Multiperiod distributionally robust portfolio selection with regime-switching under CVaR risk measures

Fei Yu; Fei Yu

doi:10.3934/math.2025456

AIMS Mathematics

2025, Volume 10, Issue 4: 9974-10001. doi: 10.3934/math.2025456

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Research article

Multiperiod distributionally robust portfolio selection with regime-switching under CVaR risk measures

Fei Yu ^,

School of Mathematics and Big Data, Chongqing University of Education, Chongqing 400065, China

Received: 01 January 2025 Revised: 02 April 2025 Accepted: 10 April 2025 Published: 27 April 2025
MSC : 9110, 91G10, 91G70

Optimal investment strategy selection has become a primary research focus in investment science and operations research. Key challenges in this field include identifying an appropriate risk measure to capture potential extreme losses, accurately modeling the impact of market volatility on investment decisions, and effectively balancing returns and risks. To handle uncertainty in return distributions, robust portfolio optimization is a more recent approach. In this study, we employ robust Conditional Value-at-Risk (CVaR) as the risk measure and propose a multi-stage robust portfolio selection model incorporating both risk-free and risky assets under a known first and second moment uncertainty set. By integrating a regime-switching framework, we derive an analytical optimal investment strategy using dynamic programming (DP) techniques. Our numerical analysis demonstrates that the optimal strategy determined by dynamic programming adjusts dynamically at each stage in response to regime switches.
- portfolio selection,
- robust optimization,
- CVaR/VaR,
- uncertainty set,
- optimal investment strategy
Citation: Fei Yu. Multiperiod distributionally robust portfolio selection with regime-switching under CVaR risk measures[J]. AIMS Mathematics, 2025, 10(4): 9974-10001. doi: 10.3934/math.2025456

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Abstract

Optimal investment strategy selection has become a primary research focus in investment science and operations research. Key challenges in this field include identifying an appropriate risk measure to capture potential extreme losses, accurately modeling the impact of market volatility on investment decisions, and effectively balancing returns and risks. To handle uncertainty in return distributions, robust portfolio optimization is a more recent approach. In this study, we employ robust Conditional Value-at-Risk (CVaR) as the risk measure and propose a multi-stage robust portfolio selection model incorporating both risk-free and risky assets under a known first and second moment uncertainty set. By integrating a regime-switching framework, we derive an analytical optimal investment strategy using dynamic programming (DP) techniques. Our numerical analysis demonstrates that the optimal strategy determined by dynamic programming adjusts dynamically at each stage in response to regime switches.

