A credibilistic mean-semivariance-PER portfolio selection model for Latin America

García, Fernando; González-Bueno, Jairo; Oliver, Javier; Tamošiūnienė, Rima Article A credibilistic mean-semivariance-PER portfolio selection model for Latin America Journal of Business Economics and Management (JBEM) Provided in Cooperation with: Vilnius Gediminas Technical University (VILNIUS TECH) Suggested Citation: García, Fernando; González-Bueno, Jairo; Oliver, Javier; Tamošiūnienė, Rima (2019) : A credibilistic mean-semivariance-PER portfolio selection model for Latin America, Journal of Business Economics and Management (JBEM), ISSN 2029-4433, Vilnius Gediminas Technical University, Vilnius, Vol. 20, Iss. 2, pp. 225-243, https://doi.org/10.3846/jbem.2019.8317 This Version is available at: https://hdl.handle.net/10419/317326 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Copyright © 2019 The Author(s). Published by VGTU Press *Corresponding author. E-mail: [email protected] This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Journal of Business Economics and Management ISSN 1611-1699 / eISSN 2029-4433 2019 Volume 20 Issue 2: 225–243 https://doi.org/10.3846/jbem.2019.8317 A CREDIBILISTIC MEAN-SEMIVARIANCE-PER PORTFOLIO SELECTION MODEL FOR LATIN AMERICA Fernando GARCÍA 1*, Jairo GONZÁLEZ-BUENO 2, Javier OLIVER 3, Rima TAMOŠIŪNIENĖ 4 1, 3Faculty of Business Administration and Management, Universitat Politècnica de València, Campus de Vera, Camino de Vera, s/n, 46022 València, Spain 2Faculty of Business Administration, Universidad Pontificia Bolivariana, Km 7 Via Piedecuesta, Bucaramanga, 681017, Santander, Colombia 4Faculty of Business Management, Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223, Vilnius, Lithuania Received 29 October 2018; accepted 09 January 2019 Abstract. Many real-world problems in the financial sector have to consider different objectives which are conflicting, for example portfolio selection. Markowitz proposed an approach to determine the optimal composition of a portfolio analysing the trade-off between return and risk. Nevertheless, this approach has been criticized for unrealistic assumptions and several changes have been proposed to incorporate investors’ constraints and more realistic risk measures. In this line of research, our proposal extends the mean-semivariance portfolio selection model to a multiobjective credibilistic model that besides risk and return, also considers the price-to-earnings ratio to measure portfolio performance. Uncertain future returns and PER ratio of each asset are approximated using L-R power fuzzy numbers. Furthermore, we consider budget, bound and cardinality constraints. To solve the constrained portfolio optimization problem, we use the algorithm NSGA-II. We assess the proposed approach generating a portfolio with shares included in the Latin American Integrated Market. Results show that this new approach is a good alternative to solve the portfolio selection problem when multiple objectives are considered. Keywords: fuzzy portfolio selection, credibility theory, L-R power fuzzy numbers, mean-semivariance-PER, evolutionary multiobjective optimization. JEL Classification: C61, G11, G17. Introduction Stock exchange investors have a variety of strategies available to allocate their wealth. Shortterm investors mainly employ technical analysis (Sobreiro et al., 2016; Zhu, Atri, & Yegen, 2016) and chartist analysis (Gerritsen, 2016; Schmitt & Westerhoff, 2017). Long-term inves- 226 F. García et al. A credibilistic mean-semivariance-PER portfolio selection model for Latin America tors widely use fundamental analysis (De Oliveira, Nobre, & Zárate, 2013; Shen, Yan, & Tzeng, 2014) or passive investment strategies (García, Guijarro, & Moya, 2013; García, Guijarro, & Oliver, 2018). Basically, all these strategies concentrate on the expected return of the assets, which are analyzed individually, not as a portfolio. Therefore, asset returns are considered to be independent. Nevertheless, when a portfolio is built, it is necessary to take into account the return correlation of the assets in the portfolio. Markowitz (1952) was the first who changed the focus of investment analysis away from individual assets selection towards the concept of diversification and shed light on the portfolio selection problem. The portfolio selection problem deals with the selection of assets among a group of candidate assets to create a portfolio which best maximizes investor’s goals regarding several criteria such as return and risk. Markowitz (1952) was the one to tackle this problem in a quantitative way and proposed as selection criteria