On quantile based co-risk measures and their estimation

Abstract: Conditional Value-at-Risk (CoVaR) is de ned as the Value-at-Risk of a certain risk given that the related risk equals a given threshold (CoVaR=) or is smaller/larger than a given threshold (CoVaR</CoVaR≥). We extend the notion of Conditional Value-at-Risk to quantile based co-risk measures that are weighted mixtures of CoVaR at di erent levels and hence involve the stochastic dependence that occurs among the risks and that is captured by copulas. We show that every quantile based co-risk measure is a quantile based risk measure and hence ful lls all related properties. We further discuss continuity results of quantile based corisk measures from which consistent estimators for CoVaR< and CoVaR≥ based risk measures immediately followwhen plugging in empirical copulas. Although estimating co-risk measures based on CoVaR= is a nontrivial endeavour since conditioning on events with zero probability is necessary we show that working with so-called empirical checkerboard copulas allows to construct strongly consistent estimators for CoVaR= and related co-risk measures under very mild regularity conditions. A small simulation study illustrates the performance of the obtained estimators for special classes of copulas.


Introduction
Conditional risk measures (co-risk measures for short) are systemic risk measures which are used to quantify a risk that may be related to another risk. This may involve quantifying the impact of a risk on another risk (for instance, the impact of an asset on another asset, the impact of a nancial institution on another nancial institution, etc.) as well as quantifying the systemic contribution of a risk to a set of risks (for instance, the impact of an asset on the whole portfolio, the impact of a nancial institution on the corresponding nancial sector, etc.).
The most prominent co-risk measure is the Co-Value-at-Risk (CoVaR) which equals the common Value-at-Risk (VaR) of the conditional distribution of a certain risk given that the related risk equals a given threshold (CoVaR = ), is smaller than a given threshold (CoVaR < ), or is larger than a given threshold (CoVaR ≥ ). The former Co-Value-at-Risk was introduced in [3], the latter so called 'modi ed' Co-Value-at-Risks were discussed in [19,29]; see also [4,21,22,34]. The stochastic dependence that occurs among the risks is captured in terms of bivariate copulas. In contrast to the former version, the modi ed Co-Value-at-Risks exhibit the advantage that they are 'dependence consistent' (see, e.g., [4,29,34]), i.e. they preserve the concordance order of copulas.
In the present paper we apply L/P (loss/pro t) notation and focus on the class of quantile based co-risk measures, which are risk measures that are weighted mixtures of the Co-Value-at-Risk at di erent levels and hence involve the stochastic dependence among the risks. This class includes the Co-Value-at-Risk itself, but also the very prominent Co-Expected-Shortfall (CoES; see, e.g., [29,34]) and the Marginal Expected-Shortfall (MES; see, e.g., [2,29]). Quantile based co-risk measures are hence de ned as conditional analogues of quantile based risk measures (which are related to distortion risk measures); for more information on this approach we refer to [1,16,20,36,41,42,44] and references therein. It turns out that every quantile based co-risk measures is a quantile based risk measure whose distortion function depends on the involved copula. Thus, quantile based co-risk measures ful ll all the properties that apply to quantile based risk measures including positive homogeneity, translativity, comonoton additivity and monotonicity with respect to the stochastic order, and properties such as subadditivity or convexity depend on the speci c shape of the related distortion function.
We further study continuity properties of quantile based co-risk measures taking into account the underlying dependence structure between the risks captured by copulas. It turns out that, depending on the type of co-risk measure, di erent notions of copula convergence need to be considered; this includes pointwise/uniform convergence for co-risk measures based on CoVaR < and CoVaR ≥ (Theorem 5.1), and the recently introduced weak conditional convergence (see [25]) for co-risk measures based on CoVaR = (Theorem 5.2). Applying the obtained continuity results strongly consistent estimators for co-risk measures based on CoVaR < and CoVaR ≥ can be achieved by simply plugging in the empirical copula and the empirical marginal distribution function.
