A Multi-Objective Portfolio Selection Model With Fuzzy Value-at-Risk Ratio

Considering nonstatistical uncertainties and/or insufficient historical data in security return forecasts, fuzzy set theory has been applied in the past decades to build portfolio selection models. Meanwhile, various risk measurements such as variance, entropy, and Value-at-Risk have been proposed in fuzzy environments to evaluate investment risks from different perspectives. Sharpe ratio, also known as the reward-to-variability ratio, which measures the risk premium per unit of the nonsystematic risk (asset deviation), has received great attention in modern portfolio theory. In this study, the Sharpe ratio in fuzzy environments is introduced, whereafter, a fuzzy Value-at-Risk ratio is proposed. Compared with Sharpe ratio, Value-at-Risk ratio is an index with dimensional knowledge that reflects the risk premium per unit of the systematic risk (the greatest loss under a given confidence level). On the basis of the two ratios, a multi-objective model is built to evaluate their joint impact on portfolio selection. Then, the proposed model is solved by a fuzzy simulation based multi-objective particle swarm optimization algorithm, where the global best of each iteration is determined by an improved dominance times based method. Finally, the algorithm superiority is justified via comparing with existing solvers on benchmark problems, and the model effectiveness is exemplified by using three case studies on portfolio selection.