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Parametric Measures of Variability Induced by Risk Measures

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Parametric Measures of Variability Induced by Risk Measures Parametric measures of variability induced by risk measures Fabio Bellini∗ Tolulope Fadina† Ruodu Wang‡ Yunran Wei§ December 10, 2020 Abstract We study general classes of parametric measures of variability with applications in risk management. Particular focus is put on variability measures induced by three classes of popular risk measures: the Value-at-Risk, the Expected Shortfall, and the expectiles. Properties of these variability measures are explored in detail, and a characterization result is obtained via the mixture of inter-ES differences. Convergence properties and asymptotic normality of their empirical estimators are established. We provide an illustration of the three classes of variability measures applied to financial data and analyze their relative advantages. Keywords: Measures of variability, quantiles, Value-at-Risk; Expected Shortfall; expec- tiles arXiv:2012.05219v1 [q-fin.RM] 9 Dec 2020 ∗Statistics and Quantitative Methods, University of Milano-Biccoca, Italy. [email protected] †Department of Mathematical Sciences, University of Essex, United Kingdom. [email protected] ‡Department of Statistics and Actuarial Science, University of Waterloo, Canada. [email protected] §Department of Statistics and Actuarial Science, Northern Illinois University, USA. [email protected] 1 1 Introduction Various measures of distributional variability are widely used in statistics, probability, economics, finance, physical sciences, and other disciplines. In this paper, we study the theory for measures of variability with an emphasis on three symmetric classes generated by popular parametric risk measures. These risk measures are the Value-at-Risk (VaR), the Expected Shortfall (ES), and the expectiles, and they respectively induce the inter-quantile difference, the inter-ES difference, and the inter-expectile difference. The great popularity and convenient properties of these risk measures lead to various features for the corresponding variability measures. The literature on risk measures is extensive, and a standard reference is F¨ollmerand Schied(2016). As classic risk measures, VaR is a quantile and ES is a coherent risk measure in the sense of Artzner et al.(1999). Both VaR and ES are implemented in current banking and insurance regulation; see McNeil et al.(2015) for a comprehensive background and Wang and Zitikis(2020) for a recent account. Expectiles, originally introduced by Newey and Powell (1987) in regression, have received an increasing attention in risk management, as they are the only elicitable coherent risk measures (Ziegel(2016)). We refer to Bellini et al.(2014) and Bellini and Di Berhardino(2015) for the theory and applications of expectiles. For a comparison of the above risk measures in the context of regulatory capital calculation, see Embrechts et al.(2014) and Emmer et al.(2015). The theory of variability measures has been studied with various different definitions; see David(1998) for a review in the context of measuring statistical dispersion. A mathe- matical formulation close to our variability measures is the deviation measure of Rockafellar et al.(2006), which is further developed by Grechuk et al.(2009, 2010). A similar notion of measures of variability was proposed by Furman et al.(2017) with an emphasis on the Gini deviation. We will explain in Section2 the differences between our definition and the ones in the literature; in particular, the inter-quantile difference does not satisfy the definition of a deviation measure of Rockafellar et al.(2006). Our main contribution is a collection of results on a theory for variability measures, with particular results on the three parametric classes mentioned above. Several novel properties are studied to discuss the special role these measures play among other variability measures. Since statistical inference for VaR, ES, and expectiles is well developed, the estimation of the corresponding variability measures is also straightforward; see e.g., Shorack and Wellner (2009) for statistical inference of VaR and Kr¨atschmer and Z¨ahle(2017) for that of expectiles. 2 The three classes of variability measures of our main interest are not new. The inter- quantile difference is a popular measure of statistical dispersion widely used in box plots, sometimes under the name inter-quartile range (e.g., David(1998)). The other two objects are, as far as we know, relatively new: the inter-ES difference appears in Example 4 of Wang et al.(2020b) as a signed Choquet integral, and the inter-expectile difference is studied by Bellini et al.