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International Research Journal of Applied and Basic Sciences © 2014 Available online at www.irjabs.com ISSN 2251-838X / Vol, 8 (7): 780-788 Science Explorer Publications

Optimization of stock portfolio based on ant Colony & grey theory

Dr.Ali Najafi Moghadam1, Dr.fraydoon Rahnama roodposhti2, Mahvash Farrokhi3

1. Professor and Faculty member of Islamic Azad University Roudehen ,Iran

2. Professor and Faculty member of Islamic Azad University Science and Research Branch of Tehran , Iran. 3. Master of Bussines Administration on Financial Management , Faculty of Management, Buien Zahra Branch,

Islamic Azad University, Iran.

Corresponding Author email: [email protected]

ABSTRACT: The objective of the portfolio management is portfolio selection, portfolio investors guide to achieve maximum efficiency.In this study, in order to select the optimal portfolio ant algorithm and Grey theory has been used and a comparison between them has been.Introducing a model for the assessment of portfolio choice for investors who are able to apply that model to select the right portfolio portfolio, our goals in this research.To this end, a review of the literature and research done and a set of indicators and criteria according to the purpose of the survey was to gather. Among the listed companies in Tehran Stock Exchange with more than 105 companies with the highest profit return on assets during the year were as sample 2005-2011 entered statistical analysis of the data. the analysis relationship grey firm prioritized and selected the model (grey& ant) Markowitz model using statistical tests were compared. Below, in order to select the optimal portfolio of ant colony algorithm, a model was derived that was very functional and minimal risk and maximum efficiency are characteristics. The findings suggest that it is associated with a comparable model of grey model and ant compared to the Markowitz model has less error in selecting the optimal investment portfolio. The most important recommendation for future research to compare with other models of ant algorithm and grey. Keywords: ant colony, Markowitz model,Optimization of stock portfolio, Risk, Return, grey theory

INTRODUCTION

Individual investments are the main actions that any person can carry out in their life and the purpose of the asset management is the determination of these variables in a way that will minimize risk and maximize returns. The optimal asset selection usually occurs in operations between risk and return. The identification of the efficient edge of the portfolios allows the investors to achieve the highest expected return from their investments based on the degree of the expected risk. The improved stock analysis methods, especially in markets with very high numbers of shares, lead to the new methods in addition to the previous methods in order to find answers to maximize profits in the financial markets. So far, many models are presented to solve the problem of the optimal assets and make the efficient edge.

Although theoretically these models are solved by using the mathematical programming methods, there are many problems in regard of the assets selection in practice. Due to the present problems in solving the non-linear assets programming model, the appropriate portfolio selection using the linear mathematical programming models is more suitable to solve the assets selection. The present study is a combination of grey theory and ant algorithm as these had less attention in the previous studies and a comprehensive model is presented for the optimal portfolio selection. This study aims to present a model for the optimal portfolio selection for investors as a decision support system (DSS) to help investors to select the optimal portfolio. The study objective is summarized as following:

―Portfolio establishment, while maximizing the returns and minimizing the investment risk.‖ Research Literature And Background

In 1952, Harry Markowitz stated the investment theory called Modern Portfolio Theory (MPT) under the uncertainty based on risk and average income of shares. This method is based on the argument that may probably have loss risk of capital or profit in one series or set of stock market shares, therefore the professional investors should not invest all their capital on one type of asset, rather they should invest in a series of stocks or assets and this series are known as stock share or portfolio. Although some investors emphasize to buy shares of an industry, while others may focus on different industry stocks (Mehregan, 2004).

Intl. Res. J. Appl. Basic. Sci. Vol., 8 (7), 780-788, 2014

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Today, we face a wide range of stocks selection methods and models, e.g. Technical, Fundamental and Modern Portfolio Theory, which have developed specific analysis assumptions (Eslami Bidgoli, 1995).

