Performance Investigation and Comparison of Two Evolutionary Algorithms in Portfolio Optimization: Genetic and Particle Swarm Optimization

Arash Talebi M.B.A. Student, dept. of Industrial Engineering and

Management Shahrood University of Technology

Shahrood, Iran E-mail: Arash.talebi.mba@gmail.com

Mohammad Ali Molaei Assistant Professor, dept. of Industrial Engineering and

Management Shahrood University of Technology

Shahrood, Iran E-mail: Malimolaei@yahoo.com

Mohammad Javad Sheikh Assistant Professor, dept. of Humanities

Shahed University Tehran, Iran

E-mail: Mjsheikh2002@yahoo.com

Abstract— Contrary to the growing use of portfolios and in spite of its rich literature, there are some problems and unanswered questions. The aim of this work is to be a useful instrument for helping finance practitioners and researchers with the portfolio selection problem. This study first reviews Modern Portfolio Theory’s literature and describes the investment selection process. Second, this paper specifically aims at applying efficient optimization methods for solving the portfolio selection problem. Hence, the genetic algorithm and the particle swarm optimization (PSO) approaches to resolve the portfolio selection problem with the objective of simultaneous risk minimization/return maximization are applied. Computational analyses are provided so as to investigate the performance of the algorithms and the input data. Therefore, at one step, four different portfolios are selected from Tehran Stock Exchange market, 50 top companies (Annual and monthly-based portfolios). Finally, this paper applies the selected portfolios to the stock market data of a 10-month period proceeding portfolios constructions. The results indicate that, the genetic annual-based portfolio has the best performance in contrast to its other counterparts; it outperforms the average return of market portfolio in a relatively short period. Moreover, PSO annual-based portfolio has a relatively good performance and could achieve the market portfolio’s return during the test period. Findings show that using annual data is more efficient than the monthly data. On the whole, the results show that the evolutionary methods of this paper with annual data, can consistently handle the practical portfolio selection problem.

Keywords- Modern Portfolio Theory (MPT); Portfolio management; Genetic algorithm; Particle Swarm Optimization (PSO). JEL classification: C61; C63; G11

I. INTRODUCTION The rationale for stock market diversification is that the

overall risk from owning many stocks is lower than the risk of holding a few stocks [1]. In a metaphor “Money is like manure – when it stacks up, it stinks; when you spread it

around, it makes things grow.” Texan Clint Murchison, Sr., one of the wealthiest investors ever. Effects of diversification on risk reduction have been studied extensively [2-4], but the importance of portfolio management lies not in the number of holdings, but rather in both the nature and degree of the combined risk of the underlying stocks [1]. Brealey [2] has shown quite dramatically what can happen when one considers both the nature and degree of risk in portfolio composition. He reported that a portfolio containing only 11 securities, which were carefully selected for their risk-diversifying characteristics, would be less risky than a portfolio of 2000 securities, selected without regard to risk! So the questions are: how should investment risk be considered? And, once it is considered, how should portfolios be optimized?

This paper intends to summarize the answers to the above questions efficiently, and also make a comparison between two outstanding methods, genetic and Particle Swarm Optimization, to answer the second question.

Since Markowitz’s article on portfolio selection [5], there has been many contributions to this important field of financial studies. Markowitz’s work has permanently changed the course of investment-related thought. Before Markowitz, it was more or less taken for granted that the proper way to construct an investment portfolio was just to select the best securities. It was erroneously assumed that this technique would maximize the expected return of the resultant portfolio. Without the benefit of Markowitz’s insights, dangerous homilies such as “put all your eggs in one basket, and watch the basket”! still prevail [6]. Finding the optimum solution to the Markowitz’s model equations has always been a challenging issue, and therefore, it has received remarkable attention of math and computer science experts. Experts have tried different optimization methods to go through this problem, many of whom have tried classics and many have tried heuristic methods.

430

___________________________________ 978-1-4244-6928-4/10/$26.00 ©2010 IEEE

Genetic algorithm (GA), an evolutionary computation technique, has been widely employed to solve the portfolio selection problem. GA searches for the optima from a lot of generations of the chromosome-represented schedules that are reproduced through cross-over and mutation.

