On multivariate extensions of the conditional Value-at-Risk measure

On Multivariate Extensions of the Conditional Value-at-Risk measure Di Bernardino, E.a, Fern´andez-Ponce, J.M.b,∗, Palacios-Rodr´ıguez, F.b, Rodr´ıguez-Gri˜nolo, M.R.c aCNAM, Paris, D´epartement IMATH, Laboratoire C´edric EA4629, 292 Rue Saint-Martin, Paris 75003, France bUniversidad de Sevilla, Departamento de Estad´ıstica e Investigaci´on Operativa, Calle Tarfia sin n´umero, 41012 Sevilla, Espa˜na cUniversidad Pablo de Olavide, Departamento de Econom´ıa, M´etodos Cuantitativos e Historia Econ´omica, Carretera Utrera, km 1, 41013 Sevilla, Espa˜na Abstract CoVaR is a systemic risk measure proposed by Adrian and Brunnermeier [1] able to measure a financial institution’s contribution to systemic risk and its contribution to the risk of other financial institutions. CoVaR stands for conditional Value-at-Risk, i.e. it indicates the Value at Risk for a financial institution that is conditional on a certain scenario. In this paper, two alternative extensions of the classic univariate Conditional Value-at-Risk are introduced in a multivariate setting. The two proposed multivariate CoVaRs are constructed from level sets of multivariate distribution functions (resp. of multivariate survival distribution functions). These vector-valued measures have the same dimension as the underlying risk portfolio. Several characterizations of these new risk measures are provided in terms of the copula structure and stochastic orderings of the marginal distributions. Interestingly, these results are consistent with existing properties on univariate risk measures. Furthermore, comparisons between existent risk measures and the proposed multivariate CoVaR are developed. Illustrations are given in the class of Archimedean copulas. Estimation procedure for the multivariate proposed CoVaRs is illustrated in simulated studies and insurance real data. Keywords: Copulas and dependence, Level sets of distribution functions, Multivariate risk measures, Stochastic orders, Value-at-Risk. Introduction A risk-based approach for supervision and regulation of the financial sector is gaining ground in both emerging and industrialized countries. As part of this approach, regulators need to measure, monitor, and manage market risk. Value-at-Risk (VaR) is one measure being explored ∗Corresponding author: ferp[email protected] Preprint submitted to Elsevier Friday 31st October, 2014 for this purpose. One of the most important sectors in which this practice has been adopted is the pension fund industry. As the recent financial crisis has shown, risks are generally difficult to measure and to manage. This becomes crucial in the case of pensions, where people rely on their savings to finance their old age. Risk is a complex notion and can take on varied forms with diverse applications. In the context of trading firms, managing risk has been traditionally achieved by the introduction of Value-at-Risk (VaR) thresholds on the portfolio risk accumulated by traders. Over recent decades, this problem has been handled mostly in a univariate version. Moreover, the risk allocation problem only involves internal risks associated with businesses in the subsidiaries. However, the solvability of financial institutions could also be affected by external risks whose sources cannot be controlled. These risks may also be strongly heterogeneous in nature and difficult to diversify away. One can think, for instance, of systemic risk or contagion effects in a strongly interconnected system of financial companies. In the last decade, much research has been devoted to risk measures and many multidimensional extensions have been investigated. On theoretical grounds, Jouini et al. [23] propose a class of set-value coherent risk measures. Unsurprisingly, the main difficulty regarding multivariate generalizations of risk measures is the fact that vector preorders are, in general, partial preorders. In order to generalize the Value-at-Risk measure, Embrechts and Puccetti [15], Nappo and Spizzichino [30], and Pr´ekopa [32] use the notion of quantile curve which is defined as the boundary of the upper-level set of a distribution function or the lower-level set of a survival function. Cousin and Di Bernardino [6] introduce two alternative extensions of the classic univariate Value-at-Risk in a multivariate setting. The proposed measures, which are based on the definitions of multivariate quantiles in Embrechts and Puccetti [15], are real-valued vectors with the same dimension as the considered portfolio of risks. This feature can be considered relevant from an operational point of view. Both measures satisfy the positivite homogeneity and translation invariance property. Cousin and Di Bernardino [7] propose two extensions of the classic univariate Conditional-Tail-Expectation (CTE ) in a multivariate setting. These multivariate extensions in Cousin and Di Bernardino [6] and Cousin and Di Bernardino [7] are constructed from level sets of multivariate distribution functions and multivariate survival distribution functions, respectively. This level sets approach is also used in this paper. Another recent interesting risk measure is that of the CoVaR, which stands for