key: cord-0589642-xo87wp20 authors: Barczy, Matyas; Ned'enyi, Fanni K.; SutHo, L'aszl'o title: Probability equivalent level of Value at Risk and higher-order Expected Shortfalls date: 2022-02-20 journal: nan DOI: nan sha: 43ec0186e7cdea9aaa524c5a2999f84af2b5cfe7 doc_id: 589642 cord_uid: xo87wp20 We investigate the probability equivalent level of Value at Risk and $n^{mathrm{th}}$-order Expected Shortfall (called PELVE_n), which can be considered as a variant of the notion of the probability equivalent level of Value at Risk and Expected Shortfall (called PELVE) due to Li and Wang (2019). We study the finiteness, uniqueness and several properties of PELVE_n, we calculate PELVE_2 of some notable distributions, PELVE_2 of a random variable having generalized Pareto excess distribution, and we describe the asymptotic behaviour of PELVE_2 of regularly varying distributions as the level tends to $0$. Some properties of $n^{mathrm{th}}$-order Expected Shortfall are also investigated. Among others, it turns out that a Gini Shortfall is a linear combination of Expected Shortfall and $2^{mathrm{nd}}$-order Expected Shortfall. The Fundamental Review of the Trading Book (FRTB) was introduced by the Basel Committee on Banking Supervision in the years following the Global Financial Crisis of 2007-2009. FRTB is expected to make a complete revision of the approach to calculating risk-based capital requirements for investments. It is supposed to be implemented in January 2023. Value at Risk (VaR, see Definition 1.1) and Expected Shortfall (ES, see Defintion 1.2) are popular risk measures used to measure portfolio risk. According to FRTB, the banks are supposed to use ES at the level 0.975 instead of VaR at the level 0.99 for the bank-wide internal models to determine market risk capital requirements. Motivated by the FRTB, Li and Wang [11] have recently introduced the notion of probability equivalent level of VaR and ES (PELVE, see Definition 1.4). Roughly speaking, for an integrable random variable X and ε ∈ (0, 1), the PELVE of X at the level ε is the infimum of those values c ∈ [1, 1 ε ] for which the ES of X at level 1 − cε is less than or equal to the VaR of X at the level 1 − ε. One can see that the level ε = 0.01 corresponds to the replacement of VaR at the level 0.99 with the ES at some appropriate level, which has particular importance due to the FRTB. Very recently, Fiori and Gianin [7] have proposed a generalization of PELVE by replacing the pair (VaR, ES) in the definition of PELVE with a general pair of monotone risk measures ( , ), where is obtained from by integration similarly as ES can be obtained from VaR by integration, for more details, see Definition 1.5. As a special case of their generalization, Fiori and Gianin [7] have also introduced the so-called conditional PELVE, where is chosen to be the ES. Just before we have finished our paper, Gianin [8] gave a talk and another generalization of PELVE, called distorted PELVE, was introduced. For more details, see the paragraph after Definition 1.6. Both the PELVE due to Li and Wang [11] and its generalizations due to Fiori and Gianin [7] are defined under the minimal assumption that the random variable representing the risk has a finite first moment. These risk measures enjoy satisfactory invariance and ordering properties, and they have interesting links to the tail index of a regularly varying random variable. Fiori and Gianin [7, Section 3.1.2] have also studied PELVE and conditional PELVE of a random variable having generalized Pareto excess distribution. In the present paper we study the probability equivalent level of VaR and a higher-order ES (see Definition 1.2), i.e., we replace ES with a higher-order ES in the definition of PELVE due to Li and Wang [11] . We note that this variant of PELVE is a special case of the newly introduced notion, called distorted PELVE, in Gianin's recent talk [8] (for more details, see the paragraph after Definition 1.6). In Appendix B, we point out that a Gini Shortfall, introduced in Furman et al. [10, formula (4.1) ], is a linear combination of Expected Shortfall and 2 nd -order Expected Shortfall, and it can underline the importance of studying this variant of PELVE in a more detailed way. Let Z + , N and R denote the set of non-negative integers, positive integers and real numbers, respectively. For a function f : R → R, its range {f (x) : x ∈ R} is denoted by Range(f ). The random variables will be defined on a probability space (Ω, F, P). The distribution function of a random variable X : Ω → R is given by F X : R → [0, 1], F X (x) := P(X x), x ∈ R. The set of random variables X satisfying E(|X|) < ∞ is denoted by L 1 . Convergence in distribution is denoted by First, we recall the notion of Value at Risk. 1.1 Definition. Let X be a random variable. The Value at Risk of X at a level p ∈ [0, 1] is defined by VaR X (p) := inf{x ∈ R : F X (x) p}, with the convention inf ∅ := ∞. Note that VaR X (p) is also called the (lower) quantile or a generalized inverse of X at a level p ∈ (0, 1). One may think about X as the loss and profit of some financial position at a given time point, and, using actuarial notation, positive values of X represent losses, while negative values profit. Roughly speaking, for each p ∈ (0, 1), VaR X (p) is the smallest value x such that the probability of a loss X greater than x is at most 1 − p. Note also that VaR X (0) = −∞ for any random variable X. Next, we recall the notion of n th -order Expected Shortfall due to Fuchs et al. [9, Example 2, part (4) ]. 