Recently, a family of optimization algorithms has been developed based on the simulation of social interactions among members of a specific species looking for food sources. When they are looking for food, in addition to searching for their targets independently and stochastically, these species also exchange information with each other so that all of them can eventually move toward the target from different directions. From this family, the two most promising algorithms are ant colony optimization, which is based on the pheromone pathways used by ants to guide other ants in a colonies towards food sources[7], and particle swarm optimization or PSO, which is based on the social behavior reflected in flock of birds, bees, and fish that adjust their physical movements to avoid predators, and to seek the best food sources[8, 9]. Like GA, PSO starts by initializing a population of random solutions and searches for the optimum by updating generations; particles fly through the problem space by following its own experience and the best experience attained by the swarm as a whole.

Because of the advantages, the portfolio selection problem is optimized via two different methods, genetic and PSO. The optimization is performed with simultaneous risk minimization/return maximization, as the objective in both methods. Moreover, to evaluate the input data, two different inputs are fed to each algorithm.

The paper is organized as follows. In section 2, an extensive literature of Markowitz portfolio theory is reviewed. Section 3 is devoted to investigating the investment selection process. Section 4 discusses function optimization and briefly talks about portfolio optimization. In section 5, methodology and data are presented. The optimization calculations of portfolios are performed next, and the results and findings are presented there, in section 6. Finally, section 7 discusses the findings, summarizes the paper, and talks about the conclusions and future works.

II. LITERATURE REVIEW

A. Markowitz Model The insight of Nobel Prize winner, Harry Markowitz,

was first published in his article entitled Portfolio Selection [5]and more extensively in his book, Portfolio Selection: Efficient Diversification of Investments [10]. As with insights from other researchers, his equation-filled presentation is over the heads of most investors. He pointed out that the goal of portfolio management is not solely to maximize the expected rate of return, but instead to maximize “expected utility” – synonymous with “satisfaction”. Markowitz began with the valid premise that rational investors maximize their utility by seeking either;

a. The highest available rate of return for a given level of risk. Or,

b. The lowest level of available risk for a given rate of return [1]. His most important innovation was using the variance of a distribution of likely returns as a measure of Risk. Therefore, Markowitz reduced the complicated and multidimensional problem of portfolio construction to a conceptually simple two-dimensional problem known as “mean-variance” analysis [6]; the selection of an efficient portfolio begins with an analysis of three estimates:

a. The expected return for each security. b. The variance of the expected return for each security. c. The possibly offsetting, or possibly complementary,

interactions, or covariance, of return with every other security consideration [1].

The expected return for an aggregate of portfolio of securities is merely the weighted average of the expected return of the individual securities:

E (Rportfolio) = 1 ( ).ni i iE R W=� (1)

Where: E (Rportfolio) = Portfolio’s expected rate of return, E (Ri) = Security i’s expected rate of return, Wi = the proportion of the portfolio’s value invested in security i [5].

The risk of a portfolio is not typically equal to the weighted average of the risks of its component securities, it is more complicated. It depends not only on the risks of securities, considered in isolation, but also on the extent to which they are affected similarly by underlying events [11]. This fact refers to the covariance (Or the correlation coefficient). The relationship between the risk of a portfolio, consisting of n securities, and the relevant variables is:

VAR (Rportfolio) = 1 1n ni j= =� � Wi.Wj.COV (Ri,Rj) (2)

Or substitution from covariance/correlation equation yields: VAR (Rportfolio) = 1 1

n ni j= =� � Wi .Wj. SRi.SRj.rij (3)

Where: VAR(Rportfolio)=Variance of portfolio’s rate of return, VAR (Ri) =Variance of security i’s rate of return, COV (Ri,Rj)= Covariance between securities i,j’s returns, n = the number of securities, Wi or Wj =the proportion of the portfolio’s value invested in security i or j, rij = Coefficient of correlation between securities i and j, Si or Sj = Standard deviation for security i or j [5].

B. Basic assumptions of Markowitz Model a. The investor does (or should) maximize the discounted (or capitalized) value of future returns [5, 12]. b. Investors behave rational in investment decisions, which means they choose to hold efficient portfolios- portfolios, which maximize each investor’s utility [5].

III. INVESTMENT SELECTION PROCESS Each investor has specific indifference curves for his/her

risk/expected-return preferences, a set of these curves is

431

named investor’s indifference map. In an efficient market, an efficient set of investment alternatives is called efficient frontier; it can be plotted using Markowitz model [13]. The matching of the available investment alternatives along the efficient frontier, with the investor’s highest indifference curve is the final step in the investment selection process. Different sets of indifference curves, representing either more defensive or more aggressive investors, would lead to the selection of different investments from the wide-ranging set of efficient alternatives [6].

