Document Type : Research Article
Authors
Department of Mathematics, Faculty of Mathematics,Statistics and Computer Sciences, Semnan University, P.O. Box 35195-363, Semnan, Iran.
Abstract
This paper investigates the complexities surrounding uncertain portfolio selection in cases where security returns are not well-represented by historical data. Uncertainty in security returns is addressed by treating them as uncertain variables. Portfolio selection models are developed using the quadratic-entropy of these uncertain variables, with entropy serving as a standard measure of diversification. Additionally, the study underscores the superior risk estimation accuracy of Average Value-at-Risk (AVaR) compared to variance. The research concentrates on the computational challenges of portfolio optimization in uncertain environments, utilizing the Mean-AVaR-Quadratic Entropy paradigm to meet investor requirements and assuage concerns. Two illustrative examples are provided to show the efficiency of the proposed models in this paper.
Keywords
[1] M. Abdi, S.B. Ebrahimi and A.A. Najafi, An online portfolio selection algorithm using
beta risk measure and fuzzy clustering, Journal of Mathematics and Modeling in Finance
(JMMF), 3:2, (2023) PP. 63–76.
[2] S. Basak and A. Shapiro, Value-at-risk based risk management: optimal policies and asset
prices, Rev. Financ. Stud., 14 (2001), PP. 371–405.
[3] A.K. Bera and S.Y. Park, Optimal portfolio diversification using the maximum entropy
principle, Economet. Rev., 27(2008), PP. 484-512.
[4] Campbell, R., Huisman, R. and Koedijk, K. Optimal portfolio selection in a value-at-risk
framework, J. Bank. Finance, 25 (2001), PP. 1789–1804.
[5] F.Y. Chen, Analytical VaR for international portfolios with common jumps, Comput. Math.
with Appl., 62 (2011), PP. 3066–3076.
[6] Y. Chen, J. Wang, and W. Zhang, Tail Distortion Risk Measure for Portfolio with Multivariate Regularly Variation, Communic in Maths and Stats., 10 (2022), PP. 263–285.
[7] D. Cuoco and H. Liu, Optimal consumption of a divisible durable good, J. Econ. Dynam.
Control., 24 (2000), PP. 561-613.
[8] P. Chou, W. Li and G. Zhou, Portfolio optimization under asset pricing nomalies, Jpn.
World Econ., 18 (2006), PP. 121-142.
[9] W. Dai, Quadratic entropy of uncertain variables, Soft Comput., 22 (2018), PP. 5699-5706.
https://doi.org/10.1007/s00500-017-2602-y
[10] X. Deng, X. He and C. Huang, A new fuzzy random multi-objective portfolio model with
different entropy measures using fuzzy programming based on artificial bee colony algorithm,
Eng Comput., 39(2) (2022), PP. 627–49.
[11] A. Dionisio, R. Menezes and D.A. Mendes, Uncertainty analysis in financial markets:
Can entropy be a solution?, In Proceedings of the 10th Annual Workshop on Economic
Heterogeneous Interacting Agents (WEHIA 2005), University of Essex, Colchester, UK, PP.
13-15, June, 2005.
[12] E. Garajova and M. Hladik, On the optimal solution set in interval linear programming,
Comput Optim Appl., 72 (2019), PP. 269–292.
[13] W.W. Hogan and J.M. Warren, Toward the development of an equilibrium capitalmarket
model based on semivariance, J. Financ. Quant. Anal., 9 (1974), PP. 1–11.
[14] R.E. Hoskisson, M.A. Hitt, R.H. Johnson and D. Moesel, Construct validity of an objective (entropy) categorical measure of diversification strategy, Strat. Manag. J., 14 (2006),
PP. 215-235.
[15] R.P. Huang, Z.S. Xu, S.J. Qu, X.G. Yang and M. Goh, Robust Portfolio Selection
with Distributional Uncertainty and Integer Constraints, J. Oper. Res. Soc. China, (2023).
https://doi.org/10.1007/s40305-023-00466-4
[16] X. Huang, Mean-risk model for uncertain portfolio selection, Fuzzy Optim Decis Making.,
10 (2011), PP. 71–89.
