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

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1 Depend Model 2017; 5:1 19 Research Article Special Issue: Recent Developments in Quantitative Risk Management Open Access Piotr Jaworski* On Conditional Value at Risk CoVaR for tail-dependent copulas DOI /demo Received September 15, 2016; accepted December 5, 2016 Abstract: The paper deals with Conditional Value at Risk CoVaR for copulas with nontrivial tail dependence We show that both in the standard and the modified settings, the tail dependence function determines the limiting properties of CoVaR as the conditioning event becomes more extreme The results are illustrated with examples using the extreme value, conic and truncation invariant families of bivariate tail-dependent copulas Keywords: Copulas, Tail dependence, Value-at-Risk VaR, Conditional Value-at-Risk CoVaR, Conditional quantiles MSC: 62H05, 60E05, 91B30, 91G40 1 Introduction This paper is based on the Profit/Loss P/L approach as for example in [1, 4, 15] We will study random variables X, Y,, which are modeling: welfare of financial institutions; financial positions; investment profits; rates of returns of stock prices and indices So generally The higher value of X, Y,, the better" We recall that Value-at-Risk, at a given significance level 0, 1, of a P/L random variable X, is defined as follows [15]: VaR X = inf{v R : PX + v < 0 } When X is modelling a financial position, VaR X is the smallest amount of capital v that ensures that X + v is solvent with probability at least equal to 1 Alternatively when X is modelling the gain from the investment, VaR X is the probabilistic" answer to the question How much may I lose?" ie it is the largest loss that one is exposed to with a confidence level of 1 compare [36] 11 The above can be expressed in terms of quantiles Namely Value-at-Risk at a level is equal to the negative upper quantile of X or lower 1 quantile of the loss X VaR X = Q + X = Q 1 X To switch to the alternative Loss/Profit L/P approach applied for example in [5, 18, 33] when random variables are modelling losses from the financial investments, actuarial risks or high water levels in hydrology, *Corresponding Author: Piotr Jaworski: Institute of Mathematics, University of Warsaw, Poland, PJaworski@mimuwedupl 2017 Piotr Jaworski, published by De Gruyter Open This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 30 License

2 2 Piotr Jaworski for example, it is is enough to change the sign of the variables L = X, and remember that, by convention, the subscript is changed The significance level is replaced by the confidence level c = 1, that is VaR c L = VaR X Now assume that we are measuring the risk basing on some extra information For example, we want to determine what size bailout would be required to keep a financial institution Y solvent with probability at least 1 β when a financial institution X would incur significant losses Conditional Value at Risk CoVaR introduced by Adrian and Brunnermeier in 2008 [1] and its later modifications proved to be very useful tools for quantifying such phenomena Let X and Y be random variables modelling positions CoVaR is defined as VaR of Y conditioned by X Specifically CoVaRY X := VaR β Y X E, where a Borel subset of the real line E represents an adverse event concerning X Most often E consists of one point a threshold or is a half-line bounded by a threshold From the definition of CoVaR, it is clear that one has to model the dependence between Y and X, and this can be achieved by means of copulas In this paper we continue the research started in [4], but we pay special attention to the case when Y and X are tail dependent, ie the tail dependence function of their copula is positive Since we are following a P/L approach, we are interested in the shape of the copula close to the origin, ie in the lower tail We will consider three families of copulas having nontrivial lower tail: the survival extreme value copulas, the survival conic copulas and left truncation invariant copulas Adrian and Brunnermeier [1] applied the construction with E consisting of one point To differentiate from their approach, we will call their CoVaR the standard one Specifically, the standard Conditional-VaR at a level, β is defined as VaR at level β of Y under the condition that X = VaR X Definition 11 CoVaR =,βy X = VaR β Y X = VaR X The above can be expressed in terms of conditional quantiles Namely CoVaR =,βy X = Q + βy X = Q + X The main technical drawback of this approach is the dependence of the CoVaR on the choice of the version of the conditional probability compare [2, Thm 331] Since we expect that a risk measure should uniquely assign a real number to a random pair, one has to extend definition 11 by providing an algorithm selecting the version of the conditional probability In section 4 we will follow the approach from [4] Under the mild assumption that the univariate distribution functions of X and Y are continuous, we will use the theory of copulas to provide a canonical way how to select a version of the conditional probability, which allows us to redefine CoVaR in a mathematically correct way, ie as a univocal risk measure Besides the issues described above, there are also some more practical drawbacks of the standard CoVaR pointed out for example by Mainik and Schaanning in [33], which motivate a modification of the original definition The main objection is due to the fact that the standard CoVaR is not compatible with concordance ordering Hence it is breaking" the paradigm: more dependence, more systemic risk The modified Conditional-VaR at a level, β is defined as the VaR at level β of Y under the condition that X VaR X Definition 12 CoVaR,βY X = VaR β Y X VaR X

