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October  2018, 14(4): 1443-1461. doi: 10.3934/jimo.2018015

Optimal liability ratio and dividend payment strategies under catastrophic risk

1. 

School of Statistics, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China

2. 

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

* Corresponding author: Lyu Chen

Received  December 2016 Revised  August 2017 Published  January 2018

Fund Project: This work was supported by National Natural Science Foundation of China (11571113,11771147), the 111 Project(B14019) and Faculty Research Grant of University of Melbourne.

This paper investigates the optimal strategies for liability management and dividend payment in an insurance company. The surplus process is jointly determined by the reinsurance policies, liability levels, future claims and unanticipated shocks. The decision maker aims to maximize the total expected discounted utility of dividend payment in infinite time horizon. To describe the extreme scenarios when catastrophic events occur, a jump-diffusion Cox-Ingersoll-Ross process is adopted to capture the substantial claim rate hikes. Using dynamic programming principle, the value function is the solution of a second-order integro-differential Hamilton-Jacobi-Bellman equation. The subsolution--supersolution method is used to verify the existence of classical solutions of the Hamilton-Jacobi-Bellman equation. The optimal liability ratio and dividend payment strategies are obtained explicitly in the cases where the utility functions are logarithm and power functions. A numerical example is provided to illustrate the methodologies and some interesting economic insights.

Citation: Linyi Qian, Lyu Chen, Zhuo Jin, Rongming Wang. Optimal liability ratio and dividend payment strategies under catastrophic risk. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1443-1461. doi: 10.3934/jimo.2018015
References:
[1]

J. Aharony and I. Swary, Quarterly dividend and earnings announcements and stockholders' returns: An empirical analysis, The Journal of Finance, 35 (1980), 1-12.   Google Scholar

[2]

L. H. R. Alvarez and J. Lempa, On the optimal stochastic impulse control of linear diffusions, SIAM Journal on Control and Optimization, 47 (2008), 703-732.  doi: 10.1137/060659375.  Google Scholar

[3]

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.  Google Scholar

[4]

F. AvramZ. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709.  Google Scholar

[5]

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.  Google Scholar

[6]

L. Bai and J. Paulsen, On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs, Stochastic Processes and Their Applications, 122 (2012), 4005-4027.  doi: 10.1016/j.spa.2012.08.009.  Google Scholar

[7]

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

[8]

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.  Google Scholar

[9]

J. C. CoxJ. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rate, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar

[10]

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

[11]

D. DuffieD. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833.  Google Scholar

[12]

D. FilipovićM. Mayerhofer and P. Schneider, Density approximations for multivariate affine jump-diffusion processes, J. Econometrics, 176 (2013), 93-111.  doi: 10.1016/j.jeconom.2012.12.003.  Google Scholar

[13]

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.  Google Scholar

[14]

Z. F. Fu and Z. H. Li, Stochastic equations of non-negative processes with jump, Stochastic Process. Appl., 120 (2010), 306-330.  doi: 10.1016/j.spa.2009.11.005.  Google Scholar

[15]

X. GuoJ. Liu and X. Y. Zhou, A constrained non-linear regular-singular stochastic control problem with applications, Stochastic Processes and their Applications, 109 (2004), 167-187.  doi: 10.1016/j.spa.2003.09.008.  Google Scholar

[16]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance policy, Insurance: Mathematics and Economics, 42 (2008), 976-983.  doi: 10.1016/j.insmatheco.2007.11.003.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

R. L. Loeffen and J. F. Renaud, De Finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108, Gerber-Shiu Functions / Longevity risk and capital markets. doi: 10.1016/j.insmatheco.2009.09.006.  Google Scholar

[20]

A. Lokka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[21]

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.  Google Scholar

[22]

M. H. Miller and F. Modigliani, Dividend policy, growth, and the valuation of shares, Journal of Business, 34 (1961), 411-433.   Google Scholar

[23]

M. H. Miller and K. Rock, Dividend policy under asymmetric information, The Journal of Finance, 40 (1985), 1031-1051.   Google Scholar

[24]

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

[25]

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

[26]

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.   Google Scholar

[27]

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.  Google Scholar

[28]

J. Zhu and H. Yang, Optimal financing and dividend distribution in a general diffusion model with regime switching, Advances in Applied Probability, 48 (2016), 406-422.  doi: 10.1017/apr.2016.7.  Google Scholar

show all references

References:
[1]

J. Aharony and I. Swary, Quarterly dividend and earnings announcements and stockholders' returns: An empirical analysis, The Journal of Finance, 35 (1980), 1-12.   Google Scholar

[2]

L. H. R. Alvarez and J. Lempa, On the optimal stochastic impulse control of linear diffusions, SIAM Journal on Control and Optimization, 47 (2008), 703-732.  doi: 10.1137/060659375.  Google Scholar

[3]

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.  Google Scholar

[4]

F. AvramZ. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative lévy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709.  Google Scholar

[5]

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.  Google Scholar

[6]

L. Bai and J. Paulsen, On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs, Stochastic Processes and Their Applications, 122 (2012), 4005-4027.  doi: 10.1016/j.spa.2012.08.009.  Google Scholar

[7]

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

[8]

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.  Google Scholar

[9]

J. C. CoxJ. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rate, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar

[10]

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

[11]

D. DuffieD. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833.  Google Scholar

[12]

D. FilipovićM. Mayerhofer and P. Schneider, Density approximations for multivariate affine jump-diffusion processes, J. Econometrics, 176 (2013), 93-111.  doi: 10.1016/j.jeconom.2012.12.003.  Google Scholar

[13]

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.  Google Scholar

[14]

Z. F. Fu and Z. H. Li, Stochastic equations of non-negative processes with jump, Stochastic Process. Appl., 120 (2010), 306-330.  doi: 10.1016/j.spa.2009.11.005.  Google Scholar

[15]

X. GuoJ. Liu and X. Y. Zhou, A constrained non-linear regular-singular stochastic control problem with applications, Stochastic Processes and their Applications, 109 (2004), 167-187.  doi: 10.1016/j.spa.2003.09.008.  Google Scholar

[16]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with proportional reinsurance policy, Insurance: Mathematics and Economics, 42 (2008), 976-983.  doi: 10.1016/j.insmatheco.2007.11.003.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

R. L. Loeffen and J. F. Renaud, De Finetti's optimal dividends problem with an affine penalty function at ruin, Insurance: Mathematics and Economics, 46 (2010), 98-108, Gerber-Shiu Functions / Longevity risk and capital markets. doi: 10.1016/j.insmatheco.2009.09.006.  Google Scholar

[20]

A. Lokka and M. Zervos, Optimal dividend and issuance of equity policies in the presence of proportional costs, Insurance: Mathematics and Economics, 42 (2008), 954-961.  doi: 10.1016/j.insmatheco.2007.10.013.  Google Scholar

[21]

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.  Google Scholar

[22]

M. H. Miller and F. Modigliani, Dividend policy, growth, and the valuation of shares, Journal of Business, 34 (1961), 411-433.   Google Scholar

[23]

M. H. Miller and K. Rock, Dividend policy under asymmetric information, The Journal of Finance, 40 (1985), 1031-1051.   Google Scholar

[24]

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

[25]

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

[26]

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.   Google Scholar

[27]

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.  Google Scholar

[28]

J. Zhu and H. Yang, Optimal financing and dividend distribution in a general diffusion model with regime switching, Advances in Applied Probability, 48 (2016), 406-422.  doi: 10.1017/apr.2016.7.  Google Scholar

Figure 1.  Optimal liability ratio values versus $c$
Figure 2.  Optimal liability ratio values versus $\mu$
Figure 3.  Optimal liability ratio values versus $\sigma$
Figure 4.  Optimal liability ratio values versus $\gamma$
Figure 5.  Optimal liability ratio values versus $\theta$ for logarithmic utility function
Figure 6.  Optimal liability ratio values versus $\theta$ for power utility function
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