[17] X. Huang, Mean-variance models for portfolio selection subject to experts estimations,
Expert Syst Appl., 39 (2012a), PP. 5887–5893.
[18] X. Huang, A risk index model for portfolio selection with returns subject to experts estimations, Fuzzy Optim Decis Making, 11 (2012b), PP. 451–463.
[19] X. Huang and L. Qiao, A risk index model for multi-period uncertain portfolio selection,
Inform Sci., 217 (2012), PP. 108–16.
[20] P. Jana, T.K. Roy and S.K. Mazumder, Multi-objective mean-variance-skewness model
for Portfolio Optimization, AMO., 9 (2007), PP. 181-193.
[21] P. Jana, T.K. Roy and S.K. Mazumder, Multi-objective possibilistic model for portfolio
selection with transaction cost, J. Comput. Appl. Math., 228 (2009), PP. 188-196.
[22] J.N. Kapur and H.K. Kesavan, Entropy Optimization Principles with Applications, Academic Press: San Diego, CA, USA, 1992.
[23] H. Konno, K. Tanaka and R. Yamamoto, Construction of a portfolio with shorter downside tail and longer upside tail, Comput Optim Appl., 48(2011), PP. 199–212.
[24] B. Li, Y. Sun, G. Aw and K.L. Teo, Uncertain portfolio optimization problem under a
minimax risk measure, Appl. Math. Model., 76 (2019), PP. 274–281.
[25] J. Li and J. Xu, Multi-objective portfolio selection model with fuzzy random returns and a
compromise approach-based genetic algorithm, Inform Sci., 220 (2013), PP. 507–521.
[26] SH. Li, J. Peng, B. Zhang and D. Ralescu, Mean-Variance-Entropy Portfolio Selection
Models with Uncertain Returns, Int. J. of Manag. Fuzzy Syst., 7(3)(2021), PP. 47–54. https:
//doi.org/10.11648/j.ijmfs.20210703.12
[27] B. Liu, Uncertainty Theory, seconded, Springer-Verlag, Berlin, 2007.
[28] B. Liu, Some research problems in uncertainty theory, J Uncertain Syst., 3(1) (2009), PP.
3–10.
[29] B. Liu, Uncertainty theory: a branch of mathematics for modeling human uncertainty,
Springer, Berlin, 2010.
[30] B. Liu, Why is there a need for uncertainty theory?, J Uncertain Syst., 6 (2012), PP. 3–10.
[31] J. Liu, J. Xie, H. Ahmadzade and M. Farahikia, A new measure of indeterminacy for
uncertain variables with application to portfolio selection, J. Intell. Fuzzy Syst., 40,3 (2021),
PP. 5289-5293. https://doi.org/10.3233/JIFS-202073
[32] R. Mehralizade and A. Mehralizade, LR mixed fuzzy random portfolio choice based on
the risk curve, Int J Uncertain Fuzz., 30(02) (2022), PP. 231–61.
[33] H. Markowitz, Portfolio selection. J Finance., 7(1) (1952), PP. 77–91.
[34] A.R. Nazemi, B. Abbasi and F. Omidi, Solving portfolio selection models with uncertain
returns using an artificial neural network scheme. Appl. Intell., 42 (2015), PP. 609–621.
[35] Y. Ning, H. Ke and Z. Fu, Triangular entropy of uncertain variables with application
to portfolio selection, Soft Comput., 19 (2015), PP. 2203-2209. https://doi.org/10.1007/
s00500-014-1402-x
[36] F. Omidi, B. Abbasi and A.R. Nazemi, An efficient dynamic model for solving a portfolio
selection with uncertain chance constraint models, J CAM., 319 (2017), PP. 43–55.
[37] G.C.Philippatos and N. Gressis, Conditions ofequivalence among E-V, SSD and E-H
portfolio selection criteria: The case for uniform, normal and lognormal Distributions,
Manag. Sci., 21 (1975), PP. 617-625.