IV. FUNCTION OPTIMIZATION Function optimization is the process of finding an

optimal solution to an objective function describing a problem; it can be either a minimization or maximization task. Optimization methods are divided into two major groups, classic optimizers, also known as gradient-based methods, and heuristic or evolutionary optimizers.

In a complicated problem, problems characterized by discontinuities, lack of derivative information, noisy function values, disjoint search spaces[14, 15], and nonlinear complicated equations, like that of Markowitz, classic solvers often stop solving as soon as reaching a local optimum; it is the problem with classics optimizers.

Evolutionary algorithms have been designed primarily to address problems that cannot be tackled through traditional optimization algorithms. Heuristic methods usually turn out to achieve better results and performances in contrast to their classic counterparts. This goes back to their nature of design; they have been designed to “jump out” of local optima and reach the global one. They are supposed not to “get stuck” in local optima; because heuristic algorithms perform a wide random search, the chance of being trapped in local optima is deeply decreased.

In recent years, several machine learning techniques such as neural networks [16, 17], evolutionary algorithms [18-20], and swarm intelligence-based algorithms [15, 21, 22], have been developed and applied successfully to solve a wide range of complex academic and practical optimization problems. Population-based search methods, a sub-group of heuristic optimizers, which present a real and viable alternative to existing numerical optimization techniques, can rapidly search large and convoluted search spaces and are not susceptible to suboptimal solutions. The genetic algorithm and the particle swarm optimization are two important and popular members of the family.

A. Portfolio Optimization Markowitz demonstrated that, once prepared, the

foregoing security descriptions could be manipulated by portfolio optimization programs to produce an explicit definition of the efficient portfolio in terms of:

a. The securities to be held. b. The proportion of funds to be allocated to each [1].

Since then, many methods have been used to optimize these equations (the earliest was quadratic programming).

V. METHODOLOGY AND DATA

A. Methodology This study aims at optimizing the portfolio selection

problem via two novel heuristic methods, genetic and PSO. • What is Genetic Algorithm and Why GA for

Optimization? The basic principles of genetic algorithms (GAs) were

first proposed by Holland [23]. GAs are general purpose search algorithms, which use principles inspired by natural genetics to evolve solutions to problems [24]. GA is easy to use, supports multi-criterion optimization and parallel usage, flexible and easily hybridized with other algorithms, which make it ideal for optimization tasks. A basic element of the Biological Genetics is the chromosome. Chromosomes cross over each other, mutate itself, and new set of chromosomes is generated. Based on the requirement, some of the chromosomes survive. This is the cycle of one generation in Biological Genetics. The above process is repeated for many generations and finally best set of chromosomes will be available. The Mathematical algorithm equivalent to the above behavior used as the optimization technique is known as Genetic Algorithm [25]. The basic idea is to maintain a population of chromosomes (representing candidate solutions to the concrete problem being solved) that evolves over time through a process of competition and controlled variation. A GA starts with a population of randomly generated chromosomes, and advances toward better chromosomes by applying genetic operators modeled on the genetic processes occurring in nature. During successive iterations, called generations, chromosomes in the population are rated for their adaptation as solutions, and based on these evaluations, a new population of chromosomes is formed using a selection mechanism and specific genetic operators such as crossover and mutation [26]. An evaluation or fitness function must be devised for each problem to be solved; then, given a particular chromosome, a possible solution, a single numerical fitness is returned.

The fundamental underlying mechanism of the basic GA consists of three operations: evaluation of individual fitness; formation of a gene pool through selection; and recombination through crossover and mutation operators.

• What is Particle Swarm Optimization and Why PSO for Optimization?

Particle swarm optimization is a stochastic population based optimization approach, first published by Kennedy and Eberhart [8, 9] as a simulation of social behavior. It is based on the premise that social sharing of information among members of a species offers evolutionary advantage. A number of advantages with respect to other algorithms make PSO an ideal candidate to be used in optimization tasks. The algorithm has shown to be an efficient, robust, flexible and simple optimization algorithm. The strength of PSO is underpinned by the fact that decentralized biological creatures can often accomplish complex goals by

432

cooperation[14]. PSO can be easily implemented and is computationally inexpensive in terms of both memory requirements and CPU speed[9] besides of advantages like rapid convergence capability. As compared to other robust design optimization methods, PSO is more efficient, requiring fewer number of function evaluations, while leading to better or the same quality of results[27, 28].

