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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.4 pp.534-541
DOI : https://doi.org/10.7232/iems.2017.16.4.534

# Portfolio Selection with Fuzzy Synthetic Evaluation and Genetic Algorithm based on Performance Perspectives

Mohsen Nazem Bokaei, Hamid Nayebpoor*
Department of Marketing Management, Shahed University, Tehran, Iran
Corresponding Author, hnayebpur@yahoo.com
20170918 20170926 20170928

## ABSTRACT

The purpose of this paper is to present a new technique to the portfolio selection using genetic algorithm and fuzzy synthetic evaluation. Portfolio selection is a multi-objective/criteria decision-making problem in financial management. The proposed approach solves the problem in two stages. In the first stage, by using genetic algorithm and fuzzy synthetic evaluation, weight of criteria will be calculated. Euclidean distance between the computed overall performance evaluation and the surveyed overall performance evaluation is used in order to determining weight of criteria. In second stage, by using genetic algorithm and fuzzy synthetic evaluation, Portfolios will be prioritized. A multi objective genetic algorithm is used to determine return and risk in the efficient frontier. A decision-making approach is based fuzzy synthetic evaluation to select the best portfolio from among the solutions obtained by multi objective genetic algorithm. The main advantage of proposed approach is help to investors to find a portfolio which have Best performance, portfolio selection doesn’t rely to expert knowledge. The value of the paper is due to it use new approach in order to determine weight of criteria and portfolio selection. It surveys firms’ performance in stock market and based on, weight of criteria will be determined and portfolio will be prioritized.

## 1.INTRODUCTION

Portfolio selection and optimization are two of the most challenging problems in the field of stock market. Selecting the weight of assets is importance for investors to invest in a stock market (Sefiane and Benbouziane, 2012: 143). Harry Markowitz (1959) proposed a quantitative model for portfolio selection. It explains relationship between risk and return in portfolio. This model minimizes an objective function risk for portfolio at a given level of return and maximizes an objective function return for portfolio at a given level of risk (Fernández and Gómez, 2007: 1178).

Portfolio selection is a multidimensional model that can be framed using a multi criteria decision making (MCDM) problem. One of the most important problem in MCDM is determining of weight of criteria. The weight values express the relative importance of criteria for decision making process. The several techniques are used for determining weight of criteria. The direct determination method (DDM), the comparative matrix method (CMM), the analytical hierarchy process (AHP), the circular comparison method (CCM), the fuzzy interval method (FIM), and the importance ordering method (IOM) (Saaty, 1990). Decision makers may select different methods for determining weighting coefficients according to their know- ledge and experience (Feng and Xu, 1999: 5).

Performance of stock market for each industry differs with other industries. On the other hand, experts (in undeveloped countries) usually don’t consider this difference among industries and weight of criteria is considered as same as among different industries for portfolio selection in stock market. This difference among industries in stock market, require advanced analytical techniques that would analyze the issue portfolio section for investors.

In this paper we will use fuzzy synthetic evaluation (FSE) and genetic algorithm (GA) for determining importance of criteria in order to portfolio selection in stock market and finally we will prioritize portfolios with FSE.

### 1.1.Related literature

With unprecedented growth of computing power in the past two decades, researchers have used new techniques of computation and optimization such as GA. It can be considered as an optimization when other techniques such as gradient descent or direct analytical methods are less effective (Cui et al., 2015: 32).

Many researchers use GA in order to portfolio selection and optimization in various stock market and it is showed in their studies. For example, Loraschi and Tettamanzi (1996) found weights of assets for portfolio selection by GA. Lin and Gen (2007), used GA in order to maximizing the return and minimizing the investing risk. reliability and efficiency of the genetic algorithm in portfolio selection was proven by their findings. Solimanpur et al. (2015) used GA to identify a set of portfolios in the efficient frontier. Punniyamoorthy and Thoppan (2012) proposed a hybrid model using advanced data mining techniques for the detection of Stock Price Manipulation. They used GA based Artificial Neural Network to classify stocks witnessing activities. Huang et al. (2011) used a GA to identify a linear stock selection model. Huang (2012) proposed a support vector machines (SVM) method with a GA, and showed that SVM and GA were effective methods for stock selection.

In considering the above-described findings, most of studies in this field by GA only have focused on methods in order to portfolio selection or techniques in order to GA optimization in portfolio selection and most of them have not been calculated weight of effective criteria on portfolio selection in their studies by GA or other artificial methods. In most of studies on portfolio selection, weight of criteria usually is determined by experts and firm’ performance in stock market have not been considered.

Thus, we need method that calculate importance of criteria in order to portfolio selection with advanced analytical tools. Therefore, we propose a FSE and GA for determining importance of criteria in order to portfolio selection in stock market.

