A review on the current applications of genetic algorithms in mean-variance portfolio optimization portföy optimizasyonunda genetik algoritma araştırması

Mean-variance portfolio optimization model, introduced by Markowitz, provides a fundamental answer to the problem of portfolio management. This model seeks an efficient frontier with the best trade-offs between two conflicting objectives of maximizing return and minimizing risk. The problem of determining an efficient frontier is known to be NP-hard. Due to the complexity of the problem, genetic algorithms have been widely employed by a growing number of researchers to solve this problem. In this study, a literature review of genetic algorithms implementations on mean-variance portfolio optimization is examined from the recent published literature. Main specifications of the problems studied and the specifications of suggested genetic algorithms have been summarized.


Introduction: Portfolio optimization and mean-variance model
Investors' desire is to have a non-decreasing fund even if the market is losing value. It is not always possible to achieve this by investing on only one security. In a financial market, it is rare that all securities gain or lose value at the same time. Therefore, an investor should use a diversification strategy, such as forming a portfolio, to spread the risk among assets. The main question in portfolio management is to decide on the assets and weights for a better investment. A fundamental answer to the problem of portfolio management was given by the mean-variance model [1], [2]. Mathematical formulation of unconstrained portfolio optimization problem (UCPO) according to Markowitz's standard mean-variance approach is given as follows where parameter represents the number of available assets, represents the expected return of asset , represents the covariance between asset and asset and asset , * represents the expected return at the desired level and variable represent the proportion of asset : Equation (1) minimizes risk of the portfolio while equation (2) ensures that expected return ( * ) is at the desired level. Equation (3) guarantees that proportions add to one while the proportion of an asset neither can be less than zero nor can be greater than one (Equation (4)). In practice, it is possible to calculate an optimal solution for a particular data set with this formulation. Solving this formulation by varying values of the expected return, an efficient frontier can be found as a non-decreasing curve. This frontier represents the balance of expected return corresponding to risk that must be accepted. Figure 1 demonstrates a standard efficient frontier. The mean-variance model has been extended throughout the decade by introducing additional real-world constraints such as the cardinality constraints that impose a predetermined limit on the number of assets ( ) to be held in the portfolio and the quantity constraints which restrict the proportion of each asset in the portfolio to satisfy lower ( ) and upper ( ) bounds. The mixed integer nonlinear programming formulation of cardinality constrained portfolio optimization (CCPO) problem is given as follows with additional parameters; representing the desired number of assets to be hold in the portfolio, representing the minimum proportion of asset , representing the maximum proportion of asset and representing a binary variable whether or not an asset is held in the portfolio [3]: Subject to: Equation (6) and equation (10) are inherited from the original Markowitz formulation. Equation (7) guarantees that exactly assets are hold in the portfolio while equation (8) restricts the proportion of an asset to be between predetermined values of minimum and maximum limits with decision variable defined in Equation (9). Equation (11) defines variable domains. The quadratic objective function given in equation (5) seeks the best trade-offs between two conflicting objectives, maximizing return and minimizing risk. In the equation (5), a parameter is used to trace the efficient frontier by gradually increasing the value of from 0 to 1. Thus, a weighted sum of two objectives is obtained. The resulting single objective aims to construct the cardinality constrained efficient frontier that represents the best balance of expected return and the risk that must be accepted since both objectives cannot be simultaneously achieved. Transaction costs are also considered as additional real life constraints in the literature [4]- [7]. It is not convenient to find an optimal efficient frontier in practice when real life constraints are taken into account. In fact, calculating an optimal portfolio for the standard mean-variance model is known to be NP-hard [8] since a classical quadratic optimization problem becomes NP-hard if a single cardinality constraint is added to the formulation [9]. Therefore, in the literature, several computationally efficient solution approaches have been developed in order to calculate the efficient frontier. Among those approaches, genetic algorithms (GA) is one of the most preferred algorithm for solving the problem. Metaxiotis and Liagkouras [10] presented a review of multi-objective evolutionary algorithms applied to portfolio management problem in a broad problem perspective. In this study, however, a comprehensive review of GA applications including single and multi-objective implementations specifically in mean-variance portfolio optimization is conducted. This aim of this review is to reveal the problem specifications considered and the key strategies of GA utilized to solve mean-variance portfolio optimization problem types. Section 2 presents the genetic algorithm implementations for mean-variance portfolio optimization while Section 3 concludes the paper with a discussion of future research directions.