PSO is initialized with a population of random particles and the algorithm searches for the optimum by updating generations via a velocity vector. The velocity vector is updated based on the ‘‘memory’’ gained by each particle, conceptually resembling an autobiographical memory, as well as the knowledge gained by the swarm as a whole[29]. Particles are ‘‘flown’’ through hyper-dimensional search space, with each particle being attracted towards the best solution found by the particle’s neighborhood and the best solution found by the particle. The effect is particles ‘‘fly’’ toward a minimum (or maximum), while still searching a wide area around the best solution. The performance of each particle is measured using a predefined fitness function which encapsulates the characteristics of the optimization problem. The search procedure of a population-based algorithm such as PSO consists of two main phases, exploration and exploitation. The former is responsible for the detection of the most promising regions in the search space, while the latter promotes convergence of the particles towards the best detected solution. Proper selection of the neighborhood size affects PSO’s trade-off between exploration and exploitation, albeit there is no formal procedure to determine the optimal size[30].

Suppose that the search space is n-dimensional, then the particle i of the swarm can be represented by an n-dimensional vector Xi = (xi1, xi2, …, xin). The velocity of this particle can be represented by Vi = (vi1, vi2, …, vin) [8, 9]. The best previously visited position of the particle i is noted as its individual best position, Pi = (pi1, pi2, …, pin). The position of the best individual of the whole swarm is noted as the global best position, G = (g1, g2, …, gn). At each step, the velocity and new position of particles, can be assigned according to the following equations: i = 1: ss, and j = 1: n:

Vij(t+1) = � * Vij(t) + c1 * r1j(t) * (Pij(t) - Xij(t)) + c2 * r2j(t) * (Gij(t) - Xij(t)) (4) Xij(t+1) = Xij(t) + Vij(t+1) (5)

Where: ss is the total number of particles in the swarm, n is the dimension of the problem, c1, c2 are positive constant parameters called acceleration coefficients, ‘‘trust’’ parameters, indicating how much confidence the current particle has in itself (c1 or cognitive parameter) and how much confidence it has in the swarm (c2 or social parameter), �, the inertia weight introduced by Shi and Eberhart [31], controls the relationship between exploration/exploitation. r1, r2 are independently uniformly distributed random variables with range (0,1),( Xi(t), Vi(t), Pi(t), and Gi(t) are explained in the previous paragraph.)

Once V is calculated by using (4), the particle flies toward a new position according to (5). This process is repeated until a user-defined stopping criterion is reached.

It can be noticed that PSO shares many common points with GA, such as random generation of initial population, search for optima by updating generations or iterations, and evaluation of a fitness or objective for possible solutions. However, they have different experience sharing mechanisms during search; all chromosomes of GA share unclassified search experience with each other, while each PSO particle shares only its own search experience (local best) and its companions’ search experience (global best).

B. Data • Population, Data gathering tools and assumptions This study concerns with portfolio selection in Tehran

stock exchange market and from the stocks of top 50 listed companies. The list of companies is extracted from the library of Tehran Stock Exchange Central Organization. It was the most updated available list up to January 21st, 2010. “Why top 50 companies”? the only way the systematic risk can be lowered is to expand the definition of the “market” to include dissimilar markets[1]. Internationally diversified portfolios are much less risky than the limited ones [32, 33]. By selecting portfolios for research purposes in limited markets, no international diversification is possible. Thus, it is conservative to select the portfolio from companies that are listed as top and successful companies and are probably outstanding in their own industry most of the time.

Raw data was gathered using library studies of Tehran Stock Exchange market, databases of its softwares, Tadbirpardaz and Rahavard Novin, and official website.

Finally, a common procedure which often happens in data gathering for portfolio selection but rarely mentioned, is using historical data as a representative of future data, because historical data is readily at hand. In some cases, historical data may not be good and real representatives of future data, ex-ante data are somehow different from historical data. Assuming historical data instead of ex-ante data is an unspoken assumption that should be taken care of and used with cautious.

