Swarm Intelligence in Portfolio Selection

Abstract

Portfolio selection problems in investments are among the most studied in modern finance, because of their computational intractability. The basic perception in modern portfolio theory is the way that upon it investors construct diversified portfolio of financial securities so as to achieve improved tradeoffs between risk and return. Portfolio optimization is a procedure for generating the composition that best achieves the portfolio manager's objectives. One of the first to apply mathematical programming models to portfolio management was the quadratic programming model of Markowitz (1952), who proposed that risk be represented as the variance of the return (a quadratic function), which is to be minimized subject to achieving a minimum expected return on investment (a linear constraint). This single-period model is explained in detail by Luenberger (1998). The inputs of this analysis are security expected returns, variances, and covariance for each pair of securities, and these are all estimated from past performances of the securities. However, it is not realistic for real ever-changing asset markets. In addition, it would be so difficult to find the efficient portfolio when short sales are not allowed. Mathematical programming (e.g., linear programming, integer linear programming, nonlinear programming, and dynamic programming) models have been applied to portfolio management for at least half a century. For a review on the application of mathematical programming models to financial markets refer to Board and Sutcliffe (1999). Several portfolio optimization strategies have been proposed to respond to the investment objectives of individuals, corporations and financial firms, where the optimization strategy is selected according to one's investment objective. Jones (2000) gives a framework for classifying these alternative investment objectives. Although the most obvious applications of portfolio optimization models are to equity portfolios, several mathematical programming methods (including linear, mixed integer, quadratic, dynamic, and goal programming) have also been applied to the solution of fixed income portfolio management problems since the early 1970s. Recently, many Evolutionary Computation (EC) techniques (Beyer, 1996) such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO) (Xu et al., 2006), (Delvalle et al., 2007) have been applied to solve combinatorial optimization problems (Angeline, 1995). These techniques use a set of potential solutions as a population, and find the optimal solution through cooperation and contest among the particles of the population. In comparison, in