Properties of Risk Measures of Generalized Entropy in Portfolio Selection

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.

Entropy and Semi-Entropies of LR Fuzzy Numbers’ Linear Function with Applications to Fuzzy Programming

“…LR be two independent triangular fuzzy numbers, and f (δ 1 , δ 2 ) = 3δ 1 + 5δ 2 . In light of Equations (16) and 21, the entropy of f (δ 1 , δ 2 ) is:…”

“…Kosko [11] developed a general entropy measure based on an intuitive ratio of distances. Pal and Pal [12,13] introduced three types of entropy along with their applications to highlight their applicability to various problems (see also [14][15][16][17]).…”

A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations

“…Second, φ-EVaR as an extension is discussed by Pichler and Schlotter (2018) by replacing the relative entropy in the dual representation with different divergences as suggested in Ahmadi-Javid (2012c) first. Third, risk measures of generalized entropy are discussed by Zhou et al (2017) who systematically investigate the properties of Information Entropy, Cumulative Residual Entropy, Fuzzy Entropy, Credibility Entropy, Sine Entropy and Hybrid Entropy. Table 1 summarizes the basic properties of all risk measures mentioned before.…”

The Entropy Complexity of an Asymmetric Dual-Channel Supply Chain with Probabilistic Selling

“…Lukáš and Hofman [31] used the classical Shannon entropy approach to discuss the operational complexity of company supplier-customer relations. Zhou et al [32] investigated the properties of six kinds of entropy based on risk measures and discovered that none of the risk measures satisfied the system's properties. Based on the 1987 and 2008 financial crises, Gençay and Gradojevic [33] provided a comparative analysis of stock market dynamics and successfuly predicted that aggregate market expectations use risk management measures based on entropy.…”