Performance evaluation of possibilistic fuzzy portfolios with different investor risk attitudes based on DEA approach

Abstract

By considering the stock market’s fuzzy uncertainty and investors’ psychological factors, this paper studies the portfolio performance evaluation problems with different risk attitudes (optimistic, pessimistic, and neutral) by the Data Envelopment Analysis (DEA) approach. In this work, the return rates of assets are characterized as trapezoidal fuzzy numbers, whose membership functions with risk attitude parameters are described by exponential expression. Firstly, these characteristics with risk attitude are strictly derived including the possibilistic mean, variance, semi-variance, and semi-absolute deviation based on possibility theory. Secondly, three portfolio models (mean-variance, mean-semi-variance, and mean-semi-absolute-deviation) with different risk attitudes are proposed. Thirdly, we prove the real frontiers determined by our models are concave functions through mathematical theoretical derivation. In addition, two novel indicators are defined by difference and ratio formulas to characterize the correlation between DEA efficiency and portfolio efficiency. Finally, numerical examples are given to verify the feasibility and effectiveness of our model. No matter what risk attitude an investor holds, the DEA can generate approximate real frontiers. Correlation analysis indicates that our proposed approach outperforms in evaluating portfolios with risk attitudes. At the same time, our model is an improvement of Tsaur’s work (2013) which did not study the different risk measures, and an extension of Chen et al.’s work (2018) which only considered risk-neutral attitude.