Abstract

Portfolio optimization involves the optimal assignment of limited capital to different available financial assets to achieve a reasonable trade-off between profit and risk. We consider an alternative Markowitz’s mean–variance model in which the variance is replaced with an industry standard risk measure, Value-at-Risk (VaR), in order to better assess market risk exposure associated with financial and commodity asset price fluctuations. Realistic portfolio optimization in the mean-VaR framework is a challenging problem since it leads to a non-convex NP-hard problem which is computationally intractable. In this work, an efficient learning-guided hybrid multi-objective evolutionary algorithm (MODE-GL) is proposed to solve mean-VaR portfolio optimization problems with real-world constraints such as cardinality, quantity, pre-assignment, round-lot and class constraints. A learning-guided solution generation strategy is incorporated into the multi-objective optimization process to promote efficient convergence by guiding the evolutionary search towards promising regions of the search space. The proposed algorithm is compared with the Non-dominated Sorting Genetic Algorithm (NSGA-II) and the Strength Pareto Evolutionary Algorithm (SPEA2). Experimental results using historical daily financial market data from S & P 100 and S & P 500 indices are presented. The results show that MODE-GL outperforms two existing techniques for this important class of portfolio investment problems in terms of solution quality and computational time. The results highlight that the proposed algorithm is able to solve the complex portfolio optimization without simplifications while obtaining good solutions in reasonable time and has significant potential for use in practice.

Introduction

Portfolio optimization is concerned with the optimal allocation of limited capital to available financial assets to achieve a trade-off between reward and risk. The classical mean-variance (MV) model [53, 54] formulates the portfolio selection problem as a bi-criteria optimization problem with a tradeoff between minimum risk and maximum expected return. In the MV model, risk is defined by a dispersion parameter and it is assumed that returns are normally or elliptically distributed. However, the distributions of returns are asymmetric and usually have excess kurtosis in practice [6, 20, 28, 45, 58]. Variance as a risk measure has thus been widely criticized by practitioners due to its symmetrical measure which equally weights desirable positive returns against undesirable negative ones. In fact, Markowitz recognized the inefficiencies embedded in the mean-variance approach and suggested the semi-variance risk measure [54] in order to measure the variability of returns below the mean. In practice, many rational investors are more concerned with under-performance rather than overperformance in a portfolio.