Selection of Portfolio by using Multi Attributed ...

American Journal of Scientific Research ISSN 1450-223X Issue 44 (2012), pp. 15-29 © EuroJournals Publishing, Inc. 2012 http://www.eurojournals.com/ajsr.htm

Selection of Portfolio by using Multi Attributed Decision Making (Tehran Stock Exchange) Mohammad Hasan Janani Department of Accounting, Islamic Azad University Boroujerd Branch, Boroujerd, Iran E-mail: [email protected] Mohammad Ehsanifar Department of Industrial Management, Islamic Azad University Tafresh Branch, Tafresh, Iran E-mail: [email protected] Sanam Bakhtiarnezhad Department of Accounting, Islamic Azad University Boroujerd Branch, Boroujerd, Iran E-mail: [email protected] Abstract In the investment topic, selection of portfolio, decision making for comparison share with each other and competence to be selected and placed in portfolio and style of capital allocation to security are important and complex subject. Theoretically, portfolio selection subject in the mode of risk minimization and stabilizing return required wide calculation and planning. In this study, the effective criteria on portfolio and the score assigned by a group of experts to criteria have been determined. In this paper, we calculate weight of each criterion by using of eigenvector, and then TOPSIS algorithm is defined by estimating distance through two solutions (i.e. positive ideal solution and negative ideal solution). Simultaneously to determine and select portfolio. The aim of this paper is to determine proper model of decision making for investment studied sample is accepted superior companies from different industry of Tehran Stock Exchange during five years period which is given as numerical example.

Keywords: Portfolio, MADM, TOPSIS method, Eigenvector

1. Introduction Investment process is decision making problem under non-confidence and decision making in financial management and resource allocation had been one of the most important issues of human being. Appropriate allocation of resource and gaining access to long term and constant economical growth require suitable field of investment and existence of instruments and technique at capital market (Panahian, 2011). Ordinarily, financial market consist of individual, establishment, instrument and procedures, which collected borrows and savers in one place (Panahian, 2011; Besley et al, 1999). In capital market, there are two groups, in one of group individuals and organizations have a money and

Selection of Portfolio by using Multi Attributed Decision Making (Tehran Stock Exchange)

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wealth while the other group with lack of financial resources and market agents are mediator between them for satisfying needs and interest acquisition from investment(Panahian, 2011; Masri et al 2010)in other words, if savings of individuals where directed to production sector by correct mechanism, in addition to return for capital owners, it could be efficient as a most important factor of capital security for putting into operation of community economical plans(Ramoz, 2003). Uncertainty and fluctuation in return and risk of investment assets cause investors worry about future of their capital. During last year, many attempts were don for directing investors so that optimization and diversification of portfolio become like a tool for understanding and developing financial market and financial decision making (Fabbozi, 2007). Portfolio formation is one of the important method for financial and investment analysis. The first significance of these analysis is associated to relationship between share in portfolio and its reciprocal effect (Raei, 2007).the main and the most important motivation for investors and financial managers of establishment to select portfolio are as follow: interest maximization, risk minimization, preservation of desired cash flow, increase and develop wealth and finally valuation of effective factors on investment return (Fabbozi, 2007; Markowitz, 1952; Homsud et al, 2009;Momani, 2008). At financial issues, portfolio is one combination or one set of investment and set of financial assets which include investment tools such as share, bonds, gold, foreign exchange, asset-backed securities, bank deposits and real-state certificates which were kept by institutes or one person (Jones, 2002; Branke et al, 2009).Main purpose of portfolio selection is to search optimized method for allocating budget and capital to a large number of financial securities and making high return and profitability and satisfaction for investors and making interaction between risk and return in different financial environment (Ma et al, 2009; Calafiore,2008; Chen and Hung, 2009). The main motivation for portfolio construction is risk dispersion. Because returns on assets which are constituted a portfolio do not flow in the same direction the risk of the portfolio will be lower than that of a single asset. According to this principle, the traditional portfolio management approach is based on the rule of increasing the number of assets in a portfolio. This approach could be defined as “not to put all the egg in the same basket “(Jones, 2002; Kapusuzoglu and Karacaer, 2009). In early years, investors discuss about risk return of the portfolio, but there was not measurable terms and then they had to define portfolio risk as a quantity in 1952, the main and the most important risk management method at financial markets was presented by Markowitz as a portfolio selection theory and to criteria model (equilibrium between yield and risk) which cause mathematical analysis and become Theoretically basis of modern financial. In addition, he concluded the formula for approximation of portfolio variance which show the significance of diversified and efficient portfolio (Davidsson, 2011; Pan and Hung, 2008; Chen et al, 2011).and obtain proper ratio of share according to investors tastes so that investors could balance between return increase and their investment risk decrease (Wenguang and Fenxia, 2011; Chang et al, 2011). The process of portfolio analysis is generally dividend into three stages as follows: 1. Information concerning stocks. 2. The criteria for better and worse portfolio that is explanatory of the objective of analysis. 3. The computing procedure (as for, data and criteria), (Markowitz, 1952). Figure A, shows a set of portfolio in a two dimensional space of the expected rate of return and standard deviation. one this bullet some point are definitely preferred to others: e.g. instead of an equal risk (standard deviation) a higher expected rate of return is preferred and instead of an equal expected rate of return a lower risk is preferable. The bullet-shape curve is named minimum variance set. The upper segment of this curve is called the efficient set. With a certain level of standard deviation which has the highest expected rate of return (Raei, 2007).