3 On Conditional Value at Risk CoVaR for tail-dependent copulas 3 The above can be expressed in terms of quantiles Namely CoVaR,βY X = Q + βy X Q + X Modified CoVaR was introduced by Girardi and AT Ergün in 2013 [16] and Mainik and Schaanning in 2014 [33], both in the L/P setting The paper is organized as follows: In section 2 we recall the basic facts about copulas, their geometric transformations and tail expansions Next we describe the tail behaviour of extreme value, conic and left truncation invariant copulas In section 3 we show how to express modified CoVaR in terms of copulas Following [4], we study the threshold w *, β, C such that CoVaR Y = VaR w* Y We discuss the compatibility of modified CoVaR with concordance ordering of copulas ie with the strength of dependence between the conditioned and conditioning variable and provide approximate bounds for the thresholds Next we deal with copulas with a nontrivial tail expansion We show that for such copulas the first order limiting properties of w * are fully determined by the tail dependence function We illustrate the above on the example of extreme value, conic and left truncation invariant copulas The last section is devoted to the standard CoVaR First we refine the definition to make it univocal even when the dependence between X and Y is described by a singular copula Following [4], we study the threshold v *, β, C such that CoVaR = Y = VaR v* Y We discuss the incoherent response of standard CoVaR to concordance ordering of copulas We show that for almost all copulas for sufficiently small β, the threshold v *, β, C is smaller than that of the comonotonic copula M Next we deal with copulas with regular nontrivial tail expansion We show that for such copulas the first order limiting properties of v * are determined by the tail dependence function We illustrate the above on the example of extreme value, conic and left truncation invariant copulas 2 Copulas 21 Basic notation To fix notation, we recall some basic facts about copulas For more details the reader is referred to standard texts such as [8, 11, 13, 24 27, 34, 35] We recall that the function C : [0, 1] 2 [0, 1] is called a copula if the following three properties hold: c1 u 1, u 2 [0, 1] Cu 1, 0 = 0, C0, u 2 = 0; c2 u 1, u 2 [0, 1] Cu 1, 1 = u 1, C1, u 2 = u 2 ; c3 u 1, u 2, v 1, v 2 [0, 1], u 1 v 1, u 2 v 2 Cv 1, v 2 Cu 1, v 2 Cv 1, u 2 + Cu 1, u 2 0 Alternatively we can characterize copulas in a more probabilistic way Namely, a function C is a copula if and only if there exist random variables U, V, which are uniformly distributed on [0, 1], such that C is a restriction to the unit square [0, 1] 2 of their joint distribution function Random variables U and V are called the representers of the copula C Proved half a century ago, Sklar s Theorem remains crucial for applications of the copula theory It states that any multivariate distribution function may be expressed as a composition of a copula and its univariate marginals It allows to split the study of the multivariate phenomena into the study of marginals and the study of dependence Since in our study of CoVaR we will rely very much on Sklar s Theorem for random pairs, we recall how it is formulated:

