3. (Positive Homogeneity) For any X and positive s , R ( s X ) = sR ( - - PDF document

Asymptotic Analysis of Multivariate Coherent Risks∗

Harry Joe† Haijun Li‡ June 2010

Multivariate coherent risks can be described as classes of portfolios consisting of extra capital reserves that are used to cover potential losses under various scenarios. Tail risk refers to the risk associated with extremal events and is often affected by extremal dependence among multivari- ate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. The fundamental idea

• f approximating multivariate tail risk via extremal dependence of multivariate extremes is high-

lighted and explicit asymptotic relations between tail risks and tail dependence functions are then

• established. Various examples involving Archimedean copulas are presented to illustrate the results.

1 Tail Estimates of Multivariate Coherent Risks

Consider a random vector X = (X1, . . . , Xd) from a multi-asset portfolio at the end of a given period, where the i-th component Xi corresponds to the loss of the position on the i-th market. A risk measure R(X) for loss vector X corresponds to a subset of Rd consisting of all the deterministic portfolios x such that the modified positions x − X is acceptable to regulators/supervisors (i.e., x cancels the risk of portfolio X from the point of view of regulators/supervisors). A risk measure is called coherent if it satisfies the following coherency axioms. Definition 1.1. A vector-valued coherent risk measure R(·) is a measurable set-valued map defined

• n the space of all random loss vectors on a probability space (Ω, F, P), satisfying that R(X) ⊂ Rd

is closed for any loss random vector X and 0 ∈ R(0) = Rd, as well as the following:

• 1. (Monotonicity) For any X and Y , X ≤ Y component-wise implies that R(X) ⊇ R(Y ).
• 2. (Subadditivity) For any X and Y , R(X + Y ) ⊇ R(X) + R(Y ).

∗Prepared for the 7th Conference on Multivariate Distributions with Applications, Maresias, August 8 - 13, 2010,

Brazil

†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 0825960.

1

• 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.

The monotonicity axiom says a portfolio with more potential loss is riskier. The subadditivity is the property of risk reduction by diversification; that is, if portfolios x and y cancel the risks of X and Y respectively, then x + y cancels the risk of X + Y . The homogeneity axiom describes how portfolio size directly influences risk, whereas the translation invariance describes how incurring additional deterministic loss would affect portfolio risk. When d = 1, ̺(X) := inf{r : r ∈ R(X)} is a univariate coherent risk measure, and thus R(X) = [̺(X), ∞). Univariate coherent risk measures have been extensively studied in the litera- ture [5, 8, 19], and it was shown that the popular risk measure VaR (i.e., value-at-risk), defined by VaRp(X) := inf{x ∈ R : Pr{X > x} ≤ 1−p}, is not coherent and violates the subadditivity axiom. Univariate coherent risk measures can be also used to analyze risk of aggregations of multivariate portfolios, in contrast, however, the main motivation for vector-valued coherent risk measures in Definition 1.1 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 differ- ent security markets (e.g., having assets in several currencies at the same time, see [13] for details). The notion of vector-valued coherent risk also provides a variety of tail conditional coherent risk measures that are useful in risk allocation/decomposition for multivariate portfolios. It follows from the duality theory that any vector-valued coherent risk measure with the Fatou property (see [13]) can be expressed as follows, R(X) = {x ∈ Rd : EQ(x − X) ≥ 0, ∀ Q ∈ P}, (1.1) where P is a closed convex set of probability measures that are absolutely continuous with respect to P, and EQ(·) denotes the expectation with respect to Q. Clearly, R(X) is a convex and upper set for any loss vector X with the Fatou property. The expression (1.1) has significant practical implication and provides a scenario-based method for measuring multivariate risks; that is, P is a set of “generalized scenarios” on physical states in Ω, and R(X) provides the set of all deterministic portfolios of extra capitals that cover potential losses under all the scenarios specified in P. For example, if P is taken to be the set of conditional probability measures given various tail events, (1.1) yields the worst conditional expectation for random vector X, defined as W CEp(X) := {x ∈ Rd : E(x − X | B) ≥ 0, ∀B ∈ F with P(B) ≥ 1 − p}, 0 < p < 1. For any continuous random loss vector X, W CEp(X) equals the tail conditional expectation (TCE) for X, defined as in [6] by, T CEp(X) := {x ∈ Rd : E(x − X | X ∈ A) ≥ 0, ∀A ∈ Qp(X)} =

