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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

* Corresponding author: Zhibin Liang

Received  January 2020 Revised  September 2020 Published  March 2022 Early access  December 2020

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

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.

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.
The influence of higher borrowing rate on the optimal investment strategies
The influence of higher borrowing rate on the optimal reinsurance strategies
The influence of risky investment on the optimal reinsurance strategies
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