1.2 Definition. Let X be a random variable such that X ∈ L 1 , and let n ∈ N. The n th -order Expected Shortfall of X at a level p ∈ [0, 1) is defined by VaR X (s) ds. In the next remark we recall some basic properties of higher-order Expected Shortfalls. For some further properties of higher-order Expected Shortfalls, see Appendix A. 1.3 Remark. (i). For X ∈ L 1 , n ∈ N, and p ∈ [0, 1), we have ES X,n (p) ∈ R, see Lemma A.1. Note also that the first order Expected Shortfall coincides with the usual Expected Shortfall (also called Conditional Value at Risk), so ES X,1 is simply denoted by ES X . (ii). By Lemma 2 in Fuchs et al. [9] , the n th -order Expected Shortfall is monotone (in the sense that if X Y , X, Y ∈ L 1 , then ES X,n (p) ES Y,n (p), p ∈ [0, 1)), positive homogeneous and translation invariant. Further, using that the distortion function corresponding to the n th -order Expected Shortfall (see the function h p in the proof of Lemma A.1) is convex, we get the n th -order Expected Shortfall is subadditive, see Fuchs et al. [9, Example 1/(4) and Theorem 4] . All in all, the n th -order Expected Shortfall is a coherent risk measure on L 1 . (iii). For X ∈ L 1 and n ∈ N, we have VaR X (p) ES X (p), p ∈ [0, 1), and ES X (p) ES X,n (p), p ∈ [0, 1). (1.1) where the second equality is a consequence of Fuchs et al. [9, Corollary 4, part (1)], since In Appendix B, we point out the fact that the 2 nd -order Expected Shortfall is nothing else but a special Gini Shortfall introduced in Furman et al. [10, formula (4.1) ], and a (general) Gini Shortfall is a linear combination of Expected Shortfall and 2 nd -order Expected Shortfall. This observation could also underline the importance of studying properties of higher-order (especially, 2 nd -order) Expected Shortfalls. Next, we recall the notion of probability equivalent level of Value at Risk and Expected Shortfall (abbreviated as PELVE) due to Li and Wang [11, formula (2) ]. 1.4 Definition. Let X be a random variable such that X ∈ L 1 . The probability equivalent level of Value at Risk and Expected Shortfall (abbreviated as PELVE) of X at a level ε ∈ (0, 1) is defined by We give a motivation why the infimum in Definition 1.4 of Π ε (X) is taken over 1, 1 ε . The level 1 − cε of ES X should be nonnegative yielding that c 1 ε ; and, by (1.1), we have Next, we recall the generalization of PELVE due to Fiori and Gianin [7, Section 3]. 1.5 Definition. For each α ∈ (0, 1), let α : L 1 → R be a risk measure such that the family { α : α ∈ (0, 1)} is monotone, i.e., if 0 < α 1 α 2 < 1, then α 1 (X) α 2 (X), X ∈ L 1 . For each p ∈ (0, 1), let us introduce the risk measure p : Given a random variable X ∈ L 1 , the generalized PELVE of X at a level ε ∈ (0, 1) corresponding to the pair (( α ) α∈(0,1) , ( p ) p∈(0,1) ) is defined by where inf ∅ = ∞. In the special case α (X) = ES X (α), α ∈ (0, 1), X ∈ L 1 , the corresponding generalized PELVE is called the conditional PELVE (abbreviated as c-PELVE). By Proposition 2 in Fiori and Gianin [7] , the family { p : p ∈ (0, 1)} is monotone, and p p , p ∈ (0, 1). Hence one can give a similar motivation why the infimum in the definition of Π g ε (X) is taken over 1, 1 ε just as we did in case of Π ε (X) (see the paragraph after Definition 1.4). Further, note that if one chooses α (X) = VaR X (α), α ∈ (0, 1), X ∈ L 1 in Definition 1.5, then the corresponding generalized PELVE is nothing else but (usual) PELVE due Li and Wang [11] given in Definition 1.4. Both Li and Wang [11, Propositions 1-2 and Theorem 1] and Fiori and Gianin [7, have studied finiteness, uniqueness, and some properties of PELVE and generalized PELVE, respectively. The PELVE values of some notable distributions, such as uniform, exponential, normal, lognormal, t and Pareto distributions, have been calculated or approximated in Li and Wang [11] . The conditional PELVE values of uniform, normal and Pareto distributions have been also calculated in Fiori and Gianin [7] , and it turned out that for uniform and Pareto distributions, the corresponding PELVE and conditional PELVE values coincide, see Fiori and Gianin [7, Subsection 3.1.1]. Li and Wang [11, Section 4.2] have described convergence of PELVE of regularly varying random variables as the level tends to 0, while Fiori and Gianin [7, Proposition 10] showed that PELVE and conditional PELVE of a random variable having generalized Pareto excess distribution coincide. In the following definition we replace the Expected Shortfall in Definition 1.4 by the n thorder Expected Shortfall, where n ∈ N. 1.6 Definition. Let X be a random variable such that X ∈ L 1 , and let n ∈ N. The probability equivalent level of Value at Risk and n th -order Expected Shortfall (abbreviated as PELVE n ) of X at a level ε ∈ (0, 1) is defined by First of all, we emphasize that the notion of PELVE n is a special case of the so-called distorted PELVE introduced in Gianin's recent talk [8] . Using the notations of [8] , one can check that PELVE n is a distorted PELVE with the function g(x) : Further, since ES X,n (p) VaR X (p), p ∈ (0, 1) (see (1.1)), we can give a similar motivation why the infimum in the definition of Π ε,n (X) is taken over 1, 1 ε just as we did in case of Π ε (X) (see the paragraph after Definition 1.4). Note also that, since ES X (p) ES X,n (p), p ∈ [0, 1) (see (1.1)), we have Π ε,n (X) Π ε (X), ε ∈ (0, 1), n ∈ N. We will prove results that can be considered as counterparts of the above mentioned results of Li and Wang [11] and Fiori and Gianin [7] . Our forthcoming Propositions 2.1, 2.3 and Theorem 2.4 are special cases of the results announced in Gianin's talk [8] for distorted PELVE. At the moment no proofs are publically available of the above mentioned results