4 4 Piotr Jaworski Theorem 21 Let F be a 2-dimensional distribution function and F 1, F 2 its marginal distribution functions, then there is a copula C such that for each x = x 1, x 2 R 2 Fx 1, x 2 = CF 1 x 1, F 2 x 2 Furthermore, the copula C is uniquely determined when the boundary distribution functions F i are continuous Conversely, if C is a 2-dimensional copula and F 1, F 2 are univariate distribution functions then the function F is a 2-dimensional distribution function and F 1, F 2 are its boundary distribution functions Remark 21 [20] Pr1 Let C be a copula of a random pair X 1, X 2 Suppose that F i s, the distribution functions of X i s, are continuous, then the random variables are representers of the copula C F 1 X 1, F 2 X 2 One more premise to use copulas to model systemic risk follows from the fact that the copulas are true measures of interdependence between random phenomena Namely they do not depend on the scale in which these phenomena are quantified Indeed, if C is a copula of a random pair X = X 1, X 2, and the functions h 1, h 2 are defined and strictly increasing on the supports of X 1, X 2, then C is also a copula of the transformed random pair Y = h 1 X 1, h 2 X 2 22 Geometrical transformations of copulas We recall that there exist eight linear isometric transformations of the unit square [0, 1] 2 : two mirror reflections with respect to the diagonals, two mirror reflections with respect to bisectors, one point reflection, two rotations ±π/2 90 and 270 degrees and identity They induce the transformations of copulas Namely let random variables U 1, U 2 be representers of a copula C and σ : [01] 2 [0, 1] 2 be an isometry, then random variables V 1, V 2 given by V 1, V 2 = σu 1, U 2 are uniformly distributed on the unit interval [0, 1] The copula C σ of the pair V 1, V 2 is called a reflection or respectively rotation of the copula C The copulas obtained by the point reflection are better known under the name survival copulas" and denoted by Ĉ Ĉp, q = p + q 1 + C1 p, 1 q Note that in this case V 1 = 1 U 1, V 2 = 1 U 2 Survival copulas are a useful tool when one is switching from P/L to L/P setting Indeed when C is a copula of gains X and Y, then the survival copula Ĉ is a copula of losses L 1 = X and L 2 = Y

5 On Conditional Value at Risk CoVaR for tail-dependent copulas 5 23 Copulas with nontrivial tail expansions In risk management one has to deal with extreme events and the interdependencies between them This leads to the study of the tail behaviour of a copula, ie of the possible approximations of a copula close to the vertices of the unit square Since applying a proper geometric transformation one may map any vertex of the unit square [0, 1] 2 to the selected one, we restrict ourselves to the vertex 0, 0 the origin Definition 21 We say that a copula C has a tail expansion at the vertex 0, 0 of the unit square if the limit exists for all nonnegative x, y The function L : [0, ] 2 [0,, Ctx, ty lim t 0 + t Ctx, ty Lx, y = lim, t 0 t is called the tail dependence function or the leading term of the tail expansion The second naming follows from the fact, proved in [21]: if L exists then we have a decomposition of a copula where R is bounded and Cu, v = Lu, v + Ru, vu + v, lim Ru, v = 0 u,v 0,0 The above can be applied to other vertices as well It is enough to reflect the copula For example for the upper tail the vertex 1,1 we get Ĉtx, ty tx + ty 1 + C1 tx, 1 ty Lx, y = lim = lim t 0 t t 0 t Note that L1, 1 and L1, 1 are equal to the lower and upper tail dependence coefficients We recall the basic properties of the tail dependence functions for details see [6, 7, 19 23, 28, 29, 31] Lemma 22 [20, 21]The tail dependence function induced by a copula C, Lu = lim t 0 + Ctu, u [0, + 2, is: t 1 homogeneous of degree 1, 2 2-nondecreasing and nondecreasing with respect to every variable, 4 nonnegative and bounded by the smaller coordinate of u: 5 Lipschitz with Lipschitz constant 1: 0 Lu minu 1, u 2 Lv Lu v 1 u 1 + v 2 u 2 6 concave: λ 1, λ 2 0, λ 1 + λ 2 = 1 Lλ 1 u + λ 2 v λ 1 Lu + λ 2 Lv Due to homogeneity, the leading term L is uniquely described by vertical sections like lt = L1, t

