بهینه سازی ازدحام ذرات (PSO) برای مشکل بهینه سازی پرتفولیو اجباری

One of the most studied problems in the financial investment expert system is the intractability of portfolios. The non-linear constrained portfolio optimization problem with multi-objective functions cannot be efficiently solved using traditionally approaches. This paper presents a meta-heuristic approach to portfolio optimization problem using Particle Swarm Optimization (PSO) technique. The model is tested on various restricted and unrestricted risky investment portfolios and a comparative study with Genetic Algorithms is implemented. The PSO model demonstrates high computational efficiency in constructing optimal risky portfolios. Preliminary results show that the approach is very promising and achieves results comparable or superior with the state of the art solvers.

Portfolio management is one of the most studied topics in finance. The problem is concerned with managing the portfolio of assets that minimizes the risk objectives subjected to the constraint for guaranteeing a given level of returns. This paper deals with the mean–variance portfolio selection, which is formulated in a similar way as Markowitz did (Elton et al., 1976, Markowitz, 1952 and Steinbach, 2001). Markowitz introduced the concepts of Modern Portfolio Theory (MPT). His theory has revolutionized the way people think about portfolio of assets, and has gained widespread acceptance as a practical tool for portfolio optimization. But in some cases, the characteristics of the problem, such as its size, real-world requirements (Campbell et al., 2001, Gennotte, 1986, Louis et al., 1999, Perold, 1984 and Zhou and Li, 2000), very limited computation time, and limited precision in estimating instance parameters, may make analytical methods not particularly suitable for tackling large instances of the constrained mean–variance model. Therefore researchers and practitioners have to resort to heuristic techniques that are able to find high-quality solutions in a reasonable amount of time. Due to the complexity and the instantaneity of the portfolio optimization model, applying meta-heuristic algorithms to portfolio selection and optimization is a good alternative to meet the challenge. Some remarkable studies have been presented to solve asset selection problem. Many meta-heuristic techniques (Chang, Meade, Beasley, & Sharaiha, 2000) have been applied in portfolio selection such as Genetic Algorithms, tabu search and simulated annealing for finding the cardinality constrained efficient frontier. Some hybrid techniques (Gaspero, Tollo, Roli, & Schaerf, 2007) have been applied in portfolio management such as local search and quadratic programming procedure. Preliminary results show that the approach is very promising and achieves results comparable or superior to the traditional solvers. Pareto Ant Colony Optimization (Doerner, Gutjahr, Hartl, Strauss, & Stummer, 2004) has been introduced as an especially effective meta-heuristic for solving the portfolio selection problem and compares its performance to other heuristic approaches (i.e., Pareto Simulated Annealing and the Non-Dominated Sorting Genetic Algorithm) by means of computational experiments with random instances. An artificial neural network model with the Particle Swarm Optimization algorithm (Giovanis, 2009) has been applied to portfolio management and shows the flexibility of hybrid models, such as the superiority in forecasting performance, in relation to the traditional econometric methodology, like Ordinary least square and ARCH-GARCH estimations. Fuzzy Analytic Hierarchy Process (AHP) (Tiryaki & Ahlatcioglu, 2009) has been combined with the portfolio selection problem to model the uncertain environments. A hybrid Genetic Algorithm approach (Jeurissen & van den Berg, 2005) has been investigated for tracking the Dutch AEX-index, it focused on building a tracking portfolio with minimal tracking error. However, these approaches have some shortcomings in solving the portfolio selection problem. For example, fuzzy approach usually lacks learning ability (Chan, Wong, Tse, Cheung, & Tang, 2002); Artificial neural network approach has over-fitting problem and is often easy to trap into local minima (Casas, 2001); while as Genetic Algorithms (Alba & Troya, 1999) are applied to harder and bigger problems there is an increase in the time required to converge for finding adequate solutions. In order to overcome these drawbacks, PSO model is introduced to solve the portfolio selection and optimization problem. PSO is a population based stochastic optimization technique developed in 1995 (Kennedy & Eberhart, 1995). The underlying biological metaphor for developing PSO algorithm is inspired by social behavior of bird flocking or fish schooling. PSO has become a popular optimization method as they often succeed in finding the best optimum by global search in contrast with most common optimization algorithms. In comparison with the dynamic programming, PSO allows the users to get the sub-optimal solution while dynamic programming cannot. It is very important for the portfolio selection and optimization problem. There are very few studies on PSO, especially all most none of them deal with the performance comparison with other approaches for solving portfolio optimization problems. The main contribution of this study is to employ a PSO algorithm for portfolio selection and optimization in investment management. Asset allocation in the selected assets is optimized using a PSO based on Markowitz’s theory. Using the PSO, an optimal portfolio can be determined. The rest of the paper is organized as follows. Section 2 describes models for portfolio optimization. In Section 3, the background of PSO and previous work are summarized. The PSO model for optimal portfolio is also discussed. In order to test the efficiency of the proposed PSO solver, a simulation and comparative study is performed in Section 4. Final conclusions and future research are drawn in Section 5.

A fundamental principle of financial investments is diversification where investors diversify their investments into different types of assets. Portfolio diversification minimizes investors’ exposure to risks, and maximizes returns on portfolios. The paper focuses on solving the portfolio optimization problem in finance investment management. A meta-heuristic Particle Swarm Optimization method has been developed to optimize investment portfolios, where the objective functions and constraints are based on both the Markowitz model and the Sharp Ratio model. The PSO algorithm bears similarity to other biologically inspired optimizing algorithms. Like the GA, it is population-based, it is typically initialized with a population (swarm) of random encodings of solutions, and search proceeds by updating these encodings over a series of generations (iterations). Unlike the GA, PSO has no explicit selection process as all particles persist over time. Instead a memory in the form of gbest/pbest is substituted for selection. In order to make a valid comparison with other methods, different test problems were solved and the results obtained when compared with the results of Genetic Algorithms (GA), Visual Basic for Applications (VBA) demonstrated the superiority of the PSO algorithm. The key learning mechanisms in the PSO algorithm are driven by a metaphor of social behavior that good solutions uncovered by one member of a population are observed and copied by other members of the population. Of course, these learning mechanisms abound in business and other social settings. Good business strategies, good product designs, and good theories stimulate imitation and subsequent adaptation. Particle swarm algorithms will be a successful optimization tool in a variety of applications, and has clear potential for application to financial modeling. Future research may be conducted to further investigate the application of some derived models or hybrid models of PSO to other investment strategy problems, for example tracking the index and so on. Another further investigation may be put on methods for improving the efficiency of the PSO solver for large portfolios in investment management.