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Mohammad Hasan Janani, Mohammad Ehsanifar and Sanam Bakhtiarnezhad Figure A:

In fact Markowitz has presented one theoretical framework for analyzing efficient investment basket which has the highest return at a certain level of risk or has the lowest risk at a certain level of return so that set of the efficient portfolio is referred as “efficient boundary “(Shing and Nagasawa, 1999 ). Since publication date of this model, we observed improvements and changes in investor’s perspectives about portfolio. In this model it is suggested that capital allocation was done at the main parameter of the decision making regarding to balance between risk and return (Fabbozi et al, 2007; Raei, 2007). Markowitz in his portfolio selection problem assumed that all of investors invest based on risk and return of the capital while numerous researches and articles criticize lack of attention to other investor’s preference at Markowitz Model (BenAbdelaziz, 2007; Raei, 2007; Lee et al, 2009). Many researchers were done by applying Means-Variance Model and quantity and quality analytic method (Tiryaki and Ahlatcioglu, 2009) but Markowitz Model was corrected because of three reasons: 1. The simplification of the type and amount of input data. 2. Introduction of alternative measures of risk. 3. The incorporation of additional criteria and or constraints (Anagnosttopoulos and Mamanis, 2010). After the Markowitz Model large number of models and corrected researches has presented such as: Mean-absolute deviation (Kono and Yamazaki, 1991), Mean-downside risk models (Speranza, 1993), Minimax-type models (Deng et al, 2005), Mean-variance-skewness (Lean et al, 2006;Chen et al, 2011), Mean-semi variance model (Markowitz, 1959), Value-At-Risk model (Hai and Zhong, 2009), Mean-variance-Skewness-Kurtosis (Lai et al, 2006). Basic and fundamental concept of Markowitz model are based on that, when the risk of stock portfolio is constant, we should pursue to maximize the return rate of stock portfolio and when the return rate of stock portfolio is constant ,we should pursue to minimize the risk of stock portfolio (Chen et al, 2011; Jones, 2002). Profitable investment require correct and logical decision making. Decision making is economic and social effort (senthil, 2008) and is a process that decision maker attempt to select proper and efficient alternative among all possible variants on a basis of it (Jahanshahloo et al, 2006; Hung, 2010). There are different criteria and factors which influence on portfolio selection. These factors and criteria are in conflict with each other and have uncertain information. So portfolio selection problem was recognized as a multi criteria decision making problem which had considerable growth at industry and business sectors regarding to abundant variation in recent decades (Jahanshahloo et al, 2006). A set of alternatives with independent and opposite attributes was screened, ranked, optimized and selected by MCDM. Main purpose of MCDM methods is to develop better decision making process and