Conditional Value-at-Risk. CoVaR is a systemic risk measure proposed by Adrian and Brunnermeier [1] that measures a financial institution’s contribution to systemic risk and its contribution to the risk of other financial institutions. In the original unidimensional model, the CoVaR (of a particular bank, portfolio of asset, etc.) indicates the Value-at-Risk for a financial institution which is conditional on a certain (stress) scenario. Assume now that Xjrepresents asset returns of the financial system (or bank j) and Xirepresents the asset returns of bank i. The CoVaRj|i αcan then be defined by: P[Xj≤CoVaRj|i α|Xi= VaRq(Xi)] = α, for α∈(0,1),(1) where VaRα(Xi) is the quantile function of the random variable Xiat risk-level α, i.e., VaRα(Xi) = 2 inf{x∈R:FXi(x)≥α}. Equation (1) implicitly defines the CoVaR of the bank jwhich is conditional on bank ibeing at its α%-VaR level (see Adrian and Brunnermeier [1]). CoVaR in (1) represents one of the major threads in the current regulatory and scientific discussion of systemic risks. In the literature, several alternative definitions of CoVaR can be found (see Girardi and Erg¨un [19] and Goodhart and Segoviano [21]). Starting from (1), we can also consider the CoVaR given by CoVaRj α(X) = VaRα(L|Xj≥VaRα(Xj)), where the financial system is represented via the total risk L=X1+. . .+Xd, i.e., the aggregated total risk of the firm network and the component jof the vector X= (X1, . . . , Xd) represents the risk exposure of the company j. In this paper, two new multivariate generalizations of CoVaR based on the multivariate quantile settings of Embrechts and Puccetti [15], Cousin and Di Bernardino [6], and Cousin and Di Bernardino [7] are introduced. These proposed CoVaR measures can be useful in the analysis of multiple financial institutions all together in the systemic context. Several properties have been obtained. In particular, the positive homogeneity and translation property are shown. The behaviour of the components of the proposed CoVaR vectors with respect to the univariate VaR of margins and to the multivariate VaR in Cousin and Di Bernardino [6] is also analysed. We also study how these measures are influenced by a change in marginal distributions, by a change in dependence structure, and by a change in risk level. Adrian and Brunnermeier [1] defined a systemic risk measure, called ∆CoVaR, as the difference between the VaR of the institution j(or financial system) conditional on the distress of a particular financial institution i(see (1)) and the VaR of the institution j. ∆CoVaR and other interesting systemic risk measures are introduced and gathered in Mainik and Schaanning [24]. The in-depth study of ∆CoVaR systemic risk measures using the multivariate CoVaR proposed in this paper goes beyond the scope of the present work. A more practical analysis on systemic risks using multivariate ∆CoVaR measures is currently in preparation. The paper is organized as follows. In Section 1, the piecewise-linear weighted loss function which, can be used to generalize several risk measures, is introduced. Moreover, some notations, tools, and technical assumptions are given. In Section 2, properties of invariance for the proposed multivariate CoVaR are shown. Furthermore, we analyse how these multivariate measures behave when the marginal risks or the copula structures increase with respect to stochastic orders (see Section 3). Illustrations and properties for the Archimedean copula class are presented in Section 4. In Section 5, estimation procedure for the multivariate proposed CoVaRs is illustrated in simulated studies and insurance real data. Conclusion discusses open problems and possible directions for future work. 3 1. Preliminaries and Definitions Let Xbe a non-negative random variable with distribution function FXand quantile function at level ωin [0,1] given by QX(ω) = inf{x:FX(x)≥ω}. Note that the quantile function is also defined as a Value-at-Risk in the economics literature and denoted as VaRω(X) (see also (1)). Let L1(Ω,A, P) be the set of all random variables with finite expectations. Assuming that Xis a random variable of L1, the Weighted Loss function (WL) is defined by LX(x;ω) = ωE[(X−x)+] + (1 −ω)E[(X−x)−] for all x∈Rand ω∈[0,1],(2) where x+= max{x, 0}and x−= max{−x, 0}. Note that if Xis a non-negative random variable, then LX(x;ω) = ωE[X] for all x < 0. This function has a key role in an actuarial context. Indeed, it represents the expected cost for the reinsurance company, called net premium, where Xdenotes the risk for the insurance company. If the insurance company prefers not to bear all the risk, passes on parts of the risk to a reinsurance company. The part retained by the original insurance company is usually called the retention. A stop-loss contract establishes a fixed retention x(see Section 8.3 in M¨uller and Stoyan [29]). This means that the maximum risk for the insurance company is x. Thus, if X > x then, the reinsurance company will take over X−x. This class of contracts is useful to protect companies from insolvency due to excessive claims. In an actuarial context, the threshold xis often called the deductible or priority (see Section 1.7.1 in Denuit et al. [11]). Certain interesting properties of the WL function in (2) are now recalled. The properties (P1)- (P6) are trivially obtained by the same arguments as those used by Mu˜noz P´erez and S´anchezG´omez [27] to prove the properties of the dispersion function. (P1) It holds that LX(x;ω) = ωZ+∞ x ¯ F(t) dt+ (1 −ω)Zx −∞ F(t) dt. (P2) Let CFdenote the set of continuity points of FXand X∈ L1. Then FX(x) = L0 X(x;ω) + ω, ∀x∈CFand x≥0 where L0 Xis the derivative of LXwith respect to x. (P3) The WL function is differentiable and its derivative has, at most, a countable number of discontinuity points. (P4) LX(x;ω) is a convex function on R+. (P5) limx→+∞L0 X(x;ω)=1−ω; and limx→−∞ L0 X(x;ω)=0. (P6) limx→+∞[LX(x;ω)−(1 −ω)x] = −(1 −ω)E[X]. 