Int. J. Industrial and Systems Engineering, Vol. 18, No. 3, 2014

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A hybrid fuzzy decision making method for a portfolio selection: a case study of Tehran Stock Exchange Abolfazl Kazemi* Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, P.O. Box 34185-1416, Qazvin, Iran Fax: +98(281)-3665275 E-mail: [email protected] *Corresponding author

Keyvan Sarrafha Young Researches and Elite Club, Qazvin Branch, Islamic Azad University, P.O. Box 34185-1416, Qazvin, Iran Fax: +98(281)-3665275 E-mail:[email protected]

Mahdi Beedel Faculty of Administrative Sciences and Economics, Department of Accounting, University of Isfahan, P.O. Box 81746-73441, Isfahan, Iran Fax: +98(311)-7932128 E-mail: [email protected] Abstract: The problem of choosing an optimal portfolio is the most significant issue for any investor in the stock exchange market. The present study seeks to prioritise seven companies through pioneer industries on Tehran Stock Exchange market by opting for suitable financial guidelines as well effective criteria in uncertain environments. To do so, a hybrid fuzzy decision making procedure was applied. Having determined the criteria, Delphi method approach was used as a process in fuzzy pairwise comparison of criteria and alternatives in fuzzy AHP. Alternatives then weighed up in fuzzy TOPSIS method to achieve a coordinated and balanced view. Finally, to evaluate the results against real conditions of Tehran exchange market, they were compared with the actual annually returns of these same companies on the market. Keywords: portfolio selection; Delphi method; decision making; fuzzy AHP; fuzzy TOPSIS; financial ratios. Reference to this paper should be made as follows: Kazemi, A., Sarrafha, K. and Beedel, M. (2014) ‘A hybrid fuzzy decision making method for a portfolio selection: a case study of Tehran Stock Exchange’, Int. J. Industrial and Systems Engineering, Vol. 18, No. 3, pp.335–354.

Copyright © 2014 Inderscience Enterprises Ltd.

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A. Kazemi et al. Biographical notes: Abolfazl Kazemi is an Assistant Professor in the Department of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran. His research interests include intelligent information systems, fuzzy set theory and supply chain management. Keyvan Sarrafha is a Master student at the Department of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran. His interests span from supply chain management, fuzzy logic, mathematical modelling and meta-heuristic algorithms. Mahdi Beedel is a Master student at Faculty of Administrative Sciences and Economics, Department of Accounting, Isfahan University, Isfahan. His research area includes optimal capital structure and portfolio selection.

1

Introduction

An active capital market is one of the marks of development on an international level. Investor’s tendency to play a more active role leads to prosperity in the market which intern contributes to companies active in different industries. To gain more profit on the capital market, investors have to choose the stocks of the companies whose stock price is on the rise due to their satisfactory performance record. However, as investors cannot be sure of the future, they have to opt for a diversified portfolio to lower risk of incurring losses (Secme et al., 2009). Portfolio selection is well known as a leading problem in finance; since the future returns of assets are not known at the time of the investment decision, the problem is one of decision-making under risk. A portfolio selection model is an ex-ante decision tool: decisions taken today can only be evaluated at a future time, once the uncertainty regarding the assets’ returns is revealed (Roman and Mitra, 2009). The main purpose of presenting portfolio selecting methods is helping the investors to choose an optimal portfolio. To do this, analysing the present and the past performance of the companies by using some important criteria can be useful. Modern portfolio theory (MPT) is based upon the classical Markowitz model (1952) which uses variance as a risk measure. He introduced it in a computational model, by measuring the risk of a portfolio via the covariance matrix associated with individual asset returns; this leads to a quadratic programming formulation. Markowitz idea on the mean-variance approach then being developed by Sharp (1966), Lintner (1965) and Mossin (1966). Since then, the model is well accepted by investors and fund managers that aimed to construct an efficient portfolio with the highest diversification benefit. Portfolio diversification is influenced by many factors by that govern the portfolio selection criteria such as the firm sizes, financial ratios, stock markets and investor’s judgment. In selecting stocks to have an optimal portfolio, the investors face various options for investing. Identifying the various criteria which affect stock prices, comparing them and identifying their relative importance throw pairwise comparison to have a better stock selection is a complex and difficult process. Thus multi criteria decision making methods can be used to select the optimal portfolio. In multiple criteria (or attribute) decision making problems, a decision maker (DM) or (DMs) often needs to select or rank alternatives associated with some usually conflicting attributes or objectives. The portfolio manager, as the DM, has a large

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set criterion for selecting stocks. These criteria depend on various interdependent primary and secondary factors. The problem is to compare the various criteria and to determine their relative importance through pairwise comparison between each pair of them. According to this rule, one can make a decision to compare each list of stocks with this criteria list and determine the amount of investment to be allocated to each stock. The DM will then be able to compare each list of stocks with this criteria list and determine the portfolio. To make the comparison, he/she will need a scale by which to compare the factors pairwise. The scale (1–9) is used in analytic hierarchy process (AHP) and reflects the DM’s belief as to which of two factors or criteria is more important (and to what degree), relative to some quality that both of them share. Saaty et al. (1980) proposed AHP to deal with the stock portfolio decision problem by evaluating the performance. Tanaka and Guo (1999) formulated portfolio selection models by quadratic programming, based on two kinds of possibility distributions. Xia et al. (2000) proposed a new model for portfolio selection using genetic algorithms. Inuiguchi and Tanino (2000) proposed a new possibility programming approach for portfolio optimisation, considering how a model yields a distribution investment solution. In real world problems, decisions are made in situations where the objectives, limitations and results are not completely known. Likewise, striking on an optimal portfolio is based on information derived from economic environments (such as company’s annual reports, inflation rate, GNP’s growth rate, Government economic policies and so on). And this information is always associated with some degree of uncertainty. In the past it was hypothesised that stock’s return can be clearly predicted by companies past performances. However, complexity of capital market does not allow this prediction. These uncertainties have made frustrated researches resort to applying fuzzy set theory. This theory was first introduced By Zadeh (1965) and has been expanded largely ever since. Classical and mathematical approaches fall short of explaining these uncertainties as related to optimisation of portfolio. Therefore using fuzzy theory helps financial DMs to overcome uncertainties. In this regard, Parra et al. (2001) formulated a fuzzy goal programming with fuzzy goal and fuzzy constraints, taking into account three criteria: return, risk and liquidity. Considering the uncertainty of investment environment, Tiryaki and Ahlatcioglu (2005) transferred expert’s linguistic value into triangle fuzzy number and used a new fuzzy ranking and weighting algorithm to obtain the investment ratio of each stock. In fact, the stock portfolio selection problem can be described as multiple criteria decision making (MCDM) problem. In recent works also, Secme et al. (2009) proposed a fuzzy multi criteria decision model to evaluate the performances of some banks. In their implementation, a fuzzy analytic hierarchy process (FAHP) and technique for order performance by similarity to ideal solution (TOPSIS) are integrated to evaluate the five largest commercial banks of Turkish banking sector in terms of several financial and non-financial indicators. Tiryaki and Ahlatcioglu (2009) used the two constrained fuzzy AHP methods, developed by Enea and Piazza method, to the problem of portfolio selection in Istanbul stock exchange. They addressed some fallacies in the first model of Enea and Piazza and corrected it. They also showed that the second model of Enea and Piazza is superior to their first model in terms of uncertainty level of the solutions obtained. Arunkumar et al. (2011) presented a new method to deal with the performance evaluation and selection of supplier for a manufacturing industry using fuzzy AHP with alpha cut analysis. The results of this paper provide valuable suggestions to managers on

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the improvement needed with respect to each criterion when selecting suppliers. GhazanfarAhari et al. (2011) also used the two fuzzy AHP methods. They have used Enea and Piazza method and van Larhoven and Pedrycz’s approach for a real world case study of pharmaceutical industry in Tehran stock exchange where all input parameters are subject to uncertainty. And so, the result of their implementation determines the asset allocation for common shares of five different companies. Liu et al. (2012) by considering various criteria, including return, transaction cost, risk and skewness of portfolio in fuzzy environment simulated the real transactions in financial market to provide investors with additional choices. Shamsuzzaman et al. (2013) described a fuzzy-logic-based MCDM approach to rank coals for industrial use. The calorific value and chemical compositions desired values of each criterion are expressed by an appropriate fuzzy number, whereas all possible values of a criterion of a given coal are captured by a numerical range. Gupta et al. (2013) presented a hybrid optimisation model of portfolio selection involving financial and ethical considerations. They proposed a comprehensive three-MCDM framework for portfolio selection based on financial and ethical criteria simultaneously. The AHP method is used to obtain the ethical performance score of each asset based on investor-preferences, and a fuzzy MCDM technique is also used to obtain the financial quality score of each asset based upon investor-ratings on the financial criteria. In the real world situations, opinions are wary. DMs consider different criteria, so the best idea should be found. Delphi logic can be applied to achieve a balanced solidarity of experts and DM’s opinion which leads to taking the right course of action. Further to confirm how reliable the results are, they should be compared against a significant index in real world. In the present paper we have tried to find an efficient way of allocating a limited amount of money to Tehran stock exchange companies in order to have an optimal portfolio. To this end after prioritising major financial ratios including: profitability ratios, activity ratios, risk ratios and market ratios in fuzzy environment, classify these companies. Now having reviewed the related literature, uncertainty can be fully resorted to all stages from criteria’s weights to comparison of approaches in order to form an optimal portfolio. The main question in this paper is then how fuzzy logic minimises the uncertainties in all stages of decision making and prioritising the company’s stocks in the portfolio. Effort was made to obtain the financial expert’s views by using a translated version of fuzzy method and FAHP to identify suitable weights of criteria and alternatives. Fuzzy TOPSIS method is also applied to obtain even more reliability in studying the portfolio. In the end, to evaluate the results against real conditions of Tehran exchange market, they were compared with the annual returns of these same companies on the market. The rest of the paper is organised as follows: Section 2, explains the methodology. Section 3 is devoted to solving methods. In Section 4, the results are presented. Finally, conclusion remarks are given in the last section to summarise the contribution of the paper in the last section.

