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A review of credibilistic portfolio selection. (English) Zbl 1187.91200
Summary: This paper reviews the credibilistic portfolio selection approaches which deal with fuzzy portfolio selection problem based on credibility measure. The reason for choosing credibility measure is given. Several mathematical definitions of risk of an investment in the portfolio are introduced. Some credibilistic portfolio selection models are presented, including mean-risk model, mean-variance model, mean-semivariance model, credibility maximization model, $$\alpha$$-return maximization model, entropy optimization model and game models. A hybrid intelligent algorithm for solving the optimization models is documented. In addition, as extensions of credibilistic portfolio selection approaches, the paper also gives a brief review of some hybrid portfolio selection models.

##### MSC:
 91G10 Portfolio theory 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming 91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
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##### References:
 [1] Arditti F.D. (1971) Another look at mutual fund performance. Journal of Financial and Quantitative Analysis 6: 909–912 [2] Arenas-Parra M., Bilbao-Terol A., Rodríguez-Uría M.V. (2001) A fuzzy goal programming approach to portfolio selection. European Journal of Oprational Research 133: 287–297 · Zbl 0992.90085 [3] Bilbao-Terol A., Pérez-Gladish B., Arenas-Parra M., Rodríguez-Uría M.V. (2006) Fuzzy compromise programming for portfolio selection. Applied Mathematics and Computation 173: 251–264 · Zbl 1138.91421 [4] Bruckley J.J., Hayashi Y. (1994) Fuzzy genetic algorithm and applications. Fuzzy Sets and Systems 61: 129–136 [5] Bruckley J.J., Feruing T. (2000) Evolution algorithm solution to fuzzy problems: Fuzzy linear programming. Fuzzy Sets and Systems 109: 35–53 · Zbl 0956.90064 [6] Carlsson C., Fullér R., Majlender P. (2002) A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems 131: 13–21 · Zbl 1027.91038 [7] Chunhachinda P., Dandapani K., Hamid S., Prakash A.J. (1997) Portfolio selection and skewness: Evidence from international stock market. Journal of Banking and Finance 21: 143–167 [8] Colubi A., Fernández-García C., Gil M.A. (2002) Simulation of random fuzzy variables: An empirical approach to statistical/probabilistic studies with fuzzy experimental data. IEEE Transactions on Fuzzy Systems 10: 384–390 [9] De Cooman G. (1997) Possibility theory I–III. International Journal of General Systems 25: 291–371 · Zbl 0955.28012 [10] Dubois D., Prade H. (1988) Possibility theory: An approach to computerized processing of uncertainty. Plenum, New York · Zbl 0703.68004 [11] Fama E. (1965) Portfolio analysis in a stable paretian market. Management Science 11: 404–419 · Zbl 0129.11903 [12] Gupta P., Mehlawat M.K., Saxena A. (2008) Asset portfolio optimization using fuzzy mathe matical programming. Information Sciences 178: 1734–1755 · Zbl 1132.91464 [13] Holland J. (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor · Zbl 0317.68006 [14] Huang, X. (2004). Portfolio selection with fuzzy returns. Proceedings of the Second Annual conference on Uncertainty (pp. 21–29). Qinhuangdao July 16–20. [15] Huang X. (2006) Fuzzy chance-constrained portfolio selection. Applied Mathematics and Computation 177: 500–507 · Zbl 1184.91191 [16] Huang X. (2007a) Two new models for portfolio selection with stochastic returns taking fuzzy information. European Journal of Operational Research 180: 396–405 · Zbl 