Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4197914, 14 pages

https://doi.org/10.1155/2017/4197914

## Large-Scale Portfolio Optimization Using Multiobjective Evolutionary Algorithms and Preselection Methods

^{1}School of Electric and Information Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China^{2}School of Information Engineering, Zhengzhou University, Zhengzhou 450001, China^{3}School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China^{4}School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798

Correspondence should be addressed to J. J. Liang; nc.ude.uzz@gnijgnail

Received 15 October 2016; Accepted 22 January 2017; Published 20 February 2017

Academic Editor: Thomas Hanne

Copyright © 2017 B. Y. Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Portfolio optimization problems involve selection of different assets to invest in order to maximize the overall return and minimize the overall risk simultaneously. The complexity of the optimal asset allocation problem increases with an increase in the number of assets available to select from for investing. The optimization problem becomes computationally challenging when there are more than a few hundreds of assets to select from. To reduce the complexity of large-scale portfolio optimization, two asset preselection procedures that consider return and risk of individual asset and pairwise correlation to remove assets that may not potentially be selected into any portfolio are proposed in this paper. With these asset preselection methods, the number of assets considered to be included in a portfolio can be increased to thousands. To test the effectiveness of the proposed methods, a Normalized Multiobjective Evolutionary Algorithm based on Decomposition (NMOEA/D) algorithm and several other commonly used multiobjective evolutionary algorithms are applied and compared. Six experiments with different settings are carried out. The experimental results show that with the proposed methods the simulation time is reduced while return-risk trade-off performances are significantly improved. Meanwhile, the NMOEA/D is able to outperform other compared algorithms on all experiments according to the comparative analysis.

#### 1. Introduction

In modern financial markets, portfolio allocation is one of the major problems faced by investors and fund managers. Investors need to choose several assets from thousands of available assets to form a single portfolio from an enormous set of possibilities in order to simultaneously maximize the return and minimize the risk. It is crucial for investors to find a perfect portfolio allocation. However, higher returns are generally associated with higher risk. Investors with varying degrees of risk aversion will demand different levels of return for taking on different degrees of risk [1, 2].

The concept of portfolio optimization has been an important tool in the development and understanding of financial markets. Portfolio optimization techniques can assist the search for the portfolio that best suits each investor’s particular objective [3, 4]. As stated by the BusinessWeek [5], the single best weapon against risk is to form portfolios with uncorrelated or negatively correlated assets because when several such assets are combined together, the overall risk of the portfolio may be less than that of the individual asset. Thus, finding a suitable combination of investments attracted attentions of investors and scholars. The major breakthrough of portfolio optimization came in 1952 with the publication of Markowitz’s theory of portfolio selection [6]. Markowitz quantified return and risk of a security using statistical measures of its expected return and standard deviation. Markowitz suggested that investors should consider return and risk together and determine the allocation of funds among investment alternatives on the basis of their return-risk trade-off [7]. This theory is popularly referred to as the modern portfolio theory and it is also the theoretical basis for this work. The details of the theory are presented in Section 2.

As there are two-conflicting objectives, there is no single optimal solution to this portfolio optimization problem. Instead, there is an efficient frontier of optimal trade-off solutions. It is often desirable to have the entire efficient frontier of optimal portfolio sets that give the average return against the possible risk so that individual investors can choose the most appropriate return-risk trade-off to suit their investment objectives. In recent years, evolutionary algorithms (EAs) have become an effective tool in handling optimization problems [8]. EAs search for the solution by initializing a population of random candidates. These candidates undergo an evolutionary process based on the survival-of-the-fittest mechanism and the individuals that have superior fitness will be passed to the next generations [9].

Using EAs to solve assets allocation problem has become a trend in recent years, as EAs are able to find multiple Pareto-optimal solutions in one single run [10]. Most of the early works transformed the optimization problem into a single-objective problem using a trade-off function [10–13]. With the development of multiobjective evolutionary algorithms (MOEAs) [8, 14–16], researchers have focused their attention on using MOEAs for solving portfolio optimization problems. The first use of MOEAs for solving portfolio optimization problem can be traced back to 1993 [17]. This work adopted lower partial moments and a genetic approach to handle the problem. In 1996, Shoaf and Foster [18] used a multiobjective GA to solve the conventional Markowitz formulation. They adopted a specific encoding scheme which was able to indicate whether the holding of the particular asset would be long or short. In [19], six different MOEAs including original VEGA [20], two versions of modified VEGA, NSGAII [15], MOGA [21], and SPEA2 [22] were examined and compared using the classical mean-variance model. A mixed binary-real encoding scheme was used for all the compared algorithms. Yan et al. [23] used the semivariance as the risk and growth rate as the return. In this work, instead of considering a single investment period, T periods were used for the optimization. The weight of each investment asset must be determined by the decision-maker at the beginning of each period. Lin and Liu [24] applied genetic algorithm on Taiwanese mutual fund for portfolio selection problem with minimum transaction lots. However, due to the computational complexity of solving a large-scale quadratic programming problem, Markowitz’s portfolio optimization model has not been used extensively in its original form to construct a large-scale portfolio [24]. As the number of assets increases, the complexity of the search space increases exponentially. The state-of-the-art evolutionary algorithms may not effectively solve the problem when there are more than 200 assets available to form portfolios. Lin and Liu [24] used up to 204 assets to form the optimal portfolios. In [25], Ghahtarani and Najafi used 20 stocks from the Tehran stock exchange in the portfolio optimization. Fernández and Gómez [26] adopted Markowitz’s model with cardinality and bounding constraints. In this work, four different markets with different number of stocks were considered and the largest number of stocks used was 225. Motivated by these observations, two asset preselection processes are proposed in this work for large-scale portfolio optimization so that the number of assets to be considered can be increased to a few thousands or more. Five different MOEAs known as Normalized Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D), Multiobjective Differential Evolution based on Summation Sorting (MODE-SS), Multiobjective Differential Evolution based on Nondomination Sorting (MODE-NDS), Multiobjective Comprehensive Learning Particle Swarm Optimizer (MOCLPSO), and NSGAII are used as the optimization tools to test the effectiveness of the preselection process.