in [8] , and hence, for completeness, we give proofs of our Propositions 2.1, 2.3 and Theorem 2.4. The paper is organized as follows. In Section 2 we study the finiteness, uniqueness and some basic properties of PELVE n such as inequalities for PELVE n of sum of comonotonic random variables (see Definition A.4), see Propositions 2.1, 2.3 and Theorem 2.4. In Section 3, under some appropriate conditions, we show that for each n ∈ N, we have Π ε,n (X m ) → Π ε,n (X) as m → ∞ whenever X m D −→ X as m → ∞. Section 4 is devoted to calculate the PELVE 2 values of some notable distributions such as uniform, exponential, normal and Pareto distributions. In particular, it turns out that, for a uniformly distributed random variable X, the PELVE 2 value Π ε,2 (X) of X equals 3 for each ε ∈ (0, 1 3 ], i.e., it is the same constant for ε ∈ (0, 1 3 ]. Similar phenomena occur in case of exponential and Pareto distributions, but not in case of normal distributions. In Section 5, we study PELVE 2 of a random variable having generalized Pareto excess distribution; and in Section 6 we describe the asymptotic behaviour of PELVE 2 of regularly varying distributions as the level tends to 0. In Sections 4, 5 and 6, we consider the PELVE 2 -values, and not PELVE n -values of the random variables in question due to the central role of 2 nd -order Expected Shortfall in the decomposition of Gini Shortfall presented in Appendix B, and the complexity of computation. Section 7 is devoted to presenting some simulations and real data analysis for PELVE 2 on S&P 500 daily returns. An interesting phenomenon occurs, PELVE 2 shows the effect of the COVID-19 pandemic via analysing S&P 500 daily returns. We close the paper with three appendices. In Appendix, A we study some properties of higherorder Expected Shortfalls given in Definition 1.2 such as finiteness, continuity, monotonicity, additivity for comonotonic random variables and connection with weak convergence. Appendix B is devoted to develop a connection between 2 nd -order Expected Shortfall and Gini Shortfall. Finally, in Appendix C, we formulate a Karamata theorem for regularly varying functions at 0 with index κ > −1. In what follows, when we write ES X,n , Π ε,n (X) and PELVE n we always mean that n ∈ N without mentioning it explicitly. The following result for PELVE n can be considered as the counterpart of the corresponding result for PELVE due to Li and Wang [11, Proposition 1] . It is a special case of a result announced for distorted PELVE in Gianin's recent talk [8] , where no proofs are presented. 2.1 Proposition. Let X be a random variable such that X ∈ L 1 , ε ∈ (0, 1) and n ∈ N. Then the following statements are equivalent: (i) There exists c 0 ∈ [1, 1/ε] such that (ii) Π ε,n (X) ∈ [1, 1/ε] and (2.1) holds for Π ε,n (X), i.e., ES X,n (1−Π ε,n (X)ε) = VaR X (1−ε). (iii) ES X,n (0) VaR X (1 − ε). (iv) Π ε,n (X) < ∞. Proof. (i) ⇒ (ii): By (i), the set {c ∈ [1, 1/ε] : ES X,n (1 − cε) VaR X (1 − ε)} is nonempty yielding that Π ε,n (X) < ∞ and Π ε,n (X) ∈ [1, 1/ε]. By the definition of infimum, there exists a sequence (c m ) m∈N in [1, 1/ε] such that c m ↓ Π ε,n (X) as m → ∞, and ES X,n (1 − c m ε) VaR X (1 − ε), m ∈ N. By Lemma A.2, the function [0, 1) p → ES X,n (p) is continuous and monotone increasing, so it is continuous at the point 1 − Π ε,n (X)ε ∈ [0, 1), and consequently, by taking the limit of both sides of ES X,n (1 − c m ε) VaR X (1 − ε) as m → ∞, we have Further, using again (i), there exists c 0 ∈ [1, 1/ε] such that ES X,n (1 − c 0 ε) = VaR X (1 − ε), and hence, by the definition of infimum, Π ε,n (X) c 0 . Since the function [0, 1) p → ES X,n (p) is continuous and monotone increasing, On the contrary to (iii), let us suppose that ES X, Consequently, by definition, Π ε,n (X) = ∞, which leads us to a contradiction, since Π ε,n (X) < ∞ (due to Π ε,n (X) ∈ [1, 1/ε]). (iii) ⇒ (iv): By (iii), choosing c = 1/ε, the set {c ∈ [1, 1/ε] : ES X,n (1 − cε) VaR X (1 − ε)} is nonempty, so Π ε,n (X) < ∞, as desired. (iv) ⇒ (i): Using (1.1), we have ES X,n (p) VaR X (p), p ∈ (0, 1). In fact, this inequality is a direct consequence of the fact that the function (0, 1) p → VaR X (p) is monotone increasing: Further, ES X,n (0) VaR X (1 − ε), since otherwise Π ε,n (X) = ∞ would hold, which can be checked similarly as in the proof of part (ii) ⇒ (iii). Consequently, by the previous two inequalities, we have In the next remark for a random variable X ∈ L 1 , we compare ES X (p) and ES X,n (p), where p ∈ [0, 1), and we also compare Π ε (X) and Π ε,n (X), where ε ∈ (0, 1). 2.2 Remark. (i). For X ∈ L 1 and ε ∈ (0, 1), the inequality ES X,n (0) VaR X (1 − ε) in part (iii) in Proposition 2.1 implies E(X) VaR X (1 − ε), which is nothing else but the inequality in part (iii) in Proposition 1 in Li and Wang [11] . Indeed, E(X) = ES X (0), since where U is a uniformly distributed random variable on (0, 1), and the distributions of VaR X (U ) and X coincide (see, e.g., Embrechts and Hofert [6, Proposition 2]). Hence, by (1.1), we have E(X) = ES X (0) ES X,n (0) VaR X (1 − ε), as desired. (ii). For X ∈ L 1 , the inequality ES X (p) ES X,n (p), p ∈ [0, 1) (see part (i)) yields Π ε,n (X) Π ε (X), ε ∈ (0, 1). 2 The following result for PELVE n can be considered as the counterpart of the corresponding result for PELVE due to Li and Wang [11, Proposition 2] . It is a special case of a result announced for distorted PELVE in Gianin's recent talk [8] , where no proofs are presented. 