A credibilistic decision support system for portfolio optimization

Abstract In this paper, a Decision Support System (DSS) for generating a suitable portfolio for an investor in an uncertain multi-criteria framework is proposed. We model uncertain parameters like return and illiquidity of various assets using L–R fuzzy numbers belonging to a power reference function family. Such usage of L–R fuzzy numbers is more generic as compared to the conventional triangular or trapezoidal fuzzy numbers and is a closer representation of uncertain behavior of the asset parameters. The credibility measure which has an advantage of being self-dual as compared to usual possibility measure marks the uncertain context of the entire setup and adds a new dimension to existing studies. We use an “Entropy-Cross Entropy (ECE) Algorithm”, for finding the solution of the optimization problem meant for finding the best fit L–R fuzzy number corresponding to uncertain asset parameters. This automates the entire subjective exercise where, otherwise, a human intervention is required for feeding the parameters required for fitting L–R fuzzy number. Once the L–R fuzzy number are created around various assets available for the portfolio formation, the portfolio optimization problem is solved using Hybrid Intelligent Algorithm (HIA). HIA is designed by embedding fuzzy simulation within the “MIBEX-SM” genetic algorithm. To demonstrate the entire solution approach, four portfolio optimization models are solved using historical data from the National Stock Exchange (NSE) of India. The performance of the models is compared using a modified Sharpe ratio in the fuzzy context, namely the “Credibilistic Sharpe Ratio (CrSR)”. To carry out the empirical study, firstly we use the data from the period of 2008–2013 for training the optimization models. Thereafter, to test the performance of the models, the data set from the two year period 2013–2015 is used. The process of validation of the results is then carried out using the data set of the one year period from 2015 to 2016.

[1]  Shouyang Wang,et al.  On Fuzzy Portfolio Selection Problems , 2002, Fuzzy Optim. Decis. Mak..

[2]  H. D. Cheng,et al.  Automatically Determine the Membership Function Based on the Maximum Entropy Principle , 1997, Inf. Sci..

[3]  Li Li,et al.  Fuzzy Chance-Constrained Project Portfolio Selection Model Based on Credibility Theory , 2014 .

[4]  E. Polak,et al.  On Multicriteria Optimization , 1976 .

[5]  Samarjit Kar,et al.  Fuzzy cross-entropy, mean, variance, skewness models for portfolio selection , 2014, J. King Saud Univ. Comput. Inf. Sci..

[6]  Zhongfeng Qin,et al.  MEAN-SEMIVARIANCE MODELS FOR PORTFOLIO OPTIMIZATION PROBLEM WITH MIXED UNCERTAINTY OF FUZZINESS AND RANDOMNESS , 2013 .

[7]  Xiaoxia Huang,et al.  Portfolio Analysis - From Probabilistic to Credibilistic and Uncertain Approaches , 2012, Studies in Fuzziness and Soft Computing.

[8]  J. Watada,et al.  Possibilistic linear systems and their application to the linear regression model , 1988 .

[9]  Pankaj Gupta,et al.  Multiobjective expected value model for portfolio selection in fuzzy environment , 2013, Optim. Lett..

[10]  Sang-Chin Yang,et al.  Portfolio optimization problems in different risk measures using genetic algorithm , 2009, Expert Syst. Appl..

[11]  Bohdan S. Butkiewicz,et al.  A Method for Automatic Membership Function Estimation Based on Fuzzy Measures , 2007, IFSA.

[12]  Xiaoxia Huang,et al.  Mean-semivariance models for fuzzy portfolio selection , 2008 .

[13]  Xiang Li,et al.  Mean-variance-skewness model for portfolio selection with fuzzy returns , 2010, Eur. J. Oper. Res..

[14]  Rupak Bhattacharyya,et al.  Possibilistic Sharpe Ratio Based Novice Portfolio Selection Models , 2013, Computer Science & Information Technology ( CS & IT ).

[15]  Konstantinos P. Anagnostopoulos,et al.  A portfolio optimization model with three objectives and discrete variables , 2010, Comput. Oper. Res..

[16]  Kusum Deep,et al.  A real coded genetic algorithm for solving integer and mixed integer optimization problems , 2009, Appl. Math. Comput..

[17]  Masahiro Inuiguchi,et al.  Multiobjective credibilistic portfolio selection model with fuzzy chance-constraints , 2013, Inf. Sci..

[18]  L. Pedersen,et al.  Asset Pricing with Liquidity Risk , 2003 .

[19]  Enriqueta Vercher,et al.  Portfolio optimization using a credibility mean-absolute semi-deviation model , 2015, Expert Syst. Appl..

