Portfolio optimization using a credibility mean-absolute semi-deviation model Enriqueta Vercher ⇑ , José D. Bermúdez Dept Statistics and Operational Research, University of Valencia, C/Dr. Moliner 50, 46100 Burjassot, Spain article info Article history: Available online 19 May 2015 Keywords: Credibility theory Fuzzy variables Portfolio selection Mean absolute semi-deviation Multi-objective optimization Genetic algorithm abstract We introduce a cardinality constrained multi-objective optimization problem for generating efficient portfolios within a fuzzy mean-absolute deviation framework. We assume that the return on a given portfolio is modeled by means of LR-type fuzzy variables, whose credibility distributions collect the con- temporary relationships among the returns on individual assets. To consider credibility measures of risk and return on a given portfolio enables us to work with its Fuzzy Value-at-Risk. The relationship between credibility expected values for LR-type fuzzy variables and possibilistic moments for LR-fuzzy numbers having the same membership function are analyzed. We apply a heuristic approach to approximate the cardinality constrained efficient frontier of the portfolio selection problem considering the below-mean absolute semi-deviation as a measure of risk. We also explore the impact of adding a Fuzzy Value-at-Risk measure that supports the investor’s choices. A computational study of our multi-objective evolutionary approach and the performance of the credibility model are presented with a data set collected from the Spanish stock market. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Financial planning has been considered one of the prototype decision making problems. A wide variety of mathematical pro- gramming models have been developed to address problems related to financial management and many optimization tech- niques have been introduced to solve them. Markowitz (1952) gave the first mathematical formulation of the portfolio selection problem, assuming that the return on every asset is a random vari- able with a given probability distribution, and that the risk of the investment is measured in terms of the variance of the portfolio. The pioneering mean–variance (MV) model by Markowitz assumes that the vector of returns on assets is multivariate-normally dis- tributed and that the investor prefers lower risks, with this approach leading to a quadratic programming problem. Alternatively, Konno and Yamazaki (1991) proposed a linear opti- mization model for portfolio selection by using the mean absolute deviation (MAD) around the expected return as a measure of risk; while Cai, Teo, Yang, and Zhou (2000) introduced the minimax rule in the portfolio selection and provided a new l 1 model based on the maximum absolute deviation as a measure of risk. Recently, the mean absolute deviation has also been used as measure of risk in a cardinality constrained portfolio selection framework, where genetic algorithms provide efficient portfolios (Chang, Yang, & Chang, 2009). In the classical modeling approach, an optimal portfolio has to satisfy a kind of balance between maximizing the expected return and minimizing the risk of the investment, respecting the inves- tor’s declared preferences. The portfolio selection problem then becomes an optimization problem with multiple objectives and/or additional constraints. Alternative statistical measures of portfolio return, like semi-variance, absolute semi-deviation, skewness and value-at-risk (the worst expected loss over a given horizon), can also be found in the multi-objective portfolio selection models and many multi-criteria tools have been used to find efficient port- folios (see, for instance, Ehrgott, Klamroth, & Schwehm (2004), Steuer, Qi, & Hirschberger (2007) and references therein). Recently, some researchers apply multi-objective evolutionary algorithms to suitably handling these more complex portfolio opti- mization problems (Anagnostopoulos & Mamanis, 2011; Liagkouras & Metaxiotis, 2014). A comprehensive survey of several heuristic approaches for portfolio management can be found in Metaxiotis and Liagkouras (2012). Solving the portfolio selection problem must provide both the assets and their corresponding proportions that are an optimal investment, in a certain way. Then, in order to determine an opti- mal portfolio both optimization techniques and uncertainty http://dx.doi.org/10.1016/j.eswa.2015.05.020 0957-4174/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: vercher@uv.es (E. Vercher), bermudez@uv.es (J.D. Bermúdez). Expert Systems with Applications 42 (2015) 7121–7131 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa