Tail Approximation of Value-at-Risk under Multivariate Regular Variation

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1 Tail Approximation of Value-at-Risk under Multivariate Regular Variation Yannan Sun Haijun Li July 00 Abstract This paper presents a general tail approximation method for evaluating the Valueat-Risk of any norm of random vectors with multivariate regularly varying distributions. The main result is derived using the relation between the intensity measure of multivariate regular variation and tail dependence function of the underlying copula, and in particular extends the tail approximation discussed in Embrechts et al [6] for Archimedean copulas. The explicit tail approximations for random vectors with Archimedean copulas and multivariate Pareto distributions are also presented to illustrate the results. Keywords and phrases: Value-at-Risk, copula, tail dependence function, multivariate regular variation, Archimedean copulas, risk management. Introduction Value-at-Risk (VaR is one of the most widely used risk measures in financial risk management [6]. Given a non-negative random variable X representing loss, the VaR at confidence level p, 0 < p <, is defined as the p-th quantile of the loss distribution: VaR p (X := inf{t R : Pr{X > t} p}. In risk analysis for multivariate portfolios, we are often more interested in calculating the VaR for aggregated data than for a single loss variable. For a non-negative random vector X = (X,..., X d representing various losses in a multivariate portfolio, we need to {ysun, lih}@math.wsu.edu, Department of Mathematics, Washington State University, Pullman, WA 9964, U.S.A.. Research is supported in part by NSF grant CMMI

2 calculate VaR p ( X where the norm determines the manner in which the data are aggregated. Calculating VaR p ( X is in general a difficult problem, but the tail estimates for VaR p ( X as p are often tractable for various multivariate loss distributions. Good tail approximations of VaR p ( X as p can be used to accurately estimate the risk, as measured by the VaR, for extreme losses. Tail asymptotics of VaR p ( d i= X i, as p, for loss vectors with Archimedean copulas and regularly varying margins are obtained in [,, 3,, 6]. These tail estimates for the VaR of sums are asymptotically linear functions of the VaR of the univariate margin, with a proportionality constant that depends on the tail dependence of the underlying Archimedean copula and marginal heavy-tail index. Such asymptotic relations can be also used to analyze the structural properties of the VaR of aggregated sums for multivariate portfolios, such as the subadditivity property of the VaR as well as the property on how the dependence and marginal parameters would affect estimates of extreme risks. In this paper, we develop a general and more unified approach to derive tail asymptotics of VaR p ( X, as p, for loss vectors that have multivariate regularly varying distributions. Our method is based on the link between multivariate regular variation and tail dependence functions of copulas, and the tail estimates obtained previously in the literature can be obtained from our tail asymptotics as special cases. The paper is organized as follows. In Section, we define copulas, tail dependence functions and multivariate regular variation. The relationship between tail dependence functions and multivariate regular variation obtained in [4] is highlighted. In Section 3, we give a tail approximation of VaRs for loss vectors that follow multivariate regularly varying distributions. Explicit approximations for Archimedean copulas, Archimedean survival copulas and multivariate Pareto distributions are also presented. For notational convenience, we denote hereafter by [a, b] the Cartesian product d i= [a i, b i ] where a, b R d := [, ] d and a i b i for each i. Also denote vector (w, p..., w p d by wp where p > 0, w i 0 for each i, and I := {,..., d} denotes the index set. The maximum and minimum of a and b are denoted as a b and a b respectively. Tail Dependence of Multivariate Regular Variation Let X = (X,..., X d be a non-negative random vector with distribution F. For any norm on R d, tail estimates for VaR p ( X as p boil down to finding the limit of Pr{ X > t}/ Pr{X i > t} as t, i d, for which the multivariate regular variation suits well. The following definition can be found in [7] pages (also see [4].