the mean and the variance of the returns of the assets in order to account for the risk-return trade-off. Nevertheless, the model by Markowitz has been criticized for several reasons. A main drawback is assuming multivariate normality, which is not the case. Furthermore, integer constraints that limit a portfolio to have a specified number of assets, or to impose limits on the proportion of the portfolio held in a given asset cannot be easily applied (Chang, Meade, Beasley, & Sharaiha, 2000). Finally, regarding the objectives which are to be simultaneously optimized, only return and risk are considered, which is not realistic. Furthermore, using the variance of returns in order to quantify the risk has been identified as a shortcoming, as well. In this context, many researches have tackled the portfolio selection problem mainly from three perspectives: i) dealing with the uncertainty of assets return; ii) including new criteria for estimating portfolio risk; iii) adding new constraints faced by investors. Regarding the uncertainty of assets return, assets’ rate of returns are assumed to be random variables and probability theory is used to select the optimal portfolio (Huang, 2009). However, as noted by Huang (2010), randomness is not the only sort of uncertainty in real life, particularly when persons are involved. Furthermore, the information available in the financial markets for investment decision making is often incomplete, ambiguous and vague (Gupta, Mittal, & Mehlawat, 2013, 2014). Due to the above considerations, as from the 90’s of last century researchers have applied fuzzy set theory (Zadeh, 1965) to describe and study the fuzziness contained in portfolio investment. There is a large body of research describing how to use possibility distributions to model the uncertainty on returns (Carlsson, Fullér, & Majlender, 2002; Saborido, Ruiz, Bermúdez, Vercher, & Luque, 2016; Vercher, Bermúdez, & Segura, 2007; Yue & Wang, 2017). However, the widely-employed possibility measure is not self-dual, which is an important drawback. In other words, it is not consistent with the law of excluded middle and the law of contradiction. For example, a fuzzy event with possibility value 1 may still not occur. Furthermore, it is possible that two fuzzy events with different chances of occurrence may have the same possibility value (Gupta et al., 2013). Therefore, the possibility value gives little information to the investor and may confuse her/him. To solve this limitation, B. Liu and Y. K. Liu (2002) suggested a self-dual credibility measure. Note that the fuzzy event will surely happen if its credibility value is 1 and fail if its credibility value is 0 (Gupta, Mehlawat, Inuiguchi, & Chandra, 2014a). Thus, credibility value is consistent with investors’ judgement and the confusion will disappear. Since then, different researchers have Journal of Business Economics and Management, 2019, 20(2): 225–243 227 used credibility distributions to approximate the uncertainty on returns (Barak, Abessi, & Modarres, 2013; Huang, 2006; Jalota, Thakur, & Mittal, 2017b; Vercher & Bermúdez, 2015). For the above reasons, this paper extends the literature on portfolio selection model by assuming that the return on each asset is an L-R power fuzzy variable whose moments are assessed employing their credibility distributions. As for the criteria to evaluate portfolio’s performance, many practitioners and academics use the variance as the risk measure to solve the portfolio selection problem (Metaxiotis & Liagkouras, 2012), despite its deficiencies. Among them, it is important to underline that not just downside deviations from the expected return, that is, losses, but gains as well (Gupta et al., 2013). Additionally, it is not an appropriate risk measure if return distributions are asymmetric (Chunhachinda, Dandapani, Hamid, & Prakash, 1997). To solve this limitation of the mean-variance model, several downside risk measures were proposed by Fishburn (1977), Morgan (1996), Markowitz (1959), Rockafellar and Uryasev (2000), and Speranza (1993), to just consider the negative deviations from a reference return level. In this context, semivariance is probably the most popular downside risk measure. In contrast to variance, semivariance is direct, clear and can easily reflect investors’ intuition about risk (Huang, 2008). Additionality, it is a more appropriate risk measure when an investor is worried about underperformance rather than over performance of portfolio (Markowitz, Todd, Xu, & Yamane, 1993). Therefore, this paper applies the semivariance to measure the risk in actual stock markets. Finally, in