6 6 Piotr Jaworski Theorem 23 Let l : [0, [0, 1], l0 = 0, be a nondecreasing, concave function, such that lt t Then the function { L : [0, + 2 xl y [0, +, Lx, y = x for x > 0, 0 for x = 0 is a leading term of some copula The function lt = L1, t will be called a generator of the leading term The proof of theorem 23 follows from the examples below see [20] s21 and [19] for detailed calculations Since l is concave, the one-sided partial derivatives of L exist everywhere in 0, + 2 For left-sided derivatives we get Lu, v v v = l l v +, u u u u Lu, v = l v v u The same is valid for right-sided derivatives, by just switching +" and " We put Obviously for u > 0 and v 0 It = lt tl t+ = L1, t u Lu, v u v = I u Lemma 24 The function is: 1 nondecreasing; 2 I0 = l0 = 0 and I+ = l+ ; 3 for t such that lt < l+ It < lt I : [0, + [0, 1] Proof Since Lu, v is two-nondecreasing, nondecreasing in u and Lipschitz with constant 1, its derivative with respect to u is nondecreasing in v, nonnegative and bounded by 1 Hence so is I It is nondecreasing, nonnegative and bounded by 1 Since Lu, 0 = 0, I0 = L1, 0 u Since Lu, v is concave and homogeneous of degree 1, its derivative with respect to the first variable is homogeneous of degree 0, and we get = 0 L1, t L, 1 I+ = lim It = lim = lim = L0+, 1 t + t + u 0 + u u L, 1 = lim = lim L1, t = lim lt = l+ 0 + t + t + To show the last point we observe that l is a concave, nondecreasing function Hence l t + > 0 if lt < l+ Therefore It = lt tl t + < lt In the following we will use a generalized inverse of I I [ 1] s = inf{t : It > s}

7 On Conditional Value at Risk CoVaR for tail-dependent copulas 7 24 Examples 241 Extreme value copulas Let l : [0, + ] [0, 1] be a concave, nondecreasing function, such that lt t, then the function C l : [0, 1] 2 lnv [0, 1], C l u, v = uv exp lnul lnu is a copula see for example [35] 334, [11] 66 or [17] with an upper tail dependence function [20] 54 y Lx, y = xl x Note that copulas C l satisfy the following property C l u n, v n = C l u, v n, n > 0 They are called extreme value copulas, and when we put we get the well known Gumbel family of copulas The survival copula is given by lt = 1 + t 1 + t θ 1/θ, θ 1, Ĉ l u, v = u + v u1 v exp v = ul + ou + v u ln1 v ln1 ul ln1 u 242 Conic copulas Let l : [0, + ] [0, 1] be a concave, nondecreasing function, such that lt t, then the function C l : [0, 1] 2 v [0, 1], C l u, v = max ul, u + v 1 u is a copula with a tail dependence function y Lx, y = xl x Copulas of this form were used in [20] to prove the existence of copulas with given lower and upper tail dependence functions The survival copulas are given by Ĉ l : [0, 1] 2 1 v [0, 1], Ĉ l u, v = max u l 1 v 1 u 1 u 1, 0 They are known under the name conic copulas" see [14, 30] 243 LTI copulas Let f : [0, + ] [0, 1] be a surjective, concave and nondecreasing function and g its right inverse f gy = y Then the function C f : [0, 1] 2 0 for x = 0, [0, 1], C f x, y = xf gy x for x > 0

8 8 Piotr Jaworski is a copula introduced and considered in [10, 12] It belongs to the class of copulas that are invariant under left truncation For a suitable generator f, the popular Clayton copulas belong to this class Namely f t = 1 + t θ 1/θ, θ > 0 Furthermore see [12] proposition 41 the leading term of C f equals Lx, y = xf g 0 + y x Note that since g is a convex increasing function its right sided derivative at 0 exists and is nonnegative Furthermore L is nonzero if and only if g 0 + > 0 Then the generator lt equals f g 0 + t For Clayton copulas with positive θ we get y θ 1 θ Lx, y = x 1 + = x θ + y θ 1 θ, x which follows as well from the general results for Archimedean copulas see [20] Th6, [21] Pr10 or [7] Th31 3 Modified CoVaR by copulas We follow the P/L approach from [4] For the L/P setting the reader is referred to [33] Th 31b Let Cu, v be a copula of random variables X and Y having continuous distribution functions F X and F Y, then Therefore PY y X Q + X = PY y X Q+ X CoVaR,βY X = VaR w* Y, where w * = w *, β, C is the largest solution of the equation C, w * = β = C, F Yy Note that: w * = CoVaR,βF Y Y F X X Furthermore, as was observed in [33] Th34 for the L/P setting, modified CoVaR is compatible with the concordance ordering of copulas The same is valid for the P/L setting Theorem 31 Let C i u, v, i = 1, 2, be a copula of random variables X i and Y i having continuous distribution functions F Xi and F Yi and, β 0, 1] some fixed thresholds If u, v [0, 1] 2 C 1 u, v C 2 u, v then w *, β, C 1 w *, β, C 2 31 If furthermore t, + F Y1 t F Y2 t then Proof Since CoVaR,βY 1 X 1 CoVaR,βY 2 X 2 32 C 1, w *, β, C 1 = β = C 2, w *, β, C 2