Selection of Portfolio by using Multi Attributed Decision Making (Tehran Stock Exchange)

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decision makers frequently select by using MCDM methods. In financial problems external and internal attributes were considered at portfolio selection: External factors are environmental attributes which influence on corporation or industry behavior but the corporation does not have direct effect on them. Internal factors are effective attributes on corporation and are considered because of evaluation of corporation ability rate at completion issues and logical decision making of corporation. External attribute are such as technical, economical, social and political factor and internal attribute are such as profitability rate, corporation size, purpose of investors like interest security and control (Tiryaki and Ahlatcioglu, 2005) .there are many possible ways to classify MCDM methods: Elton and Stewart (2002) grouped them in three broad categories: value measurement model such as multi-attribute utility theory (MAUT) and analytical hierarchy process (AHP) .out ranking model such as Elimination and (ET) choice translating reality (ELECTRE) and preference ranking organization method for Enrichment (PROMETITEE)and at last good aspirate and reference level model such as technique of order preference by similarity to deal solution (TOPSIS) (Saremi, 2008). For the advantages of TOPSIS and eigenvector, these events are presented for portfolio selection problem in this paper. In the following, in the selection2, we review the literature on portfolio selection and other cases. In selection 3, AHP and TOPSIS model are described and finally our input data is related to information from accepted corporations at Tehran stock exchange.

2. Literature Review In the literature, there are approach and method to constructing a portfolio. There are a lot of models for portfolio optimization problem. As noted before, first model was Markowitz model in 1952 And then black model, tow factor model, sharp - lintner model, CAPM are presented (Carlo, 2009; Tiryaki and Ahlatcioglu, 2005). Amiri et al (2010) have used eigenvector-DEA-TOPSIS methodology for portfolio risk evaluation in the FOREX market and their index for considering are technology, trading psychology, trading system, capital management, technical analysis, trading tools, broker. Chang et al (2009) considered 6 financial product by using criteria such as liquidity, safety of principle, profit stability, capital growth, tax advantage, inflation amount (inflation stability) and selected the best measures with fuzzy numbers and knowledge and fuzzy AHP and ranked financial products. In 2009, complex system of portfolio was analyzed at Istanbul stock exchange. In this analysis, applying technique was fuzzy AHP and its measures were external and internal factors and investor’s purpose. This article was present by Tiryaki and Ahlatcioglo (2009). Chen and Hung (2009), in 2006 presented new model by combining two method of MADM i.e EIECTRE, TOPSIS. They considered different multi variables for expressing expert’s ideas and evaluating behavior of each share with respect to criteria. In MODM techniques which consist of goal planning, compromise planning and fuzzy ideal planning we can refer to papers by Para et al (2001) with FGM technique and Abdelaziz et al (2007) with CM technique and GM at Tunes market. Applied criteria by these three researchers had been for portfolio selection of yield, risk and liquidity. Anagnostapoules et al (2010), have formulated portfolio selection with equilibrium between risk and return and the number of security in portfolio. Class limitation and quantity are also considered. This model include Decision variables of complex integer number and multipurpose and also covered the newest evolutionary multipurpose optimization techniques named as non-dominated sortin Genetic Algorithm (NSGA-2),Pareto envelope-based selection Algorithm (PESA) and strength Pareto Evolutionary Algorithm 2 (SPEA 2). Chang and et al (2009) considered risk portfolio in optimization problem by using three different measurement of Markowitz Mean-variance, Semi-variance and average deviation and variance with skewness and they have used of Genetic Algorithm. Also Huang (2008) constructed portfolio with three different definition of risk(variance, semi-variance and probability of an adverse outcome) In addition, a hybrid intelligent algorithm employed to solve the optimization problem.

