Particle Swarm Optimization for entropy-based risk measures in portfolio selection problems: a mean – Entropic-VaR application

Abstract

Portfolio selection is a cornerstone of economics and finance. The problem consists in the minimization of a risk measure, while taking into account a series of constraints. Portfolio selection problem was introduced by Markowitz in 1952. His model was the first one to face the problem of how to efficiently invest a given amount of capital. Known also as Modern Portfolio Theory, Markowitz model revolutionized financial market investments. However, this model does present some serious limits and a set of assumptions rather utopic in the real world. In this dissertation we will create a portfolio model trying to include some of the aspects that were avoided at that time, as for instance the presence of transaction costs and the allowance to buy or sell only determined quantities of assets. Concurrently, some model’s assumptions will be modified, in particular the concept of risk measure. According to the most recent literature, in fact, only the so-called coherent risk measures can be employed as real financial risk measures. The first and more common coherent risk measures, that have been used as alternatives to variance, are Value-at-Risk (VaR) and Expected Shortfall (ES) or Conditional Value-at-Risk (CVaR). The risk measure chosen for the portfolio model developed in this work belongs to the class of entropy-based risk measures and is called Entropic Value at Risk (EVaR). EVaR is a coherent risk measure introduced by Ahmadi-Javid , it corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the Value-at-Risk (VaR) as well as the Conditional Value-at-Risk (CVaR). Conversely to what proposed by Markowitz, the model developed in this work will not be based on the mean-variance criterion, but rather on mean-entropic VaR. A system of characteristic Markowitz model’s constraints will be applied, as budget constraint and desired minimum return. We will successively introduce mixed-integer constraints, as the possibility to buy only maximum or minimum quantities of a specific asset, useful for transaction costs management. The difficulty of the constrained minimization problem is so elevated that currently it does not exist an algorithm able to provide exact results. Hence, the method proposed in the present dissertation will be metaheuristic-based (the concept of metaheuristic will be largely discussed in what follows): the metaheuristic employed, Particle Swarm Optimization, will not give an exact result to the problem, but an optimum level of approximation. This method consists in the employment of a bio-inspired metaheuristic algorithm able to search for an optimal solution to the problem, while not exact, exploiting the dynamics of exploration of groups of animals in the nature, like ant or bee swarms, birds’ flocks or shoals of fish. The application of the model will be conducted on the Matlab software and the results will be compared with a mean-variance portfolio selection model.