2

Problem definition

As previously mentioned, the portfolio manager as the DM, face a large set of criteria for selecting stocks. In addition, DMs are often misled by the large amount of information that is associated with uncertainty in financial environment. However, financial

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statements are reliable sources for evaluating the performance of companies and choosing the stocks of companies with better performances (Brigham et al., 1996). From the perspective of an investor predicting future is why financial statements are analysed. Financial statement analysis is assessed by financial ratios. These ratios can not only help an investor to understand strengths and weaknesses of different companies but they can also be used for comparing economic performances of companies. In different studies different ratios are considered as effective criteria in order to select optimal stocks (Edirisinghe and Zhang, 2008). In the present study, eight ratios from four main groups of ratios are considered as decision making criteria. The four main groups of financial ratios are: profitability criteria, activity criteria, risk criteria, market criteria (Brigham et al., 1996).

2.1 Category profitability criteria 2.1.1 Operating profit margin (OPM) The profit margin reveals the profitability of each dollar of sale (Brickley, 1983).

2.1.2 Return on equity Return on equity (ROE) measures a corporation’s profitability by revealing how much profit a company generates with the money shareholders have invested (Edirisinghe and Zhang, 2008).

2.2 Category activity ratio 2.2.1 Inventory turnover ratio (ITR) A ratio showing how many times a company’s inventory is sold and replaced over a period (Reilly and Brown, 2003).

2.2.2 Receivables collection period (RCP) This reveals how many days it takes to collect all accounts receivable (Brigham et al., 1996).

2.3 Category risk criteria 2.3.1 Standard deviation (SD) A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation (Edirisinghe and Zhang, 2008).

2.3.2 Interest coverage ratio A ratio used to determine how easily a company can pay interest on outstanding debt. The interest coverage ratio (ICR) is calculated by dividing a company’s earnings before (Brigham et al., 1996).

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2.4 category market criteria ⎛P⎞ 2.4.1 Market value to book value ratio ⎜ ⎟ ⎝B⎠ A ratio used to compare a stock’s market value to its book value. A lower P/B ratio could mean that the stock is undervalued (Brickley, 1983).

⎛P⎞ 2.4.2 Price to earnings ratio ⎜ ⎟ ⎝E⎠ A valuation ratio of a company’s current share price compared to its per-share earnings. In general, a high P/E suggests that investors are expecting higher earnings growth in the future compared to companies with a lower P/E (Brigham et al., 1996).

3

Methods

In this section, we use two popular methods namely fuzzy AHP and fuzzy TOPSIS to select the proposed portfolio. AHP is a systematic method for comparing a list of metrics or alternatives. TOPSIS is based on the concept that the chosen alternative should have the shortest geometric distance from the positive ideal alternative and the longest geometric distance from the negative ideal alternative. Here, we considered Delphi method as a structured communication technique for all stages of paired comparisons in fuzzy AHP and rating companies in fuzzy TOPSIS.

3.1 Delphi method Delphi is a tool for qualitative researches. This method is used in macro subjects, especially qualitative matters (Sarokhani, 2004). The Delphi is a structured process for predicting and assisting to make decisions during survey rounds. Gathering information and finally grouping agreement are its other usages. While most surveys attempt to respond to the question: ‘What is it?’ the Delphi tries to answer to the question: ‘What could or should it be?’ (Powell, 2003). The Delphi method is the most important technique to detect and study subjects, which are mixtures of academic bases and social values (Sarokhani, 2004). Therefore, this is an appropriate method to recognise judgment issues like risk assessment. Delphi could be used to form a group communication, which facilitates thinking and being involved as a whole to complex issues (Linstone and Turoff, 2002). Although this group judgment is form different mental view points, it is more trustable from individual and personal opinions and its results are more objective and precise (Masini, 1993). Participants in the new round could compare their personal opinions with others in prior round and may adjust or change them in next round. Hence, the final results are group judgments and there are no opinions belonging to a single person (Sarokhani, 2004). That is why we selected this qualitative method for the identification important criteria and alternatives in both methods. The population was TSE experts that they are ranked according to several measures by TSE. The sample is formed with 20 experts (Delphi panel). Delphi questionnaires are

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sent by email messages to where to sure consistency in the responses found. Here, is the conclusion reached in the second round. In each round, the goals and processes of the research are explained completely to the participants. Figure 1 presents Delphi method flowchart. Figure 1

Proposed hybrid method flowchart (see online version for colours)

3.2 Extent analysis method on fuzzy AHP In the following, first the outlines of the extent analysis method (Chang, 1996) on fuzzy AHP are given and then the method is applied to a portfolio selection problem. Let X = {x1, x2, …, xn} be an object set, and U = {u1, u2, …, un} be a goal set. According to the method of Chang’s extent analysis, each object is taken and extent analysis for each goal is performed respectively. Therefore, m extent analysis values for each object can be obtained, with the following signs: M 1g1 , M g22 , ..., M gmi

i = 1, 2, ..., n

(1)

where all the M gji (j = 1, 2, …, n) are triangular fuzzy numbers. The value of fuzzy synthetic extent with respect to the ith object is defined as: m

si =

∑ j =1

M gji

⎡ n ⊗⎢ ⎢⎣ i =1

m

∑∑ j =1

M gji

⎤ ⎥ ⎥⎦

−1

(2)

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The degree of possibility of M1 ≥ M2 is defined as: V ( Si ≥ Sk ) = Sup ⎡⎣ min ( μsi ( x), μsk ( y ) ) ⎤⎦

When a pair (x, y) exists such that x ≥ y and μM1 ( x), V(M1 ≥ M2) = 1. Since M1 and M2 are convex fuzzy numbers we have that:

(3) μM 2 ( y ), then we have

V ( Si ≥ Sk ) = 1 if mi ≥ mk and if lk ≥ ui

(4)

V ( Si ≥ Sk ) = hgt ( Si ∩ Sk ) = μsi (d )

(5)

where d is the ordinate of the highest intersection point D between μM1 and μM 2 that is shown in Figure 2. When Si = (li, mi, ui) and Sk = (lk, mk, uk), the ordinate of D is given by equation (6):

V ( Si ≥ Sk ) = hgt ( Si ∩ Sk ) = Figure 2

lk − ui

( m i −ui ) − ( m k −lk )

(6)

Intersection point between μSi and μSk (see online version for colours)

To compare M1 and M2, we need both the values of V(Si ≥ Sk) and V(Sk ≥ Si). The degree possibility for a convex fuzzy number to be greater than k convex fuzzy number Si = (i = 1, 2, …, k) can be defined by: V ( S ≥ S1 , S2 ,..., Sk ) = V ⎡⎣( S ≥ S1 ) , ( S ≥ S2 ) , ..., ( S ≥ Sk ) ⎤⎦ = min V ( S ≥ Si ) i = 1, 2, ..., k

(7)

Assume that: d ' ( Ai ) = min V ( Si ≥ Sk ) , k = 1, 2, ..., n, i ≠ k

Then the weight vector is given by:

(8)

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W ' = ( d ' ( A1 ), d ' ( A2 ), ..., d ' ( An ) )

T

(9)

where Ai = (i = 1, 2, …, n) have n elements. Via normalisation, the normalised weight vectors are: W = ( d ( A1 ) , d ( A2 ) , ..., d ( An ) )

T

(10)

where W is a non-fuzzy number. So, overall stages fuzzy AHP method by Chang’s extent analysis method, as follow: 1

Create a hierarchical structure for problem.

2

determine pairwise comparisons matrix and apply judgments by using linguistic variable given in Table 1. Membership function for linguistic variable is shown in Figure 3.

3

Calculate the relative weights of criteria and alternative.

4

Calculate the final weights alternative that obtained by combination the relative weights.

Table 1

Triangular fuzzy number values

Linguistic variable Just equal

Fuzzy pairwise comparison value (1, 1, 1)

weak

(2/3, 1, 3/2)

Fairly strong

(3/2, 2, 5/2)

Very strong

(5/2, 3, 7/2)

Absolute

(7/2, 4, 9/2)

Figure 3

Membership function for linguistic variable (see online version for colours)

3.3 Fuzzy TOPSIS TOPSIS is one of the well-known MCDM methods of adaptive subtype that too, is a subgroup of the compensation. This way every option considers as a point in space. The basic principle of the TOPSIS method is that the chosen alternative should have the

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shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). It is an effective method to determine the total ranking order of decision alternatives. In fuzzy environment, the matrix elements of the decision or weights of indicators or both are expressed as fuzzy number. Several fuzzy TOPSIS methods are presented. All of these methods with little change in the method of Chen et al. (1992) have obtained (Wang and Chang, 2007). The steps are as follows: Step 1

Determine the weighting of evaluation criteria.

