doi: 10.3934/jimo.2021218
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Distributionally robust multi-period portfolio selection subject to bankruptcy constraints

1. 

School of Management, Guangzhou University, Guangzhou, 400065, China

2. 

Department of Mathematics and Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia

*Corresponding author: Changzhi Wu (changzhiwu@gzhu.edu.cn)

Received  April 2021 Revised  October 2021 Early access December 2021

An optimization problem with moments information which suffers from distributional uncertainty can be handled through distributionally robust optimization. In this paper, we will consider distributionally robust multi-period portfolio selection since only moment information of portfolios can be gathered in practice. We will consider two different scenarios. One is that moments information can be obtained exactly and the other one is that the moments information is also uncertain. For the two scenarios, we will show how to transform the corresponding distributionally robust optimization problem into a second order cone problem (SOCP) which can be easily solved by existing methods. Some numerical experiments are presented to demonstrate the effectiveness of our proposed method.

Citation: Lin Jiang, Changzhi Wu, Song Wang. Distributionally robust multi-period portfolio selection subject to bankruptcy constraints. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021218
References:
[1]

S. AlexanderT. Coleman and Y. Li, Minimizing CVaR and VaR for a portfolio of derivatives, Journal of Banking and Finance, 30 (2006), 583-605. 

[2]

J. Blanchet, L. Chen and X. Zhou, Distributionally robust mean-variance portfolio selection with Wasserstein distances, arXiv: 1802.02885.

[3]

L. ChenS. He and S. Zhang, Tight bounds for some risk measures, with applications to robust portfolio selection, Oper. Res., 59 (2011), 847-865.  doi: 10.1287/opre.1110.0950.

[4]

Y. ChenH. Sun and H. Xu, Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems, Comput. Optim. Appl., 78 (2021), 205-238.  doi: 10.1007/s10589-020-00234-7.

[5]

X. CuiJ. GaoX. Li and D. Li, Optimal multi-period mean–variance policy under no-shorting constraint, European J. Oper. Res., 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.

[6]

X. CuiJ. GaoY. Shi and S. Zhu, Time-consistent and self-coordination strategies for multi-period mean-conditional value-at-risk portfolio selection, European J. Oper. Res., 276 (2019), 781-789.  doi: 10.1016/j.ejor.2019.01.045.

[7]

X. CuiX. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Trans. Automat. Control, 59 (2014), 1833-1844.  doi: 10.1109/TAC.2014.2311875.

[8]

E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res., 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.

[9]

L. GhaouiM. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res., 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101.

[10]

B. LiY. ZhuY. SunG. Aw and K. Teo, Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint, Appl. Math. Model., 56 (2018), 539-550.  doi: 10.1016/j.apm.2017.12.016.

[11]

D. Li and W. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Math. Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.

[12]

A. LingJ. Sun and M. Wang, Robust multi-period portfolio selection based on downside risk with asymmetrically distributed uncertainty set, European J. Oper. Res., 285 (2020), 81-95.  doi: 10.1016/j.ejor.2019.01.012.

[13]

B. Liu, A new risk measure and its application in portfolio optimization: The SPP–CVaR approach, Economic Modelling, 51 (2015), 383-390. 

[14]

J. LiuZ. ChenA. Lisser and Z. Xu, Closed-form optimal portfolios of distributionally robust mean-CVaR problems with unknown mean and variance, Appl. Math. Optim., 79 (2019), 671-693.  doi: 10.1007/s00245-017-9452-y.

[15]

K. LwinR. Qu and B. MacCarthy, Mean-VaR portfolio optimization: A nonparametric approach, European J. Oper. Res., 260 (2017), 751-766.  doi: 10.1016/j.ejor.2017.01.005.

[16]

H. Markowitz, Portfolio Selection, Cowles Foundation for Research in Economics at Yale University, Monograph 16 John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1959.

[17]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-42. 

[18]

H. Sun and H. Xu, Convergence analysis for distributionally robust optimization and equlibrium problems, Math. Oper. Res., 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.

[19]

Y. SunG. AwR. Loxton and K. L. Teo, Chance-constrained optimization for pension fund portfolios in the presence of default risk, European J. Oper. Res., 256 (2017), 205-214.  doi: 10.1016/j.ejor.2016.06.019.

[20]

H. YaoZ. Li and D. Li, Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability, European J. Oper. Res., 252 (2016), 837-851.  doi: 10.1016/j.ejor.2016.01.049.

[21]

H. YaoZ. Li and Y. Lai, Mean–CVaR portfolio selection: A nonparametric estimation framework, Comput. Oper. Res., 40 (2013), 1014-1022.  doi: 10.1016/j.cor.2012.11.007.

[22]

Y. ZhangR. Jiang and S. Shen, Ambiguous chance-constrained binary programs under mean-covariance information, SIAM J. Optim., 28 (2018), 2922-2944.  doi: 10.1137/17M1158707.

[23]

S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684.

