The survey of the relevant literature showed that there have been many studies for portfolio optimization problem and that the number of studies which have investigated the optimum portfolio using heuristic techniques is quite high. But almost none of these studies deals with particle swarm optimization (PSO) approach. This study presents a heuristic approach to portfolio optimization problem using PSO technique. The test data set is the weekly prices from March 1992 to September 1997 from the following indices: Hang Seng in Hong Kong, DAX 100 in Germany, FTSE 100 in UK, S&P 100 in USA and Nikkei in Japan. This study uses the cardinality constrained mean-variance model. Thus, the portfolio optimization model is a mixed quadratic and integer programming problem for which efficient algorithms do not exist. The results of this study are compared with those of the genetic algorithms, simulated annealing and tabu search approaches. The purpose of this paper is to apply PSO technique to the portfolio optimization problem. The results show that particle swarm optimization approach is successful in portfolio optimization.
The particle swarm optimization (PSO) approach is a heuristic technique introduced comparatively recently by Kennedy and Eberhart [1]. There are very few studies on PSO in the literature, and almost none of them deals with portfolio optimization (PO). This study presents a new approach to PO using PSO.
PO consists of the portfolio selection problem in which we want to find the optimum way of investing a particular amount of money in a given set of securities or assets [5]. Although the task of yielding minimum risk and maximum return looks simple, there is more than one way of establishing an optimum portfolio. Markowitz [2] and [6] formulated the fundamental theorem of a mean–variance portfolio framework, which explains the trade-off between mean and variance, representing expected returns and risk of a portfolio, respectively. An advanced model was introduced by Konno and Yamazaki [3] in which a mean-absolute deviation (MAD) model and absolute deviation are utilized as a measure of risk. However, it was insensitive to some extremes, which could be the source of serious error, contrary to the suggestion that the MAD model is suitable under all circumstances [7]. As Mansini and Sprenza stated [8], most of the portfolio selection models assume a perfect fractionability of the investments; however, securities are negotiated as multiples of a minimum transaction lot in the real world, and they suggested a mixed integer programming model with minimum lot constraint for portfolio selection.
Some researchers have investigated the multi-period PO case, in which investors invest continuously rather than at intervals or only once. Celikyurt and Ozekici [15] accomplished this, assuming that there are some economic, social, political and other factors affecting the asset returns. They formed their stochastic market with respect to these factors, and they used a Markov chain approach in their study.
This study basically employs the Markowitz mean–variance model. However, the standard model does not contain any cardinality or bounding constraints, which restrict the number of assets and, the upper and the lower bounds of proportion of each asset in the portfolio, respectively. Chang et al. [9] and Fernandez and Gomez [5] used a modified Markowitz model called a “cardinality constrained mean–variance (CCMV) model”. Here, the CCMV model is used and is solved by a PSO approach.
There are some reports of solving the PO problem using heuristic methods. These methods consist of genetic algorithms (GA) [9], [6] and [10], tabu search (TS) [9], simulated annealing (SA) [9], [11] and [12], neural networks [5] and others [13], [8] and [14]. The results of this study are compared with those of the GA, SA and TS approaches [9]. The test data set is the weekly prices from March 1992 to September 1997 from the following indices: Hang Seng in Hong Kong, DAX 100 in Germany, FTSE 100 in UK, S&P 100 in USA and Nikkei in Japan. The number of different assets for each of the test problems is 31, 85, 89, 98, and 225, respectively.
This study was focused on solving the portfolio selection problem and tracing out its efficient frontier. A Markowitz-based cardinality constrained mean–variance model that includes cardinality and bounding constraints was used to develop a particle swarm optimization-based heuristic method. The results were compared to those obtained from heuristic methods based on (1) genetic algorithms, (2) tabu search and (3) simulated annealing.
The experimental results have shown that none of the four heuristics has clearly outperformed the others in all kinds of investment policies. However, Fig. 7 shows that, when dealing with problem instances that demand portfolios with a low risk of investment, the particle swarm optimization model gives better solutions than the other heuristic methods.