• Samples and Raw Data Rates of returns for 50 companies’ stocks and for six

periods are extracted, starting from March 20th, 2004 and ending in March 20th, 2009. Each period is considered 1 total financial year (Note that Iranian New Year, Norouz, and so the financial year begins on March 21st. This is important to be considered, because of the effects of New Year on financial issues; known as Farvardin effect- in the name of the 1st month in Iranian’s calendar- the same as January effect) Six periods were selected for two reasons: 1. the most updated and possible information could be extracted considering six recent periods; any consideration more than six periods could damage the data, because some companies were not present in the market in the preceding periods. 2. Fig.’s 1, 2 show the Index diagram for Tehran

433

stock market and the 50 top companies, respectively. To extract a rigorous covariance, both prosperity and depression economic periods, should be considered. Thus, period of at least six years is conservative.

To make a comprehensive comparison between different input data as well, raw monthly data for the mentioned six years were gathered, too; 61 available monthly return data were also considered to construct monthly-based portfolios, via two algorithms, versus annual-based portfolios.

VI. CALCULATIONS, RESULTS AND FINDINGS Two different 50×50 covariance matrices between each

pair of stocks are approximated using different data of rates of return and covariance/correlation equations. Let’s refer to them as Annual-COV.matrix and Monthly-COV.matrix. Both algorithms are implemented in MATLAB[34] via codes prepared by authors. Because MATLAB uses minimization objective as the default procedure, the portfolio optimization problem was to be coded as minimization of the fitness function with simultaneous risk minimization/return maximization. Finally, table 1 is yielded using related equations and codes. Table 1 is the optimum portfolio selected via Genetic and PSO algorithms by Annual/Monthly-COV.matrices. Percentages in table 1, known as Wi, are the proportion of the portfolio’s value invested in security i. Fig.’s 3, 4, 5 and 6 illustrate the amounts of Mean/Best fitness functions vs. the algorithms’ generation/iteration number, for annual and monthly portfolios, respectively.

Substituting the weights of tables 1 in (1), (3), table 2 is achieved. In this table, expected risks/returns for the portfolios are summarized. In order to have a comparison between the input data (Annual vs. Monthly) and also the algorithms, all portfolios are applied to the market data of proceeding months of historical data, hence; securities were selected and hold, accordingly. The holding was for a period of 10 months, since March 21st 2009 up to January 21st 2010. The return results of portfolios are summarized in the second section of table 2. Market portfolio’s return is also presented for comparison purposes. Moreover, table 3 summarizes the best algorithms’ parameters found to be consistent with the portfolio problem for GA and PSO.

Figure 1. Market Index, six periods of Tehran Stock Market

(Source: Tehran Stock exchange market official website - TSEMOW -)

Figure 2. 50 top companies’ Index, six periods of Tehran Stock Market (Source: TSEMOW: http://www.irbourse.com)

VII. DISCUSSION, SUMMARY AND CONCLUSIONS Markowitz demonstrated that the two relevant

characteristics of a portfolio are its expected return and some measure of its risk. Rational investors will choose to hold efficient portfolios. The identification of efficient portfolios would require information on each security’s expected return, variance of return and covariance of returns. Finally, once prepared, the foregoing security descriptions could be manipulated by portfolio optimization programs. Optimization methods are divided into two major groups, classics and heuristics (or evolutionary algorithms). Evolutionary algorithms have been designed primarily to address problems that cannot be tackled through traditional optimization algorithms, problems like the portfolio selection problem. Population-based search methods, a sub-group of heuristic optimizers, present a real and viable alternative to existing numerical optimization techniques. Population based optimization techniques can rapidly search large and convoluted search spaces and are not susceptible to suboptimal solutions. Genetic algorithm belongs to them and is applied to the problem. Besides, in recent years, evolutionary-stochastic-based search algorithms have been widely used for optimization. The particle swarm optimization (PSO) algorithm is a late member of the family and is used throughout this paper.

