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

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

Yan Zeng 1, and  Zhongfei Li 2,

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

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 and 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), 299-324.

[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), 26-34. doi: 10.1016/j.insmatheco.2008.09.006.

[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), 968-975. doi: 10.1016/j.insmatheco.2007.11.002.

[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), 181-205. doi: 10.1007/s00186-007-0195-4.

[5]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.

[6]

F. Black and A. F. Perold, Theory of constant proportion portfolio insurance, Journal of Economic Dynamics and Control, 16 (1992), 403-426. doi: 10.1016/0165-1889(92)90043-E.

[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), 937-958. doi: 10.1287/moor.20.4.937.

[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), 157-162. doi: 10.1016/j.insmatheco.2009.05.006.

[9]

Ł. Delong and R. Gerrard, Mean-variance portfolio selection for a non-life insurance company, Mathematical Methods of Operations Research, 66 (2007), 339-367. doi: 10.1007/s00186-007-0152-2.

[10]

S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bound downside risks, Working paper, (2000).

[11]

S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bounded captial at risk, Mathematical Finance, 11 (2001), 365-384. doi: 10.1111/1467-9965.00121.

[12]

P. Gänssler and W. Stute, "Wahrscheinlichkeitstheorie," Springer-Verlag, Berlin-New York, 1977.

[13]

J. Grandlle, "Aspects of Risk Theory," Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.

[14]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228. doi: 10.1016/S0167-6687(00)00049-4.

[15]

D. Iglehart, Diffusion approximations in collective risk theory, Journal of Applied Probability, 6 (1969), 285-292.

[16]

C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investment for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51. doi: 10.1016/j.insmatheco.2004.04.004.

[17]

R. Korn, "Optimal Portfolios," World Scientific, Singapore, 1997.

[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), 35-49.

[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.

[20]

Z.-B. Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion, Acta Mathematicae Applicatae Sinica, English Series, 23 (2007), 477-488.

[21]

Z.-B. Liang and J.-Y. Guo, Optimal proportional reinsurance and ruin probability, Stochastic Models, 23 (2007), 333-350. doi: 10.1080/15326340701300894.

[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), 109-128. doi: 10.1002/asmb.694.

[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), 11-31.

[24]

S.-Z. Luo, Ruin minimization for insurers with borrowing constraints, North American Actuarial Journal, 12 (2009), 143-174.

[25]

S.-Z. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002.

[26]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model, Review of Economics and Statistics, 51 (1969), 247-256. doi: 10.2307/1926560.

[27]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.

[28]

A. F. Perold and W. F. Sharpe, Dynamic strategies for asset allocation, Financial Analyst Journal, 44 (1988), 16-27. doi: 10.2469/faj.v44.n1.16.

[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), 109-128.

[30]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68.

[31]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.

[32]

M. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7 (2003), 97-121.

[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), 322-334. doi: 10.1016/j.insmatheco.2006.05.003.

[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), 801-815.

[35]

H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009.

[36]

Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance: Mathematics and Economics, 49 (2011), 145-154. doi: 10.1016/j.insmatheco.2011.01.001.

[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), 483-496.

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), 299-324.

[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), 26-34. doi: 10.1016/j.insmatheco.2008.09.006.

[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), 968-975. doi: 10.1016/j.insmatheco.2007.11.002.

[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), 181-205. doi: 10.1007/s00186-007-0195-4.

[5]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.

[6]

F. Black and A. F. Perold, Theory of constant proportion portfolio insurance, Journal of Economic Dynamics and Control, 16 (1992), 403-426. doi: 10.1016/0165-1889(92)90043-E.

[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), 937-958. doi: 10.1287/moor.20.4.937.

[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), 157-162. doi: 10.1016/j.insmatheco.2009.05.006.

[9]

Ł. Delong and R. Gerrard, Mean-variance portfolio selection for a non-life insurance company, Mathematical Methods of Operations Research, 66 (2007), 339-367. doi: 10.1007/s00186-007-0152-2.

[10]

S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bound downside risks, Working paper, (2000).

[11]

S. Emmer, C. Klüppelberg and R. Korn, Optimal portfolio with bounded captial at risk, Mathematical Finance, 11 (2001), 365-384. doi: 10.1111/1467-9965.00121.

[12]

P. Gänssler and W. Stute, "Wahrscheinlichkeitstheorie," Springer-Verlag, Berlin-New York, 1977.

[13]

J. Grandlle, "Aspects of Risk Theory," Springer Series in Statistics: Probability and its Applications, Springer-Verlag, New York, 1991.

[14]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228. doi: 10.1016/S0167-6687(00)00049-4.

[15]

D. Iglehart, Diffusion approximations in collective risk theory, Journal of Applied Probability, 6 (1969), 285-292.

[16]

C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investment for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51. doi: 10.1016/j.insmatheco.2004.04.004.

[17]

R. Korn, "Optimal Portfolios," World Scientific, Singapore, 1997.

[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), 35-49.

[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.

[20]

Z.-B. Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion, Acta Mathematicae Applicatae Sinica, English Series, 23 (2007), 477-488.

[21]

Z.-B. Liang and J.-Y. Guo, Optimal proportional reinsurance and ruin probability, Stochastic Models, 23 (2007), 333-350. doi: 10.1080/15326340701300894.

[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), 109-128. doi: 10.1002/asmb.694.

[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), 11-31.

[24]

S.-Z. Luo, Ruin minimization for insurers with borrowing constraints, North American Actuarial Journal, 12 (2009), 143-174.

[25]

S.-Z. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444. doi: 10.1016/j.insmatheco.2007.04.002.

[26]

R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model, Review of Economics and Statistics, 51 (1969), 247-256. doi: 10.2307/1926560.

[27]

R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.

[28]

A. F. Perold and W. F. Sharpe, Dynamic strategies for asset allocation, Financial Analyst Journal, 44 (1988), 16-27. doi: 10.2469/faj.v44.n1.16.

[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), 109-128.

[30]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68.

[31]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, The Annals of Applied Probability, 12 (2002), 890-907.

[32]

M. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7 (2003), 97-121.

[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), 322-334. doi: 10.1016/j.insmatheco.2006.05.003.

[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), 801-815.

[35]

H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634. doi: 10.1016/j.insmatheco.2005.06.009.

[36]

Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance: Mathematics and Economics, 49 (2011), 145-154. doi: 10.1016/j.insmatheco.2011.01.001.

[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), 483-496.

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