Tail Risk of Multivariate Regular Variation

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1 Tail Risk of Multivariate Regular Variation Harry Joe Haijun Li Third Revision, May 21 Abstract Tail risk refers to the risk associated with extreme values and is often affected by extremal dependence among multivariate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. Asymptotic expressions for tail risk are established in terms of the intensity measure that characterizes multivariate regular variation. Tractable bounds for tail risk are derived in terms of the tail dependence function that describes extremal dependence. Various examples involving Archimedean copulas are presented to illustrate the results and quality of the bounds. Key words and phrases: Coherent risk, tail conditional expectation, regularly varying, copula, tail dependence. MSC2 classification: 62H2, 91B3. 1 Introduction The performance (gain or loss, etc.) of a financial portfolio at the end of a given period is often evaluated by a real-valued random variable X. A risk measure ϱ is defined as a measurable mapping, with some coherency principles, from the space of all the performance variables into R [28], and these coherency principles provide a set of operational axioms that ϱ should satisfy in order to accurately characterize risky behaviors of portfolios. The coherent risk measure, introduced in [5] for analyzing economic risk of financial portfolios, is an example of such an axiomatic approach. Let L be the convex cone 1 consisting of all the performance variables which represent losses of financial portfolios at the end of a given period. Note that X, where X L, represents the net worth of a financial position. A mapping ϱ : L R is called a coherent risk measure if ϱ satisfies the following four economically coherent axioms: harry@stat.ubc.ca, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada. This author is supported by NSERC Discovery Grant. lih@math.wsu.edu, Department of Mathematics, Washington State University, Pullman, WA 99164, U.S.A. This author is supported in part by NSF grant CMMI A subset L of a linear space is a convex cone if x 1 L and x 2 L imply that λ 1x 1 + λ 2x 2 L for any λ 1 > and λ 2 >. A convex cone is called salient if it does not contain both x and x for any non-zero vector x. 1

2 1. (monotonicity) For X 1, X 2 L with X 1 X 2 almost surely, ϱ(x 1 ) ϱ(x 2 ). 2. (subadditivity) For all X 1, X 2 L, ϱ(x 1 + X 2 ) ϱ(x 1 ) + ϱ(x 2 ). 3. (positive homogeneity) For all X L and every λ >, ϱ(λx) = λϱ(x). 4. (translation invariance) For all X L and every l R, ϱ(x + l) = ϱ(x) + l. The interpretations of these axioms have been well documented in the literature (see, e.g., [28] for details), and risk ϱ(x) for loss X corresponds to the amount of extra capital requirement that has to be invested in some secure instrument so that the resulting position ϱ(x) X is acceptable to regulators/supervisors. The general theory of coherent risk measures was developed for arbitrary real random variables in [12], and the convex measures that combine subadditivity and positive homogeneity into the convexity property were extended to càdlàg processes in [9], and to abstract spaces in [14] that include deterministic, stochastic, single or multi-period cash-stream structures. It follows from the duality theory that any coherent risk measure ϱ(x) arises as the supremum of expected values of X, taken over with respect to a convex set of probability measures on environmental states, all of them being absolutely continuous with respect to the underlying physical measure. If the set is taken to be the set of all conditional probability measures conditioning on events with probability greater than or equal to p, < p < 1, then the corresponding coherent risk measure is known as the worst conditional expectation W CE p (X), which, in the case that loss variable X is continuous, equals to the tail conditional expectation (TCE) defined as follows, TCE p (X) := E(X X > VaR p (X)), (1.1) where VaR p (X) := inf{x R : Pr{X > x} 1 p} is known as the Value-at-Risk (VaR) with confidence level p (i.e., p-quantile). The VaR has been widely used in risk management, but it violates the subadditivity of coherency on convex cone L and often underestimates risks. Although VaR is coherent on a much smaller convex cone consisting of only linearized portfolio losses from elliptically distributed risk factors, the non-subadditivity of VaR can occur in the situations where portfolio losses are skewed or heavy-tailed with asymmetric dependence structures [28]. It can be shown that for continuous losses, TCE is the average of VaR over all confidence levels greater than p, focusing more than VaR does on extremal losses. Thus, TCE is more conservative than VaR at the same level of confidence (i.e., TCE p (X) VaR p (X)) and provides an effective tool for analyzing tail risks. The TCE is also related to the expected residual lifetime, a performance measure widely used in reliability theory and survival analysis. For light-tailed loss distributions, such as normal distributions, TCE and VaR at the same level p of confidence are asymptotically equal as p 1. Another example of light-tailed losses is the phase-type distribution 2. The explicit relation between TCE and VaR for the phase-type loss distributions was obtained in [8], from which asymptotic equivalence of TCE and VaR as p 1 2 That is, the hitting time distribution of a finite-state Markov chain. 2

3 is evident. It is precisely the heavy-tails of loss distributions that make TCE more effective in analyzing tail risks. Formally, a non-negative loss variable X with distribution function (df) F has a heavy or regularly varying right tail at with heavy-tail index α if its survival function is of the following form (see, e.g., [7] for detail), F (r) := 1 F (r) = r α L(r), r >, α >, (1.2) where L is a slowly varying function; that is, L is a positive function on (, ) with property L(cr) = 1, for every c >. (1.3) r L(r) For example, the Pareto distribution with survival function F (r) = (1+r) α, r, has a regularly varying tail. It can be easily verified that if α > 1 for Pareto loss variable X, then TCE p (X) α α 1 VaR p(x), as p 1. (1.4) In fact, (1.4) holds for any loss distribution (1.2) with heavy-tail index α > 1. Observe that TCE p (X) = E(XI{X > VaR p(x)}) Pr{X > VaR p (X)} ( 1 = VaR p (X) Pr{X > VaR p (X)} + Pr{X > VaR p (X)} VaR p(x) Pr{X > x}dx ), (1.5) where I(A) hereafter denotes the indicator function of set A. By the Karamata theorem (see, e.g., [31]), we have VaR p(x) Pr{X > x}dx 1 α 1 VaR p(x) Pr{X > VaR p (X)}, as p 1. (1.6) Plug this estimate into (1.5), we obtain (1.4) for any regularly varying distribution with α > 1. The asymptotic formula (1.4) of TCE for univariate tail risks is fairly straightforward, but the multivariate case remains unsettled and is the focus of this paper. Consider a random vector X = (X 1,..., X d ) from a multi-assets portfolio at the end of a given period, where the i-th component X i corresponds to the loss of the financial position on the i-th market. A risk measure R(X) for loss vector X corresponds to a subset of R d consisting of all the deterministic portfolios x such that the modified positions x X is acceptable to regulators/supervisors. The coherency principles that are similar to the univariate case were formulated in [2] for multivariate risk measure R(X), and it was further shown in [6] that for continuous loss vectors, multivariate TCEs are coherent in the sense of [2]. Note, however, that multivariate TCEs, to be formally defined in Section 2, are subsets of R d, which lack tractable expressions even for some widely used multivariate distributions, such as multivariate normals. The effect of dependence among losses X 1,..., X d in different assets on the multivariate TCE also remains difficult to understand. In this paper, we 3