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Mohammad Hasan Janani, Mohammad Ehsanifar and Sanam Bakhtiarnezhad

Best et al (2006) considered transaction cost at portfolio optimization process and used Quadratic planning. But chen et al (2010) studied transaction cost regarding to risk and return measure and they employed Particle Swarm Optimization (PSO) Algorithm. Panahian (2011) studied weekly risk and return at Tehran Stock Exchange of ten company from each industry. he enjoy Neural Artificial Network model and compared it with Markowitz model and concluded that portfolio with Neural Artificial Network model is better and more suitable. Powers et al (2000) have employed DEA and considered several index such as one, three, five and ten years return of capital rate, EPS,P/E ratio, Beta, standard deviation of return and efficiency of 185 corporation . Aranha et al (2009) optimized traditional solution approach as different levels of risk-return and formulated Memetic three-based Genetic Algorithm model. Hai xiang (2009) start to select portfolio by utility maximization model and with different rate of Borrowing-Lending, return and risk measurement with Value-At-Risk (VAR). In 2006, portfolio selection was done on the basis of dynamic covariance matrices evaluation by cross-covariance matrices. Proposed method was compared with vector auto regression model and it concluded that Recurrent Neural Network model performed more precise and its paper was given by Chi-Ming et al (2006). Decision making is one of the other important functions of manager, individual, and organization and rarely decision making was performed just on the basis of one criterion. Decision making techniques not only are applied in financial problem but also play important role in management problem, investment plan selection, selection of the management technique and strategies, total quality control plans, information technology evaluation, development and production new products, marketing and selection of operational and computer systems. Scholar| Name Toloie & Ehsanifar,(2011) Saremi et al,(2008) Monavvarian et al,(2011 Bayazit et al.,(2007) Kengpol et al.,(2006) Lee et al, (2008) Hallikainen et al, (2009) Mahmoodzade et al.,(2007) Salehi et al., (2008) Khozein et al.,(2011)

Decision making Product Selection TQM Consultant Selection Selecting knowledge Management Strategies TQM project Evaluation of Information Technology Developing new products ERP implementation process Project Selection Criterion: Net present value, Rate of return, Benefit cost analysis, Payback period Project Selection Criterion: Net present value, Rate of return, Benefit cost analysis, Payback period Effectiveness Implementing and Lunching the Target Costing System

Toloie & Ehsanifar,(2011)

Supplier selection

Sharma et al.,(2009) Balli et al.,(2009) Dincer, (2011)

Credit Union Portfolio Management Operating system selection Analysis of Economic Activity

MADM Fuzzy AHP,TOPSIS Fuzzy TOPSIS ANP & TOPSIS ANP ANP ANP ANP TOPSIS Fuzzy AHP Fuzzy TOPSIS AHP Fuzzy Fuzzy AHP,SAW,TOPSIS,L.A. Fuzzy Goal Programming TOPSIS, Fuzzy AHP TOPSIS and WSA

3. Problem Definition and Proposed Methodology 3.1. Portfolio Selection Portfolio selection problem has been one of the important issues and the core of the field of financial management, which involves computing the proportion of the initial budge that should be allocated in the financial assets.

Selection of Portfolio by using Multi Attributed Decision Making (Tehran Stock Exchange)

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3.2. The Eigenvector – TOPSIS Methodology In order to select and rank the portfolio according to their criteria we use an MADM method called the eigenvector-TOPSIS methodology. The concrete steps of constructing this methodology or as follows: • Select suitable decision criteria and construct a hierarchical structure for the portfolio problem and consider m criteria and n decision alternatives. • Determine the weights of criteria using the eigenvector method. • Describe a group of evaluation grades for each criterion and ask experts from different financial domains to determine decision alternative using the defined evaluation with respect to each criterion. • Collect the weights of each alternative with respect to different criteria. • Rank or select alternatives by TOPSIS methods. Figure 1: Structure the eigenvector-TOPSIS methodology Introduction of suitable criteria for portfolio selection and construct a hierarchical structure for the decision problem

Construct pair wise comparison matrix between criteria and ask experts to assess weights of criteria