## 2.FORMALIZATION OF PORTFOLIO SELECTION

### 2.1.Markowitz’ Model

Markowitz’ model uses the mean and variance of historical returns to measure the expected return and risk of a portfolio in stock market. It can be proposed such as a multi objective problem:

$R p = ∑ i = 1 i = N ( r i × x i )$
(1)

$σ = ∑ i = 1 i = N ∑ j = 1 j = N ( x i × x j × c o v i j )$
(2)

$∑ i = 1 i = N ( x i = 1 ) 0 ≤ x i ≤ 1 i = 0.1 ⋯ N$
(3)

$∑ j = 1 j = N ( x j = 1 ) 0 ≤ x j ≤ 1 j = 0.1 ⋯ N$
(4)

Rp is Return of portfolio and is defined as the weighted average of the returns of stocks being in the portfolio, σ is Risk of portfolio and is defined as a tolerance of stock’s return from the mean, i is index for stocks, N total number of stocks, ri return of stock i, xi percentage of stock i in portfolio, rj return of stock j and xj percentage of stock j in portfolio.

Process of portfolio selection is included two steps

• (1) Using of FSE and GA for determining weight of criteria.

• (2) Using of FSE for portfolio selection

### 2.1.FSE and GA for Determining Weight of Criteria

#### 2.1.1.FSE System

To illustrate multidimensional model of portfolio selection, we use the model proposed by Edirisinghe and Zhang (2008) and it was based on performance perspective. This model can analyze a three-layer system (hierarchical process) for portfolio selection (Figure 1). This criteria(O0) considers the overall portfolio selection by six major criteria: the combination criteria for profitability (O1), the combination criteria for asset utilization (O2), the combination criteria for liquidity (O3), the combination criteria for leverage (O4), the combination criteria for valuation (O5) and the combination criteria for growth perspectives (O6). This hierarchical process decomposes from the final objective (General criteria) to more specific objectives (Macro layer criteria). Every Macro layers are determined by sets of decision questions u (Micro layer). For example, the criteria of (O1) is determined by survey 4 questions (u11, u12, u13 , u14). Finally, each question (micro layer) can be measured by a set of evaluation scores representing by typical 5-point Likert scale “very low,” “low,” “average,” “high,” and “Very high,” or typical 9- point Likert scale. According to Edirisinghe and Zhang (2008), we use range criteria as profitability, asset utilization, liquidity, leverage, valuation, and growth perspectives. each of these criteria have several sub criteria (Figure 2).

We consider the problem of portfolio selection as a problem of FSE. Fuzzy theory (Zadeh, 1965) was developed to deal with uncertainty problems. It has been widely applied to many fields such as decision-making and evaluation processes in imprecise situations or uncertainty of information (Dahiya et al., 2007: 940). It better reflects the uncertainty among persons regarding the same responses (Hao et al., 2015: 233). Success of this theory has been proven by relevant literature. FSE uses fuzzy mathematics to transform unclear information and has various attributes concerning evaluation related criteria (Kuo and Chen, 2006: 614).

#### 2.1.2.FSE Process

The FSE process for determining importance of criteria is divided into six main steps.

• Step 1: Determination an evaluation criteria set U

The first step is generally determining of criteria for study. In this study. we select model proposed by Edirisinghe and Zhang (2008)$U = { n i } , i = 1 , 2 , ⋯ , m$.

• Step 2: Evaluating of criteria grade set V

Evaluation grades are of a typical 5-point Likert scale as “very low,” “low,” “average,” “high,” and “Very high,” $V = { v j } , j = 1 , 2 , ⋯ , n$

• Step 3: Evaluating to obtain membership functions matrix $R ˜$

If N person make appraisals of criteria ui in U separately, the xij denotes the number of person who determine ui as vj , the sum of each list of numbers equals N, rij = xij/N, then we can derive the membership function ($R ˜$). As a result, we obtain the appraisal matrix

$R ˜ = r i j = [ r 11 ⋯ r 1 n r 21 ⋯ r 2 n ⋮ ⋱ ⋮ r m 1 ⋯ r n m ]$

• Step 4: Determination weight of criteria $W ˜$

We use GA for determining weights of effective criteria $w ˜$ on portfolio selection

• Step 5: Using the fuzzy operator to obtain vector $W ˜$ We use the fuzzy method to obtain vector $W ˜$ . It is the critical factor in FSE, which would affect the final evaluation results.

The overall evaluation score is obtained by the following equation:

$B ˜ = W ˜ ∘ R ˜$
(5)

where “◦” is the fuzzy composition operator. According to Feng and Xu (1999), four fuzzy composite operators are widely used.

Type 1 of operator: Considers only those important criteria.

According to type 1 of operator, only those criteria with the largest value determine the evaluation result. It operator is suitable to the evaluation in which single items are emphasized.

