Assistant reference point guided evolutionary algorithm for many-objective fuzzy portfolio selection

Abstract

Portfolio selection is dedicated to assigning limited capital to available financial assets to achieve an optimal trade-off between profit and risk. This work is targeted at a portfolio selection problem simultaneously considering five optimization objectives namely mean, variance, skewness, kurtosis, and entropy of return. An expert’s acknowledge-based fuzzy number variable, rather than a random variable, is used to model the return of a risk asset, such that a more accurate estimation of the return of a risky asset is achieved. An assistant reference point guided evolutionary algorithm (ARPGEA) is proposed to solve this five-objective portfolio problem. Assistant reference points are generated for the non-dominated solutions in all dimensions of the objective space. The generated reference points and a comprehensive fitness evaluation method are used to guide the search towards promising regions leading to well-distributed and well-converged solutions. ARPGEA is tested on publicly available data sets collected from six market indices including up to 1,203 assets. Experiment results demonstrate the efficiency and effectiveness of ARPGEA in comparison with other five state-of-the-art many-objective evolutionary algorithms.

Introduction

Portfolio selection is an important problem in the practice and theory of finance. The goal of Portfolio selection is to assign limited capital among different available risky assets to achieve the possible optimal future return. The modern portfolio selection is derived from the Markowitz mean-variance model [1], which maximizes the expected return and minimizes the variance of return simultaneously. However, Markowitz mean-variance model is likely to underestimate the risk without considering high-order moments, such as skewness and kurtosis. Moreover, the assumption that the return of risky asset obeys normal distribution in the model is challenged [2], [3], [4], [5]. Many studies have discovered that the distributions of asset returns tend to be heavy-tailed and asymmetric leptokurtic, i.e., not subject to normal distribution [2]. To deal with these issues, researchers have turned their attention to higher-order moments [2], [3], [4], [5]. Three and/or four-order moment asset returns have been introduced to the Markowitz mean-variance model in [3], [4], [5]. For example, the extensions of the classical Markowitz model (i.e., mean-variance-skewness model) take the high-order moments into consideration in works [6], [7] and showed the benefit of incorporating higher-order moments into portfolio selection. A mean-variance-skewness-kurtosis model for portfolio selection was proposed in [8] and a multi-objective evolutionary algorithm (MOEA) was introduced to find the optimal portfolios. The introduction of high-order moments shows promising results yet it arise other problems, i.e., the high-order moments portfolio models suffer from estimation error in the corner solutions and a low diversity in the portfolio. The low diversity of the portfolio may lead to capital loss [9]. Shannon’s entropy was further introduced to address the problem for generating well-diversified portfolios in the last few years [10]. For example, A mean-variance-skewness-entropy model [11] was proposed to increase the diversity of the selected portfolio. Huang [12] applied Shannon’s entropy as a constraint in the mean-variance model to avoid the concentrative portfolio allocation. Introducing different entropy measures can yield a higher economic value of diversification [9], [13], [14].

Besides the higher-order moments, uncertainty is another key factor in the portfolio selection model as investors usually face uncertain, imprecise and vague data. For decades, the majority of the existing portfolio selection models are based on probability theory. However, the return of a risky asset is not completely random and they are very likely affected by the politic change, economic status, profitability of the related company, and other factors. Compared with a random variable, a fuzzy variable relied on expert knowledge is more suitable to measure the uncertain return of a risky asset [4]. Along with this idea, more attention has been paid to fuzzy portfolio selection models [4], [5], [13]. For example, the study [7] proposed a fuzzy mean-variance-skewness portfolio selection model considering transaction cost and solved it with an elitist genetic algorithm (GA) [15]. Kocadagli et al. [16] developed a fuzzy portfolio selection model based on the fuzzy goal programming technique and the model achieves better performance than the Markowitz mean-variance model. Li et al. [17] proposed a new fuzzy portfolio selection model with background risk and a GA was used to solve the proposed model. Wang et al. [18] presented a multi-period portfolio selection problem with security returns as fuzzy random variables. Sadati et al. [19] modeled the portfolio problem with fuzzy random variables considering both degrees of possibility and necessity, and proposed a linear programming method to search for the optimum solution. A fuzzy portfolio mean-variance-skewness model with assets limitations and liquidity requirement was considered and solved by a GA in [20]. The work [5] proposed a mean-variance-efficiency model solved by the second version of the non-dominated sorting genetic algorithm (NSGA-II) [21].

Portfolio selection problem is naturally a multi-objective optimization problem (MOP) [22], [23], [24] as it pursues the optimal trade-off between profit and risk. Modeling the portfolio selection problem as an MOP can obtain a set of the optimal trade-off solutions in one run facilitating a posteriori selection of the decision-maker. Different types of methods, e.g., the weight-sum methods [20], Pareto-dominance-based methods [5], [25], and decomposition-based methods [8], [26], have been proposed to solve multi-objective or many-objective (the number of objectives is greater than three) portfolio selection problems. For example, Barak et al. [20] used a weight-sum method and GA to solve the tri-objective mean-variance-skewness model. Mashayekhi et al. [5] incorporated the data envelopment analysis cross-efficiency into Markowitz mean-variance model and applied the NSGA-II to the proposed modal. Yue et al. [8] formulated the fuzzy portfolio selection problem as a five-objective optimization problem and proposed an MOEA to solve the problem. The existing multi-objective optimization methods have achieved successes in various portfolio selection problems, yet they also suffer from some limitations. For example, the weight-sum methods require a priori weight vector or preference, which is heavily problem-dependent. Traditional Pareto-dominance-based methods might have difficulty in dealing with many-objective optimization problems (MaOPs) due to the failure of diversity measurement [27], [28], [29]. The performance of traditional decomposition-based methods is usually sensitive to the shape of the Pareto-optimal front [30], [31].

Taking all the aforementioned issues into consideration, we propose an assistant reference point-guided evolutionary algorithm (ARPGEA) in this study to handle the five-objective mean-variance-skewness-kurtosis-entropy portfolio selection problem [8]. The assistant reference points in ARPGEA are generated for the non-dominated solutions in all dimensions of objective space. The generated reference points are then used for the environmental selection to find a well-convergent and well-diversified solution set. Experimental results on six well-known stock data sets show the superior performance of ARPGEA to other state-of-the-art many-objective evolutionary algorithms (MaOEAs). Note that this work is an extension of our previous conference paper [32] with substantial improvement. Particularly, the algorithm is improved by incorporating a new comprehensive fitness evaluation method to balance the population convergence and diversity for MaOP. Much more extensive experimental studies are performed to demonstrate the effectiveness of the proposed method. The writing is also considerably expanded to cover more background acknowledge, literature review, and details of the methodologies. The main contributions of this work are summarized as follows:

• An assistant reference point-guided evolutionary algorithm ARPGEA is proposed to solve a five-objective portfolio selection problem. The work represents a good application of MaOEA to solve complex problems in the practice of finance.

• A set of assistant reference points together with comprehensive fitness evaluation are designed to obtain well-distributed and -converged solutions which is critical to the success of portfolio selection.

• Extensive experimental studies on real-world data sets are carried out to reveal the strengths and weaknesses of MaOEAs.

In the rest of this paper, Section 2 provides the related background and literature review. Section 3 details the proposed ARPGEA. Section 4 presents the experimental results. Finally, Section 5 concludes this work.