9 On Conditional Value at Risk CoVaR for tail-dependent copulas 9 and C 1 C 2 we get w 1 = w *, β, C 1 w *, β, C 2 = w 2 Now since F Y1 F Y2 and w 1 w 2, we get CoVaR,βY 1 X 1 = VaR w1 Y 1 VaR w2 Y 1 VaR w2 Y 2 = CoVaR,βY 2 X 2 Theorem 31 implies the rough bounds for the threshold w * Due to the Fréchet-Hoeffding bounds see [11, 35], we have Mu, v Cu, v Wu, v, where Mu, v = minu, v is the comonotonic copula and Wu, v = u + v 1 + the countermonotonic one Since w *, β, M = β and w *, β, W = 1 + β [4] 31, we get: Corollary 32 Let C be any bivariate copula Then β w *, β, C 1 1 β If furthermore we assume that copula C is PQD positively quadrant dependent ie C dominates the independence copula Πu, v = uv u, v [0, 1] 2 Cu, v uv = Πu, v, we may improve the upper bound Indeed, since w *, β, C = β, we get: Corollary 33 Let C be a PQD copula then β w *, β, C β 31 Copulas with nontrivial tail expansions Theorem 34 Let the copula C have a nonzero tail dependence function L Ctu, tv v lim = Lu, v = ul t 0 t u Then for β < l Proof We have to solve the equation lim w *, β, C = 0, 0 w lim *, β, C = l 1 β 0 C, w * = β 33 First we show that for sufficiently small, w *, β, C is bounded by some linear function of We choose β 1 from the interval β, l+ We obtain So, for smaller than sufficiently small 1 C, l 1 β lim 1 = L1, l 1 β 0 1 = β 1 > β C, l 1 β 1 > β

10 10 Piotr Jaworski Since C is continuous and monotonic in the second variable, we get that for 0, 1 the solution w * of 33 is between 0 and l 1 β 1 Hence lim 0 w *, β, C = 0 To show the second equality we decompose C Cu, v = Lu, v + Ru, vu + v As was shown in [21], R is bounded and has a limit at zero lim Ru, v = 0, u,v 0,0 ie for any two sequences of numbers u n and v n from the unit interval, which are tending to 0 when n, the sequence R n = Ru n, v n tends to 0 as well We can rewrite equation 33 as L, w * + R, w * + w * = β We divide both sides by Hence Since for < 1 w * < l 1 β 1, we get l w* = β R, w * w * = l 1 β R, w * 1 + w * 1 + w * lim R, w * 1 + w * = 0 0 Since l 1 is continuous we obtain w lim *, β, C = l 1 β Survival conic copulas We need to solve the equation w max l, 1 + w 1 = β We get w * = { minl 1 β, 1 1 β for β < l, 1 1 β for β l, Hence for β < l and 1 1+l 1 β β w * = l 1 β 312 LTI copulas We need to solve the equation We get [4] f gw = β w * = f gβ,

