The lower partial moments risk measure in a novel fuzzy framework based on possibility density function

Abstract

As a comprehensive and flexible risk measure, the lower partial moments (LPM) can be converted into a wide range of the downside risk measures by modifying its target parameter and degree of moment. However, the distribution function of the return rate is hard to recognize, and the existing calculation methods have great calculation difficulties, so it has limited application. In previous studies, the historical data was mostly used to simulate the distribution function of return rate for approximate calculation of LPM risk, resulting in a huge computational complexity. In this paper, based on the possibility density function, the LPM risk measure is creatively defined in a novel fuzzy framework instead of a traditional random framework. By deriving the trapezoidal fuzzy number of return rate from the historical data, the LPM risk can be easily calculated according to its possibility density function, which can significantly improve the calculation process. Based on this idea, this paper derives the specific expression when the rate of return is a trapezoidal fuzzy number under the definition of fuzzy LPM. Furthermore, stock-driven and portfolio-driven methods are used to calculate the portfolio risk, and then two fuzzy mean-LPM models are constructed respectively. Finally, a numerical example based on global stock market index is used to illustrate the feasibility and validity of the proposed methods and models. Compared with the classic mean–variance model, our proposed model with adjustable target parameter and degree of moment is more flexible in reflecting investors’ attitude towards risk.