Worst-case conditional value-at-risk and conditional expected shortfall based on covariance information

We construct the above new ambiguity set, then propose the optimization problem of CoVaR and CoES based on this ambiguity set


Introduction
Modern risk management often requires the evaluation of risks under multiple scenarios.For instance, in the fundamental review of the trading book of Basel Ⅳ [1] , banks need to evaluate the risk of their portfolios under stressed scenarios including the model generated from data during the 2007 financial crisis.In the aftermath of the financial crisis, there has been growing interest in measuring systemic risk, which refers to the risk that an event at the company level could trigger severe instability or collapse an entire industry or economy.Capital requirements are closely linked to an institution's contribution to the overall risk of the financial system and not merely to its individual risk.
Measuring the contribution of each institution to the overall systemic risk can help regulators inhibit the tendency to generate systemic risk by identifying institutions that make significant contributions to systemic risk.Starting with the seminal paper of Adrian and Brunnermeier [2] (first published online in 2008), many methods have been proposed for measuring systemic risk [3,4] .While the conditional value-at-risk (CoVaR) proposed by Ref. [2] described the of the financial system conditional on an institution being in financial distress, Girardi and Ergün [5] modified the computation of in Ref. [2] by changing the definition of financial distress from the loss of an institution being exactly its to being no less than its .Specifically, for a set of financial institutions (or portfolio) , we denote by as the total systemic risk.The value-at-risk (VaR) of an institution at level is defined as the -quantile function of , that is, .The notation is defined as the VaR of the systemic risk at level , conditional on one of the institutions beyond its VaR at level , that is, We use the right-continuous version of in this study, that is, instead of in Eq. ( 1).In the case of the right-continuous version, the worst-case value is reachable.Both definitions have the same worst-case value, which has no effect on our study.Huang and Uryasev [6] linked the systemic risk contribution of an institution to the increase in the CoVaR of the entire financial system while the institution is under distress.Acharya et al. [4] proposed the marginal expected shortfall (ES) to measure the contributions of financial institutions to systemic risk, whose mathematical expression has been generalized to the following conditional expected shortfall (CoES) [2] , at level , conditional on one of the institutions beyond its VaR at level , Notably, CoVaR and CoES, defined by Eqs. ( 1) and ( 2), and their transformers have played an essential role in measuring the system risk.Detailed discussions and their applications in economics, finance, and other fields can be found in Refs.[2, 5, 7, 8], as well as the references therein.
Measuring systemic risk requires the knowledge of its probability distribution.In most practices, the exact form of the distribution is often lacking, and only sample data are available for estimating the distribution, which is inevitably prone to sampling error.This situation, wherein the probability distribution of uncertain outcomes cannot be uniquely identified, is referred to as distributional uncertainty.The question of how to account for distributional uncertainty in decision-making has been of central interest in several fields, including economics, finance, control system, and operations research/ management science.One modeling paradigm that F F F has been successfully adopted in all these fields to address this issue is distributionally robust optimization (DRO).In the standard form of DRO, we characterize one's (partial) information by specifying an uncertainty set , which is also known as an ambiguity set, instead of an underlying probability distribution that is known exactly.Various types of uncertainty sets have been proposed in the literature.One common way of defining the set is by specifying the moments of the distribution.The earlier works of Popescu [9] , Bertsimas et al. [10] , Delage and Ye [11] , and Natarajan et al. [12] have considered the case where the uncertainty set is specified in terms of the first two moments.More recently, Wiesemann et al. [13] considered a case in which the uncertainty set was described through supports and higher-order moments.In this study, we consider the worst-case CoVaR and CoES under moment uncertainty.Specifically, we consider the following optimization problems: where is an uncertainty set specifying the mean vector and the covariance of X , and and represent the calculation for and under the joint distribution , respectively.While the current stduy focuses on moment-based uncertainty sets, we should point out here that the uncertainty set can also be defined according to a certain distance over distributions, such as KL divergence and Wasserstein metric.

Let
be an atomless probability space, and as its n-dimensional product space, where is a set of possible states of nature and is a σ-algebra on .The random variable is a measurable real-valued functional on .For a random vector (random variable) , its distribution function is defined by for , and we denote the distribution function of the random vector (random variable) X by .The notation represents the point mass at .The left and right quantile function of a univariate distribution are denoted by and , respectively.For a mapping , the notation indicates that it has the same value as , where X is a random vector (random variable) in and is its distribution.In this study, both notation, and , are used.For a random variable , we denote the mean and the variance of X by and , respectively, and for a random vector , we denote the mean vector and the covariance matrix by and , respectively.
Notably, VaR and ES are two popular and important risk measures in financial practice.The left and right VaRs of a random variable at level are defined by and , respectively.The ES of a random variable at level is defined by .Furthermore, CoVaR and CoES are defined by Eqs. ( 1) and ( 2) in the Introduction, respectively.