The stock market behaviors are nonlinear similar to many natural phenomena. The linear models are incapable of correct diagnosis of nonlinear and linear parts and can only recognize the good behavior linearity. Thus they need the non-linear models to predict the future behavior of the effective equity stake and make appropriate decisions. In order to clarify the issue broadly, we define and explain the variables: Investment Management Investment management includes two main topics of ―portfolio analysis‖ and ―portfolio management‖. The analysis of the securities relies on the estimation of the each investment benefits, while the portfolio management includes the investments components analysis and investments retention management. In recent decade, the discussion of choosing the stock for investment (stock portfolio analysis) has moved to the portfolio management (Strong, 2000). The investment process includes a series of activities in which they eventually lead to purchase of the real properties or securities that can have risk and return. Return

Return is composed of two parts: dividends and profits and losses of capital. The most important component of the return is the interest in the form of the periodic cash flows and can be in the form of the interest or dividends. The capital gains or losses are the important component dedicated to the return of the common stock for on long term bonds and it is consistent for the other securities' fixed income and caused by increase (or decrease) of the asset price. The sum of these two components makes the total returns of the securities (P. Jones, 2012). Risk

Risk is called hazard in Farsi that has the potential to cause injury or damage. The encyclopedia defines venture capital investment as the calculated potential loss of the investment. The subject of risk is a combination of hazard and return that can be examined by the different approaches. The different risk sources are defined and interpreted in different ways:

Harry Markowitz proposed the quantitative definitions on Numerical Index of Risk for the first time. He defined risk as the standard deviation of a multi period variable. Galitz defines risk as every fluctuation on every income. The mentioned definition clarifies that the possible changes in a particular index, whether positive or negative poses us with risks. Therefore it is possible that these changes make us losses or beneficiary. In Webster dictionary, the term risk is defined as danger or hazard or exposure to loss or damage. Hence, the risk is the probability of an adverse event (Parsaeian and Farahani, 2012). Risk (probability), change (standard deviation), and outcome (result) are defined as the expected risk (Roudposhti Guide, 2004).

The relationship between expected return and risk is positive in modern stock share portfolio theory, which means the higher expected return is related to the higher accepted risk. This diagram shows these relationships, even if the different exchange rates of return are shown on the outcome risks as a line drawn in the diagram that is called the capital market line (CML).

Figure1.

Portfolio

Portfolio means the investment share as translated and the investment share has a transcendental concept than stock share which also includes other non-stock investments. Technically, a portfolio includes total real and financial assets in investment. The issue of choosing the optimal set of assets is one of the capital market theories subject to particular interest in the microeconomics and macroeconomics. The optimization is seeking the optimal quality and the process of finding the best answer in the scientific view (Sharifi, 2008).

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Grey Theory

Analysis of Grey Theory (GRA: Grey Relational Analysis) was first developed by Deng. This theory is used to solve the fuzzy problems and issues of the discrete data and deficient information. This theory uses the deficient least information with high fickleness in criteria to produce satisfactory and ideal outputs. Grey Theory similar to Fuzzy Theory is an effective mathematical model to solve the uncertain and fuzzy problems. This theory has been used in many fields and solves the multi-criteria decision making via Grey Theory analysis. Grey Theory Analysis is some part of Grey Theory used to solve the complex relations between factors and variables )Hsia,1997). The grey theory systems is an algorithm that analyzes the uncertain relationships between the members of a system and a reference member and has the capability of solving problems in multi-criteria decision making (Royaee, 2013). Grey System Theory

In 1982, Deng in Harmung Science and Technology University of China presented the first paper on the grey system theory titled the ―grey systems control‖ and then it has become known as the grey system theory. In the real world, there are many different abundant systems, as each one has its own subsystems and components. If the evident and clear information of a system are imagined with white color and the completely unknown information of a system are imagined with black color, most of the information in the present systems in nature are not white and black information, but a combination of both in grey color )Deng,1982(.

