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doi: 10.3934/jimo.2021145
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Optimal per-loss reinsurance and investment to minimize the probability of drawdown

1. 

Department of Statistics and Actuarial Science, University of Waterloo, Ontario, N2L 3G1, Canada

2. 

School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu, 210023 China

3. 

School of Management Science and Engineering, Nanjing University of Information Science and Technology, Jiangsu, 210044, China

4. 

School of Finance, Nanjing University of Finance and Economics, Jiangsu, 210023, China

*Corresponding author: Zhibin Liang

Received  January 2021 Revised  May 2021 Early access September 2021

Fund Project: This research was supported by the National Natural Science Foundation of China (Grant No.12071224)

In this paper, we study an optimal reinsurance-investment problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. We assume that the insurer can purchase per-loss reinsurance for each line of business and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. Under the criterion of minimizing the probability of drawdown, the closed-form expressions for the optimal reinsurance-investment strategy and the corresponding value function are obtained. We show that the optimal reinsurance strategy is in the form of pure excess-of-loss reinsurance strategy under the expected value principle, and under the variance premium principle, the optimal reinsurance strategy is in the form of pure quota-share reinsurance. Furthermore, we extend our model to the case where the insurance company involves $ n $ $ (n\geq3) $ dependent classes of insurance business and the optimal results are derived explicitly as well.

Citation: Xia Han, Zhibin Liang, Yu Yuan, Caibin Zhang. Optimal per-loss reinsurance and investment to minimize the probability of drawdown. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021145
References:
[1]

B. AngoshtariE. Bayraktar and V. R. Young, Optimal investment to minimize the probability of drawdown, Stochastics, 88 (2016), 946-958.  doi: 10.1080/17442508.20H.155590.

[2]

B. AngoshtariE. Bayraktar and V. R. Young, Minimizing the probability of lifetime drawdown under constant consumption, Insurance Math. Econom., 69 (2016), 210-223.  doi: 10.1016/j.insmatheco.2016.05.007.

[3]

L. BaiJ. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance Math. Econom., 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008.

[4]

M. Brachetta and C. Ceci, Optimal proportional reinsurance and investment for stochastic factor models, Insurance Math. Econo., 87 (2019), 15-33.  doi: 10.1016/j.insmatheco.2019.03.006.

[5]

X. ChenD. LandriaultB. Li and D. Li, On minimizing drawdown risks of lifetime investments, Insurance Math. Econom., 65 (2015), 46-54.  doi: 10.1016/j.insmatheco.2015.08.007.

[6]

A. GuX. GuoZ. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003.

[7]

S. J. Grossman and Z. Zhou, Optimal investment strategies for controlling drawdowns, Mathematical Finance, 3 (1993), 241-276. 

[8]

X. HanZ. Liang and V. R. Young, Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle, Scand. Actuar. J., 2020 (2020), 873-903.  doi: 10.1080/03461238.2020.1788136.

[9]

X. HanZ. Liang and K. C. Yuen, Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure, Scand. Actuar. J., 2018 (2018), 863-889.  doi: 10.1080/03461238.2018.1469098.

[10]

X. HanZ. Liang and C. Zhang, Optimal proportional reinsurance with common shock dependence to minimize the probability of drawdown, Annals of Actuarial Science, 13 (2019), 268-294. 

[11]

C. Hipp and M. Taksar, Optimal non-proportional reinsurance control, Insurance Math. Econom., 47 (2010), 246-254.  doi: 10.1016/j.insmatheco.2010.04.001.

[12]

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

[13]

C. Jaksa and K. Ioannis, On portfolio optimization under drawdown constraints, IMA Lecture Notes in Mathematical Applications, 65 (1995), 77-88. 

[14]

B. Karl, An attempt to determine the optimum amount of stop loss reinsurance, Transaction of the $16$th International Congress of Actuaries, (1960), 597–610.

[15]

D. LiX. Rong and H. Zhao, Equilibrium excess-of-loss reinsurance-investment strategy for a mean-variance insurer under stochastic volatility model, Comm. Statist. Theory Methods, 46 (2017), 9459-9475.  doi: 10.1080/03610926.20I.112071.

[16]

D. LiY. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679.

[17]

X. Liang and V. R. Young, Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance Math. Econom., 82 (2018), 181-190.  doi: 10.1016/j.insmatheco.2018.07.005.

[18]

X. Liang and V. R. Young, Minimizing the discounted probability of exponential Parisian ruin via reinsurance, SIAM J. Control Optim., 58 (2020), 937-964.  doi: 10.1137/19M1282714.

[19]

Z. LiangJ. BiK. C. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Math. Methods Oper. Res., 84 (2016), 155-181.  doi: 10.1007/s00186-016-0538-0.

[20]

Z. Liang and J. Guo, newblock Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, J. Appl. Math. Comput., 36 (2011), 11–25., doi: 10.1007/s12190-010-0385-8.

[21]

S. Luo and M. Taksar, On absolute ruin minimization under a diffusion approximation model, Insurance Math. Econom., 48 (2011), 123-133. 

[22]

V. C. Pestien and W. D. Sudderth, Continuous-time red and black: How to control a diffusion to a goal, Math. Oper. Res., 8 (1985), 599-611. 