In this paper the importance weights of various criteria is obtained and using Chang method in fuzzy AHP in before section. Step 2

Construct the fuzzy decision matrix and choose the appropriate linguistic variables for the alternative with respect to criteria.

In this paper, rating alternative considered as linguistic variables as Table 2 (Chen et al., 2006). These linguistic variables shown as triangular fuzzy number shown Figure 4. Table 2

Linguistic scales for the importance of each criterion

Linguistic variable

Corresponding triangular fuzzy number

Very very low (VVL)

(0, 0, 1)

Very low (VL)

(0, 1, 3)

Low (L)

(1, 3, 5)

Medium (M)

(3, 5, 7)

High (H)

(5, 7, 9)

Very high (VH)

(7, 9, 10)

Very very high (VVH)

(9, 10, 10)

Figure 4

Membership function for linguistic variables (see online version for colours)

A hybrid fuzzy decision making method for a portfolio selection C1

Cj

Cn

A1 ⎡ r11 ⎢ ⎢ D = Ai ⎢ ri1 ⎢ ⎢ Am ⎢⎣ rm1

r1 j

r1n ⎤ ⎥ ⎥ rin ⎥ i = 1, 2, ..., m; j = 1, 2, ..., n ⎥ ⎥ rmn ⎥⎦

rij rmj

345

(11)

where rij is the rating of alternative Ai with respect to criterion Cj evaluated by experts by using Delphi method and rij = ( aij , bij , cij ) . Step 3

Normalise the fuzzy decision matrix.

The normalised fuzzy decision matrix denoted by R is shown as formula:

R = [ rij ]m×n ; i = 1, 2, ..., m;

j = 1, 2, ..., n

(12)

Then the normalisation process can be performed by following formula: ⎛ aij bij cij ⎞ rij = ⎜ + , + , + ⎟ ; C +j = max Cij ⎜ Cj Cj Cj ⎟ i ⎝ ⎠

(13)

⎛ C ij C ij C ij ⎞ rij = ⎜ , , ⎟ ; C ij = min Cij i ⎝ aij bij cij ⎠

(14)

The normalised rij are still triangular fuzzy numbers. For trapezoidal fuzzy numbers, the normalisation process can be conducted in the same way. The weighted fuzzy normalised decision matrix is shown as following matrix V : C = [ vij ]m×n ; i = 1, 2, ..., m;

j = 1, 2, ..., n

vij = rij ⊗ W j Step 4

(15) (16)

Determine the fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS).

According to the weighed normalised fuzzy decision matrix, we know that the elements vij are normalised positive triangular fuzzy numbers and their ranges belong to the closed interval [0, 1]. Then, we can define the FPIS A+ and FNIS A– by using of Lee and Li ranking method as following formula: M ( vij ) =

− aij2 − aij .bij + bij .cij + cij2 3 ( cij − aij )

(17)

After obtaining M (vij ) for each column j A+ = ( v1+ , v2+ , ..., vn+ )

(18)

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(19)

where v +j = max( M (vij )) and v −j = min( M (vij )); j = 1, 2, ..., n.

Step 5

Calculate the distance of each alternative from FPIS and FNIS

The distance ( di+ and di− ) of each alternative A+ from and A– can be currently calculated by the area compensation method. n

di+ =

∑ d ( v , v ); i = 1, 2, ..., m ; j = 1, 2, ..., n ij

+ j

(20)

j =1 n

di− =

∑ d ( v , v ); i = 1, 2, ..., m ; j = 1, 2, ..., n ij

− j

(21)

j =1

Step 6

Obtain the closeness coefficient and rank the order of alternatives.

The CCi is defined to determine the ranking order of all alternatives once the di+ and di− of each alternative have been calculated. Calculate similarities to ideal solution. This step solves similarities to an ideal solution by formula: CCi =

di− ; i = 1, 2, ..., m di− + di+

(22)

According to the CCi, we can determine the ranking order of all alternatives and select the best one from among a set of feasible alternatives.

4

Results

In this section, considered methods are applied to rank seven companies through pioneer industries in Tehran stock exchange. The constructed hierarchy consists of eight most important criteria which is expressed in Section 2. A comprehensive fundamental analysis have been performed in TSE and seven companies of Iranian chemical industry, Magsal agricultural products, Shahrood mining industry, Iran glucose, pars oil, Iranian zinc industry, and Parsian commerce selected for the process of assets allocation. Figure 5 gives an overall view of the fuzzy AHP hierarchy used in this paper. The eight mentioned criteria are compared with respect to the goal portfolio selection. Then, corresponding fuzzy pairwise comparison matrix is solicited by DM using Delphi method in Table 3. In Table 3, for example, C1 (OPM) has an absolute priority in comparison with C6 (ICR) (7/2, 4, 9/2). Or C7 (P/B) has a fairly strong priority in comparison with C8 (P/E) (3/2, 2, 5/2). The seven considered stocks are now compared with respect to all the criteria in the hierarchy. The corresponding eight fuzzy pairwise comparison matrices are solicited by DM using Delphi method. Here, for example an RCP criterion (C4) pairwise comparison is reported in Table 4.

C8

C7

C6

C5

C4

C3

2⎞ ⎟ 3⎠

3⎞ ⎟ 2⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎟ 2⎠

2⎞ ⎟ 7⎠

1 2⎞ , ⎟ 3 5⎠ 3⎞ 1, ⎟ 2⎠

⎛2 1 ⎜ , , ⎝9 4 ⎛2 ⎜ , 1, ⎝3

⎛2 ⎜ , ⎝7 ⎛2 ⎜ , ⎝3

⎛2 ⎜ , 1, ⎝3 ⎛2 1 ⎜ , , ⎝5 2

(1, 1, 1)

C1

C2

C1

1 2⎞ , ⎟ 2 3⎠ 1 2⎞ , ⎟ 3 5⎠

⎛2 ⎜ , ⎝5 ⎛2 ⎜ , ⎝7

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

1 2⎞ , ⎟ 3 5⎠ 1 2⎞ , ⎟ 3 5⎠

⎛2 ⎜ , ⎝7 ⎛2 ⎜ , ⎝7

1 2⎞ , ⎟ 2 3⎠ 3⎞ 1, ⎟ 2⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

⎛2 ⎜ , ⎝5 ⎛2 ⎜ , ⎝3

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠ ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

(1, 1, 1)

C3

C2

2⎞ ⎟ 3⎠

3⎞ ⎟ 2⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

⎛2 ⎜ , 1, ⎝3 ⎛2 1 ⎜ , , ⎝5 2

(1, 1, 1)

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

C4

1 2⎞ , ⎟ 2 3⎠ 1 2⎞ , ⎟ 2 3⎠ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

⎛2 ⎜ , ⎝5 ⎛2 ⎜ , ⎝5

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

C5

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

9⎞ ⎛7 ⎜ , 4, ⎟ 2⎠ ⎝2 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

C6

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

C7

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2 2⎠ ⎝

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

C8

Table 3

Goal

A hybrid fuzzy decision making method for a portfolio selection Fuzzy pairwise comparison matrix for criteria with respect to goal

347

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

⎛2 1 7⎞ ⎜ , , ⎟ ⎝9 4 7⎠

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝5 2 3⎠ ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝5 2 3⎠ ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠

A2

A3

A4

A5

A6

A7

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

9⎞ ⎛7 ⎜ , 4, ⎟ 2⎠ ⎝2

(1, 1, 1)

A1

A3

A2

A1

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

A4

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

A5

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

A6

(1, 1, 1)

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

A7

Table 4

C4

348 A. Kazemi et al.

Fuzzy pairwise comparison matrix for alternatives with respect to C4

A hybrid fuzzy decision making method for a portfolio selection Figure 5

349

Hierarchy of the problem (see online version for colours)

The other ratios comparisons are also paired as well as Table 4. The relative weights of criteria and alternatives are implemented using expert choice software and Chang extend analysis method: ⎡ n ⎢ ⎣⎢ i =1

m

∑∑ j =1

M gji

⎤ ⎥ ⎦⎥

−1

= (58.63, 76.08, 98.89) −1 = (0.01, 0.013, 0.017)

For example, for criteria of C1: (11.17, 14, 17.5) ⊗ (0.01, 0.013, 0.017) = (0.11, 0.13, 0.18). So, fuzzy weight of ratios is reported in Table 5. Table 5

Fuzzy weight of criteria

Criteria

Fuzzy triangular number

C1: OPM

(0.11, 0.18, 0.3)

C2: ROE

(0.12, 0.2, 0.32)

C3: ITR

(0.08, 0.13, 0.21)

C4: RCP

(0.06, 0.09, 0.16)

C5: SD

(0.07, 0.12, 0.2)

C6: ICR

(0.04, 0.07, 0.11)

C7: MVBVR

(0.06, 0.11, 0.2)

C8: P/E

(0.05, 0.1, 0.18)

