Mean-AVaR-Skewness-Kurtosis Optimization Portfolio Selection Model in Uncertain Environments
Subject Areas : Financial Mathematics
Farahnaz Omidi
1
,
Leila Torkzadeh
2
,
Kazem Nouri
3
1 -
2 -
3 -
Keywords: Portfolio optimization, Uncertain variables, Skewness, Kurtosis, Average Value-at-Risk, Mean AVaR-skewness-kurtosis Model,
Abstract :
Several research investigations have indicated that asset returns exhibit notable skewness and kurtosis, which have a substantial impact on the utility function of investors. Additionally, it has been observed that Average Value-at-Risk (AVaR) provides a more accurate estimation of risk compared to variance. This study focuses on the computational challenge associated with portfolio optimization in an uncertain context, employing the Mean-AVaR-skewness-kurtosis paradigm.The uncertainty around the total return is con-sidered and analyzed in the context of the challenge of selecting an optimal portfolio. The concepts of Value-at-Risk (VaR), Average Value-at-Risk (AVaR), skewness, and kurtosis are initially introduced to describe uncertain variables. These concepts are then further explored to identify and analyse relevant aspects within specific distributions. The outcomes of this study will convert the existing models into deterministic forms and uncertain mean-AVaR-skewness-kurtosis optimization models for portfolio selection. These models are designed to cater to the demands of investors and mitigate their apprehensions.
[1] Markowitz, H., Portfolio selection. J Finance. 1952; 7(1): 77-91. Doi:10.2307/2975974
[2] Takano, Y., Sotirov, R., A polynomial optimization approach to constant rebalanced portfolio selection. Comput Optim Appl, 2012; 52: 645-666 Doi:10.1007/s10589-011-9436-9
[3] Basak, S., Shapiro, A., Value-at-risk based risk management: optimal policies and asset prices, Rev. Financ. Stud. 2001; 14: 371-405.
[4] Cai, X., Teo, K.L., Yang, X., Zhou, X., Portfolio optimization under a minimax rule, Manag. Sci. 2000, 46: 957-972. Doi:10.1287/mnsc.46.7.957.12039
[5] Campbell, R., Huisman, R., Koedijk, K., Optimal portfolio selection in a value-at-risk framework, J. Bank. Finance, 2001; 25: 1789-1804. Doi:10.1016/S0378-4266(00)00160-6
[6] Chen, F.Y., Analytical VaR for international portfolios with common jumps, Comput. Math. with Appl. 2011; 62: 3066-3076. Doi:10.1016/j.camwa.2011.08.018
[7] Hogan, W.W., Warren, J.M., Toward the development of an equilibrium capital market model based on semivariance, J. Financ. Quant. Anal. 1974; 9: 1-11. Doi:10.2307/2329964
[8] Konno, H., Piecewise linear risk function and portfolio optimization, J. Oper. Res. Soc. Jpn. 1990; 33: 139-156. http://www.jstor.org/stable/2632458.
[9] Li, B., Sun, Y., Aw, G., Teo, K.L., Uncertain portfolio optimization problem under a minimax risk measure, Appl. Math. Model. 2019; 76: 274-281. Doi:10.1016/j.apm.2019.06.019
[10] Polak, G.G., Rogers, D.F., Sweeney, D.J., Risk management strategies via minimax portfolio optimization, Eur. J. Oper. Res. 2010; 207: 409-419.