• A∈Qp(X)

(E(X | X ∈ A) + Rd

+), 0 < p < 1,

(1.2) 2

where Qp(X) = {A ⊆ Rd : A is upper, Pr{X ∈ A} ≥ 1 − p}. When d = 1, T CEp(X) = E(X|X > VaRp(X)) becomes the univariate coherent risk measure of tail VaR. Observe that T CEp(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) for all the tail events that X ∈ A with probability exceeding 1 − p. Tail risk can be described by W CEp(X) or T CEp(X) when p → 1. Note that multivariate coherent risk measures discussed in [13, 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 [8] 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 [8] for details). As a matter of fact, it can be easily verified that T CEp(X) is coherent in the sense of Definition 1.1 if X, which may not be bounded, has a continuous density function. The extreme value analysis of TCE T CEp(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 [21]) distribution F if there exists a Radon measure µ (i.e., finite on compact sets), called the intensity measure, on R

d +\{0} such that

lim

r→∞

Pr{X ∈ rB} Pr{||X|| > r} = µ(B), (1.3) for any relatively compact set B ⊂ R

d +\{0}1 with µ(∂B) = 0, where || · || denote a norm on Rd.

Note that intensity measure µ(·) satisfies the homogeneity property that µ(rB) = r−αµ(B) for any r > 0 and any Borel-measurable set B that is bounded away from the origin. We assume that the heavy-tail index α > 1 to ensure the existence of expectations. The examples and properties

• f MRV distributions, including the relation between MRV distributions and multivariate extreme

value distributions with identical Fr´ echet margins can be found in [21]. Since margins F1, . . . , Fd of F are usually assumed to be tail equivalent [21], we have that the margins are regularly varying with survival function F j(x) = x−αLj(x), 1 ≤ j ≤ d, where Li(x)’s are slowly varying [7], and Li(x)/Lj(x) → cij as x → ∞, 0 < cij < ∞. We assume hereafter that cij = 1 for notational convenience. If cij = 1 for some i = j, we can properly rescale the margins and the results still follow. Observe from (1.3) that the probability of any joint tail event can be estimated proportionally by tail probabilities of the margins: for any subset B bounded away from

1Here R d + = [0, ∞]d is compact and the punctured version R d +\{0} is modified via the one-point uncompactification

(see, e.g., [21]).

3

the origin, Pr{X ∈ rB} ≈ µ(B) µ([1, ∞] × R

d−1 +

) Pr{Xi > r}, as r → ∞. (1.4) That is, multivariate heavy-tails can be approximated proportionally by univariate heavy-tails where the proportionality constants µ(B)/µ([1, ∞] × R

d−1 +

) encode the extremal dependence infor- mation of multivariate extremes. In particular, Pr{||X|| > r} ≈ µ({x : ||x|| > 1}) µ([1, ∞] × R

d−1 +

) Pr{Xi > r}, as r → ∞. (1.5) Thus, any quantile-based risk measure, such as VaR, for the tail risk of portfolio aggregations can be approximated proportionally by marginal tail risk. For example, the computation of VaR for the norm ||X|| is difficult in general, but the tail estimate of VaRp(||X||), when p → 1, is relatively simple in light of (1.5). The tail estimates of VaR of the sum are obtained in [2, 1, 4, 15, 9] in a similar spirit. For the maximum norm of loss vector (X1, . . . , Xd) with identical margins, the VaR can be estimated from the asymptotic relation Pr{max1≤i≤d Xi > r} ≈ Pr{X1 > r}/µ([1, ∞] × R

d−1 +

) for sufficiently large r. The tail estimates for vector-valued TCE T CEp(X) in (1.2) is more difficult, because they involve tail estimates of integrals of joint tail probabilities. The Karamata’s theorem (see [21]) has to be used to control the convergence of tail integrals so that the limiting process can be passed through integration. Theorem 1.2. (Joe and Li [11]) 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 0.

