Figure 1.
The influence of higher borrowing rate on the optimal investment strategies
-
Previous Article
Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices
- JIMO Home
- This Issue
-
Next Article
A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet
March
2022, 18(2): 933-967.
doi: 10.3934/jimo.2021003
Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, China |
We study the optimal investment and reinsurance problem in a risk model with two dependent classes of insurance businesses, where the two claim number processes are correlated through a common shock component and the borrowing rate is higher than the lending rate. The objective is to minimize the probability of drawdown, namely, the probability that the value of the wealth process reaches some fixed proportion of its maximum value to date. By the method of stochastic control theory and the corresponding Hamilton-Jacobi-Bellman equation, we investigate the optimization problem in two different cases and divide the whole region into four subregions. The explicit expressions for the optimal investment/reinsurance strategies and the minimum probability of drawdown are derived. We find that when wealth is at a relatively low level (below the borrowing level), it is optimal to borrow money to invest in the risky asset; when wealth is at a relatively high level (above the saving level), it is optimal to save more money; while between them, the insurer is willing to invest all the wealth in the risky asset. In the end, some comparisons are presented to show the impact of higher borrowing rate and risky investment on the optimal results.
Keywords:
Optimal investment,
proportional reinsurance,
common shock dependence,
borrowing constraint,
stochastic optimal control,
probability of drawdown.
Mathematics Subject Classification:
Primary: 93E20; Secondary: 62P05, 91G05, 91G10.
Citation:
Yu Yuan, Zhibin Liang, Xia Han. Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs. Journal of Industrial and Management Optimization,
2022, 18
(2)
: 933-967.
doi: 10.3934/jimo.2021003
![]()
References:
| [1] |
B. Angoshtari, E. Bayraktar and V. R. Young,
Optimal investment to minimize the probability of drawdown, Stochastics, 88 (2016), 946-958.
doi: 10.1080/17442508.2016.1155590. |
| [2] |
B. Angoshtari, E. Bayraktar and V. R. Young,
Minimizing the probability of lifetime drawdown under constant consumption, Insurance: Mathematics and Economics, 69 (2016), 210-223.
doi: 10.1016/j.insmatheco.2016.05.007. |
| [3] |
N. B$\ddot{a}$uerle,
Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.
doi: 10.1007/s00186-005-0446-1. |
| [4] |
E. Bayraktar and V. R. Young,
Minimizing the probability of lifetime ruin under borrowing constraints, Insurance: Mathematics and Economics, 41 (2007), 196-221.
doi: 10.1016/j.insmatheco.2006.10.015. |
| [5] |
E. Bayraktar and V. R. Young,
Minimizing the probability of ruin when consumption is ratcheted, North American Actuarial Journal, 12 (2008), 428-442.
doi: 10.1080/10920277.2008.10597535. |
| [6] |
L. Bo and A. Capponi, Optimal credit investment with borrowing costs, Mathematics of Operations Research, 42 (2017), 546-575.
doi: 10.1287/moor.2016.0818. |
| [7] |
S. Brown,
Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probaiblity of ruin, Mathematics of Operations Research, 20 (1995), 937-958.
doi: 10.1287/moor.20.4.937. |
| [8] |
X. Chen, D. Landriault, B. Li and D. Li,
On minimizing drawdown risks of lifetime investments, Insurance: Mathematics and Economics, 65 (2015), 46-54.
doi: 10.1016/j.insmatheco.2015.08.007. |
| [9] |
J. Cvitanić and I. Karatzas,
On portfolio optimization under drawdown constrainsts, IMA Volumes in Mathematics and its Applications, 65 (1995), 77-88.
|
| [10] |
C. Deng, X. Zeng and H. Zhu,
Non-zero-sum stochastic differential reinsurance and investment games with default risk, European Journal of Operational Research, 264 (2018), 1144-1158.
doi: 10.1016/j.ejor.2017.06.065. |
| [11] |
R. Elie and N. Touzi,
Optimal lifetime consumption and investment under a drawdown constrainst, Finance and Stochastics, 12 (2008), 299-330.