Determine the weights of criteria

Eigenvector technique

Calculate the overall weights, rank and prioritize companies by TOPSIS method

Extraction of data of sample companies in Tehran Stock Exchange TOPSIS method

Selection of effective criteria and construction of hierarchical structure: There are different criteria for evaluation of portfolio in many research, but in this paper, nine criteria is used: C1:Return (average return 5 year period) ,C2: Earnings Per Share (EPS), C3:Price/Earnings ratio(P/E), C4:Beta (Systematic Risk measurement), C5:Turnover rates (number of shares traded as a fraction of the number of shares outstanding), C6:Trade times, C7:Return on Assets (ROA), C8:Return on Equity, C9:Current Ratio. And given alternative for considering and ranking which are extracted for Tehran Stock Exchange includes eighteen superior companies which are as follow: A1:Electric Khodro Com., A2: Iran Argham Com., A3:Iran Transfo Com., A4:Azarab Com.,A5: Khark Petrochemical Com. ,A6: Takinkoo Com. ,A7: Tose sanaye behshahr Com., A8:Tooli pers Com. ,A9: Chadormaloo Com., A10:Khadamate Anformatic Com., A11:Jaberebnehayan Drug Com., A12:Saipa Azin Com., A13:Shahid Ghandi Com., A14:Kalsimin Com., A15:Bahman Group Com. ,A16: Golgohar Com., A17:Sahand Tire Com., A18:Mes Bahonar Com.

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Mohammad Hasan Janani, Mohammad Ehsanifar and Sanam Bakhtiarnezhad Figure 2: Hierarchical model for example

Selection of Portfolio

C1

A1

C2

A2

C3

A3

C4

A4

C5

……

C6

A15

C7

A16

C8

C9

A17

A18

3.3. Determination of the Weights of Criteria To make pair wise comparison matrix of criteria, we transfer expert’s judgments into quantitative value from 1 to 9 (Table 1).this matrix controls and conducted by answering different experts and by using of Group AHP, we have standardized pairwise comparison matrix because experts do not have agreement in answering. for example EPS criterion is slightly more important with regard to the P/E. that numerical value for slightly more important is 3. Table 1:

Scales for pairwise comparisons

Importance Level 1 3 5 7 9 2,4,6,8 Reciprocals

Description Equally important Slightly more important Strongly more important Very Strongly more important Extremely more important Intermediate values To reflect dominates of the second alternative over the first

Pair wise comparison matrix (A) is defined as follows, with using of this matrix and eigenvector, the weight vector W can be determined.  a11 a12 … a1m  a a22 … a2 m   (1) A = ( aij ) =  21 m× m    …      am1 am 2 … amm  In matrix A, aij is quantitative judgment between criteria and aii = 1 ,and aij = 1 a ji i , j = 1,..., m ,

3.3.1. The Eigenvector Method Eigenvector method, first introduced by Saaty in 1980,and has been used to solve MCDM problems, Eigenvector is one method for determining and generating weights, this technique results in final weights that are an average of all possible ways of comparing the alternatives. We calculate criteria weights (W) by using Eq. (2), and we continue computation of (W) until the both (W) to be equal and result to convergence (Amiri et al, 2010).

Selection of Portfolio by using Multi Attributed Decision Making (Tehran Stock Exchange)

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Ak e k →∞ eT Ak e After calculating the weight of criteria, ( λmax ) investigate by using Eq.(3) W = lim

(2)

AW = λmaxW (3) Next we investigate the consistency of matrix A, through calculating the in consistency of this matrix in Eq.(4) C .I C .R = R.I (4) Where C.I shows inconsistency index and R.I shows inconsistency index of random matrix ,( n) is rank of matrix ,and C.I calculate through Eq.(5) Table 2: n R.I