4 (P7) Finally, VaRω(X) = arg min x∈R+ LX(x;ω),for w∈[0,1], with VaR0(X) = xF−and VaR1(X) = xF+, where xF+and xF−are, respectively, the right and left endpoints of F, such that xF+= sup{x∈R:F(x)<1}and xF−= inf{x∈ R:F(x)>0}. It is easy to see that Properties (P1)-(P7) uniquely characterize a WL function, i.e., if LX(x;ω) is a function that satisfies Properties (P1)-(P7) above, then there exits a unique distribution function which has LX(x;ω) as its WL function. Therefore, it uniquely determines a probability measure PFon B(the σ-field of Borel set on R). An interesting interpretation of the WL function is that 2 LX(x; 1/2) is the L1-distance between FXand Fx, where Fxis the distribution function of the degenerate random variable at the point x∈R(Mu˜noz P´erez and S´anchez-G´omez [27]). It is also interesting to remark that LX(x; 1) is the well-known stop-loss function of X, and that LX(x; 0) could be interpreted as the stop-gain function of X. Consequently, the WL function is a weighting of both functions in terms of x. Now, let X= (X1, . . . , Xd) be a non-negative d-dimensional random vector1. Cousin and Di Bernardino [6] defined, under certain regularity conditions, the multivariate Lower-Orthant Value-at-Risk at probability level αas the d-dimensional vector VaRα(X) = E[X|F(X) = α],for α∈(0,1), where Fis the distribution function of X. Particularly, the i-th component of this vector trivially verifies VaRi α(X) = LXi|F(X)=α(0; 1).(3) Using Property (P7), our purpose is now to give a new multivariate approach of the classic Conditional Value-at-Risk model (see CoVaR in (1)) which, as introduced previously, is defined as the VaR of a financial institution, conditional on a certain scenario (see Adrian and Brunnermeier [1]). In this case, the approach is based on the conditional scenario being a restriction for both financial institutions. Thus, in general, no relationship exists between the two CoVaRs. From now on, assume that X= (X1, . . . , Xd) is a non-negative absolutely-continuous random vector (with respect to Lebesgue measure λon Rd) with distribution function Fand survival function F. Furthermore, the multivariate distribution function Fis assumed to be partially strictly-increasing2such that E(Xi)<∞for i= 1, . . . , d. Such Fis said to verify the regularity 1We restrict ourselves to Rd +because, in our applications, components of d−dimensional vectors correspond to random losses and are then valued in R+. 2A function F(x1,...,xn) is partially strictly-increasing on Rd +\0if the function of one variable g(·) = F(x1,...,xj−1,·, xj+1,...,xd) are strictly-increasing. 5 conditions. Note that if Fis the survival function of X, and Fverifies the regularity conditions, then Fis a partially strictly-decreasing function. Unless stated otherwise, the dimension of the vectors is d, and the null vector of dimension dwill be denoted by 0, and the unity vector of dimension dby 1. Therefore, the order ≤between vectors will be considered component-wise. Throughout the paper, given a random variable or a vector Xand any event A,X|Ais denoted as the random variable or vector whose distribution is the conditional distribution of Xgiven A. Eventually, the equality in law is given by d =. Several useful definitions of stochastic orders are now recalled. Further details, equivalent definitions and applications may be found in Shaked and Shanthikumar [37], M¨uller [28], and Joe [22]. Definition 1.1. Let Xand Ybe two random variables with distribution functions FXand FY respectively. Xis said to be smaller than Yin the usual stochastic order, denoted by X≤st Y, if FX(x)≥FY(x),for all x∈R. Definition 1.2 (Supermodular function).A function f:Rd→Ris said to be supermodular if, for any x,y∈Rd, it satisfies f(x) + f(y)≤f(x∧y) + f(x∨y), where the operators ∧and ∨denote coordinate-wise minimum and maximum respectively. Definition 1.3 (Supermodular Order).Let Xand Ybe two d−dimensional random vectors. Xis said to be smaller than Ywith respect to the supermodular order (denoted by X≤sm Y) iff E(f(X)) ≤E(f(Y)), for all supermodular functions f:Rd→R, provided the expectations exist. In Definition 1, from the discussion above, a multivariate generalization of the CoVaR measure is now introduced. Definition 1 (Multivariate Lower-Orthant CoVaR).Consider a random vector Xwhich satisfies the regularity conditions. For α∈(0,1), we define the multivariate lower-orthant CoVaR at probability level αby CoVaRα,ω(X) = VaRω(X|X∈∂L(α)) =    VaRω1(X1|X∈∂L(α)) . . . VaRωd(Xd|X∈∂L(α))   ,(4) 6 where ω= (ω1, . . . , ωd)is a marginal risk vector with ωi∈[0,1], for i= 1, . . . , d, and ∂L(α)is the boundary of the set L(α) := {x∈Rd +:F(x)≥α}. Therefore, CoVaRα,ω(X) =    VaRω1(X1|F(X) = α) . . . VaRωd(Xd|F(X) = α)   .