The situation turns out to be more challenging when co-risk measures based on CoVaR = are to be estimated. In the literature several approaches have been discussed for the estimation of Co-Value-at-Risk CoVaR = (and related co-risk measures) including quantile regression and GARCH estimation methods, but also using rather restrictive assumptions on the underlying bivariate distribution; see, e.g., [5,8,19]. In the current paper we present a quite general estimation procedure using checkerboard aggregations of the empirical copula. In Theorem 6.1 we show that under some very mild regularity conditions on the copula or the marginal distribution function, the estimator based on checkerboard approximations is strongly consistent. As an additional feature of our approach we nally present a more sophisticated estimation approach that involves information about the underlying dependence structure using results from [23] and, in an Extreme Value copula setting, we compare the performance of the estimators incorporating or ignoring information about the dependence structure.
The rest of this contribution is organized as follows: Section 2 gathers preliminaries and notations that will be used throughout the paper. In Section 3 we start with the unconditional setting and de ne quantile based risk measures that are weighted mixtures of the Value-at-Risk at di erent levels and quantify single risks. In Section 4 we then proceed with the conditional setting and introduce the notion of quantile based corisk measures that involve the stochastic dependence among the risks. Section 5 derives the afore-mentioned continuity results and in Section 6 we discuss consequences of these results for the estimation of quantile based co-risk measures.

Notation and preliminaries
In the sequel, let I := [ , ] and let C denote the family of all bivariate copulas. For each copula C the corresponding probability measure (also known as doubly stochastic measure or copula measure) will be denoted by µ C , i.e. µ C ([ , u] × [ , v]) = C(u, v) for all u, v ∈ I. For more background on copulas we refer to [11,32]. For every metric space (S, d) the Borel σ-eld on S will be denoted by B(S).
In what follows Markov kernels will play a prominent role. A Markov kernel from R to R is a mapping K : R × B(R) → I such that for every xed E ∈ B(R) the mapping x → K(x, E) is (Borel-)measurable and for every xed x ∈ R the mapping E → K(x, E) is a probability measure. Given two real-valued random variables X, Y on a probability space (Ω, A, P) we say that a Markov kernel K is a regular conditional distribution of Y given holds P X -almost surely for every E ∈ B(R). It is well-known (see, e.g., [24,26]) that for X, Y as above, a regular conditional distribution of Y given X always exists and is unique for P X -a.e. x ∈ R, i.e. any two versions may di er only on a P X null set. If (X, Y) has distribution function H (in which case we will also write (X, Y) ∼ H and let µ H denote the corresponding probability measure on B(R )) we will let K H denote (a version of) the regular conditional distribution of Y given X and simply refer to it as Markov kernel of H. If C is a copula then we will consider the Markov kernel of C automatically as mapping K C : I × B(I) → I. De ning the x-section of a set G ∈ B(R ) as Gx := {y ∈ R : (x, y) ∈ G} the so-called disintegration theorem (see [24,26]) yields As a direct consequence, for every C ∈ C we get for every E ∈ B(I), whereby λ denotes the Lebesgue measure on R. For more background on conditional expectation and general disintegration we refer to [24,26]. We denote by M the comonotonicity copula, by Π the independence copula and by W the countermonotonicity copula (see, e.g., [11,32]). As copulas are Lipschitz continuous, Rademacher's theorem (see, e.g., [35]) implies that every copula is di erentiable λ-a.e. It hence follows that ∂ C(u, v) := ∂ ∂u C(u, v) exists a.e. and, for every v ∈ I, the identity ∂ C(u, v) = K C (u, [ , v]) holds for a.e. u ∈ I; see, e.g., [11,Theorem 3.4.4].

Quantile based risk measures
We start with the unconditional setting and provide a brief overview of quantile based risk measures with which single risks can be quanti ed.
A function D : I → I is said to be a distortion function if it is increasing, right-continuous and ful lls D( ) = and sup u∈( , ) D(u) = (and hence D( ) = ). Thus, a distortion function corresponds to the distribution function of a probability measure µ on B(I) ful lling µ(( , )) = .
Throughout this paper, we consider pairs (D, Q) consisting of a distortion function D and the probability measure Q corresponding to D, and we use identical sub-or superscripts for both, D and Q, in the case of a particular choice of D or Q. Note that, due to the properties of D we have Q[( , )] = .