The rest of the paper is outlined as follows. Section 2 presents the models of portfolio optimization. Section 3 briefly reviews the concept of multiobjective optimization and the four multiobjective optimization algorithms used in this work. Sections 4 and 5 introduce the preselection process and the constraint handling method, respectively. The experimental settings and results are presented and analyzed in Section 6. Finally, the relevant conclusions and directions for future work are discussed in Section 7.

#### 2. Portfolio Optimization

The mathematical representation of portfolio optimization was introduced by Markowitz in 1952 and he was rewarded with a Nobel Prize in Economics in 1990 [27]. The Markowitz model assumes that investors would like to maximize return under a certain risk level or minimize the risk with a certain return level [6] and this model makes use of the mean and variance of normalized historical asset prices to compute the expected portfolio return and risk [24], respectively. The model can be expressed as a biobjective problem as follows:where is the number of assets in the portfolio which is also the dimensionality of the optimization problem. is the weight of the th asset to be optimized. stands for the portfolio risk while is the covariance between asset and asset . If , is just the variance of that particular asset. is the average portfolio return and is the average individual return of asset . Due to the determined efforts of various researchers including Sharpe [28], Pang [29], Best and Hlouskova [30], and others, Markowitz’s work has been widely extended. Recent works also include various constraints for the portfolio optimization problem. Speranza [31] proposed a model that takes into account the characteristics of the portfolio optimization. Chang et al. [32] extended the standard model to include cardinality constraints that limit a portfolio to have a specified number of assets and to impose limits on the proportion of the portfolio held in a given asset. In this paper, the original form of Markowitz model is adopted.

#### 3. Multiobjective Optimization

##### 3.1. Mathematical Model of Multiobjective Optimization Problem

Optimization refers to finding one or more feasible solutions which correspond to extreme values of one or more objectives [14, 33]. Typically, a multiobjective optimization problem can be defined as follows: where is the th objective function, is the number of objectives, and are decision variables. and are inequality and equality constraint functions, respectively. and are the lower and upper bounds of the decision variables. If an optimization problem involves more than one objective function, it is considered as a multiobjective optimization problem and the task of this problem is to find a set of solutions which can represent the trade-off among different objectives.

##### 3.2. Normalized Multiobjective Evolutionary Algorithm Based on Decomposition

###### 3.2.1. Original MOEA/D

The Multiobjective Evolutionary Algorithm based on Decomposition (MOEA/D) was developed by Zhang et al. in 2007 [34] and it is considered as one of the most effective multiobjective optimization algorithms at present. MOEA/D provides a new framework by combining decomposition method and evolutionary algorithm for MOPs [35]. It explicitly decomposes the MOP into a number of single optimization subproblems and solves these subproblems simultaneously by evolving a population of solutions.

In MOEA/D, various decomposition methods can be used to decompose an MOP into a set of subproblems [34]. However, the weighted Tchebycheff approach is the most prominent method as it is less sensitive to the shape of Pareto Front. Therefore, this approach is also adopted in this work. The objective of each subproblem can be represented as follows:where is the number of objectives, is the reference point for the th objective, and is the weight vector. Note that the reference point can be represented as . The details of the original MOEA/D can be found in [34].

###### 3.2.2. Normalized MOEA/D with Normalized Objectives

The MOEA/D algorithm decomposes the MOPs into a number of subproblems by a set of evenly spread weight vectors. The main idea is to narrow the gap between the objective value and the reference value in (the number of objectives) dimensions with the assigned weight vector. In this way, MOEA/D forces the objective function to evolve in the direction of minimizing the maximum difference. However, the scales of different objectives are generally not the same for real-world applications. In such cases, most of the searching resources will be spent on the most significant objective (with largest value) while the other objectives are barely evolved. Considering portfolio optimization as an example, the Pareto Front of optimizing 100 stocks by the original MOEA/D is plotted in Figure 1. As can been seen from this plot, the solutions in the low risk area are relatively sparse. To overcome this problem, a normalization method is proposed as follows: where , are the original and the normalized value of the th objective, respectively. , are the maximum and minimum value of corresponding objective function. In each generation, the objectives of all the individuals are normalized using the above formula. To demonstrate the effect of normalization, the PFs obtained by MOEA/D with and without normalization are plotted in Figure 2. It is clear that a more integrated front is obtained by using the normalization method.