4 study asymptotic behaviors of multivariate TCEs for multivariate regularly varying distributions. Our method, based on tail dependence functions developed in [29, 18], not only yields explicit asymptotic expressions of multivariate TCEs for various multivariate distributions, but also leads to better insights into how the dependence among extreme losses would affect analysis on tail risks. The rest of the paper is organized as follows. In Section 2, we briefly discuss the multivariate coherent risk measures, and then obtain the tail estimates of TCEs for multivariate regular variation in terms of intensity measures and their asymptotic bounds in terms of tail dependence functions. In Section 3, we present several examples to examine the quality of the bounds. Section 4 concludes the paper with some remarks and Appendix in Section 5 details two lengthy proofs. Throughout this paper, measurability of functions and sets are often assumed without explicitly mention, and the maximum operator is denoted by. 2 Tail Risks of Multivariate Regular Variation To explain the vector-valued coherent risk measures, we use the notations from [2]. Let K be a closed, salient convex cone 1 of R d such that R d + K. The convex cone K induces a partial order on R d : x K y if and only if y x + K. Note that a convex cone K must be an upper set 3 with respect to partial order K induced by itself. Moreover, if A is an upper set with respect to partial order K, then for any x A and k K, x + k K x, leading to x + k A and thus A + K A. Observe that we always have A + K A due to the fact that any closed convex cone must contain the origin. Conversely, if A + K = A for some subset A, then for any y K x with x A, y x + K A + K = A, implying that A must be upper with respect to partial order K. Hence, A is an upper set with respect to partial order K if and only if A + K = A. If K = R d +, then the K -order becomes the usual component-wise order. For any two loss random vectors X and Y on the probability space (Ω, F, P), define X K Y if and only if Y X K, P-almost surely. Using the partial order K rather than the usual component-wise partial order can account for some financial market frictions such as transaction cost, etc.. Definition 2.1. Consider random loss vectors on a probability space (Ω, F, P). A vector-valued coherent risk measure R( ) is a measurable set-valued map satisfying that R(X) R d is closed for any loss random vector X and R() R d, as well as the following axioms: 1. (Monotonicity) For any X and Y, X K Y implies that R(X) R(Y ). 2. (Subadditivity) For any X and Y, R(X + Y ) R(X) + R(Y ). 3. (Positive Homogeneity) For any X and positive s, R(sX) = sr(x). 4. (Translation Invariance) For any X and any deterministic vector l, R(X + l) = R(X) + l. 3 A set S is called upper (lower) with respect to partial order K if s K ( K) s and s S imply that s S. 4

5 Note that the risk set R(X) consists of all the deterministic portfolios x such that the multivariate portfolio x X is acceptable to the regulator/supervisor. The motivation for set-valued risk measures is that investors are sometimes not able to aggregate their multivariate portfolios on various security markets because of liquidity problems and/or transaction costs between the different security markets (e.g., having assets in several currencies at the same time). See [2] for details. When d = 1, ϱ(x) := inf{r : r R(X)} is a univariate coherent risk measure satisfying the four axioms discussed in Section 1, and thus R(X) = [ϱ(x), ). It was shown in [2] that the worst conditional expectation for random vector X, defined as W CE p (X) := {x R d : E(x X B) K, B F with P(B) 1 p}, < p < 1, is a vector-valued coherent risk measure. Since W CE p (X) = B F with P(B) 1 p (E(X B) + K) and K is an upper set, W CE p (X) is also an upper set. For any continuous random vector X, W CE p (X) equals the tail conditional expectation (TCE) for X, defined as in [6] by, T CE p (X) := {x R d : E(x X X A) K, A Q p (X)} = (E(X X A) + K), < p < 1, (2.1) A Q p(x) where Q p (X) = {A R d : A is Borel-measurable and A + K = A, Pr{X A} 1 p} is the set of all the upper sets (with respect to K ) with probability mass greater than or equal to 1 p. Observe that T CE p (X) is a convex and upper set that consists of all the portfolios x of capital reserves that can be used to cover the expected losses E(X X A) in the events that X A. Note that multivariate coherent risk measures discussed in [2, 6] are defined for essentially bounded random vectors. To discuss asymptotic properties, these measures have to be extended to the set of all random vectors on R d = [, ] d. This can be done using the idea in [12] that allows vectors in R(X) to have components taking the value of ; that is, the positions corresponding to these components are so risky, whatever that means, that no matter what the capital added, the positions will remain unacceptable. We need also to exclude the situations where components of the vectors in R(X) take the value of, which would mean that arbitrary amounts of capitals could be withdrawn without endangering the portfolios (see [12] for details). As a matter of fact, it can be easily verified that T CE p (X) is coherent in the sense of Definition 2.1 if X, which may not be bounded, has a continuous density function. The extreme value analysis of TCE T CE p (X) as p 1 boils down to analyzing asymptotic behaviors of E(X X rb) as r for various upper set B, for which multivariate regular variation suits well. A non-negative random vector X with joint df F is said to have a multivariate regularly varying (MRV, see [3]) distribution F if there exists a Radon measure µ (i.e., finite on compact sets), called the intensity measure, on R d +\{} such that r Pr{X rb} = µ(b), (2.2) Pr{ X > r} 5

