A Review and Some Complements on Quantile Risk Measures and Their Domain

In the present paper, we study quantile risk measures and their domain. Our starting point is that, for a probability measure Q on the open unit interval and a wide class LQ of random variables, we define the quantile risk measure $Q as the map that integrates the quantile function of a random variable in LQ with respect to Q. The definition of LQ ensures that $Q cannot attain the value +∞ and cannot be extended beyond LQ without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. We also present a particularly short proof of the subadditivity of expected shortfall.


Introduction
In the present paper, we study quantile risk measures and their domain.Our starting point is that, for a probability measure Q on the open unit interval and a wide class L Q of random variables, we define the quantile risk measure Q as the map that integrates the quantile function of a random variable in L Q with respect to Q.The definition of L Q ensures that Q cannot attain the value +∞ and cannot be extended beyond L Q without losing this property.The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval.
Quantile risk measures are thus mixtures of the values at risk at different levels and hence mixtures of a parametric family of risk measures.Such mixtures have already been considered by Acerbi (2002), who, however, gave little attention to the domain on which a given risk measure can be defined; he argued that in a real-world risk management application, the integral (defining a risk measure) will always be well-defined and finite.Nevertheless, Acerbi (2002) proposed a maximal class of random variables on which a given spectral risk measure is well-defined and finite.In the case of a spectral risk measure, the domain of a quantile risk measure proposed in the present paper contains the class proposed by Acerbi (2002) and turns out to be a convex cone, which is of interest with regard to the subadditivity of the risk measure.
In this paper, we review and partly extend known results on quantile risk measures, with particular attention to spectral risk measures and, in particular, expected shortfall, with emphasis on their maximal domain mentioned before.We deliberately adopt arguments from the literature, with appropriate modifications if necessary, but some of our proofs and results are new.
The literature on risk measures is vast and rapidly growing.A substantial part of the theory can be found in the monographs by Föllmer andSchied (2016), McNeil et al. (2015), Pflug and Römisch (2007) and Rüschendorf (2013) and in the references given in these books.Since the theory of risk measures is inspired by two sources, finance and insurance, the definitions of financial and insurance risk measures are slightly different, and the terminology is not fully consistent; for example, the use of the term expected shortfall is not generally agreed upon.In the present paper, we consider insurance risk measures, which are closely related to premium principles, and to avoid more ponderous expressions, we employ the short-term quantile risk measure for a well-defined class of risk measures.
This paper is organized as follows: We first fix some notation, recall some basic properties of the quantile function and present a couple of examples of distortion functions (Section 2).We then introduce quantile risk measures and provide several alternative representations of quantile risk measures and their domain, as well as conditions under which certain quantile risk measures can be compared (Section 3).In the next step, we consider spectral risk measures and characterize spectral risk measures within the class of all quantile risk measures (Section 4).We then present a particularly short proof of the subadditivity of expected shortfall and use this result to show that a quantile risk measure is subadditive if and only if it is spectral (Section 5).As a major issue of this paper, we proceed with a detailed comparison of the domain of a quantile risk measure with the classes of random variables proposed by Acerbi (2002) and Pichler (2013) in the spectral case (Section 6).Finally, and as a complement, we briefly discuss related integrated quantile functions occurring in the measurement of economic inequality (Section 7).