$M ( Λ , V ) , b j = m a x { m i n ( a k r k j ) } < k < m$
(6)

Type 2 of operator: Emphasizes important criteria (i).

Type 2 of operator provides a finer evaluation than Type 1 of operator, because some non-major criteria are included in the evaluation through the induced multiplication operation.

$M ( ⋅ , V ) , b j = m a x { a k × r k j } < k < m$
(7)

Type 3 of operator: Emphasizing important criteria (ii).

Type 3 of operator considers non-major criteria in the evaluation similar to Type 2 of operator with a slightly different way and is suitable to the evaluation in which the result obtained from type 2 is indistinguishable

$M ( Λ , ⊕ ) , b j = ⊕ ( a k Λ r k j ) = ∑ k = 1 m m i n ( a k ⋅ r k j ) . 1 < k < m$
(8)

Type 4 of operator: Considers every single criterion overall.

Type 4 of operator requires the involves all criteria that are based on the weighting coefficients. and it is suitable to the evaluation in which all values must be accommodated.

$M ( ⋅ , + ) , b j = ∑ k = 1 M ( a k × r k j ) . 1 < k < m$
(9)

• Step 6: Determination best operator

For determining best operator, with compare compute overall performance with survey overall performance for each operator, we select operator that minimize fitness score as following:

Fitness score = compute overall performance evaluation - survey overall performance evaluation.

#### 2.1.3.GA for Determining Importance of Criteria

GA is an optimization technique based on the principles of Genetics that first proposed by John Holland in the 1960s and developed by Holland, his students and colleagues in the 1960s and the 1970s (Mitchell, 1998). GA process takes four steps:

• 1) generating Initial population is the starting point for GA.

• 2) fitness value evaluation, it measures the quality of a solution and provide best solution for optimum detection. The evolution of the population is based on the principle of survival, we used Eq. (10) for determining weight of criteria in each layer (micro layer or macro layer) for segments of customers. S denote a set of parameters, including the values of weight of criteria, oi denote to compute overall evaluation and d denote to survey overall evaluation for determining weight of criteria, we use following format:

$Min e ( s ) = ∑ ( d × w i − o i ) 2 ∑ i = 1 i = m ( w i = 1 ) 0 ≤ w i ≤ 1 i = 0.1 ⋯ m$
(10)

e(s) is the Euclidean distance between the computed overall performance evaluation and the surveyed overall performance evaluation.

• 3) Termination test, GA will be stopped if termination condition is satisfied, else it go to next steps.

• 4) Generating new population with selection, crossover and mutation operators. We with selection operator ensure that the better members of the current population have a high probability for selection in order to create children for next generation. The crossover operator aims to generate new individuals that maintain some characteristics of their parents and at the same time extend the search space. The mutation operator maintains the diversity of the population and prevents premature convergence.

### 2.2.GA and FSE for Portfolio Selection

Markowitz’ model is considered as multi objective decision making (MODM). This model is included two function, return and risk. We will minimize risk function and will maximize return function. Therefore, we use GA based on MODM for model optimization and determining possible portfolios on efficient frontier. We use FSE for sorting possible portfolios and selecting best portfolio according to type of operators.

## 3.DATA COLLECTION AND MEASURES

### 3.1.Data Collection for Determining Weight of Criteria

To evaluate the performance perspective of the proposed GA and FSE for determining weight of criteria, we designed questionnaire according to Edirisinghe and Zhang (2008). A questionnaire was designed to determine weight of effective criteria portfolio selection. The questionnaire was divided into three main parts.

Part one consisted of a series of respondents’ demographic characteristics such as sex, age and education. Part two began with the general information about to dimension performance perspectives on stock market and Part three dealt with the respondents’ assessment of the performance perspectives. Questionnaires were sent to target respondents. The study population were consisted of persons with more than one year of experience on Tehran stock market. We obtained 210 questionnaires to be analyzed. Table 1 shows the demographic information on the respondents.

### 3.2.Measures

In this section, we report the results that obtained from the assessments of reliability and validity for the measurement instrument. Our study adapted the performance perspectives scale Edirisinghe and Zhang (2008) to measure weight of criteria. All scales were measured by using a 5-point Likert scale. For the measurement of survey scale reliability, we used Cronbach’s alpha value. If Cronbach reliability coefficient was higher than 0.7, indicating adequate internal consistency (Nunnally, 1978). Table 2 lists the measurements with their reliability indicators. Cronbach reliability coefficient for all constructs had a value above 0.7, indicating adequate internal consistency of the measures.

To make the extracted factors more interpretable, we used the Warimax with Kaiser Normalization method for determining the number of factors. According to Table 3, the exploratory factor results show that six factors with an eigenvalue greater than 1 emerge. These six factors explain 80.891% of the total variance in the latent variable. Thus, The results show that the factor structure fits with our research model.

