability by investment and reinsurance,, The Annals of Applied Probability, 12 (2002), 890. Google Scholar
[32]	M. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios,, Finance and Stochastics, 7 (2003), 97. Google Scholar
[33]	Z. W. Wang, J. M. Xia and L. H. Zhang, Optimal investment for an insurer: The martingale approach,, Insurance: Mathematics and Economics, 40 (2007), 322. doi: 10.1016/j.insmatheco.2006.05.003. Google Scholar
[34]	L. Xu, R. M. Wang and D. J. Yao, On maximizing the expected terminal utility by investment and reinsurance,, Journal of Industrial and Management Optimization, 4 (2008), 801. Google Scholar
[35]	H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process,, Insurance: Mathematics and Economics, 37 (2005), 615. doi: 10.1016/j.insmatheco.2005.06.009. Google Scholar
[36]	Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers,, Insurance: Mathematics and Economics, 49 (2011), 145. doi: 10.1016/j.insmatheco.2011.01.001. Google Scholar
[37]	Y. Zeng, Z. F. Li and J. J. Liu, Optimal strategies of benchmark and mean-variance portfolio selection problems for insurers,, Journal of Industrial and Management Optimization, 6 (2010), 483. 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]	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
[3]	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
[4]	Yan Zhang, Peibiao Zhao. Optimal reinsurance-investment problem with dependent risks based on Legendre transform. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-23. doi: 10.3934/jimo.2019011
[5]	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
[6]	Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial & Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044
[7]	Yong Ma, Shiping Shan, Weidong Xu. Optimal investment and consumption in the market with jump risk and capital gains tax. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1937-1953. doi: 10.3934/jimo.2018130
[8]	Gongpin Cheng, Rongming Wang, Dingjun Yao. Optimal dividend and capital injection strategy with excess-of-loss reinsurance and transaction costs. Journal of Industrial & Management Optimization, 2018, 14 (1) : 371-395. doi: 10.3934/jimo.2017051
[9]	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
[10]	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
[11]	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
[12]	Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019080
[13]	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
[14]	Fengjun Wang, Qingling Zhang, Bin Li, Wanquan Liu. Optimal investment strategy on advertisement in duopoly. Journal of Industrial & Management Optimization, 2016, 12 (2) : 625-636. doi: 10.3934/jimo.2016.12.625
[15]	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
[16]	Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009
[17]	Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control & Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003
[18]	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
[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

American Institute of Mathematical Sciences