Degree of possibility is calculated using these vectors: V ( S1 ≥ S2 , S3 , S4 , S5 , S6 , S7 , S8 ) = 0.86, V ( S 2 ≥ S1 , S3 , S 4 , S5 , S6 , S7 , S8 ) = 1 V ( S3 ≥ S1 , S 2 , S 4 , S5 , S6 , S7 , S8 ) = 0.58,

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A. Kazemi et al. V ( S4 ≥ S1 , S2 , S3 , S5 , S6 , S7 , S8 ) = 0.29 V ( S5 ≥ S1 , S2 , S3 , S4 , S6 , S7 , S8 ) = 0.51, V ( S6 ≥ S1 , S 2 , S3 , S 4 , S5 , S7 , S8 ) = 0.12

V ( S7 ≥ S1 , S2 , S3 , S4 , S5 , S6 , S8 ) = 0.48, V ( S8 ≥ S1 , S2 , S3 , S4 , S5 , S6 , S7 ) = 0.38

Then, ratios weight vector have as: W ' = (0.86,1, 0.58, 0.29, 0.51, 0.12, 0.48, 0.38)T W = (0.204, 0.237, 0.137, 0.069, 0.121, 0.028, 0.114, 0.09)T Therefore, these processes are also implemented for all alternatives. So, the final weight alternatives obtained in Table 6. Table 6

The final weight stocks in portfolio selection

Company

The final weight stocks

Iranian chemical industry

0.34

Magsal agricultural products

0.114

Shahrood mining industry

0.162

Iran glucose

0.135

Pars oil

0.119

Iranian zinc industry

0.072

Parsian commerce

0.058

Table 6 indicates that Iranian chemical industry has highest weight, and other weight of alternatives are 11.4%, 16.2%, 13.5%, 11.9%, 7.2% and 5.8%, respectively. In fuzzy TOPSIS, Delphi method is used to rank the alternatives after the obtaining ratios weight step. Table 7 indicates linguistic variables used to rank the alternatives. Table 8 demonstrates the normalised weighted decision matrix. Finally, Table 9 shows the distance of each alternative from the ideal of positive and negative. Table 7

Linguistic variables used to rank the alternatives according to criteria C1

C2

C3

C4

A1

H

A2

M

VVH

VH

M

VVL

VVL

H

A3

VH

A4

L

H

H

VH

VH

VVH

H

C5

C6

C7

C8

M

H

VVH

H

VH

VVL

VH

H

M

H

VL

M

L

VH

H

VH

A5

VL

L

M

H

H

VVH

L

VL

A6

VVL

M

L

H

M

L

M

VVH

A7

VVH

VL

VL

H

M

VL

VVL

VVL

A hybrid fuzzy decision making method for a portfolio selection Table 8

351

Normalised weighted decision matrix C1

C2

C3

C4

A1

(0.55, 0.13, 0.3)

(0.11, 0.2, 0.36)

(0.05, 0.12, 0.23)

(0.02, 0.09, 0.38)

A2

(0.33, 0.09, 0.23)

(0, 0, 0.036)

(0, 0, 0.023)

(0.02, 0.06, 0.23)

A3

(0.08, 0.16, 0.33)

(0.06, 0.14, 0.32)

(0.04, 0.09, 0.21)

(0.02, 0.05, 0.16)

A4

(0.01, 0.05, 0.17)

(0.09, 0.18, 0.36)

(0.07, 0.13, 0.23)

(0.02, 0.06, 0.23)

A5

(0, 0.02, 0.1)

(0.01, 0.06, 0.18)

(0.02, 0.07, 0.16)

(0.02, 0.06, 0.23)

A6

(0, 0, 0.03)

(0.04, 0.1, 0.25)

(0.01, 0.04, 0.12)

(0.02, 0.06, 0.23)

A7

(0.1, 0.18, 0.33)

(0, 0.02, 0.11)

(0, 0.013, 0.07)

(0.02, 0.06, 0.23)

C5

C6

C7

C8

A1

(0.01, 0.07, 0.33)

(0.02, 0.05, 0.11)

(0.06, 0.11, 0.23)

(0.03, 0.07, 019)

A2

(0.01, 0.04, 0.14)

(0, 0, 0.012)

(0.05, 0.1, 0.23)

(0.03, 0.07, 019)

A3

(0.01, 0.07, 0.33)

(0.02, 0.05, 0.11)

(0, 0.01, 0.07)

(0.02, 0.05, 0.14)

A4

(0.01, 0.12, 1)

(0.03, 0.06, 0.12)

(0.03, 0.08, 0.2)

(0.04, 0.09, 0.21)

A5

(0.01, 0.05, 0.2)

(0.04, 0.07, 0.12)

(0.01, 0.03, 0.11)

(0, 0.01, 0.06)

A6

(0.01, 0.07, 0.33)

(0.004, 0.02, 0.06)

(0.02, 0.06, 0.16)

(0.05, 0.1, 0.21)

A7

(0.01, 0.07, 0.33)

(0, 0.01, 0.04)

(0, 0, 0.02)

(0, 0, 0.02)

According to the results of Table 9, A1 is the best alternative, and the investors can allocate their capital in A1, A4, A3, A6, A5, A2 and A7, respectively. Table 9

A1

Calculation of d i− , d i+ and CCi d i−

d i+

CCi

1.87

0.84

0.693

A2

0.7

1.21

0.367

A3

1.37

0.84

0.621

A4

2.24

1.3

0.635

A5

0.8

1.13

0.408

A6

1.02

1.16

0.467

A7

0.77

1.4

0.356

High profitability of companies, which is the result of production of high quality goods at low prices, determines their profitability on the exchange market. So it can be concluded that the stock returns of companies reflex the past performance of them in comparison with other companies active in the same field. The annual returns of company ranked in the period of study as Table 10. Table 10

The annual stock returns of the companies during the period of study

Company Annual stock returns

A1

A2

A3

A4

A5

A6

A7

74%

42%

12%

29%

18%

–4%

4%

Table 10 indicates the close correspondence between the results of the methods and reality available in the market. Figures 6 and 7 shows this comparison.

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Figure 6

The final weight alternatives in fuzzy AHP (see online version for colours)

Figure 7

Ranking of alternatives in fuzzy TOPSIS (see online version for colours)

A hybrid fuzzy decision making method for a portfolio selection

5

353

Conclusions and future research

In the MCDM field, there are several crisp or fuzzy ranking methods that only provide DMs with a ranking order of alternatives. Yet, in financial applications, it is evident that DMs or investors would like to know not only the ranking order of the stocks but also at which proportions they should invest in particular stocks. The models used in this paper provide both ranking and weighting information to the investors by fuzzy AHP. The present study prioritises seven companies through pioneer industries on the Tehran stock exchange to select an optimal portfolio. Having reviewed the related literature and opinions of the experts, four classes of financial ratios were selected. From each class then two more effective ratios were considered as a decision making criteria. The criteria selected were used in a hierarchical structure to be weighed through Chang extend analysis. Delphi method was resorted to use linguistic variables in the pairwise comparison matrix. Also efforts were made to prioritise companies using fuzzy AHP method. Delphi method was then used to classify alternatives in fuzzy TOPSIS. Close correlation between the situations of the companies on the Tehran Stock Exchange during the period of study and the paper results, can be one of positive aspects of this study. For future research we can focus be a fuzzy expert system to achieve better decisions and take advantage of the base rules if-then offered.

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Lintner, J. (1965) ‘The valuation of risk assets and the selection of risky investment in stock portfolio and capital budgets’, Review of Economic and Statistics, Vol. 47, No. 1, pp.13–37. Liu, Y.J., Zhang, W.G. and Xu, W.J. (2012) ‘Fuzzy multi-period portfolio selection optimization models using multiple criteria’, Automatica, Vol. 48, No. 12, pp.3042–3053. Markowitz, H. (1952) ‘Portfolio selection’, Journal of Finance, Vol. 7, No. 1, pp.77–91. Masini, E. (1993) Why Future Studies?, Grey Seal, London. Mossin, J. (1966) ‘Equilibrium in capital asset market’, Econometrica, Vol. 34, No. 4, pp.768–783. Parra, M.A., Terol, A.B. and Uria, M.V.R. (2001) ‘A fuzzy goal programming approach to portfolio selection’, European Journal of Operational Research, Vol. 113, No. 2, pp.287–297. Powell, C. (2003) ‘The Delphi technique: myths and realities’, Journal of Advanced Nursing, Vol. 41, No. 4, pp.376–382. Reilly, F.K. and Brown, K.C. (2003) Investment Analysis and Portfolio Selection, 6th ed. Roman, D. and Mitra, G. (2009) ‘Portfolio selection models: a review and new directions’, Wilmott Journal, Vol. 1, No. 2, pp.69–85. Saaty, T.L., Rogers, P.C. and Bell, R. (1980) ‘Portfolio selection through hierarchies’, The Journal of Portfolio Management, Vol. 6, No. 3, pp.16–21. Sarokhani, B. (2004) ‘Qualitative Delphi: a tool to research in social science’, The Women Studies, Vol. 2, No. 4, pp.85–120. Secme, N.Y., Bayrakdaro, Lu, A. and Kahraman, C. (2009) ‘Fuzzy performance evaluation in Turkish banking sector using analytic hierarchy process and TOPSIS’, Expert Systems with Applications, Vol. 36, No. 9, pp.11699–11709. Shamsuzzaman, M., Sharif Ullah, A.M.M. and Dweiri, F.T. (2013) ‘A fuzzy decision model for the selection of coals for industrial use’, Int. J. of Industrial and Systems Engineering, Vol. 14, No. 2, pp.230–244. Sharp, W.F. (1966) ‘Mutual fund performance’, Journal of Business, Vol. 39, No. 1, pp.119–138. Tanaka, H. and Guo, P. (1999) ‘Portfolio selection based on upper and lower exponential possibility distributions’, Fuzzy Sets and Systems, Vol. 111 No. 3, pp.387–397. Tiryaki, F. and Ahlatcioglu, B. (2009) ‘Fuzzy portfolio selection using fuzzy analytic hierarchy process’, Information Sciences, Vol. 179, No. 1, pp.53–69. Tiryaki, F. and Ahlatcioglu, M. (2005) ‘Fuzzy stock selection using a new fuzzy ranking and weighting algorithm’, Applied Mathematical and Computation, Vol. 170, No. 1, pp.144–157. Linston, H.A. and Turoff, M. (2002) The Delphi method: Technique and Application, Addison-Wesley Pub.Co., Advanced Book Program. Wang, T.C. and Chang, T.H. (2007) ‘Application of TOPSIS in evaluating initial training aircraft under a fuzzy environment’, Expert systems with Applications, Vol. 33, No. 4, pp.870–880. Xia, Y., Liu, B., Wang, S. and Lai, K.K. (2000) ‘A model for portfolio selection with order of expected returns’, Computers and Operation research, Vol. 27, No. 5, pp.409–422. Zadeh, L.A. (1965) ‘Fuzzy sets’, Information and Control, Vol. 8, No. 3, pp.338–353.

