Active Portfolio Management With Cardinality Constraints: An Application Of Particle Swarm Optimization

This paper considers the task of forming a portfolio of assets that outperforms a benchmark index, while imposing a constraint on the tracking error volatility. We examine three alternative formulations of active portfolio management. The first one is a typical set up in which the fund manager myopically maximizes excess return. The second formulation is an attempt to set a limit on the total risk exposure of the portfolio by adding a constraint that forces a priori the risk of the portfolio to be equal to the benchmark’s. The third formulation, presented in this paper, directly maximizes the efficiency of active portfolios, while setting a limit on the maximum tracking error variance. In determining optimal active portfolios, we incorporate additional constraints on the optimization problem, such as a limit on the maximum number of assets included in the portfolio (i.e. the cardinality of the portfolio) as well as upper and lower bounds on asset weights. From a computational point of view, the incorporation of these complex, though realistic, constraints becomes a challenge for traditional numeric optimization methods, especially when one has to assemble a portfolio from a big universe of assets. To deal properly with the complexity and the “roughness” of the solution space, we use particle swarm optimization, a population-based evolutionary technique. As an application, we select portfolios of different cardinality that actively reproduce the performance of the FTSE/ATHEX 20 Index of the Athens Stock Exchange. Our empirical study reveals important results as concerns the efficiency of common practices in active portfolio management and the incorporation of cardinality constraints.