Enhanced Index Tracking Portfolio Based On Uncertainty Theory

Enhanced index tracking portfolio is concerned with selecting a tracking portfolio to beat the benchmark on returns.Most of the existing relevant studies and conclusions are based on the fact that securities returns are random variables and probability theory is the main mathematical tool.However,in the real financial market,there exist the situations that people have no data or invalid data,e.g.,the newly listed stocks lack historical data;emergencies such as COVID-19 or war make the historical data unable to effectively reflect the future financial market.In this case,people have to use human’s estimations to help investment decision-making.Human’s estimations may deviate from the real situation in the future.At this time,the use of probability theory will amplify the artificial estimation deviation.In view of the above situation,this paper will use uncertainty theory,adopt uncertain variables to describe securities returns,to study the enhanced index tracking portfolio problem based on uncertainty theory.The main research contents and results of this paper are as follows:(1)This paper uses variance to measure tracking error and constructs an uncertain mean-variance enhanced index tracking model.The analytical solution of the proposed model is given when the securities returns take normal uncertainty distributions.Based on the analytical solution,the return and risk of the optimal tracking portfolio are analyzed,the boundary of the optimal tracking portfolio is given,and a function including investor’s risk aversion coefficient,portfolio’s return and risk is defined to represent the performance of the optimal tracking portfolio in the eye of an investor.It is found that the performance of the benchmark index,investor’s risk aversion coefficient and tracking error tolerance will affect the performance of the optimal tracking portfolio.Investors can adjust parameters according to their preferences to get the tracking portfolio that they think performs best.(2)The uncertain mean-variance enhanced index tracking portfolio with portfolio risk constraint is studied.The uncertain mean-variance enhanced index tracking models with portfolio risk constraint measured by variance and risk index are constructed where risk index is defined as the average loss lower than zero return.From the perspective of return and risk,this paper compares the differences between uncertain enhanced index tracking models with and without portfolio risk constraint.From the perspective of tracking portfolio performance in the eye of an investor,this paper analyzes the impact of portfolio risk constraint on investors’ decision-making.The results show that the optimal tracking portfolio determined by the model with portfolio risk constraint has less risk and expected return;Investors with relatively small risk aversion coefficient prefer to choose the model with portfolio risk constraint.(3)This paper uses the absolute dowside deviation between tracking portfolio and benchmark returns to measure the tracking error and constructs an uncertain mean-absolute downside deviation enhanced index tracking model.The analytical solution is derived when securities returns take linear uncertainty distributions.The return and risk of the optimal tracking portfolio are analyzed and the boundary of the optimal tracking portfolio in the mean-absolute downside deviation space is given.The effects of the changes of benchmark return distribution and tracking error tolerance level on the optimal tracking portfolio are analyzed,which helps investors make decisions.The experimental results show that under the same expected excess return level,compared with the models using absolute deviation and standard deviation to measure the tracking error,the uncertain mean-absolute downside deviation enhanced index tracking model has a smaller tracking error.(4)The uncertain enhanced index tracking model considering higher-order moment of downside is studied.Compared to the model with lower-order moment of downside,the advantages of the proposed model are analyzed.The results show that the optimal tracking portfolio generated by the uncertain enhanced index tracking model considering lower-order moment of downside has either less expected return or higher tracking error than that generated by the uncertain enhanced index tracking model considering higher-order moment of downside.This implies that the uncertain enhanced index tracking model considering higher-order moment of downside has a wider range of application.