These systems are known as grey systems whose main feature is the deficient information. The aim of the grey systems theory and its applications provide a bridge between social science and natural science as the grey quality means deficiency and unreliability. Each grey system is described by grey numbers, grey equations, and grey matrices, as the grey numbers resemble atoms and cells in this system. The grey number can be defined as a number with uncertain information. These numerical ranges include unreliable information.Generally, we can say that grey number is a number whose exact value is unknown and the comprised range is known. Grey numbers can only be in the form of lower bound or upper bound or be written as a number in this way (Jabbari, 2000).

Ant Colony Algorithm

Ants are social insects that live in colonies and their behavior is comprised for more survival of the colony and they have work division and the ability to solve complex problems in their lives and their colony is called the ultra organization. Ants despite blindness and low intelligence find the shortest route from nest to food. This is one of the most important and interesting behavior of ants and this type of ants behavior is some kind of swarm intelligence with the random behavior components (probability) and there is no direct relationship between them (each other) and they contact with each other only indirectly by signs.

Ants leave the chemical pheromone when they walk regarded as positive feedback, but this substance evaporates quickly, but it remains as ants trace on the ground in the short term, although ants take positive feedback when choosing the optimal route. When an ant is drawn to the optimal route, it puts some pheromone on the route and it increases the pheromone volume in the route and draws more ants, thus the other ants increase the intensity of the pheromone in the route in turn and this process is repeated to reach saturation and select the optimal route. Thus, the positive feedback increases the probability that an ant chooses a route that the previous ants have chosen (Grasse, 1952). Grass named this routing intelligence the ―evaporation‖. The positive and negative feedbacks of the ant algorithm make very high flexibility in the ant algorithm to solve any optimization problem. If the conditions change, the algorithm aligns itself with the new conditions rapidly. In fact, three characteristics of the secreted pheromone in the evaporation route and the probability and random moves give the ants the shortest route on route selection.

In choosing the best portfolio of stocks, the purpose is the best arrangement of the various stocks in the market to get the maximum and the minimum risk return portfolio. In this algorithm, each share is a vector with coefficients capable to get a value from 0 to k that is the weight per share in the portfolio.

Here we face the pheromone matrix that contains (n) columns for the number of the different types of the market shares and (m) row for the number of the selected coefficients of each share. The number 0 means there is no share in the related portfolio. This algorithm begins to find the best possible arrangement from the first coefficient of the first share combined with the weights of the other shares and the search starts based on the defined functions and continues until the best combination is found. Markowitz Model

One of the problem solving methods was offered by Markowitz in 1952. Markowitz method begins with the assumption that the investors in the current time have a certain sum of money and the money would be invested for a certain period of time, which is called the investors’ hold period. Markowitz argues that such


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method is unreasonable, because the investors typically tend to have high return, meanwhile they want to be sure about the return as much as possible. This means that the investor seeks to maximize their return and minimize the uncertainty (risk) that follows two conflicting aims and the balance should be established between them to decide at zero time, while Markowitz method considers achieving both aims to decide.

Background Of Study

In this context, many studies are performed by Denver (2001), Wang and Yang (2009), Forqandoost and Kazemi (2012). Jia and Dyer found about Markowitz model that the characteristics of this model cannot meet investors’ needs. Also, the return-risk function (variance) cannot serve as the best tool to measure the risk for capital owners. Thus, the other conditions and restrictions should be considered in the model, e.g. the limits on buying and selling stocks, size (capacity), portfolio, etc. Thereby, the model is converted from linear to non-linear mode, which would be very difficult to solve (Jia & Dyer, 1996). Taghavifard (2007) presented a meta-heuristic algorithm using genetic algorithm in study. Yansen (2008) simplified the portfolio optimization process through the single-index model, in which the Sharpe Single-Index Model is one simplification way to calculate the optimization. for (int i=0;i<N_CITY_COUNT);i )++) cc[i].x=x_Ary[ i ;[ cc[i].y=y_Ary[i[; cc[i].num=i ; for(int i=0;i<N_CITY_COUNT);i )++) for (int j=0;j<N_CITY_COUNT);j)++) Map.distance[i][j]=(int)(sqrt(pow((cc[i].x-cc[j].x),2)+pow((cc[i].y-cc[j].y),2))+ 0.5 ;) ) int city= 0 ; for (int i=0;i<N_ANT_COUNT);i )++) city=rnd(N_CITY_COUNT) ; ants[i].addcity(city (; void project:StartSearch )( begin to find best solution int max=0;//every ant tours times double temp ; int temptour[N_CITY_COUNT] ; double dbMin= 0.0 ; while (max < N_IT_COUNT ( dbMin= 1 00000000.0 for(int j=0;j<N_ANT_COUNT);j )++) for (int i=0;i<N_CITY_COUNT-1);i )++) ants[j].move last ;)( ants[j].UpdateResult;)( find out the best solution of the step and put it into temp temp=ants[0].m_dLength ;