Random inconsistency index for pair wise comparison matrix 1 0.0

C .I =

2 0.0

3 0.58

4 0.90

5 1.12

6 1.24

7 1.32

8 1.41

9 1.45

10 1.49

11 1.51

12 1.48

13 1.56

14 1.57

λmax − n

(5) n −1 If C.R ≤ 0.1 , the pair wise comparison matrix have an acceptable consistency. Otherwise, it need to be revised. 3.4. TOPSIS Method TOPSIS is a multi-criteria method to identify solutions from a finite set of alternative based on simultaneous minimization of distance from an ideal point and maximization of distance from nadir point and one can incorporate relative weights of criterion importance (Olson, 2004) and at the first presented by Hwang and Yoon. In the recent years, TOPSIS has been desired technique for solving MCDM problems and it is effective method for determining total ranking order of decision making. This is because of two reasons which are as follow: Its concept is reasonable and easy to understand and requires less computational works in comparison with MCDM method such as AHP and therefore it can be applied easy. TOPSIS is a based on the concept that the optimal alternative should have shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS) (Hwang and Yoon, 1981; Shahnagi and Yazdian, 2008; Mahdavi et al, 2008). This method considers three types of criteria: a) qualitative benefit criteria, b) quantitative benefit criteria, c) cost criteria (Dodangeh, 2009). The input for TOPSIS is N candidate objects and M evaluation factors (criteria), a matrix of evaluating factor forms:  x11 x12 … x1m  x x22 … x2 m   (6) X =  21    …      xn1 xn 2 … xnm  The procedure of TOPSIS model or as follows :(Chuan et al, 2009; Huang, 2010; Jahanshahloo, 2006). Step1: to construct the normalized data matrix. The normalized value rij ,is calculated as :

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Mohammad Hasan Janani, Mohammad Ehsanifar and Sanam Bakhtiarnezhad rij =

xij

(7)

n

∑x

2 ij

i =1

So we can get the normalized decision matrix R :  r11 r12 … r1m  r r22 … r2 m  21   (8) R=   …      rn1 rn 2 … rnm  Step2: to calculate the weighted normalized matrix. We multiply matrix R and Unit matrix W*. The weighted normalized value is called the weighted and normalized decision matrix Vij . Step3: determine the positive ideal point and the negative ideal point.

{( max v i ∈ I ) , ( min v = {v ,..., v } = {( min v i ∈ I ) , ( max v

A+ = {v1+ ,..., vn+ } = A−

− 1

ij

− n

ij

ij

ij

} i ∈ J )} i∈J )

(9)

Where I is benefit criteria, and J is cost criteria. Step4: calculate the distance di+ and distance di− . The separation of each alternative from the ideal point is given as: 1

2  n + 2 d = ∑ ( vij − v j )  , i = 1,..., m  j =1  Similarly, the separation from the negative ideal point is given as:

+ i

(10)

1

2  n 2 d = ∑ ( vij − v −j )  , i = 1,..., m (11) j = 1   Step5: calculate the relative closeness to the ideal point. The Eq. (12) is defined as: d− CL = − i + , CL ∈ [ 0,1] (12) di + di Step6: finally we can rank alternatives using this index, in decreasing order. Higher the closeness means the better the rank. Fig. (1) Shows the structure of the eigenvector-TOPSIS methodology.

− i

4. Numerical Example in the Tehran Stock Exchange Standardized pair wise comparison matrix for nine evaluation criteria by Group AHP, is as follow: 2.825 2.324 3.011 3.151 3.294 3.201 2.587 3.290   1  0.353 1 2.643 3.180 3.245 3.209 2.583 2.419 3.180    0.430 0.378 1 3.230 2.126 3.282 2.428 2.993 3.072    1 2.075 3.277 2.205 2.404 2.485  0.332 0.314 0.309 A = 0.317 0.308 0.470 0.481 (12) 1 3.259 2.509 3.168 3.242    1 2.621 2.684 3.194   0.303 0.311 0.304 0.305 0.306  0.312 0.387 0.411 0.453 0.398 0.381 1 2.334 3.137    1 3.254   0.386 0.413 0.334 0.415 0.315 0.419 0.428  0.303 0.314 0.325 0.402 0.308 0.313 0.318 0.307 1  

Selection of Portfolio by using Multi Attributed Decision Making (Tehran Stock Exchange)

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Next, weights of criteria is calculated by eigenvector method Eq.(2) as: W1 = ( 0.1979 − 0.1749 − 0.1518 − 0.1154 − 0.1183 − 0.0860 − 0.0706 − 0.0558 − 0.0287) W2 = ( 0.2353 − 0.1949 − 0.1532 − 0.1097 − 0.1020 − 0.0681− 0.0576 − 0.0468 − 0.0320)  (13)

