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doi: 10.3934/jimo.2022054
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## $\alpha$-robust portfolio optimization problem under the distribution uncertainty

 1 School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China 2 School of International Education, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China

*Corresponding author: Peibiao Zhao

Received  October 2021 Revised  February 2022 Early access April 2022

Fund Project: This research was funded by NNSF of China (no.11871275)

In this paper, we investigate the $\alpha$-robust portfolio optimization problem under the distribution uncertainty of returns. We establish the model associated with the safety-first criterion and a generalized Chebyshev's inequality. Then we achieve the optimal investment strategy to this model. At the last part, an empirical analysis is carried out.

Citation: Shihan Di, Dong Ma, Peibiao Zhao. $\alpha$-robust portfolio optimization problem under the distribution uncertainty. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022054
##### References:
 [1] C. Acerbi and D. Tasche, On the coherence of expected shortfall, Journal of Banking and Finance, 26 (2002), 1487-1503. doi: 10.1016/S0378-4266(02)00283-2. [2] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068. [3] L. Chen, S. 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] A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Ops. Research Letters, 25 (1999), 1-13. doi: 10.1016/S0167-6377(99)00016-4. [5] M. A. Bhat, G. Kosuru, A generalization of chebyshev's inequality and its applications, arXiv, (2021), 1-11. doi: https://arXiv.org/abs/2108.01479. [6] C. Cong and P. Zhao, Non-cash risk measure on nonconvex sets, Mathematics, 6 (2018), 186.  doi: 10.3390/math6100186. [7] C. Cong and P. Zhao, Nonconvex noncash risk measures, Journal of Risk Model Validation, 15 (2021), no.2, 23-38. doi: 10.21314/JRMV.2021.004. [8] D. Ellsberg, Risk, ambiguity, and the Savage axioms, The Quarterly Journal of Economics, 75 (1961), 643-669. doi: 10.2307/1884324. [9] J. E. Dawson, Solution I: A sum of Chebyshev inequalities, Amer. Math, 107 (2000), 282-283. doi: 10.2307/2589335. [10] 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. [11] H. Föllmer and A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics, 6 (2002), 429-447.  doi: 10.1007/s007800200072. [12] H. Föllmer and A. Schied, Robust preferences and convex measures of risk, In Advances in Finance and Stochastics, (2002), 39-56. doi: 10.1007/97836620479032. [13] L. E. Ghaoui, M. Oks and F. Oustry, Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach, Operations Research, 51 (2003), 543-556. doi: 10.1287/opre.51.4.543.16101. [14] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econ., 18 (1989), 141-153. doi: 10.1016/0304-4068(89)90018-9. [15] W. Guo, C. Gao and S. O. Finance, Optimal Portfolio Selection Bounded by Safety-first Criterion for Insurer, Journal of Henan Normal University(Natural Science Edition), 43 (2015), 14-19. [16] C. Heath and A. Tversky, Preference and belief: Ambiguity and competence in choice under uncertainty, Journal of Risk and Uncertainty, 4 (1991), 5-28. doi: 10.1007/BF00057884. [17] Z. Kang, X. Li and Z. Li, Mean-CVaR portfolio selection model with ambiguity in distribution and attitude, Journal of Industrial and Management Optimization, 16 (2017), 3065-3081. doi: 10.3934/jimo.2019094. [18] Z. Kang, X. Li, Z. Li and S. Zhu, Data-driven robust mean-CVaR portfolio selection under distribution ambiguity, Quant. Finance, 19 (2019), 105-121. doi: 10.1080/14697688.2018.1466057. [19] S. Kataoka, A stochastic programming model, Econometricah, 31 (1963), 181-196. doi: 10.2307/1910956. [20] J. Liu, Z. Chen, A. Lisser and Z. Xu, Closed-Form optimal portfolios of distributionally robust mean-CVaR problems with unknown mean and variance, Appl. Math. and Optimization, 79 (2019), 671-693. doi: 10.1007/s00245-017-9452-y. [21] P. Ghirardato, F. Maccheroni and M. Marinacci, ADifferentiating ambiguity and ambiguity attitude, Journal of Economic Theory, 118 (2004), 133-173. doi: 10.1016/j.jet.2003.12.004. [22] H. Markowitz, Portfolio selection, J. of Finance, 7 (1952), 77-91. doi: 10.12987/9780300191677. [23] K. Natarajan, D. Pachamanova and M. Sim, AIncorporating asymmetric distributional information in robust value-at-risk optimization, Management Science, 54 (2008), 573-585. [24] P. Y. Qian, Z. Z. Wang and Z. W. Wen, A Composite Risk Measure Framework for Decision Making under Uncertainty, Journal of the Operations Research Society of China, (2015), 1-26. doi: 10.1007/s40305-018-0211-9. [25] K. Natarajan, D. Pachamanova and M. Sim, Constructing risk measures from uncertainty sets, Operations Research, 57 (2009), 1129-1141. doi: 10.1287/opre.1080.0683. [26] D. H. Pyle and S. J. Turnovsky, Safety-First and Expected Utility Maximization in Mean-Standard Deviation Portfolio Analysis, Review of Economics and Statistics, 52 (1970), 75-81. doi: 10.1016/B978-0-12-780850-5.50025-3. [27] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-Atrisk, Journal of Risk, 2 (2000), 21-41. doi: 10.1007/978-1-4757-6594-6_17. [28] R. T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1007/978-1-4757-6594-6_17. [29] A. D. Roy, Safety first and the holding of assets, Econometrica, 20 (1952), 431-449. doi: 10.2307/1907413. [30] L. Telser, Safety first and hedging, Review of Economic Studies, 23 (1956), 1-16. doi: 10.2307/2296146. [31] W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Ops. Research, 62 (2014), 1358-1376. doi: 10.1287/opre.2014.1314. [32] 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.