[11] Rockafellar, R.T., Uryasev, S., Conditional value-at-risk for general loss distributions, J. Bank. Finance, 2002; 26: 1443-1471. Doi:10.1016/S0378-4266(02)00271-6
[12] Sun, Y., Aw, G., Teo, K.L., Zhou, G., Portfolio optimization using a new probabilistic risk measure, J. Ind. Manag. Optim. 2015; 11: 1275-1283. Doi: 10.3934/jimo.2015.11.1275
[13] Yoshida, Y., Maximization of Returns under an Average Value-at-Risk Constraint in Fuzzy Asset Management, Procedia Com. Sci., 2017; 112: 11-20. Doi:10.1016/j.procs.2017.08.001
[14] Konno, H., Tanaka, K., Yamamoto, R., Construction of a portfolio with shorter downside tail and longer upside tail, Comput Optim Appl, 2011; 48: 199-212. Doi:10.1007/s10589-009-9255-4
[15] Liu, B., Uncertainty Theory, seconded, Springer-Verlag, Berlin, 2007. Doi:10.1007/978-3-540-73165-8
[16] Esmaeili B., Souri, A., Mirlohi, S.M., Higher moments portfolio Optimization with unequal weights based on Generalized Capital Asset pricing model with independent and identically asymmetric Power Distribution. Adv. Math. Fin. App., 2021; 6(2): 263-283. Doi: 10.22034/amfa.2020.1909590.1484
[17] Amini, A., Khalili Araghi, M., Nikoomaram, H., Investigating portfolio performance with higher moment considering entropy and rolling window in banking, insurance, and leasing industries. Adv. Math. Fin. App., 2024; 9(1): 67-83. Doi: 10.22034/AMFA.2022.1954904.1720
[18] Briec, W., Kerstens, K., Jokung, O., Mean-Variance-Skewness Portfolio Performance Gauging: A General Shortage Function and Dual Approach, Manag. Sci., 2007; 53(1):135-149. Doi:10.1287/mnsc.1060.0596
[19] Chunhachinda, P., Dandapani, K., Hamid, S., Prakash, A.J., Portfolio Selection And Skewness: Evidence From International Stock Markets, J. Bank. Finance, 1997; 21: 143-167. Doi:10.1016/S0378-4266(96)00032-5
[20] Cremers, J.H., Kritzman, M., Page, S., Portfolio Formation With Higher Moments And Plausible Utility, Revere Street Working Paper Series, Finan. Econ., 2003; 1-25. https://api.semanticscholar.org/CorpusID:150668448
[21] Dinh, T. P., Niu, Y. S., An efficient DC programming approach for portfolio decision with higher moments, Comput Optim Appl. 2011; 50: 525-554. Doi: 10.1007/s10589-010-9383-x
[22] Foroozesh, N., Tavakkoli-Moghaddam, R., Mousavi, SM., et al., A new comprehensive possibilistic group decision approach for resilient supplier selection with mean-variance-skewness-kurtosis and asymmetric information under interval-valued fuzzy uncertainty, Neural Comput Appl. 2019; 31(11):6959-6979. Doi:10.1007/s00521-018-3506-1
[23] Harvey, C.R., Liechty, J.C., Liechty, MW., Muller, P., Portfolio Selection With Higher Moments, Social Sci. Res. Net. W. P. S., 2004; 294-2745. Doi:10.1080/14697681003756877
[24] Jondeau, E., Rockinger, M., Optimal Portfolio Allocation Under Higher Moments, EFMA 2004 Basel Meetings Paper, 2004. Doi:10.1111/j.1354-7798.2006.00309.x
[25] Konno, H., Shirakawa, H., Yamazaki, H., A Mean-Absolute Deviation-Skewness Portfolio Optimization Model, J. Annals of Operat. Res., 1993; 45(1): 205-220. Doi:10.1007/BF02282050
[26] Lai, TY., Portfolio Selection with Skewness: A Multiple-Objective Approach, Review of Quant Finan Accou, 1991; 1(3): 293-305. Doi:10.1007/BF02408382
[27] Lai, K.K., Lean, Y., Shouyang, W., Mean-Variance-Skewness-Kurtosis-Based Portfolio Optimization, Proceedings of The First International Multi-Symposiums on Computer and Computational Sciences, 1-6, 2006. Doi:10.1109/IMSCCS.2006.252
[28] Li, X., Qin, Z., Kar, S., Mean-variance-skewness model for portfolio selection with fuzzy returns. Eur J. Oper Res. 2010; 202(1):239-247. Doi:10.1016/j.ejor.2009.05.003
[29] Liu, S., Wang, S.Y., Qiu, W., Mean-Variance-Skewness Model for Portfolio Selection With Transaction Costs, IJSS, 2003; 34(4): 255-262. Doi:10.1080/0020772031000158492
[30] Lu, Y., Li, J., Mean-variance-skewness portfolio selection model based on RBF-GA. Manage Sci Eng. 2017; 11(1):47-53. Doi:10.3968/n
[31] Theodossiou, P., Savva, C.S., Skewness and the relation between risk and return. Manage Sci. 2015; 62(6): 1598-1609.