6 for any relatively compact set B R d +\{} 4 with µ( B) =, where denote a norm on R d. Any MRV df F with support in R d + admits the following spectral representation: for all continuous points x of µ, 1 F (rx) r 1 F (r1) = Pr{X/r [, x] c } r Pr{X/r [, 1] c } = kµ([, x]c ), (2.3) where k > is a constant and µ([, x] c ) = S d 1 + S d 1 max 1 j d (u j /x j ) α S(du) for a finite measure S on + := {x Rd + : x = 1}. Non-degenerate margins F j, 1 j d, of an MRV df F are regularly varying in the sense of (1.2). Since F 1,..., F d are usually assumed to be tail equivalent [31], we have that F j (x) = L j (x)/x α, 1 j d, where L i (x)/l j (x) c ij as x, < c ij <. We assume hereafter that c ij = 1 for notational convenience. If c ij 1 for some i j, we can properly rescale the margins and the results still follow. We also assume that the heavy-tail index α > 1 to ensure the existence of expectations. The examples and properties of MRV distributions, including the relation between MRV distributions and multivariate extreme value distributions with identical Fréchet margins can be found in [3, 31]. The asymptotic relation between T CE p (X) and intensity measure µ is given below and its proof is detailed in Appendix in Section 5. Theorem 2.2. Let X be a non-negative loss vector that has an MRV df with intensity measure µ. 1. Let B be an upper set bounded away from. Then r r 1 E(X j X rb) = µ(a j (w) B) µ(b) dw =: u j (B; µ), where A j (w) := {(x 1,..., x d ) R d : x j > w}, 1 j d. 2. Let Q := {B R d : B + K = B, B S d 1 +, B (Bd ) c }, and B d := {x R d : x < 1} denote the open unit ball in R d with respect to the norm. As p 1, T CE p (X) VaR 1 (1 p)/µ(b) ( X ) ((u 1 (B; µ),..., u d (B; µ)) + K). B Q Remark Theorem 2.2 provides the multivariate extension of (1.4) and shows how extremal dependence, as described by the intensity measure, would quantitatively affect tail risks. It also provides a unified tool to analyze the structural properties of tail asymptotics of TCEs for various portfolio and risk aggregations of loss vector (X 1,..., X d ). For example, the tail asymptotics of TCEs of the portfolio aggregation d i=1 X i can be obtained from Theorem 2.2 (1) by taking B = {x : d i=1 x i > 1} (also see [3]). The tail estimate obtained in Theorem 2.2 (2) can be also applied to analyzing coherent aggregations [2] of extremal risks. 2. Theorem 2.2 (1) can be used in analyzing portfolio tail risk decomposition. For example, for any 1 j d, d ( E (X d ) ) ( d ) j X i > VaR p X i VaR p X i uj (B; µ), as p 1, i=1 i=1 4 Here R d + = [, ] d is compact and the punctured version R d +\{} is modified via the one-point uncompactification (see, e.g., [31]). i=1 6

7 where B = {x : d i=1 x i > 1}. The tail estimate of E ( X j d i=1 X i > VaR p ( d i=1 X i) ) provides the contribution to the total tail risk attributable to risk j, as measured by TCEs. The risk allocation/decomposition with TCE for elliptically distributed loss vectors can be found in [24]. 3. The computation of VaR for the norm X is difficult in general, but the tail estimate of VaR p ( X ), when p 1, is relatively simple in light of (2.2). The tail estimates of VaR of the sum are obtained in [2, 1, 4, 22, 13] in a similar spirit. For the maximum norm of loss vector (X 1,..., X d ) with identical margins, the VaR can be estimated from the asymptotic relation Pr{max 1 i d X i > r} Pr{X 1 > r}/µ(b) for sufficiently large r, where B = (1, ) R d 1. In the situations that the asymptotic expression obtained in Theorem 2.2 may be intractable, we can utilize the method of tail dependence functions introduced in [29, 18] to derive tractable bounds for TCE. For notational convenience, we only consider the case where K = R d + in the remainder of this paper. The idea is to separate the margins from the dependence structure of df F, so that TCE s can be expressed asymptotically in terms of the marginal heavy-tail index and tail dependence of the copula of F. Assume that df F of random vector X = (X 1,..., X d ) has continuous margins F 1,..., F d, and then from [32], the copula C of F can be uniquely expressed as C(u 1,..., u d ) = F (F 1 1 (u 1 ),..., F 1 d (u d)), (u 1,..., u n ) [, 1] d, where Fj 1, 1 j d, are the quantile functions of the margins. The extremal dependence of a df F can be described by various tail dependence parameters of its copula C. The upper tail dependence parameters, for example, are the conditional probabilities that random vector (U 1,..., U d ) := (F 1 (X 1 ),..., F d (X d )) with standard uniform margins belongs to upper tail orthants given that a univariate margin takes extreme values: C(1 u,..., 1 u) λ U = Pr{U 1 > 1 u,..., U d > 1 u U d > 1 u} =, (2.4) u u u where C denotes the survival function of C. Bivariate tail dependence has been widely studied [16], and various multivariate versions of tail dependence parameters have also been introduced and studied in [21, 25]. In fact, various upper tail dependence parameters can be represented by the upper tail dependence function [21, 29, 18], defined as follows, b C(1 uw j, 1 j d) (w) :=, w = (w 1,..., w d ) R d u u +. (2.5) The lower tail dependence can be similarly studied but we focus only on upper tail dependence in this paper. It was shown in [18] that b (w) > for all w R d + if and only if λ U >. Unlike λ U, however, the tail dependence function provides all the extremal dependence information [29, 18, 26]. 7

8 Using the inclusion-exclusion principle, we define the upper exponent function of C as follows a (w) := ( 1) S 1 b S(w i, i S; C S ), (2.6) S {1,...,d},S where b S (w i, i S; C S ) denotes the upper tail dependence function of the margin C S of C with component indexes in S. The intensity measure µ and tail dependence function b of an MRV distribution F are uniquely determined from each other and their detailed relations can be found in [26]. In particular, b (w) = µ( d i=1 [w 1/α i µ([1, ] R d 1 + ), ]) µ([w, ]), and µ([, 1] c ) = b (w1 α,..., w α d ) a. (2.7) (1,..., 1) Using this equivalence and Theorem 2.2 (1), E(X X rb) can be asymptotically expressed in terms of the tail dependence function b for sufficiently large r. But the asymptotic estimation of T CE p (X) via Theorem 2.2 (2) is still cumbersome because B Q can be quite arbitrary. More tractable bounds for T CE p (X) can be established directly using the tail dependence, as shown in the next theorem whose proof is detailed in Appendix in Section 5. Theorem 2.4. Let X be a non-negative loss vector with an MRV df F and heavy-tail index α > 1. Assume that the copula C of F has a positive upper tail dependence function b (w) >. Let max denote the maximum norm. 1. For 1 j d, 2. Let S j (b, α) := 1 r r E(X j X r(x, ]) = b (1,...,1,(w j 1) α,1,...,1) T CE p (X) VaR a 1 (1 p) (1,...,1) ( X max ) b (1,...,1) 3. For sufficiently small 1 p, where, for 1 j d, VaR p ( X max ) s j (b, α) := α 1 α 1 b (1,..., 1) + b (x α 1,..., (w j x j ) α,..., x α d ) b (x α 1,..., x α d ) dw j. b (1,...,1) dw j, 1 j d. For sufficiently small 1 p, ( ) (S 1 (b, α),..., S d (b, α)) + R d +. ( ) (s 1 (b, α),..., s d (b, α)) + R d + T CE p (X) =S {i:i j} 1 ( 1) S w jd b {j} S (w α j, 1,..., 1; C {j} S ) b, (1,..., 1) and b {j} S (w α j, 1,..., 1; C {j} S ) denotes the upper tail dependence function of the multivariate margin C {j} S evaluated with the j-th argument being wj α and others being one. Observe that if d = 1, then Theorem 2.4 (2) and (3) reduce to (1.4). In multivariate risk management, the upper (subset) bound presented in Theorem 2.4 (3) is more important, because it provides a set of portfolios of conservative reserves so that even in worst case scenarios the resulting positions are still acceptable to regulators/supervisors. 8

9 3 Illustrative Examples of Bounds for Tail Risks We have some examples to examine the quality of the results in Theorem 2.4 when used as approximations. The examples show that they are better with more tail dependence and a larger ζ, where ζ is in the exponent of the second order expansion C(1 uw j, 1 j d) u b (w) + u 1+ζ b 2(w), u. (3.1) It is intuitive that if ζ is larger (especially if ζ 1), then the second order term is less important. Note that for the Fréchet upper bound copula, C U (1 uw) = u min{w 1,..., w d }, and there is no second order term. Example 3.1. (a) Analysis of complete dependence (the Fréchet upper bound). Let C U be the Fréchet upper bound copula of dimension d. Then b (w) = min{w 1,..., w d } and b (1) = 1, a (1) = 1. In part (2) of Theorem 2.4, 1 (1 p)a /b = p, and for α > 1, S j (b, α) = min{1, w α }dw = 1 + (α 1) 1 = α/(α 1). In part (3) of Theorem 2.4, for α > 1, s j (b, α) = α/(α 1) + =S {i:i j} ( 1) S = α/(α 1). That is, the expressions in parts (2) and (3) coincide. (b) Analysis of near independence. As the d-variate copula C (with tail dependence) moves towards independence, b (1) and a (1) d and 1 (1 p)a (1)/b (1) > only if p > 1 b (1)/a (1) so that for small b (1), the result in part (2) of Theorem 2.4 is non-trivial only for large p near 1. This is a hint that all of the iting results of Theorem 2.4 are worse for weak tail dependence. In this case, one has to use Theorem 2.2 to approximate the multivariate TCE. Example 3.2. We show some details for two copula families to illustrate Theorem 2.4. The first copula is the exchangeable MTCJ copula (or Mardia-Takahasi-Cook-Johnson copula, see [27, 33, 11]), and the second is a mixture of the MTCJ copula and the independence copula. Second order expansions of the tail dependence functions are obtained and the approximation from part (1) of Theorem 2.4 is summarized in Tables for some special cases. (a) The MTCJ copula in dimension d, with dependence increasing in δ, is: C(u; δ) = [ u δ u δ d (d 1) ] 1/δ, δ >. (3.2) Let w j > for j = 1,..., d, and let W := w1 δ + + w δ. Then C(uw; δ) = u[w1 δ + + w δ d (d 1)u δ ] 1/δ = uw 1/δ [1 (d 1)u δ /W] 1/δ uw 1/δ[ 1 + (d 1)δ 1 u δ /W] = ub (w; δ) + u 1+δ b 2(w; δ), as u, where b (w; δ) = W 1/δ = (w1 δ + +w δ d ) 1/δ, b 2 (w; δ) = (d 1)δ 1 (w1 δ + +w δ d ) 1/δ 1. The second order term of C(uw; δ) is O(u 1+ζ ), where ζ = δ increases with more dependence. 9 d

10 Suppose (X 1,..., X d ) is multivariate Pareto of the form used in [27]; the univariate survival function is x α for x > 1 for all d margins and the survival copula is given in (3.2). That is, [ ] 1/δ, F (x) = C(x α 1,..., x α d ; δ) = x δα x δα d (d 1) xj > 1, j = 1,..., d. (3.3) An expression for the conditional expectation (given for the first component only because of symmetry) is: leading to TCE E [X 1 X 1 > x 1,..., X d > x d ] = x 1 + r 1 E [X 1 X 1 > rx 1,..., X d > rx d ] = x 1 + The above expectations exist for α > 1. F (x 1 + z 1, x 2,..., x d ) dz 1, F (x 1,..., x d ) Exact calculation of the last summand in (3.4): C ( (r[x 1 + w 1 ]) α, (rx 2 ) α,..., (rx d ) α ; δ ) dw 1 C ( (rx 1 ) α,..., (rx d ) α ; δ ) F (rx 1 + rw 1, rx 2,..., rx d ) dw 1. (3.4) F (rx) = [ (r[x1 + w 1 ]) αδ + (rx 2 ) αδ + (rx d ) αδ (d 1) ] 1/δ dw1 [ (rx1 ) αδ + + (rx d ) αδ (d 1) ]. 1/δ First order approximation of the last summand in (3.4): b ( (x 1 + w 1 ) α, x α 2,..., x α d ; δ) dw 1 b ( x α 1,..., x α d ; δ) = ( (x1 + w 1 ) αδ + x αδ ( x αδ xαδ d ) 1/δdw xαδ d ) 1/δ. This can be computed via numerical integration. Let the numerator and denominator of the above be denoted as N 1 := N 1 (x; α, δ) and D 1 := D 1 (x; α, δ). Second order approximation of the last summand in (3.4): r α N 1 + r α(1+δ) b ( 2 (x1 + w 1 ) α, x α r α D 1 + r α(1+δ) b 2 ( x α 1,..., x α d ; δ) 2,..., x α d ; δ) dw 1 = N 1 + (d 1)r αδ δ 1 ( (x1 + w 1 ) αδ + x αδ ) 1/δ 1dw1 xαδ d D 1 + (d 1)r αδ δ 1( x αδ ) 1/δ 1. xαδ d Table 1 has some (representative) results to show how the approximations compare; we take r = (1 p) 1/α, d = 2, x 1 = x 2 = 1, p =.999, α = 2 and 5, and δ [.1, 1.9]. The table shows that the first order approximation is worse only when the dependence is weak and the exponent ζ of the second order term is much less than 1; in these cases, the second order term of the expansion is useful. 1