Genetic algorithm and PSO are applied to the portfolio selection problem with the objective of simultaneous risk minimization/return maximization. Security weights in a portfolio are manipulated to primarily satisfy three constraints concerned with the portfolio selection; upper, lower bounds and a linear constraint that requires sum of weights equal 1. Four different portfolios were selected from Tehran Stock Exchange market, from 50 top companies (Annual-based and monthly-based portfolios with GA and PSO). To investigate their performances, portfolios were applied to the return data of companies for a test-period of 10-month, which proceeded the portfolio construction historical data. The results indicate that the Genetic annual-based portfolio has the best performance in contrast to the other portfolios. The Genetic annual-based portfolio outperforms the market portfolio’s average return in a relatively short period of 10 month. PSO annual-based portfolio stands in the second level and has a relatively fast and good performance. It approximately achieves market portfolio’s return. On the whole, the results indicate that heuristic approaches with annual input data can consistently handle the practical portfolio selection problem with good efficiency.

More work is currently underway to consider different time steps for reporting rates of returns to construct the covariance and hence the portfolio. Trying other heuristic algorithms to solve the problem and making comparisons could be another subject for further investigations.

434

TABLE I. OPTIMUM PORTFOLIO, PROPORTION OF THE PORTFOLIO’S VALUE INVESTED IN SECURITY I, SELECTED VIA GENETIC AND PSO ALGORITHMS USING ANNUAL & MONTHLY-COV MATRICES. (THE NUMBERS ARE EXPRESSED AS PERCENTAGE) (SOURCE: CALCULATED BY AUTHORS)

Company Name

Percentage of

portfolio’s Value

invested in stock by Annual-Genetic

Percentage of

portfolio’s Value

invested in stock by Annual-

PSO

Percentage of

portfolio’s Value

invested in stock by Monthly-Genetic

Percentage of

portfolio’s Value

invested in stock by Monthly-

PSO

Company Name

Percentage of

portfolio’s Value

invested in stock by Annual-Genetic

Percentage of

portfolio’s Value

invested in stock by Annual-

PSO

Percentage of

portfolio’s Value

invested in stock by Monthly-Genetic

Percentage of

portfolio’s Value

invested in stock by Monthly-

PSO1.Iran Khodro 0.5 0.12

6.26 1.12 26.Metals & Min. 1.86 1.42 1.4 0

2. Iran Khodro D.

1.88 1.1

1.41 0 27. Melli 1.9 0 1.4 1.41

3. EN Bank 1.92 0 1.49 2.76 28. Oil Ind. Inv. 1.9 0.71 1.51 3.1 4.Pars Darou 1.95 3.13

4.61 1.98 29.Tehran Cement 1.89 0 1.41 0.32

5.Abadan Petr. 1.92 3.53

1.4 1.92 30. Fars Cement 1.89 0.18 1.48 4.38

6.Khark Petr. 1.92 5.61

2.75 2.89 31. Shahdiran Inc. 1.86 0 1.51 0

7.Farabi Petr. 1.93 0 1.5 0 32.Behshahr Ind. 1.49 5.33 4.5 3.41 8. Zangan

Equip. 2.12 6.07

4.6 1.36 33. Sanati Daryayi 1.85 0 1.4 0

9. Iran Tractor 1.89 1.01

3.86 2.06 34. Doroud Farsit 4.58 6.48 1.41 2.46

10. Chadormalu

1.94 6.48

1.5 4.1 35.Azar Refract. 1.9 0 1.4 0.62

11.Jaber Hayan P.

1.95 0.02

1.52 6.04 36.Calcimine 2.02 6.48 1.41 0

12.Zamyad 1.89 0.04

1.52 0 37.Iran Carbon 1.95 0 1.4 2.23

13.Saipa 1.85 0.08

1.51 6.66 38.Kaf 1.95 1.77 1.51 4.03

14.Saipa Diesel

1.88 0 1.41 1.85 39.Gazlouleh 1.9 0 1.4 0

15.Pension Fund

1.85 0.01

1.4 2.96 40.Bahman Group 1.85 0 1.51 5.23

16. Buali Inv. 1.89 0.34

1.4 0 41.Sadid Group 1.85 0 4.63 2.31

17.Pars Tousheh

2.02 3.14

1.45 3.06 42.Looleh o Mashin Sazi

2.11 6.48 1.4 0.61

18.Petro. Inv. 1.93 0 1.4 0.68 43.Niromohareke M.

1.92 0 1.85 1.79

19. Iran Ind. Dev.

1.94 0 1.4 0 44.Mehvar Sazan 2.2 0 1.52 1.29

20. Rena Investment

1.9 0.73

4.51 3.36 45.Iran Zinc Mines 1.94 0.08 1.4 0.71

21.Insurance Inv.

