Abstract

     In order to solve the problem of portfolio optimization, this paper proposes a method that combines multi-objective optimization and multi-attribute decision-making to solve the dual-objective portfolio optimization model with conditional value-at-risk (CVaR) measuring risk and including transaction costs. First, in the multi-objective optimization stage, a multi-population parallel NSGA-II based on sparsity strategy (SMP-NSGA-II) is proposed to obtain multiple Pareto optimal solutions of the model. Second, in the multi-attribute decision-making stage, in order to reflect different investment preferences, the Pareto optimal set obtained is clustered through the fuzzy C-means, and then the grey relational projection method is used to evaluate the solutions belonging to the same cluster to select the optimal compromise solution. Finally, a case study of 9 semiconductor stocks in China’s Shanghai and Shenzhen stock markets is carried out, and the optimal compromise portfolio under different investment preferences is given. At the same time, the proposed algorithm is compared with the other six multi-objective evolutionary algorithms (MOEAs), which verifies that the algorithm in this paper has certain competitiveness.

Introduction

     Nowadays, in the field of securities investment, the indicators used to quantify risk mainly include value-atrisk (VaR) and conditional value-at-risk (CVaR). Among them, VaR is represented by nonlinear, non-convex and nondifferentiable function with multiple local optima, making it difficult to calculate. To solve these problems, Rockafellar et al. [1] introduced the CVaR, which is a coherent risk measure that considers risk as the most serious loss in a given scenario, taking into account a certain degree of confidence. Since CVaR is a convex function, it can effectively solve the optimization problem that uses CVaR as a minimization goal or constraint [2, 3]. At the same time, Yu et al. [4] compared five different risk models and verified through experiments that using CVaR to measure risk is a good choice.

     As the complexity of practical applications continues to increase, scholars have developed various heuristic algorithms to solve portfolio optimization problems. The application of heuristic algorithms in portfolio optimization problems is divided into two categories. The first category simplifies portfolio objectives through the setting of weight coefficients [5–۷], and obtains a risk-return curve by continuously changing the risk avoidance parameters of representative investors. This method has a certain degree of subjectivity. The second type uses multi-objective evolutionary algorithm (MOEA) to directly optimize risks and benefits simultaneously [8–۱۱], and can obtain a complete effective frontier in one operation. Obviously, it is more convenient to use MOEAs to solve portfolio optimization problems.

Conclusion

     In order to solve the dual-objective portfolio optimization model with conditional value-at-risk (CVaR) as a measure of risk and including transaction costs, this paper proposes a method combining multi-objective optimization and multi-attribute decision-making. In the multi-objective optimization stage, this paper proposes a multi-population parallel NSGA-II based on sparsity strategy (SMP-NSGAII). In the case studies of 9 stocks in the semiconductor industry, we compared SMP-NSGA-II with the other six MOEAs through two performance evaluation indicators (HV and SP) and running time, then verified the feasibility of the SMP-NSGA-II algorithm. In the multi-attribute decision-making stage, this paper adopts the FCM-GRP hybrid method to give the optimal compromise investment portfolio under different DM preferences.

     There are a couple of drawbacks in the present study. Firstly, the model studied in this paper is relatively simple and does not take into account the many unstable factors of the real securities market; Secondly, the space complexity of the proposed SMP-NSGA-II algorithm is also high.