Genetic algorithms for mean-variance portfolio optimization
GA, firstly introduced by Holland [11], is a search method that can be modified to solve complex optimization problems. In GA, a set of iterative search procedures based on biological natural selection and genetic inheritance principals is executed. A population of solutions is updated over generations using selection, crossover and mutation strategies. Each individual that is evaluated in the population represents a potential solution to the problem in hand. Individuals form new individuals a stochastic transformation of individuals is achieved by genetic operators such as crossover and mutation. Crossover provides better solutions to be constructed from good solutions by a random, yet structured change of genetic materials. The role of mutation is to obtain lost or unexplored genetic materials, thereby preventing premature convergence and stuck in local optima.
After several iterations, the algorithm converges a (near) optimal solution. Basic steps of the GA are given in Table 1. Table 1: Basic steps of GA.
Step Procedure 1 Generate initial population. 2 Evaluate fitness of each individual in the population. 3 Select the set of individuals for applying genetic operators. 4 Apply genetic operators and evaluate new fitness values. 5 Form new generation according to fitness values. 6 Go to step 3 if termination criteria are not satisfied. 7 End evolution and report the results.
Components of a typical GA are summarized below: -Genetic representation (Encoding strategy): The solution of the problem that is formed by binary, integer or real numbers, -Chromosome: A solution of encoding, -Population: A set of chromosomes, -Fitness: A function that evaluates how good a solution is, -Genetic operators: Procedures such as crossover and mutation that provide to obtain new population from the current population, -Control parameters: Input parameters such as population size, crossover and mutation rates.
Goldberg [12] pointed out search and optimization applications of GA in different areas. Efficient portfolio selection is one of the main concerns of researchers who practice in financial optimization domain. One of the most preferred solution approach for portfolio optimization is GA. Several researchers applied GA variants for solving portfolio optimization problems since 1998 [13]. In this study, 44 articles published in conferences and refereed journals between 1998-2016 are examined. Figure    Studies in the literature can be classified in many ways. In this review, two classification schemes for studies that apply GA to mean-variance portfolio optimization are used. First classification is formed according to the problem specifications such as data type, compared methods, problem type and coded programming language. Table 2 provides an up-to-date list of problem specifications. As summarized in Table 2, experimental settings are generally carried out on either real world applications or hypothetical data sets for benchmarking purposes. Methods reported in the literature are generally compared against other metaheuristics either taken from the literature or coded by the authors themselves. Literature analysis show that most of the problems types considered so far consist of UCPO, CCPO and POTC. Most of recent studies focused on CCPO and POTC while UCPO provided a basis for other types of problems.
The second classification scheme is formed according to applied algorithm specifications such as the generation methodology of initial population, size of the population, chromosome representation, crossover type and rate, mutation type and rate, type of selection mechanism, survival type, feasibility construction and termination criteria. Table 3 provides an up-to-date list of algorithm specifications. As summarized in Table 3, single objective GA are widely applied while some multi-objective GA are also suggested for meanvariance portfolio optimization. A two-stage GA is employed in [14] that firstly identifies good quality assets in terms of asset ranking and then optimizes investment allocation in the selected good quality assets. Some hybrid strategies are also suggested as in [15] that utilize quadratic programming approach with GA, in [16] that combines GA with simulated annealing approach and in [17] that utilizes a position displacement strategy of the particle swarm optimization methodology with GA.
GA implementations in the literature shows that initial population is widely preferred to be randomly generated while just a few studies [15], [18] employed heuristic approaches for the construction of initial population. In the studies examined, population size (PS) parameter is set to be 20 as minimum and 2000 as maximum. However, most of the researchers zoomed in the range of 100 and 300 for PS parameter. Binary and real valued chromosome representation are observed to be popular although some other representation strategies such as integer based, tree based are also utilized. Several studies differentiate from each other with the use of crossover operators such as uniform, BLX-Alpha, one point, n-point and mutation operators such as swap, one-point, Gaussian, guided, bit-flipping strategies. Tournament selection is mostly utilized while roulette wheel selection is also used. Elitism and ranking strategies are generally employed for survival of population. Feasibility of chromosomes are ensured by repair or penalty functions. It is observed that iteration number is used as the termination criterion in all of the studies examined.
Although several authors used the data as downloaded from the mentioned OR-Library to test their proposed algorithm, unfortunately, there is a limited number of papers that provide a performance comparison against other published papers in the literature.
In terms of evaluation approaches, there are two types of methodologies in the literature, namely weighted sum and pareto based approaches. As for weighted sum approach, studies make use of equation (5) given in Section 1 by combining two conflicting objectives: risk minimization and return maximization. On the other hand, pareto based methodology, especially used in multi-objective evolutionary algorithms, considers two objectives separately by systematically removing dominated solutions from the heuristic frontier during the search in solution space. Table 3 summarizes the classification.

Introduction
Portfolio optimization is a significant problem that intrigues investors and challenges researchers. As GA was established to be a popular technique in the optimization field, the application of GA to optimization problems related to portfolio selection has expanded since 2000 as the problem is known to be NP-Hard. Two different classifications are introduced. Firstly, the main specifications of the problems were summarized, and then implemented GAs with chromosome representations, genetic operators and the fitness functions used for performance evaluation were discussed. 44 articles were examined and grouped in chronological order.
Although, there are several implementations of GA for mean-variance portfolio optimization problem, unfortunately, the improvements and enhancements made to the algorithms' main framework is not evidently noticeable since there is a limited number of papers that provide a benchmark based comparison against other published studies in the literature.
Therefore, future studies should definitely consider such a comparison that may lead the way towards a better algorithmic design and related software implementations.

Acknowledgment
This research is funded by the Scientific and Technological Research Council of Turkey (TUBITAK) with the grant number 214M224.