2.3 Proposition. Let X be a random variable such that X ∈ L 1 , ε ∈ (0, 1), and n ∈ N. Let us suppose that the function (0, 1) p → VaR X (p) is not constant on the interval [1−ε, 1), and let us assume that ES X,n (0) VaR X (1 − ε). Then there exists a unique c 0 ∈ [1, 1/ε] such that (2.1) holds. The following result for PELVE n can be considered as the counterpart of the corresponding result for PELVE due to Li and Wang [11, Theorem 1] . It is a special case of a result announced for distorted PELVE in Gianin's recent talk [8] , where no proofs are presented. 2.4 Theorem. Let X be a random variable such that X ∈ L 1 , ε ∈ (0, 1), and n ∈ N. Let us suppose that ES X,n (0) VaR X (1 − ε) holds. Then the following statements hold: (i) scale invariance: Π ε,n (λX + a) = Π ε,n (X) for each λ > 0 and a ∈ R. (ii) Π ε,n (f (X)) Π ε,n (X) for each monotone increasing and concave function f : (iii) Π ε,n (X) Π ε,n (g(X)) for each strictly monotone increasing and convex function g : (iv) quasi-convexity and quasi-concavity for comonotonic random variables: Proof. First, we prove that (ii) yields (i). Let λ > 0, a ∈ R, f : R → R, f (x) := λx + a, x ∈ R, and g : R → R, g(y) := 1 λ (y − a), y ∈ R. Then f and g are monotone increasing, linear (hence convex and concave) functions, and they are inverses of each other. Further, yielding Π ε,n (f (X)) = Π ε,n (X), i.e., (i), as desired. Now we prove (ii). Let f : R → R be a monotone increasing and concave function with f (X) ∈ L 1 . Since f is concave and defined on (the open interval) R, we have f is continuous. Further, it is known that for any monotone increasing and continuous function h : R → R, we have VaR h(X) (p) = h(VaR X (p)), p ∈ (0, 1), see, e.g., Shorack Let c 1 := Π ε,n (X) and c 2 := Π ε,n (f (X)). Since, by assumption, ES X,n (0) VaR X (1 − ε), using the equivalence of (ii), (iii) and (iv) in Proposition 2.1, we have c 1 < ∞ and , is a distortion function in the sense of Definition 3.6 in Pflug and Pichler [13] , since it is nonnegative, monotone increasing and 1 0 σ p (s) ds = 1. Hence for each p ∈ [0, 1) one can apply Corollary 3.19 in Pflug and Pichler [13] with the distortion function σ p , and we have Further, one can check that for any random variable U which is uniformly distributed on [0, 1], we have Q(A) := A (σ p (U ))(ω) P(dω), A ∈ F, is a probability measure on (Ω, F) such that Q is absolutely continuous with respect to P and E(f (X)σ p (U )) = E Q (f (X)). Hence, for any random variable U which is uniformly distributed on [0, 1], by Jensen's inequality, we get Consequently, using also that f is monotone increasing, we have Since the functions f , [0, 1) p → ES X,n (p) and [0, 1) p → ES f (X),n (p) are continuous (see Lemma A.2), by taking the limit of both sides of the inequality (2.7) as p ↓ 0, we have where, for the second inequality, we used that f is monotone increasing and ES X,n (0) VaR X (1 − ε) (by assumption), and, for the equality, (2.5). So, by Proposition 2.1, we have . Using (2.7) with the choice of p := 1 − c 1 ε, (2.6) and (2.5), we get and, by Definition 1.6 of PELVE n , we have Π ε,n (f (X)) c 1 , i.e., Π ε,n (f (X)) Π ε,n (X), as desired. (iii). Since g is a convex function defined on R, it is continuous. Further, due to our assumptions, g −1 : R → R is a strictly monotone increasing and concave function with Range(g −1 ) = R, and since g(X) ∈ L 1 and g −1 (g(X)) = X ∈ L 1 , part (ii) yields that Π ε,n (X) = Π ε,n (g −1 (g(X))) Π ε,n (g(X)), as desired. (iv). First, we check that for any c, d ∈ [1, 1 ε ] and Y ∈ L 1 , we have If Π ε,n (Y ) = c, then, again by the definition of infimum, there exists a sequence (c m ) m∈N in Using that the function [0, 1) p → ES Y,n (p) is continuous (see Lemma A.2), by taking the limit of both sides of the inequality above as and, by the definition of Π ε,n (Y ), we get Π ε,n (Y ) . This leads us to a contradiction, since, by assumption, VaR Y (1 − ε) < ES Y,n (1 − dε), and consequently, d > Π ε,n (Y ) cannot hold. If d = Π ε,n (Y ) would hold, then similarly as in proving part =⇒ of (2.8), by definition of infimum and the continuity of the function [0, . This leads us to a contradiction, since, by assumption, Since f, g : R → R are monotone increasing and λ ∈ [0, 1], we have λf (X) and (1 − λ)g(X) are comonotonic random variables, so, by the additivity of VaR for comonotonic random variables (see, e.g., McNeil et al. [ Let c 1 := Π ε,n (f (X)) and c 2 := Π ε,n (g(X)). Let us suppose that c 1 = ∞ and c 2 = ∞. In this case min(c 1 , c 2 ) = ∞ and max(c 1 , c 2 ) = ∞, so it is enough to check that Π ε,n (λf (X)+(1−λ)g(X)) = ∞. By Proposition 2.1, the positive homogeneity and comonotonic additivity of Value at Risk and n th -order Expected Shortfall (see Proposition A.5), we have Here the last inequality is satisfied, since using again Proposition 2.1, c 1 = Π ε,n (f (X)) = ∞, c 2 = Π ε,n (f (X)) = ∞, and f (X), g(X) ∈ L 1 , we have and Let us suppose now that at least one of c 1 and c 2 is finite. Then for each d < min{c 1 , c 2 } with d ∈ [1, 1 ε ], using (2.8) and Proposition 2.1, we get Indeed, if both c 1 and c 2 are finite, then it readily follows by (2.8); and if c 1 < ∞ and c 2 = ∞, then, by (2.8) , , and, by Proposition 2.1 and the monotone increasing property of the function [0, 1) , as desired. The case c 1 = ∞ and c 2 < ∞ can be handled similarly. As a consequence, for each λ ∈ [0, 1], we have and using again the positive homogeneity and comonotonic additivity of Value at Risk and n th -order Expected Shortfall, we have Hence, by Definition 1.6 and the continuity and monotone increasing property of the function and d < min{c 1 , c 2 }. By taking the limit d ↑ min{c 1 , c 2 }, we have as desired. If max(c 1 , c 2 ) = ∞, then the second inequality in (iv) automatically holds. If max(c 1 , c 2 ) < ∞, i.e., both c 1 