Figure3.


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METHODOLOGY

The present study is quantitative type and it is descriptive and innovative in regard to the objective. Since the past performance of the companies is examined in the study, the study is considered causational in terms of study plan. The statistic population includes all of the listed companies in Tehran Stock Exchange during 2005-2011 for period of 7 years with the following conditions: Those companies that were active before 2006 and their membership continued until the end of 2011. The companies should not have the financial year of the same 29/12/_. The companies should not be investment companies, banks, leasing companies, holdings, financial or insurance institutions that have the specific mediated nature. The companies should not be those companies that did not provide the financial data required for this study. The companies should not be those companies that had been active in the period of 2006-2011.

After the volume of the statistic population became determined by filtration method using Cochran method of 105 companies as the sample volume with the most achievement on the highest return on assets (ROA) that is the trait of these companies. The data collection is done through the company's financial statements. Also the data analysis and hypothesis tests are used for the multi-criteria decision making to weight the study indices. Grey theory would be used to rank each target stock in the portfolio. Also Matlab software is used as data analysis tool. Research Hypothesis Hypothesis 1 The ant algorithm based model and Grey theory have better performance compared to Markowitz model. Hypothesis 2 The average performance of the ant algorithm is better than the other optimization methods with multi-objective functions. Data Analysis

On the analysis of the collected data, the descriptive statistics, the inferential statistics, and the drawn tables are used. The purpose of the descriptive statistics is summarizing the collected data and more appreciation on the purpose of the inferential statistics, the interpretation of the population parameters by analyzing the information in the data samples and the measurement of the uncertainty in the interpretations. According to the hypotheses, the performance of the optimization functions is evaluated and compared in support of the data. Normality Test Of Data Time Series

During the portfolio optimization, some hypotheses are considered as the most important hypothesis is the normality of the distribution function portfolio. In this test, the hypotheses are defined as above (data distribution is normal and/or data distribution is not normal) and the results of the Kolmogorov-Smirnov Test are calculated using the daily returns at the certainty level of 95%.

Table 1. Kolmogorov-Smirnov Test Variable Kolmogorov-Smirnov Z Significance level Outcome

Daily return 1.091 0.464 Normal distribution

According to Table 1 of Kolmogorov-Smirnov Statistic Z, since the significance level for the daily stock

return is more than 0.05, H0 hypothesis is confirmed, thus we can say with 95% certainty that the distribution function of the portfolio returns has been normally distributed. Feedback Test

Table 2 presents two test results of the feedback test and the calculated risk of the portfolios in the under study period. In this test, the hypothesis H0 (risk) is correct in calculation of (β) using the daily returns and H1 (risk) is not correct in calculation of (β) using the daily returns as following:

According to Table 2, the variance values are calculated for each portfolio and if these variances are larger than 2, we can certainly argue that the hypothesis H0 is confirmed and the calculations show that the hypothesis H0 is confirmed in all 7 portfolios, which means the calculated risks using the daily returns is correct and has been stable over time and the tests are valid.


