Properties of Risk Measures of Generalized Entropy in Portfolio Selection

Rongxi Zhou, Xiao Liu, Mei Yu, Kyle Huang
2017 Entropy  
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:
more » ... y, 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 2017, 19, 657 2 of 17 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. Zhang et al. [16] contributed the possibilistic mean semivariance entropy model, in which the degree of diversification in a portfolio was measured by its possibilistic entropy. Zhou et al. [17] developed a new portfolio selection model, in which the portfolio risk was measured using Information Entropy and the expected return was expressed using incremental entropy. Implementing a proportional entropy constraint as the divergence measure of a portfolio, Zhang et al. [18] studied a multiperiod portfolio selection problem in a fuzzy investment environment. Yao [19] presented another type of entropy, named Sine Entropy, as a measure of the variable uncertainty in portfolio selection. Yu [20] compared the mean variance efficiency, portfolio values, and diversity of the models incorporating different entropy measures. Zhou et al. [21] defined risk as Hybrid Entropy and proposed a mean variance Hybrid Entropy model with both random and fuzzy uncertainty. Gao and Liu [22] put forward a risk-free protection index model with an entropy constraint under an uncertainty framework. In the aforementioned studies, different concepts of entropy were used to measure portfolio risk. However, the properties of these entropy-based measures of risk in portfolio selection were not discussed as substantially. In fact, Ramsay [23] introduced the idea that an effective risk measure function should satisfy the five properties of Risklessness, Non-negativity, Sub-additivity, Consistency, and Objectivity. Artzner et al. [24] defined the concept of coherent risk measures and asserted that a rational risk measure should satisfy the four axioms of Translation Invariance, Sub-additivity, Positive Homogeneity and Monotonicity. Follmer and Schied [25] introduced the notion of convex risk measures, taking into consideration the fact that the risk of a position may increase in a nonlinear fashion with the size of the position. Bali et al. [26] proposed a generalized measure of risk based on the risk-neutral return distribution of financial securities. The theories associated with the risk measures examined in these studies [23] [24] [25] [26] can provide a useful methodology for studying entropy-based measures of risk. Therefore, this paper systematically investigates the properties of Information Entropy, Cumulative Residual Entropy, Fuzzy Entropy, Credibility Entropy, Sine Entropy and Hybrid Entropy, which, together, make up generalized entropy. The first two methods are in the probability space, the next three methods are in the fuzzy space, and Hybrid Entropy is in the uncertainty of both fuzziness and randomness. The rest of this paper is organized as follows: Section 2 presents some basic properties of risk measures. We comprehensively discuss properties of risk measures based on generalized entropy in Section 3. In Section 4, we develop seven different portfolio selection models and make empirical comparisons using samples from industries in the Shenzhen Stock Exchange of China and the New York Stock Exchange. Finally, Section 5 details the conclusions of this paper. Some Basic Properties of Risk Measures Let X be a random variable describing outcomes of a risky asset, and let Ω be the set of all X. ρ is a mapping from Ω onto R, i.e., ρ : Ω → R . X ∈ Ω is considered risk-less if and only if X is a constant with a probability of one, that is, there exists a constant a such that P[X = a] = 1. ρ(X) denotes the risk value for the asset outcomes, X. The properties of ρ(X) in [23] [24] [25] can be defined as follows: (1) Sub-additivity. For X 1 , X 2 ∈ Ω, we have ρ(X 1 + X 2 ) ≤ ρ(X 1 ) + ρ(X 2 ). (2) Consistency. For X ∈ Ω, a ∈ R, we have ρ(X + a) = ρ(X). (3) Monotonicity. For X, Y ∈ Ω, with X ≤ Y, we have ρ(X) ≥ ρ(Y). (4) Translation Invariance. For X ∈ Ω , a ∈ R, we have ρ(X + a · r) = ρ(X) − a where the particular risk-free asset is modeled as having an initial price of 1 and a strictly positive price, r(or total return), in any state at date, T. Entropy 2017, 19, 657 3 of 17 (5) Positive Homogeneity. For X ∈ Ω, λ ≥ 0, we have ρ(λX) = λρ(X). (6) Convexity. For λ ∈ [0, 1], and X, Y ∈ Ω, we have ). Definition 1. A risk measure ρ(X) is called a monetary risk measure if ρ(0) is finite, and if ρ(X) satisfies the axioms of Monotonicity and Translation Invariance [27]. Definition 2. A risk measure ρ(X) satisfying the four axioms of Translation Invariance, Sub-additivity, Positive Homogeneity, and Monotonicity is called a coherent measure of risk [24]. Definition 3. A risk measure ρ(X) satisfying the axioms of Positive Homogeneity, Consistency and Sub-additivity is called a deviation measure of risk [28]. Definition 4. A risk measure ρ(X) satisfying the axioms of Translation Invariance, Monotonicity and Convexity is called a convex measure of risk [25]. Properties of Risk Measures of Generalized Entropy We will sequentially explore the properties of the six kinds of entropy-based risk measures: Information Entropy, Cumulative Residual Entropy, Fuzzy Entropy, Credibility Entropy, Sine Entropy and Hybrid Entropy. Information Entropy Definition 5. Suppose that X is a continuous random variable with a probability density function f (x). Then, its Information Entropy is defined as follows [29]:
doi:10.3390/e19120657 fatcat:njciqjpydzcbxe4z5ejsnqxtq4