[23]

S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128. 

[24]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907. 

[25]

Y. Yuan, Z. Liang and X. Han, Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs, Journal of Industrial and Management Optimization, (2021). doi: 10.3934/jimo.2021003.

[26]

X. ZhangH. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance Math. Econom., 67 (2016), 125-132. 

[27]

X. ZhangM. Zhou and J. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Appl. Stoch. Models Bus. Ind., 23 (2007), 63-71. 

show all references

References:
[1]

B. AngoshtariE. Bayraktar and V. R. Young, Optimal investment to minimize the probability of drawdown, Stochastics, 88 (2016), 946-958.  doi: 10.1080/17442508.20H.155590.

[2]

B. AngoshtariE. Bayraktar and V. R. Young, Minimizing the probability of lifetime drawdown under constant consumption, Insurance Math. Econom., 69 (2016), 210-223.  doi: 10.1016/j.insmatheco.2016.05.007.

[3]

L. BaiJ. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance Math. Econom., 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008.

[4]

M. Brachetta and C. Ceci, Optimal proportional reinsurance and investment for stochastic factor models, Insurance Math. Econo., 87 (2019), 15-33.  doi: 10.1016/j.insmatheco.2019.03.006.

[5]

X. ChenD. LandriaultB. Li and D. Li, On minimizing drawdown risks of lifetime investments, Insurance Math. Econom., 65 (2015), 46-54.  doi: 10.1016/j.insmatheco.2015.08.007.

[6]

A. GuX. GuoZ. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003.

[7]

S. J. Grossman and Z. Zhou, Optimal investment strategies for controlling drawdowns, Mathematical Finance, 3 (1993), 241-276. 

[8]

X. HanZ. Liang and V. R. Young, Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle, Scand. Actuar. J., 2020 (2020), 873-903.  doi: 10.1080/03461238.2020.1788136.

[9]

X. HanZ. Liang and K. C. Yuen, Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure, Scand. Actuar. J., 2018 (2018), 863-889.  doi: 10.1080/03461238.2018.1469098.

[10]

X. HanZ. Liang and C. Zhang, Optimal proportional reinsurance with common shock dependence to minimize the probability of drawdown, Annals of Actuarial Science, 13 (2019), 268-294. 

[11]

C. Hipp and M. Taksar, Optimal non-proportional reinsurance control, Insurance Math. Econom., 47 (2010), 246-254.  doi: 10.1016/j.insmatheco.2010.04.001.

[12]

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

[13]

C. Jaksa and K. Ioannis, On portfolio optimization under drawdown constraints, IMA Lecture Notes in Mathematical Applications, 65 (1995), 77-88. 

[14]

B. Karl, An attempt to determine the optimum amount of stop loss reinsurance, Transaction of the $16$th International Congress of Actuaries, (1960), 597–610.

[15]

D. LiX. Rong and H. Zhao, Equilibrium excess-of-loss reinsurance-investment strategy for a mean-variance insurer under stochastic volatility model, Comm. Statist. Theory Methods, 46 (2017), 9459-9475.  doi: 10.1080/03610926.20I.112071.

[16]

D. LiY. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679.

[17]

X. Liang and V. R. Young, Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance Math. Econom., 82 (2018), 181-190.  doi: 10.1016/j.insmatheco.2018.07.005.

[18]

X. Liang and V. R. Young, Minimizing the discounted probability of exponential Parisian ruin via reinsurance, SIAM J. Control Optim., 58 (2020), 937-964.  doi: 10.1137/19M1282714.

[19]

Z. LiangJ. BiK. C. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Math. Methods Oper. Res., 84 (2016), 155-181.  doi: 10.1007/s00186-016-0538-0.

[20]

Z. Liang and J. Guo, newblock Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, J. Appl. Math. Comput., 36 (2011), 11–25., doi: 10.1007/s12190-010-0385-8.

[21]

S. Luo and M. Taksar, On absolute ruin minimization under a diffusion approximation model, Insurance Math. Econom., 48 (2011), 123-133. 

[22]

V. C. Pestien and W. D. Sudderth, Continuous-time red and black: How to control a diffusion to a goal, Math. Oper. Res., 8 (1985), 599-611. 

[23]

S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128. 

[24]

H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907. 

[25]

Y. Yuan, Z. Liang and X. Han, Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs, Journal of Industrial and Management Optimization, (2021). doi: 10.3934/jimo.2021003.

[26]

X. ZhangH. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance Math. Econom., 67 (2016), 125-132. 

[27]

X. ZhangM. Zhou and J. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Appl. Stoch. Models Bus. Ind., 23 (2007), 63-71. 

Figure 1.  The function $ G(7,d) $
Figure 2.  The effect of $ u $ on optimal strategy
Figure 3.  The effect of η1 on optimal strategy (η2 = 0.25)
Figure 4.  The effect of $ \eta_2 $ on optimal strategy ($ \eta_1 = 0.4 $)
Figure 5.  The effect of $ \lambda $ on optimal strategy
Figure 6.  The effect of $ \lambda $ on correlation coefficient
Figure 7.  Comparison between the optimal strategies
Figure 8.  Comparison between the value functions
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