11 On Conditional Value at Risk CoVaR for tail-dependent copulas 11 lim w * = f 0 = 0, 0 w lim * 0 = lim 0 w * = f 0 + gβ Note that since f is concave, nondecreasing its derivative may be finite or infinite In the first case C f has a nontrivial leading term and w * = f 0 + gβ + o 313 Survival extreme value copulas We get the equation ln1 w* + w * w * exp ln1 l = β ln1 Solving this with respect to l gives ln1 w* l = ln1 + β w * ln1 ln1 w * ln1 ln1 = β + O Next we invert l For β < l we obtain w * = 1 exp ln1 l 1 ln1 + β w* ln1 ln1 w * ln1 = l 1 β + O 2 To get a better approximation of w *, one may apply the following recurrence: Lemma 35 Let the functions w 1, w 2 : [0, 1] [0, 1] satisfy w 2 = 1 exp ln1 l 1 ln1 + β w1 ln1 ln1 w 1 ln1 If w * w 1 = O k, k 2, then w * w 2 = O k+1 Proof We have for sufficiently small and some constants C and C 1 w * w 2 = exp ln1 l 1 ln1 + β w* ln1 ln1 w * ln1 exp ln1 l 1 ln1 + β w1 ln1 ln1 w 1 ln1 C ln1 w * w 1 C 1 O k = O k+1 4 Standard CoVaR by copulas The discrepancy following the non-uniqueness of the conditional probability is usually overcome by an assumption that the pair X, Y has a continuous density f x, y, which allows us to select, as the density of Y X = x, the following function φ x y = { f x,y f X x when f X x > 0, f Y y when f X x = 0 or f X x = +, 41 where the density of X, denoted by f X x, is given by a formula f X x = f x, ydy

12 12 Piotr Jaworski and similarly the density of Y Note that, since the versions of the density of a given random pair may differ only at the set of Lebesgue measure 0, there may exist at most one continuous version Therefore the above determines uniquely a version of the conditional density But when every version of the density f x, y is discontinuous, there is no such canonical choice Furthermore, the choice of the version of the density f in formula 41 may affect CoVaR significantly We illustrate this by the following simple example: Example 41 For 0, 1, consider a pair of normally distributed random variables X and Y, X, Y N0, 1 coupled by a copula being the ordinal sum see [11, 35] of two copies of the independence copula with respect to the intervals [0, ] and [, 1] The resulting probability distribution has a discontinuous density We select two versions of the density, left+down continuous" and right+up continuous": and f 1 x, y = f 2 x, y = 1 1 2π exp π exp 1 1 2π exp 1 1 2π 1 exp x2 +y 2 2 x2 +y 2 2 when x, y Φ 1, when x, y > Φ 1, 0 otherwise x2 +y 2 2 x2 +y 2 2 when x, y < Φ 1, when x, y Φ 1, 0 otherwise, where Φ denotes the distribution function of the standard normal probability law The first choice implies via formula 41 CoVaR =,βx, Y = Φ 1 β, while the second CoVaR =,βx, Y = Φ 1 + β β As we see when + β = 2β, the results are quite different, which is not acceptable for risk measures In what follows, we will show how to avoid the discrepancy from the above example and moreover how to deal with singular distributions We keep only the assumption that random variables X and Y have continuous distribution functions F X and F Y We will redefine the standard CoVaR without any assumptions concerning the linking copula We start with so called technicalities" There is a well known relation between conditional probabilities and partial derivatives of copulas But in general copulas are only Lipschitz functions, which may not be differentiable Therefore we apply the notion of Dini derivatives see [32] From four possibilities we choose a left-sided upper one By D u C we denote the partial left-sided upper Dini derivative of copula Cu, v with respect to the first variable Cu, v Cu h, v D u Cu, v = lim sup h 0 + h We recall that for u 0, 1] D u Cu, 0 = Cu, 0 Cu h, lim = lim = 0, h 0 + h h 0 + h D u Cu, 1 = Cu, 1 Cu h, 1 u u h lim = lim = 1 h 0 + h h 0 + h Furthermore D u Cu, v is nondecreasing in v compare [9] The Dini derivative D u may be rewritten in the following way D u Cu, v = lim sup ν 0 + 0<h ν Cu, v Cu h, v, h