Preliminaries
We use the terms positive and increasing in the weak sense which admits equality in the inequalities defining these terms.For B ⊆ R, we denote by χ B the indicator function of B (such that . Furthermore, we denote: -by B(R) the σ-field of all Borel sets of R, -by B((0, 1)) the σ-field of all Borel sets of (0, 1) and -by λ the Lebesgue measure on B(R) or its restriction to B((0, 1)).
By the correspondence theorem, there exists a bijection between the distribution functions on R and the probability measures on B(R) such that the probability measure Correspondingly, there exists a bijection between the distribution functions on (0, 1) and the probability measures on B((0, 1)).
Throughout this paper, we consider a fixed probability space (Ω, F , P) and random variables (Ω, F ) → (R, B(R)), and we denote: -by L 0 the vector lattice of all random variables, -by L 1 the vector lattice of all integrable random variables, -by L 2 the vector lattice of all square integrable random variables and -by L ∞ the vector lattice of all almost surely bounded random variables.
Then, we have L ∞ ⊆ L 2 ⊆ L 1 ⊆ L 0 .For a random variable X ∈ L 0 , we denote by F X its distribution function R → [0, 1] given by: F X (x) and by F ← X its (lower) quantile function (0, 1) → R given by: For u ∈ (0, 1) and x ∈ R, the quantile function satisfies F ← X (u) ≤ x if and only if u ≤ F X (x).Moreover, the quantile function is increasing and has the following properties: Lemma 1.Consider X, Y ∈ L 0 .Then: A function D : [0, 1] → [0, 1] is said to be a distortion function if it is increasing and continuous from the right and satisfies D(0) = 0 and sup u∈(0,1) D(u) = 1 (and hence, D(1) = 1).The restriction of a distortion function D to (0, 1) is a distribution function on (0, 1), and for simplicity, the probability measure corresponding to the restriction of D to (0, 1) will be referred to as the probability measure corresponding to D.
Example 1.The terms attached to the following examples are the names of the risk measures resulting from the respective distortion functions.
Throughout this paper, we consider pairs (D, Q) consisting of a distortion function D : [0, 1] → [0, 1] and the probability measure Q : B((0, 1)) → [0, 1] corresponding to D, and we use identical subor super-scripts for both, D and Q, in the case of a particular choice of D or Q.

Quantile Risk Measures
Define: given by: is said to be a quantile risk measure.
For every X ∈ L 0 , we have X ∈ L Q if and only if X + ∈ L Q , by Lemma 1.This implies that, for every Z ∈ L 0 satisfying Z ≤ X for some X ∈ L Q , we have Z ∈ L Q .Lemma 1 also yields the following properties of a quantile risk measure: To obtain alternative representations of a quantile risk measure and its domain, we need the following Lemma: Lemma 3. The identities: and: hold for every X ∈ L 0 .
The following result is immediate from Lemma 3: Theorem 1.The domain of Q satisfies: and the identities: hold for every X ∈ L Q .
Because of the previous result, the quantile risk measure generated by the probability measure Q corresponds to the distortion risk measure generated by the distortion function D; the latter is also known as Wang's premium principle.

Example 2.
(1) Expectation: The distortion function D E satisfies D E • F X = F X .Because of Theorem 1, this yields: (2) Value at risk: For α ∈ (0, 1), the probability measure Q VaR α corresponding to D VaR α is the Dirac measure at α.This yields: for every X ∈ L Q VaRα ; in particular, Q VaRα is finite.The quantile risk measure Q VaRα is called value at risk at level α and is usually denoted by VaR α .
(4) Expected shortfall of higher degree: For n ∈ N and α ∈ [0, 1), the probability measure Q ES n;α corresponding to D ES n;α satisfies: for every X ∈ L Q ESn;α .In particular, Q ES 1;α = Q ESα , and Q ESn;α is finite for every n ∈ N and α ∈ (0, 1).The quantile risk measure Q ESn;α is called expected shortfall of degree n at level α.
The examples show that the domains of different quantile risk measures may be distinct.
Lemma 3 and Theorem 1 have several applications.For example, they provide a condition on D under which Q is finite: Corollary 1. Assume that there exists some δ ∈ (0, 1) such that D(u) = 0 holds for every u ∈ (0, δ).Then: and Q is finite.
Proof.For every X ∈ L 0 , the assumption yields: Since F ← X (δ) is finite, this proves the assertion.
Theorem 1 also provides a condition for the comparison of the domains of quantile risk measures: Corollary 2. Assume that there exists some δ ∈ (0, 1) such that D 1 (u) ≤ D 2 (u) holds for every u ∈ [δ, 1).
Proof.For every X ∈ L 0 , we have: Corollary 3. Assume that there exist some n ∈ N and α, δ ∈ (0, 1) such that: Proof.Because of Corollary 2, we have Combining Corollaries 1 and 3 yields a condition under which L Q = L Q E and Q is finite.Corollary 2 also yields some further results on the comparison of quantile risk measures and their domains: Proof.Assume first that D 1 ≤ D 2 .Then Corollary 2 yields L Q 1 ⊆ L Q 2 and Theorem 1 yields consider u ∈ (0, 1).Then, for any choice of a, b ∈ R such that a < b and for every random variable X satisfying P for all i ∈ {1, 2}, and hence, D 1 (u) ≤ D 2 (u).Since u ∈ (0, 1) was arbitrary, it follows that D 1 ≤ D 2 .This proves (1).Assertions (2)-( 4) are immediate from (1), and Assertion (5) follows from the dominated convergence theorem.
Assertion (1) of Corollary 4 extends a result of Wang et al. (2015), who considered risk measures that are defined on a common convex cone containing L ∞ .