3 Definition.. A random vector X or its distribution F is said to be multivariate regularly varying (MRV if there exists a Radon measure µ (i.e., the measure is finite on compact sets, called the intensity measure, on R d \{0} such that lim t Pr{X tb} Pr{ X > t} = µ(b, (. for any relatively compact set B R d \{0} that satisfies µ( B = 0. Note that here R d is compact and the punctured version R d \{0} is modified via one-point uncompactification. It must be emphasized that the intensity measure µ in (. depends on the choice of norm, but intensity measures for any two different norms are proportional and thus equivalent. Observe also that for any non-negative MRV random vector X, its non-degenerate univariate margins X i have regularly varying right tails, that is, F i (t := Pr{X i > t} = t β L(t, t 0, (. where β > 0 is the marginal heavy-tail index and L(t is a slowly varying function with L(xt/L(t as t for any x > 0. The multivariate regular variation defined in Definition. describes multivariate heavy tail phenomena. The detailed discussions on multivariate regular variation and its various applications can be found in [9, 0]. All the extremal dependence information of an MRV vector X is encoded in the intensity measure µ, which can be further decomposed, using the copula approach, into the rank-invariant tail dependence and marginal heavy-tail index. A copula C is a distribution function, defined on the unit cube [0, ] d, with uniform one-dimensional margins. Given a copula C, if one defines F (t,..., t d := C(F (t,..., F d (t d, (t,..., t d R d, (.3 then F is a multivariate distribution with univariate margins F,..., F d. Given a distribution F with margins F,..., F d, there exists a copula C such that (.3 holds. If F,..., F d are all continuous, then the corresponding copula C is unique and can be written as C(u,..., u d = F (F (u,..., F d (u d, (u,..., u d [0, ] d. Thus, for multivariate distributions with continuous margins, the univariate margins and the multivariate rank-invariant dependence structure (as described by their copulas can be separated [7]. 3

4 Using copulas we can define tail dependence functions. Let X = (X,..., X d have continuous margins F,..., F d and copula C. The lower and upper tail dependence functions, denoted by b( and b ( respectively, are introduced in [0,, 8] as follows, C(uw j, j I b(w := lim, w = (w,..., w d R d u 0 u ; b C( uw j, j I (w := lim, w = (w,..., w d R d u 0 u, (.4 provided that the limits exist, where C denotes the survival function of C. Since the lower tail dependence function of a copula is the upper tail dependence function of its survival copula, we only focus on upper tail dependence in this paper. Define the upper exponent function a ( of copula C as a (w := S I,S ( S b S(w i, i S, (.5 where b S (w i, i S denotes the upper tail dependence function of the margin of C with component indices in S. It is shown in [0] that tail dependence functions {b S (w i, i S} and the exponent function a (w are uniquely determined from each other. The link between tail dependence functions and multivariate regular variation is obtained in [4] as follows. Theorem.. Let X = (X,..., X d be a non-negative MRV random vector with intensity measure µ, copula C and continuous margins F,..., F d. If the margins are tail equivalent (i.e. F i (t/f j (t as t for any i j with heavy-tail index β > 0 (see (., then the upper tail dependence function b ( exists and. b (w = µ([w /β, ] and a (w = µ(([0,w /β ]c µ([, ] R d µ(([0,] R d c ;. µ([w, ] = b (w β and µ([0,w]c = a (w β. µ([0,] c a ( µ([0,] c a ( Therefore, the tail dependence function and intensity measure are equivalent in the sense that the Radon measure generated by the tail dependence function is a rescaled version of the intensity measure with marginal scaling functions being of Pareto type. In contrast, the tail dependence function describes the rank-invariant extremal dependence extracted from the intensity measure. Since intensity measures for any two different norms are proportionally related, the rescaled intensity measures, such as µ( /µ([, ] R d and µ( /µ([0, ] c, do not depend on the choice of norms. 4

