• Previous Article
    Two-agent integrated scheduling of production and distribution operations with fixed departure times
  • JIMO Home
  • This Issue
  • Next Article
    Analysis of dynamic service system between regular and retrial queues with impatient customers
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  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 & Management Optimization, doi: 10.3934/jimo.2021003
References:
[1]

B. AngoshtariE. Bayraktar and V. R. Young, Optimal investment to minimize the probability of drawdown, Stochastics, 88 (2016), 946-958.  doi: 10.1080/17442508.2016.1155590.  Google Scholar

[2]

B. AngoshtariE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X. ChenD. LandriaultB. 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.  Google Scholar

[9]

J. Cvitanić and I. Karatzas, On portfolio optimization under drawdown constrainsts, IMA Volumes in Mathematics and its Applications, 65 (1995), 77-88.   Google Scholar

[10]

C. DengX. 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.  Google Scholar

[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.  Google Scholar

[12]

C. FuA. 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.  Google Scholar

[13]

J. Grandell, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.  doi: 10.1080/03461238.1977.10405071.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

X. HanZ. 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.  Google Scholar

[17]

X. HanZ. 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.  Google Scholar

[18]

X. HanZ. 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.  Google Scholar

[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.  Google Scholar

[20]

X. LiangZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. Luo, Ruin minimization for insurers with borrowing constrainsts, North American Actuarial Journal, 12 (2008), 143-174.  doi: 10.1080/10920277.2008.10597508.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

K. C. YuenZ. 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.  Google Scholar

[30]

X. ZhangH. 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.  Google Scholar

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: 10.1080/17442508.2016.1155590.  Google Scholar

[2]

B. AngoshtariE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

X. ChenD. LandriaultB. 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.  Google Scholar

[9]

J. Cvitanić and I. Karatzas, On portfolio optimization under drawdown constrainsts, IMA Volumes in Mathematics and its Applications, 65 (1995), 77-88.   Google Scholar

[10]

C. DengX. 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.  Google Scholar

[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.  Google Scholar

[12]

C. FuA. 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.  Google Scholar

[13]

J. Grandell, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal, 1977 (1977), 37-52.  doi: 10.1080/03461238.1977.10405071.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

X. HanZ. 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.  Google Scholar

[17]

X. HanZ. 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.  Google Scholar

[18]

X. HanZ. 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.  Google Scholar

[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.  Google Scholar

[20]

X. LiangZ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. Luo, Ruin minimization for insurers with borrowing constrainsts, North American Actuarial Journal, 12 (2008), 143-174.  doi: 10.1080/10920277.2008.10597508.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

K. C. YuenZ. 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.  Google Scholar

[30]

X. ZhangH. 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.  Google Scholar

Figure 1.  The influence of higher borrowing rate on the optimal investment strategies
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]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[2]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[3]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[4]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[5]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[6]

Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014

[7]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207

[8]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004

[9]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[10]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[11]

Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020127

[12]

Ziteng Wang, Shu-Cherng Fang, Wenxun Xing. On constraint qualifications: Motivation, design and inter-relations. Journal of Industrial & Management Optimization, 2013, 9 (4) : 983-1001. doi: 10.3934/jimo.2013.9.983

[13]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

[14]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313

[15]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208

[16]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[17]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329

[18]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[19]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909

[20]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (29)
  • HTML views (73)
  • Cited by (0)

Other articles
by authors

[Back to Top]