13 On Conditional Value at Risk CoVaR for tail-dependent copulas 13 which implies that it is a Borel measurable function Therefore the composition D u CF X X, v is a well defined σx measurable random variable Since a Dini derivative of a Lipschitz function is almost everywhere equal to the classical" derivative, the random variable D u CF X X, F Y y is a version of the conditional expected value see [2] 34 of the indicator function D u CF X X, F Y y = E1l Y y X as Putting F Y X=x = PY y X = x = lim η y + D ucf X x, F Y η we fix the version of the conditional probability This leads to the following definition of CoVaR see [4] and [33] Th 31a for L/P approach, compare [3] for conditional quantile setting where CoVaR,β Y X = sup{y : F Y y = v * } = Q + v * Y = VaR v* Y, When C is continuously differentiable we get Note that: v * = inf{v : D u C, v > β} 42 C u, F Y CoVaR,β Y X = β v * = CoVaR,β F Y Y F X X The L/P versions of the above two formulas can be found in [33] and [18] The next proposition is in line with the approach presented in [33] We provide conditions under which copula C, for given 0, 1 and sufficiently small β, is more stress testing sensitive" than the comonotonic copula M, although the latter dominates in concordance ordering We recall that v *, β, M = see for example [4] 31 Proposition 41 For any pair, β 0, 1, if then β < D u C, = lim η D uc, η, v *, β, C < = v *, β, M Proof We fix and β From 42 we have D u, v * β < D u C, Since the Dini derivative D u Cu, v is nondecreasing in the second variable, v * is smaller than Proposition 41 has interesting consequences Let X and Y be linked by a copula C, such that for a given 0, 1 D u C, > 0 Proposition implies that for sufficiently small β CoVaR =,βy X > CoVaR =,βy c X c, where X c and Y c are comonotonic versions of X and Y, which is quite intuitive A more precise information may bound the risk

14 14 Piotr Jaworski 41 Copulas with regular tail dependence In this section we will discuss the case of heavy tails We add to the assumption of the convergence of the copula C to its leading part L the assumption that the partial derivative of C with respect to the first variable is converging to the partial derivative of L We will base on the notation from section 23 Lu, v = ul v u, L v u u, v = I u Theorem 42 Let the copula C have a nonzero tail dependence function L and v lim D u Cu, v I = 0 u,v 0,0 u Then for β < l+ and lim v *, β, C = 0 0 v lim η β I[ 1] η lim inf *, β, C v lim sup *, β, C I [ 1] β 0 0 Proof We have to find the infimum v * = infv : D u C, v > β 43 First we show that for sufficiently small, v *, β, C is bounded by some linear function of We choose β 1 from the interval β, l+ ], such that I [ 1] β 1 > I [ 1] β note that I = l We obtain So, for smaller than sufficiently small 1 On the other hand lim D uc, I [ 1] β 1 = II [ 1] β 1 β 1 > β 0 D u C, I [ 1] β 1 > β lim D uc, l 1 β = Il 1 β = β l 1 βl l 1 β < β 0 Hence l 1 β does not belong to the half-line {v : D u C, v > β} and l 1 β inf{v : D u C, v > β} = v * Since D u C is nondecreasing in the second variable, we get that for 0, 1 the infimum v * from 43 lies between l 1 β and I 1 β 1 Hence lim 0 v *, β, C = 0 To show the first order estimate we decompose D u C We rewrite 43 Hence where D u Cu, v = Iv/u + R 1 u, v v * / = inf{v/ : Iv/ + R 1, v > β} = inf{v : Iv + R 1, v > β} I [ 1] β r = inf{v : Iv > β r} v * / inf{v : Iv > β + r} = I [ 1] β + r, r = sup{ R 1, v : l 1 β < v < I [ 1] β 1 } Since Ru, v tends to 0 with u, v 0, 0, we get that v lim η β I[ 1] η lim inf *, β, C 0