and c 2 are finite, then let c := max{c 1 , c 2 }. Then Π ε,n (f (X)) = c 1 c and Π ε,n (g(X)) = c 2 c, so, by (2.8), and and then the positive homogeneity and comonotonic additivity of VaR and n th -order Expected Shortfall yield that In the next remark, we point out that part (iii) of Theorem 2.4 does not hold for a general monotone increasing and convex function g : R → R. Then for any random variable X, n ∈ N, and p ∈ (0, 1), we have VaR g(X) (p) = A and and hence for each ε ∈ (0, 1), Consequently, if X is random variable such that Π ε,n (X) > 1, then Π ε,n (g(X)) Π ε,n (X) cannot hold. Hence part (iii) of Theorem 2.4 does not hold for a general monotone increasing and convex function g : R → R. 2 The following result for PELVE n can be considered as the counterpart of the corresponding result for PELVE due to Li and Wang [11, Theorem 2]. 3.1 Theorem. Let X m , m ∈ N, and X be random variables such that X m ∈ L 1 , m ∈ N, and X ∈ L 1 . Let ε ∈ (0, 1) and n ∈ N. If Proof. By the second part of (ii) and (iii), the quantile convergence theorem (see, e.g., Shorack and Wellner [15, Exercise 5, page 10]) yields that Using (iii), (iv) and Lemma A.6, we have By (3.1) and (3.2), we have f m converges pointwise to f on [0, 1) as m → ∞, and, using also Lemma A.2, we get that f m , m ∈ N, and f are continuous and monotone increasing functions. Further, the first part of (ii) and Lemma A.3 yield that f is strictly monotone increasing on the interval [0, 1 − ε]. Let us recall the following result from calculus: given a < b, a, b ∈ R, and a sequence of monotone increasing real-valued functions on [a, b] converging pointwise to a continuous function, we have that the convergence holds uniformly on [a, b] as well. The above recalled result yields that f m converges uniformly on any interval [0, 1 − δ] to f as m → ∞, where δ ∈ (0, 1). Let us consider the reparametrizations g m : [1, 1 ε ] → R, m ∈ N, and g : [1, 1 ε ] → R of f m , m ∈ N, and f , respectively, given by Then g m , m ∈ N, and g are continuous and monotone decreasing functions, and g is strictly monotone decreasing as well. Further, g m converges uniformly on [1, 1 ε ] to g as m → ∞. Using since g m converges uniformly on [1, 1 ε ] to g as m → ∞. So lim m→∞ g(Π ε,n (X m )) = 0. Further, if c ∈ [1, 1 ε ] is a limit point of the sequence (Π ε,n (X m )) m∈N , then there exists a subsequence (Π ε,n (X m k )) k∈N in [1, 1 ε ] such that Π ε,n (X m k ) = c, k ∈ N, and Π ε,n (X m k ) → c as k → ∞. Since g is continuous, we have g(Π ε,n (X m k )) → g( c) as k → ∞, where g( c) = 0 due to lim m→∞ g(Π ε,n (X m )) = 0. Hence c ∈ [1, 1 ε ] is a root of g, and using that g has a unique root Π ε,n (X) on [1, 1 ε ] (since we already checked that Π ε,n (X) is a root of g and g is continuous and strictly monotone decreasing), we get c = Π ε,n (X). All in all, for any limit point c of (Π ε,n (X m )) m∈N , we have c = Π ε,n (X). Since (Π ε,n (X m )) m∈N is a bounded sequence in [1, 1 ε ], it has a limit point, and taking into account our previous considerations, Π ε,n (X m ) converges to its unique limit point Π ε,n (X) as m → ∞. 2 In this section, we calculate PELVE 2 of some notable distributions such as uniform distributions, exponential distributions, normal distributions, and Pareto distributions. Let X be a random variable with uniform distribution on the interval [0, 1]. Then VaR X (p) = p, p ∈ (0, 1), and So for each ε ∈ (0, 1), the inequality ES X,2 (0) VaR X (1 − ε) is equivalent to 2 3 1 − ε, and hence, by Proposition 2.1, if ε ∈ (0, 1 3 ], then Π ε,2 (X) is a solution of the equation Hence Π ε,2 (X) = 3 for ε ∈ (0, 1 3 ]. If ε ∈ ( 1 3 , 1), then, by Definition 1.6, Π ε,2 (X) = ∞. Note that, by Li and Wang [11, part (i) of Example 5], Π ε (X) = 2 for ε ∈ (0, 1 2 ], and hence Π ε,2 (X) > Π ε (X) for ε ∈ (0, 1 3 ] (as it is expected, see part (ii) of Remark 2.2). Let Y be a random variable with uniform distribution on the interval [a, b], where a < b, a, b ∈ R. Using that the distribution of Y coincides with that of (b − a)X + a, part (i) of 4.2 Example. (Exponential distribution) Let X be an exponentially distributed random variable with parameter 1. Then VaR X (p) = − ln(1 − p), p ∈ (0, 1), and Here, by partial integration, one can check that where C 1 , C 2 ∈ R, which yields that Consequently, by some algebraic calculations, So for each ε ∈ (0, 1), the inequality ES X,2 (0) VaR X (1 − ε) is equivalent to 3 2 − ln(ε), i.e., ε ∈ (0, e −3/2 ], and hence, by Proposition 2.1, if ε ∈ (0, e −3/2 ], then Π ε,2 (X) is a solution of the equation Hence Π ε,2 (X) = e 3 2 ≈ 4.482 for ε ∈ (0, e −3/2 ]. If ε ∈ (e −3/2 , 1), then, by Definition 1.6, Π ε,2 (X) = ∞. Note that, by Li and Wang [11, Example 5, part (ii)], Π ε (X) = e for ε ∈ (0, e −1 ], so Π ε,2 (X) > Π ε (X) for ε ∈ (0, e −3/2 ] (as it is expected, see part (ii) of Remark 2.2). Let Y be an exponentially distributed random variable with parameter λ > 0. Using that the distribution of Y coincides with that of 1 Example. (Normal distribution) Let X be a standard normally distributed random variable. Let Φ and ϕ denote the distribution function and density function of X, respectively. Then VaR X (p) = Φ −1 (p), p ∈ (0, 1), and, by substitution s = Φ(x), we have Further, by partial integration and then substitution x = y √ 2 , we get Hence So for each ε ∈ (0, 1), the inequality ES X,2 (0) Let Y be a normally distributed random variable with mean m and variance σ 2 , where m ∈ R and σ > 0. Using that the distribution of Y coincides with that of σX + m, part (i) of Theorem 2.4 yields Π ε,2 (Y ) = Π ε,2 (σX + m) = Π ε,2 (X). Using the software R, for levels ε ∈ {0.1, 0.05, 0.01, 