doi: 10.1007/s00780-008-0066-8. |
| [12] |
C. Fu, A. Lari-Lavassani and X. Li,
Dynamic mean-variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 312-319.
doi: 10.1016/j.ejor.2009.01.005. |
| [13] |
J. Grandell,
A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.
doi: 10.1080/03461238.1977.10405071. |
| [14] |
J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4613-9058-9. |
| [15] |
S. Grossman and Z. Zhou,
Optimal investment strategies for controlling drawdowns, Mathematical Finance, 3 (1993), 241-276.
doi: 10.1111/j.1467-9965.1993.tb00044.x. |
| [16] |
X. Han, Z. Liang and K. C. Yuen,
Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure, Scandinavian Actuarial Journal, 2018 (2018), 863-889.
doi: 10.1080/03461238.2018.1469098. |
| [17] |
X. Han, Z. Liang and V. R. Young,
Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle, Scandinavian Actuarial Journal, 2020 (2020), 879-903.
doi: 10.1080/03461238.2020.1788136. |
| [18] |
X. Han, Z. Liang and C. Zhang,
Optimal proportional reinsurance with common shock dependence to minimise the probability of drawdown, Annals of Actuarial Science, 13 (2019), 268-294.
doi: 10.1017/S1748499518000210. |
| [19] |
C. Hipp and M. Taksar,
Optimal non-proportional reinsurance, Insurance: Mathematics and Economics, 47 (2010), 246-254.
doi: 10.1016/j.insmatheco.2010.04.001. |
| [20] |
X. Liang, Z. Liang and V. R. Young,
Optimal reinsurance under the mean-variance premium principle to minimize the probability of ruin, Insurance: Mathematics and Economics, 92 (2020), 128-146.
doi: 10.1016/j.insmatheco.2020.03.008. |
| [21] |
X. Liang and V. R. Young,
Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance: Mathematics and Economics, 82 (2018), 181-190.
doi: 10.1016/j.insmatheco.2018.07.005. |
| [22] |
Z. Liang and E. Bayraktar,
Optimal proportional reinsurance and investment with unobservable claim size and intensity, Insurance: Mathematics and Economics, 55 (2014), 156-166.
doi: 10.1016/j.insmatheco.2014.01.011. |
| [23] |
Z. Liang and K. C. Yuen,
Optimal dynamic reinsurance with dependent risks: variance premium principle, Scandinavian Actuarial Journal, 2016 (2016), 18-36.
doi: 10.1080/03461238.2014.892899. |
| [24] |
S. Luo,
Ruin minimization for insurers with borrowing constrainsts, North American Actuarial Journal, 12 (2008), 143-174.
doi: 10.1080/10920277.2008.10597508. |
| [25] |
R. C. Merton,
Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.
doi: 10.2307/1926560. |
| [26] |
R. C. Merton,
Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.
doi: 10.1016/0022-0531(71)90038-X. |
| [27] |
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), 110-128.
doi: 10.1080/10920277.2005.10596214. |
| [28] |
V. R. Young,
Optimal investmet strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126.
doi: 10.1080/10920277.2004.10596174. |
| [29] |
K. C. Yuen, Z. Liang and M. Zhou,
Optimal proportional reinsurance with common shock dependence, Insurance: Mathematic and Economics, 64 (2015), 1-13.
doi: 10.1016/j.insmatheco.2015.04.009. |
| [30] |
X. Zhang, H. Meng and Y. Zeng,
Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance: Mathematic and Economics, 67 (2016), 125-132.
doi: 10.1016/j.insmatheco.2016.01.001. |
show all references
References:
| [1] |
B. Angoshtari, E. Bayraktar and V. R. Young,
Optimal investment to minimize the probability of drawdown, Stochastics, 88 (2016), 946-958.
doi: 10.1080/17442508.2016.1155590. |
| [2] |
B. Angoshtari, E. Bayraktar and V. R. Young,
Minimizing the probability of lifetime drawdown under constant consumption, Insurance: Mathematics and Economics, 69 (2016), 210-223.
