Abstract
This paper systematically investigates the properties of six kinds of entropy-based risk measures: Information Entropy and Cumulative Residual Entropy in the probability space, Fuzzy Entropy, Credibility Entropy and Sine Entropy in the fuzzy space, and Hybrid Entropy in the hybridized uncertainty of both fuzziness and randomness. We discover that none of the risk measures satisfy all six of the following properties, which various scholars have associated with effective risk measures: Monotonicity, Translation Invariance, Sub-additivity, Positive Homogeneity, Consistency and Convexity. Measures based on Fuzzy Entropy, Credibility Entropy, and Sine Entropy all exhibit the same properties: Sub-additivity, Positive Homogeneity, Consistency, and Convexity. These measures based on Information Entropy and Hybrid Entropy, meanwhile, only exhibit Sub-additivity and Consistency. Cumulative Residual Entropy satisfies just Sub-additivity, Positive Homogeneity, and Convexity. After identifying these properties, we develop seven portfolio models based on different risk measures and made empirical comparisons using samples from both the Shenzhen Stock Exchange of China and the New York Stock Exchange of America. The comparisons show that the Mean Fuzzy Entropy Model performs the best among the seven models with respect to both daily returns and relative cumulative returns. Overall, these results could provide an important reference for both constructing effective risk measures and rationally selecting the appropriate risk measure under different portfolio selection conditions.
Highlights
Portfolio selection has always been an important part of the financial field, and at its core is the development of effective risk measures
In order to analyze the effect of generalized entropy on actual portfolio selection problems, we developed seven portfolio models based on different risk measures under the standard risk/return framework
Mean Hybrid Entropy Model (MHEM) has the lowest means for daily returns (DR) and Mean Sine Entropy Model (MSEM) has the lowest means for relative cumulative returns (RCR)
Summary
Portfolio selection has always been an important part of the financial field, and at its core is the development of effective risk measures. In 1952, Markowitz [1] first proposed using variance to measure risk and developed the famous mean variance model (MVM) for solving portfolio selection problems. Entropy 2017, 19, 657 established two types of credibility-based fuzzy mean entropy models. Xu et al [14] developed a λ Mean-Hybrid Entropy model to study portfolio selection problems with both random and fuzzy uncertainty. Usta and Kantar [15] presented a multiobjective approach based on a mean variance skewness entropy portfolio selection model. Zhou et al [21] defined risk as Hybrid Entropy and proposed a mean variance Hybrid Entropy model with both random and fuzzy uncertainty. The properties of these entropy-based measures of risk in portfolio selection were not discussed as substantially.
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