(5) In a similar way, the multivariate upper-orthant CoVaR can be defined. Definition 2 (Multivariate Upper-Orthant CoVaR).Consider a random vector Xwhich satisfies the regularity conditions. For α∈(0,1), we define the multivariate upper-orthant CoVaR at probability level αby CoVaRα,ω(X) = VaRω(X|X∈∂L(α)) =    VaRω1(X1|X∈∂L(α)) . . . VaRωd(Xd|X∈∂L(α))   ,(6) where ω= (ω1, . . . , ωd)is a marginal risk vector with ωi∈[0,1], for i= 1, . . . , d, and ∂L(α)is the boundary of the set L(α) := {x∈Rd +:F(x)≤1−α}. Therefore, CoVaRα,ω(X) =    VaRω1(X1|F(X)=1−α) . . . VaRωd(Xd|F(X) = 1 −α)   .(7) Remark 1.1. Using the same notation and framework of Definitions 1 and 2, we can also consider a modified version of the multivariate upper and lower CoVaR proposed in Equations (4) and (6). Indeed, consider a financial institution Xiand the firm network without Xi, i.e., (X1, . . . , Xi−1, Xi+1, . . . , Xd) := Xd−1. The following modified version of the lower CoVaR in Definition 1 can therefore be proposed: CoVaRi α,ω(X) = VaRωi(Xi|F(Xd−1) = α), where Fd−1is the (d−1)-dimensional distribution function associated to the vector Xd−1. Analogously, a modified version of the upper CoVaR in Definition 2 can be : CoVaRi α,ω(X) = VaRωi(Xi|F(Xd−1)=1−α), where Fd−1is the survival (d−1)-dimensional distribution function associated to the vector Xd−1. It should be borne in mind that, using this modified versions, when d= 2 and ωi=α, CoVaRα,ω(X)and CoVaRα,ω(X)become the classic CoVaR in (1). 7 The following interpretation of our measures can be considered. The ith component of multivariate lower-orthant CoVaR of X(resp. multivariate upper-orthant CoVaR of X) corresponds to the point x∗that minimizes the WL function of the associated ith marginal given that X stands in the α−level curve of its multivariate distribution function (resp. multivariate survival distribution function). It is worth mentioning that under regularity conditions, ∂L(α) (resp. ∂L(α)) is the α-level curve (resp. (1 −α)-level curve) of F(resp. F) (see for instance Di Bernardino et al. [12], Cuevas et al. [8]). This means that there is no plateau in the graph of Ffor each level α. Therefore, regularity conditions guarantee that the minimizer x∗is unique for each component i= 1, . . . , d. Trivially, given that our CoVaRs are the minimizers of suitable expected losses (see (P7)), they therefore verify the elicitability property. This property was studied by Gneiting [20], while Bellini and Bignozzi [4] suggested a slightly more restrictive definition. Recently, Embrechts and Hofert [13] stated that elicitability is a very important property of a risk measure since it provides a natural methodology to perform backtesting. Ziegel [41] has also studied the connections between elicitability and coherence properties of risk measures. Moreover, the solvency of an insurance company depends on the frequency of large claims. One of the advantages of working with the quantile function is that this function is more robust to extreme values than other central tendency measures. 2. Properties of the multivariate CoVaR In this section, the aim is to analyse the lower-orthant and upper-orthant CoVaR introduced in Definitions 1 and 2 in terms of classic suitable properties of risk measures (see, for instance, Artzner et al. [2], Denuit et al. [11]). We focus on invariance properties (see Section 2.1). Furthermore, in Section 2.2, the relationships between our CoVaR, the univariate VaR, and the multivariate VaR introduced by Cousin and Di Bernardino [6] are analysed. In Section 2.3, some comonotonic dependence properties for our measures are investigated. 2.1. Invariance properties The following results (Proposition 2.1 and Corollary 2.1) are now introduced, which will be central in proving invariance properties of our risk measures. Proposition 2.1. Let the function hbe such that h(x1, . . . , xd)=(h1(x1), . . . , hd(xd)). Let ω be a vector in [0,1]dand α∈(0,1). (1) If h1, . . . , hdare non-decreasing functions, then, for i= 1, . . . , d, CoVaRi α,ω(h(X)) = VaRωi(hi(Xi)|F(X) = α). 8 (2) If h1, . . . , hdare non-increasing functions, then, for i= 1, . . . , d, CoVaRi α,ω(h(X)) = VaRωi(hi(Xi)|F(X) = α). Proof. By Definition 1, CoVaRi α,ω(h(X)) = VaRωi(hi(Ti)) = arg min x∈[hi(VaRα(Xi)),+∞)ωiE[(hi(Ti)−x)+] + (1 −ωi)E[(hi(Ti)−x)−], where hi(Ti)=[hi(Xi)|Fh(X)(h(X)) = α], for i= 1, . . . , d. Since Fh(X)(y1, . . . , yd) = F(h−1 1(y1), . . . , h−1 d(yd)) if h1, . . . , hdare non-decreasing functions, F(h−1 1(y1), . . . , h−1 d(yd)) if h1, . . . , hdare non-increasing functions, then CoVaRi α,ω(h(X)) = VaRωi(hi(Xi)|F(X) = α) if h1, . . . , hdare non-decreasing functions, VaRωi(hi(Xi)|F(X) = α) if h1, . . . , hdare non-increasing functions. As in Proposition 2.1, a similar result can also be obtained for the multivariate upper-orthant CoVaR, by interchanging Fwith F. From Proposition 2.1, one can trivially obtain the following property which links the multivariate upper-orthant CoVaR and lower-orthant CoVaR. Corollary 2.1. Let hbe a linear function such that h(x1, . . . , xd) = (h1(x1), . . . , hd(xd)). Let ωbe a vector in [0,1]dand α∈(0,1). (1) If h1, . . . , hdare non-decreasing functions, then CoVaRα,ω(h(X)) = h(CoVaRα,ω(X)) and CoVaRα,ω(h(X)) = h(CoVaRα,ω(X)). (2) If h1, . . . , hdare non-increasing functions, then CoVaRα,ω(h(X)) = h(CoVaR1−α,1−ω(X)) and CoVaRα,ω(h(X)) = h(CoVaR1−α,1−ω(X)). The following result proves the positive homogeneity and invariance translation properties for risk measures in Definitions 1 and 2. 