show all references

##### References:
 [1] C. Acerbi and D. Tasche, On the coherence of expected shortfall, Journal of Banking and Finance, 26 (2002), 1487-1503. doi: 10.1016/S0378-4266(02)00283-2. [2] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068. [3] L. Chen, S. 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] A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Ops. Research Letters, 25 (1999), 1-13. doi: 10.1016/S0167-6377(99)00016-4. [5] M. A. Bhat, G. Kosuru, A generalization of chebyshev's inequality and its applications, arXiv, (2021), 1-11. doi: https://arXiv.org/abs/2108.01479. [6] C. Cong and P. Zhao, Non-cash risk measure on nonconvex sets, Mathematics, 6 (2018), 186.  doi: 10.3390/math6100186. [7] C. Cong and P. Zhao, Nonconvex noncash risk measures, Journal of Risk Model Validation, 15 (2021), no.2, 23-38. doi: 10.21314/JRMV.2021.004. [8] D. Ellsberg, Risk, ambiguity, and the Savage axioms, The Quarterly Journal of Economics, 75 (1961), 643-669. doi: 10.2307/1884324. [9] J. E. Dawson, Solution I: A sum of Chebyshev inequalities, Amer. Math, 107 (2000), 282-283. doi: 10.2307/2589335. [10] 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. [11] H. Föllmer and A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics, 6 (2002), 429-447.  doi: 10.1007/s007800200072. [12] H. Föllmer and A. Schied, Robust preferences and convex measures of risk, In Advances in Finance and Stochastics, (2002), 39-56. doi: 10.1007/97836620479032. [13] L. E. Ghaoui, M. Oks and F. Oustry, Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach, Operations Research, 51 (2003), 543-556. doi: 10.1287/opre.51.4.543.16101. [14] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econ., 18 (1989), 141-153. doi: 10.1016/0304-4068(89)90018-9. [15] W. Guo, C. Gao and S. O. Finance, Optimal Portfolio Selection Bounded by Safety-first Criterion for Insurer, Journal of Henan Normal University(Natural Science Edition), 43 (2015), 14-19. [16] C. Heath and A. Tversky, Preference and belief: Ambiguity and competence in choice under uncertainty, Journal of Risk and Uncertainty, 4 (1991), 5-28. doi: 10.1007/BF00057884. [17] Z. Kang, X. Li and Z. Li, Mean-CVaR portfolio selection model with ambiguity in distribution and attitude, Journal of Industrial and Management Optimization, 16 (2017), 3065-3081. doi: 10.3934/jimo.2019094. [18] Z. Kang, X. Li, Z. Li and S. Zhu, Data-driven robust mean-CVaR portfolio selection under distribution ambiguity, Quant. Finance, 19 (2019), 105-121. doi: 10.1080/14697688.2018.1466057. [19] S. Kataoka, A stochastic programming model, Econometricah, 31 (1963), 181-196. doi: 10.2307/1910956. [20] J. Liu, Z. Chen, A. Lisser and Z. Xu, Closed-Form optimal portfolios of distributionally robust mean-CVaR problems with unknown mean and variance, Appl. Math. and Optimization, 79 (2019), 671-693. doi: 10.1007/s00245-017-9452-y. [21] P. Ghirardato, F. Maccheroni and M. Marinacci, ADifferentiating ambiguity and ambiguity attitude, Journal of Economic Theory, 118 (2004), 133-173. doi: 10.1016/j.jet.2003.12.004. [22] H. Markowitz, Portfolio