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Table 2 . Result of Feedback Test

Result of Feedback Test

Amount of Variance Accounted for e Number of Portfolio

Hypothesis H0 is confirmed 0.93816 1 Hypothesis H0 is confirmed 0.98199 2 Hypothesis H0 is confirmed 0. 9713 3 Hypothesis H0 is confirmed 0.95729 4 Hypothesis H0 is confirmed 1.02434 5 Hypothesis H0 is confirmed 1.06512 6 Hypothesis H0 is confirmed 1.0 8054 7

Algorithm Convergence

The convergence means that all points move around one point and the algorithm gets close answers by repetitions. The applied algorithm in this study is single phased and developed by Matlab software.

Consequently, the results of the convergence analysis are presented in Fig 1, which show the complete convergence in one route around specific points and there is no significant dispersion resulted from the convergence of the algorithm.

Figure1. Exploring the convergence of the algorithm

Answers Stability Evaluation On Algorithm

Another important test of the algorithm is the stability check of the algorithm, whether the algorithm repetition has the same optimal unique answer. For this purpose, the portfolio with 15 stock shares is considered with the return and risk criterion. The results are presented in Table 3. In each run, the algorithms are repeated for thousand times and there are some series of errors every time and if the series of errors are close to each other, it means that there is convergence and the results show the insignificant difference of the answers from the repetition of the algorithm in several times. As it can be seen, the variance of the algorithm responses is close to zero (0.0000051975) in 20 repetitions.

Table 3. Answers stability evaluation on algorithm The Optimal Value of the Target Function Number of Algorithm Implemention

0.10206 1 0.10102 2 0.09993 3 0.10411 4 0.09772 5 0.10229 6 0.10336 7 0.10481 8 0.10204 9 0.10372 10 0.10355 11 0.09986 12 0.10231 13 0.10276 14 0.10056 15 0.10234 16 0.10171 17 0.10487 18 0.09797 19 0.10423 20 0.0000051975 Variance


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CONCLUSION

RESULTS OF PORTFOLIO OPTIMIZATION

According to the objective of this study, the abovementioned 3 methods formed the 7 portfolios using

as each portfolio consists of 15 companies to select and optimize the portfolio by using the 3 theoretical methods of the ants theory, the Grey theory and Markowitz model during the 2005-2010 based on risk and return. According to the hypothesis, these three models are compared separately on the model performance of the two models of the ants and the Gray theory related to Markowitz and the final results are related to the weight per share in the portfolio (optimization), the optimal value of the target function, the returns, and the variance of each portfolio by each method separately and, when these tables are compared, we can examine the risk and return of each model and check how they perform.

Table 4. Results of portfolio optimization based on Ants theory 2010 2009 Number of

Portfolio Variance Return Optimal Vakue of the Target Function

Weight Variance Return Optimal Vakue of the Target Function

Weight

0.2518 0.4046 0.4459 0.1382 0.2518 0.3343 0.4916 0.1087 1 0.2494 0.4569 0.4000 0.1383 0.2207 0.3414 0.3454 0.1070 2 0.2201 0.2194 0.4910 0.0983 0.2126 0.2988 0.5082 0.1202 3 0.2027 0.3517 0.3916 0.1346 0.2521 0.2257 0.4613 0.1401 4 0.2442 0.5014 0.4815 0.1056 0.2349 0.4618 0.4400 0.1049 5 0.2358 0.0940 0.3853 0.1089 0.2070 0.1670 0.3562 0.0972 6 0.2181 0.4730 0.5596 0.2761 0.2300 0.3839 0.3638 0.3221 7

Table 5. Results of portfolio optimization based on Grey theory

2010 2009 Number of Portfolio Variance Return Optimal

Vakue of the Target Function


Weight

0.2134 0.5671 0.3338 0.1139 0.2387 0.0855 0.3577 0.1328 1 0.2593 0.3737 0.3660 0.1128 0.2029 0.1636 0.5602 0.1120 2 0.2388 0.3693 0.3883 0.1023 0.2390 0.3969 0.3681 0.1016 3 0.2311 0.2212 0.3033 0.0978 0.2137 0.2324 0.5103 0.1150 4 0.2231 0.3721 0.3138 0.1355 0.2336 0.5203 0.3820 0.1075 5 0.2685 0.5150 0.3933 0.1206 0.2317 0.5939 0.5633 0.0980 6 0.2500 0.3511 0.5383 0.3171 0.2172 0.3112 0.3332 0.3331 7