W7 = ( 0.2399 − 0.1918 − 0.1491− 0.1053 − 0.0993 − 0.0690 − 0.0603 − 0.0508 − 0.0341) W8 = ( 0.2399 − 0.1917 − 0.1491− 0.1053 − 0.0993 − 0.0690 − 0.0603 − 0.0508 − 0.0341) W9 = ( 0.2399 − 0.1917 − 0.1491− 0.1053 − 0.0993 − 0.0690 − 0.0603 − 0.0508 − 0.0341)

Then by using Eq.(3) maximum eigenvalue is λmax = 10.1094 .and by using Eq.(5) C.I = 0.1386 that is inconsistency index for matrix and the inconsistency ratio of matrix is C.R = 0.095 .due to that C.R ≤ 0.1 ,we conclude that the pair wise comparison marix is consistent. In next step, by using TOPSIS method and extraction of the data with respect to 18 accepted companies in Tehran Stock Exchange, normalized decision matrix is as follows: Table 3:

Normalized decision matrix C1 0.0122 0.316 0.4519 0.0635 0.214 0.106 0.0354 0.0104 0.6145 0.0560 0.0239 0.0082 0.302 0.0617 0.1093 0.5872 0.0010 0.0149

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18

C2 0.1257 0.1346 0.2660 0.0064 0.4638 0.3119 0.0530 0.1711 0.1553 0.4124 0.2039 0.0803 0.0488 0.4296 0.0724 0.3245 0.0851 0.0585

C3 0.0582 0.1407 0.1092 0.1323 0.0984 0.0799 0.1054 0.0867 0.1895 1.0026 0.1152 0.0580 0.0815 0.8840 0.0576 0.1987 0.0699 0.1191

C4 0.2194 0.2492 0.1878 0.2071 0.4212 0.2861 0.1439 0.1808 0.3440 0.2088 0.2475 0.2159 0.2282 0.1913 0.1948 0.1158 0.1895 0.2299

C5 0.1913 0.1642 0.1112 -0.0097 0.5285 0.2588 0.1027 0.0832 0.3641 0.3213 0.2770 0.1295 0.0524 0.2603 0.1651 0.3076 0.2043 0.0955

C6 0.2581 0.1474 0.2298 -0.0520 0.3245 0.2530 0.0929 0.2451 0.1989 0.2482 0.2378 0.1629 0.0941 0.5490 0.1387 0.2866 0.1634 0.1159

C7 0.2997 -0.0522 0.0046 0.2338 0.2770 -0.0181 0.1317 0.3338 0.3179 0.1067 0.0908 0.0023 0.1930 0.3678 0.2747 0.2407 0.1975 0.5154

C8 0.0672 0.3933 0.0259 0.0169 0.1625 0.222 0.0119 0.0624 0.1657 0.0603 0.1873 0.0351 0.0445 0.7609 0.0326 0.3265 0.0599 0.0199

C9 0.1857 0.3948 0.1183 -0.66 0.1615 0.0457 -0.0423 0.0492 0.273 0.1451 0.1079 0.0734 0.1295 0.3585 -0.0052 0.2315 -0.1123 0.0570

After multiplication matrix (R) and matrix (W*), we can calculate matrix (V) as follows: Table 4:

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13

Weighted and normalized decision matrix C1 0.0029 0.0075 0.1084 0.0152 0.0513 0.0025 0.0084 0.0024 0.1474 0.0134 0.0057 0.0019 0.0072