11 (b) Mixture model with MTCJ and independence copulas. Now, the second order term is between O(u) and O(u 2 ), depending on the amount of dependence in the copula. Let C(u; δ, β) = (1 β) d j=1 u j + β[u δ u δ d (d 1)] 1/δ, δ >, < β < 1 so that dependence increases as δ and β increase. Let W := w1 δ + + w δ. Then where d C(uw; δ, β) (1 β)u d w j + βuw 1/δ[ 1 + (d 1)δ 1 u δ /W ] j=1 = u b (w; δ, β) + u 1+ζ b 2(w; δ, β), b (w; δ, β) = βw 1/δ = β(w δ w δ b 2(w; δ, β) = d ) 1/δ, { (d 1)βδ 1 (w1 δ + + w δ d ) 1/δ 1 if δ < d 1, (1 β) d j=1 w j + (d 1)βδ 1 (w1 δ + + w δ d ) 1/δ 1 if δ = d 1, (1 β) d j=1 w j if δ > d 1, and ζ = δ if δ < d 1 and ζ = d 1 if δ d 1. The second order term is not far from the first order term if δ is near (i.e., weak dependence). Similar to part (a), we list the exact TCE and the first/second order approximations for the last summand in (3.4). Exact (assuming α > 1 as before): with P x = d j=1 x α i, β { (r[x1 + w 1 ]) αδ + (rx 2 ) αδ + + (rx d ) αδ (d 1) } 1/δ dw1 + (1 β)r dα P x x 1 /(α 1) β { (rx 1 ) αδ + + (rx d ) αδ (d 1) } 1/δ + (1 β)r dα P x since (x 1 + w) α dw = x α+1 1 /(α 1). First order approximation: this is the same as in part (a) because β cancels from the numerator and denominator. Second order approximation: this is the same as in part (a) for δ < d 1. For δ d 1, one gets b ( (x 1 + w 1 ) α, x α 2,..., x α d ; δ, β) dw 1 + r α(d 1) b ( 2 (x1 + w 1 ) α, x α 2,..., x α d ; δ, β) dw 1 b ( x α 1,..., x α d ; δ, β) ( + r α(d 1) b 2 x α 1,..., x α d ; δ, β) Table 2 has some (representative) results to show how the approximations compare; we take r = (1 p) 1/α, d = 2, x 1 = x 2 = 1; p =.999, β =.25, α = 2 and 5, δ [.1, 1.9]. The conclusions are similar to Table 1, except the first and second order approximations are slightly off in the last decimal place shown, even for δ > 1. The accuracy is of order O(u d ) = O(u 2 ) for δ > 1 rather than the order O(u 1+δ ) in part (a). d 11

12 Table 1: Values of exact TCE minus x 1, together with first/second order approximations for the bivariate MTCJ copula with Pareto survival margins; r = (1 p) 1/α, x 1 = x 2 = 1, p =.999. α = 2 α = 5 δ exact appr1 appr2 exact appr1 appr Table 2: Values of exact TCE minus x 1, together with first/second order approximations for the bivariate mixture of independence and MTCJ copulas, with Pareto survival margins; r = (1 p) 1/α, x 1 = x 2 = 1, p =.999, β =.25. α = 2 α = 5 δ exact appr1 appr2 exact appr1 appr

13 Table 3: Bounds for parts (2) and (3) of Theorem 2.4 for the MTCJ copula, with Pareto survival margins; p =.999, (1 p) 1/α α/(α 1) = and 4.98 provides an intermediate value for α = 2 and 5 respectively. α = 2 α = 5 δ LB 2 UB 2 LB 3 UB 3 LB 2 UB 2 LB 3 UB Example 3.3. We show the quality of the approximations in parts (2) and (3) of Theorem 2.4 for (3.3) with survival copula (3.2). Since b (w) = (w1 δ + + w δ d ) 1/δ, the margins are given by b S (w j : j S) = ( j S w δ j ) 1/δ, and these can be used to compute s j (b, α) and S j (b, α) via numerical integrations. The exponent function a is in (2.6). If (X 1,..., X d ) has the distribution in (3.3), the distribution of X max = max{x 1,..., X d } is F Xmax (x) = F (x,..., x) = 1 + d ( ) d ( 1) j (jx αδ j + 1) 1/δ, x >. j j=1 Based on this distribution, expressions of the form VaR g(p) ( X max ) can be computed numerically. Because of exchangeability, parts (2) and (3) have the form UB d [1 d, ] T CE p (X) LB d [1 d, ]. Table 3 lists the values of LB d and UB d for d = 2, 3 with α = 2 and 5. As might be expected, the ratio UB d /LB d decreases as δ and α increase, and increases as d increases. Example 3.4. We consider general Archimedean copulas which satisfy a regular variation condition. Consider a loss vector (X 1,..., X d ) that has regularly varying margins with heavy-tail index α > 1, and the Archimedean survival copula Ĉ(u; φ) = φ( d i=1 φ 1 (u i )) where the Laplace transform φ is regularly varying at in the sense of (1.2) with tail index β >. It follows from 13

14 Proposition 2.8 of [18] that b (w 1,..., w d ) = (w 1/β w 1/β d ) β. Observe that (X 1,..., X d ) is more tail dependent as β decreases. Thus, for 1 j d, s j (b, α) = S j (b, α) = 1 + d β α α 1 dβ + d β =S {i:i j} 1 ( w α/β + d 1) β dw. ( 1) S [( S + 1) β It follows from Theorem 2.4 that computable asymptotic bounds are given by Since 1 (w α/β + S ) β dw]. (S 1 (b, α),..., S d (b, α)) + R d T CE p (X) + p 1 VaR p ( X max ) (s 1(b, α),..., s d (b, α)) + R d +. β 1 ( w α/β + d 1 ) β dw = 1 w α dw = 1, and α 1 β 1 (w α/β + S ) β dw = 1, we obtain that for fixed α > 1, β s j (b, α)/s j (b, α) = 1, for 1 j d. That is, asymptotic subset and superset bounds are approximately identical for small β. Remark 3.5. With the point process approach applied to data in the tails, the tail dependence function b can be estimated, and then the results in the theorems can be used. An outline of the steps is as follows. 1. For risk variable j, a heavy-tail index estimation method, such as the Hill estimation, can be applied to data values above a threshold to get an estimated univariate tail index ˆα j. If any risk variable shows a thin tail (i.e., exponentially decayed), it can be removed from calculations of multivariate extremal risk. For the remaining risks with possibly different heavy-tail indexes, make appropriate power transforms and rescale the data so that exceedances above the threshold have a Pareto distribution with tail index α in the middle of the range of the ˆα j s (see page 31 of [31]). 2. Transform to Fréchet margins and use the point process likelihood approach for the joint tails of the risk variables [1, 19, 16]. The exponent function a corresponds to the intensity measure of the point process. For example, the simplest exchangeable Gumbel model discussed in Example 3.2 (a), nested Gumbel models [15], scale mixture models [17, 25], and several other parametric models of a all have tractable forms so that the point process likelihood can be easily optimized numerically; included are models with flexible dependence (labeled as MM1, MM2, MM3 in [16]) which have a parameter denoting a minimal dependence level and additional parameters for each pair that add onto the minimal dependence. An estimated b can be obtained from a using the inclusion-exclusion relation. 14