References

[1]	P. Artzner, F. Delbaen, E. Jean-Marc, Coherent measures of risk, Math. Finance, 9 (1999), 203–228. https://doi.org/10.1111/1467-9965.00068 doi: 10.1111/1467-9965.00068
[2]	C. Acerbi, D. Tasche, On the coherence of expected shortfall, J. Bank. Finance, 26 (2002), 1487–1503. https://doi.org/10.1016/S0378-4266(02)00283-2 doi: 10.1016/S0378-4266(02)00283-2
[3]	R. T. Rockafellar, S. Uryasev, Conditional value-at-risk for general loss distributions, J. Bank. Finance, 26 (2002), 1443–1471. Available from: https://www.sciencedirect.com/science/article/pii/S0378426602002716
[4]	C. Acerbi, Spectral measures of risk: A coherent representation of subjective risk aversion, J. Bank. Finance, 26 (2002), 1505–1518. https://doi.org/10.1016/S0378-4266(02)00281-9 doi: 10.1016/S0378-4266(02)00281-9
[5]	T. Fischer, Risk capital allocation by coherent risk measures based on one-sided moments, Math. Economics, 32 (2003), 135–146. https://doi.org/10.1016/S0167-6687(02)00209-3 doi: 10.1016/S0167-6687(02)00209-3
[6]	A. L. Soyster, Technical note-convex programming with set-inclusive constraints and applications to inexact linear programming, Operat. Research, 21 (1973), 1154–1157. https://doi.org/10.1287/opre.21.5.1154 doi: 10.1287/opre.21.5.1154
[7]	A. Ben-Tal, T. Margalit, A. Nemirovski, robust modeling of multi-stage portfolio problems, High Perf. Optim., 33 (2000), 303–328. https://doi.org/10.1007/978-1-4757-3216-0_12 doi: 10.1007/978-1-4757-3216-0_12
[8]	L. E. Ghaoui, M. Oks, F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Research, 51 (2003), 543–556. https://doi.org/10.1287/opre.51.4.543.16101 doi: 10.1287/opre.51.4.543.16101
[9]	L. Garlappi, R. Uppal, T. Wang, Portfolio selection with parameter and model uncertainty: A multi-prior approach, Rev. Financ. Stud., 20 (2007), 41–81. https://doi.org/10.1093/rfs/hhl003 doi: 10.1093/rfs/hhl003
[10]	O. L. V. Costa, A. C. Paiva, Robust portfolio selection using linear-matrix inequalities, J. Economic Dyn. Control, 26 (2002), 889–909. https://doi.org/10.1016/S0165-1889(00)00086-5 doi: 10.1016/S0165-1889(00)00086-5
[11]	D. Goldfarb, G. Iyengar, Robust portfolio selection problems, Math. Operat. Research, 28 (2003), 1–38. https://doi.org/10.1287/moor.28.1.1.14260 doi: 10.1287/moor.28.1.1.14260
[12]	Z. Lu, A new cone programming approach for robust portfolio selection, Opti. Methods Soft., 26 (2006), 89–104. Available from: https://optimization-online.org/wp-content/uploads/2006/12/1554.pdf
[13]	B. V. Halldórsson, R. H. Tütüncü, An interior-point method for a class of saddle-point problems, J. Opt. Theory Appl., 116 (2003), 559–590. https://doi.org/10.1023/A:1023065319772 doi: 10.1023/A:1023065319772
[14]	I. Popescu, Robust mean-covariance solutions for stochastic optimization, Oper. Research, 55 (2007), 98–112. https://doi.org/10.1287/opre.1060.0353 doi: 10.1287/opre.1060.0353
[15]	K. Natarajan, M. Sim, J. Uichanco, Tractable robust expected utility and risk models for portfolio optimization, Math. Finance, 20 (2010), 695–731. https://doi.org/10.1111/j.1467-9965.2010.00417.x doi: 10.1111/j.1467-9965.2010.00417.x
[16]	S. Zhu, M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Oper. Research, 57 (2009), 1155–1168. https://doi.org/10.1287/opre.1080.0684 doi: 10.1287/opre.1080.0684
[17]	N. Du, Y. Liu, Y. Liu, A new data-driven distributionally robust portfolio optimization method based on wasserstein ambiguity set, IEEE Access, 9 (2021), 3174–3194. https://doi.org/10.1109/ACCESS.2020.3047967 doi: 10.1109/ACCESS.2020.3047967
[18]	C. S. Pun, T. Wang, Z. Yan, Data-driven distributionally robust cvar portfolio optimization under a regime-switching ambiguity set, Manufac. Service Oper. Mana., 25 (2023), 1779–1795. https://doi.org/10.1287/msom.2023.1229 doi: 10.1287/msom.2023.1229
[19]	D. Huang, S. S. Zhu, F. J. Fabozzi, M. Fukushima, Portfolio selection with uncertain exit time: A robust CVaR approach, J. Economic Dyn. Control, 32 (2008), 594–623. https://doi.org/10.1016/j.jedc.2007.03.003 doi: 10.1016/j.jedc.2007.03.003
[20]	R.H. Tütüncü, M. Koenig, Robust asset allocation, Annal. Oper. Research, 132 (2004), 157–187. https://doi.org/10.1023/B:ANOR.0000045281.41041.ed doi: 10.1023/B:ANOR.0000045281.41041.ed
[21]	B. Rustem, R. G. Becker, W. Marty, Robust min-max portfolio strategies for rival forecast and risk scenarios, J. Economic Dyn. Control, 24 (2000), 1591–1621. https://doi.org/10.1016/S0165-1889(99)00088-3 doi: 10.1016/S0165-1889(99)00088-3