C2 0.0240 0.0258 0.0509 0.0012 0.0889 0.0597 0.0101 0.0327 0.0297 0.0790 0.039 0.0153 0.0093

C3 0.0086 0.0209 0.0162 0.0197 0.0146 0.0119 0.0157 0.0129 0.0282 0.1422 0.0171 0.0086 0.0121

C4 0.0231 0.0262 0.0197 0.0218 0.0443 0.0301 0.0151 0.0190 0.0362 0.0219 0.026 0.0227 0.024

C5 0.0189 0.0163 0.0110 -0.0009 0.0524 0.0256 0.0101 0.0082 0.0361 0.0319 0.0275 0.0128 0.0052

C6 0.0178 0.0101 0.0158 -0.0035 0.0223 0.0174 0.0064 0.0169 0.0137 0.1712 0.0164 0.0112 0.0064

C7 0.018 -0.0031 0.0003 0.0140 0.0167 -0.001 0.0079 0.0201 0.0191 0.0064 0.0054 0.0001 0.0116

C8 0.0034 0.0199 0.0013 0.0008 0.0082 0.0112 0.0006 0.0031 0.0084 0.003 0.0095 0.0017 0.0022

C9 0.0063 0.0134 0.004 -0.0225 0.0055 0.0015 -0.0014 0.0016 0.0093 0.0049 0.0036 0.0025 0.0044

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Mohammad Hasan Janani, Mohammad Ehsanifar and Sanam Bakhtiarnezhad

Table 4:

Weighted and normalized decision matrix - continued 0.148 0.0262 0.1408 0.0002 0.0035

A14 A15 A16 A17 A18

0.0823 0.0138 0.0622 0.0163 0.0112

0.1318 0.0085 0.0296 0.0104 0.0177

0.0201 0.0205 0.0121 0.0199 0.0242

0.0258 0.0163 0.0305 0.0202 0.0094

0.0378 0.0095 0.0197 0.0112 0.0079

0.0221 0.0165 0.0145 0.0119 0.031

0.0386 0.0016 0.0165 0.003 0.001

0.0122 -0.0001 0.0078 -0.0038 0.0019

The positive ideal point and the negative ideal point for each criterion are: Table 5:

V j+ Table 6:

V j− Table 7:

Positive ideal point C1 Max

C2 Min

C3 Max

C4 Max

C5 Max

C6 Min

C7 Max

C8 Max

C9 Max

0.1474

0.0012

0.1422

0.0443

0.0524

-0.0035

0.0310

0.0386

0.0134

Negative ideal point C1 Min

C2 Max

C3 Min

C4 Min

C5 Min

C6 Max

C7 Min

C8 Min

C9 Min

0.0002

0.0889

0.0085

0.0121

-0.0009

0.1712

-0.0031

0.0006

-0.0225

Computations of d i+ , d i− , d i , CLi and priority ranking of companies

Companies A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18

d i+

d i−

d i+ + d i−

CLi

0.206719 0.195497 0.158053 0.199414 0.187256 0.211351 0.199894 0.207999 0.124522 0.239574 0.200266 0.209532 0.202898 0.165371 0.191112 0.138888 0.207469 0.200590

0.17189 0.179364 0.195635 0.197353 0.173006 0.161996 0.184888 0.168067 0.232123 0.142009 0.168492 0.178763 0.186220 0.193171 0.183619 0.215279 0.178786 0.186575

0.378609 0.374861 0.353688 0.396767 0.360262 0.373347 0.384782 0.376066 0.356645 0.381583 0.368758 0.388295 0.389118 0.358542 0.374731 0.354167 0.386255 0.387165

0.4540 0.4784 0.5531 0.4974 0.4802 0.4339 0.4805 0.4469 0.6508 0.3721 0.4569 0.4603 0.4785 0.5387 0.4900 0.6078 0.4628 0.4819

Priority Ranking 15 11 3 5 9 17 8 16 1 18 14 13 10 4 6 2 12 7

After considering and analyzing CL of each of the 18 superior companies, we can be ranked and select the suitable company for construction of portfolio. This is real example related to Tehran Stock Exchange.

5. Conclusion In this study we have introduced a new eigenvector –TOPSIS technique to consider the key criteria of construction of portfolio and to rank the companies in Tehran Stock Exchange. We use and consider experiences and estimations of experts (DM) in the portfolio selection process, eigenvector method

Selection of Portfolio by using Multi Attributed Decision Making (Tehran Stock Exchange)

26

determine the weights of criteria and the flexible TOPSIS method rank the companies under different criteria. The proposed method can be easily used to analyze other Stock Exchange and criteria, other managerial and financial decision making, operating issues, costing systems (Target Costing, ABC & TDABC Costing).

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