Table 6. Results of portfolio optimization based on Markowitz model

2010 2009 Number of Portfolio Variance Return Optimal

Vakue of the Target Function


Weight

0.2396 0.4713 0.3776 0.1397 0.2211 0.2229 0.4263 0.1133 1 0.2702 0.3293 0.4934 0.1239 0.2506 0.3567 0.3920 0.1146 2 0.2147 0.4293 0.3795 0.1347 0.2437 0.0987 0.3462 0.1317 3 0.2655 0.5814 0.5335 0.1328 0.2140 0.3997 0.3706 0.0991 4 0.2196 0.0563 0.5624 0.1312 0.2388 0.1148 0.5284 0.1326 5 0.2203 0.4336 0.5439 0.1203 0.2235 0.3190 0.3573 0.1342 6 0.2430 0.3459 0.3500 0.2175 0.2669 0.5052 0.4062 0.2744 7

The nature of the portfolio optimization models is based on risk and return, so that we can say that its

predictions are different from other prediction models in many respects. The most obvious difference is that there is no constant or correct observation or period with β index, therefore the estimations vary, so the only available criterion for the comparison is the actual observations. Also the concept of the prediction error is different in the model. The main concern on the normal prediction models is that how much are the predictions close to the real data. But the difference in the risk-oriented models is that the amount of the actual loss is greater than the predicted loss by the model for several times. Therefore, many common criteria are not used in assessing the accuracy of the risk-oriented prediction models, e.g. mean square error or mean absolute percentage error. An important criterion in this context is considering the number or the proportion of failures (the deviation of the expected value).


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When the actual results of the daily income and loss are compared with the calculated risk, the result would be a binomial distribution. If the actual loss exceeds the estimated losses in the model, then this event is considered as a failure. Inversely, if the actual loss is less than the expected loss, this event is registered as success. If the daily risk and returns are assumed independent, the binomial results indicate the number of the independent Bernoulli tests, at which the failure probability of each test is equal to one minus the target certainty level in the model. According to the above descriptions, the performance of the three models is estimated and compared in this study (Markowitz model, Ants theory and Grey theory). The Kopek Probable Failure Test is used to check whether the models are proper or not. In other words, the validation test checks the models. In this test, when LR (the expected failure) based on the data in the model is calculated more than the extracted critical value of the chi square distribution, we could claim about the target certainty level that the prediction error percentage of the model would be maximum at the determined error level (α) and the model has appropriate validity.

Table 7. Results of The Kopek Probable Failure Test critical value of the chi square distribution

LR Statistics Certainty Level Model

4.041 6.981 95% Markowitz 4.567 8.647 95% Ant Theory 3.352 5.347 95% Grey Theory

According to Table 7, the certainty level is calculated LR = 95% based on the optimization of the above

three models, which higher than the extracted critical value from the chi square distribution. On the target certainty level of 95%, it can be argued that the prediction error percentage of the model would be maximum equal to the error level (α) and the model has appropriate validity. Therefore it can be argued that the fuzzy optimization method can be used to enhance the portfolio performance in the real-world problems, which have constant uncertainty.

Therefore, the first hypothesis of the study is confirmed based on the Ants algorithm and the Grey Theory which perform better than Markowitz model. In addition, the capability of the above three models is compared by the number of the Kopek Probable Failure Test. If the success ratio of the portfolio optimization model is greater than the other models in a period, that model has higher measurement and prediction capability. The results of these tests are presented in Table 8. As it is observed the Ants theory based model has less failure and more success than the other models and afterward the Grey theory is in the second rank. Thus, the second hypothesis of this study is confirmed based on the average performance of the Ants algorithm, which is better than the other optimization methods with multi-objective functions.

