An integrated multi-objective Markowitz–DEA cross-efficiency model with fuzzy returns for portfolio selection problem
Graphical abstract
Introduction
Markowitz [1] introduced a quantitative framework for portfolio selection and took a major step in portfolio optimization problem. His model is well known as mean–variance model and considers maximizing return of the portfolio and minimizing risk of the portfolio at the same time. Following to Markowitz, many researchers worked based on his model and extended it. They added a variety of assumption, constraints or objectives such as cardinality constraint, transaction cost, skewness and kurtosis to his model to make it more realistic. On the other hand, some researchers employed data envelopment analysis (DEA) which was introduced by Charnes et al. [2] in portfolio selection. Although there are many studies for developing Markowitz model, the efficiency of the portfolio is not incorporated in the mean–variance model. In this paper, the efficiency of the portfolio is added to the Markowitz basic model. Some studies which extended the Markowitz model are listed as follows.
Cardinality constraint which is introduced by Chang et al. [3] is the most popular constraint which is added to the mean–variance model. This constraint limits the number of assets in the portfolio. Some studies which applied cardinality constraint and added some more criteria to mean–variance model are as follows.
Schaerf [4] considered the quantity of individual shares constraints in addition to cardinality constraint. Crama and Schyns [5] considered cardinality, turnover (purchase and sale) and trading constraints. Chang et al. [6] presented portfolio selection models with different risk measures semi-variance, mean absolute deviation and variance with skewness. Soleimani et al. [7] applied Markowitz model with cardinality and minimum transaction lots constraints. They also proposed market (sector) capitalization as a new constraint in their model. Anagnostopoulos and Mamanis [8] considered minimization the number of securities in the portfolio as an objective in addition to return and risk. They added the quantity and class constraints into the model. Ehrgott et al. [9] considered five sub-objective related to return and risk (short term expected return, long term expected return, annual dividend, standard and poor star ranking and volatility).
Several studies did not consider cardinality constraint. Yoshimoto [10] added transaction costs to Markowitz model. Yu and Lee [11] considered skewness, kurtosis and short selling in addition to return and risk in portfolio selection and proposed five portfolio rebalancing models by using some or all of these criteria. Xidonas et al. [12] proposed an integrated multiple-criteria approach for portfolio selection. First, they applied two multiple-criteria methods to gain the initial appraisal of the stocks. Then, they selected a portfolio that includes the stocks which have optimal characteristics in the first stage.
The above mentioned studies did not consider uncertainty in the mean–variance model. Uncertainty is an important issue of employing quantitative models for portfolio selection. The mean–variance model is very sensitive to perturbation in the input data. If we could estimate future expected returns of each asset and the covariance matrix accurately, the Markowitz model would produce optimum portfolios [13]. We live in an uncertain world and there are many uncertain factors which influence on asset returns. So, obtaining accurate prediction of input data needed for models is difficult. Fuzzy theory is one of the most popular approaches to dealing with uncertainty. In portfolio selection literature, many researchers employed fuzzy theory in order to consider uncertainty in their models. Gupta et al. [14] proposed a fuzzy multi-objective portfolio selection. Their model considered maximization of the portfolio AHP weighted score of suitability as a new objective in addition to return, risk and liquidity. They used this additional objective in order to consider suitability and optimality at the same time in portfolio selection. Jana et al. [15] added the entropy maximization as an objective to achieve a well-diversified portfolio. They used trapezoidal fuzzy numbers in order to consider uncertainty. Barak et al. [16] proposed a mean–variance–skewness model with cardinality constraint and also considered liquidity in their model. They considered the return of an asset as trapezoidal fuzzy number. Li and Xu [17] proposed a multi-objective portfolio selection with considering return, risk and liquidity. They considered return as fuzzy random variable. Huang [18] considered return as random fuzzy variable. For studying more works on this issue(see [19], [20], [21], [22], [23]).
The above studies did not consider the efficiency issue in portfolio selection. One of the most important model for considering the efficiency in portfolio selection is the DEA technique. Some studies which applied DEA for portfolio selection are as follows.