5 3 Tail Approximation In this section, we derive the tail asymptotics for VaR p ( X, as p, for any fixed norm on R d. The results discussed in [6] can be obtained by taking the l -norm and the tail dependence function of Archimedean copulas. Consider a non-negative MRV random vector X = (X,..., X d with intensity measure µ, joint distribution F and margins F,..., F d that are tail equivalent with heavy-tail index β > 0. Without loss of generality, we use F to define the following limit: q (β, b Pr{ X > t} := lim, (3. t F (t where b denotes the upper tail dependence function of X. This limiting constant depends on the intensity measure µ, which in turn depends on heavy-tail index β, tail dependence function b and norm. Theorem 3.. If F is absolutely continuous and the partial derivative d b (v/ v v d exists everywhere, then q (β, b has the following representation q (β, b = ( d d b (w β d b (v dw = dv, (3. w w d v v d W W /β where W = {w 0 : w > } and W /β = {v 0 : v /β > }. Proof. It follows from (. that Pr{ X > t} Pr{X > t} Pr{X tw } = Pr{X tw } µ(w, as t, µ(w where W = {w : w > }, and W = {w : w > }. Let µ( := µ( /µ(w, then by Theorem. (, µ( is the measure generated by µ([w, ] = b (w β. Thus the partial derivative µ (w := ( d d d µ([w, ] dw dw d µ([, ] R d = db (w β ( d w w d exists everywhere for w > 0. Since F is absolutely continuous, µ( is absolutely continuous with respect to the Lebesgue measure, it follows from the Radon-Nikodym Theorem that q (β, b = µ(w µ(w = µ (w dw, and by the uniqueness of the Radon-Nikodym derivative, µ (w 0 is the Radon-Nikodym derivative of the intensity measure µ with respect to the Lebesgue measure. Therefore, using Theorem. ( we obtain the first expression in (3.. The second expression in (3. follows by taking variable substitutions. 5 W

6 Remark 3.. µ (w that. It follows from the non-negativity of the Radon-Nikodym derivative 0 β d d i= w β i µ (w = d b (v v v d, with v i = w β i, w i > 0, i d, which shows that d b (v/ v v d 0 for all v 0. It is easy to see that W /β W /β for any β β, and thus by (3., q (β, b is non-decreasing in β. This extends Theorem.5 in [] to multivariate regular variation with respect to any norm.. If we choose to be the l -norm, i.e., w = max{w i, i =,..., d}, then W /β = {w 0 : d i=w i < }. In this case, q (β, b is independent of β and depends on the tail dependence function only. The limiting proportionality constant q (β, b takes particularly tractable forms for Archimedean copulas as the following corollaries show. Corollary 3.3. (Archimedean Copula Consider a random vector (X,..., X d which satisfies the assumptions of Theorem 3.. Assume that (X,..., X d has an Archimedean copula C(u,..., u d = ψ ( d i= ψ(u i, where the generator ψ is regularly varying at with tail index α >. Then [ ] q (β, b = W d ( S ( v α α j dv dv d, (3.3 v /β v d where W /β = {v : v β > }. S I,S Proof. From [0], the upper tail dependence function b (v can be expressed by the upper exponent functions of the margins: b (v = S I,S j S ( S a S(v i, i S, where a S (v i, i S = lim vi 0,i/ S a (v is the upper exponent function of the margin of C with component indices in S. It is given in [8] that the upper exponent function of Archimedean copula is a (v = ( d j= vα j α. Therefore, a S (v i, i S = ( j S vα j α and the result follows. Corollary 3.4. (Archimedean Survival Copula Consider a random vector (X,..., X d which satisfies the assumptions of Theorem 3.. Assume that ( X,..., X d has an 6

7 Archimedean copula C(u,..., u d = φ ( d i= φ(u i, where the generator φ is regularly varying at 0 with tail index α > 0. Then q (β, b = where B = {x : x > }. d ( d B x x d i= x αβ α i dx dx d, (3.4 Proof. It is given in [0] that the upper tail dependence function of ( X,..., X d is b (w = ( d α. Therefore, j= w α j q (β, b = ( d W d b (w β dw = ( d w w d W d ( d w w d j= w αβ α j dw where W = {w : w > }. If we substitute x i for w i, then we obtain (3.4. Remark In Corollary 3.3, if we choose to be the l -norm, then the explicit expression of the limiting constant can be easily computed for Archimedean copulas. Take the bivariate case for example, q (β, b = W /β v v [v v (v α v α α ] dv dv where W /β = {v : v β v β > } = {v : 0 < v v < }. Hence, q (β, b = ( v v [v v (v α v α α ] dv dv = [v v (v α v α α ] v =0 v =0 [v v (v α v α α ] v =0 v =0 = α which only depends on the tail dependence of the Archimedean copula.. In Corollary 3.4, if we choose to be the l -norm, i.e., x = d i= x i for nonnegative x i, i =,..., d, then the result is reduced to the limiting constant obtained in Proposition. of [6]. Using the l -norm is the most common way to aggregate data. The monotonicity property of q (β, b with respect to α for d is also given in [6]: q (β, b is increasing in α when β > ; q (β, b is decreasing in α when β < ; q (β, b = d when β =. It is further shown that for α > 0 and β > (β <, VaR is asymptotically subadditive (superadditive, i.e., VaR p ( d i= X i < d i= VaR p(x i (VaR p ( d i= X i > d i= VaR p(x i for p near. 7

8 Heavy-tailed scale mixtures of multivariate exponential distributions also have tractable tail dependence functions [3, 4] and thus the limiting constant q (β, b can be calculated using Theorem 3. for these distributions. Consider the following multivariate Pareto distribution: X = (X,..., X d = ( T Z,..., T d, (3.5 Z where Z is a positive random variable and the random vector (T,..., T d, independent of Z, has a min-stable multivariate exponential distribution with identical margins, that is, the distribution G of (T,..., T n satisfies that for all w R d, min{t /w,..., T d /w d } has an exponential distribution (see pages of [9] for details. The well-known Pickands representation shows that the survival function G can be written as G(t = exp{ P (t}, where P (t := d i=s i t i U(ds, S d is called the Pickands dependence function, and S d := {s 0 : d i= s i = }, U is a finite measure on S d and S d s i U(ds = λ for i d. Observe that P (t is homogeneous of order, that is, P (ct = cp (t for any c 0. Let Z have the Laplace transform ϕ, and then the survival function F of (3.5 can be written as F (x = E(e P (Zx = E(e ZP (x = ϕ(p (x, with the marginal survival function F i (x = ϕ(λx, i d. Assume that ϕ(x, and thus F i (x, is regularly varying at with heavy-tail index β > 0 in the sense of (.. The upper tail dependence function of X in (3.5 can be calculated explicitly as follows, b F (tw (w = lim t F i (t = lim ϕ(tp (w t ϕ(λt ( β P (w = (3.6 λ For example, if Z in (3.5 has the gamma distribution with shape parameter (Pareto index β > 0 and scale parameter, then ϕ(x = (x β. Note that in this case R = /Z has an inverse gamma distribution with shape parameter β > 0 and scale parameter. It is known that R is regularly varying with heavy-tail index β and its survival function is given by Pr{R > r} = Γ(β, r, r > 0, β > 0, Γ(β where Γ(β, = t β e t dt is known as the upper incomplete gamma function and Γ(β = r r 0 t β e t dt is the gamma function. 8

9 If the distribution G of (T,..., T d in (3.5 is absolutely continuous, then the d-th order partial derivative of the Pickands dependence function P (w exists, and thus the limit q (β, b for such X can be obtained by using Theorem 3.. Note that if T,..., T d in (3.5 are independent, then the survival copula of X is Archimedean, and thus (3.5 includes Archimedean copulas as a special case. In contrast to Archimedean copulas, (3.5 models local dependence of T,..., T d on top of the global dependence described by the mixture. Example 3.6. Let (T,..., T d have the following joint distribution G(t := i<j d where H i (t = exp{ αt}, i d, and K ij (H i (t i, H j (t j d H i (t i, K ij (u, u ; α ij = exp { [( log u α ij ( log u α ij ] /α ij }, 0 u, u is known as the extreme-value copula with dependence parameter α ij [8]. Marginally, T i has the exponential distribution G i (t = exp{ λt} where λ = αd. Jointly, T i and T j are coupled with copula K ij. Clearly, the Pickands dependence function of G is given by [ ] d P (t = log G(t = α (t α ij t α ij j /α ij t i. It then follows from (3.6 that b (w = d β [ i<j d i<j d (w α ij i i i= w α ij j /α ij i= ] β d w i. i= The limit q (β, b can be obtained by using Theorem 3.. Note that the absolute continuity of F in Theorem 3. is also necessary for (3.. Here we give the bivariate case as an example. Example 3.7. (Bivariate Pareto Distribution of Marshall-Olkin Type The Marshall-Olkin distribution with rate parameters {λ, λ, λ } is the joint distribution of T := E E, T := E E, where {E S, S {, }} is a sequence of independent exponentially distributed random variables, with E S having mean /λ S. In the reliability context, T, T can be viewed as the lifetime of two components operating in a random shock environment where a fatal shock governed by the Poisson process with rate λ S destroys all the components with indices in S {, } simultaneously [5]. In credit-risk modeling, 9

