Uncertain Entropy as a Risk Measure in Multi-Objective Portfolio Optimization

Subject Areas : Financial Mathematics

Mahsa  mahmoodvandgharahshiran ¹ (Department of Statistics, Science and Research Branch, Islamic Azad University, Tehran, Iran)
Gholamhossein  Yari ² (Department of Applied Mathematics, Faculty of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran)
Mohammad Hassan  Behzadi ³ (Department of Statistics, Science and Research Branch, Islamic Azad University, Tehran, Iran)

Received: 2022-10-25 Accepted : 2023-01-10 Published : 2023-12-01

Keywords: Uncertainty theory, Uncertain Entropy, Multi-Objective Optimization, Information Theory, Uncertain Mean-Entropy Portfolio Optimization (UMEPO),

Abstract :

As we are looking for knowledge of stock future returns in portfolio optimization, we are practically faced with two principal concepts: Uncertainty and Information about variables. This paper attempts to introduce a pragmatic bi-objective investment model based on uncertainty, instead of probability space and information theory, instead of variance and other moments as a risk measure for portfolio optimization. Not only is uncertainty space expected to be more in line with investment theory, but also, applying and learning this approach seems more straightforward and practical for novice investors. The proposed model simultaneously maximizes the uncertain mean of stock returns and minimizes uncertain entropy as a measure of portfolio risk. The uncertain zigzag distribution has been used for variables to avoid the complexity of fitting distributions for data. This uncertain mean-entropy portfolio optimization (UMEPO) has been solved by three meta-heuristic methods of multi-objective optimization: NSGA-II, MOPS, and MOICA. Finally, it was observed that the optimal portfolio obtained from the proposed model has a higher return and a lower entropy as a risk measure compared to the same model in the probability space.

References:

[1] Atashpaz-Gargari, E., and Lucas, C., Imperialist Competitive Algorithm, An Algorithm for Optimization Inspired by Imperialistic Competition, IEEE Congress on Evolutionary Computation, 2007, 7, P.4661–4667.

[2] Abtahi, S.H., Yari, GH., Hosseinzadeh Lotfi, F., and Farnoosh, R., Multi-objective Portfolio Selection Based on Skew-Normal Uncertainty Distribution and Asymmetric Entropy, International Journal of Fuzzy Log-ic and Intelligent Systems, 2021, 21(1), P.38-48. Doi: 10.5391/IJFIS.2021.21.1.38

[3] Babaei, S., Sepehri, M.M., and Babaei, E., Multi-objective portfolio optimization considering the depend-ence structure of asset returns, European Journal of Operational Research, 2015, 244, P. 525–539. Doi: 10.1016/j.ejor.2015.01.025.

[4] Bhattacharyya, R., Chatterjee, A., and Kar, S., Uncertainty Theory Based Multiple Objective Mean-Entropy-Skewness Stock Portfolio Selection Model with Transaction Costs, Journal of Uncertainty Analysis and Applications, 2013, 1, P. 1-16.

[5] Bhattacharyya, R., and Kar, S., Multi-Objective Fuzzy Optimization for Portfolio Selection, An Embed-ding Theorem Approach, Turkish journal of fuzzy systems, 2011, 2(1), P. 14–35. Doi: 10.1186/2195-5468-1-16.

[6] Black, F., Global Asset Allocation with Equities, Bonds, and Currencies, Fixed Income, United States, Goldman Sachs, 1991. Doi: 10.2307/1910098.

[7] Black, F., and Litterman, R, Global Portfolio Optimization, Financial Analysts Journal, 1992, 48(5), P. 28–43.

[8] Cesarone, F., Scozzari, A., and Tardella, F., An optimization diversification approach to portfolio selec-tion, Journal of Global Optimization, 2020, 76(2), P.245–265.

[9] Chen, L., Gao, R., Bian, Y., and Di, H., Elliptic entropy of uncertain random variables with application to portfolio selection, Soft Computing, 2021, 25, P. 1925–1939 . Doi:10.1007/s00500-020-05266-z.

[10] Dai, W., and Chen, X.W., Entropy of Function of Uncertain Variables, Mathematical and Computer Mod-elling, 2012, 55, P. 754-760.

[11] Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T., A Fast and Elitist Multiobjective Genetic Algorithm, NSGA-II, IEEE Transactions on Evolutionary Computation, 2002, 6(2), P. 182–197.

[12] Fang, Y., Lai, KK., and Wang, SY., Portfolio Rebalancing Model with Transaction Costs Based on Fuzzy Decision Theory, European Journal of Operational Research, 2006, 175(2), P. 879–893. Doi:10.1016/j.ejor.2005.05.020.

[13] Garner, W., The Relation between Information and Variance Analyses, Psychometrika, 1956, 21(3), P. 219–228. Doi:10.1007/BF02289132.