15 On Conditional Value at Risk CoVaR for tail-dependent copulas 15 and lim sup 0 v *, β, C I [ 1] β Corollary 43 Let the copula C have a nonzero tail dependence function L and v lim D u Cu, v I = 0 u,v 0,0 u Then for β < l+ and sufficiently small If furthermore then for sufficiently small Proof We compare the limits of v * / and of w * / Since β < l, from Lemma 24 we get v *, β, C w *, β, C β < I1 = lim η 1 Iη v *, β, M = > v *, β, C Il 1 β < ll 1 β = β On the other hand, since I is right-sided continuous, I lim η β I[ 1] η β Hence Finally, from Theorems 34 and 42 we get that l 1 β < lim η β I[ 1] η v lim inf *, β, C w *, β, C > lim 0 + η β I[ 1] η l 1 β > 0, which concludes the proof of the first inequality The condition β < lim η 1 Iη implies that I [ 1] β < 1 Therefore, due to Theorem 42, for sufficiently small v *, β, C < Corollary 43 has interesting consequences Let X and Y be linked by a tail dependent copula C Then the first inequality implies that for sufficiently small and β, modified CoVaR dominates the standard one CoVaR =,βy X CoVaR,βY X The adverse event X VaR X is worse for Y than X = VaR X The second inequality implies that for C = M and for sufficiently small and β CoVaR =,βy X > CoVaR =,βy c X c, where X c and Y c are comonotonic versions of X and Y Although M dominates C in concordance ordering, C is more stress testing sensible" as regards standard CoVaR Note, that when we add one more assumption to those in Corollary 43, namely that the copula C has a density which is bounded from 0 on some neighbourhood of the diagonal = {t, t : t [0, 1]}, then there exists a global" bound β 0, such that for all 0, 1 and all β 0, β 0 ] v *, β, M = > v *, β, C

16 16 Piotr Jaworski 42 Examples 421 Clayton copulas, θ > 0 We recall that the Clayton copula with positive θ is given by C Cl u, v; θ = u θ + v θ 1 θ 1 From this we derive We get [4] Note that if then for all 0, 1 C Cl u, v; θ u = u θ + v θ 1+θ θ 1 u θ 1 v * = β θ 1+θ 1 + θ 1 θ = β θ 1 θ 1+θ 1 + O 1+θ β < 2 θ+1 θ, v *, β, C Cl ; θ < = v *, β, M Since for fixed and β lim v *, β, C Cl ; θ =, θ we observe that v *, β, C Cl ; θ is not a decreasing function with respect to θ Moreover for sufficiently large θ it is increasing Thus we observe for Clayton copulas the same phenomenon as described in [33] for normal distributions The standard CoVaR is decreasing when the increasing in concordance ordering family of copulas is approaching the maximal copula M 422 Survival conic copulas Since v C l u, v = max u l, u + v 1, u it can be easily derived that D u C l, v = { I v when l v + v 1, 1 when l v < + v 1, where It = lt tl t + is a nondecreasing function introduced in section 23 1 Thus, we get for β < I and I [ 1] β Furthermore if β < I1 than for all 0, 1 v * = I [ 1] β β < D u C l, and Proposition 41 implies v *, β, C l < = v *, β, M

17 On Conditional Value at Risk CoVaR for tail-dependent copulas LTI copulas Since we get where We have to solve the equation C f x, y = { D u C f x, y = f xf 0 for x = 0, for x > 0, gy x gy x gv I f = β, I f t = f t tf t +, f + gy gy x x We obtain [4] For fixed β 0, 1 we get v * = f I [ 1] f β, lim v * = f 0 = 0, 0 v lim * 0 = f 0 + I [ 1] f β Furthermore when g 0 + > 0 and β < I f g 0 +, then due to Proposition 41, for all 0, 1 v *, β, C f < = v *, β, M 424 Survival extreme value copulas Since we get ln1 v Ĉ l u, v = u + v u1 v exp ln1 ul, ln1 u D u Ĉ l u, v = v exp ln1 ul = I ln1 v + Ou + v, ln1 u ln1 v 1 + I ln1 u ln1 v ln1 u where It = lt tl t + is a nondecreasing function introduced in section 23 We get the following formula determining v * { } ln1 v ln1 v v * = inf v : v exp ln1 l 1 + I > β ln1 ln1 We rearrange this to give v * = inf { v : I ln1 v > 1 1 β ln1 1 v exp ln1 l } ln1 v ln1 For simplicity we assume that generator l is strictly concave and twice differentiable with positive l t This implies that I is continuous and strictly increasing, hence invertible In such a case we get for β < I v * = 1 exp ln1 I β ln1 v* exp ln1 l 1 v * ln1 = I 1 β + O 2

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