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Open AccessArticle

Properties of Risk Measures of Generalized Entropy in Portfolio Selection

by Rongxi Zhou 1, Xiao Liu 2,*, Mei Yu 1 and Kyle Huang 3
School of Finance and Banking, University of International Business and Economics, Beijing 100029, China
School of Economics and Management, Beijing University of Chemical Technology, Beijing 100029, China
Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08540, USA
Author to whom correspondence should be addressed.
Entropy 2017, 19(12), 657;
Received: 21 September 2017 / Revised: 20 November 2017 / Accepted: 30 November 2017 / Published: 1 December 2017
(This article belongs to the Special Issue Entropic Applications in Economics and Finance)
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. View Full-Text
Keywords: generalized entropy; properties of risk measures; portfolio selection model generalized entropy; properties of risk measures; portfolio selection model
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Zhou, R.; Liu, X.; Yu, M.; Huang, K. Properties of Risk Measures of Generalized Entropy in Portfolio Selection. Entropy 2017, 19, 657.

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