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A hybrid fuzzy decision making method for a portfolio selection: a case study of Tehran Stock Exchange Abolfazl Kazemi* Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, P.O. Box 34185-1416, Qazvin, Iran Fax: +98(281)-3665275 E-mail: [email protected] *Corresponding author

Keyvan Sarrafha Young Researches and Elite Club, Qazvin Branch, Islamic Azad University, P.O. Box 34185-1416, Qazvin, Iran Fax: +98(281)-3665275 E-mail:[email protected]

Mahdi Beedel Faculty of Administrative Sciences and Economics, Department of Accounting, University of Isfahan, P.O. Box 81746-73441, Isfahan, Iran Fax: +98(311)-7932128 E-mail: [email protected] Abstract: The problem of choosing an optimal portfolio is the most significant issue for any investor in the stock exchange market. The present study seeks to prioritise seven companies through pioneer industries on Tehran Stock Exchange market by opting for suitable financial guidelines as well effective criteria in uncertain environments. To do so, a hybrid fuzzy decision making procedure was applied. Having determined the criteria, Delphi method approach was used as a process in fuzzy pairwise comparison of criteria and alternatives in fuzzy AHP. Alternatives then weighed up in fuzzy TOPSIS method to achieve a coordinated and balanced view. Finally, to evaluate the results against real conditions of Tehran exchange market, they were compared with the actual annually returns of these same companies on the market. Keywords: portfolio selection; Delphi method; decision making; fuzzy AHP; fuzzy TOPSIS; financial ratios. Reference to this paper should be made as follows: Kazemi, A., Sarrafha, K. and Beedel, M. (2014) ‘A hybrid fuzzy decision making method for a portfolio selection: a case study of Tehran Stock Exchange’, Int. J. Industrial and Systems Engineering, Vol. 18, No. 3, pp.335–354.

Copyright © 2014 Inderscience Enterprises Ltd.

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A. Kazemi et al. Biographical notes: Abolfazl Kazemi is an Assistant Professor in the Department of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran. His research interests include intelligent information systems, fuzzy set theory and supply chain management. Keyvan Sarrafha is a Master student at the Department of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran. His interests span from supply chain management, fuzzy logic, mathematical modelling and meta-heuristic algorithms. Mahdi Beedel is a Master student at Faculty of Administrative Sciences and Economics, Department of Accounting, Isfahan University, Isfahan. His research area includes optimal capital structure and portfolio selection.

1

Introduction

An active capital market is one of the marks of development on an international level. Investor’s tendency to play a more active role leads to prosperity in the market which intern contributes to companies active in different industries. To gain more profit on the capital market, investors have to choose the stocks of the companies whose stock price is on the rise due to their satisfactory performance record. However, as investors cannot be sure of the future, they have to opt for a diversified portfolio to lower risk of incurring losses (Secme et al., 2009). Portfolio selection is well known as a leading problem in finance; since the future returns of assets are not known at the time of the investment decision, the problem is one of decision-making under risk. A portfolio selection model is an ex-ante decision tool: decisions taken today can only be evaluated at a future time, once the uncertainty regarding the assets’ returns is revealed (Roman and Mitra, 2009). The main purpose of presenting portfolio selecting methods is helping the investors to choose an optimal portfolio. To do this, analysing the present and the past performance of the companies by using some important criteria can be useful. Modern portfolio theory (MPT) is based upon the classical Markowitz model (1952) which uses variance as a risk measure. He introduced it in a computational model, by measuring the risk of a portfolio via the covariance matrix associated with individual asset returns; this leads to a quadratic programming formulation. Markowitz idea on the mean-variance approach then being developed by Sharp (1966), Lintner (1965) and Mossin (1966). Since then, the model is well accepted by investors and fund managers that aimed to construct an efficient portfolio with the highest diversification benefit. Portfolio diversification is influenced by many factors by that govern the portfolio selection criteria such as the firm sizes, financial ratios, stock markets and investor’s judgment. In selecting stocks to have an optimal portfolio, the investors face various options for investing. Identifying the various criteria which affect stock prices, comparing them and identifying their relative importance throw pairwise comparison to have a better stock selection is a complex and difficult process. Thus multi criteria decision making methods can be used to select the optimal portfolio. In multiple criteria (or attribute) decision making problems, a decision maker (DM) or (DMs) often needs to select or rank alternatives associated with some usually conflicting attributes or objectives. The portfolio manager, as the DM, has a large

A hybrid fuzzy decision making method for a portfolio selection

337

set criterion for selecting stocks. These criteria depend on various interdependent primary and secondary factors. The problem is to compare the various criteria and to determine their relative importance through pairwise comparison between each pair of them. According to this rule, one can make a decision to compare each list of stocks with this criteria list and determine the amount of investment to be allocated to each stock. The DM will then be able to compare each list of stocks with this criteria list and determine the portfolio. To make the comparison, he/she will need a scale by which to compare the factors pairwise. The scale (1–9) is used in analytic hierarchy process (AHP) and reflects the DM’s belief as to which of two factors or criteria is more important (and to what degree), relative to some quality that both of them share. Saaty et al. (1980) proposed AHP to deal with the stock portfolio decision problem by evaluating the performance. Tanaka and Guo (1999) formulated portfolio selection models by quadratic programming, based on two kinds of possibility distributions. Xia et al. (2000) proposed a new model for portfolio selection using genetic algorithms. Inuiguchi and Tanino (2000) proposed a new possibility programming approach for portfolio optimisation, considering how a model yields a distribution investment solution. In real world problems, decisions are made in situations where the objectives, limitations and results are not completely known. Likewise, striking on an optimal portfolio is based on information derived from economic environments (such as company’s annual reports, inflation rate, GNP’s growth rate, Government economic policies and so on). And this information is always associated with some degree of uncertainty. In the past it was hypothesised that stock’s return can be clearly predicted by companies past performances. However, complexity of capital market does not allow this prediction. These uncertainties have made frustrated researches resort to applying fuzzy set theory. This theory was first introduced By Zadeh (1965) and has been expanded largely ever since. Classical and mathematical approaches fall short of explaining these uncertainties as related to optimisation of portfolio. Therefore using fuzzy theory helps financial DMs to overcome uncertainties. In this regard, Parra et al. (2001) formulated a fuzzy goal programming with fuzzy goal and fuzzy constraints, taking into account three criteria: return, risk and liquidity. Considering the uncertainty of investment environment, Tiryaki and Ahlatcioglu (2005) transferred expert’s linguistic value into triangle fuzzy number and used a new fuzzy ranking and weighting algorithm to obtain the investment ratio of each stock. In fact, the stock portfolio selection problem can be described as multiple criteria decision making (MCDM) problem. In recent works also, Secme et al. (2009) proposed a fuzzy multi criteria decision model to evaluate the performances of some banks. In their implementation, a fuzzy analytic hierarchy process (FAHP) and technique for order performance by similarity to ideal solution (TOPSIS) are integrated to evaluate the five largest commercial banks of Turkish banking sector in terms of several financial and non-financial indicators. Tiryaki and Ahlatcioglu (2009) used the two constrained fuzzy AHP methods, developed by Enea and Piazza method, to the problem of portfolio selection in Istanbul stock exchange. They addressed some fallacies in the first model of Enea and Piazza and corrected it. They also showed that the second model of Enea and Piazza is superior to their first model in terms of uncertainty level of the solutions obtained. Arunkumar et al. (2011) presented a new method to deal with the performance evaluation and selection of supplier for a manufacturing industry using fuzzy AHP with alpha cut analysis. The results of this paper provide valuable suggestions to managers on

