In this paper, we discuss the portfolio selection problem with transaction costs under the assumption that there exist admissible errors on expected returns and risks of assets. We propose a new admissible efficient portfolio selection model and design an improved particle swarm optimization (PSO) algorithm because traditional optimization algorithms fail to work efficiently for our proposed problem. Finally, we offer a numerical example to illustrate the proposed effective approaches and compare the admissible portfolio efficient frontiers under different constraints.
In 1952, Markowitz [1] published his pioneering work which laid the foundation of modern portfolio analysis. Markowitz’s mean–variance model has served as a basis for the development of modern financial theory over the past five decades. With the continuous effort of various researchers, Markowitz’s seminal work has been widely extended. In the mean–variance portfolio selection problem, previous research includes Refs. [2], [3] and [4]. More researches on portfolio selection may be found in Refs. [5], [6] and [7].
Recently, a number of researchers have investigated fuzzy portfolio selection problems. Tanaka et al. [8] proposed the portfolio selection model based on fuzzy probabilities, which can be regarded as a natural extension of Markowitz’s model because of extending probability into fuzzy probability. By using fuzzy approaches, the experts’ knowledge and investors’ subjective opinions can be better integrated into a portfolio selection model. Ida [9], Lai et al. [10] and Giove et al. [11] constructed interval programming models of portfolio selection. Zhang and Nie [12], Zhang et al. [13], and Zhang and Wang [14] discussed the admissible efficient portfolio selection under the assumption that the expected return and risk of assets have admissible errors to reflect the uncertainty in real investment actions and gave an analytic derivation of admissible efficient frontier when short sales are not allowed on all risky assets.
It is well known that transaction cost is one of the main concerns for portfolio managers. It has an important effect on the portfolio optimization and the frequency of time rebalancing the portfolio. Arnott and Wagner [15] found that ignoring transaction costs would result in an inefficient portfolio. The experimental analysis done by Yoshimoto [16] also verified this fact. Moreover, when some more realistic constraints such as transaction costs, bounded constraints, liquidity constraints, minimum transaction lots constraints, and cardinality constraints are considered, the portfolio selection problem becomes a complex programming problem and traditional optimization algorithms fail to find the optimal solution efficiently. Therefore, many researchers solve the complex constrained portfolio problems by using heuristic algorithms. For example, Xia et al. [17] designed a genetic algorithm for portfolio selection problem with order of expected returns. Chang et al. [18] used heuristics algorithms based upon genetic algorithms, tabu search and simulated annealing for the cardinality constrained mean–variance model. Speranza [19] used linear programming based heuristics algorithms for a portfolio optimization model with transaction costs, minimum transaction units and limits on minimum holdings. Jobst et al. [20] designed two heuristic solution procedures, ‘integer restart’ and a two-stage ‘reoptimization’ heuristic, for the mean–variance model with buy-in thresholds, cardinality and round lots constraints. Fernández and Gómez [21] presented heuristics algorithms based upon neural networks for the standard Markowitz mean–variance model with cardinality and bounding constraints. Crama and Schyns [22] applied simulated annealing to a portfolio problem with cardinality, turnover and trading constraints.
However, there are few researches in modelling and solving portfolio selection problems by using the particle swarm optimization (PSO) algorithm proposed by Kennedy and Eberhart [23] and [24] in previous work. In this paper, we will discuss the admissible portfolio selection problem with transaction costs and bounded constraints. We also present an improved PSO algorithm for the portfolio selection problem.
The organization of this paper is as follows. We present the admissible portfolio selection model with transaction costs and bounded constraints in Section 2. An improved PSO algorithm is designed to solve the corresponding portfolio problem in Section 3. A numerical example is given to illustrate our proposed effective means and approaches, and the admissible portfolio efficient frontiers under different constraints are compared in Section 4. Some concluding remarks are given in Section 5.
Inthispaper,wehavediscussedtheadmissibleportfolioselectionproblemwhichincludestransactioncostsandbounded
constraints.Althoughtherearesometraditionalalgorithmsforourproposedproblem,theycannotbehavewell.Therefore,basedonthestandardPSOwehavedesignedanimprovedPSOalgorithmtosolveourproposedportfolioselectionproblem.
Wehavealsogivenanumericalexampleoftheportfolioselectionproblemtoillustrateourproposedeffectivemeansand
approaches.ExperimentresultsshowthattheimprovedPSOalgorithmiseffectiveforsolvingcomplexconstrainedportfolio
selectionproblemandtransactioncostsandboundedconstraintshaveagreatimpactontheportfolioselectiondecision.