1.9 0 1.52 5.78 46.Iran Mn. Mines 1.97 6.48 1.4 0

22.Ind. & Mine Inv.

1.88 0.55 1.46 1.73 47.Mehrcam Pars 1.98 6.48 1.43 6.02

23.Ghadir Inv. 1.89 0 1.4 0 48.Motogen 1.91 3.84 1.52 1.74 24.Behshahr

Group 2.02 6.48 1.46 0.03 49.Behran Oil 4.75 6.48 4.55 1.11

25.Housing Inv.

1.9 3.16 1.5 4.34 50.Pars Oil 1.92 6.19 1.4 2.55

Figure 3. Mean/Best fitness functions value versus the GA’s generation

(Annual) (Source:Calculated by Authors)

Figure 4. Mean/Best fitness functions value versus the PSO’s iteration

number (Annual) (Source: Calculated by Authors)

435

Figure 5. Mean/Best fitness functions value versus the GA’s generation (Monthly) (Source: Calculated by Authors)

Figure 6. Mean/Best fitness functions value versus the PSO’siteration Number(Monthly) (Source: Calculated by Authors)

TABLE II. SUMMARY OF EXPECTED/ACHIEVED PORTFOLIOS’ RETURN AND RISK, AND A COMPARISON OF PORTFOLIOS PERFORMANCES WITH MARKET PORTFOLIO’S RETURN (SOURCE: CALCULATED BY AUTHORS)

Algorithm/Input data

Expected Return

(percentage)

Expected Risk

(percentage)

Real achieved rate of return

during test period

(percentage)

Market portfolio’s

return during the same test

period (percentage)

Genetic/Annual 29.5293 50.4426 4.2400 4.159 Genetic/Monthly 0.7990 4.2704 3.6917 4.159

PSO/Annual 33.8522

30.2732

4.1211 4.159

PSO/Monthly 0.9651 3.8290 2.8213 4.159

TABLE III. GA/PSO’S BEST PARAMETERS, CONSISTENT WITH PORTFOLIO SELECTION PROBLEM (SOURCE: CALCULATED BY AUTHORS)

Population type (GA) Double Vector Damping factor start/end points (PSO)

0.98 /0.1326

Selection function (GA)

Roulette Damping factor and �(t), inertia weight (PSO)

Exponential, descending

Crossover fraction (GA)

0.8 Population/Swarm size (GA/PSO)

500/500

Crossover function (GA)

Scattered Number of generation/Iteration Number (GA/PSO)

150/100

Mutation function (GA)

Gaussian Function tolerance (GA/PSO)

10-6/10-6

c1,Cognitive parameter (PSO)

2 Stall time limit(GA/PSO)

Infinite/ Infinite

C2,Social parameter (PSO)

2 Stall generations (GA/PSO)

Infinite/ Infinite

REFERENCES [1] Hagin, R.L., The Dow Jones-Irwin Guide to Modern Portfolio Theory,

Homewood, Illinois: Dow Jones-Irwin, 1979. [2] Brealey, R.A., An Introduction to Risk and Return from Common

Stock Prices, Massachusetts: M.I.T Press, 1969. [3] Evans, J.L., Stephen H. Archer, "Diversification and the Reduction of

Dispersion: An Empirical Analysis," Journal of Finance, vol.23, no.(12), 1968, pp. 761-767.

[4] Gaumnitz, J.E., "Investment Diversification under uncertainty: An Examination of the Number of Securities in a Diversified Portfolio," Stanford University, 1967, unpublished thesis.

[5] Markowitz, H., "Portfolio Selection," Journal of finance, vol.7, no.1, 1952, pp. 77-91.

[6] Hagin, R.L., Investment Management - Portfolio Diversification, Risk, and Timing - Fact and Fiction, ed.1st, W. finance. Hoboken, New Jersey: John Wiley & Sons, Inc, 2004.

[7] Dorigo, M., V. Maniezzo, Colorni A, "The ant system: optimization by a colony of cooperating agents," IEEE Trans Syst Man Cybernet B, vol. 26, no. 1, 1996, pp. 29–41.

[8] Kennedy, J., R.C. Eberhart, "Particle swarm optimization," in The IEEE International Joint Conference on Neural Networks, Piscataway, NJ., IEEE Press, 1995.