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the improvement needed with respect to each criterion when selecting suppliers. GhazanfarAhari et al. (2011) also used the two fuzzy AHP methods. They have used Enea and Piazza method and van Larhoven and Pedrycz’s approach for a real world case study of pharmaceutical industry in Tehran stock exchange where all input parameters are subject to uncertainty. And so, the result of their implementation determines the asset allocation for common shares of five different companies. Liu et al. (2012) by considering various criteria, including return, transaction cost, risk and skewness of portfolio in fuzzy environment simulated the real transactions in financial market to provide investors with additional choices. Shamsuzzaman et al. (2013) described a fuzzy-logic-based MCDM approach to rank coals for industrial use. The calorific value and chemical compositions desired values of each criterion are expressed by an appropriate fuzzy number, whereas all possible values of a criterion of a given coal are captured by a numerical range. Gupta et al. (2013) presented a hybrid optimisation model of portfolio selection involving financial and ethical considerations. They proposed a comprehensive three-MCDM framework for portfolio selection based on financial and ethical criteria simultaneously. The AHP method is used to obtain the ethical performance score of each asset based on investor-preferences, and a fuzzy MCDM technique is also used to obtain the financial quality score of each asset based upon investor-ratings on the financial criteria. In the real world situations, opinions are wary. DMs consider different criteria, so the best idea should be found. Delphi logic can be applied to achieve a balanced solidarity of experts and DM’s opinion which leads to taking the right course of action. Further to confirm how reliable the results are, they should be compared against a significant index in real world. In the present paper we have tried to find an efficient way of allocating a limited amount of money to Tehran stock exchange companies in order to have an optimal portfolio. To this end after prioritising major financial ratios including: profitability ratios, activity ratios, risk ratios and market ratios in fuzzy environment, classify these companies. Now having reviewed the related literature, uncertainty can be fully resorted to all stages from criteria’s weights to comparison of approaches in order to form an optimal portfolio. The main question in this paper is then how fuzzy logic minimises the uncertainties in all stages of decision making and prioritising the company’s stocks in the portfolio. Effort was made to obtain the financial expert’s views by using a translated version of fuzzy method and FAHP to identify suitable weights of criteria and alternatives. Fuzzy TOPSIS method is also applied to obtain even more reliability in studying the portfolio. In the end, to evaluate the results against real conditions of Tehran exchange market, they were compared with the annual returns of these same companies on the market. The rest of the paper is organised as follows: Section 2, explains the methodology. Section 3 is devoted to solving methods. In Section 4, the results are presented. Finally, conclusion remarks are given in the last section to summarise the contribution of the paper in the last section.

2

Problem definition

As previously mentioned, the portfolio manager as the DM, face a large set of criteria for selecting stocks. In addition, DMs are often misled by the large amount of information that is associated with uncertainty in financial environment. However, financial

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statements are reliable sources for evaluating the performance of companies and choosing the stocks of companies with better performances (Brigham et al., 1996). From the perspective of an investor predicting future is why financial statements are analysed. Financial statement analysis is assessed by financial ratios. These ratios can not only help an investor to understand strengths and weaknesses of different companies but they can also be used for comparing economic performances of companies. In different studies different ratios are considered as effective criteria in order to select optimal stocks (Edirisinghe and Zhang, 2008). In the present study, eight ratios from four main groups of ratios are considered as decision making criteria. The four main groups of financial ratios are: profitability criteria, activity criteria, risk criteria, market criteria (Brigham et al., 1996).

2.1 Category profitability criteria 2.1.1 Operating profit margin (OPM) The profit margin reveals the profitability of each dollar of sale (Brickley, 1983).

2.1.2 Return on equity Return on equity (ROE) measures a corporation’s profitability by revealing how much profit a company generates with the money shareholders have invested (Edirisinghe and Zhang, 2008).

2.2 Category activity ratio 2.2.1 Inventory turnover ratio (ITR) A ratio showing how many times a company’s inventory is sold and replaced over a period (Reilly and Brown, 2003).

2.2.2 Receivables collection period (RCP) This reveals how many days it takes to collect all accounts receivable (Brigham et al., 1996).

2.3 Category risk criteria 2.3.1 Standard deviation (SD) A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation (Edirisinghe and Zhang, 2008).

2.3.2 Interest coverage ratio A ratio used to determine how easily a company can pay interest on outstanding debt. The interest coverage ratio (ICR) is calculated by dividing a company’s earnings before (Brigham et al., 1996).

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2.4 category market criteria ⎛P⎞ 2.4.1 Market value to book value ratio ⎜ ⎟ ⎝B⎠ A ratio used to compare a stock’s market value to its book value. A lower P/B ratio could mean that the stock is undervalued (Brickley, 1983).

⎛P⎞ 2.4.2 Price to earnings ratio ⎜ ⎟ ⎝E⎠ A valuation ratio of a company’s current share price compared to its per-share earnings. In general, a high P/E suggests that investors are expecting higher earnings growth in the future compared to companies with a lower P/E (Brigham et al., 1996).

3

Methods

In this section, we use two popular methods namely fuzzy AHP and fuzzy TOPSIS to select the proposed portfolio. AHP is a systematic method for comparing a list of metrics or alternatives. TOPSIS is based on the concept that the chosen alternative should have the shortest geometric distance from the positive ideal alternative and the longest geometric distance from the negative ideal alternative. Here, we considered Delphi method as a structured communication technique for all stages of paired comparisons in fuzzy AHP and rating companies in fuzzy TOPSIS.

3.1 Delphi method Delphi is a tool for qualitative researches. This method is used in macro subjects, especially qualitative matters (Sarokhani, 2004). The Delphi is a structured process for predicting and assisting to make decisions during survey rounds. Gathering information and finally grouping agreement are its other usages. While most surveys attempt to respond to the question: ‘What is it?’ the Delphi tries to answer to the question: ‘What could or should it be?’ (Powell, 2003). The Delphi method is the most important technique to detect and study subjects, which are mixtures of academic bases and social values (Sarokhani, 2004). Therefore, this is an appropriate method to recognise judgment issues like risk assessment. Delphi could be used to form a group communication, which facilitates thinking and being involved as a whole to complex issues (Linstone and Turoff, 2002). Although this group judgment is form different mental view points, it is more trustable from individual and personal opinions and its results are more objective and precise (Masini, 1993). Participants in the new round could compare their personal opinions with others in prior round and may adjust or change them in next round. Hence, the final results are group judgments and there are no opinions belonging to a single person (Sarokhani, 2004). That is why we selected this qualitative method for the identification important criteria and alternatives in both methods. The population was TSE experts that they are ranked according to several measures by TSE. The sample is formed with 20 experts (Delphi panel). Delphi questionnaires are

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sent by email messages to where to sure consistency in the responses found. Here, is the conclusion reached in the second round. In each round, the goals and processes of the research are explained completely to the participants. Figure 1 presents Delphi method flowchart. Figure 1

Proposed hybrid method flowchart (see online version for colours)

3.2 Extent analysis method on fuzzy AHP In the following, first the outlines of the extent analysis method (Chang, 1996) on fuzzy AHP are given and then the method is applied to a portfolio selection problem. Let X = {x1, x2, …, xn} be an object set, and U = {u1, u2, …, un} be a goal set. According to the method of Chang’s extent analysis, each object is taken and extent analysis for each goal is performed respectively. Therefore, m extent analysis values for each object can be obtained, with the following signs: M 1g1 , M g22 , ..., M gmi

i = 1, 2, ..., n

(1)

where all the M gji (j = 1, 2, …, n) are triangular fuzzy numbers. The value of fuzzy synthetic extent with respect to the ith object is defined as: m

si =

∑ j =1

M gji

⎡ n ⊗⎢ ⎢⎣ i =1

m

∑∑ j =1

M gji

⎤ ⎥ ⎥⎦

−1

(2)

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The degree of possibility of M1 ≥ M2 is defined as: V ( Si ≥ Sk ) = Sup ⎡⎣ min ( μsi ( x), μsk ( y ) ) ⎤⎦

When a pair (x, y) exists such that x ≥ y and μM1 ( x), V(M1 ≥ M2) = 1. Since M1 and M2 are convex fuzzy numbers we have that:

(3) μM 2 ( y ), then we have

V ( Si ≥ Sk ) = 1 if mi ≥ mk and if lk ≥ ui

(4)

V ( Si ≥ Sk ) = hgt ( Si ∩ Sk ) = μsi (d )

(5)

where d is the ordinate of the highest intersection point D between μM1 and μM 2 that is shown in Figure 2. When Si = (li, mi, ui) and Sk = (lk, mk, uk), the ordinate of D is given by equation (6):

V ( Si ≥ Sk ) = hgt ( Si ∩ Sk ) = Figure 2

lk − ui

( m i −ui ) − ( m k −lk )

(6)

Intersection point between μSi and μSk (see online version for colours)

To compare M1 and M2, we need both the values of V(Si ≥ Sk) and V(Sk ≥ Si). The degree possibility for a convex fuzzy number to be greater than k convex fuzzy number Si = (i = 1, 2, …, k) can be defined by: V ( S ≥ S1 , S2 ,..., Sk ) = V ⎡⎣( S ≥ S1 ) , ( S ≥ S2 ) , ..., ( S ≥ Sk ) ⎤⎦ = min V ( S ≥ Si ) i = 1, 2, ..., k

(7)

Assume that: d ' ( Ai ) = min V ( Si ≥ Sk ) , k = 1, 2, ..., n, i ≠ k

Then the weight vector is given by:

(8)

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W ' = ( d ' ( A1 ), d ' ( A2 ), ..., d ' ( An ) )

T

(9)

where Ai = (i = 1, 2, …, n) have n elements. Via normalisation, the normalised weight vectors are: W = ( d ( A1 ) , d ( A2 ) , ..., d ( An ) )

T

(10)

where W is a non-fuzzy number. So, overall stages fuzzy AHP method by Chang’s extent analysis method, as follow: 1

Create a hierarchical structure for problem.