[22]	M. J. Best, R. R. Grauer, On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results, Rev. Fina. Studies, 4 (1991), 315–342. https://doi.org/10.1093/rfs/4.2.315 doi: 10.1093/rfs/4.2.315
[23]	F. Black, R. Litterman, Global portfolio optimization, Fina. Anal. J., 48 (1992), 28–43. https://doi.org/10.2469/faj.v48.n5.28 doi: 10.2469/faj.v48.n5.28
[24]	Z. Dai, X. Chen, F. Wen, A modified Perry's conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations, Appl. Math. Comput., 270 (2015), 378–386. https://doi.org/10.1016/j.amc.2015.08.014 doi: 10.1016/j.amc.2015.08.014
[25]	C. Peng, Y. S. Kim, S. Mittnik, Portfolio optimization on multivariate regime-switching garch model with normal tempered stable innovation, J. Risk Fina. Manag., 15 (2022), 230. https://doi.org/10.3390/jrfm15050230 doi: 10.3390/jrfm15050230
[26]	S. Wan, Stochastic differential portfolio games based on utility with regime switching model, 2007 IEEE Int. Conf. Control Autom., 2007, 2302–2305. https://doi.org/10.1109/ICCA.2007.4376772
[27]	S. Wan, Stochastic control in optimal portfolio with regime switching model, Rob. Vision 2006 9th Int. Conf. Control, 2006, 1–4. https://doi.org/10.1109/ICARCV.2006.345087
[28]	Z. Dai, H. Zhu, X. Chang, F. Wen, Forecasting stock returns: The role of VIX-based upper and lower shadow of Japanese candlestick, Fina. Innov., 11 (2025), 19. https://doi.org/10.1186/s40854-024-00682-8 doi: 10.1186/s40854-024-00682-8
[29]	W. Deng, J. Wang, A. Guo, H. Zhao, Quantum differential evolutionary algorithm with quantum-adaptive mutation strategy and population state evaluation framework for high-dimensional problems, Inf. Sci., 676 (2024), 120787. https://doi.org/10.1016/j.ins.2024.120787 doi: 10.1016/j.ins.2024.120787
[30]	Z. Zhu, X. Li, H. Chen, X. Zhou, W. Deng, An effective and robust genetic algorithm with hybrid multi-strategy and mechanism for airport gate allocation, Inf. Sci., 654 (2024), 119892. https://doi.org/10.1016/j.ins.2023.119892 doi: 10.1016/j.ins.2023.119892
[31]	Y. Lin, M. Ruan, K. Cai, D. Li, Z. Zeng, F. Li, B. Yang, Identifying and managing risks of AI-driven operations: A case study of automatic speech recognition for improving air traffic safety, Chinese J. Aeronautics, 36 (2023), 366–386. https://doi.org/10.1016/j.cja.2022.08.020 doi: 10.1016/j.cja.2022.08.020
[32]	A. Ling, J. Sun, M. Wang, Robust multi-period portfolio selection based on downside risk with asymmetrically distributed uncertainty set, Europ. J. Operat. Research, 285 (2020), 81–95. https://doi.org/10.1016/j.ejor.2019.01.012 doi: 10.1016/j.ejor.2019.01.012
[33]	J. Liu, Z. Chen, Time consistent multi-period robust risk measures and portfolio selection models with regime-switching, Europ. J. Oper. Research, 268 (2018), 373–385. https://doi.org/10.1016/j.ejor.2018.01.009 doi: 10.1016/j.ejor.2018.01.009
[34]	R.T. Rockafellar, S. Uryasev, Optimization of conditional value-at-risk, J. Risk, 2 (2000), 21–41. https://doi.org/10.21314/JOR.2000.038 doi: 10.21314/JOR.2000.038
[35]	A. Shapiro, Topics in stochastic programming, CORE lecture series, universite catholique de louvain, 2011. Available from: https://bpb-us-e1.wpmucdn.com/sites.gatech.edu/dist/4/1470/files/2021/03/core.pdf
[36]	L. Chen, S. He, S. Zhang, Tight bounds for some risk measures, with applications to robust portfolio selection, Oper. Research, 59 (2011), 847–865. https://doi.org/10.1287/opre.1110.0950 doi: 10.1287/opre.1110.0950
[37]	Q. Zhang, P. Zhang, T. Li, Information fusion for large-scale multi-source data based on the Dempster-Shafer evidence theory, Inf. Fusion, 115 (2025), 102754. https://doi.org/10.1016/j.inffus.2024.102754 doi: 10.1016/j.inffus.2024.102754
[38]	P. Zhang, D. Wang, Z. Yu, Y. Zhang, T. Jiang, T. Li, A multi-scale information fusion-based multiple correlations for unsupervised attribute selection, Inf. Fusion, 106 (2024), 102276. https://doi.org/10.1016/j.inffus.2024.102276 doi: 10.1016/j.inffus.2024.102276
[39]	R. Yao, H. Zhao, Z. Zhao, C. Guo, W. Deng, Parallel convolutional transfer network for bearing fault diagnosis under varying operation states, IEEE Trans. Instr. Measur., 73 (2024), 1–13. https://doi.org/10.1109/TIM.2024.3480212 doi: 10.1109/TIM.2024.3480212

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