[9] Eberhart, R.C., J. Kennedy, "A new optimizer using particle swarm theory," in The Sixth International Symposium on Micromachine and Human Science, Nagoya, Japan, 1995.

[10] Markowitz, H., Portfolio Selection: Efficient Diversification of Investments, New York: John Wiley & Sons, 1959.

[11] Sharpe, W.F., Investments, Englewood Cliffs, U.S.A: Prentice-Hall, Inc, 1978.

[12] Williams, J.B., The Theory of Investment Value, Cambridge, Mass.: Harvard University Press, 1938.

[13] Parker Jones, C., Investments: Analysis and Management, New York, Wiley finance, 1999.

[14] Bonabeau, E., M. Dorigo, G. Th´eraulaz, From Natural to Artificial Swarm Intelligence, New York: Oxford University Press, 1999.

[15] Kennedy, J., R.C. Eberhart, Swarm Intelligence, Morgan Kaufman, 2001.

[16] Bishop, C.M., Neural Networks for Pattern Recognition, Oxford University Press, 1995.

[17] Fiesler, E., R. Beale, Handbook of Neural Computation, IOP Publishers and Oxford University Press, 1996.

[18] Back, T., Evolutionary Algorithms in Theory and Practice, Evolution Strategies, Evolutionary Programming, Genetic Algorithms, New York, Oxford: OXFORD UNIVERSITY PRESS, 1996.

[19] Back, T., David �. Fogel, and Zbigniew Michalewicz, Handbook of Evolutionary Computation, IOP Publishers and Oxford University Press, 1997.

[20] Back, T., David �. Fogel, and Zbigniew Michalewicz, Evolutionary Computation 1 Basic Algorithms and Operators, Bristol and Philadelphia: INSTITUTE OF PHYSICS PUBLISHING, 2000.

[21] Dorigo, M., T. Stu¨ tzle, Ant Colony Optimization, Massachusetts: MIT Press, 2004.

[22] Engelbrecht, A.P., Fundamentals of Computational Swarm Intelligence, New York: John Wiley & Sons, 2005.

[23] Holland, J., Adaptation in natural and artificial systems, Massachusetts: MIT press, 1975.

[24] G´omez, S.F., Jim´enez F, "Fuzzy modeling with hybrid systems," Fuzzy Sets and Systems, vol. 104, 1999, pp. 199-208.

[25] Gopi, E.S., Algorithm Collections for Digital Signal Processing Applications Using Matlab, Dordrecht, The Netherlands: Springer, 2007.

[26] Pourzeynali, S., H.H. Lavasani, and A.H. Modarayi, "Active control of high rise building structures using fuzzy logic and genetic algorithms," Engineering Structures, vol.29, 2006, pp. 346–357.

[27] Hu, X., R. Eberhart , and Y. Shi, "Engineering optimization with particle swarm," in IEEE swarm intelligence symposium(SIS 2003), Indianapolis, IN., 2003, pp. 53–70.

[28] Hassan R., B.C., O. de Weck, and G. Venter, "A comparison of particle swarm optimization and the genetic algorithm," in 1st AIAA multidisciplinary design optimization specialist conference, Austin, TX , 2005, doi No.: AIAA-2005-1897.

[29] Perez, R.E., K. Behdinan, "Particle swarm approach for structural design optimization," Computers and Structures, vol. 85, 2007, pp. 1579–1588.

436

[30] Parsopoulos, K.E., M.N. Vrahatis, "Parameter selection and adaptation in Unified Particle Swarm Optimization," Mathematical and Computer Modelling, vol. 46, 2007, pp. 198–213.

[31] Eberhart, R.C., Y. Shi, "A modified particle swarm optimizer," in Proceedings of the IEEE World Conference on Computational Intelligence, Anchorage, Alaska, 1998.

[32] Solnik, B.H., "The International Pricing of Risk: An Empirical Investigation of the World Capital Market Structure," Journal of Finance, vol. 29, no. 2, 1974, pp. 364-378.

[33] Solnik, B.H., "Why Not Diversify Internationally Rather Than Domestically?," Financial Analysis Journal, vol. 30, no.4, 1974, pp. 45-54.

[34] The MathWorks, I., Matrix laboratory (MATLAB). 1984-2009 ©The MathWorks, Inc.

437