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

Abstract

y v.vassiliadis@fme.aegean.gr z g.dounias@aegean.gr 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¯rst one is a typical setup 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. In this paper, we also propose a third formulation that directly maximizes the e±ciency 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 numerical optimization methods, especially when one has to assemble a portfolio from a big universe of assets. To deal * We are thankful to the two anonymous referees for their valuable comments and suggestions.New Mathematics and Natural Computation Vol. 5, No. 3 (2009) 535À555 # . c World Scienti¯c Publishing Company 535 New Math. and Nat. Computation 2009.05:535-555. Downloaded from www.worldscientific.com by UNIVERSITY OF QUEENSLAND on 08/15/15. For personal use only.properly with the complexity and the \roughness" of the solution space, we use particle swarm optimization, a population-based evolutionary technique. As an empirical application of the methodology, we select portfolios of di®erent cardinality that actively reproduce the performance of the FTSE/ATHEX 20 Index of the Athens Stock Exchange. Our empirical study reveals important results concerning the e±ciency of common practices in active portfolio management and the incorporation of cardinality constraints.

An Evolutionary Approach to Preference Disaggregation in a MURAME-Based Credit Scoring Problem

“…To avoid this problem, different strategies have been proposed in the literature, and most of them involve the repositioning of the particles ( [22]) or the introduction of some external criteria to rearrange the components of the particles ( [8] and [20]). In this paper we follow the same approach adopted in [6], which consists in keeping PSO as in its original formulation and reformulating the optimization problem into an unconstrained one: min w,q,t,s P (w, q, t, s; ε) (17) where the objective function P (w, q, t, s; ε) is defined as follows:…”

Particle Swarm Optimization with Non-Smooth Penalty Reformulation for a Complex Portfolio Selection Problem

“…When constraints are included, different strategies were proposed in the literature (see also [3]) to ensure that at any step of PSO, feasible positions are generated. Most of them involve repositioning of the particles, as for example the bumping and the random positioning strategies proposed in [32], or introducing some external criteria to rearrange the components of the particles, as the ones specific for cardinality constraints proposed in [20,10,28]. However, in this paper we decided to use PSO coherently with its original formulation, that is as a tool for the solution of unconstrained optimization problems.…”

“…The latter approach is known in the literature of constrained optimization as exact penalty method, where the term exact refers to the correspondence between the minimizers of the original constrained problem and the minimizers of the unconstrained (penalized) one. Some applications presented in the literature in which PSO is applied to portfolio selection problems for minimizing penalty functions or seemingly penalty functions are given in [6,28] respectively. Nevertheless, unlike our solution approach, the solution method proposed in [6] is not based on exact penalty functions, and it does not consider integer unknowns.…”

“…Moreover, the rule used in that paper for updating the parameter values appears to be far too problem dependent. Conversely, the solution method proposed in [28] has only the appearance of a penalty approach, but to a large extent it is based on so-called Lagrangian methods, inasmuch as the entire vector of the Lagrangian multipliers associated to the constraints is estimated at each iterate. With regard to that, generally Lagrangian methods have been proven to be effective in practice, but they require an accurate estimation of the Lagrangian multipliers, which is unnecessary for penalty-based solution approaches.…”

Cooperating swarms: A paradigm for collective intelligence and its application in finance.

“…Recent developments in such advanced computational methods, such as neural network and evolutionary computation have opened new doors in computational finance. In [19], the authors had effectively applied PSO to select active portfolios under a constraint on tracking error volatility. Their work considers the task of forming a portfolio of assets that outperforms a benchmark index, while imposing a constraint on the tracking error volatility.…”