9 As a result, any random vector U= (U1, . . . , Ud) which follows an Archimedean copula with generator φcan be represented as a deterministic function of C(U) and an independent random vector S= (S1, . . . , Sd) uniformly distributed on the unit simplex, i.e., (U1, . . . , Ud)d = (φ−1(S1φ(C(U))), . . . , φ−1(Sdφ(C(U)))).(10) Corollary 4.1. Let Xbe a d-dimensional random vector with an Archimedean copula with generator φand α∈(0,1). Therefore, CoVaRi α,ω(X) = VaRωihF−1 Xi(φ−1(Siφ(α)))i,for i= 1, . . . , d, (11) where ω∈[0,1]dand Siis a random variable with Beta(1, d −1) distribution. Proof. Note that Xis distributed as (F−1 X1(U1), . . . , F−1 Xd(Ud)), where U= (U1, . . . , Ud) follows an Archimedean copula Cwith generator φ. Consequently, each component i= 1, . . . , d of the multivariate risk measure introduced in Definition 1 can be expressed as CoVaRi α,ω(X) = arg min x∈[VaRα(Xi),+∞)ωiE[(Ti−x)+] + (1 −ωi)E[(Ti−x)−], where Ti= [F−1 Xi(Ui)|C(U) = α]. Moreover, from representation (10), the following relation is verified [U|C(U) = α]d = (φ−1(S1φ(α)), . . . , φ−1(Sdφ(α))),(12) since Sand C(U) are stochastically independent. The result comes from the fact that the random vector Sfollows a symmetric Dirichlet distribution. Note that, by using (12), the marginal distributions of Ugiven C(U) = αcan be expressed in a very simple way, that is, P(Uk≤u|C(U) = α) = 1−φ(u) φ(α)d−1 for 0 <α<u<1,and any k= 1, . . . , d. (13) Corollary 4.2. Let Xbe a d-dimensional random vector with an Archimedean survival copula with generator φand α∈(0,1). Therefore, CoVaRi α,ω(X) = VaRωihF−1 Xi(φ−1(Siφ(1 −α)))ifor i= 1, . . . , d, (14) where ω∈[0,1]dand Siis a random variable with Beta(1, d −1) distribution. The proof is similar to Corollary 4.1 and is therefore omitted here. From (11) and (14), analytical expressions of the lower-orthant and the upper-orthant CoVaR for a vector X= (X1, . . . , Xd) with a particular Archimedean copula are now derived. Assume 16 that Xiis uniformly-distributed on [0,1], for i= 1, . . . , d. Since Archimedean copulas are exchangeable, the components of CoVaRα,ω(X) (resp. CoVaRα,ω(X)) are equal in the case where ω1=. . . =ωd. Furthermore, it is also possible to obtain expressions for the upperorthant CoVaRα,ωfor ˜ X= (1 −X1,...,1−Xd) since, by using Corollary 2.1: CoVaRi α,ω(˜ X)=1−CoVaRi 1−α,1−ω(X). 4.1. Analytical expressions of CoVaR measures for Archimedean copulas In the following, Corollary 4.1 is illustrated for some commonly used Archimedean copula families (see Example 4.1, 4.2, 4.3). Example 4.1 (Bivariate Clayton family).In Table 1 (left), the bivariate random vector (X, Y ) is considered with uniform marginal distributions and a Clayton copula with parameter θ≥ −1 is considered. One can readily show that ∂CoVaR1 α,ω ∂θ ≤0and ∂CoVaR1 α,ω ∂θ ≥0,for θ≥ −1, α ∈(0,1) and ω∈[0,1]. Hence, the components of the multivariate CoVaR (resp. CoVaR) are decreasing (resp. increasing) functions of the dependence parameter θ. Interestingly, in the comonotonic case, both multivariate risk measures CoVaR and CoVaR correspond to the vector composed of the univariate VaR at level αassociated with each component. These properties are illustrated in Figure 1 where upper and lower CoVaR are plotted as functions of the risk level ωfor different values of dependence parameter θand for a fixed level α. Note that, when the parameter θincreases, the lower CoVaR tends to decrease. Conversely, the upper bound for the upper CoVaR is represented by the perfect positive dependence case. The latter empirical behaviours will be formally confirmed in the following (see Corollary 4.4). θCoVaR1 α,ω,θ(X, Y ) (−1,∞)1 + 1 αθ−1(1 −ω1)−1/θ −1 1 −(1 −ω1)(1 −α) 0α1−ω1 1α (1−α)(1−ω1)+α ∞α θCoVaR1 α,ω,θ(X, Y ) [−1,1) 1−θ 1−θ(1−α) α(1−ω1)−θ 0α1−ω1 Table 1: CoVaR1 α,ω(X, Y ), for a bivariate Clayton copula (left) and a bivariate Ali-Mikhail-Haq copula (right). 17 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Lower−orthant CoVaR ω θ=−1 θ=0 θ=1 θ=5 θ=∞ 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Upper−orthant CoVaR ω θ=−1 θ=0 θ=1 θ=5 θ=∞ Figure 1: Behaviour of CoVaR1 α,ω(X, Y ) (left) and CoVaR1 α,ω(1 −X, 1−Y) (right) with respect to the risk level ωfor different values of dependence parameter θand for α= 0.7. Here, (X, Y ) is a bivariate random vector with uniform marginal distributions and a Clayton copula with parameter θ≥ −1. Example 4.2 (Bivariate Ali-Mikhail-Haq family).Table 1 (right) illustrates the analytical expressions of CoVaR for the first component of a bivariate random vector with uniform marginal distributions and a Ali-Mikhail-Haq copula, for θ∈[−1,1). Recall that bivariate Archimedean copulas can be extended to d−dimensional copulas, with d > 2, on the condition that the generator φis a d−monotone function in [0,∞) (see McNeil and Neˇslehov´a [26]). The bivariate Gumbel family can be generalized in dimension d, for θ≥1 (see Example 4.25 in Nelsen [31]). Example 4.3 (3−dimensional Gumbel family).In this case, analytical expressions of the first component of lower CoVaR of a 3−dimensional random vector (X1, X2, X3)with uniform marginal distributions and a Gumbel copula, for θ≥1are provided in Table 2. 18 θCoVaR1 α,ω,θ(X1, X2, X3) [1,∞)α(1−√ω1)1/θ 1α(1−√ω1) ∞α Table 2: CoVaR1 α,ω(X1, X2, X3) for a 3−dimensional Gumbel copula. 4.2. Illustrations of some properties for Archimedean copulas In the following, some theoretical properties presented in Section 2 are illustrated in the large class of d−dimensional Archimedean copula. Firstly, using Corollary 4.1, an illustration of Proposition 2.4 in the Clayton copula case is provided. Example 4.4. Assume that Xis a bivariate random vector with uniform marginal distributions and Clayton copula. The distribution function of Xis therefore given by: F(x1, x2) = hmax{x−θ 1+x−θ 2−1,0}i−1/θ ,for θ∈[−1,∞)\{0}and (x1, x2)∈[0,1]2. Then, by straightforward computation, one can obtain, for α∈(0,1) and ω1∈[0,1], VaR1 α(X) = θ θ−1 αθ−α αθ−1,and CoVaR1 α,ω(X) = 1 + 1 αθ−1(1 −ω1)−1/θ , where VaR1 α(X)is the first-component lower VaR proposed by Cousin and Di Bernardino [6]. Consequently, both measures coincide in ω∗=α−θ−θ θ−1 αθ−α αθ−1−θ[α−θ−1]−1. For a fixed α= 0.6we obtain the results gathered in Figure 2. VaRα(X)represents the case that the complete risk of the insurance company is reinsured by another company (x= 0) (see Cousin and Di Bernardino [6]). The insurance company gives the total weight to the expected cost of the reinsurance company, that is, establishes ω= 1. By contrast, CoVaR defines the minimum retention of the insurance company given a weight ω∈[0,1] for the expected cost of the reinsurance company. For instance, for θ= 2, it can be observed in Figure 2 that VaR1 0.6(X) = 0.75 and the cut-off point is ω∗= 0.56. Similarly, analytical expressions for multivariate upper CoVaR and comparisons with the associated VaRα(X)(see Cousin and Di Bernardino [6]) can be obtained. Corollary 4.3 proves that assumptions of Proposition 2.7 are automatically satisfied in the large class of d-dimensional Archimedean copulas. 19 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.7 0.8 0.9 1.0 Lower−orthant CoVaR ω θ=−0.99 θ=0.01 θ=2 θ=4 θ=10 Figure 2: VaR1 α(X) and CoVaR1 α,ω(X). Here, (X, Y ) is a bivariate random vector with uniform marginal distributions and a Clayton copula with parameter θ≥ −1, and α= 0.6. Corollary 4.3. Consider a d-dimensional random vector X, which satisfies the regularity conditions, with marginal distributions FXi, for i= 1, . . . , d, copula Cand survival copula C. (1) If Cis a d-dimensional Archimedean copula, then CoVaRi α,ω(X)is a non-decreasing function of αwith ω∈[0,1]d. (2) If Cis a d-dimensional Archimedean copula, then CoVaRi α,ω(X)is a non-decreasing function of αwith ω∈[0,1]d. Proof. Let Ui=FXi(Xi), U= (U1, . . . , Un), Vi=FXi(Xi) and V= (V1, . . . , Vn). Since C is the copula of X, then Uis distributed as C. If Cis an Archimedean copula, from (13), P(Ui> u|C(U) = α) is a non-decreasing function of α. Similarly, P(Vi> u|C(V) = 1 −α) is a non-decreasing function of α. The results are therefore trivially derived from Proposition 2.7. In the following, an illustration of Proposition 3.1 is provided in the Archimedean case. Example 4.5. Three different random vectors (X, Yi), for i= 1,...,3are considered with the same bivariate Clayton copula with dependence parameter 2, such that X∼Exp(1),Y1∼Exp(2),Y2∼Burr(5,1),Y3∼Fr´echet(4). Since Y1≤st Y2≤st Y3, from Proposition 3.1, then CoVaR2 α,ω(X, Y1)≤CoVaR2 α,ω(X, Y2)≤CoVaR2 α,ω(X, Y3), 20 for any ω∈[0,1]2and α∈(0,1). The results are gathered in Figure 3. It should also be emphasised that, by Corollary 3.1, the first components of the multivariate lower-orhant CoVaR and upper-orthant CoVaR for the four vectors coincide. 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Distribution Functions x F(x) Exp(2) Fréchet(4) Burr(5, 1) 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Lower−Orthant CoVaR ω Exp(2) Fréchet(4) Burr(5, 1) Figure 3: Left: Distribution functions of random variables Yi, for i= 1,...,3, with Y1∼Exp(2), Y2∼Burr(5,1) and Y3∼Fr´echet(4). Right: CoVaR2 α,ω(X, Yi) for i= 1,...,3, with the same copula Clayton with parameter 2, X∼Exp(1), Y1∼Exp(2), Y2∼Burr(5,1), Y3∼Fr´echet(4) and α= 0.8. The following remark will be useful in Corollary 4.4. Remark 4.1. Let Uand U∗be two random vectors with copula Cand C∗, respectively, and with uniform marginal distributions. It is easy to prove that U≤sm U∗implies C(u)≤C∗(u), for u∈[0,1]d(Section 6.3.3 in Denuit et al. [11]). In addition, for Gumbel, Frank, Clayton, and Ali-Mikhail-Haq families, it can be shown that an increase of θyields an increase of dependence in the sense of the supermodular order (see examples in Wei and Hu [39], Joe [22]). As a consequence, in these cases, θ≤θ∗⇒C(u)≤C∗(u), for u∈[0,1]d.