### 3.3.Empirical Specifications

We used MATLAB R2016a with Optimization Toolbox for determining weight of criteria. We set the initial parameter specifications as shown in Table 4 with the parameter that Hao et al. (2015) have used. The other parameters were the default values in MATLAB toolbox.

### 3.4.Data Collection for Determining Possible Portfolios

We selected 38 firms of Tehran stock market among chemical industry as petrochemicals, refineries and cleaning. 8 firms were eliminated of our study due to the shortage of data. So, we analyzed data of 30firms for determining possible portfolios. We surveyed data in period 2010-2014.

## 4.RESULTS AND DISCUSSION

### 4.1.Results for Determining of Weight of Criteria

We analyzed the data on performance perspective with FSE and GA. Table 5 shows weight of criteria (in micro layer) of performance perspective.

For determining the best operator, we used Table 6. In according to Table 6, operator M(Λ, ) had lowest fitness score among all operators. So we selected M(Λ, ) operator for determining weight of criteria of performance perspective and we determined Profitability criteria (weight = 0.239), Asset Utilization criteria (weight = 0.032), Liquidity criteria (weight = 0.170), Leverage condition criteria (weight = 0),Valuation perspective criteria (weight = 0.320), Growth potential criteria (weight = 0.240) respectively.

### 4.2.Result for Portfolio Selection

Generated efficient frontier by GA is showed in Figure 3. we obtained 12 portfolios with multi objective GA.

According to Figure 3, in least risk, return is obtained 0.227. In fact, investors with minimum risk obtain it. We normalized information on 12 portfolios and it is showed in Table 7.

The following are the detailed evaluation procedures:

• Step 1: Evaluate the 12 portfolio in terms of the micro layer Ui (i = 1, 2 … 17).

• 1) Choose a membership grade calculation method. it is proposed in Table 7.

• 2) Using of GA and FSE for determining weight of criteria. it is proposed in Table 5.

• 3) Using the fuzzy operators for determining overall evaluation $B ˜ = W ˜ ∘ R ˜$.

Results are proposed in Table 8.

• Step 2: Evaluate the 12 portfolio in terms of the macro layer Oi (i = 1, 2 … 6).

• 1) Choose membership grade calculation method. It is proposed in Table 8.

• 2) Using of GA and FSE for determining weight of criteria (its proposed in Table 6).

• 3) Using the fuzzy operators for determining overall evaluation, B0 is as follows:

$B ˜ = W ˜ ∘ R ˜$

Now, we prioritize portfolios according to overall evaluation. Results of selecting composition operators for prioritizing portfolio are showed Table 9.

According to Table 9, same results approximately are obtained with M(·, +) and M(·, V) operators. But different results are obtained with other operators. In fact, selecting type of operators depend on operators defining of experts’ perspectives. They may consider only important criteria, so M(Λ, V) operator is selected. On the other hand, they may consider every single criteria overall, M(·, +) composition operator that uses a weight average to all the factors. Each factor ui is considered in the model. so this operator is selected.

## 4.CONCLUSION

We use FSE and GA for portfolio selection of chemical industries in Tehran stock market. The proposed process is composed of two step. In step1, determining weight of criteria with GA and FSE and in step 2, selecting optimum portfolio with FSE. We consider performance perspectives for our study. In perspectives are included 6 macro criteria (Figure 1) and 17 micro criteria (Figure 2). results show that Valuation perspective criteria, Growth potential, Profitability criteria, Liquidity criteria, Asset Utilization and Leverage condition have highest important, respectively of investors perspectives. This study had three main limitations

First limitation was selection of artificial methods. Several methods existed in order for solving fitness function Eq. (10) as GA, NN, colony optimization and other methods. We used GA and FSE to solve Eq. (10). It is better to be considered comparison among different methods in order for determining the best method in future research. Second, we considered Tehran stock market for our study. It is better to be considered comparison among different stocks markets and sorting them in future researches. Last limitation was selection of fitness function. Many fitness functions have been proposed in other researches such as Euclidean distance, Manhattan distance, Cosine distance and Pearson correlation distance metric, but unfortunately there is no widely accepted rule for selecting fitness functions. We select Euclidean distance in our study. It is better to be considered comparison among different fitness functions.

## Figure

Conceptual model of performance perspective.

Three-layer evaluation system.

Efficient frontier in chemical industries (Tehran stock market).

## Table

Characterization of the final sample

Measurements with their reliability indicators

Results of exploratory factor analysis

Parameter specifications

Results of optimized criteria weights

Results of selecting composition operators, with weights of macro layer

Normalizing information on 12 portfolio based on 17criteria (micro layer)

Using fuzzy operators for evaluation microlayer

Sorting of portfolio with FSE

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