Table 8. comparison the capability of models Failures Ratio Success Ratio Certainty Level Model

139 1121 95% Markowitz 101 1159 95% Grey Theory 88 1172 95% Ant Theory

RESULTS

In general, the analyses and the tests in relation with the study hypotheses indicate that the hypotheses

are confirmed and the study model is designed based on Grey Theory and Ants algorithm, which have better performance compared to Markowitz model. In other words, the average performance of Ants algorithm is better than the other optimization methods with multi-objective functions.

The Ants theory based model has less failure and more success than the other models. Markowitz model has better average performance on the multi-objective functions according to the research background and the other researches and the hypotheses indicate our expectation on the research based on the better performance of Ants theory compared to Markowitz theory.

Jia and Dyer evaluated Markowitz model and found that the characteristics of this model cannot meet the investors’ needs in practice. Also the target function, the return, and the risk (variance) could not be settled as the best tool to measure risks for the capital owners. Therefore, the other conditions and restrictions, e.g. limits on buying and selling stocks, stocks portfolio size (capacity), etc. should be considered in the model, therefore the model is converted from linear to non-linear, though it would be difficult to solve it (Jia & Dyer, 1996). Taghavifard has presented a meta-heuristic algorithm to select the stocks portfolio in regard to the integer constraints. Therefore, the genetic algorithm is compared with Markowitz and the results indicate that the proposed algorithm of both samples could optimize (Taghavifard ,2007 ).


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The Optimal Portfolio Sharp Model indicated the capability to select the optimized portfolio that has a slight difference with Markowitz model and its proposed hypothesis test is confirmed similar to the hypotheses in this study and it states that the single-index model estimates Markowitz model (Yansen,2008 ).

The ROA index is used as variable in this study and it is also suggested to apply the other relevant economic and financial variables for more efficiency in future studies.

REFRANCES Alam Tabriz A, Afshari M, Maleki M, Siahkali Moradi J. 2010 .Optimal Portfolio Selection using Artificial Neural Networks Model and

Markowitz Model on Tehran Stock Exchange .First Annual Conference on Management, Innovation and Entrepreneurship, Shiraz. Bazargan A .2002. Research Methods in Behavioral Sciences, First Edition. Bowen WM, Klosterment R, et al. AMP:Multiple Criteria Evaluatin,in EDS ,Spreadesh Model for Urban & Regional Analysis. Davani Gh .2012. Accounting Industry in the World and Looking at its Condition in Iran, Accounting Journal. Deng JL. 1988. Properties of relational space for grey system. In: Deng, J.L. (Ed.), Essential Topics on Grey System—Theory and

Applications. China Ocean, Beijing, pp. 113. Deng JL.1982. Control problems of grey system. Systems and Control letters 1, 288–294 Deng Y, Xu Y, Li H, Dai M.2008. A comparative study of the numerical scales and the prioritization methods in AHP. European Journal of

Operational Research 186, 229–242. Eslami Bidgoli Gh, Heibati F .1995 .Portfolio management using index model. Tehran, organizational research. Forman H. 2005.Ernest,the Analytical Hierarchy process an Exposition, school of Business &public management,Gorge washangton

university & Gass, I.Saul,college of business & management,University of Maryland, Ghalibaf Asl H .2010. Financial Management . Puran-Pazhohesh Publication, Emad Print. Gholami S . Stocks Portfolio Selection using Fuzzy Genetic Algorithms. MS Thesis, Central Branch. Tehran. Gondzio J, Grothey A. 2007. Solving non-linear portfolio optimization problems with the primal-dual interior point method. European Journal