the classical portfolio selection problem, the main decision criteria employed by investors are return and risk. However, other criteria might generate an equal or greater satisfaction level for the investor. Following Omidi, Abbasi, and Nazemi (2017), when other criteria are considered, it may be possible to obtain portfolios in which lower return or higher risk are compensated by other criteria, which may produce more satisfaction to investors seeking, not just to maximize return and minimize risk, but to consider other variables. A review of the literature shows that numerous papers have been devoted to develop the original mean-variance model by Markowitz into a new multicriteria framework in order to account for additional decision criteria (Fang, Chen, & Fukushima, 2008; García et al., 2013; Li, Zhu, Sun, Aw, & Teo, 2018; Xia, Liu, Wang, & Lai, 2000). Following this trend, the price-to-earnings ratio (P/E ratio) is one important criterion applied by practitioners to select stocks, due to its ability to capture the current expectations of the market about the companies (Pouya, Solimanpur, & Rezaee, 2016). As far as we know, no previous research has evaluated the performance of the portfolio price to earnings ratio (PER) in a credibility environment. Thus, this study makes a contribution to the literature by proposing a fuzzy multiobjective model, where besides return and risk, also PER ratio is included to measure the performance of a portfolio. The aim and contribution of our research is to extend the mean-semivariance portfolio selection model from a stochastic environment into a multi-criteria portfolio model in a fuzzy environment. Besides return and risk, the proposed approach also considers the PER ratio to measure portfolio performance. Uncertain future returns and PER of each asset is calculated by means of L-R power fuzzy numbers. To make a more realistic model, budget, bound and cardinality constraints are included, as well. The inclusion of these constraints 228 F. García et al. A credibilistic mean-semivariance-PER portfolio selection model for Latin America in the portfolio optimization model change it into a constrained NP-hard multi-objective problem, for which traditional optimization methods cannot be used to find efficient portfolios. In order to solve this problem we apply the Non-dominated Sorting Genetic Algorithm II (NSGA-II), which is the most commonly used multiobjective evolutionary algorithm (MOEAs) for solving similar portfolio optimization problems. The performance of this approach is assessed using the stocks included in the Latin American Integrated Market, which integrates the stock exchange markets of Chile, Colombia, Mexico, and Peru for the period from June 2011 to December 2016. The remainder of the research is organized as follows: First, the basics of L-R fuzzy numbers and the credibility theory are introduced. Then, the multiobjective credibilistic meansemivariance-PER portfolio selection model is described and the methodology to solve the proposed approach is discussed. The next section presents an actual empirical study to stress the advantages of our model. Finally, Section 5 concludes. 1. L-R fuzzy number and the credibility theory: basic background In this section, some basic definitions from the relevant literature on L-R fuzzy number and the credibility theory are presented so the model introduced in section 2 can properly be understood. 1.1. L-R power fuzzy number Following Jalota, Thakur, and Mittal (2017a), using L–R fuzzy numbers for individual stocks makes it possible to capture the information about their behavior more precisely, so their proper contribution to the portfolio is calculated in a more appropriate way. Definition 1. The functions L, R:[0,1]→[0,1] are reference functions of a fuzzy number they satisfy the following conditions A = (x, µA(x)), they satisfy the following conditions (Dubois & Prade, 1987): i) L(1) = R(1) = 0, L(0) = R(0) = 1; ii) L(x) and R(x) are strictly decreasing and upper semicontinuous functions. Definition 2. A fuzzy number M = (a, b, c, d)LπRρ is said to be an LR-type fuzzy number if its membership function has the following form (Dubois & Prade, 1980): ( ) –,  – 1,  –, – 0, Otherwise M bx L Sí a x b ba Sí b x c xxc R Sí c x d dc π ρ  ≤<    ≤≤  µ=  <≤      where (b – a) and (d – c) show the left and right spreads of M, respectively. Lπ and Rρ are Journal of Business Economics and Management, 2019, 20(2): 225–243 229 the reference functions that define the left and right shapes of M, respectively. In this paper, left and right shapes of L-R fuzzy numbers are defined by Lπ(k) = 1 – xπ, and Lρ(k) = 1 – xρ, respectively. Throughout this study, L-R power fuzzy numbers will be denoted by M = (a, b, c, d)πρ. In the case of LR-fuzzy numbers with linear reference functions or with the same shape for L and R, the aggregation provides fuzzy numbers of the same shape (Vercher et al., 2007). However, when the shape of the fuzzy numbers is not the same, their aggregation will not result in fuzzy numbers with the same shape (Inuiguchi, Ichihashi, & Tanaka, 1990; León & Vercher, 2004). Throughout this paper, the LR-fuzzy numbers employed will have the same reference functions L and R. In this way, following arithmetical rules hold according to the extension principle of Zadeh: Theorem 1. Let A = (a1, b1, c1, d1)LR and B = (a2, b2, c2, d2)LR be two LR-fuzzy numbers and λ ∈ ℜ be a real number (Vercher et al., 2007). Then, a) ( ) 12 12 12 1 2 , , , LR AB a a b b c c d d+= + + + + ; b) ( ) ( ) 111 1 11 1 1 ,  , , , ,   0, , ,  ,    0, LR LR a b c d if Aa b c d if λ λ λ λ λ≥  λ= λ λ λ λ λ<   where the addition and multiplication by a scalar is defined by means of the sup-min extension principle. 1.2. The credibility theory In the capital markets investors deal with different fuzzy phenomena, not just randomness. In order to describe fuzziness, Zadeh (1965) proposed for the first time the concept of fuzzy set via membership function. Additionally, to measure a fuzzy event, Zadeh (1978) proposed the possibility measure. This measure has been widely accepted and applied, but it is not consistent with the law of excluded middle and the law of contradiction, that is, it has no the self-duality property. Following Huang (2010), by using possibility, investors knowing the possibility level of a portfolio reaching a target return, cannot know the possibility level of the opposite event. Therefore, this situation may be confusing for investors. To avoid this situation, B. Liu and Y. K. Liu (2002) defined a self-dual measure, a credibility measure. The credibility theory was introduced later by Liu (2004) and extended in Liu (2007). Definition 3. Let ξ be a fuzzy variable with membership function µ, and x a real number. The credibility measure of a fuzzy event, characterized by ξ ≤ x, is defined by B. Liu and Y. K. Liu (2002): { } ( ) ( ) 1  sup 1 –sup , 2 yx yx Cr x y y x R ≤>  ξ≤ = µ + µ ∀ ∈    . The value of credibility takes values in [0, 1] (Liu, 2004). It is easy to verify that the credibility is self-dual, that is, { } { }  1Cr x Cr x ξ≤ + ξ> = . 230 F. García et al. A credibilistic mean-semivariance-PER portfolio selection model for Latin America Definition 4. Let ξ be a fuzzy variable, then the expected value of ξ is defined by B. Liu and Y. K. Liu (2002): ( ) { } { } 0 0- –E Cr x dx Cr x dx +∞ ∞ ξ = ξ≥ ξ≤ ∫∫ provided that at least one of the two integrals is finite. The crisp equivalent expression for credibilistic expected value of an L-R power fuzzy ξ= (a, b, c, d)πρ, it is obtained by deriving the expected value of a fuzzy variable (Jalota etal., 2017b): () 1 (–) (–) – 2 11 dc ba E bc  ρπ ξ= ++  ρ+ π+  . (1) Definition 5. Let ξ be a fuzzy variable with finite expected value e = E[ξ]. Then the semivariance of ξ is defined by B. Liu and Y. K. Liu (2002): () 2 - –, SV E e  ξ= ξ         where -–,  –0,  e Sí e eSí e ξ ξ≤   ξ=    ξ>   . The crisp equivalent expression for the credibility measure based semivariance value of an L-R power fuzzy number ξ = (a, b, c, d)πρ, was derived by (Jalota et al., 2017b): ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) 2 22 22 2 22 – –– – – – , if  21 12 – 12 – –– – – , if  21 12 – –– – – – – , if  21 12 – 12 0 otherwise ea eaba ba ec ced dc ea eaba ba bec SV ea eaba be ba aeb ba ρ+ ρ π+ π  + + <≤ π+ π+ π+ ρ+ ρ+   + <≤  ξ= π+ π+ π+  + <≤ π+ π+ π+ π+ π+    (2) 2. Multiobjective credibilistic mean-semivariance-P/E portfolio selection model Th is section discusses the proposed credibilistic multi-objective portfolio selection model and the methodology used to solve it. The parameters and decision variables used in the mathematical model are the following: Parameters ξri: fuzzy rate of return of the asset i expressed as L-R power fuzzy number ξri = (ari, bri, cri, dri)πriρri , i: 1, 2, ..., n. Journal of Business Economics and Management, 2019, 20(2): 225–243 231 ξP/Ei: fuzzy P/E of the asset i expressed as L-R power fuzzy number ξP/Ei = (aP/Ei, bP/Ei, cP/Ei, dP/Ei)πP/EiρP/Ei , i: 1, 2, ..., n. ξrp: fuzzy expected return of the portfolio expressed as L-R power fuzzy number ξrp = (arp, brp, crp, drp)πrpρrp , e: expected return of the portfolio, ui: maximal fraction of the capital budget allocated to asset i, i: 1, 2, ..., n. li: minimal fraction of the capital budget allocated to asset i, i: 1, 2, ..., n. k: number of assets held in the portfolio. Decision variables i ω : proportion of the total funds invested in the asset i, i: 1, 2, ..., n, yi: binary variable indicating whether the asset i is contained in the portfolio, i: 1, 2, ..., n, that is 1, if asset is contained in the portfolio, 0, otherwise. i i y  =   2.1. Objective functions i) Return Considering that in the capital markets information available for the decision maker is often incomplete, ambiguous and vague, this paper assumes that an investor allocates his capital among n assets that have fuzzy returns. The return on the i-th asset is expressed by L-R power Figure 1. Parameters of the membership function of ξri by using the sample percentiles of the returns of the asset i 1 0 Support Percentile 3thPercentile 97th Core Percentile 45th Percentile 55th t Left Shape ParameterRightShape Parameter µ aribricridri Ln0.5 – Percentile 25th Ln – i i ii r r rr b ba π=      Ln0.5 Percentile 75th – Ln – i i ii r r rr c dc ρ=      232 F. García et al. A credibilistic mean-semivariance-PER portfolio selection model for Latin America fuzzy numbers (i.e. ξri = (ari, bri, cri, dri)πriρri, i: 1, 2, ..., n., T: 1, 2, …, n), where its α-level cuts are [ξri]α = [{bri – (bri – ari)}(1 – α),{cri – (dri – cri)}(1 – α)] for α ∈ [0, 1]. The core, support and shape parameters of the fuzzy return of every asset are obtained from the empirical percentiles of its historical returns (Vercher & Bermúdez, 2012, 2013, 2015), asshown in theFigure 1. Anoter way to obtain the mentioned parameter values is from the expertise of professional investors. The maximization of the expected return of the portfolio can be expressed by the following crisp objective: ()() 1 1 1  ( ) – 21 – 1 – i ii i ii ii ii nr rr r rr i rr i rr i dc ba Max F b c =    ρπ    ω= + + ω    ρ+ π+       ∑ . ii) Risk Semivariance is better suited to properly capture risk than the variance (Markowitz, 1959), as semivariance is a downside risk, which means that it only deals with adverse deviations. The minimization of the semivariance of the portfolio can be expressed by the following crisp objective: 2 2 22 22 2 () (–)(–)(–) (–) (–) – ,; 2 1 ( 1)( 2) ( – ) ( 1)( 2) (–)(–)(–) (–) – 2 1 ( 1)( 2) (– ) (– – 2 r r i r rrr rr r rr r rr rr r r r rrr rr rr r rr rr MinF ea ea b a b a ec if c e d dc ea ea b a b a if b e c ea ea ρ ρ ρρρ ρρ ρ ρρ ρ ρ ρρ ρρ ρ ρ ρ ρρρ ρρ ρρ ρ ρρ ρ ρ+ ρ ω= + + << π+ π+ π+ ρ+ ρ+ + << π+ π+ π+ 22 )(–) (–) (–) – ,; 1 ( 1)( 2) ( – ) ( 1)( 2) 0, r r rr r rr rr r rr rr r r ba be ba if a e b ba ρ ρρρ ρ ρρ ρρ ρ ρ ρρ ρρ ρ ρ π+ π+ << π+ π+ π+ π+ π+ otherwise.                iii) Price-to-earnings ratio Price-to-earnings (PER) is a valuation tool of market’s confidence in the shares of a firm (Masa’deh, Tayeh, Al-Jarrah, & Tarhini, 2015). The practical common way to compute the PER is dividing the current stock’s price by its current earnings per share. Generally, most investors consider the PER criterion in their analysis and investment decisions (Pouya et al., 2016). Therefore, it is reasonable to include the PER as an additional criterion in the meansemivariance base model to adapt it to investors’ requirements. However, because of incomplete information in the capital markets, P/E ratios are only vague estimates. Therefore, this paper considers that the P/E ratio of the i-th asset is expressed by an L-R power fuzzy number (i.e. ξP/Ei = (aP/Ei, bP/Ei, cP/Ei, dP/Ei)πP/Ei ρP/Ei, i: 1, 2, ..., n. T: 1, 2, …, n), where its α-level cuts are [ξP/Ei]α = [{bP/Ei – (bP/Ei – aP/Ei)}(1 – α), {cP/Ei – (dP/Ei – cP/Ei)}(1 – α)] for α ∈ [0, 1]. Journal of Business Economics and Management, 2019, 20(2): 225–243 239 Results show that the model can be apply under real world conditions. The model generates a three-dimensional Pareto-front that illustrates the trade-off between return, risk and PER. Portfolios on this frontier cannot be improved by any other portfolio in terms of return, risk and PER simultaneously. It is interesting to underline that the different interpretations of PER by market participants, as a “buy” or “sell” signal, are captured by the model. This lack of consensus in the interpretation leads portfolios with extreme PER values to being riskier. Finally, several future research possibilities exist to overcome the limitations of our study. Such research lines include the use of other selection criteria. In fact, the research context could be expanded in order to include other criteria considered by contemporary investors, who also care about companies’ performance evaluation issues, especially