(15) Corollary 4.4. Let Xbe a d−dimensional random vector satisfying the regularity conditions with copula Cand survival copula C. If Cis a d−dimensional Archimedean copula that satisfies Property (15) in Remark (4.1), each component of CoVaRα,ω(X)is a decreasing function of θ, with α∈(0,1) and ω∈[0,1]d. If Cis a d−dimensional Archimedean copula that satisfies Property (15) in Remark (4.1), each component of CoVaRα,ω(X)is a increasing function of θ, with α∈(0,1) and ω∈[0,1]d. It should be noted that, for instance for Gumbel, Frank, Clayton and Ali-Mikhail-Haq families, assumptions of Corollary 4.4 are satisfied. The reader is referred, for instance, to the behaviour of the lower and upper CoVaR with respect to the copula parameter θpresented in Figure 1. 21 Proof. We consider two Archimedean copulas of the same family, Cθ(associated to vector U) and Cθ∗(associated to vector U∗) with generator φθand φθ∗such that θ≤θ∗. By Proposition 3.2, we have to prove that [U∗ i|Cθ∗(U∗) = α]≤st [Ui|Cθ(U) = α] holds for i= 1, . . . , d. On the other hand, from Eq. (13), it is readily obtained that [U∗ i|Cθ∗(U∗) = α]≤st [Ui|Cθ(U) = α] for any α∈(0,1) ⇔φθ∗ φθ is a decreasing function. Finally, by taking into account Remark 4.1 for Clayton, Frank, Gumbel and Ali-Mikhail-Haq families, the function φθ∗ φθis decreasing when θ≤θ∗. Therefore, from Proposition 3.2, an increase of the parameter θyields a decrease in each component of CoVaRα,ω(X). The second statement is obtained trivially using the same arguments. 4.3. A weak subadditivity tail property in the Archimedean cases The additivity of our CoVaR is provided in Section 2.3 in a comonotonic dependence vectorial case (see Proposition 2.6 for π-comonotonic vectors). In the following, the aim is to study the condition for a copula to obtain subadditivity inequalities for our lower CoVaR . To this end, as in the univariate case (see Dan´ıelsson et al. [9]), we focus on the tails of the considered multivariate distribution. In the following, two notions of regular variation are applied. A measurable function U:R→R is regularly varying at ∞with index ρ(denoted by U∈RVρ), if it holds that lim t→∞ U(tx) U(t)=xρ, for any real number x > 0. Also, a random vector Xwith joint distribution function Fis said to be multivariate regularly varying (X∈MRV ) if there exists a Radon measure νon [0,∞]\{0}, such that lim t→∞ 1−F(tx) 1−F(t1)=ν([0,x]c), for all points x∈[0,∞)\{0}, which are continuity points of the function ν([0,·]c). Observe also that for any non-negative MRV random vector X, its non-degenerate univariate margins Xi have regularly varying right tails, that is, Fi(t) := t−βL(t), t ≥0, where β > 0 is the marginal heavy-tail index and L(t) is a slowly varying function, i.e. L(x t)/L(t)→1 as t→ ∞ for any x > 0. Further details about regular variation can be found in Resnick [34], Resnick [35] and Embrechts et al. [14]. Therefore in this setting, the following result can be obtained. From now on, the following notation is considered. Let Xbe a bivariate random vector with distribution function F, Archimedean copula Cand with same margins FXi,i= 1,2. Let us denote Ti= [Xi|F(X) = α], for α∈(0,1), i= 1,2. Theorem 4.1. Assume that φis twice differentiable and that (φ◦FX1)∈RV−β,β > 0. Then T:= (T1, T2)∈MRV . 22 Proof. Firstly, the copula of random vector Tis computed. Note that F(x1, x2) = φ−1(φ(FX1(x1)) + φ(FX2(x2))). For simplicity, the univariate random variable F(X1, X2) is denoted by V. Similarly to Theorem 1 in Wang and Oakes [38], we obtain P[V≤α, X1≤x1, X2≤x2] = (α−φ(α) φ0(α)+φ(F(x1,x2)) φ0(α),if 0 < α ≤F(x1, x2); 0,if α > F(x1, x2). (16) By straightforward calculation, it can be shown that the distribution function of Tis defined as FT(x1, x2) = (P[V=α,X1≤x1,X2≤x2] P(V=α),if 0 < α ≤F(x1, x2); 0,if α > F(x1, x2), =(1−φ(F(x1,x2)) φ(α),if 0 < α ≤F(x1, x2); 0,if α > F(x1, x2), (17) where P(V=α) is the density in αof random variable V. On the other hand, for i= 1,2, FTi(xi) = (1−φ(FXi(xi)) φ(α),if α≤FXi(xi); 0,if α > FXi(xi), and F−1 Ti(wi) = ((φ◦FXi)−1(φ(α)(1 −wi)),if 0 < wi≤1; 0,if wi= 0. Therefore, the copula of the random vector Tis CT(u1, u2) = FT(F−1 T1(u1), F−1 T2(u2)) = (u1+u2−1,if u1+u2≥1; 0,otherwise. It is now shown that T∈MRV by Theorem 3.2 in Weng and Zhang [40]. Therefore, conditions (C1) and (C2) of Theorem 3.2 in Weng and Zhang [40] are proved. As a result of that (φ◦FX1)∈ RV−β,β > 0, we trivially obtain FT1∈RV−β,β > 0 (C1). In addition, since Xhas the same margins then, lim t→∞ FT2(t) FT1(t)= 1, 23 that is, FT1and FT2have equivalent tails. (C2) Finally, the lower tail dependence function of the survival copula of T, λ2(u1, u2) = lim t→0+ CT(tu1, tu2) t, is equal to 0. Due to that and considering (C1) and (C2), by Theorem 3.2 in Weng and Zhang [40], T∈MRV . Remark 4.2. Note that, if (φ◦FX1)∈RV−β,β > 1, by applying Theorem 4.1 and Proposition 1 in Dan´ıelsson et al. [9] for T, then the VaR of Tis subadditive sufficiently deep in the tail regions. In this case, a weak subadditivity of the proposed multivariate lower CoVaR is obtained, that is, since VaRω(Ti) = CoVaRi α,ω(X), then VaRω(T1+T2)<CoVaR1 α,ω(X) + CoVaR2 α,ω(X) (18) sufficiently deep in tail regions. Now, an illustration of Remark 4.2 is presented (see Figure 4 and Example 4.6 below). 