Solving Cardinality Constrained Portfolio Optimization Problem by Binary Particle Swarm Optimization Algorithm

Mathematical programming methods dominate in the portfolio optimization problems, but they cannot be used if we introduce a constraint limiting the number of different assets included in the portfolio. To solve this model some of the heuristics methods (such as genetic algorithm, neural networks and particle swarm optimization algorithm) must be used. In this paper we utilize binary particle swarm optimization algorithm and quadratic programming method to find an efficient frontier in portfolio optimization problem. Two datasets are utilized. First dataset consists of the stocks incorporated in the Dow Jones Industrial Average, second dataset contains stocks from the Standard & Poor's 500. The comparison of found efficient frontiers for different limitation on the number of stock held is made at the close of the paper. Introduction Proper allocation of the funds is nowadays getting more and more important. With the increasing amount of the money fund managers administer, their responsibility is increasing and quantitative approaches get more attention than qualitative. In the field of the portfolio optimization the pioneer work was Markowitz mean-variance model [1]. Assuming that assets returns follow a multivariate normal distribution, we are concerned only in the portfolio expected return and variance. We are thus looking for the portfolios with maximum expected return and minimal variance. This Pareto efficient set of portfolios is called an efficient frontier (EF). Selection of one optimal portfolio from the efficient frontier then depends only on the risk attitude.