selection, J. of Finance, 7 (1952), 77-91. doi: 10.12987/9780300191677. [23] K. Natarajan, D. Pachamanova and M. Sim, AIncorporating asymmetric distributional information in robust value-at-risk optimization, Management Science, 54 (2008), 573-585. [24] P. Y. Qian, Z. Z. Wang and Z. W. Wen, A Composite Risk Measure Framework for Decision Making under Uncertainty, Journal of the Operations Research Society of China, (2015), 1-26. doi: 10.1007/s40305-018-0211-9. [25] K. Natarajan, D. Pachamanova and M. Sim, Constructing risk measures from uncertainty sets, Operations Research, 57 (2009), 1129-1141. doi: 10.1287/opre.1080.0683. [26] D. H. Pyle and S. J. Turnovsky, Safety-First and Expected Utility Maximization in Mean-Standard Deviation Portfolio Analysis, Review of Economics and Statistics, 52 (1970), 75-81. doi: 10.1016/B978-0-12-780850-5.50025-3. [27] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-Atrisk, Journal of Risk, 2 (2000), 21-41. doi: 10.1007/978-1-4757-6594-6_17. [28] R. T. Rockafellar and S. Uryasev, Conditional Value-at-Risk for general loss distributions, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1007/978-1-4757-6594-6_17. [29] A. D. Roy, Safety first and the holding of assets, Econometrica, 20 (1952), 431-449. doi: 10.2307/1907413. [30] L. Telser, Safety first and hedging, Review of Economic Studies, 23 (1956), 1-16. doi: 10.2307/2296146. [31] W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Ops. Research, 62 (2014), 1358-1376. doi: 10.1287/opre.2014.1314. [32] 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.
Closing price trend figure of four stocks
Daily return rate fluctuation figure of four stocks
$\alpha$-robust portfolio efficient frontier based on generalized Chebyshev's inequality of four stocks
The proportion of $\alpha$-robust portfolio efficient frontier based on generalized Chebyshev's inequality of four stocks
Annual return rate, price earnings ratio and price-to-book ratio of the four stocks
 Stock name Annual return rate price earnings ratio price-to-book ratio Longi Stock 2.7286 42.6588 8.5426 Jinlei Stock 1.391 21.6987 3.341 Great Wall Motor 3.3006 83.3346 3.2168 Sunshine Power Supply 5.8708 68.6777 4.1329
 Stock name Annual return rate price earnings ratio price-to-book ratio Longi Stock 2.7286 42.6588 8.5426 Jinlei Stock 1.391 21.6987 3.341 Great Wall Motor 3.3006 83.3346 3.2168 Sunshine Power Supply 5.8708 68.6777 4.1329
Expected return rate of each stock
 Stock code 601012 300443 601633 300274 Expected return rate 0.1289 0.0874 0.1517 0.1909
 Stock code 601012 300443 601633 300274 Expected return rate 0.1289 0.0874 0.1517 0.1909
Covariance matrix of each stock
 Stock code 601012 300443 601633 300274 601012 0.0298 0.0201 0.0186 0.0229 300443 0.0201 0.0306 0.0279 0.0193 601633 0.0186 0.0279 0.0569 0.0352 300274 0.0229 0.0193 0.0352 0.0418
 Stock code 601012 300443 601633 300274 601012 0.0298 0.0201 0.0186 0.0229 300443 0.0201 0.0306 0.0279 0.0193 601633 0.0186 0.0279 0.0569 0.0352 300274 0.0229 0.0193 0.0352 0.0418
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