October  2018, 14(4): 1323-1348. doi: 10.3934/jimo.2018009

Optimal investment and dividend payment strategies with debt management and reinsurance

1. 

School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai 201620, China

2. 

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia

3. 

School of Statistics, Faculty of Economics and Management, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China

* Corresponding author

Received  August 2016 Revised  October 2017 Published  October 2018 Early access  January 2018

Fund Project: This work was supported by Program of Shanghai Subject Chief Scientist (14XD1401600), the 111 Project (B14019), National Natural Science Foundation of China (11601157,11601320,11571113,11231005,11501211), Research Grants Council of the Hong Kong Special Administrative Region (project No. HKU 17330816) and Faculty Research Grant by The University of Melbourne.

This paper derives the optimal debt ratio, investment and dividend payment strategies for an insurance company. The surplus process is jointly determined by the reinsurance strategies, debt levels, investment portfolios and unanticipated shocks. The objective is to maximize the total expected discounted utility of dividend payments in finite-time period subject to three control variables. The utility functions are chosen as the logarithmic and power utility functions. Using dynamic programming principle, the value function is the solution of a second-order nonlinear Hamilton-Jacobi-Bellman equation. The explicit solution of the value function is derived and the corresponding optimal debt ratio, investment and dividend payment strategies are obtained. In addition, the investment borrowing constraint, dividend payment constraint and impacts of reinsurance policies are considered and their impacts on the optimal strategies are analyzed. Further, to incorporating the interest rate risk, the problem is studied under a stochastic interest rate model.

Citation: Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009
References:
[1]

H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, RACSAM-Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 103 (2009), 295-320.  doi: 10.1007/BF03191909.

[2]

S. AsmussenB. Høgaard 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.  doi: 10.1007/s007800050075.

[3]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0.

[4]

P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, The Annals of Applied Probability, 20 (2010), 1253-1302.  doi: 10.1214/09-AAP643.

[5]

Y. C. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 5 (2014), 424-438. 

[6]

T. ChoulliM. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quant. Finance, 1 (2001), 573-596.  doi: 10.1088/1469-7688/1/6/301.

[7]

B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. 

[8]

W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM Journal of Control and Optimization, 43 (2004), 502-531.  doi: 10.1137/S0363012902419060.

[9]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.  doi: 10.1080/10920277.2004.10596125.

[10]

H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.  doi: 10.1080/10920277.2006.10596249.

[11]

Z. JinH. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329.  doi: 10.1016/j.automatica.2013.04.043.

[12]

Z. JinH. Yang and G. Yin, Optimal debt ratio and dividend payment strategies with reinsurance, Insurance: Mathematics and Economics, 64 (2015), 351-363.  doi: 10.1016/j.insmatheco.2015.07.005.

[13]

N. Kulenko and H. Schimidli, An optimal dividend strategy in a Craḿer Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013.

[14]

Z. F. LiY. Zeng and Y. Z. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.

[15]

H. Meng and T. K. Siu, Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279.  doi: 10.1137/090773167.

[16]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992.

[17]

Stein and L. Jerome, Stochastic Optimal Control and the U. S. Financial Debt Crisis Springer, New York, 2012.

[18]

J. WeiH. Yang and R. Wang, Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching, Journal of Optimization Theory and Applications, 147 (2010), 358-377.  doi: 10.1007/s10957-010-9726-x.

[19]

D. YaoH. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576.  doi: 10.1016/j.ejor.2011.01.015.

[20]

G. YinH. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints, J. Computational Appl. Math., 233 (2009), 564-581.  doi: 10.1016/j.cam.2009.08.055.

[21]

X. Y. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

[22]

M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modeling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.

[23]

J. Zhu, Dividend optimization for a regime-switching diffusion model with restricted dividend rates, ASTIN Bulletin, 44 (2014), 459-494.  doi: 10.1017/asb.2014.2.

show all references

References:
[1]

H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, RACSAM-Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 103 (2009), 295-320.  doi: 10.1007/BF03191909.

[2]

S. AsmussenB. Høgaard 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.  doi: 10.1007/s007800050075.

[3]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0.

[4]

P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, The Annals of Applied Probability, 20 (2010), 1253-1302.  doi: 10.1214/09-AAP643.

[5]

Y. C. Chi and H. Meng, Optimal reinsurance arrangements in the presence of two reinsurers, Scandinavian Actuarial Journal, 5 (2014), 424-438. 

[6]

T. ChoulliM. Taksar and X. Y. Zhou, Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction, Quant. Finance, 1 (2001), 573-596.  doi: 10.1088/1469-7688/1/6/301.

[7]

B. De Finetti, Su unimpostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. 

[8]

W. H. Fleming and T. Pang, An application of stochastic control theory to financial economics, SIAM Journal of Control and Optimization, 43 (2004), 502-531.  doi: 10.1137/S0363012902419060.