of Operational Research,181(3), 1019–1029. Hall J.2003. Translator: Sayah S. Financial Engineering and Risk Management , Rayaneh Tadbirpardaz Publication Group. Hsia KH, Wu JH.1997. A study on the data preprocessing in Grey relational analysis. Journal of Chinese Grey System 1, 47–53. Jabbari F. 2000. Ranking of Industries in Iran based on the Selected Economic Indicators of Stock (AHP). MS degree at Tehran University. Jahankhani A, Abdeh Tabrizi H .1993. Theory of Efficient Capital Markets. Journal of Financial Researches.Tehran. Jahankhani AP.1995. Tehran Stock Exchange . Tehran Management College Press. Jia J, Dyer JS.1996 .A Standard measure of risk & risk value models. Management sience,42(12),1691-1705. Karimi .2007. Portfolio optimization using value-at-risk models at Tehran Stock Exchange, Master's thesis, university. Khaki Gh .2005. Research Method with Dissertation Approach. Tehran, Baztab Publication. Latifi .2005. Evaluation and performance of a portfolio of shares of listed companies on the Stock Exchange. MS Thesis, Tehran. Lee sang M, Chasser Dalton L. 1980.Goal Programming for portfolio S.Lection.The Journal The journal of portfolio management Lee Sang M, Chesser Dalton L. 1980.Goal programing for Protfolio selection, the journal of portfplio management. Markowitz H.1952.Portfolio Selection, Journal of Finance, vol.7 , pp:105. Markowitz H.1987. Mean-Variance Analysis in Portfolio Choice & Capital Markets. Combridge,Ma:Black Well. Mc Nulty T, Florackis Ch, Ormrod Ph. 2013.Board of Directors and Financial Risk during the Credit Crisis,University of Liverpool,

Managment Scool.Magazine: Corporate Governance; An Internantional Review. 21(1): 58-78,. Journay 2013. Mc Nulty T, Florackis Ch, Ormrod Ph. 2012 ,Corporate Governance and Risk: A Study of Board Structure and Process,University of

Liverpool, Managment Scool,. Mehdizadeh.2012. optimal portfolio selection model of Markowitz listed oil company pension fund Semnan. Mehregan M .2004. Operational research, Ketab-e-Daneshgahi Publication. Montazeri R. 2009. Specified Investment Portfolio Optimization model for enhancing investment in securities of listed companies. MS

Thesis, Tehran. A model for portfolio selection with regard to the integer constraints, Journal of Economic Research, Issue IV. P. Jones Ch.2012. Translated by Tehrani R, Nourbakhsh A . Investment Management, Negah-e-Danesh Publication. Raee R, Talangi A.2004. Advanced Investment Management, First Edition, Human Science University Books Organization. Tehran. Raee R, Fallahpour S.2011. Designing a Model for Active Portfolio Management using Var and Genetic Algorithm, Accounting

Journal.Iran. Rahnameye Roodposhti F, Nikoomaram H .2008. Markets and Financial Institutions. Islamic Azad University, Science and Research

Branch. Tehran. Rezai F. Financial applications of ant colony algorithm, Master's thesis, Qazvin Azad. Royaee R .Selecting Optimal Stocks Portfolio using Combined Approach of Grey Theory and Linear Programming Model. MS Thesis,

Tehran, Central Branch. Sharif S.2008. A model for portfolio selection in the stock market and securities with multiple criteria decision making (Case study: top 50

companies), MS thesis, Beheshti University.Iran. Tanaka H, Guo P, Turksen IB.2000. Portfolio selection based on fuzzy probabilities and possibility distributions, Fuzzy Sets and Systems

111 (3) 387–397. Tayebi K, Movahednia N, Kazemini M.2008. Using Artificial Networks in Predicting Economic Variables and Comparing it with Econometric

Methods. Prediction of Currency Exchange Rate in Iran, Sharif Scientific and Research Journal, No. 43, pp. 40-99. Tehrani R .2003. analysis of financial statements. First published in Tehran, Iran, at school. Weston B.1997. Translator: Abdeh Tabrizi; Hanifi, Farhad . Financial Management, Volume II, First Edition. Yansen A. 2008. Simplifying the Portfolio Optimization Process via Single Index Model. Zolhouri Q.1999. Application of Social Sciences Research Methodologies in Management, Tehran, Mir Publication Institute.

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