We denote by L the vector lattice of all random variables, by L the vector lattice of all integrable random variables, and by L ∞ the vector lattice of all almost surely bounded random variables. Then we have L ∞ ⊆ L ⊆ L . For a random variable Y, we further denote by F Y its univariate distribution function.
De ne [11]. Note that L Q may fail to be a vector space. Then we have L ∞ ⊆ L Q and the map ϱ Q : L Q → R given by is said to be a quantile based risk measure. The notion of a quantile based risk measure naturally generalizes that of a spectral risk measure that has been introduced in [1] in P/L (pro t/loss) setting.
The quantile risk measure RVaR α,β is called Range-Value-at-Risk at level (α, β); see [9,12]. In particular, for α, β ∈ ( , ) with α < β The composition D • F of a distortion function D and a univariate distribution function F is again a distribution function, and we denote by µ D•F the probability measure corresponding to D • F. We note in passing that quantile based risk measures may also be represented in terms of the composition D • F, as is well-known in the literature: For every distortion function D and every distribution function F the identity holds for every Y ∈ L Q ; see, e.g., [14,16,43].

Quantile based co-risk measures
We proceed with the conditional setting and introduce the notion of quantile based co-risk measures which are risk measures that involve the stochastic dependence among risks.
In what follows, we consider two random variables X and Y on (Ω, A, P) with continuous distribution functions F X and F Y , respectively.
1. For every γ ∈ ( , ) we denote by F Y|X<F ← X (γ) the distribution function corresponding to the probability measure P Y (· | {X < F ← X (γ)}), 2. for every γ ∈ ( , ) we denote by F Y|X≥F ← X (γ) the distribution function corresponding to the probability measure P Y (· | {X ≥ F ← X (γ)}), and 3. for every γ ∈ ( , ) we denote by F Y|X=F ← X (γ) the distribution function corresponding to the probability measure P Y (· | X = F ← X (γ)) (notice that as soon as a Markov kernel of Y given X is chosen F Y|X=F ← X (γ) is de ned for every γ ∈ ( , )).
We assume that the random variables X and Y are connected via the unique copula C (see, e.g., [11]), i.e. the identity H(x, y) = C(F X (x), F Y (y)) holds for all (x, y) ∈ R . In this case, since ( γ holds for every y ∈ R and every γ ∈ ( , ), 2. the identity holds for every y ∈ R and every γ ∈ ( , ), and 3. the identity holds for every y ∈ R and a.e. γ ∈ ( , ). Indeed, since λ F ← X = P X and (F X • F ← X )(u) = u for every u ∈ I, we obtain [ ,u] for every u ∈ I. The assertion hence follows from the Radon-Nikodym theorem. The previous identity was also discussed in [31]. Note that, in Equation (3), we can choose any Markov kernel that is a regular conditional distribution of Y given X (which may be the "canonical version" constructed in [4,22]). Thus, the Markov kernel on the right hand side of Equation (3) is de ned for every γ ∈ ( , ) but is unique only for a.e. γ ∈ ( , ). Therefore, although the results related to Markov kernels are valid only a.s., we may de ne quantities related to the Markov kernel for arbitrary γ ∈ ( , ).
For C ∈ C and γ ∈ ( , ), we now de ne the maps δ < γ,C , δ ≥ γ,C , δ = γ,C : I → I by letting Then, for every * ∈ {<, ≥, =}, δ * γ,C is a distribution functions on I satisfying Note that the two above equations for * ∈ {=} are valid only a.s. Due to Lipschitz continuity of copulas, the maps δ < γ,C and δ ≥ γ,C are Lipschitz continuous. In contrast to that, δ = γ,C may fail to be continuous.

Remark 4.1.
Note that the maps δ < γ,C and δ ≥ γ,C are related to each other via δ ≥ γ,C = δ < −γ,ν (C) where ν (C) denotes the re ection of C in the rst coordinate given by (ν (C))(u, v) [10,15,17] for more information on copula re ections. The rst approach applying conditional distribution functions F Y|X<F ← X (γ) is usually considered when X and Y are interpreted as nancial positions describing the wealth of an institution or the pro t of an asset portfolio; see, e.g., [4]. The second approach using conditional distribution functions F Y|X≥F ← X (γ) is common for instance when considering X and Y as random loss variables; see, e.g., [29]. Thus, in either case we can restrict our consideration to the functions δ < and δ = .