1114.90166 [17] Huang X. (2007b) A new perspective for optimal portfolio selection with random fuzzy returns. Information Sciences 177: 5404–5414 · Zbl 1304.91195 [18] Huang X. (2007c) Portfolio selection with fuzzy returns. Journal of Intelligent & Fuzzy Systems 18: 383–390 · Zbl 1130.91337 [19] Huang X. (2008a) Mean-semivariance models for fuzzy portfolio selection. Journal of Computational and Applied Mathematics 217: 1–8 · Zbl 1149.91033 [20] Huang X. (2008b) Risk curve and fuzzy portfolio selection. Computers and Mathematics with Applications 55: 1102–1112 · Zbl 1142.91527 [21] Huang X. (2008c) Expected model for portfolio selection with random fuzzy returns. International Journal of General Systems 37: 319–328 · Zbl 1155.91384 [22] Huang X. (2008d) Portfolio selection with a new definition of risk. European Journal of Operational Research 186: 351–357 · Zbl 1138.91450 [23] Huang X. (2008e) Mean-entropy models for fuzzy portfolio selection. IEEE Transactions on Fuzzy Systems 16: 1096–1101 · Zbl 05516375 [24] Huang, X. (2008f). Minimax mean-variance models for fuzzy portfolio selection, Technical Report. · Zbl 1149.91033 [25] Huang, X. (2008g). Fuzzy minimax chance constrained models for portfolio selection, Technical Report. [26] Huang, X. (2008h). Mean-variance model for hybrid portfolio selection with randomness and fuzziness, Technical Report. [27] Ida M. (2002) Mean-variance portfolio optimization model with uncertain coefficients. Proceedings of IEEE International Conference on Fuzzy Systems 3: 1223–1226 [28] Inuiguchi M., Ramík J. (2000) Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets and Systems 111: 3–28 · Zbl 0938.90074 [29] Katagiri H., Ishii H. (1999) Fuzzy portfolio selection problem. Proceedings of the IEEE International Conference on Systems, Man and Cybernetics 3: III-973–III-978 [30] Kaufman A., Gupta M.M. (1985) Introduction to fuzzy arithmetic: Theory and applications. Van Nostrand Reinhold, New York [31] Klir G.J., Yuan B. (1995) Fuzzy sets and fuzzy logic: Theory and applications. Prentice-Hall, New Jersey · Zbl 0915.03001 [32] Lacagnina V., Pecorella A. (2006) A stochastic soft constraints fuzzy model for a portfolio selection problem. Fuzzy Sets and Systems 157: 1317–1327 · Zbl 1132.90357 [33] León T., Liern V., Vercher E. (2002) Viability of infeasible portfolio selection problems: A fuzzy approach. European Journal of Operational Research 139: 178–189 · Zbl 1007.90082 [34] Li J., Xu J. (2009) A novel portfolio selection model in a hybrid uncertain environment. Omega 37: 439–449 [35] Li P., Liu B. (2008) Entropy of credibility distributions for fuzzy variables. IEEE Transactions on Fuzzy Systems 16: 123–129 · Zbl 05516418 [36] Li X., Liu B. (2007) Maximum entropy principle for fuzzy variable. International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems 15(Supp. 2): 43–52 [37] Lin C.C., Liu Y.T. (2008) Genetic algorithm for portfolio selection problems with minimum transaction lots. European Journal of Operational Research 185: 393–404 · Zbl 1137.91461 [38] Liu B. (1998) Minimax chance constrained programming models for fuzzy decision systems. Information Sciences 112: 25–38 · Zbl 0965.90058 [39] Liu B. (2002) Theory and practice of uncertain programming. Physica-Verlag, Heidelberg · Zbl 1029.90084 [40] Liu B. (2004) Uncertainty theory: An introduction to its axiomatic foundations. Springer–Verlag, Berlin · Zbl 1072.28012 [41] Liu B. (2006a) A survey of credibility theory. Fuzzy Optimization and Decision Making 5: 387–408 · Zbl 1133.90426 [42] Liu B. (2007) Uncertainty