(2020) via a connection to option prices. Nevertheless, this paper is the first unifying study of these measures, in particular focusing on their qualitative and quantitative advanatges. The rest of the paper is organized as follows. In the remainder of this section, we introduce some notation. The definitions of the three classes of variability measures induced by VaR, ES, and the expectiles is presented in Section2, with some basic properties. In Section3, we summarize many properties of some common variability measures which are arguably desirable in practice. A characterization result of these measures established. In Section4 we discuss non-parametric estimation of the three classes of variability measures. We obtain the asymptotic normality and the asymptotic variances explicitly for the empirical estimators. It may be undesirable and financial unjustifiable to choose the same probability level for the three classes of variability measures induced by VaR, ES, and the expectiles; see Li and Wang(2019) for a detailed analysis on plausible equivalent probability levels when ES is to replace VaR. A simple analysis of a cross-comparison of an equivalent probability level for the variability measures using different distributions is carried out in Section5.A small empirical analysis using the variability measures on the S&P 500 index is conducted in Section6, where we observe the differences between these variability measures during different economic regimes. In Section7 we conclude the paper with some discussions on the suitability of the three classes in different situations. AppendixA contains a list of classic variability measures, and proofs of all results are put in AppendixB. Notation Throughout, Lq is the set of all random variables in an atomless probability space 1 (Ω; A; P) with finite q-th moment, q 2 (0; 1) and L is the set of essentially bounded random 0 variables. X = L is the set of all random variables, and M is the set of all distributions on R. 0 −1 For any X 2 L , FX represents the distribution function of X, FX its left-quantile function, −1 and UX is a uniform [0; 1] random variable such that FX (UX ) = X almost surely. The exis- tence of such uniform random variable UX for any X is given, for example, in Lemma A.32 of F¨ollmerand Schied(2016). Two random variables X and Y are comonotonic if there exist 3 increasing functions f and g such that X = f(X + Y ) and Y = g(X + Y ). We write X =d Y if X and Y have the same distribution. In this paper, terms \increasing" and \decreasing" are in the non-strict sense. 2 Definitions 2.1 Basic requirements for variability measures A variability measure is a functional ν : X! [0; 1] which quantifies the magnitude of variability of random variables. In order for our definition of variability measures to be as general as possible, we only require three natural properties on a functional X! R to be a variability measure. Definition 1. A variability measure is a functional ν : X! [0; 1] satisfying the following properties. (A1) Law invariance: if X 2 X and Y 2 X have the same distribution under P, denoted as X =d Y , then ν(X) = ν(Y ). (A2) Standardization: ν(m) = 0 for all m 2 R. (A3) Positive homogeneity: there exists α 2 [0; 1) such that ν(λX) = λαν(X) for any λ > 0 and X 2 X . The number α is called the homogeneity index of ν. Some examples of classic variability measures are given in AppendixA. There are some relative measures of variability in the literature which are only formulated for positive random variables, such as the Gini coefficient or the relative deviation (see AppendixA). In this paper, we omit these variability measures, although our definition can be easily amended to include them by replacing X with a positive convex cone. We call the set Xν = fX 2 X : ν(X) < 1g the effective domain of ν. Remark 1. A deviation measure ν of Rockafellar et al.(2006) satisfies, in addition to stan- dardization and positive homogeneity with index 1, that ν(X) > 0 for all non-constant X and ν is subadditive. The latter two properties are not satisfied by the inter-quantile difference, which we will see later in Table1. Thus, to study the three parametric classes of variabil- ity measures, our set of definitions is more suitable than that of Rockafellar et al.(2006). Alternatively, Furman et al.(2017) defined measures of variability via a location-invariance property instead of positive homogeneity. This property is not satisfied by the relative vari- ability measures in AppendixA. In view of the above discussions, we use the three properties 4 in Definition1 as the defining properties of variability measures, and all other properties (such as location invariance and subadditivity) will be additional properties that may or may not be satisfied by a variability measure; see Section3 for a detailed analysis. 2.2 Three parametric families of risk measures Below we list three parametric families of risk measures which are popular in risk man- agement (e.g., Embrechts et al.(2014) and Emmer et al.(2015)). We define two versions of quantiles and ES for notational convenience. (i) The right-quantile (right VaR): for p 2 (0; 1), Qp(X) = inffx 2 R : P(X 6 x) > pg;X 2 X : The left-quantile (left VaR): for p 2 (0; 1), − Qp (X) = inffx 2 R : P(X 6 x) > pg;X 2 X : (ii) The ES: for p 2 (0; 1), 1 Z 1 ESp(X) = Qr(X)dr; X 2 X : 1 − p p The left-ES: for p 2 (0; 1), Z p − 1 ESp (X) = Qr(X)dr; X 2 X : p 0 (iii) The expectile: for p 2 (0; 1), 1 exp(X) = minfx 2 R : pE[(X − x)+] 6 (1 − p)E[(X − x)−]g;X 2 L : − − In the above risk measures, Qp and Qp are finite on X , and ESp, ESp and exp are finite on L1.
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