It turns out that the maps δ < γ,C and δ = γ,C themselves are distortion functions and may be used to construct distortion functions for co-risk measures. The next result is straightforward: For every γ ∈ ( , ) and every copula C the maps δ < γ,C and δ = γ,C are distortion functions.
We now apply δ < γ,C and δ = γ,C to the comonotonicity copula M, the independence copula Π and the countermonotonicity copula W: Corollary 4.2 re ects the idea of considering δ < γ,C (and possibly also δ = γ,C ) as a distortion function whose corresponding quantile based risk measure is based on contagion from an external scenario whereas the dependence is modeled with a copula having horizontal concave sections; see, e.g., [7,44]. As can be seen from the previous table, the Expected-Shortfall possesses such an interpretation.
The quantile functions of δ < γ,C and δ = γ,C satisfy: we may conclude that the map C → δ < γ,C is order preserving and that the map C → (δ < γ,C ) ← is order reversing with respect to PLOD order on copulas: The following result extends Corollary 4.2: for every t ∈ I.

Co-Range-Value-at-Risk:
for every t ∈ I.
Representations for the above distortion functions in the case of comonotonic, independent and countermonotonic random variables, respectively, may be obtained from the following table: Now, for γ ∈ ( , ) and C ∈ C, de ne are said to be a quantile based co-risk measure.
3. Co-Expected-Shortfall: For α ∈ ( , ), the quantile co-risk measure ϱ * γ,C,Q ESα satis es for every Y ∈ L * γ,C,Q ESα and is called Co-Expected-Shortfall at level α; see, e.g., [29]. 4. Co-Range-Value-at-Risk: For α, β ∈ [ , ] with α < β, the quantile co-risk measure ϱ * γ,C,Q RVaR α,β satis es , every quantile based co-risk measure is a CoVaR-based risk measure. It even turns out that every quantile based co-risk measure is a quantile based risk measure for which the distortion function depends on γ ∈ ( , ) and C ∈ C: Theorem 4.8. Let γ ∈ ( , ), C ∈ C and consider * ∈ {<, ≥, =}. Then which is immediate from Corollary 4.5. Then the identities hold for every distortion function D and every Y ∈ L . Thus, This proves the assertion for * ∈ {<}. The same reasoning applies to * ∈ {≥} and * ∈ {=} where the latter follows from the fact that δ = γ,C is right-continuous at and left-continuous at . As a rst consequence of Theorem 4.8 we obtain the following corollary which is immediate from the identity δ ≥ γ,C = δ < −γ,ν (C) discussed in Remark 4.1: Thus, in what follows we may restrict our consideration to the quantile co-risk measures ϱ < γ,C,Q and ϱ = γ,C,Q . Nevertheless, we add results for ϱ ≥ γ,C,Q where appropriate. For C ∈ {M, Π, W} we immediately obtain the following identities: Remarkably, no matter how Q is chosen, ϱ = γ,M,Q (which refers to the case when the random variables are comonotonic) equals the Value-at-Risk at level γ, and ϱ = γ,W ,Q (which refers to the case when the random variables are countermonotonic) equals the Value-at-Risk at level − γ. The copulas M and W belong to the class of all completely dependent copulas: A copula C is said to be completely dependent if there exists some λ-preserving transformation h : I → I (i.e., a transformation ful lling λ(h − (F)) = λ(F) for every F ∈ B(I)) such that K(γ, E) := 1 E (h(γ)) is a Markov kernel of C. For more properties of complete dependence we refer to [27] as well as to [13] and the references therein.   γ ∈ ( , ).
Additional results including order properties of Y can be derived from [29]. The next result is a consequence of Corollary 4.4 and yields relations for the domain of a co-risk measure: Corollary 4.13. Consider γ ∈ ( , ).
(1) The inclusion holds for every C ∈ C.
Proof. We rst prove (1). To this end, consider Y ∈ L . Corollary 4.4 together with the fact that the pseudo inverse is increasing then yields which proves (1). Analogously, we obtain (2).