2

determine pairwise comparisons matrix and apply judgments by using linguistic variable given in Table 1. Membership function for linguistic variable is shown in Figure 3.

3

Calculate the relative weights of criteria and alternative.

4

Calculate the final weights alternative that obtained by combination the relative weights.

Table 1

Triangular fuzzy number values

Linguistic variable Just equal

Fuzzy pairwise comparison value (1, 1, 1)

weak

(2/3, 1, 3/2)

Fairly strong

(3/2, 2, 5/2)

Very strong

(5/2, 3, 7/2)

Absolute

(7/2, 4, 9/2)

Figure 3

Membership function for linguistic variable (see online version for colours)

3.3 Fuzzy TOPSIS TOPSIS is one of the well-known MCDM methods of adaptive subtype that too, is a subgroup of the compensation. This way every option considers as a point in space. The basic principle of the TOPSIS method is that the chosen alternative should have the

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shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). It is an effective method to determine the total ranking order of decision alternatives. In fuzzy environment, the matrix elements of the decision or weights of indicators or both are expressed as fuzzy number. Several fuzzy TOPSIS methods are presented. All of these methods with little change in the method of Chen et al. (1992) have obtained (Wang and Chang, 2007). The steps are as follows: Step 1

Determine the weighting of evaluation criteria.

In this paper the importance weights of various criteria is obtained and using Chang method in fuzzy AHP in before section. Step 2

Construct the fuzzy decision matrix and choose the appropriate linguistic variables for the alternative with respect to criteria.

In this paper, rating alternative considered as linguistic variables as Table 2 (Chen et al., 2006). These linguistic variables shown as triangular fuzzy number shown Figure 4. Table 2

Linguistic scales for the importance of each criterion

Linguistic variable

Corresponding triangular fuzzy number

Very very low (VVL)

(0, 0, 1)

Very low (VL)

(0, 1, 3)

Low (L)

(1, 3, 5)

Medium (M)

(3, 5, 7)

High (H)

(5, 7, 9)

Very high (VH)

(7, 9, 10)

Very very high (VVH)

(9, 10, 10)

Figure 4

Membership function for linguistic variables (see online version for colours)

A hybrid fuzzy decision making method for a portfolio selection C1

Cj

Cn

A1 ⎡ r11 ⎢ ⎢ D = Ai ⎢ ri1 ⎢ ⎢ Am ⎢⎣ rm1

r1 j

r1n ⎤ ⎥ ⎥ rin ⎥ i = 1, 2, ..., m; j = 1, 2, ..., n ⎥ ⎥ rmn ⎥⎦

rij rmj

345

(11)

where rij is the rating of alternative Ai with respect to criterion Cj evaluated by experts by using Delphi method and rij = ( aij , bij , cij ) . Step 3

Normalise the fuzzy decision matrix.

The normalised fuzzy decision matrix denoted by R is shown as formula:

R = [ rij ]m×n ; i = 1, 2, ..., m;

j = 1, 2, ..., n

(12)

Then the normalisation process can be performed by following formula: ⎛ aij bij cij ⎞ rij = ⎜ + , + , + ⎟ ; C +j = max Cij ⎜ Cj Cj Cj ⎟ i ⎝ ⎠

(13)

⎛ C ij C ij C ij ⎞ rij = ⎜ , , ⎟ ; C ij = min Cij i ⎝ aij bij cij ⎠

(14)

The normalised rij are still triangular fuzzy numbers. For trapezoidal fuzzy numbers, the normalisation process can be conducted in the same way. The weighted fuzzy normalised decision matrix is shown as following matrix V : C = [ vij ]m×n ; i = 1, 2, ..., m;

j = 1, 2, ..., n

vij = rij ⊗ W j Step 4

(15) (16)

Determine the fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS).

According to the weighed normalised fuzzy decision matrix, we know that the elements vij are normalised positive triangular fuzzy numbers and their ranges belong to the closed interval [0, 1]. Then, we can define the FPIS A+ and FNIS A– by using of Lee and Li ranking method as following formula: M ( vij ) =

− aij2 − aij .bij + bij .cij + cij2 3 ( cij − aij )

(17)

After obtaining M (vij ) for each column j A+ = ( v1+ , v2+ , ..., vn+ )

(18)

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(19)

where v +j = max( M (vij )) and v −j = min( M (vij )); j = 1, 2, ..., n.

Step 5

Calculate the distance of each alternative from FPIS and FNIS

The distance ( di+ and di− ) of each alternative A+ from and A– can be currently calculated by the area compensation method. n

di+ =

∑ d ( v , v ); i = 1, 2, ..., m ; j = 1, 2, ..., n ij

+ j

(20)

j =1 n

di− =

∑ d ( v , v ); i = 1, 2, ..., m ; j = 1, 2, ..., n ij

− j

(21)

j =1

Step 6

Obtain the closeness coefficient and rank the order of alternatives.

The CCi is defined to determine the ranking order of all alternatives once the di+ and di− of each alternative have been calculated. Calculate similarities to ideal solution. This step solves similarities to an ideal solution by formula: CCi =

di− ; i = 1, 2, ..., m di− + di+

(22)

According to the CCi, we can determine the ranking order of all alternatives and select the best one from among a set of feasible alternatives.

4

Results

In this section, considered methods are applied to rank seven companies through pioneer industries in Tehran stock exchange. The constructed hierarchy consists of eight most important criteria which is expressed in Section 2. A comprehensive fundamental analysis have been performed in TSE and seven companies of Iranian chemical industry, Magsal agricultural products, Shahrood mining industry, Iran glucose, pars oil, Iranian zinc industry, and Parsian commerce selected for the process of assets allocation. Figure 5 gives an overall view of the fuzzy AHP hierarchy used in this paper. The eight mentioned criteria are compared with respect to the goal portfolio selection. Then, corresponding fuzzy pairwise comparison matrix is solicited by DM using Delphi method in Table 3. In Table 3, for example, C1 (OPM) has an absolute priority in comparison with C6 (ICR) (7/2, 4, 9/2). Or C7 (P/B) has a fairly strong priority in comparison with C8 (P/E) (3/2, 2, 5/2). The seven considered stocks are now compared with respect to all the criteria in the hierarchy. The corresponding eight fuzzy pairwise comparison matrices are solicited by DM using Delphi method. Here, for example an RCP criterion (C4) pairwise comparison is reported in Table 4.

C8

C7

C6

C5

C4

C3

2⎞ ⎟ 3⎠

3⎞ ⎟ 2⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎟ 2⎠

2⎞ ⎟ 7⎠

1 2⎞ , ⎟ 3 5⎠ 3⎞ 1, ⎟ 2⎠

⎛2 1 ⎜ , , ⎝9 4 ⎛2 ⎜ , 1, ⎝3

⎛2 ⎜ , ⎝7 ⎛2 ⎜ , ⎝3

⎛2 ⎜ , 1, ⎝3 ⎛2 1 ⎜ , , ⎝5 2

(1, 1, 1)

C1

C2

C1

1 2⎞ , ⎟ 2 3⎠ 1 2⎞ , ⎟ 3 5⎠

⎛2 ⎜ , ⎝5 ⎛2 ⎜ , ⎝7

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

1 2⎞ , ⎟ 3 5⎠ 1 2⎞ , ⎟ 3 5⎠

⎛2 ⎜ , ⎝7 ⎛2 ⎜ , ⎝7

1 2⎞ , ⎟ 2 3⎠ 3⎞ 1, ⎟ 2⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

⎛2 ⎜ , ⎝5 ⎛2 ⎜ , ⎝3

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠ ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

(1, 1, 1)

C3

C2

2⎞ ⎟ 3⎠

3⎞ ⎟ 2⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

⎛2 ⎜ , 1, ⎝3 ⎛2 1 ⎜ , , ⎝5 2

(1, 1, 1)

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

C4

1 2⎞ , ⎟ 2 3⎠ 1 2⎞ , ⎟ 2 3⎠ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

⎛2 ⎜ , ⎝5 ⎛2 ⎜ , ⎝5

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

C5

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

9⎞ ⎛7 ⎜ , 4, ⎟ 2⎠ ⎝2 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2 7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

C6

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

C7

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2 2⎠ ⎝

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

C8

Table 3

Goal

A hybrid fuzzy decision making method for a portfolio selection Fuzzy pairwise comparison matrix for criteria with respect to goal

347

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

⎛2 1 7⎞ ⎜ , , ⎟ ⎝9 4 7⎠

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠ 3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3 ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝5 2 3⎠ ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝5 2 3⎠ ⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠

A2

A3

A4

A5

A6

A7

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

9⎞ ⎛7 ⎜ , 4, ⎟ 2⎠ ⎝2

(1, 1, 1)