[38] P.Y. Qian, Z.Z. Wang and Z.W. Wen, A Composite Risk Measure Framework for Decision
Making Under Uncertainty. J. Oper. Res. Soc. China., 7 (2019), PP. 43–68. https://doi.
org/10.1007/s40305-018-0211-9
[39] R.T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, J. Bank. Finance., 26 (2002), PP. 1443–1471.
[40] M. Salahi, T. Khodamoradi and A. Hamdi, Mean-standard deviation-conditional valueat-risk portfolio optimization, Journal of Mathematics and Modeling in Finance (JMMF),
3:1, (2023) PP. 83–98.
[41] B. Samanta and T.K. Roy, Multi-objective Portfolio Optimization Model, Tamsui Oxf. J.
Math. Sci., 21 (2005), PP. 55-70.
[42] Sh. Shahbeyk, Robustness in Mean-Variance Portfolio Optimization, Journal of Mathematics and Modeling in Finance (JMMF), 2 (2022), PP. 195-204.
[43] N. Shi, M. Lai, S. Zheng and B. Zhang, Optimal algorithms and intuitive explanations
for Markowitzs portfolio selection model and Sharpes ratio with no short-selling, Sci. China
Math. Phys. Astron. Technol. Sci., 51 (2008), PP. 2033-2042.
[44] Y. Sun, G. Aw, K.L. Teo and G. Zhou, Portfolio optimization using a new probabilistic
risk measure, J. Ind. Manag. Optim., 11 (2015), PP. 1275–1283.
[45] Y. Takano and R. Sotirov, A polynomial optimization approach to constant rebalanced
portfolio selection. Comput Optim Appl., 52 (2012), PP. 645–666.
[46] I. Usta and Y.M. Kantar, Analysis of multi-objective portfolio models for the istanbul
stock exchange, In Proceedings of the 2nd International Workshop on Computational and
Financial Econometrics (CFE08), Neuchatel, Switzerland, PP. 19-21 June, 2008.
[47] J. Yang, H. Lian and W. Zhang, A Class of Structured High-Dimensional Dynamic Covariance Matrices, Communic in Maths and Stats., 2023.
[48] M. Yin, W. Qian and W. Li, Portfolio selection models based on Cross-entropy of uncertain
variables, J. of Intell. & Fuzzy Sys., 31(2)(2016), PP. 737–747, https://doi.org/10.3233/
JIFS-169006
[49] Y. Yoshida, Maximization of Returns under an Average Value-at-Risk Constraint in Fuzzy
Asset Management, Procedia Com. Sci., 112 (2017), PP. 11–20.
[50] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybern
Syst., 41 (2010), PP. 535–547.
beta risk measure and fuzzy clustering, Journal of Mathematics and Modeling in Finance
(JMMF), 3:2, (2023) PP. 63–76.
[2] S. Basak and A. Shapiro, Value-at-risk based risk management: optimal policies and asset
prices, Rev. Financ. Stud., 14 (2001), PP. 371–405.
[3] A.K. Bera and S.Y. Park, Optimal portfolio diversification using the maximum entropy
principle, Economet. Rev., 27(2008), PP. 484-512.
[4] Campbell, R., Huisman, R. and Koedijk, K. Optimal portfolio selection in a value-at-risk
framework, J. Bank. Finance, 25 (2001), PP. 1789–1804.
[5] F.Y. Chen, Analytical VaR for international portfolios with common jumps, Comput. Math.
with Appl., 62 (2011), PP. 3066–3076.
[6] Y. Chen, J. Wang, and W. Zhang, Tail Distortion Risk Measure for Portfolio with Multivariate Regularly Variation, Communic in Maths and Stats., 10 (2022), PP. 263–285.
[7] D. Cuoco and H. Liu, Optimal consumption of a divisible durable good, J. Econ. Dynam.
Control., 24 (2000), PP. 561-613.
[8] P. Chou, W. Li and G. Zhou, Portfolio optimization under asset pricing nomalies, Jpn.
World Econ., 18 (2006), PP. 121-142.