15 3. Combining α in step 1 and b in step 2, Theorem 2.4 can be used to obtain bounds on the scaled risks, with one-dimensional numerical integration. With the rescaled risk variables X 1,..., X d, let the thresholds T j satisfy ˆF Xj (T ) = q for all j, where q might be in the range [.5,.8]. Let ˆF Xj (x j ) = q + (1 q)[1 (1 + x j T j ) α ] for x j > T j with the estimated common α. With a parametric a, we use the copula C(u 1,..., u d ) = e a ( log u 1,..., log u d ) in the tail region. For x 1 > T 1,..., x d > T d, the estimated tail distribution is: ˆF X1,...,X d (x 1,..., x d ) = C( ˆF X1 (x 1 ),..., ˆF X1 (x 1 )). (3.5) The tail conditional expectation E(X j X r(x, ]) can be evaluated using the estimated α and b or using (3.5) through one-dimensional numerical integration, like in Example 3.2, provided rx j > T j for all j. 4. For non-rectangular upper sets B such that rb d j=1 [T j, ), E(X j X rb) can be evaluated by simulation of the tail of (3.5). This is better than fitting a distribution to the entire data for simulation, to avoid extrapolation from a fit that is dominated from the middle of the data, and to reduce the simulation sample size. 5. Parts (2) and (3) of Theorem 2.4 are useful as a way to more quickly give insight on the effect of the univariate tail index α and the amount of tail dependence (represented by b ) on the size of T CE p (X) for p near 1. The pattern in the it should carry over to the non-it in the tail region. 4 Concluding Remarks Our results illustrate how tail risk is quantitatively affected by extremal dependence and also show how the tool of tail dependence functions can be used to estimate such an asymptotic relation. Similar to the univariate case (1.4), the multivariate tail conditional expectation T CE p (X) as p 1 is essentially linearly related to the value-at-risk of an aggregated norm of X. In contrast to the univariate case where the asymptotic proportionality constant is related to the heavy-tail index α, the asymptotic proportionality constants in the multivariate case depend not only on the heavy-tail index α but also on the tail dependence structure. As illustrated in the paper, the lower and upper bounds for multivariate TCEs become approximately equal for highly tail dependent distributions, and thus our method is especially effective for analyzing extremal risks for loss variables with significant tail dependence. For example, nonoverlapping aggregations of large numbers of loss variables in high-dimensional portfolios can have strong tail dependence even though loss variables themselves only demonstrate weak tail dependence; see [23]. When the lower and upper bounds are far apart, reducing the class of relevant upper sets is suggested. The quality of the bounds presented in Theorem 2.4 might be poor for the distributions with weaker tail dependence. In this situation, one may aggregate loss variables with weak tail depen- 15

16 dence, which also corresponds to choosing some reduced class of specific upper sets B in Theorem 2.2, so that better bounds can be obtained. One can also use the higher order expansions such as (3.1) to reveal the dependence structure at sub-extreme levels so that more accurate, tractable bounds can be developed. Our numerical examples via the second order expansion show some significant improvements in the presence of weak tail dependence, but more theoretical studies are indeed needed in this area. Acknowledgments. The authors would like to thank two referees, an Editor and Editor-in-Chief for their comments that lead to an improvement of the presentation of this paper. 5 Appendix: Proofs 5.1 Proof of Theorem 2.2 Proof. To estimate E(X X rb) for any upper set B bounded away from, consider, E(X j X rb) = Pr{X j > x X rb}dx = r Pr{X j > rw, X rb} dw. (5.1) Pr{X rb} for any 1 j d. We first argue that we can pass the it through the integration (5.1). Since Pr{X j > rw, X rb} Pr {X j > rw}, (5.2) it follows from the Karamata theorem (1.6) that for any fixed c >, r c Pr {X j > rw} dw = Pr{X rb} r rc Pr {X j > x} r Pr{X rb} dx = Let A j (w) := {(x 1,..., x d ) R d : x j > w}, then via (2.2), we have, r c Pr {X j > rw} Pr{X rb} dw = c µ(a j (c)) = α 1 µ(b) c α 1 Pr {X j > rc} r Pr{X rb}. c µ(a j (w)) dw, (5.3) µ(b) where the last equality follows from the direct calculation via (2.3). Because of (5.2), (5.3) and the generalized dominated convergence theorem, we have from (2.2) that for any c >, r c Pr{X j > rw, X rb} dw = Pr{X rb} ɛ/3 ɛ/3 c r Pr{X j > rw, X rb} dw = Pr{X rb} c µ(a j (w) B) dw, µ(b) which implies that for any small ɛ >, there exists r ɛ such that for all r r ɛ, Pr{X j > rw, X rb} µ(a j (w) B) ɛ/3 Pr{X j > rw, X rb} dw dw dw Pr{X rb} µ(b) Pr{X rb} Pr{X j > rw, X rb} µ(a j (w) B) ɛ/3 µ(a j (w) B) + dw dw + dw Pr{X rb} µ(b) µ(b) ɛ/3 Pr{X j > rw, X rb} dw + ɛ ɛ/3 Pr{X rb} µ(a j (w) B) dw ɛ µ(b) 3 + ɛ 3 + ɛ 3 = ɛ,

17 where the last inequality follows due to the fact that Pr{X j > rw, X rb} Pr{X rb} and µ(a j (w) B) µ(b). Therefore, we have from (5.1) that 1 r r E(X j X rb) = r This concludes the proof of statement (1). Pr{X j > rw, X rb} dw = Pr{X rb} µ(a j (w) B) dw. (5.4) µ(b) For statement (2), we simplify (2.1) asymptotically. For any upper set A Q p (X), there exists an upper set B with B S d 1 + and a positive number r B such that A = r B B. Since Pr{X rb} is decreasing in r, we can find r B,p r B for any A = r B B such that Pr{X A} Pr{X r B,p B} = 1 p, as p 1. It follows from (5.4) that E(X j X r B,p B) is asymptotically increasing for sufficiently small 1 p and goes to + as p 1, and thus we have E(X X A) E(X X r B,p B) for sufficiently small 1 p. Since E(X X A) + K E(X X r B,p B) + K for sufficiently small 1 p, and r B,p B Q p (X), we have, [( ) ] (E(X X r B,p B) + K) \ T CE p (X) p 1 = p 1 B Q A Q p(x) (E(X X A) + K) \ T CE p (X) =, where Q := {B R d : B + K = B, B S d 1 +, B is bounded away from } and Pr{X r B,p B} = 1 p. That is, (2.1) can be rewritten as follows, for sufficiently small 1 p, T CE p (X) (E(X X r B,p B) + K). (5.5) B Q For any B Q, there exists a real number r B with r B 1 such that r B B Q = {B R d : B + K = B, B S d 1 +, B (Bd ) c }. That is, for any B Q with Pr{X r B,p B} = 1 p, we can find a B Q and a real number r B,p (e.g., r B,p = r B,p /r B ) such that r B,p B = r B,pB. Thus (5.5) can be rewritten further as T CE p (X) B Q,Pr{X r B,p B}=1 p (E(X X r B,p B) + K), (5.6) for sufficiently small 1 p. Observe that as p 1, r B,p, and thus it follows from (2.2) that for sufficiently small 1 p, µ(b) Pr{ X > r B,p } 1 p, implying that r B,p VaR 1 (1 p)/µ(b) ( X ) as p 1. Therefore, (5.4) and (5.6) imply that T CE p (X) as p 1, where u j (B; µ) = B Q VaR 1 (1 p)/µ(b) ( X ) ((u 1 (B; µ),..., u d (B; µ)) + K) µ(a j (w) B) µ(b) dw, 1 j d. 17