0.005} we calculate an approximated value of the unique root Π ε,2 (X) of the equation (4.3), see Table 1 . Let X be a random variable with Pareto distribution having parameters k > 0 and α > 0, i.e., the distribution function of X takes the form F X : R → [0, 1], In what follows we suppose that α > 1, yielding that X ∈ L 1 . In this case we have Here, by partial integration, where C ∈ R, and hence we get So for each ε ∈ (0, 1), the inequality ES X,2 (0) VaR X (1 − ε) is equivalent to and hence, by Proposition 2.1, if ε ∈ 0, (α−1)(2α−1) , 1 , then, by Definition 1.6, Π ε,2 (X) = ∞. Note that x is monotone decreasing. Indeed, x ∈ (1, ∞), and the logarithm of the function (1, ∞) x → (x/(x − 1)) x , i.e., the function (1, ∞) x → x ln(x/(x − 1)), is monotone decreasing. Hence, by (4.5), for each ε ∈ 0, (α−1)(2α−1) where, by Example 4.2, the limit e 3 2 is nothing else but the PELVE 2 -value (at the given level ε) of an exponentially distributed random variable. The inequality (4.6) for PELVE 2 can be considered as the counterpart of the corresponding inequality for PELVE of X due to Li and Wang [11, inequality (8) ]. For levels ε ∈ {0.1, 0.05, 0.01, 0.005} and parameters α ∈ {2, 10, 30}, we calculate Π ε,2 (X), see Table 2 . In this section, we calculate the PELVE 2 -values of random variables with some generalized Pareto distribution and random variables of which the excess distribution function over a threshold is given by the distribution function of a generalized Pareto distribution with tail index less than 1. It will turn out that the PELVE 2 -value at a level of such random variables depends on the tail index but not on ε (below some threshold). Such a result was already established for the PELVE-values of the random variables in question by Fiori and Gianin [7, Proposition 10]. Both results might be useful for estimating the tail index of the random variables in question. First, we recall the notion of a generalized Pareto distribution. 5.1 Definition. Let κ ∈ R and β > 0. We say that a random variable X has a generalized Pareto distribution with parameters κ and β if its distribution function F X takes the form: • in case of κ > 0, • in case of κ = 0, • in case of κ < 0, The parameter κ is sometimes called the tail index of X. The distribution function of a random variable having generalized Pareto distribution with parameters κ ∈ R and β > 0 is denoted by G κ,β . If X is a random variable having a generalized Pareto distribution with parameters κ > 0 and β > 0, then X + β κ has a (usual) Pareto distribution with parameters β κ and 1 κ (recalled in Example 4.4). Further, if X is a random variable having a generalized Pareto distribution with parameters κ = 0 and β > 0, then X is in fact exponentially distributed with parameter 1 β . 2 Next, to give an application of Theorem 2.4, we calculate the PELVE 2 -value of a generalized Pareto distribution with parameters κ ∈ (0, 1) and β := 1. Let κ ∈ (0, 1) and let X be a random variable with distribution function Then X has a generalized Pareto distribution with parameters κ and β := 1, and X has the unbounded support [0, ∞). Consequently, X + 1 κ has a (usual) Pareto distribution with parameters 1 κ and 1 κ . This yields that the calculations of VaR, 2 nd -order ES and PELVE 2 of X can be traced back to those of X + 1 κ , for which we can use Example 4.4. Using Example 4.4 and the translation invariance of VaR, we have Since κ ∈ (0, 1), we have X + 1 κ ∈ L 1 , and hence X ∈ L 1 . Consequently, using again Example 4.4 and the translation invariance of the 2 nd -order ES (being a coherent risk measure on L 1 ), we have Consequently, for each ε ∈ (0, 1), the inequality ES X,2 (0) VaR X (1 − ε) is equivalent to Using part (i) of Theorem 2.4 and (4.5) yield that κ , 1 , then, by Definition 1.6, Π ε,2 (X) = ∞. Finally, note that, by L'Hospital's rule, we get where − ln(1 − p) + 3 2 is nothing else but the 2 nd -order Expected Shortfall of an exponentially distributed random variable at a level p, see Example 4.2. This is in accordance with in accordance with Lemma A.6, since a generalized Pareto distribution having parameters κ > 0 and 1 converges in distribution to the exponential distribution with parameter 1 as κ ↓ 0. 2 For a random variable X with distribution function F X , let x F X denote the right endpoint of F X , i.e., x F X := sup{x ∈ R : F X (x) < 1}. If X has a generalized Pareto distribution with parameters κ ∈ R and β > 0, then 5.4 Definition. Let X be a nonnegative random variable with distribution function F X . The excess distribution function corresponding to F X over a threshold u ∈ [0, x F X ) is given by F X,u : [0, ∞) → R, For the forthcoming Propositions 5.5, 5.6 and 5.7, one can refer for example to Example 5.19, Lemma 5.22, formulae (5.18), (5.19 ) and (5.20) in McNeil et al. [12] . Let X be a random variable having a generalized Pareto distribution with parameters κ ∈ R and β > 0. For the excess distribution function F X,u corresponding to F X over a threshold u, we have F X,u (x) = G κ,β+κu (x) for x 0 and u 0 in case of κ 0; and for x 0 and u < − β κ in case of κ < 0. 5.6 Proposition. Let X be a nonnegative random variable, and assume that there exist Let X be a nonnegative random variable, and assume that there exist If, in addition κ < 1, then we have The next result can be considered as a counterpart of Proposition 10 in Fiori and Gianin [7] and that of (5.2). Let X be a nonnegative random variable, and assume that there exist and in case of κ = 0, we have Π ε,2 (X) = e 3 2 for 0 < ε < (1 − F X (u))e − 3 2 . Further, if κ < 0. Proof. Recall that VaR X (p), p ∈ (F X (u), 1), is given in (5.1). Next, we calculate ES X,2 (p) for p ∈ (F X (u), 1). First, we consider the case of κ = 0. Using (4.1) and (4.2), for p ∈ (F X (u), 1) we have Note that if u = 0, κ = 0 and β = 1, then ES X,2 (p) = − ln(1−p)+ 3 2 , which is nothing else but the 2 nd -order Expected Shortfall at the level p of an exponentially distributed random variable (see Example 4.2) . This is expected, since if u = 0, κ = 0 and β = 1, then x 0, where G 0,1 is the distribution function of an exponentially distributed random variable with parameter 1. Let ε ∈ (0, 1 − F X (u)). Then the inequality This inequality holds if and only if c e as desired. Next, we consider the case of κ < 1 with κ = 0. Using (4.4), for p ∈ (F X (u), 1) we have Let ε ∈ (0, 1 − F X (u)). Then the inequality Consequently, using that the function [1, 1 ε ] c → ES X,2 (1 − cε) is monotone decreasing (see Now, we turn to prove (5.3). In case of κ = 0, we have lim p↑1 VaR X (p) = ∞ and In case of κ ∈ (0, 1), we have lim p↑1 VaR X (p) = ∞ and . In case of κ < 0, we have lim p↑1 VaR X (p) = u − β κ and lim p↑1 ES X,2 (p) VaR X (p) First, we recall the notion of regularly varying functions. In case of ρ = 0, we call U slowly varying at infinity. A measurable function V : (0, x 0 ) → (0, ∞) (where x 0 > 0) is called regularly varying at 0 with index κ ∈ R if for all q > 0, we have Next, we recall the notion of regularly varying non-negative random variables. A non-negative random variable X is called regularly varying with index α 0 if F X (x) := P(X > x) > 0 for all x > 0, and F X is regularly varying at infinity with index −α. Since the notion of a non-negative regularly varying random variable X is defined via the asymptotic behaviour of its survival function F X at infinity, we will study the asymptotic behaviour of the PELVE 2 of X at a level ε tending to 0. Recall that if X is a non-negative regularly varying random variable with index α > 1, then E(X) < ∞, i.e., X ∈ L 1 , and hence in this case PELVE 2 of X at any level ε ∈ (0, 1) is well-defined. 6.3 Theorem. Let X be a non-negative regularly varying random variable with index α > 1 such that (0, 1) p → VaR X (p) is continuous. Then i.e., the function (0, 1) p → VaR X (p) is regularly varying at 0 with index − 1 α , see, e.g., Li and Wang [11, formula (A.9) ]. For completeness, we present a proof of (6.1). Since X is regularly varying with index α > 1, we have 1 F X is a monotone increasing and regularly varying function at infinity with index α satisfying lim x→∞ Hence and consequently lim ε↓0 VaR X 1 − tε Recall also that see, e.g., Li and Wang [11, formula (A.10) ]. For completeness, we present a proof of (6.2). For each ε ∈ (0, 1), we have By (6.1), the function (0, 1) v → VaR X (1−v) is regularly varying at 0 with index − 1 α > −1, and hence, by Karamata theorem for regularly varying functions at zero (see Lemma C.2), we have yielding (6.2). Next, we prove that . For each ε ∈ (0, 1), we have Here the function (0, 1) v → v VaR X (1 − v) is regularly varying at 0 with index α−1 α > −1, since, using (6.1), we have So, by Karamata theorem for regularly varying functions at zero (see Lemma C.2), we have Hence, using (6.2), we get . Consequently, using (6.1), we have Note that for each α > 1, the function (0, ∞) t → 2α 2 (α−1)(2α−1) t − 1 α is strictly monotone decreasing and it takes value 1 if and only if Let t 1 and t 2 be such that Hence, using (6.3), for sufficiently small ε > 0, we have Since X is non-negative and regularly varying, we have VaR X (1 − v) → ∞ as v ↓ 0, and hence for sufficiently small ε > 0, we get VaR X (1 − ε) > 0 and Using again VaR X (1 − v) → ∞ as v ↓ 0, and ES X,2 (0) < ∞, we have ES X,2 (0) < VaR X (1 − ε) for sufficiently small ε > 0. Using Proposition 2.1, for sufficiently small ε > 0, we have Π ε,2 (X) ∈ [1, 1 ε ] is a solution of the equation ES X,2 (1 − cε) = VaR X (1 − ε), c ∈ [1, 1 ε ]. Consequently, using (6.4), the definition of PELVE 2 , and the continuity and monotone increasing property of ES X,2 (see Lemma A.2), we get Π ε,2 (X) ∈ (t 1 , t 2 ] for sufficiently small ε > 0. By taking the limits the statement follows. 2 6.4 Remark. In Theorem 6.3, the limit is nothing else but the PELVE 2 of a Pareto distributed random variable with parameters k > 0 and α > 1 at any level less then or equal to , see Example 4.4. So Theorem 6.3 is in accordance with Theorem 3 in Li and Wang [11] . Note that a Pareto distributed random variable with parameters k > 0 and α > 0 is regularly varying with index −α, and a random variable with generalized Pareto distribution having shape parameter λ > 0 is regularly varying with index − 1 λ . 2 7 Simulations and real data analysis for PELVE 2 First, we present an empirical estimator of PELVE 2 of a random variable. Let X ∈ L 1 , m ∈ N, and let X 1 , . . . , X m be independent and identically distributed random variables such that their common distribution coincides with that of X, i.e., X 1 , . . . , X m is a sample of length n for X. Let X * 1 X * 2 . . . X * m be the corresponding ordered sample. Given p ∈ (0, 1), an empirical estimator of VaR X (p) based on X 1 , . . . , X m is given by Following Acerbi [1, Section 7] (where one can find a construction of empirical estimators of spectral risk measures), given p ∈ [0, 1), an empirical estimator of the 2 nd -order Expected Shortfall ES X,2 (p) based on X 1 , . . . , X m is given by a weighted sum of X * 1 , . . . , X * m : We check that if p ∈ [0, m−1 m ), then Given ε ∈ (0, 1), an empirical estimator of the PELVE 2 value Π ε,2 (X) based on X 1 , . . . , X m can be defined as In principle, the empirical estimator (7.3) of the PELVE 2 value Π ε,2 (X) can be used even if the random variables X 1 , . . . , X m are not independent or identically distributed. We will do so in analyzing real data. For illustrative purposes, we present a simulation result for calculating the PELVE 2 value of a standard normally distributed random variable at the level ε = 0.05. We generated 10000 samples of length m = 5000 for a standard normally distributed random variable X. For each generated sample, we calculated the empirical estimator Π ε,2 (X) (given in (7. 