We now resume Example 4.7: holds for every C ∈ C and every Y ∈ C∈C L < γ,C,Q E , and the inclusion L ⊆ L < γ,C,Q E holds for every holds for every C ∈ C and every Y ∈ C∈C L ≥ γ,C,Q E , and the inclusion L ⊆ L ≥ γ,C,Q E holds for every C ∈ C. 2. Co-Value-at-Risk: For α ∈ ( , ), the Co-Value-at-Risk satis es (see [29,Theorem 3 for every Y ∈ L * γ,C,Q VaRα = L . In particular, every Co-Value-at-Risk at level α is a Value-at-Risk at level (δ * γ,C ) ← (α). Moreover, (a) the inequality holds for every C ∈ C and every Y ∈ L .
holds for every C ∈ C and every Y ∈ C∈C L ≥ γ,C,Q ESα , and the inclusion L ,+ ⊆ L ≥ γ,C,Q ESα holds for every C ∈ C. 4. Co-Range-Value-at-Risk: For α, β ∈ ( , ) with α < β, the Co-Range-Value-at-Risk satis es for every Y ∈ L * holds for every C ∈ C and every Y ∈ L .

Remark 4.15.
Although every quantile co-risk measure is a quantile risk measure (Theorem 4.8), it needs not to be a quantile risk measure of the same type. For instance, even though every Co-Value-at-Risk at level α is a Value-at-Risk at level (δ * γ,C ) ← (α), a Co-Expected-Shortfall may fail to be of Expected-Shortfall type.

Continuity results
In this section we study continuity properties of quantile based co-risk measures taking into account the underlying dependence structure. It turns out that, depending on the type of co-risk measure, di erent notions of copula convergence need to be considered including pointwise/uniform convergence (for co-risk measures based on CoVaR < and CoVaR ≥ ) and weak convergence of the corresponding Markov kernels known as weak conditional convergence (for co-risk measures based on CoVaR = ).
We start with two results that are key to this section: Theorem 5.1. Consider a sequence of copulas (Cn) n∈N converging pointwise to C and a sequence of continuous univariate distribution functions (F Y ,n ) n∈N converging weakly to F Y . Then, for every γ ∈ ( , ), the identities hold for every y ∈ R.
Proof. As F Y is continuous, weak convergence of (F Y ,n ) n∈N to F Y coincides with pointwise convergence. Moreover, as pointwise convergence (Cn) n∈N to C equals uniform and hence continuous convergence, we have that the sequence (δ < γ,Cn • F Y ,n ) n∈N converges pointwise to δ < γ,C • F Y . The same reasoning yields the second identity.
Viewing bivariate copulas in terms of their conditional distributions and considering weak convergence leads to the notion of weak conditional convergence introduced in [25]: Suppose that C, C , C , . . . are copulas and let K C , K C , K C , . . . be (versions of) the corresponding Markov kernels. We will say that (Cn) n∈N converges weakly conditional to C if and only if for λ-almost every u ∈ I we have that the sequence (K Cn (u, ·)) n∈N of probability measures on B(I) converges weakly to the probability measure K C (u, ·). In the latter case we will write Cn wcc − − → C (where 'wcc' stands for 'weak conditional convergence'). Following the construction in [22] it is straightforward to verify that weak conditional convergence coincides with almost sure convergence of the partial derivatives on a dense set. As is well known, many standard parametric classes {C θ : θ ∈ Θ} of copulas depend on the parameter θ weakly conditional in the sense that if θn → θ then Cn wcc − − → C: (among many others) -the family of EFGM copulas: see [11,Section 6.3].
ful ll this property (see [25]). In [25], the authors have also proved that within the class of Archimedean copulas and the class of Extreme Value copulas standard pointwise convergence and weak conditional convergence are equivalent. The equivalence between the two notions of convergence reveals its potential when studying risk measures based on CoVaR = (as we will see in Section 6).

Theorem 5.2. Consider a sequence of copulas (Cn) n∈N converging weakly conditional to C and a sequence of continuous univariate distribution functions (F Y ,n ) n∈N converging weakly to F Y . Then there exists a set Λ ∈ B(( , )) with λ(Λ) = such that for every
is continuous, then Equation (4) holds for all but at most countable in nitely many y ∈ R (interpreting the expressions as conditional distribution functions on R we hence have weak convergence).
