On Conditional Value at Risk (CoVaR) for tail-dependent copulas

Jaworski, Piotr

doi:10.1515/demo-2017-0001

Cited by 19 publications

(8 citation statements)

References 31 publications

(22 reference statements)

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“…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].…”

Section: Continuity Resultsmentioning

confidence: 99%

“…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 3is de ned for every γ ∈ ( , ) but is unique only for a.e.…”

Section: Quantile Based Co-risk Measuresmentioning

confidence: 99%

“…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 [4,22].…”

Section: Marginal Expected-shortfall or Co-expectationmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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On quantile based co-risk measures and their estimation

Fuchs

Trutschnig

2020

Dependence Modeling

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Conditional Value-at-Risk (CoVaR) is defined 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 different 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 fulfills all related properties. We further discuss continuity results of quantile based co-risk measures from which consistent estimators for CoVaR< and CoVaR≥ based risk measures immediately follow when 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.

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Section: Continuity Resultsmentioning

confidence: 99%

Section: Quantile Based Co-risk Measuresmentioning

confidence: 99%

Section: Marginal Expected-shortfall or Co-expectationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On quantile based co-risk measures and their estimation

Fuchs

Trutschnig

2020

Dependence Modeling

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“…Therefore its inverse function d − u can be obtained easily (by using analytical and/or numerical methods). Properties for the conditional quantile functions (also known as Conditional Value at Risk or CoVaR) of di erent copulas can be seen in, e.g., [2,3,9]. Note that here we are xing one of the in nitely many versions of the conditional distribution Y|X.…”

Section: Proposition 21 If C Is the Copula Function Of (X Y) Thenmentioning

confidence: 99%

Bivariate box plots based on quantile regression curves

Navarro

2020

Dependence Modeling

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In this paper, we propose a procedure to build bivariate box plots (BBP). We first obtain the theoretical BBP for a random vector (X, Y). They are based on the univariate box plot of X and the conditional quantile curves of Y|X. They can be computed from the copula of (X, Y) and the marginal distributions. The main advantage of these BBP is that the coverage probabilities of the regions are distribution-free. So they can be selected by the users with the desired probabilities and they can be used to perform fit tests. Three reasonable options are proposed. They are illustrated with two examples from a normal model and an exponential model with a Clayton copula. Moreover, several methods to estimate the theoretical BBP are discussed. The main ones are based on linear and non-linear quantile regression. The others are based on empirical estimators and parametric and non-parametric (kernel) copula estimations. All of them can be used to get empirical BBP. Some extensions for the multivariate case are proposed as well.

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Conditional risk based on multivariate hazard scenarios

Bernardi

Durante

Jaworski

et al. 2017

Stoch Environ Res Risk Assess

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We present a novel methodology to compute conditional risk measures when the conditioning event depends on a number of random variables. Specifically, given a random vector (X,Y), we consider risk measures that express the risk of Y given that X assumes values in an extreme multidimensional region. In particular, the considered risky regions are related to the AND, OR, Kendall and Survival Kendall hazard scenarios that are commonly used in environmental literature. Several closed formulas are considered (especially in the AND and OR scenarios). An application to spatial risk analysis involving real data is discussed

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On Conditional Value at Risk (CoVaR) for tail-dependent copulas

Cited by 19 publications

References 31 publications

On quantile based co-risk measures and their estimation

On quantile based co-risk measures and their estimation

Bivariate box plots based on quantile regression curves

Conditional risk based on multivariate hazard scenarios

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