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 CoVaR1ω, α(X)+CoVaR2ω, α(X) and VaRω(T1+T2) ω CoVaR1ω, α(X)+CoVaR2ω, α(X) VaRω(T1+T2) 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 CoVaR1ω, α(X)+CoVaR2ω, α(X) and VaRω(T1+T2) ω CoVaR1ω, α(X)+CoVaR2ω, α(X) VaRω(T1+T2) Figure 4: CoVaR1 α,ω(X) + CoVaR2 α,ω(X) and VaRω(T1+T2) for Xwith X1∼X2∼Pareto(2) and a Gumbel copula with θ= 2, as in Example 4.6, for α=ω(left panel) and for α= 0.75 (right panel). Example 4.6. In this example, a bivariate random vector, X, with X1∼X2∼Pareto(2) and a Gumbel copula, θ= 2, is considered. Analytical expressions of CoVaRi α,ω(X),i= 1,2are 24 obtained. In addition, VaRω(T1+T2)is calculated by numeric approximation. The obtained results are gathered in Figure 4: for ω=α∈(0,1) (see Figure 4, left) and for α= 0.75, ω∈(0,1) (see Figure 4, right). It can be easily observed that (18) is verified for large ω. 5. Estimation Semiparametric estimators by assuming Archimedean copula for the proposed multivariate CoVaRs are given in this section. Moreover, illustrations with simulated and insurance real data are provided. Firstly, let assume that Xhas an Archimedean copula structure. The generator of an Archimedean copula depends on the dependence parameter θof the copula (see, e.g., Table 4.1. in Nelsen [31]). Consequently, a semiparametric estimator of the generator is obtained by considering a maximum pseudo-likelihood estimator of the dependence parameter θassociated with this generator. Following these considerations and using Equation (11), we introduce a semiparametric estimator for the multivariate lower CoVaR (see Definition 5.1) by using a semiparametric estimation for θand the empirical quantile estimation. Definition 5.1. Let Xbe a d−dimensional random vector with Archimedean copula with generator φθand α∈(0,1). A semiparametric estimator of the i−component of the multivariate lower CoVaR is defined as \ CoVaRi α,ω(X) = d VaRωihˆ F−1 Xi(φ−1 ˆ θn(Siφˆ θn(α)))i,for i= 1, . . . , d, (19) where ω∈[0,1]d,Siis a random variable with Beta(1, d −1) distribution, d VaRω(X)is the empirical estimator of VaRω(X),φˆ θnis the semiparametric estimator of φθand ˆ F−1 Xiis the empirical estimator of F−1 Xifor i= 1, . . . , d. Secondly, let assume that Xhas an Archimedean survival copula structure. From Equation (14), we introduce a semiparametric estimation of multivariate upper CoVaR (see Definition 5.2) using the semiparametric estimation of the generator of the Archimedean survival copula and the empirical estimation of the quantile functions. Definition 5.2. Let Xbe a d−dimensional random vector with Archimedean survival copula with generator φθand α∈(0,1). A semiparametric estimator of the i−component of the multivariate upper CoVaR is defined as \ CoVaR i α,ω(X) = d VaRωiˆ F−1 Xi(φ−1 ˆ θn(Siφˆ θn(1 −α))),for i= 1, . . . , d, (20) where ω∈[0,1]d,Siis a random variable with Beta(1, d −1) distribution, d VaRω(X)is the empirical estimator of VaRω(X),φˆ θnis the semiparametric estimator of φθand ˆ F−1 Xithe empirical estimator of F−1 Xifor i= 1, . . . , d. 25 2 4 6 8 10 12 14 4 6 8 10 12 Loss ALAE data in logarithmic scale Loss ALAE 2 4 6 8 10 12 14 4 6 8 10 12 Loss ALAE data in logarithmic scale Loss ALAE 2 4 6 8 10 12 14 4 6 8 10 12 Loss ALAE data in logarithmic scale Loss ALAE Figure 9: Loss ALAE data in log scale, boundary of estimated level sets (∂L(α), red line), boundary of estimated level sets (∂L(α), blue line), empirical quantile of Loss data (dotted black line), empirical quantile of ALAE data (dotted black line), \ CoVaRα,ω (stars) and \ CoVaRα,ω (solid circles) with (α= 0.75, ω = 0.9) (left panel); (α= 0.9, ω = 0.95) (center panel); (α= 0.95, ω = 0.98) (right panel). The positive homogeneity and translation invariance properties are shown for the two proposed multivariate CoVaR. The relations between the univariate VaR and our CoVaR are also analysed as well as the relations between the multivariate VaR proposed by Cousin and Di Bernardino [6] and our multivariate CoVaR. Interestingly, both multivariate CoVaRs coincide with the univariate VaR when a comonotonic random vector is considered, and they verify the additivity property under π-comonotonic conditions. The behaviour of the multivariate CoVaR with respect to the risk level, the usual stochastic order of marginal distributions, and the dependence structure are studied. Unsurprisingly, the effect in the multivariate lower CoVaR (resp. upper CoVaR) with respect to a change in the risk level, a change in the dependence structure, or the usual stochastic order of marginal distributions, tends to be the same as for the multivariate lower VaR (resp. upper VaR) proposed in Cousin and Di Bernardino [6]. Important results and analytical expressions for our multivariate risk measures are obtained for random vectors with Archimedean copulas. In particular, certain subadditivity inequality is presented in the Archimedean case under regular variation conditions. Moreover, under Archimedean copula condition, estimators of the two proposed multivariate CoVaRs are provided in simulated data and insurance real data. In a future perspective, quantile regression estimations in extreme theory for the two multivariate CoVaRs can be studied by adapting the works by Di Bernardino et al. [12] and by Daouia et al. [10]. 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