[20]  Xiaoxia Huang,et al.  Risk curve and fuzzy portfolio selection , 2008, Comput. Math. Appl..

[21]  W. Sharpe.  The Sharpe Ratio , 1994 .

[22]  G. Klir,et al.  Fuzzy Measure Theory , 1993 .

[23]  Xiang Li.  Fuzzy cross-entropy , 2015 .

[24]  Xiang Li,et al.  A hybrid intelligent algorithm for portfolio selection problem with fuzzy returns , 2009, J. Comput. Appl. Math..

[25]  Rico von Wyss,et al.  Measuring and Predicting Liquidity in the Stock Market , 2004 .

[26]  Zhongfeng Qin,et al.  Portfolio selection based on fuzzy cross-entropy , 2009 .

[27]  S. Satchell,et al.  On the foundation of performance measures under asymmetric returns , 2002 .

[28]  Manoj Thakur,et al.  A modified real coded genetic algorithm for constrained optimization , 2014, Appl. Math. Comput..

[29]  David W. Coit,et al.  Multi-objective optimization using genetic algorithms: A tutorial , 2006, Reliab. Eng. Syst. Saf..

[30]  Robert LIN,et al.  NOTE ON FUZZY SETS , 2014 .

[31]  Teresa León,et al.  Solving a class of fuzzy linear programs by using semi-infinite programming techniques , 2004, Fuzzy Sets Syst..

[32]  Yian-Kui Liu,et al.  Expected value of fuzzy variable and fuzzy expected value models , 2002, IEEE Trans. Fuzzy Syst..

[33]  Rubén Saborido,et al.  Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection , 2016, Appl. Soft Comput..

[34]  K. Deb.  An Efficient Constraint Handling Method for Genetic Algorithms , 2000 .

[35]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[36]  C. Zopounidis,et al.  Multicriteria decision systems for financial problems , 2013, TOP.

[37]  Kathrin Klamroth,et al.  An MCDM approach to portfolio optimization , 2004, Eur. J. Oper. Res..

[38]  Hideki Katagiri,et al.  Constructing membership function based on fuzzy shannon entropy and human's interval estimation , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[39]  Constantin Zopounidis,et al.  Multicriteria Portfolio Management , 2012 .

[40]  D. Dubois,et al.  A NOTE ON MEASURES OF SPECIFICITY FOR FUZZY SETS , 1985 .

[41]  Xiaoxia Huang,et al.  Fuzzy chance-constrained portfolio selection , 2006, Appl. Math. Comput..

[42]  Hashem Omrani,et al.  An integrated multi-objective Markowitz-DEA cross-efficiency model with fuzzy returns for portfolio selection problem , 2016, Appl. Soft Comput..

[43]  Manoj Thakur,et al.  Modelling and constructing membership function for uncertain portfolio parameters: A credibilistic framework , 2017, Expert Syst. Appl..

[44]  Aparna Mehra,et al.  Financial analysis based sectoral portfolio optimization under second order stochastic dominance , 2017, Ann. Oper. Res..

[45]  Zbigniew Michalewicz,et al.  Computational Intelligence for Evolving Trading Rules , 2009, IEEE Transactions on Evolutionary Computation.

[46]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[47]  D. Dubois,et al.  Operations on fuzzy numbers , 1978 .

[48]  Hisao Ishibuchi,et al.  Multi-objective genetic local search algorithm , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

[49]  Enriqueta Vercher,et al.  A Possibilistic Mean-Downside Risk-Skewness Model for Efficient Portfolio Selection , 2013, IEEE Transactions on Fuzzy Systems.

[50]  Michael Haliassos,et al.  Portfolio Choice and Liquidity Constraints , 2000, SSRN Electronic Journal.

[51]  Rebel A. Cole,et al.  Changes in REIT Liquidity 1988–2007: Evidence from Daily Data , 2010 .

[52]  Paul D. Kaplan,et al.  Semivariance in Risk-Based Index Construction , 1997 .

[53]  Saeid Nahavandi,et al.  Fuzzy Portfolio Allocation Models Through a New Risk Measure and Fuzzy Sharpe Ratio , 2015, IEEE Transactions on Fuzzy Systems.

[54]  Manoj Thakur,et al.  Fuzzy Classification using Self-Adaptive Algorithm to Generate Membership Function , 2016 .

[55]  Y. Amihud,et al.  Illiquidity and Stock Returns II: Cross-Section and Time-Series Effects , 2018, The Review of Financial Studies.

[56]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[57]  Suzana Baresa,et al.  Measuring Liquidity on Stock Market: Impact on Liquidity Ratio , 2012 .