[9] W. Dai, Quadratic entropy of uncertain variables, Soft Comput., 22 (2018), PP. 5699-5706.
https://doi.org/10.1007/s00500-017-2602-y
[10] X. Deng, X. He and C. Huang, A new fuzzy random multi-objective portfolio model with
different entropy measures using fuzzy programming based on artificial bee colony algorithm,
Eng Comput., 39(2) (2022), PP. 627–49.
[11] A. Dionisio, R. Menezes and D.A. Mendes, Uncertainty analysis in financial markets:
Can entropy be a solution?, In Proceedings of the 10th Annual Workshop on Economic
Heterogeneous Interacting Agents (WEHIA 2005), University of Essex, Colchester, UK, PP.
13-15, June, 2005.
[12] E. Garajova and M. Hladik, On the optimal solution set in interval linear programming,
Comput Optim Appl., 72 (2019), PP. 269–292.
[13] W.W. Hogan and J.M. Warren, Toward the development of an equilibrium capitalmarket
model based on semivariance, J. Financ. Quant. Anal., 9 (1974), PP. 1–11.
[14] R.E. Hoskisson, M.A. Hitt, R.H. Johnson and D. Moesel, Construct validity of an objective (entropy) categorical measure of diversification strategy, Strat. Manag. J., 14 (2006),
PP. 215-235.
[15] R.P. Huang, Z.S. Xu, S.J. Qu, X.G. Yang and M. Goh, Robust Portfolio Selection
with Distributional Uncertainty and Integer Constraints, J. Oper. Res. Soc. China, (2023).
https://doi.org/10.1007/s40305-023-00466-4
[16] X. Huang, Mean-risk model for uncertain portfolio selection, Fuzzy Optim Decis Making.,
10 (2011), PP. 71–89.
[17] X. Huang, Mean-variance models for portfolio selection subject to experts estimations,
Expert Syst Appl., 39 (2012a), PP. 5887–5893.
[18] X. Huang, A risk index model for portfolio selection with returns subject to experts estimations, Fuzzy Optim Decis Making, 11 (2012b), PP. 451–463.
[19] X. Huang and L. Qiao, A risk index model for multi-period uncertain portfolio selection,
Inform Sci., 217 (2012), PP. 108–16.
[20] P. Jana, T.K. Roy and S.K. Mazumder, Multi-objective mean-variance-skewness model
for Portfolio Optimization, AMO., 9 (2007), PP. 181-193.
[21] P. Jana, T.K. Roy and S.K. Mazumder, Multi-objective possibilistic model for portfolio
selection with transaction cost, J. Comput. Appl. Math., 228 (2009), PP. 188-196.
[22] J.N. Kapur and H.K. Kesavan, Entropy Optimization Principles with Applications, Academic Press: San Diego, CA, USA, 1992.
[23] H. Konno, K. Tanaka and R. Yamamoto, Construction of a portfolio with shorter downside tail and longer upside tail, Comput Optim Appl., 48(2011), PP. 199–212.
[24] B. Li, Y. Sun, G. Aw and K.L. Teo, Uncertain portfolio optimization problem under a
minimax risk measure, Appl. Math. Model., 76 (2019), PP. 274–281.
[25] J. Li and J. Xu, Multi-objective portfolio selection model with fuzzy random returns and a
compromise approach-based genetic algorithm, Inform Sci., 220 (2013), PP. 507–521.
[26] SH. Li, J. Peng, B. Zhang and D. Ralescu, Mean-Variance-Entropy Portfolio Selection
Models with Uncertain Returns, Int. J. of Manag. Fuzzy Syst., 7(3)(2021), PP. 47–54. https:
//doi.org/10.11648/j.ijmfs.20210703.12
[27] B. Liu, Uncertainty Theory, seconded, Springer-Verlag, Berlin, 2007.
[28] B. Liu, Some research problems in uncertainty theory, J Uncertain Syst., 3(1) (2009), PP.
3–10.
[29] B. Liu, Uncertainty theory: a branch of mathematics for modeling human uncertainty,
Springer, Berlin, 2010.
[30] B. Liu, Why is there a need for uncertainty theory?, J Uncertain Syst., 6 (2012), PP. 3–10.