after losses caused by the financial crisis (Ahmed, Ali, Ejaz, & Ahmad, 2018; Narkunienė & Ulbinaitė, 2018; Zemguliene & Valukonis, 2018). Additionally, transaction costs are one of the main concerns for investors and portfolio managers as the utility of the portfolio depends heavily on them. Therefore, a future research opportunity would be to extend the proposed portfolio optimization model to incorporate transaction cost constraints. Another limitation of this research is to treat portfolio optimization as a single-period problem, that is, assuming that the allocation decision made at the beginning is static until the end of an investment. The single-period framework is a myopic portfolio strategy which does not capture intertemporal effects and hedging demands. Thus, a future research option is to develop an approach to consider a multi-period portfolio optimization model. Moreover, in this research, we consider the semivariance to measure portfolio risk. However, portfolio risk quantification has been widely studied in the literature and several downside risk measures have been proposed. In this sense, a future interesting research would be to consider other coherent risk measures, such as value at risk and mean-absolute semi-deviation. Author contribution FG and JGB conceived the study and the methodology, JGB obtained the data, JO was responsible for the data curation, analysis and editing, FG, JGB and JO wrote the first draft of the article and RT made proofreading and supervision. Disclosure statement Authors declare that they do not have any competing financial, professional, or personal interests from other parties. References Ahmed, A., Ali, R., Ejaz, A., & Ahmad, M. I. (2018). Sectoral integration and investment diversification opportunities: evidence from Colombo Stock Exchange. Entrepreneurship and Sustainability Issues, 5(3), 514-527.https://doi.org/10.9770/jesi.2018.5.3(8) Arora, R., Kaushik, S. C., & Arora, R. (2015). Multi-objective and multi-parameter optimization of twostage thermoelectric generator in electrically series and parallel configurations through NSGA-II. Energy, 91, 242-254. https://doi.org/10.1016/j.energy.2015.08.044 240 F. García et al. A credibilistic mean-semivariance-PER portfolio selection model for Latin America Barak, S., Abessi, M., & Modarres, M. (2013). Fuzzy turnover rate chance constraints portfolio model. European Journal of Operational Research, 228(1), 141-147. https://doi.org/10.1016/j.ejor.2013.01.036 Carlsson, C., Fullér, R., & Majlender, P. (2002). A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets and Systems, 131(1), 13-21. https://doi.org/10.1016/S0165-0114(01)00251-2 Chang, T. J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302. https://doi.org/10.1016/S0305-0548(99)00074-X Chunhachinda, P., Dandapani, K., Hamid, S., & Prakash, A. J. (1997). Portfolio selection and skewness: Evidence from international stock markets. Journal of Banking & Finance, 21(2), 143-167. https://doi.org/10.1016/S0378-4266(96)00032-5 De Oliveira, F. A., Nobre, C. N., & Zárate, L. E. (2013). Applying Artificial Neural Networks to prediction of stock price and improvement of the directional prediction index – Case study of PETR4, Petrobras, Brazil. Expert Systems with Applications, 40(18), 7596-7606. https://doi.org/10.1016/j.eswa.2013.06.071 Deb, K., Agrawal, K., Pratap, A., & Meyarivan, T. (2002). A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimization: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2), 182-197. https://doi.org/10.1109/4235.996017 Dubois, D. J., & Prade, H. (1980). Fuzzy sets and systems: theory and applications (Vol. 144). New York: Academic Press. Dubois, D. J., & Prade, H. (1987). Fuzzy numbers: an overview. In J. Bezdek (Ed.), Analysis of fuzzy information (pp. 3-39). Boca Raton: CRC Press. Fang, Y., Chen, L., & Fukushima, M. (2008). A mixed R&D projects and securities portfolio selection model. European Journal of Operational Research, 185(2), 700-715. https://doi.org/10.1016/j.ejor.2007.01.002 Fishburn, P. C. (1977). Mean-risk analysis with risk associated with below-target returns. The American Economic Review, 67(2), 116-126. García, F., Guijarro, F., & Moya, I. (2013). A multiobjective model for passive portfolio management: an application on the S&P 100 index. Journal of Business Economics and Management, 14(4), 758-775. https://doi.org/10.3846/16111699.2012.668859 García, F., Guijarro, F., & Oliver, J. (2018). Index tracking optimization with cardinality constraint: a performance comparison of genetic algorithms and tabu search heuristics. Neural Computing and Applications, 30(8), 2625-2641. https://doi.org/10.1007/s00521-017-2882-2 Gerritsen, D. F. (2016). Are chartists artists? The determinants and profitability of recommendations based on technical analysis. International Review of Financial Analysis, 47, 179-196. https://doi.org/10.1016/j.irfa.2016.06.008 Gupta, P., Mehlawat, M. K., Inuiguchi, M., & Chandra, S. (2014a). Portfolio optimization using credibility. In Fuzzy Portfolio optimization. Studies in fuzziness and soft computing (Vol. 316, pp. 127-160). Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-54652-5_5 Gupta, P., Mehlawat, M. K., Inuiguchi, M., & Chandra, S. (2014b). Portfolio optimization with interval coefficients. In Fuzzy portfolio optimization. Studies in fuzziness and soft computing (Vol. 316, pp. 33-59). Berlin, Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-54652-5_2 Gupta, P., Mittal, G., & Mehlawat, M. K. (2013). Expected value multiobjective portfolio rebalancing model with fuzzy parameters. Insurance: Mathematics and Economics, 52(2), 190-203. https://doi.org/10.1016/j.insmatheco.2012.12.002 Gupta, P., Mittal, G., & Mehlawat, M. K. (2014). A multicriteria optimization model of portfolio rebalancing with transaction costs in fuzzy environment. Memetic Computing, 6(1), 61-74. https://doi.org/10.1007/s12293-012-0102-2 Journal of Business Economics and Management, 2019, 20(2): 225–243 241 Huang, X. (2006). Fuzzy chance-constrained portfolio selection. Applied Mathematics and Computation, 177(2), 500-507. https://doi.org/10.1016/j.amc.2005.11.027 Huang, X. (2008). Mean-semivariance models for fuzzy portfolio selection. Journal of Computational and Applied Mathematics, 217(1), 1-8. https://doi.org/10.1016/j.cam.2007.06.009 Huang, X. (2009). A review of credibilistic portfolio selection. Fuzzy Optimization and Decision Making, 8(3), 263-281. https://doi.org/10.1007/s10700-009-9064-3 Huang, X. (2010). Portfolio analysis, from probabilistic to credibilistic and uncertain approaches (Vol.250). Berlin, Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-11214-0 Inuiguchi, M., Ichihashi, H., & Tanaka, H. (1990). Fuzzy programming: a survey of recent developments. In R. Slowinski & J. Teghem (Eds.), Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty (Vol. 6, pp. 45-68). Dordrecht: Springer-Verlag. https://doi.org/10.1007/978-94-009-2111-5_4 Jalota, H., Thakur, M., & Mittal, G. (2017a). A credibilistic decision support system for portfolio optimization. Applied Soft Computing, 59, 512-528. https://doi.org/10.1016/j.asoc.2017.05.054 Jalota, H., Thakur, M., & Mittal, G. (2017b). Modelling and constructing membership function for uncertain portfolio parameters: A credibilistic framework. Expert Systems with Applications, 71, 40-56. https://doi.org/10.1016/j.eswa.2016.11.014 León, T., & Vercher, E. (2004). Solving a class of fuzzy linear programs by using semi-infinite programming techniques. Fuzzy Sets and Systems, 146(2), 235-252. https://doi.org/10.1016/j.fss.2003.09.010 Li, B., Zhu, Y., Sun, Y., Aw, G., & Teo, K. L. (2018). Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint. Applied Mathematical Modelling, 56, 539-550. https://doi.org/10.1016/j.apm.2017.12.016 Liagkouras, K., & Metaxiotis, K. (2015). Efficient portfolio construction with the use of multiobjective evolutionary algorithms: best practices and performance metrics. International Journal of Information Technology & Decision Making, 14(3), 535-564. https://doi.org/10.1142/S0219622015300013 Liu, B. (2004). Uncertainty theory: an introduction to its axiomatic foundations (Vol. 154). Berlin, Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-540-39987-2 Liu, B. (2007). Uncertainty theory an introduction to its axiomatic foundations (2nd ed.). Berlin, Heidelberg: Springer-Verlag. Liu, B., & Liu, Y. K. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 10(4), 445-450. https://doi.org/10.1109/TFUZZ.2002.800692 Lwin, K., Qu, R., & Kendall, G. (2014). A learning-guided multi-objective evolutionary algorithm for constrained portfolio optimization. Applied Soft Computing, 24, 757-772. https://doi.org/10.1016/j.asoc.2014.08.026 Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x Markowitz, H. (1959). Portfolio selection: efficient diversification of investments. New York: John Wiley & Sons, Inc. Markowitz, H., Todd, P., Xu, G., & Yamane, Y. (1993). Computation of mean-semivariance efficient sets by the Critical Line Algorithm. Annals of Operations Research, 45(1), 307-317. https://doi.org/10.1007/BF02282055 Masa’deh, R., Tayeh, M., Al-Jarrah, I. M., & Tarhini, A. (2015). Accounting vs. market-based measures of firm performance related to information technology investments. International Review of Social Sciences and Humanities, 9(1), 129-145. Metaxiotis, K., & Liagkouras, K. (2012). Multiobjective Evolutionary Algorithms for portfolio management: a comprehensive literature review. Expert Systems with Applications, 39(14), 11685-11698. https://doi.org/10.1016/j.eswa.2012.04.053 242 F. García et al. A credibilistic mean-semivariance-PER portfolio selection model for Latin America Morgan, J. P. (1996). Riskmetrics technical document (4th ed.). New York: Morgan Guaranty Trust Company of New York. Narkunienė, J., & Ulbinaitė, A. (2018). Comparative analysis of company performance evaluation methods. Entrepreneurship and Sustainability Issues, 6(1), 125-138. https://doi.org/10.9770/jesi.2018.6.1(10) Omidi, F., Abbasi, B., & Nazemi, A. (2017). An efficient dynamic model for solving a portfolio selection with uncertain chance constraint models. Journal of Computational and Applied Mathematics, 319, 43-55. https://doi.org/10.1016/j.cam.2016.12.020 Pouya, A. R., Solimanpur, M., & Rezaee, M. J. (2016). Solving multi-objective portfolio optimization problem using invasive weed optimization. Swarm and Evolutionary Computation, 28, 42-57. https://doi.org/10.1016/j.swevo.2016.01.001 Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21-41. https://doi.org/10.21314/JOR.2000.038 Saborido, R., Ruiz, A. B., Bermúdez, J. D., Vercher, E., & Luque, M. (2016). Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection. Applied Soft Computing, 39, 48-63. https://doi.org/10.1016/j.asoc.2015.11.005 Schmitt, N., & Westerhoff, F. (2017). On the bimodality of the distribution of the S&P 500’s distortion: Empirical evidence and theoretical explanations. Journal of Economic Dynamics and Control, 80, 34-53. https://doi.org/10.1016/j.jedc.2017.05.002 Shen, K.-Y., Yan, M.-R., & Tzeng, G.-H. (2014). Combining VIKOR-DANP model for glamor stock selection and stock performance improvement. Knowledge-Based Systems, 58, 86-97. https://doi.org/10.1016/j.knosys.2013.07.023 Sobreiro, V. A., Cruz Cacique da Costa, T. R., Farias Nazário, R. T., Lima e Silva, J., Moreira, E. A., Lima Filho, M. C., & Arismendi Zambrano, J. C. (2016). The profitability of moving average trading rules in BRICS and emerging stock markets. The North American Journal of Economics and Finance, 38, 86-101. https://doi.org/10.1016/j.najef.2016.08.003 Speranza, M. (1993). Linear programming models for portfolio optimization. Finance, 14, 107-123. Torre, A., & Schmukler, S. (2007). Emerging Capital markets and globalization. The Latin American experience. Washington: Stanford University Press & World Bank. Vercher, E., & Bermúdez, J. D. (2012). Fuzzy Portfolio selection models: a numerical study. In M.Doumpos, C. Zopounidis, & P. M. Pardalos (Eds.), Springer optimization and its applications (Vol.70, pp. 253-280). Boston, MA: Springer. https://doi.org/10.1007/978-1-4614-3773-4_10 Vercher, E., & Bermúdez, J. D. (2013). A Possibilistic mean-downside risk-skewness model for efficient portfolio selection. IEEE Transactions on Fuzzy Systems, 21(3), 585-595. https://doi.org/10.1109/TFUZZ.2012.2227487 Vercher, E., & Bermúdez, J. D. (2015). Portfolio optimization using a credibility mean-absolute semideviation model. Expert Systems with Applications, 42(20), 7121-7131. https://doi.org/10.1016/j.eswa.2015.05.020 Vercher, E., Bermúdez, J. D., & Segura, J. V. (2007). Fuzzy portfolio optimization under downside risk measures. Fuzzy Sets and Systems, 158(7), 769-782. https://doi.org/10.1016/j.fss.2006.10.026 Xia, Y., Liu, B., Wang, S., & Lai, K. K. (2000). A model for portfolio selection with order of expected returns. Computers & Operations Research, 27(5), 409-422. https://doi.org/10.1016/S0305-0548(99)00059-3 Yue, W., & Wang, Y. (2017). A new fuzzy multi-objective higher order moment portfolio selection model for diversified portfolios. Physica A: Statistical Mechanics and its Applications, 465, 124-140. https://doi.org/10.1016/j.physa.2016.08.009 Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X Journal of Business Economics and Management, 2019, 20(2): 225–243 243 Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3-28. https://doi.org/10.1016/S0165-0114(99)80004-9 Zemguliene, J., & Valukonis, M. (2018). Structured literature review on business process performance analysis and evaluation. Entrepreneurship and Sustainability Issues, 6(1), 226-252. https://doi.org/10.9770/jesi.2018.6.1(15) Zhu, M., Atri, S., & Yegen, E. (2016). Are candlestick trading strategies effective in certain stocks with distinct features? Pacific-Basin Finance Journal, 37, 116-127. https://doi.org/10.1016/j.pacfin.2015.10.007