[32] Zhao, S., Lu, Q., Han, L., et al., A mean-CVaR-skewness portfolio optimization model based on asymmetric Laplace distribution. Ann Oper Res. 2015; 226(1): 727-739. Doi: 10.1007/s10479-014-1654-y
[33] Deng, X., He, X., Huang, C., A new fuzzy random multi-objective portfolio model with different entropy measures using fuzzy programming based on artificial bee colony algorithm, Eng Comput. 2022; 39(2): 627-49. Doi:10.1108/ec-11-2020-0654.
[34] Garajova, E., Hladik, M., On the optimal solution set in interval linear programming, Comput Optim Appl. 2019; 72: 269-292. Doi:10.1007/s10589-018-0029-8
[35] Li, J., Xu, J., Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm. Inform Sci., 2013; 220:507-521. Doi:10.1016/j.ins.2012.07.005
[36] Mehralizade, R., Mehralizade, A., LR mixed fuzzy random portfolio choice based on the risk curve. Int J Uncertain Fuzz. 2022; 30(02): 231-61. Doi:10.1142/S0218488522500106
[37] Huang, X., Qiao, L., A risk index model for multi-period uncertain portfolio selection. Inform Sci. 2012; 217: 108-116. Doi:10.1016/j.ins.2012.06.017
[38] Liu, B., Why is there a need for uncertainty theory?. J Uncertain Syst. 2012; 6: 3-10. Online at: www.jus.org.uk
[39] Huang, X., Mean-risk model for uncertain portfolio selection. Fuzzy Optim Decis Making, 2011; 10: 71-89. Doi:10.1007/s10700-010-9094-x
[40] Mahmoodvandgharahshiran, M., Yari, GH., Behzadi, M.H., Uncertain Entropy as a Risk Measure in Multi-Objective Portfolio Optimization. Adv. Math. Fin. App., 2024, 9(1): 337-354.
Doi: 10.22034/AMFA.2023.1971454.1815
[41] Huang, X., A risk index model for portfolio selection with returns subject to experts estimations. Fuzzy Optim Decis Making 2012b; 11: 451-463. Doi:10.1007/s10700-012-9125-x
[42] Nazemi, A.R., Abbasi, B., Omidi, F., Solving portfolio selection models with uncertain returns using an artificial neural network scheme. Appl. Intell. 2015; 42: 609-621. Doi:10.1007/s10489-014-0616-z
[43] Omidi, F., Abbasi, B., Nazemi, A.R., An efficient dynamic model for solving a portfolio selection with uncertain chance constraint models, J CAM, 2017; 319: 43-55. Doi:10.1016/j.cam.2016.12.020
[44] Zhu, Y., Uncertain optimal control with application to a portfolio selection model. Cybern Syst. 2010; 41: 535-547. Doi:10.1080/01969722.2010.511552
[45] Zhai, J., Bai, M., Hao, J., Uncertain random mean-variance-skewness models for the portfolio optimization problem. OPTIMIZATION, Taylor & francis, 2021. Doi:10.1080/02331934.2021.1928122
[46] Liu B., Some research problems in uncertainty theory. J Uncertain Syst, 2009; 3(1): 3-10.
https://api.semanticscholar.org/CorpusID:4999177
[47] Liu B., Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin, 2010. Doi: 10.1007/978-3-642-13959-8
[48] Zamanpour,A., Zanjirdar, M., Davodi Nasr,M, Identify and rank the factors affecting stock portfolio optimization with fuzzy network analysis approach, Financial Engineering And Portfolio Management,2021;12(47): 210-236