3)) of the PELVE 2 value of X. Then we made a density histogram based on the 10000 estimated PELVE 2 values, see Figure 1 . On this figure, we also plotted the density function of the fitted normally distribution in red. The theoretical PELVE 2 value of a standard normally distributed random variable at the level ε = 0.05 is approximately 4.040815. The sample mean of the 10000 estimated PELVE 2 values is 4.046066, which is quite close to the theoretical value. Based on Figure 1 , one could conjecture that a kind of central limit theorem might hold for PELVE 2 in case of a standard normal distribution. We do not study this question here, we only note that in case of PELVE, such a result is available due to Li and Wang [11, Theorem 4] . As real data applications, we calculate the empirical estimator of PELVE 2 for S&P 500 daily returns based on two data sets: (i) ranging from 4th January 2020 to 4th January 2022, and (ii) Density histogram of PELVE_2 for N(0,1) ranging from 6th April 2020 to 4th January 2022. The data are obtained from Investing.com. Note that in case (i) the data set contains approximately four months before the COVID-19 crisis started in Europe (i.e., before April 2020), and in case (ii) the data set just starts when the the COVID-19 crisis started in Europe. The S&P 500 historical data sets were downloaded from Investing.com. Recall that, given some asset prices S t , t = 0, 1, . . . , N , where N ∈ N, the one-period (linear) return at time t = 1, . . . , N is defined by S t /S t−1 − 1. In the used data sets, the daily returns are rounded off to two decimal places. For both data sets in question, we calculated the empirical PELVE 2 estimator (7.3) and the empirical PELVE estimator at levels ranging from 0.001 to 0.56, see Figure 2 . The empirical PELVE estimator is not presented in the present paper, we used the same empirical estimators as Li and Wang [11, Section 5] and Fiori and Gianin [7, Section 4]. On Figure 2 , the red horizontal lines correspond to the PELVE 2 value of an exponential distribution, i.e., e 3/2 ≈ 4.482 (see Example 4.2); and the green horizontal lines correspond to the PELVE value of an exponential distribution, i.e., e ≈ 2.718 (see Li and Wang [11, Example 5, part (ii)]). On Figure 2 , one can see that there are levels for which no empirical PELVE 2 or PELVE values are plotted. It just means that the corresponding empirical PELVE 2 or PELVE values are infinity. For example, on the right subfigure of Figure 2 , no PELVE 2 values are plotted at levels greater than (approximately) 0.27. As shown in the left subfigure of Figure 2 , most of the empirical PELVE 2 values are above e 3/2 , but it is not the case for the right subfigure of Figure 2 . Further, the empirical PELVE 2 values on the left subfigure of Figure 2 are greater than the corresponding empirical PELVE 2 values on the right subfigure of Figure 2 . A possible explanation for this phenomenon is that the data set used for the left subfigure of Figure 2 contains four months daily returns of S&P 500 before the start of the COVID-19 crisis in Europe (i.e., before April 2020), while the data set used for the right subfigure of Figure 2 does not include these four months, it just starts at April 2020. Our real data applications may suggest that PELVE 2 might be an indicator for structural changes in stock prices. We used the open software R for making the simulations and real data analysis. This appendix is devoted to study some properties of higher-order Expected Shortfalls given in Definition 1.2 such as finiteness, continuity, monotonicity, additivity for comonotonic random variables and connection with weak convergence. These results generalize the corresponding known properties for (the usual, i.e., first order) Expected Shortfall. A.1 Lemma. For each X ∈ L 1 , n ∈ N, and p ∈ [0, 1), we have ES X,n (p) ∈ R. where we recall that F X (x) = 1 − F X (x) = P(X > x), x ∈ R. Formula (A.2) is also a special case of Theorem 1 in Fuchs et al. [9] . We check that both integrals on the right hand side of (A.2) are finite. Namely, we have Note that for each x ∈ R and p ∈ [0, 1), we have that F (1 − F X (x)) dx. Using that P(X > x) = P(X + > x), x 0, we have ∞ 0 (1 − h p (F X (x))) dx max(0, VaR X (p)) + n 1 − p ∞ max(0,VaR X (p)) P(X + > x) dx max(0, VaR X (p)) + n 1 − p ∞ 0 P(X + > x) dx = max(0, VaR X (p)) + n 1 − p E(X + ) < ∞. Further, using that h p is monotone increasing, we get A.3 Lemma. Let X be a random variable such that X ∈ L 1 , let ε ∈ (0, 1), n ∈ N, and let us suppose that the function (0, 1) p → VaR X (p) is not constant on the interval [1 − ε, 1). Then the function [0, 1 − ε] p → ES X,n (p) is strictly monotone increasing. Proof. For each p ∈ [0, 1), let us consider the function h p defined in (A.1). For each 0 p 1 < p 2 < 1, we have (1 − h p 1 (F X (x))) dx (1 − h p 2 (F X (x))) dx, where we used (A.3), the fact that h p (1) = 1, p ∈ [0, 1), and that the assumption that the function (0, 1) p → VaR X (p) is not constant on the interval [1 − ε, 1) yields that the set {x 0 : p 2 F X (x) < 1} has a positive Lebesgue measure. Consequently, we get ES X,n (p 1 ) < {x 0:x