18 5.2 Proof of Theorem 2.4 Proof. Since margins F 1,..., F d of F are tail equivalent [31], we have that F j (x) = L j (x)/x α, 1 j d, where L i (x)/l j (x) 1 as x. (1) Without loss of generality, let j = 1. The straightforward calculation shows Pr{X 1 > x, X 1 > rx 1,..., X d > rx d } E(X 1 X > rx) = dx Pr{X 1 > rx 1,..., X d > rx d } Pr{X 1 > x, X 2 > rx 2,..., X d > rx d } = rx 1 + dx rx 1 Pr{X 1 > rx 1,..., X d > rx d } ( ) Pr{X 1 > rw, X 2 > rx 2,..., X d > rx d } = r x 1 + dw x 1 Pr{X 1 > rx 1,..., X d > rx d } ( ) Pr{U 1 > F 1 (rw), U 2 > F 2 (rx 2 ),..., U d > F d (rx d )} = r x 1 + dw. x 1 Pr{U 1 > F 1 (rx 1 ),..., U d > F d (rx d )} Applying the Karamata theorem and generalized dominated convergence theorem, we are allowed to pass the it through the integral. Since L j, 1 j d, are slowly varying and the margins are tail equivalent, we have, 1 r r E(X 1 X > rx) Pr{U 1 > 1 L 1 (rw)/(rw) α,..., U d > 1 L d (rx d )/(rx d ) α } = x 1 + r x 1 Pr{U 1 > 1 L 1 (rx 1 )/(rx 1 ) α,..., U d > 1 L d (rx d )/(rx d ) α } dw = x 1 + = x 1 + = x 1 + x 1 x 1 x 1 Pr{U 1 > 1 w α L 1 (r)r α, U 2 > 1 x α 2 L 1(r)r α,..., U d > 1 x α d L 1(r)r α } r Pr{U 1 > 1 x α 1 L 1(r)r α,..., U d > 1 x α d L dw 1(r)r α } Pr{U 1 > 1 w α u, U 2 > 1 x α 2 u,..., U d > 1 x α d u} u Pr{U 1 > 1 x α 1 u,..., U d > 1 x α d u} dw b (w α α, x2,..., x α d ) b ((w 1 x 1 ) α, x α 2,..., x α d ) b (x α 1, x α 2,..., x α d ) dw = (2) It follows from (5.6) that as p 1, T CE p (X) x S d 1 + b (x α 1,..., x α d ) dw 1. (E(X X r x,p (x, ]) + R d +) where r x,p satisfies Pr{X r x,p (x, ]} = 1 p. Since b (1) >, it follows from Theorem 2.4 of [25] that µ((1, ]) >. Since X max is regularly varying at, we have for sufficiently small 1 p, there exists r 1,p, such that µ((1, ]) Pr{ X max > r 1,p } = 1 p, which implies that r 1,p VaR 1 (1 p)/µ((1, ]) ( X max ) as p 1. Observe that as p 1, r 1,p, and thus it follows from (2.2) that for sufficiently small 1 p, Pr{X r 1,p (1, ]} µ((1, ]) Pr{ X max > r 1,p } = 1 p. 18

19 Therefore, as p 1, T CE p (X) x S d 1 + (E(X X r x,p (x, ]) + R d +) E(X X r 1,p (1, ]) + R d +. (5.7) Since X max > r 1,p if and only if X r 1,p [, 1] c, the constant k in (2.3) equals 1 and µ([, 1] c ) = 1. It then follows from (2.7) that µ((1, ]) = b (1,..., 1)/a (1..., 1), and thus from (1) that as p 1, E(X X r 1,p (1, ]) VaR a 1 (1 p) (1,...,1) ( X max )(S 1 (b, α),..., S d (b, α)) b (1,...,1) where S j (b, α) = b (1,...,1,(w j 1) α,1,...,1) b (1,...,1) dw j, 1 j d. Plug this into (5.7), we obtain (2). (3) In light of (5.6), consider, for any B Q max with Pr{X r B,p B} = 1 p, E(X j X r B,p B) = E(X ji{x r B,p B}). Pr{X r B,p B} Since (1, ] d B [, 1] c for any B Q max, we have E(X j X r B,p B) E(X ji{x r B,p [, 1] c }) Pr{X r B,p (1, ] d } If x > r B,p then If x r B,p then = Pr{{X j > x} {X r B,p [, 1] c }} = Pr{X j > x}. Pr{{X j > x} {X r B,p [, 1] c }} Pr{X r B,p (1, ] d dx.(5.8) } Pr{{X j > x} {X r B,p [, 1] c }} = Pr{{X j > x} ( d i=1{x i > r B,p })} = Pr{ d i=1({x j > x} {X i > r B,p })} = Pr{( i j {X j > x, X i > r B,p }) {X j > r B,p }} = ( 1) S Pr{X j > r B,p, X i > r B,p, i S} ( 1) S Pr{X j > x, X i > r B,p, i S} S {i:i j} =S {i:i j} = Pr{X j > r B,p } + ( 1) S (Pr{X j > r B,p, X i > r B,p, i S} Pr{X j > x, X i > r B,p, i S}). (5.9) =S {i:i j} Since the margins are tail equivalent and slowly varying, we have, for any w j 1, and any S {i : i j}, Pr{X j > r B,p w j, X i > r B,p, i S} p 1 Pr{X r B,p (1, ] d } Pr{U j > 1 wj α = p 1 = r B,p r α B,p L j(r B,p w), U i > 1 r α B,p L i(r B,p ), i S} Pr{U i > 1 r α B,p L i(r B,p ), 1 i d} Pr{U j > 1 wj α r α B,p L 1(r B,p ), U i > 1 r α B,p L 1(r B,p ), i S} Pr{U i > 1 r α B,p L 1(r B,p ), 1 i d} = b {j} S (w α j, 1,..., 1; C {j} S )/b (1,..., 1), 19