Branda [24] suggested new efficiency tests based on traditional DEA models and considered diversification in the portfolio. He applied deviation measures as the inputs and return measures as the outputs in the proposed model. Lamb and Tee [25] used DEA for investment funds and considered risk and return measures justifiably. They also considered diversification and shown how to handle it. Joro and Na [26] evaluated portfolio performance in a mean–variance–skewness framework by using DEA. Lim et al. [27] proposed a new model for portfolio selection which used DEA cross-efficiency evaluation under a mean–variance (MV) framework and called it DEA MV cross-efficiency model. But none of these researchers combined Markowitz model with the DEA in their investigations. Also they did not consider uncertainty in their studies. Edirisinghe and Zhang [28] developed a generalized DEA model to analyze a firm's financial statements and determined a relative financial strength indicator (RFSI). First, they applied a selection process based on RFSI to select desirable stocks as potential candidates in portfolio and then assigned weights to choose candidate stocks by using the mean–variance model. But they did not consider constraints such as cardinality or bounded constraints and they did not consider uncertainty in their model. Also they did not consider DEA and Markowitz model, simultaneously and used Markowitz model only to assigned weights to the chosen assets.
By considering different constraints and assumptions in Markowitz model, it becomes more complicated and solving it by using exact approaches is impossible. Employing heuristic and meta-heuristic methods is appropriate way to solve such problems. There are many researches in which heuristic and meta-heuristic approaches such as genetic algorithm, PSO, tabu search, simulated annealing and, etc. have been used to solve extended portfolio selection models (e.g. [4], [17], [29], [30], [31], [32], [33], [34], [35], [36]).
In addition to return and risk, this paper considers efficiency for the portfolio selection problem. The proposed multi objectives model combines Markowitz mean–variance model and DEA MV cross-efficiency model which is introduce by Lim et al. [27]. The DEA MV cross-efficiency model is selected for considering efficiency due to its significant advantages. Standard DEA model has a well-known deficiency. It is so flexible in choosing weights of inputs and outputs, thus a DMU can attain high efficiency by choosing extremely high weights on some factors and extremely low weights on other factors. DEA cross-efficiency evaluation prevents selection such DMUs. Also, Lim et al. [27] mentioned two advantages for their proposed model. They indicated that DEA MV cross-efficiency model eliminate two problems arising in the simple use of DEA cross-efficiency in portfolio selection: One is the lack of portfolio diversification and another is the “ganging-together” phenomenon (for more detail see [27]). The DEA MV cross-efficiency model is described in Section 2.
In this paper, the asset returns are considered as the trapezoidal fuzzy numbers. Also, since the model is NP-hard, so a meta-heuristic algorithm named NSGA-II is applied to solve it. In order to illustrate the proposed model, the model is applied for a case study including 52 firms listed in Iran stock exchange market.
The rest of this paper is as follows: the fuzzy multi objective proposed model based on Markowitz model and DEA cross-efficiency model is presented in Section 2. In Section 3, NSGA-II is described. In Section 4, the proposed model is applied for actual data and the results are shown in Section 5. The conclusion of the paper is summarized in Section 6.
Section snippets
Methodology
The methodology of this paper is based on Markowitz and DEA cross-efficiency models. The models are described as follows.
Solution approach
Non-dominated sorting genetic algorithm II (NSGA-II) which is introduced by Deb et al. [40] is one of the most practical algorithms in multi objective optimization problems. This paper employs NSGA-II to solve the suggested model. Briefly, the steps of NSGA-II are as follows:
- 1.
Representation structure and create initial population: we consider a string with N genes as a chromosome as x = (x1, x2, …, xN), where N is the number of available assets. Each gene is representative of one asset. The
Case study: stock exchange market in Iran
In this section, the stock exchange market in Iran is chosen as data source. In the end of each season, department of Information of Tehran stock exchange corporation market declare the name of 50 best firms which are chosen by certain criteria. By considering lists from 2009 to 2013, firms with the most attained are selected. Also, the firms of the latest list are selected. We exclude financial institutions since their accounting standards are far different from other industries. This method
Results
We selected portfolio three times: (I) by applying only Markowitz model without consideration efficiency (model (9) except objectives (54), (55)), (II) by applying DEA MV cross-efficiency model (model (9) except objectives (52), (53)) and (III) using our proposed model (model (9)). We then compared their results. In all three models, the same constraints (56), (57), (58), (59) are considered. Also, the assets rate of return are considered as fuzzy numbers. The parameters of models are set the
Conclusion
This paper developed a fuzzy multi objective model based on Markowitz mean–variance and DEA MV cross-efficiency models. The proposed model considered the efficiency, expected return and risk of the portfolio at same time. The DEA MV cross-efficiency was selected for measuring efficiency due to some advantages. Also, the assets rate of return was considered as trapezoidal fuzzy numbers to deal with uncertainty in the proposed model. To solve the proposed model, the second version of
Acknowledgements
The authors would like to thank the anonymous referees for their constructive comments and suggestions on the earlier version of this paper.
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