theory, 2nd edn. Springer–Verlag, Berlin · Zbl 1141.28001 [43] Liu B., Iwamura K. (1998) Chance constrained programming with fuzzy parameters. Fuzzy Sets and Systems 94: 227–237 · Zbl 0923.90141 [44] Liu B., Liu Y.K. (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems 10: 445–450 [45] Liu Y.K. (2006b) Convergent results about the use of fuzzy simulation in fuzzy optimization problems. IEEE Transactions on Fuzzy Systems 14: 295–304 · Zbl 05452693 [46] Markowitz H. (1952) Portfolio selection. Journal of Finance 7: 77–91 [47] Markowitz H. (1959) Portfolio selection: efficient diversification of investments. Wiley, New York [48] Nahmias S. (1978) Fuzzy variables. Fuzzy Sets and Systems 1: 97–110 · Zbl 0383.03038 [49] Oh K.J., Kim T.Y., Min S.H., Lee H.Y. (2006) Portfolio algorithm based on portfolio beta using genetic algorithm. Expert Systems with Applications-Intelligent Information Systems for Financial Engineering 30: 527–534 [50] Rahib H.A., Mustafa M. (2007) Fuzzy portfolio selection using genetic algorithm. Soft Computing–A Fusion of Foundations, Methodologies and Applications 11: 1157–1163 · Zbl 1147.91022 [51] Rom B.M., Ferguson K.W. (1994) Post-modern portfolio theory comes of age. Journal of Investing 3: 11–17 [52] Shoaf, J., & Foster, J. A. (1996). The efficient set GA for stock portfolios. Proceedings of the Decision Science Institute, Orlando (pp. 571–573). [53] Simkowitz M., Beedles W. (1978) Diversification in a three moment world. Journal of Financial and Quantitative Analysis 13: 927–941 [54] Skolpadungket, P., Dahal, K., & Harnpornchai, N. (2007). Portfolio optimization using multi-objective genetic algorithms. Proceeding of 2007 IEEE Congress on Evolutionary Computation (pp. 516–523). [55] Tanaka H., Guo P. (1999) Portfolio selection based on upper and lower exponential possibility distributions. European Journal of Operational Research 114: 115–126 · Zbl 0945.91017 [56] Tanaka H., Guo P., Türksen B. (2000) Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets and Systems 111: 387–397 · Zbl 1028.91551 [57] Tiryaki F., Ahlatcioglu M. (2005) Fuzzy stock selection using a new fuzzy ranking and weighting algorithm. Applied Mathematics and Computation 170: 144–157 · Zbl 1151.91547 [58] Unser M. (2000) Lower partial moments as measures of perceived risk: An experimental study. Journal of Economic Psychology 21: 253–280 [59] Vercher E., Bermúdez J.D., Segura J.V. (2007) Fuzzy portfolio optimization under downside risk measures. Fuzzy Sets and Systems 158: 769–782 · Zbl 1190.91140 [60] Watada J. (1997) Fuzzy portfolio selection and its applications to decision making. Tatra Mountains Mathematical Publication 13: 219–248 · Zbl 0915.90008 [61] Yan W., Miao R., Li S. (2007) Multi-period semi-variance portfolio selection: Model and numerical solution. Applied Mathematics and Computation 194: 128–134 · Zbl 1193.91146 [62] Zadeh L. (1965) Fuzzy sets. Information and Control 8: 338–353 · Zbl 0139.24606 [63] Zadeh L. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3–28 · Zbl 0377.04002 [64] Zhang W.G., Nie Z.K. (2004) On admissible efficient portfolio selection problem. Applied Mathematics and Computation 159: 357–371 · Zbl 1098.91065 [65] Zhang W.G., Wang Y.L., Chen Z.P., Nie Z.K. (2007) Possibilistic mean-variance models and efficient frontiers for portfolio selection problem. Information Sciences 177: 2787–2801 · Zbl 1286.91131 [66] Zimmermann H.J. (1985) Fuzzy set theory and its applications. Kluwer Academic Publisher, Boston · Zbl 0578.90095
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