Another direct consequence of Theorems 5.1 and 5.2 can be obtained for co-risk measures ϱ * γ,C,Q , * ∈ {<, ≥, =}, for which Q is absolutely continuous; the next result focusses on Co-Range-Value-at-Risk and hence Marginal Expected-Shortfall and Co-Expected-Shortfall: Corollary 5.5. Consider a sequence of copulas (Cn) n∈N , a copula C and a sequence of continuous univariate distribution functions (F Y ,n ) n∈N converging weakly to F Y . Further assume that Q is absolutely continuous with respect to λ.
weakly. Because (δ * γ,C •F Y ) ← has at most countably many discontinuity points and Q is absolutely continuous with respect to λ, the previous identity holds Q-a.s. The assertion hence follows from dominated convergence theorem.
The convergence results stated in Corollaries 5.4 and 5.5 for co-risk measures based on CoVaR = are valid whenever the limiting distribution function F Y is strictly increasing, a condition which might seem acceptable in most applications. Also the conditions to the copula seem not to be too restrictive: Remark 5.6. Notice that, if the copula C is absolutely continuous such that its Lebesgue density c satis es c(u, v) > for all (u, v) ∈ ( , ) , then 1. the map v → C(γ, v) is strictly increasing for every γ ∈ ( , ); 2. the map v → v − C(γ, v) is strictly increasing for every γ ∈ ( , ); 3. the map v → K C (γ, [ , v]) is absolutely continuous and strictly increasing for every γ ∈ ( , ).
In this case, all the necessary and additional conditions to the limiting copula used in Corollaries 5.4 and 5.5 are ful lled.

Consequences for the estimation
In this section we focus on consequences of the results presented in Section 5 to the estimation of co-risk measures. Since consistent estimators for co-risk measures based on CoVaR < and CoVaR ≥ can be obtained by simply plugging in the empirical copula and the empirical distribution function (compare Corollaries 5.4 and 5.5), we restrict our consideration to the more challenging situation when co-risk measures based on CoVaR = are to be estimated.
Suppose that (X , Y ), (X , Y ), . . . is a random sample from (X, Y) ∼ H; recall that H is continuous and that there exists some unique copula C satisfying H(x, y) = C(F X (x), F Y (y)) for all (x, y) ∈ R . In the following, we denote by F Y ,n the empirical distribution function given Y , Y , . . . , Yn, and by En the empirical copula (by which we mean the unique copula determined by bilinear interpolation of the empirical subcopula, i.e., a checkerboard copula with a quite speci c structure) given (X , Y ), (X , Y ), . . . , (Xn , Yn). Then En ful lls for n → ∞ with probability one; see, e.g., [23]. Plugging in the empirical distribution function and the empirical copula into the co-quantile risk measures ϱ * , * ∈ {<, ≥}, by Corollaries 5.4 and 5.5, we obtain consistency of these estimators. In particular, we obtain consistent estimators for CoVaR * , MES * , ES * and RVaR * in the case * ∈ {<, ≥}.
For the reasons discussed in [23] it is not possible to estimate risk measures based on CoVaR = by simply plugging in the empirical copula. One possibility to overcome this problem is to use (general) checkerboard aggregations instead of empirical copulas. Following [23], we let CB N (C) be the N-checkerboard approximation of C ∈ C and we will write N(n) := n s for some xed s ∈ ( , ). The next result shows that it is possible to estimate δ = γ,C • F Y in almost full generality. weakly.
Proof. According to Theorem 5.2 it su ces to prove that (CB N(n) ( En)) n∈N converges weakly conditional to C with probability one. Since the empirical copula En ful lls (5) we may assume from now on that d∞( En , C) = O( ln (ln (n))/n) holds. Consider Then according to [28] there exists a set Γ ∈ B([ , ] ) with λ (Γ) = such that limn→∞ Jn = holds for all (x, y) ∈ Γ. For In we can proceed as follows: Let Λ denote the set of all γ ∈ ( , ) such that λ(Γγ) = and suppose that γ ∈ Λ and (γ, y) ∈ Γ holds. Further, de ne the squares R ij for i, j ∈ { , . . . , N(n)} by Considering (see [23]) that for every γ ∈ i− N(n) , i N(n) and every copula A ∈ C we have for every j ∈ { , , . . . , N(n)} it follows immediately that In ≤ N(n) · · d∞( En , C) and therefore lim sup n→∞ In = . This completes the proof since λ(Γγ) = and γ ∈ Γ was arbitrary.