doi: 10.1016/j.insmatheco.2016.05.007. |
| [3] |
N. B$\ddot{a}$uerle,
Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.
doi: 10.1007/s00186-005-0446-1. |
| [4] |
E. Bayraktar and V. R. Young,
Minimizing the probability of lifetime ruin under borrowing constraints, Insurance: Mathematics and Economics, 41 (2007), 196-221.
doi: 10.1016/j.insmatheco.2006.10.015. |
| [5] |
E. Bayraktar and V. R. Young,
Minimizing the probability of ruin when consumption is ratcheted, North American Actuarial Journal, 12 (2008), 428-442.
doi: 10.1080/10920277.2008.10597535. |
| [6] |
L. Bo and A. Capponi, Optimal credit investment with borrowing costs, Mathematics of Operations Research, 42 (2017), 546-575.
doi: 10.1287/moor.2016.0818. |
| [7] |
S. Brown,
Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probaiblity of ruin, Mathematics of Operations Research, 20 (1995), 937-958.
doi: 10.1287/moor.20.4.937. |
| [8] |
X. Chen, D. Landriault, B. Li and D. Li,
On minimizing drawdown risks of lifetime investments, Insurance: Mathematics and Economics, 65 (2015), 46-54.
doi: 10.1016/j.insmatheco.2015.08.007. |
| [9] |
J. Cvitanić and I. Karatzas,
On portfolio optimization under drawdown constrainsts, IMA Volumes in Mathematics and its Applications, 65 (1995), 77-88.
|
| [10] |
C. Deng, X. Zeng and H. Zhu,
Non-zero-sum stochastic differential reinsurance and investment games with default risk, European Journal of Operational Research, 264 (2018), 1144-1158.
doi: 10.1016/j.ejor.2017.06.065. |
| [11] |
R. Elie and N. Touzi,
Optimal lifetime consumption and investment under a drawdown constrainst, Finance and Stochastics, 12 (2008), 299-330.
doi: 10.1007/s00780-008-0066-8. |
| [12] |
C. Fu, A. Lari-Lavassani and X. Li,
Dynamic mean-variance portfolio selection with borrowing constraint, European Journal of Operational Research, 200 (2010), 312-319.
doi: 10.1016/j.ejor.2009.01.005. |
| [13] |
J. Grandell,
A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.
doi: 10.1080/03461238.1977.10405071. |
| [14] |
J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4613-9058-9. |
| [15] |
S. Grossman and Z. Zhou,
Optimal investment strategies for controlling drawdowns, Mathematical Finance, 3 (1993), 241-276.
doi: 10.1111/j.1467-9965.1993.tb00044.x. |
| [16] |
X. Han, Z. Liang and K. C. Yuen,
Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure, Scandinavian Actuarial Journal, 2018 (2018), 863-889.
doi: 10.1080/03461238.2018.1469098. |
| [17] |
X. Han, Z. Liang and V. R. Young,
Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle, Scandinavian Actuarial Journal, 2020 (2020), 879-903.
doi: 10.1080/03461238.2020.1788136. |
| [18] |
X. Han, Z. Liang and C. Zhang,
Optimal proportional reinsurance with common shock dependence to minimise the probability of drawdown, Annals of Actuarial Science, 13 (2019), 268-294.
doi: 10.1017/S1748499518000210. |
| [19] |
C. Hipp and M. Taksar,
Optimal non-proportional reinsurance, Insurance: Mathematics and Economics, 47 (2010), 246-254.
doi: 10.1016/j.insmatheco.2010.04.001. |
| [20] |
X. Liang, Z. Liang and V. R. Young,
Optimal reinsurance under the mean-variance premium principle to minimize the probability of ruin, Insurance: Mathematics and Economics, 92 (2020), 128-146.
doi: 10.1016/j.insmatheco.2020.03.008. |
| [21] |
X. Liang and V. R. Young,
Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance: Mathematics and Economics, 82 (2018), 181-190.
doi: 10.1016/j.insmatheco.2018.07.005. |
| [22] |
Z. Liang and E. Bayraktar,
Optimal proportional reinsurance and investment with unobservable claim size and intensity, Insurance: Mathematics and Economics, 55 (2014), 156-166.
doi: 10.1016/j.insmatheco.2014.01.011. |
| [23] |
Z. Liang and K. C. Yuen,
Optimal dynamic reinsurance with dependent risks: variance premium principle, Scandinavian Actuarial Journal, 2016 (2016), 18-36.
doi: 10.1080/03461238.2014.892899. |
| [24] |
S. Luo,
Ruin minimization for insurers with borrowing constrainsts, North American Actuarial Journal, 12 (2008), 143-174.
doi: 10.1080/10920277.2008.10597508. |
| [25] |
R. C. Merton,
Lifetime portfolio selection under uncertainty: The continuous-time case, The Review of Economics and Statistics, 51 (1969), 247-257.
doi: 10.2307/1926560. |
| [26] |
R. C. Merton,
Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.
doi: 10.1016/0022-0531(71)90038-X. |
| [27] |
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), 110-128.
doi: 10.1080/10920277.2005.10596214. |
| [28] |
V. R. Young,
Optimal investmet strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126.
doi: 10.1080/10920277.2004.10596174. |
| [29] |
K. C. Yuen, Z. Liang and M. Zhou,
Optimal proportional reinsurance with common shock dependence, Insurance: Mathematic and Economics, 64 (2015), 1-13.
doi: 10.1016/j.insmatheco.2015.04.009. |
| [30] |
X. Zhang, H. Meng and Y. Zeng,
Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance: Mathematic and Economics, 67 (2016), 125-132.
doi: 10.1016/j.insmatheco.2016.01.001. |
Figure 2.
The influence of higher borrowing rate on the optimal reinsurance strategies
Figure 3.
The influence of risky investment on the optimal reinsurance strategies
| [1] |
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 |
| [2] |
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, 2022, 18 (6) : 4011-4041. doi: 10.3934/jimo.2021145 |
| [3] |
Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1915-1934. doi: 10.3934/jimo.2021049 |
| [4] |
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 |
| [5] |
Bo Yang, Rongming Wang, Gongpin Cheng. Optimal dividend and capital injection strategies in common shock dependence model with time-inconsistent preferences. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022034 |
| [6] |
Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143 |
| [7] |
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 |
| [8] |
Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2195-2211. doi: 10.3934/jimo.2019050 |
| [9] |
Jingzhen Liu, Lihua Bai, Ka-Fai Cedric Yiu. Optimal investment with a value-at-risk constraint. Journal of Industrial and Management Optimization, 2012, 8 (3) : 531-547. doi: 10.3934/jimo.2012.8.531 |
| [10] |
Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial and Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743 |
| [11] |
Zuo Quan Xu, Fahuai Yi. An optimal consumption-investment model with constraint on consumption. Mathematical Control and Related Fields, 2016, 6 (3) : 517-534. doi: 10.3934/mcrf.2016014 |
| [12] |
Pengxu Xie, Lihua Bai, Huayue Zhang. Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022020 |
| [13] |
Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial and Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008 |
| [14] |
Yan Zhang, Peibiao Zhao. Optimal reinsurance-investment problem with dependent risks based on Legendre transform. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1457-1479. doi: 10.3934/jimo.2019011 |
| [15] |
Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial and Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044 |
| [16] |
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 |
| [17] |
Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control and Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003 |
| [18] |
Xiaoyu Xing, Caixia Geng. Optimal investment-reinsurance strategy in the correlated insurance and financial markets. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3445-3459. doi: 10.3934/jimo.2021120 |
| [19] |
Xia Zhou, Peimin Chen, Jiawei Zhang, Jingwen Tu, Yong He. The optimal investment-reinsurance strategies for ambiguity aversion insurer in uncertain environment. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022141 |
| [20] |
Meng Wu, Jiefeng Yang. The optimal exit of staged investment when consider the posterior probability. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1105-1123. doi: 10.3934/jimo.2016064 |
2021 Impact Factor: 1.411
Tools
Metrics
Other articles
by authors
[Back to Top]