PSO and BFO: two alternative metaheuristics for portfolio optimization problem

Abstract

As part of Modern Portfolio Theory, Portfolio management is a subject introduced by Markowitz back in 1950s. In Chapter 1 of this paper, we describe the Mean- Variance portfolio selection model proposed by Markowitz. Since its introduction, this model has been considered as the standard model. Although it has many advantages, with the years passing and the increased complexity of the markets, the model showed all its limits. The main drawbacks arises from the unrealistic assumptions at the base of the model. In other words, the assumptions are not able to present the real world and the risk measure used. Therefore, there was the need of a new class of risk measures, coherent risk measure, suitable for financial portfolios. The coherent measure of risk that belongs to this class chosen for this paper is the two-sided risk measure introduced by Chen and Wang in 2008. Chapter 2 describes the metaheuristics and in particular focuses on those chosen for this paper. Metaheuristic can be describe as trial and error optimization techniques able to find high level solutions to complex problems. Those high-level solutions, although high quality solutions, are not the optimal ones. However, metaheuristics find good solutions in a reasonable amount of time. In this paper we decided to choose bio-inspired metaheuristics, in particular Particle Swarm Optimization and Bacterial Foraging Optimization. In Chapter 3 we presented an alternative model to the one introduced by Markowitz, that is the realistic portfolio proposed by Corazza, Fasano and Gusso. This strategy allows to make the analysis more realistic by overcoming the limits of the model described in Chapter 2. However, in order to effectively solve the NP-hard problem that arises from the use of the two-sided risk measure combined with the realistic portfolio chosen, we applied an exact penalty method, which allows to transform the constrained problem into an unconstrained one. Finally, in Chapter 4, we applied PSO and BFO to solve the portfolio selection problem presented in the previous chapter. For this application, the data used are the daily closing prices of DAX 30 index from March 2014 to November 2018. The periods considered are eight, and each one consists of 8 in-sample months and 3 out-of-sample. In addition, we also analyzed the respect of the monotonicity property of the risk measure. Lastly, we carried out a comparison between the given respectively between PSO and BFO.