[31] J. Liu, J. Xie, H. Ahmadzade and M. Farahikia, A new measure of indeterminacy for
uncertain variables with application to portfolio selection, J. Intell. Fuzzy Syst., 40,3 (2021),
PP. 5289-5293. https://doi.org/10.3233/JIFS-202073
[32] R. Mehralizade and A. Mehralizade, LR mixed fuzzy random portfolio choice based on
the risk curve, Int J Uncertain Fuzz., 30(02) (2022), PP. 231–61.
[33] H. Markowitz, Portfolio selection. J Finance., 7(1) (1952), PP. 77–91.
[34] A.R. Nazemi, B. Abbasi and F. Omidi, Solving portfolio selection models with uncertain
returns using an artificial neural network scheme. Appl. Intell., 42 (2015), PP. 609–621.
[35] Y. Ning, H. Ke and Z. Fu, Triangular entropy of uncertain variables with application
to portfolio selection, Soft Comput., 19 (2015), PP. 2203-2209. https://doi.org/10.1007/
s00500-014-1402-x
[36] F. Omidi, B. Abbasi and A.R. Nazemi, An efficient dynamic model for solving a portfolio
selection with uncertain chance constraint models, J CAM., 319 (2017), PP. 43–55.
[37] G.C.Philippatos and N. Gressis, Conditions ofequivalence among E-V, SSD and E-H
portfolio selection criteria: The case for uniform, normal and lognormal Distributions,
Manag. Sci., 21 (1975), PP. 617-625.
[38] P.Y. Qian, Z.Z. Wang and Z.W. Wen, A Composite Risk Measure Framework for Decision
Making Under Uncertainty. J. Oper. Res. Soc. China., 7 (2019), PP. 43–68. https://doi.
org/10.1007/s40305-018-0211-9
[39] R.T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, J. Bank. Finance., 26 (2002), PP. 1443–1471.
[40] M. Salahi, T. Khodamoradi and A. Hamdi, Mean-standard deviation-conditional valueat-risk portfolio optimization, Journal of Mathematics and Modeling in Finance (JMMF),
3:1, (2023) PP. 83–98.
[41] B. Samanta and T.K. Roy, Multi-objective Portfolio Optimization Model, Tamsui Oxf. J.
Math. Sci., 21 (2005), PP. 55-70.
[42] Sh. Shahbeyk, Robustness in Mean-Variance Portfolio Optimization, Journal of Mathematics and Modeling in Finance (JMMF), 2 (2022), PP. 195-204.
[43] N. Shi, M. Lai, S. Zheng and B. Zhang, Optimal algorithms and intuitive explanations
for Markowitzs portfolio selection model and Sharpes ratio with no short-selling, Sci. China
Math. Phys. Astron. Technol. Sci., 51 (2008), PP. 2033-2042.
[44] Y. Sun, G. Aw, K.L. Teo and G. Zhou, Portfolio optimization using a new probabilistic
risk measure, J. Ind. Manag. Optim., 11 (2015), PP. 1275–1283.
[45] Y. Takano and R. Sotirov, A polynomial optimization approach to constant rebalanced
portfolio selection. Comput Optim Appl., 52 (2012), PP. 645–666.
[46] I. Usta and Y.M. Kantar, Analysis of multi-objective portfolio models for the istanbul
stock exchange, In Proceedings of the 2nd International Workshop on Computational and
Financial Econometrics (CFE08), Neuchatel, Switzerland, PP. 19-21 June, 2008.
[47] J. Yang, H. Lian and W. Zhang, A Class of Structured High-Dimensional Dynamic Covariance Matrices, Communic in Maths and Stats., 2023.
[48] M. Yin, W. Qian and W. Li, Portfolio selection models based on Cross-entropy of uncertain
variables, J. of Intell. & Fuzzy Sys., 31(2)(2016), PP. 737–747, https://doi.org/10.3233/
JIFS-169006
[49] Y. Yoshida, Maximization of Returns under an Average Value-at-Risk Constraint in Fuzzy
Asset Management, Procedia Com. Sci., 112 (2017), PP. 11–20.
[50] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybern
Syst., 41 (2010), PP. 535–547.
)