Further assume that Q is absolutely continuous with respect to λ. Then there exists a set Λ
for every γ ∈ Λ. Remark 6.3. At this point note that Corollary 5.5 and Corollary 6.2 can be extended in a straightforward way to functionals of the form where µ is a signed measure satisfying µ = a Q − a Q with a , a ∈ [ , ∞) and probability measures Q and Q corresponding to distortion functions as de ned in Section 3. Such functionals belong to the class of distortion risk metrics discussed in [38].
We now illustrate the performance of our estimators CoVaR = α,γ,CB N(n) ( En) (Yn|X) for Co-Value-at-Risk and CoES = α,γ,CB N(n) ( En) (Yn|X) for Co-Expected Shortfall discussed in Corollary 6.2 in a Marshall-Olkin copula setting; recall that the Marshall-Olkin family of copulas (M θ ,θ ) θ ,θ ∈I is de ned by [11,Section 6.4]).  Even though the co-risk measures CoVaR = , MES = , ES = and RVaR = can be consistently estimated by means of checkerboard aggregations of the empirical copula as shown above, in what follows we discuss a more sophisticated approach also incorporating information about the dependence structure between X and Y. We then compare the performance of the estimators incorporating or ignoring information about the dependence structure.
Extreme Value copulas A copula C ∈ C is called an Extreme Value copula if there exists a convex function A : I → I satisfying A( ) = A( ) = and max( − t, t) ≤ A(t) ≤ for all t ∈ I such that for all u, v ∈ ( , ) the copula C can be expressed in terms of A as ln (uv) (see [11,33]). The function A is called Pickands dependence function. Now, suppose that C A is an Extreme Value copula with Pickands dependence function A and suppose that (X , Y ), (X , Y ), . . . is a random sample from (X, Y) ∼ H with underlying copula C A . Following [25] and lettingÂn denote the CFG estimator according to [6,18] it can be shown that, for suitable weight functions, (Ân) n∈N is uniformly, strongly consistent (see [6,Proposition 4.1]). Although the estimatorÂn may fail to be convex in general, following an idea from [18] it can be used to construct a convex estimatorÂ * n given bŷ A * n := greatest convex minorant of max{min{Ân , }, id, − id} where id denotes the identity function on I.Â * n is a Pickands dependence function (see [18,Section 3.3]) and the estimatorÂ * n is uniformly, strongly consistent (which follows from [30]). Hence [25, Theorem 5.1] directly yields weak conditional convergence of the sequence of corresponding Extreme Value copulas (CÂ * n ) n∈N to C A . Therefore, as a consequence of Theorem 5.2, for estimating co-risk measures based on CoVaR = in an Extreme Value setting, it is possible to replace the estimator based on checkerboard aggregations by an estimator based onÂ * n . The bene t of including information about the dependence structure is illustrated in a Galambos copula setting; the Galambos family of copula (C A θ ) θ∈( ,∞) is de ned by means of its Pickands function A θ (t) := − t −θ + ( − t) −θ − /θ Example 6.5. Consider a random variable Y following a Beta-distribution Beta( , ) and a Galambos copula C A with parameter θ = . Figure 3 depicts a boxplot summarizing the obtained estimates (left panel: estimator based onÂ * n , right panel: estimator based on checkerboard aggregations) for CoVaR = α,γ,C A (Y|X) at level γ = . ; instead of comparing Co-Value-at-Risk values for speci c choices of α, we use the Ldistance between the functions α → CoVaR = α,γ,CB N(n) ( En) (Yn|X) and α → CoVaR = α,γ,C A (Y|X) to illustrate the estimators' performance. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q n=10 n=50 n=100 n=500 n=1000 Not surprisingly, Figures 3 and 4 show that the plug-in estimator based onÂ * n (using the Extreme Value information) outperforms the empirical checkerboard estimator (ignoring the Extreme Value information).