[9]

H. U. Gerber and E. S. W. Shiu, Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.  doi: 10.1080/10920277.2004.10596125.

[10]

H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.  doi: 10.1080/10920277.2006.10596249.

[11]

Z. JinH. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329.  doi: 10.1016/j.automatica.2013.04.043.

[12]

Z. JinH. Yang and G. Yin, Optimal debt ratio and dividend payment strategies with reinsurance, Insurance: Mathematics and Economics, 64 (2015), 351-363.  doi: 10.1016/j.insmatheco.2015.07.005.

[13]

N. Kulenko and H. Schimidli, An optimal dividend strategy in a Craḿer Lundberg model with capital injections, Insurance: Mathmatics and Economics, 43 (2008), 270-278.  doi: 10.1016/j.insmatheco.2008.05.013.

[14]

Z. F. LiY. Zeng and Y. Z. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance: Mathematics and Economics, 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002.

[15]

H. Meng and T. K. Siu, Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279.  doi: 10.1137/090773167.

[16]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum Press, New York, 1992.

[17]

Stein and L. Jerome, Stochastic Optimal Control and the U. S. Financial Debt Crisis Springer, New York, 2012.

[18]

J. WeiH. Yang and R. Wang, Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching, Journal of Optimization Theory and Applications, 147 (2010), 358-377.  doi: 10.1007/s10957-010-9726-x.

[19]

D. YaoH. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576.  doi: 10.1016/j.ejor.2011.01.015.

[20]

G. YinH. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints, J. Computational Appl. Math., 233 (2009), 564-581.  doi: 10.1016/j.cam.2009.08.055.

[21]

X. Y. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

[22]

M. Zhou and K. C. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modeling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.

[23]

J. Zhu, Dividend optimization for a regime-switching diffusion model with restricted dividend rates, ASTIN Bulletin, 44 (2014), 459-494.  doi: 10.1017/asb.2014.2.

[1]

Farai Julius Mhlanga, Lesiba Charles Galane, Nicholas Mwareya, Eriyoti Chikodza, Calisto Guambe. Stochastic differential game strategies in the presence of reinsurance and dividend payout. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022099

[2]

Linyi Qian, Lyu Chen, Zhuo Jin, Rongming Wang. Optimal liability ratio and dividend payment strategies under catastrophic risk. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1443-1461. doi: 10.3934/jimo.2018015

[3]

Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022068

[4]

Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062

[5]

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

[6]

Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2781-2797. doi: 10.3934/jimo.2019080

[7]

Xiaoming Yan, Minghui Zhang, Ke Liu, Yong Wang. Optimal ordering policies and sourcing strategies with supply disruption. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1147-1168. doi: 10.3934/jimo.2014.10.1147

[8]

Dingjun Yao, Kun Fan. Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1055-1083. doi: 10.3934/jimo.2017090

[9]

Jingzhen Liu, Shiqi Yan, Shan Jiang, Jiaqin Wei. Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022040

[10]

Dingjun Yao, Hailiang Yang, Rongming Wang. Optimal financing and dividend strategies in a dual model with proportional costs. Journal of Industrial and Management Optimization, 2010, 6 (4) : 761-777. doi: 10.3934/jimo.2010.6.761

[11]

Liming Zhang, Rongming Wang, Jiaqin Wei. Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021140

[12]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial and Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[13]

Hiroaki Hata, Li-Hsien Sun. Optimal investment and reinsurance of insurers with lognormal stochastic factor model. Mathematical Control and Related Fields, 2022, 12 (2) : 531-566. doi: 10.3934/mcrf.2021033

[14]

Ning Li, Zheng Wang. Optimal pricing and ordering strategies for dual-channel retailing with different shipping policies. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2021227

[15]

Jesús-Javier Chi-Domínguez, Francisco Rodríguez-Henríquez. Optimal strategies for CSIDH. Advances in Mathematics of Communications, 2022, 16 (2) : 383-411. doi: 10.3934/amc.2020116

[16]

Chao Xu, Yimeng Dong, Zhigang Ren, Huachen Jiang, Xin Yu. Sensor deployment for pipeline leakage detection via optimal boundary control strategies. Journal of Industrial and Management Optimization, 2015, 11 (1) : 199-216. doi: 10.3934/jimo.2015.11.199

[17]

Cristiana J. Silva, Delfim F. M. Torres. Optimal control strategies for tuberculosis treatment: A case study in Angola. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 601-617. doi: 10.3934/naco.2012.2.601

[18]

Holly Gaff, Elsa Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences & Engineering, 2009, 6 (3) : 469-492. doi: 10.3934/mbe.2009.6.469

[19]

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

[20]

Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1393-1423. doi: 10.3934/jimo.2021025

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (481)
  • HTML views (1217)
  • Cited by (2)

Other articles
by authors

[Back to Top]