10 T, T can be viewed as the times-to-default of various different counterparties or types of counterparty, for which the Poisson shocks might be a variety of underlying economic events [5]. Let (RT, RT be defined as in (3.5, where R = /Z and Z follows a gamma distribution with shape parameter β > 0 and scale parameter. From [4], the bivariate distribution of (RT, RT is regularly varying, and its upper tail dependence function is given by ( b w T β w T β (w, w = E. (3.7 E(T β E(T β Since T i is exponentially distributed with mean /(λ i λ for i =,, we have ET β i = β!(/(λ i λ β for any positive integer β, thus b (w, w = β! E[w/β (λ λ T w /β (λ λ T ] β. Consider { } Pr [w /β (λ λ T w /β (λ λ T ] β > t { } = Pr (λ λ T > (t/w β, (λ λ T > (t/w β = Pr { T > (λ λ (t/w β, T > (λ λ (t/w β = e ( λ λ (λ λ w /β (λ λ w /β λ (λ λ w /β (λ λ w /β t β. } Thus, for any positive integer β, = β! ( = b (w, w = β! 0 βs β e λ (λ λ w /β ( e λ (λ λ w /β 0 ( λ λ (λ λ w /β (λ λ w /β λ (λ λ w /β λ (λ λ w /β λ (λ λ w /β (λ λ w /β λ (λ λ w /β (λ λ w /β λ s ds (λ λ w /β (λ λ w /β t β dt β. (3.8 Observing the fact that b is not differentiable at (w, w with (λ λ w /β = (λ λ w /β. Consider a simple case where λ = λ = 0 and λ > 0 and β > 0. In this case T = T and b (w, w = w w. Observe that b (w, w / w w = 0 for any w w, and thus 0

11 it follows from (3. that q (β, b = 0 for any norm. On the other hand, however, it follows from (3. that q (β, b = lim t Pr{RT > t} Pr{RT > t} = β 0, and thus (3. does not hold. Remark 3.8. In Example 3.7, if we set λ = 0, then T i = E i, i =,. In this case, it is known that the survival copula of (RT, RT is a bivariate Clayton copula. On the other hand, equation (3.8 becomes b (w, w = (w /β w /β β. This is the upper tail dependence function of ( X, X from the bivariate case of Corollary 3.4, where the dependence parameter is /β. The next theorem, an application of Theorem 3., provides a method of obtaining VaR approximations in higher dimensions from one-dimensional VaR. Theorem 3.9. Consider a non-negative random vector X = (X,..., X d with MRV distribution function F and continuous margins F,..., F d, d. Assume that the margins are tail equivalent with heavy-tail index β > 0, and the tail dependence function b > 0. Then VaR p ( X lim p VaR p (X Proof. Let G be the distribution function of X. From (3., = q (β, b β. (3.9 G(t F (t q (β, b as t, i.e. F (t q (β, b G(t, where denotes tail equivalence. Hence we have t F (q (β, b G(t. Define u := G(t. Then G (u F (q (β, b u. Since F is regularly varying at with heavy-tail index β > 0, we have from Proposition.6 of [0] that F (t is regularly varying at 0, or more precisely, F (uc/f (u c β as u 0 for any c > 0. Thus F (q (β, b u/f (u q (β, b β. Therefore, G (u q (β, b β F (u, i.e., lim u 0 G (u/f (u = q (β, b β. Replace u by p, then we have (3.9. This result gives an asymptotic estimate of VaR p ( X which does not generally have a closed form expression. The closed forms of tail dependence functions occur more often than for explicit copula expressions. This is because we can obtain tail dependence functions from either explicit copula expressions or the closure properties of the related conditional distributions whose parent distributions do not have explicit copulas. Using Theorems 3. and 3.9, when the confidence level is close to and the tail dependence functions exist, we can take q (β, b β VaRp (X as an approximation to the VaR of the normed (aggregated loss.

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