[14] Gao, R., Ahmadzade, H., Rezaei, K., Rezaei, H., and Naderi, H., Partial similarity measure of uncertain random variables and its application to portfolio selection, Journal of Intelligent and Fuzzy Systems, 2020, 39(1), P. 155-166.

[15] Guo, C., and Gao, J., Optimal Dealer Pricing Under Transaction Uncertainty, Journal of Intelligent Manu-facturing, 2017, 28(3), P. 657–665.

[16] Gupta, P., Mehlawat, M., Yadav, S., and Kumar, A., Intuitionistic fuzzy optimistic and pessimistic multi-period portfolio optimization models, Soft Computing, 2020, 24(16), P.11931–11956.

[17] Huang, X., A Risk Index Model for Portfolio Selection with Return Subject to Experts’ Evaluations, Fuzzy Optimization and Decision Making, 2012, 11(4), P. 451–463. Doi: 10.1007/s10700-012-9125-x.

[18] Huang, X., Fuzzy Chance-Constrained Portfolio Selection, Applied Mathematics and Computation, 2006, 177(2), P. 500–507.

[19] Huang, X., Mean-Risk Model for Uncertain Portfolio Selection, Fuzzy Optimization and Decision Making, 2011, 10(1), P. 71–89.

[20] Huang, X., Mean-Semi variance Models for Fuzzy Portfolio Selection, Journal of Computational and Ap-plied Mathematics, 2008, 217(1), P. 1–8.

[21] Huang, X., Mean-Variance Models for Portfolio Selection Subject to Expert’s Estimations, Expert Systems with Applications, 2012, 39(5), P. 5887–5893.

[22] Huang, X., Portfolio Analysis, From Probabilistic to Credibilistic and Uncertain Approaches, Germany, Springer, 2010.

[23] Huang, X., and Di, H., Uncertain Portfolio Selection with Background Risk, Applied Mathematics and Computation, 2016, 276, P. 284–296.

[24] Kar, M.B., Majumder, S., Kar, S., and Pal, T., Cross-Entropy Based Multi-Objective Uncertain Portfolio Selection Problem, Journal of Intelligent and Fuzzy Systems, 2017, 32(6), P. 4467–4483. Doi: 10.3233/JIFS-169212.

[25] Lassance, N., Vrins, F., Minimum Rényi Entropy Portfolios, Annals of Operations Research, 2021, 299(1), P. 23–46.

[26] Leon, T., Liern, V., and Vercher, E., Viability of Infeasible Portfolio Selection Problems, a Fuzzy Ap-proach, European Journal of Operational Research, 2002, 139(1), P. 178–189. Doi: 10.1016/S0377-2217(01)00175-8.

[27] Li, B., and Zhang, R., A New Mean-Variance-Entropy Model for Uncertain Portfolio Optimization with Liquidity and Diversification, Chaos, Solitons and Fractals, 2021, 146, 110842. Doi:10.1016/j.chaos.2021.110842.

[28] Li, B., Li, X., Teo, K.L. and Zheng, P., A new uncertain random portfolio optimization model for complex systems with downside risks and diversification, Chaos, Solitons & Fractals, 2022, 160, 112213.

[29] Li, B., Shu, Y., Sun, Y., and Teo, K.L., An Optimistic Value–Variance–Entropy Model of Uncertain Portfo-lio Optimization Problem under Different Risk Preferences, Soft Computing, 2021, 25(5), P. 3993–4001. Doi: 10.1007/s00500-020-05423-4.

[30] Li, Y., Wang, B., Fu, A., and Watada, J., Fuzzy portfolio optimization for time-inconsistent investors: a multi-objective dynamic approach, Soft Computing, 2020, 24(13), P.9927–9941.

[31] Liu, B., Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems, 2009, 3(1), P. 3-10.

[32] Liu, B., Uncertainty Theory, 2th ed, Germany, Springer, 2007.

[33] Liu, B., Uncertainty Theory, 5th ed, Uncertainty Theory Laboratory, 2018.

[34] Liu, B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer, 2018.

[35] Liu, B., Why Is There a Need for Uncertainty Theory?, Journal of Uncertain Systems, 2012, 6(1), P. 3–10.

[36] Liu, J., Ahmadzade, H., Zarei, H., Portfolio Optimization of Uncertain Random Returns based on Partial Exponential Entropy, Journal of Uncertain Systems, 2022, 15(1), 2250004.

[37] Liu, Y., Ahmadzade, H., and Farahikia, M., Portfolio selection of uncertain random returns based on val-ue at risk, Soft Computing, 2021, 25, P. 6339-6346. Doi:10.1007/s00500-021-05623-6.

[38] Lv, L., Zhang, B., Peng, J., and Ralescu, D.A., Uncertain Portfolio Selection with Borrowing Constraint and Background Risk, Mathematical Problems in Engineering, 2020, 1249829. Doi: 10.1155/2020/1249829.

[39] Majumder, S., Kar, S., and Pal, T., Mean-Entropy Model of Uncertain Portfolio Selection Problem, Multi-Objective Optimization, 2018, P. 25-54.

[40] Markowitz, H., Portfolio Selection, Journal of Finance, 1952, 7(1), P. 77–91.

[41] McGill, W., Multivariate Information Transmission, Psychometrika, 1954, 19(2), P. 97–116. Doi: 10.1007/BF02289159.

[42] Mehralizade, R., Amini, M., Gildeh, B.S., and Ahmadzade, H., Uncertain random portfolio selection based on risk curve, Soft Computing, 2020, 24(17), P.13331–13345.

[43] Mercurio, P., Wu, Y., and Xie, H., An Entropy-Based Approach to Portfolio Optimization, Entropy, 2020, 22(3), P. 332. Doi: 10.3390/e22030332.

[44] Moore, J., and Chapman, R., Application of Particle Swarm to Multiobjective Optimization, Computer Systems, Science and Engineering, 1999.

[45] Ning, Y., Ke, H., and Fu, Z., Triangular Entropy of Uncertain Variables with Application to Portfolio Se-lection, Soft Computing, 2015, 19(8), P. 2203-2209. Doi: 10.1007/s00500-014-1402-x.

[46] Peng, Z.X., and Iwamura, K., A Sufficient and Necessary Condition of Uncertainty Distribution, Journal of Interdisciplinary Mathematics, 2010, 13(3), P. 277-285.

[47] Philippatos, G., Entropy, Market Risk, and the Selection of Efficient Portfolios, Applied Economics, 1972, 4, P. 209–220. Doi: 10.1080/00036847200000017.

[48] Qin, Z., Kar, S., and Zheng, H., Uncertain Portfolio Adjusting Model Using Semiabsolute Deviation, Soft Computing, 2016, 20(2), P. 717–725.

[49] Rom, B., Post-Modern Portfolio Theory Comes of Age, Journal Investing, 1993, 2(4), P. 27–33. Doi:10.3905/joi.2.4.27.

[50] Rompolis, L., Retrieving Risk Neutral Densities from European Option Prices Based On the Principle of Maximum Entropy, Journal of Empirical Finance, 2010, 17(5), P. 918-937.

[51] Sajedi, A., and Yari, Gh., Order ʋ Entropy and Cross Entropy of Uncertain Variables for Portfolio Selec-tion, International Journal of Fuzzy Logic and Intelligent Systems, 2020, 20(1), P.35-42. Doi: 10.5391/IJFIS.2020.20.1.35.

[52] Shannon, C., A Mathematical Theory of Communication, Part 1, Bell System Technical Journal, 1948, 27(3), P. 379–423. Doi:10.1002/j.1538-7305.1948.tb01338.x.

[53] Shannon, C., A Mathematical Theory of Communication, Part 2, Bell System Technical Journal, 1948, 27(4), P. 623–656. Doi:10.1002/j.1538-7305.1948.tb00917.x.

[54] Wu, X., Ralescu, D., and Liu, Y., A new quadratic deviation of fuzzy random variable and its application to portfolio optimization, Iranian Journal of Fuzzy System, 2020, 17(3), P. 1–18.

[55] Xie, F., Zhao, H., Ahmadzade, H., and GhasemiGol, M., Uncertain Portfolio Optimization based on Tsallis Entropy of Uncertain Sets , Research Square, 2022. Doi: 10.21203/rs.3.rs-1925999/v1.

[56] Yan, L., Optimal Portfolio Selection Models with Uncertain Returns, Modern Applied Science, 2009, 3(8), P. 76–81. Doi:10.5539/mas.v3n8p76.

[57] Yang, X., and Gao, J., Bayesian Equilibria for Uncertain Bimatrix Game with Asymmetric Information, Journal of Intelligent Manufacturing, 2017, 28(3), P. 515–525.

[58] Yang, X., and Gao, J., Linear Quadratic Uncertain Differential Game with Application to Resource Ex-traction Problem, IEEE Transactions on Fuzzy Systems, 2016, 24(4), P. 819–826.

[59] Zhai, J., and Bai, M., Mean-risk model for uncertain portfolio selection with background risk, Journal of Computational and Applied Mathematics, 2018, 330, P. 59-69. Doi:10.1016/j.cam.2017.07.038.

[60] Zhai, J., and Bai, M., Uncertain Portfolio Selection with Background Risk and Liquidity Constraint, Mathematical Problems in Engineering, 2017, P. 1–10. Doi:10.1155/2017/8249026.