A1

A3

A2

A1

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

A4

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝ 5 2 3⎠

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 2⎠ ⎝3

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

A5

⎛ 2 1 2⎞ ⎜ , , ⎟ ⎝7 3 5⎠

(1, 1, 1)

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

A6

(1, 1, 1)

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

5⎞ ⎛3 ⎜ , 2, ⎟ 2⎠ ⎝2

3⎞ ⎛2 ⎜ , 1, ⎟ 3 2⎠ ⎝

7⎞ ⎛5 ⎜ , 3, ⎟ 2⎠ ⎝2

A7

Table 4

C4

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Fuzzy pairwise comparison matrix for alternatives with respect to C4

A hybrid fuzzy decision making method for a portfolio selection Figure 5

349

Hierarchy of the problem (see online version for colours)

The other ratios comparisons are also paired as well as Table 4. The relative weights of criteria and alternatives are implemented using expert choice software and Chang extend analysis method: ⎡ n ⎢ ⎣⎢ i =1

m

∑∑ j =1

M gji

⎤ ⎥ ⎦⎥

−1

= (58.63, 76.08, 98.89) −1 = (0.01, 0.013, 0.017)

For example, for criteria of C1: (11.17, 14, 17.5) ⊗ (0.01, 0.013, 0.017) = (0.11, 0.13, 0.18). So, fuzzy weight of ratios is reported in Table 5. Table 5

Fuzzy weight of criteria

Criteria

Fuzzy triangular number

C1: OPM

(0.11, 0.18, 0.3)

C2: ROE

(0.12, 0.2, 0.32)

C3: ITR

(0.08, 0.13, 0.21)

C4: RCP

(0.06, 0.09, 0.16)

C5: SD

(0.07, 0.12, 0.2)

C6: ICR

(0.04, 0.07, 0.11)

C7: MVBVR

(0.06, 0.11, 0.2)

C8: P/E

(0.05, 0.1, 0.18)

Degree of possibility is calculated using these vectors: V ( S1 ≥ S2 , S3 , S4 , S5 , S6 , S7 , S8 ) = 0.86, V ( S 2 ≥ S1 , S3 , S 4 , S5 , S6 , S7 , S8 ) = 1 V ( S3 ≥ S1 , S 2 , S 4 , S5 , S6 , S7 , S8 ) = 0.58,

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A. Kazemi et al. V ( S4 ≥ S1 , S2 , S3 , S5 , S6 , S7 , S8 ) = 0.29 V ( S5 ≥ S1 , S2 , S3 , S4 , S6 , S7 , S8 ) = 0.51, V ( S6 ≥ S1 , S 2 , S3 , S 4 , S5 , S7 , S8 ) = 0.12

V ( S7 ≥ S1 , S2 , S3 , S4 , S5 , S6 , S8 ) = 0.48, V ( S8 ≥ S1 , S2 , S3 , S4 , S5 , S6 , S7 ) = 0.38

Then, ratios weight vector have as: W ' = (0.86,1, 0.58, 0.29, 0.51, 0.12, 0.48, 0.38)T W = (0.204, 0.237, 0.137, 0.069, 0.121, 0.028, 0.114, 0.09)T Therefore, these processes are also implemented for all alternatives. So, the final weight alternatives obtained in Table 6. Table 6

The final weight stocks in portfolio selection

Company

The final weight stocks

Iranian chemical industry

0.34

Magsal agricultural products

0.114

Shahrood mining industry

0.162

Iran glucose

0.135

Pars oil

0.119

Iranian zinc industry

0.072

Parsian commerce

0.058

Table 6 indicates that Iranian chemical industry has highest weight, and other weight of alternatives are 11.4%, 16.2%, 13.5%, 11.9%, 7.2% and 5.8%, respectively. In fuzzy TOPSIS, Delphi method is used to rank the alternatives after the obtaining ratios weight step. Table 7 indicates linguistic variables used to rank the alternatives. Table 8 demonstrates the normalised weighted decision matrix. Finally, Table 9 shows the distance of each alternative from the ideal of positive and negative. Table 7

Linguistic variables used to rank the alternatives according to criteria C1

C2

C3

C4

A1

H

A2

M

VVH

VH

M

VVL

VVL

H

A3

VH

A4

L

H

H

VH

VH

VVH

H

C5

C6

C7

C8

M

H

VVH

H

VH

VVL

VH

H

M

H

VL

M

L

VH

H

VH

A5

VL

L

M

H

H

VVH

L

VL

A6

VVL

M

L

H

M

L

M

VVH

A7

VVH

VL

VL

H

M

VL

VVL

VVL

A hybrid fuzzy decision making method for a portfolio selection Table 8

351

Normalised weighted decision matrix C1

C2

C3

C4

A1

(0.55, 0.13, 0.3)

(0.11, 0.2, 0.36)

(0.05, 0.12, 0.23)

(0.02, 0.09, 0.38)

A2

(0.33, 0.09, 0.23)

(0, 0, 0.036)

(0, 0, 0.023)

(0.02, 0.06, 0.23)

A3

(0.08, 0.16, 0.33)

(0.06, 0.14, 0.32)

(0.04, 0.09, 0.21)

(0.02, 0.05, 0.16)

A4

(0.01, 0.05, 0.17)

(0.09, 0.18, 0.36)

(0.07, 0.13, 0.23)

(0.02, 0.06, 0.23)

A5

(0, 0.02, 0.1)

(0.01, 0.06, 0.18)

(0.02, 0.07, 0.16)

(0.02, 0.06, 0.23)

A6

(0, 0, 0.03)

(0.04, 0.1, 0.25)

(0.01, 0.04, 0.12)

(0.02, 0.06, 0.23)

A7

(0.1, 0.18, 0.33)

(0, 0.02, 0.11)

(0, 0.013, 0.07)

(0.02, 0.06, 0.23)

C5

C6

C7

C8

A1

(0.01, 0.07, 0.33)

(0.02, 0.05, 0.11)

(0.06, 0.11, 0.23)

(0.03, 0.07, 019)

A2

(0.01, 0.04, 0.14)

(0, 0, 0.012)

(0.05, 0.1, 0.23)

(0.03, 0.07, 019)

A3

(0.01, 0.07, 0.33)

(0.02, 0.05, 0.11)

(0, 0.01, 0.07)

(0.02, 0.05, 0.14)

A4

(0.01, 0.12, 1)

(0.03, 0.06, 0.12)

(0.03, 0.08, 0.2)

(0.04, 0.09, 0.21)

A5

(0.01, 0.05, 0.2)

(0.04, 0.07, 0.12)

(0.01, 0.03, 0.11)

(0, 0.01, 0.06)

A6

(0.01, 0.07, 0.33)

(0.004, 0.02, 0.06)

(0.02, 0.06, 0.16)

(0.05, 0.1, 0.21)

A7

(0.01, 0.07, 0.33)

(0, 0.01, 0.04)

(0, 0, 0.02)

(0, 0, 0.02)

According to the results of Table 9, A1 is the best alternative, and the investors can allocate their capital in A1, A4, A3, A6, A5, A2 and A7, respectively. Table 9

A1

Calculation of d i− , d i+ and CCi d i−

d i+

CCi

1.87

0.84

0.693

A2

0.7

1.21

0.367

A3

1.37

0.84

0.621

A4

2.24

1.3

0.635

A5

0.8

1.13

0.408

A6

1.02

1.16

0.467

A7

0.77

1.4

0.356

High profitability of companies, which is the result of production of high quality goods at low prices, determines their profitability on the exchange market. So it can be concluded that the stock returns of companies reflex the past performance of them in comparison with other companies active in the same field. The annual returns of company ranked in the period of study as Table 10. Table 10

The annual stock returns of the companies during the period of study

Company Annual stock returns

A1

A2

A3

A4

A5

A6

A7

74%

42%

12%

29%

18%

–4%

4%

Table 10 indicates the close correspondence between the results of the methods and reality available in the market. Figures 6 and 7 shows this comparison.

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A. Kazemi et al.

Figure 6

The final weight alternatives in fuzzy AHP (see online version for colours)

Figure 7

Ranking of alternatives in fuzzy TOPSIS (see online version for colours)

A hybrid fuzzy decision making method for a portfolio selection

5

353

Conclusions and future research

In the MCDM field, there are several crisp or fuzzy ranking methods that only provide DMs with a ranking order of alternatives. Yet, in financial applications, it is evident that DMs or investors would like to know not only the ranking order of the stocks but also at which proportions they should invest in particular stocks. The models used in this paper provide both ranking and weighting information to the investors by fuzzy AHP. The present study prioritises seven companies through pioneer industries on the Tehran stock exchange to select an optimal portfolio. Having reviewed the related literature and opinions of the experts, four classes of financial ratios were selected. From each class then two more effective ratios were considered as a decision making criteria. The criteria selected were used in a hierarchical structure to be weighed through Chang extend analysis. Delphi method was resorted to use linguistic variables in the pairwise comparison matrix. Also efforts were made to prioritise companies using fuzzy AHP method. Delphi method was then used to classify alternatives in fuzzy TOPSIS. Close correlation between the situations of the companies on the Tehran Stock Exchange during the period of study and the paper results, can be one of positive aspects of this study. For future research we can focus be a fuzzy expert system to achieve better decisions and take advantage of the base rules if-then offered.

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