[24]

S. ZymlerD. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Math. Program., 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.

show all references

References:
[1]

S. AlexanderT. Coleman and Y. Li, Minimizing CVaR and VaR for a portfolio of derivatives, Journal of Banking and Finance, 30 (2006), 583-605. 

[2]

J. Blanchet, L. Chen and X. Zhou, Distributionally robust mean-variance portfolio selection with Wasserstein distances, arXiv: 1802.02885.

[3]

L. ChenS. He and S. Zhang, Tight bounds for some risk measures, with applications to robust portfolio selection, Oper. Res., 59 (2011), 847-865.  doi: 10.1287/opre.1110.0950.

[4]

Y. ChenH. Sun and H. Xu, Decomposition and discrete approximation methods for solving two-stage distributionally robust optimization problems, Comput. Optim. Appl., 78 (2021), 205-238.  doi: 10.1007/s10589-020-00234-7.

[5]

X. CuiJ. GaoX. Li and D. Li, Optimal multi-period mean–variance policy under no-shorting constraint, European J. Oper. Res., 234 (2014), 459-468.  doi: 10.1016/j.ejor.2013.02.040.

[6]

X. CuiJ. GaoY. Shi and S. Zhu, Time-consistent and self-coordination strategies for multi-period mean-conditional value-at-risk portfolio selection, European J. Oper. Res., 276 (2019), 781-789.  doi: 10.1016/j.ejor.2019.01.045.

[7]

X. CuiX. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Trans. Automat. Control, 59 (2014), 1833-1844.  doi: 10.1109/TAC.2014.2311875.

[8]

E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res., 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.

[9]

L. GhaouiM. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res., 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101.

[10]

B. LiY. ZhuY. SunG. Aw and K. Teo, Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint, Appl. Math. Model., 56 (2018), 539-550.  doi: 10.1016/j.apm.2017.12.016.

[11]

D. Li and W. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Math. Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.

[12]

A. LingJ. Sun and M. Wang, Robust multi-period portfolio selection based on downside risk with asymmetrically distributed uncertainty set, European J. Oper. Res., 285 (2020), 81-95.  doi: 10.1016/j.ejor.2019.01.012.

[13]

B. Liu, A new risk measure and its application in portfolio optimization: The SPP–CVaR approach, Economic Modelling, 51 (2015), 383-390. 

[14]

J. LiuZ. ChenA. Lisser and Z. Xu, Closed-form optimal portfolios of distributionally robust mean-CVaR problems with unknown mean and variance, Appl. Math. Optim., 79 (2019), 671-693.  doi: 10.1007/s00245-017-9452-y.

[15]

K. LwinR. Qu and B. MacCarthy, Mean-VaR portfolio optimization: A nonparametric approach, European J. Oper. Res., 260 (2017), 751-766.  doi: 10.1016/j.ejor.2017.01.005.

[16]

H. Markowitz, Portfolio Selection, Cowles Foundation for Research in Economics at Yale University, Monograph 16 John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1959.

[17]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-42. 

[18]

H. Sun and H. Xu, Convergence analysis for distributionally robust optimization and equlibrium problems, Math. Oper. Res., 41 (2016), 377-401.  doi: 10.1287/moor.2015.0732.

[19]

Y. SunG. AwR. Loxton and K. L. Teo, Chance-constrained optimization for pension fund portfolios in the presence of default risk, European J. Oper. Res., 256 (2017), 205-214.  doi: 10.1016/j.ejor.2016.06.019.

[20]

H. YaoZ. Li and D. Li, Multi-period mean-variance portfolio selection with stochastic interest rate and uncontrollable liability, European J. Oper. Res., 252 (2016), 837-851.  doi: 10.1016/j.ejor.2016.01.049.

[21]

H. YaoZ. Li and Y. Lai, Mean–CVaR portfolio selection: A nonparametric estimation framework, Comput. Oper. Res., 40 (2013), 1014-1022.  doi: 10.1016/j.cor.2012.11.007.

[22]

Y. ZhangR. Jiang and S. Shen, Ambiguous chance-constrained binary programs under mean-covariance information, SIAM J. Optim., 28 (2018), 2922-2944.  doi: 10.1137/17M1158707.

[23]

S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Res., 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684.

[24]

S. ZymlerD. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Math. Program., 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.

Figure 1.  The optimal $ \mathbb{E}(x_t) $ with $ \underline{x} = 1.15 $ and $ \underline{x} = 1.196 $
Figure 2.  The optimal $ u_2(t) $ with $ \underline{x} = 1.15 $ and $ \underline{x} = 1.196 $
Figure 3.  The optimal $ u_3(t) $ with $ \underline{x} = 1.15 $ and $ \underline{x} = 1.196 $
Figure 4.  The optimal $ \textbf{u}_{t} $ with $ \gamma_1 = 0.0001 $ and $ \gamma_2 = 1.2 $
Figure 5.  The optimal investment return under different $ \gamma_2 $
Figure 6.  The optimal investment return under different $ \gamma_1 $
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