20 where b {j} S (w α j, 1,..., 1; C {j} S ) denotes the upper tail dependence function of the multivariate margin C {j} S evaluated with the j-th argument being wj α and others being one. Similarly, Pr{X j > r B,p, X i > r B,p, i S} p 1 Pr{X r B,p (1, ] d } p 1 = b {j} S (1,..., 1; C {j} S) b, (1,..., 1) Pr{X j > r B,p } Pr{X r B,p (1, ] d } = 1 b (1,..., 1). (5.1) Using the bounded convergence theorem, we then have, for sufficiently small 1 p, 1 ( 1) S Pr{X j > r B,p, X i > r B,p, i S} Pr{X j > r B,p w j, X i > r B,p, i S} Pr{X r B,p (1, ] d dw j } =S {i:i j} =S {i:i j} ( 1) S b {j} S (1,..., 1; C {j} S) 1 b {j} S (w α j b (1,..., 1) Plug (5.1) and (5.11) into (5.9), and we have, for sufficiently small 1 p, rb,p r B,p Pr{{X j > x} {X r B,p [, 1] c }} Pr{X r B,p (1, ] d dx } =S {i:i j} ( 1) S b {j} S (1,..., 1; C {j} S) 1, 1,..., 1; C {j} S )dw j. (5.11) r B,p b (1,..., 1) + b {j} S (w α j b (1,..., 1), 1,..., 1; C {j} S )dw j.(5.12) On the other hand, using the Karamata theorem (1.6), we have, for sufficiently small 1 p, Pr{{X j > x} {X r B,p [, 1] c }} r B,p Pr{X r B,p (1, ] d } 1 Pr{X j > r B,p } r B,p α 1 Pr{X r B,p (1, ] d } dx = Pr{X j > x} r B,p Pr{X r B,p (1, ] d } dx r B,p (α 1)b (1,..., 1). (5.13) Combining (5.12) and (5.13) into (5.8), we have, for sufficiently small 1 p, E(X j X r B,p B) α r B,p α 1 b (1,..., 1) + r B,p ( 1) S b {j} S (1,..., 1; C {j} S) 1 b {j} S (w α j, 1,..., 1; C {j} S )dw j b. (1,..., 1) =S {i:i j} As p 1, r B,p VaR 1 (1 p)/µ(b) ( X max ) VaR 1 (1 p)/µ([,1] c )( X max ) = VaR p ( X max ) due to the fact that µ([, 1] c ) = 1. Thus, for sufficiently small 1 p, + E(X j X r B,p B) VaR p ( X max ) =S {i:i j} α 1 α 1 b (1,..., 1) ( 1) S b {j} S (1,..., 1; C {j} S) 1 b {j} S (w α j b (1,..., 1), 1,..., 1; C {j} S )dw j for any B Q max, where the equality follows from the integration by parts. Therefore, ( ) T CE p (X) VaR p ( X max ) (s 1 (b, α),..., s d (b, α)) + R d +, = s j (b, α), for sufficiently small 1 p. 2

21 References [1] Albrecher, H., Asmussen, S. and Kortschak, D. (26). Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes, 9: [2] Alink, S., Löwe, M. and Wüthrich, M. V. (24). Diversification of aggregate dependent risks. Insurance: Math. Econom., 35: [3] Alink, S., Löwe, M. and Wüthrich, M. V. (25). Analysis of the expected shortfall of aggregate dependent risks, ASTIN Bulletin, 35(1): [4] Alink, S., Löwe, M. and Wüthrich, M. V. (27). Diversification for general copula dependence. Statistica Neerlandica, 61: [5] Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999). Coherent measures of risks. Mathematical Finance 9: [6] Bentahar, I. (26). Tail conditional expectation for vector-valued risks. Discussion paper 26-29, Technische Universität Berlin, Germany. [7] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press, Cambridge, UK. [8] Cai, J. and Li, H. (25). Conditional tail expectations for multivariate phase-type distributions. J. Appl. Prob. 42: [9] Cheridito, P., Delbaen, F. and Klüppelberg, C. (24). Coherent and convex monetary risk measures for bounded càdlàg processes. Stochastic Processes and their Applications, 112:1 22. [1] Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. R. Statist. Soc., B, 53: [11] Cook, R.D. and Johnson, M.E. (1981). A family of distributions for modelling non-elliptically symmetric multivariate data. J. Roy. Statist. Soc. B, 43: [12] Delbaen, F. (22). Coherent risk measure on general probability spaces. Advances in Finance and Stochastics-Essays in Honour of Dieter Sondermann, Eds. K. Sandmann, P. J. Schönbucher, Springer-Verlag, Berlin, [13] Embrechts, P., Neslehová, J. and Wüthrich, M. V., (29). Additivity properties for valueat-risk under Archimedean dependence and heavy-tailedness. Insurance: Mathematics and Economics, 44(2): [14] Föllmer, H. and Schied, A. (22). Convex measures of risk and trading constraints. Finance and Stochastics, 6:

22 [15] Joe, H. (1993). Parametric family of multivariate distributions with given margins. J. Multivariate Anal., 46: [16] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London. [17] Joe, H. and Hu, T. (1996). Multivariate distributions from mixtures of max-infinitely divisible distributions. J. Multivariate Anal., 57: [18] Joe, H., Li, H. and Nikoloulopoulos, A.K. (21). Tail dependence functions and vine copulas. Journal of Multivariate Analysis, 11: [19] Joe, H., Smith, R. L. and Weissman, I. (1992), Bivariate threshold methods for extremes. J. R. Statist. Soc. B. 54: [2] Jouini, E., Meddeb, M. and Touzi, N. (24). Vector-valued coherent risk measures. Finance and Stochastics 8: [21] Klüppelberg, C., Kuhn, G. and Peng, L. (28). Semi-parametric models for the multivariate tail dependence function the asymptotically dependent. Scandinavian Journal of Statistics, 35(4): [22] Kortschak, D. and Albrecher, H. (29). Asymptotic results for the sum of dependent nonidentically distributed random variables. Methodol. Comput. Appl. Probab. 11: [23] Kousky, C. and Cooke, R. M. (29). Cate Change and Risk Management: Challenges for insurance, adaptation and loss estimation. Discussion paper RFF DP 9-3-Rev, Resources For the Future ( [24] Landsman Z. and Valdez, E. (23). Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7: [25] Li, H. (29). Orthant tail dependence of multivariate extreme value distributions. Journal of Multivariate Analysis, 1: [26] Li, H. and Sun, Y. (29). Tail dependence for heavy-tailed scale mixtures of multivariate distributions. J. Appl. Prob. 46 (4): [27] Mardia, K.V. (1962). Multivariate Pareto distributions. Ann. Math. Statist., 33: [28] McNeil, A. J., Frey, R., Embrechts, P. (25). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton, New Jersey. [29] Nikoloulopoulos, A.K., Joe, H. and Li, H. (29). Extreme value properties of multivariate t copulas. Extremes, 12: