American Institute of Mathematical Sciences

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July  2018, 14(3): 1055-1083. doi: 10.3934/jimo.2017090

Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle

Dingjun Yao 1, and  Kun Fan 2,,

1. 

School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China
2. 

School of Statistics, East China Normal University, Shanghai 200241, China

* Corresponding author

Received  March 2017 Revised  June 2017 Published  July 2018 Early access  September 2017

Fund Project: This work was supported by National Natural Science Foundation of China (71671082,71471081,11501211), Humanities and Social Sciences Project of the Ministry Education of China (15YJC910008), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (15KJB110009), Shanghai Pujiang Program (15PJC026), Shanghai Philosophy Social Science Planning Office Project (2015EJB002), China Postdoctoral Science Foundation (2015M581564), Shanghai Chenguang Plan (15CG22).

This paper assumes that an insurer can control the dividend, refinancing and reinsurance strategies dynamically. Particularly, the reinsurance is provided by two reinsurers and the variance premium principle is applied in pricing insurance contracts. Using the optimal control method, we identify the optimal strategies for maximizing the insurance company's value. Meanwhile, the effects of transaction costs and terminal value at bankruptcy are investigated. The results turn out that the insurer should consider refinancing when and only when the transaction costs and terminal value are relatively low. Also, it should buy less reinsurance when the surplus increases, while the proportion of risk allocation between two reinsurers remains constant. When the dividend rate is unbounded, dividends should be paid according to the barrier strategy. When the dividend rate is restricted, dividends should be distributed according to the threshold strategy. Some examples are provided to illustrate the implementation of our results.

Keywords: Dividend, refinancing, reinsurance with two reinsurers, variance premium principle, terminal value.
Mathematics Subject Classification: Primary: 97M30, 93E20; Secondary: 91G80.
Citation: 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
References:
[1]

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.

[2]

L. Bai, J. Guo and H. Zhang, Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes, Quantitative Finance, 10 (2010), 1163-1172.  doi: 10.1080/14697680902968005.

[3]

A. Barth and S. Moreno-Bromberg, Optimal risk and liquidity management with costly refinancing opportunities, Insurance: Mathematics and Economics, 57 (2014), 31-45.  doi: 10.1016/j.insmatheco.2014.05.001.

[4]

A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202.  doi: 10.1111/j.1467-9965.2006.00267.x.

[5]

M. Chen, X. Peng and J. Guo, Optimal dividend problem with a nonlinear regular-singular stochastic control, Insurance: Mathematics and Economics, 52 (2013), 448-456.  doi: 10.1016/j.insmatheco.2013.02.010.

[6]

M. Chen and K. C. Yuen, Optimal dividend and reinsurance in the presence of two reinsurers, Journal of Applied Probability, 53 (2016), 554-571.  doi: 10.1017/jpr.2016.20.

[7] B. De Finetti, Su un'impostzione alternativa della teoria collettiva del rischio, in Transactions of the XVth International Congress of Actuaries, Congrès Internationald'Actuaires, New York, 1957. 
[8]

W. Fleming and H. Soner, Controlled Markov Process and Viscosity Solutions, Springer-Verlag, 1993.

[9]

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.

[10]

J. Grandell, Aspects of Risk Theory, Springer-Verlag, 1991. doi: 10.1007/978-1-4613-9058-9.

[11]

H. Guan and Z. Liang, Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs, Insurance: Mathematics and Economics, 54 (2014), 109-122.  doi: 10.1016/j.insmatheco.2013.11.003.

[12]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94.  doi: 10.1016/j.insmatheco.2008.10.001.

[13]

B. H$\phi $gaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.

[14]

Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005.

[15]

W. Liu and Y. Hu, Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy, Statistics and Probability Letters, 84 (2014), 121-130.  doi: 10.1016/j.spl.2013.09.034.

[16]

R. L. Loeffen, An optimal dividends problem with transaction costs for spectrally negative L$\acute{e}$vy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.  doi: 10.1016/j.insmatheco.2009.03.002.

[17]

A. L$φ$kka 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.

[18]

H. Meng, Optimal impulse control with variance premium principle, Science China Mathematics (in Chinese), 43 (2013), 925-939. 

[19]

H. Meng and T. K. Siu, On optimal reinsurance, dividend and reinvestment strategies, Economic Modelling, 28 (2011), 211-218.  doi: 10.1016/j.econmod.2010.09.009.

[20]

X. Peng, M. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.

[21]

M. Taksar, Dependence of the optimal risk control decisions on the terminal value for a financial corporation, Annals of Operations Research, 98 (2000), 89-99.  doi: 10.1023/A:1019239920624.

[22]

J. Xu and M. Zhou, Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.  doi: 10.1016/j.mcm.2011.12.041.

[23]

D. Yao, H. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.

[24]

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

show all references

References:
[1]

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.

[2]

L. Bai, J. Guo and H. Zhang, Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes, Quantitative Finance, 10 (2010), 1163-1172.  doi: 10.1080/14697680902968005.

[3]

A. Barth and S. Moreno-Bromberg, Optimal risk and liquidity management with costly refinancing opportunities, Insurance: Mathematics and Economics, 57 (2014), 31-45.  doi: 10.1016/j.insmatheco.2014.05.001.

[4]

A. Cadenillas, T. Choulli, M. Taksar and L. Zhang, Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm, Mathematical Finance, 16 (2006), 181-202.  doi: 10.1111/j.1467-9965.2006.00267.x.

[5]

M. Chen, X. Peng and J. Guo, Optimal dividend problem with a nonlinear regular-singular stochastic control, Insurance: Mathematics and Economics, 52 (2013), 448-456.  doi: 10.1016/j.insmatheco.2013.02.010.

[6]

M. Chen and K. C. Yuen, Optimal dividend and reinsurance in the presence of two reinsurers, Journal of Applied Probability, 53 (2016), 554-571.  doi: 10.1017/jpr.2016.20.

[7] B. De Finetti, Su un'impostzione alternativa della teoria collettiva del rischio, in Transactions of the XVth International Congress of Actuaries, Congrès Internationald'Actuaires, New York, 1957. 
[8]

W. Fleming and H. Soner, Controlled Markov Process and Viscosity Solutions, Springer-Verlag, 1993.

[9]

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.

[10]

J. Grandell, Aspects of Risk Theory, Springer-Verlag, 1991. doi: 10.1007/978-1-4613-9058-9.

[11]

H. Guan and Z. Liang, Viscosity solution and impulse control of the diffusion model with reinsurance and fixed transaction costs, Insurance: Mathematics and Economics, 54 (2014), 109-122.  doi: 10.1016/j.insmatheco.2013.11.003.

[12]

L. He and Z. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance: Mathematics and Economics, 44 (2009), 88-94.  doi: 10.1016/j.insmatheco.2008.10.001.

[13]

B. H$\phi $gaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.

[14]

Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005.

[15]

W. Liu and Y. Hu, Optimal financing and dividend control of the insurance company with excess-of-loss reinsurance policy, Statistics and Probability Letters, 84 (2014), 121-130.  doi: 10.1016/j.spl.2013.09.034.

[16]

R. L. Loeffen, An optimal dividends problem with transaction costs for spectrally negative L$\acute{e}$vy processes, Insurance: Mathematics and Economics, 45 (2009), 41-48.  doi: 10.1016/j.insmatheco.2009.03.002.

[17]

A. L$φ$kka 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.

[18]

H. Meng, Optimal impulse control with variance premium principle, Science China Mathematics (in Chinese), 43 (2013), 925-939. 

[19]

H. Meng and T. K. Siu, On optimal reinsurance, dividend and reinvestment strategies, Economic Modelling, 28 (2011), 211-218.  doi: 10.1016/j.econmod.2010.09.009.

[20]

X. Peng, M. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.

[21]

M. Taksar, Dependence of the optimal risk control decisions on the terminal value for a financial corporation, Annals of Operations Research, 98 (2000), 89-99.  doi: 10.1023/A:1019239920624.

[22]

J. Xu and M. Zhou, Optimal risk control and dividend distribution policies for a diffusion model with terminal value, Mathematical and Computer Modelling, 56 (2012), 180-190.  doi: 10.1016/j.mcm.2011.12.041.

[23]

D. Yao, H. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.

[24]

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

Figure 1.  The influences of $\beta_1$ on $u(x)$ and $u_1(x)$
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Figure 2.  The influences of the refinancing costs on u1(x)
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Figure 3.  The influences of P on u(x) and u1(x)
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Figure 4.  The influences of M on v(x) and v1(x)
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Table 1.  The influence of β1 on optimal strategy
$\beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, P=4, K=0.2$
${ \beta_1}\uparrow$ $q^{\pi^*}(0)\downarrow$ $h_0\uparrow$ $q_1^{\pi^*}(0)\uparrow$ $h_1\downarrow$ $\tilde{\xi}_0\uparrow$ $u'(0)\uparrow$ $I(\xi_0)\uparrow$ Re
$0.70$ 0.6409 3.2729 - - - 1.2438 0.0184 n
$0.75$ 0.6116 3.4762 - - - 1.4845 0.1096 n
$0.80$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.2610 y
$0.85$ 0.5719 3.7665 0.6236 3.3919 0.9531 2.0110 0.4598 y
$0.90$ 0.5582 3.8730 0.6455 3.2418 1.0286 2.2903 0.6979 y
$0.95$ 0.5472 3.9621 0.6695 3.0787 1.1255 2.5775 0.9700 y
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$\beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, P=4, K=0.2$
${ \beta_1}\uparrow$ $q^{\pi^*}(0)\downarrow$ $h_0\uparrow$ $q_1^{\pi^*}(0)\uparrow$ $h_1\downarrow$ $\tilde{\xi}_0\uparrow$ $u'(0)\uparrow$ $I(\xi_0)\uparrow$ Re
$0.70$ 0.6409 3.2729 - - - 1.2438 0.0184 n
$0.75$ 0.6116 3.4762 - - - 1.4845 0.1096 n
$0.80$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.2610 y
$0.85$ 0.5719 3.7665 0.6236 3.3919 0.9531 2.0110 0.4598 y
$0.90$ 0.5582 3.8730 0.6455 3.2418 1.0286 2.2903 0.6979 y
$0.95$ 0.5472 3.9621 0.6695 3.0787 1.1255 2.5775 0.9700 y
Table 2.  The influence of β2 on optimal strategy
$\beta_1=0.8, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, P=4, K=0.2$
${ \beta_2}\uparrow$ $q^{\pi^*}(0)\equiv$ $h_0\equiv$ $q_1^{\pi^*}(0)\downarrow$ $h_1\uparrow$ $\tilde{\xi}_0\downarrow$ $u'(0)\equiv$ $I(\xi_0)\downarrow$ Re
$1.05$ 0.5892 3.6368 - - - 1.7414 0.3145 n
$1.10$ 0.5892 3.6368 - - - 1.7414 0.2610 n
$1.15$ 0.5892 - - - 3.6368 1.7414 0.2146 n
$1.20$ 0.5892 3.6368 0.6160 3.4445 0.9557 1.7414 0.1743 y
$1.25$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.1395 y
$1.30$ 0.5892 3.6368 0.5927 3.6116 0.8387 1.7414 0.1095 y
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$\beta_1=0.8, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, P=4, K=0.2$
${ \beta_2}\uparrow$ $q^{\pi^*}(0)\equiv$ $h_0\equiv$ $q_1^{\pi^*}(0)\downarrow$ $h_1\uparrow$ $\tilde{\xi}_0\downarrow$ $u'(0)\equiv$ $I(\xi_0)\downarrow$ Re
$1.05$ 0.5892 3.6368 - - - 1.7414 0.3145 n
$1.10$ 0.5892 3.6368 - - - 1.7414 0.2610 n
$1.15$ 0.5892 - - - 3.6368 1.7414 0.2146 n
$1.20$ 0.5892 3.6368 0.6160 3.4445 0.9557 1.7414 0.1743 y
$1.25$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.1395 y
$1.30$ 0.5892 3.6368 0.5927 3.6116 0.8387 1.7414 0.1095 y
Table 3.  The influence of K on optimal strategy
$\beta_1=0.8, \beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, P=4$
$K\uparrow$ $q^{\pi^*}(0)\equiv$ $h_0\equiv$ $q_1^{\pi^*}(0)\downarrow$ $h_1\uparrow$ $\tilde{\xi}_0\uparrow$ $u'(0)\equiv$ $I(\xi_0)\equiv$ Re
$0.05$ 0.5892 3.6368 0.6625 3.1266 0.4850 1.7414 0.2610 y
$0.10$ 0.5892 3.6368 0.6364 3.3034 0.6619 1.7414 0.2610 y
$0.15$ 0.5892 3.6368 0.6180 3.4308 0.7893 1.7414 0.2610 y
$0.20$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.2610 y
$0.25$ 0.5892 3.6368 0.5916 3.6195 0.9779 1.7414 0.2610 y
$0.30$ 0.5892 - - - 3.6368 1.7414 0.2610 n
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$\beta_1=0.8, \beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, P=4$
$K\uparrow$ $q^{\pi^*}(0)\equiv$ $h_0\equiv$ $q_1^{\pi^*}(0)\downarrow$ $h_1\uparrow$ $\tilde{\xi}_0\uparrow$ $u'(0)\equiv$ $I(\xi_0)\equiv$ Re
$0.05$ 0.5892 3.6368 0.6625 3.1266 0.4850 1.7414 0.2610 y
$0.10$ 0.5892 3.6368 0.6364 3.3034 0.6619 1.7414 0.2610 y
$0.15$ 0.5892 3.6368 0.6180 3.4308 0.7893 1.7414 0.2610 y
$0.20$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.2610 y
$0.25$ 0.5892 3.6368 0.5916 3.6195 0.9779 1.7414 0.2610 y
$0.30$ 0.5892 - - - 3.6368 1.7414 0.2610 n
Table 4.  The influence of P on optimal strategy
$\beta_1=0.8, \beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, K=0.2$
$P\uparrow$ $q^{\pi^*}(0)\uparrow$ $h_0\downarrow$ $q_1^{\pi^*}(0)\equiv$ $h_1\equiv$ $\tilde{\xi}_0\equiv$ $u'(0)\downarrow$ $I(\xi_0)\downarrow$ Re
$-4$ 0.4368 5.5029 0.6035 3.5331 0.8915 9.6961 6.2083 y
$-2$ 0.4449 5.2679 0.6035 3.5331 0.8915 7.4376 4.4668 y
$0$ 0.4600 4.9497 0.6035 3.5331 0.8915 5.2780 2.8168 y
$2$ 0.4940 4.4728 0.6035 3.5331 0.8915 3.3073 1.3414 y
$4$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.2610 y
$6$ 0.8209 2.0027 - - - 0.9351 - n
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$\beta_1=0.8, \beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, K=0.2$
$P\uparrow$ $q^{\pi^*}(0)\uparrow$ $h_0\downarrow$ $q_1^{\pi^*}(0)\equiv$ $h_1\equiv$ $\tilde{\xi}_0\equiv$ $u'(0)\downarrow$ $I(\xi_0)\downarrow$ Re
$-4$ 0.4368 5.5029 0.6035 3.5331 0.8915 9.6961 6.2083 y
$-2$ 0.4449 5.2679 0.6035 3.5331 0.8915 7.4376 4.4668 y
$0$ 0.4600 4.9497 0.6035 3.5331 0.8915 5.2780 2.8168 y
$2$ 0.4940 4.4728 0.6035 3.5331 0.8915 3.3073 1.3414 y
$4$ 0.5892 3.6368 0.6035 3.5331 0.8915 1.7414 0.2610 y
$6$ 0.8209 2.0027 - - - 0.9351 - n
Table 5.  The influence of M on optimal strategy
$\beta_1=0.8, \beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, K=0.2, P=4$
$M\uparrow$ $q^{\pi^*}(0)\downarrow$ $d_0\uparrow$ $q_1^{\pi^*}(0)\uparrow$ $d_1\uparrow$ $\tilde{\zeta}_0\uparrow$ $v'(0)\uparrow$ $J(\zeta_0)\uparrow$ Re
$0.5$ 0.6036 2.3715 - - - 1.5666 0.1530 n
$1$ 0.5918 3.0719 0.6011 3.0056 0.8841 1.7066 0.2380 y
$5$ 0.5893 3.5344 0.6035 3.4318 0.8913 1.7403 0.2602 y
$10$ 0.5892 3.5862 0.6035 3.4827 0.8915 1.7411 0.2608 y
$100$ 0.5892 3.6318 0.6035 3.5280 0.8915 1.7414 0.2610 y
$1000$ 0.5892 3.6363 0.6035 3.5325 0.8915 1.7414 0.2610 y
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$\beta_1=0.8, \beta_2=1.1, \lambda=0.5, \theta_0=0.24, \theta_1=1, \theta_2=0.8, \sigma^2=2, \delta=0.05, K=0.2, P=4$
$M\uparrow$ $q^{\pi^*}(0)\downarrow$ $d_0\uparrow$ $q_1^{\pi^*}(0)\uparrow$ $d_1\uparrow$ $\tilde{\zeta}_0\uparrow$ $v'(0)\uparrow$ $J(\zeta_0)\uparrow$ Re
$0.5$ 0.6036 2.3715 - - - 1.5666 0.1530 n
$1$ 0.5918 3.0719 0.6011 3.0056 0.8841 1.7066 0.2380 y
$5$ 0.5893 3.5344 0.6035 3.4318 0.8913 1.7403 0.2602 y
$10$ 0.5892 3.5862 0.6035 3.4827 0.8915 1.7411 0.2608 y
$100$ 0.5892 3.6318 0.6035 3.5280 0.8915 1.7414 0.2610 y
$1000$ 0.5892 3.6363 0.6035 3.5325 0.8915 1.7414 0.2610 y
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