Worst-case systemic risk measures
In this study, we examine the worst-case CoVaR and CoES with an uncertainty set based on moment constraints.Specifically, for a portfolio , let , and we consider the following optimization problems: F where the uncertainty set is defined by moment information, that is , is a given semi-positive matrix.Our aim is to investigate the optimization problems in (4).
The objective function of (4) only depends on and .The uncertainty constraint can be replaced by the following set: where is the joint distribution function of .Applying the general projection property in Ref. [9] (see also Ref. [14, Lemma 2.4]), equals to the following uncertainty set: where , and is the vector whose ith element equals to and the other elements are all zero.These arguments inspire us to consider the following general optimization problems: CoVaR F α, β (Y|X) and sup F (µ, Σ) where is a mean-variance uncertainty set of twodimensional random vectors defined by where , and is a semi-positive matrix.It is easy to verify that the original optimization problems in (4) are a special case of optimization problems in (6) with and , where is the covariance of the random vector .
In the remainder of this paper, we aim to solve the optimization problems in (6).We always assume that all considered random variables in (6) satisfy that is continuous at so that .Some preliminaries on the copula theory are needed, and these will be introduced in the next subsection.

Copula
In this section, we recall the definition of two-dimensional copula that is a tool for separating dependence and marginal distributions.A two-dimensional copula is a function that satisfies (ii) for any such that and ,

C
For a two-dimensional random vector with joint distribution , its copula is denoted by .By Sklar's theorem [15] , the joint distribution can be expressed as An extremal copula is the comonotonicity copula, defined as C C ⩽ C M For any copula , it holds that pointwisely.We refer the readers to Ref. [15] for more details on copula.
By the relation between joint distribution function and the copula in Eq. ( 7), it is natural to denote thus: where and are the marginal distributions of and , respectively, and is the copula of .Hence, we have The above-stated formula illustrates that the value of does not depend on the marginal distribution .The following proposition, which collects the results of Ref. [16, Theorem 3.4] shows that pointwisely implies and for all , and it is useful throughout the paper.
, and let , and be three univariate distributions.For any copula , we have Moreover, if are two copulas such that pointwisely, then we have Proof.By Eq. ( 8), we can immediately obtain .Because CoES is formulated as the integral of CoVaR, we have .To see the "Moreover" part, if pointwisely, then we have for all .It follows from Eq. ( 8) that .Noting that CoES is formulated as the integral of CoVaR, we obtain .Hence, we complete the proof.

C
C(u, v) ⩽ C M Since for any copula , we have .The next proposition is a direct result followed by Proposition 2.1.
Proposition 2.2.Let be a copula function, and Consider two optimization problems Both maximization problems can both be attained at .Specifically, if , then the optimization problems (i) and (ii) can be attained at , and the values of (i) and (ii) are and , respectively, where and has distribution .
Proof.The maximizer can be attained at , immediately following Proposition 2.1.If , and for all copulas , the maximizer can be attained at .Moreover, in this case, we obtain Hence, we complete the proof.
and for , and The following lemma, which plays an important role in the proof of the main theorems presented in this paper, is a direct result of Refs.[17, Theorem 1] and [14, Theorem 2.9].
For , and , it holds that sup The next result shows the value and the closed-form solution of the worst-case CoVaR and CoES defined by (9).
Theorem 3.1.For , , , and defined by (10), we have where , and the supremum can be attained at the joint distribution where the mean and variance of is and , respectively, and is a twopoint distribution, defined as where the equalities in the two formulas given above follow from Lemma 3.1.On the other hand, one can verify that the worst-case value of CoVaR and CoES can be attained at the distribution given in the theorem.Thus, we have This completes the proof.X (X, Y) X Theorem 3.1 illustrates that the closed-form solution of ( 9) does not depend on the form of the distribution of but requires a comonotonicity copula structure of .In other words, we can derive different forms of the closed-form solution by constructing different forms of the distribution of .The following is a special case of the closed-form solution defined in Theorem 3.1.
Example 3.1.Define where , satisfy , and represents a uniform distribution on .It is easy to verify that the mean and variance of are and ; hence, is a closed-form solution of (9).Moreover, suppose that . Then, we have and the correlation coefficient of with joint distribution is , then reduces to a two-point distribution.In this case, the correlation coefficient equals to 1.

F
In this subsection, we consider the optimization problems in (6) when the uncertainty set contains the information of the mean vector and covariance with a fixed correlation coefficient, that is, where , for , and represents the correlation coefficient of .It noteworthy that for all , where is defined by Eq. (10).By Theorem 3.1, we have sup F∈F (µ1 ,µ2 ,σ1 ,σ2,ρ) where .Recall Example 3.1, we know that the equalities of (12) hold if . A natural question is that what is the range of that will make (12) become equalities?To answer this question, we first present the following lemma, which shows that the values of the worstcase CoVaR and CoES with the uncertainty set increase in .
Lemma 3.2.For , , , , let defined by Eq. (11).Both and increase in with the comonotonicity copula and denote the correlation coefficient of by ρ 0 .We consider the following two cases.in case of model uncertainty with a known mean and covariance of the portfolio , that is, the uncertainty set is defined by (11).When the correlation coefficient , the values of the worst-case and are equal to a constant .To calculate the value of the worst-case CoVaR and CoES with uncertainty set when is beyond current technology, and it could be an open question.

= 1 f ( 1 ) 1 )
λ → f (λ) := corr(X c 2,λ , Y c 1 ) f (0) = corr(X c , Y c 1 ) = corr(X c 1 , = corr(X c 2,λ * , Y c 1 ) = ρicity copula are positive.By Theorem 3.2, in the cases of these two copulas, the value of the worst-case CoVaR and CoES are also equal to(15) if the partial information is derived from the first two moments of the marginal distribution and the correlation coefficient of the given copula.5ConclusionsCoVaR α, β CoES α, β In this paper, we study the worst-case and (X, Y)