Spectral Risk Measures
A map s : (0, 1) → R + is said to be a spectral function if it is increasing and satisfies (0,1) s(u) dλ(u) = 1.
The quantile risk measure Q is said to be a spectral risk measure if there exists a spectral function s such that: Thus, if Q is a spectral risk measure with spectral function s, then the domain of Q satisfies: and the identity: holds for every X ∈ L Q .Note that the spectral function of a spectral risk measure is unique almost everywhere, by the Radon-Nikodym theorem.
(1) Expectation: Since D E (u) = u, we have: Q E = λ and the function s E : (0, 1) → R + given by: s E (u) := 1 is a spectral function.Therefore, Q E is a spectral risk measure.
(2) Value at risk: For every α ∈ (0, 1), Q VaR α is the Dirac measure at α and hence does not have a density with respect to λ.Therefore, Q VaRα is not a spectral risk measure.
(4) Expected shortfall of higher degree: For every n ∈ N and α ∈ [0, 1), we have: and the function s ES n;α : (0, 1) → R + given by: is a spectral function.Therefore, Q ESn;α is a spectral risk measure.
Our aim is to characterize the spectral risk measures within the class of all quantile risk measures.The following result is inspired by Gzyl and Mayoral (2008), who considered distortion risk measures on the positive cone of L 2 : Theorem 2. The following are equivalent: (a) D is convex.(b) There exists a spectral function s such that Q = s(u) dλ(u).(c) Q is a spectral risk measure.
In this case, every spectral function s representing Q satisfies s = D almost everywhere (with respect to λ).
Assume first that (a) holds.The following arguments are taken from Aliprantis and Burkinshaw (1990, chp. 29).Since D is increasing, D is differentiable almost everywhere, and since D is convex, its derivative D is increasing.Consider now an arbitrary interval [u, v] ⊆ (0, 1).Since D is convex, the restriction of D to [u, v] is Lipschitz continuous, hence absolutely continuous and, thus, continuous and of bounded variation.Therefore, the restriction of Q to the σ-field of all Borel sets in [u, v] is absolutely continuous with respect to the restriction of λ, and its Radon-Nikodym derivative agrees with D .Since [u, v] ⊆ (0, 1) was arbitrary, it follows that Q is absolutely continuous with respect to λ, and since the Radon-Nikodym derivative s : (0, 1) → R + of Q with respect to λ is unique almost everywhere, it follows that s = D almost everywhere.This yields the existence of an increasing function s : (0, 1) → R + satisfying Q = s(u) dλ(u).Therefore, (a) implies (b).
Assume now that (b) holds.Since s is increasing, we have, for any u, v, w ∈ (0, 1) such that u < v < w, The following result is inspired by Kusuoka (2001), who studied risk measures on L ∞ : holds for every X ∈ L Q .
Proof.Without loss of generality, we may and do assume that s is continuous from the right.Define s(0) := inf u∈(0,1) s(u).Then, there exists a unique σ-finite measure ν : ) is measurable and its positive part is integrable with respect to the product measure ν ⊗ λ, Fubini's theorem yields: This proves the assertion.

Subadditivity of Spectral Risk Measures
In the present section, we show that a quantile risk measure is subadditive if and only if its distortion function is convex.To prove that the convexity of the distortion function is sufficient for subadditivity of the quantile risk measure, we use Theorem 3. Since the expectation is additive and hence subadditive, it remains to show that the expected shortfall at any level is subadditive.
To establish subadditivity of the expected shortfall, we need the following lemma, which provides another representation of the values of the expected shortfall: Lemma 4. For every α ∈ (0, 1), the identity: Lemma 4 is well-known and is frequently used to establish the subadditivity of expected shortfall on L ∞ ; see, e.g., Embrechts and Wang (2015), who used a general extension procedure to extend this result beyond L ∞ .Here, we use Lemma 4 to establish the subadditivity of expected shortfall on its (maximal) domain L Q ESα in a single step: Lemma 5.For every α ∈ [0, 1), L Q ESα is a convex cone and Q ESα is subadditive.
The previous result provides the key for proving the main implication of the following theorem; see also Wang and Dhaene (1998), who considered distortion risk measures on the positive cone of L 1 and used a proof based on comonotonicity.
Theorem 4. The following are equivalent: Proof.Assume first that (a) holds, and consider a spectral function s representing Q and the measure ν constructed in the proof of Theorem 3. Consider X, Y ∈ L Q and a ∈ R + .Then, we have aX ∈ L Q .Moreover, since D is convex, Corollary 4 yields X, Y ∈ L Q E .For every α ∈ [0, 1), this yields X, Y ∈ L Q ESα ; hence, X + Y ∈ L Q ESα , by Lemma 5; and thus, X + , Y + , (X + Y) + ∈ L Q ESα .Proceeding as in the proof of Theorem 3 and using Lemma 5 again, we obtain: This yields (X + Y) + ∈ L Q , and hence, X + Y ∈ L Q .Thus, L Q is a convex cone, and Theorem 3 together with Lemma 5 implies that Q is subadditive.Therefore, (a) implies (c).Obviously, (c) implies (b), and it follows from Example 4 below that (b) implies (a).
For the discussion of the subsequent Example 4, we need the following lemma: Lemma 6.The following are equivalent: holds for all u ∈ (0, 1) and ε ∈ (0, min{u, 1−u}).
Proof.Assume that (b) holds.Then, the inequality: holds for all u, v ∈ (0, 1), and this implies that D is continuous on (0, 1).Since D is a distortion function, it follows that D is continuous on [0, 1], and now, the previous inequality implies that D is convex.Therefore, (b) implies (a).The converse implication is obvious.
The bivariate distribution discussed in the following example was proposed by Wirch and Hardy (2002).
Example 4. Assume that D is not convex.Then, Lemma 6 yields the existence of some u ∈ (0, 1) and ε ∈ (0, min{u, 1−u}) such that: Consider random variables X, Y ∈ L ∞ whose joint distribution is given by the following table with a ∈ (0, ∞): Then, the distribution of the sum X + Y is given by the table: and hence: Therefore, Q fails to be subadditive.

On the Domain of a Quantile Risk Measure
In this section, we compare the domain: of the quantile risk measure Q with two other classes of random variables.Define: In the case where Q is represented by a spectral function, these classes were introduced by Acerbi (2002) and Pichler (2013), respectively.We have L Acerbi Q ⊆ L Q , and Corollary 1 provides a sufficient condition for L To this end, we need the following lemma: , we obtain: To prove that the integral (0,1) (F ← X (u)) − dQ(u) is finite, as well, we need the upper quantile function F → X : (0, 1) → R given by: F → X (u) := sup x ∈ R F X (x) ≤ u The lower and upper quantile functions satisfy F ← X ≤ F → X , and we have: (F ← X (u)) − = − F ← X (u) χ (0,F X (0)] (u) and: almost everywhere with respect to λ.Since D is convex and hence continuous, Q is absolutely continuous with respect to λ.This yields: 0 ≤ (0,1) χ {F X (x)} (u) dQ(u) dλ(x) = 0 and hence.F → X = F ← X almost everywhere with respect to Q. Consider now a spectral function s representing Q.Since s is positive and increasing, we obtain: (1/2,1) , the last expression is finite, and this yields: The following examples provide some further insight into the relationships between these three classes of random variables: Example 5. (1) Because of these examples, it appears to be reasonable to extend the notion of a quantile risk measure Q to the case of an arbitrary integrating measure or even an integrating signed measure Q : B((0, 1)) → R, although in the latter case, Property (1) of Lemma 2, would be lost.