Then limr→∞ r−1E(Xj | X ∈ rB) = ∞

µ(Aj(w)∩B) µ(B)

dw =: uj(B; µ), where Aj(w) := {(x1, . . . , xd) ∈ Rd : xj > w}, 1 ≤ j ≤ d.

• 2. Let Q||·|| := {B ⊆ Rd : B is upper, B ∩ Sd−1

+

= ∅, B ⊆ (Bd)c}, and Bd := {x ∈ Rd : ||x|| < 1} denote the open unit ball in Rd with respect to the norm || · ||. As p → 1, T CEp(X) ≈

• B∈Q||·||

VaR1−(1−p)/µ(B)(||X||)

• (u1(B; µ), . . . , ud(B; µ)) + Rd

. Remark 1.3.

• 1. Theorem 1.2 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

• f loss vector (X1, . . . , Xd).

For example, the tail asymptotics of TCEs of the portfolio aggregation d

i=1 Xi can be obtained from Theorem 1.2 (1) by taking B = {x : d i=1 xi > 1}

(also see [3]). The tail estimate obtained in Theorem 1.2 (2) can be also applied to analyzing coherent aggregations [13] of extremal risks. 4

• 2. Theorem 1.2 (1) can be used in analyzing portfolio tail risk decomposition. For example, for

any 1 ≤ j ≤ d, E

• Xj
• d
• i=1

Xi > VaRp

• d
• i=1

Xi

• ≈ VaRp
• d
• i=1

Xi

• uj(B; µ), as p → 1,

where B = {x : d

i=1 xi > 1}. The tail estimate of E

• Xj | d

i=1 Xi > VaRp(d i=1 Xi)

• 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 [17]. In the situations that the asymptotic expression obtained in Theorem 1.2 (2) may be intractable, we can utilize the method of tail dependence functions introduced in [20, 12] to derive tractable bounds for TCE. 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 = (X1, . . . , Xd) has continuous margins F1, . . . , Fd, and then from [22], the copula C of F can be uniquely expressed as C(u1, . . . , ud) = F(F −1

1 (u1), . . . , F −1 d (ud)), (u1, . . . , un) ∈ [0, 1]d,

where F −1

j

, 1 ≤ j ≤ d, are the quantile functions of the margins. The extremal dependence of a df F can be described the upper tail dependence function [14, 20, 12], defined as follows, b∗(w) := lim

u↓0

C(1 − uwj, 1 ≤ j ≤ d) u , ∀w = (w1, . . . , wd) ∈ Rd

+.

(1.6) Similar to (1.4), (1.6) provides an approximation scheme for upper orthant tail probabilities via univariate margins, that is, C(1−uwj, 1 ≤ j ≤ d) ≈ ub∗(w), as u → 0. Using the inclusion-exclusion principle, we define the upper exponent function of C as follows a∗(w) :=

• S⊆{1,...,d},S=∅

(−1)|S|−1b∗

S(wi, i ∈ S; CS),

(1.7) where b∗

S(wi, i ∈ S; CS) denotes the upper tail dependence function of the margin CS 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 [18]. In particular, b∗(w) = µ(d

i=1[w−1/α i

, ∞]) µ([1, ∞] × R

d−1 +

) , and µ([w, ∞]) µ([0, 1]c) = b∗(w−α

1 , . . . , w−α d )

a∗(1, . . . , 1) . (1.8) Using this equivalence and Theorem 1.2 (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 CEp(X) via Theorem 1.2 (2) is still cumbersome because B ∈ Q||·|| can be quite arbitrary. More tractable bounds for T CEp(X) can be established directly using the tail dependence. 5

Theorem 1.4. (Joe and Li [11]) 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) > 0. Let || · ||max denote the maximum norm.

• 1. For 1 ≤ j ≤ d,

lim

r→∞

1 rE(Xj | X ∈ r(x, ∞]) = ∞ b∗(x−α

1 , . . . , (wj ∨ xj)−α, . . . , x−α d )

b∗(x−α

1 , . . . , x−α d )

dwj.

• 2. Let Sj(b∗, α) :=

∞

b∗(1,...,1,(wj∨1)−α,1,...,1) b∗(1,...,1)

dwj, 1 ≤ j ≤ d. For sufficiently small 1 − p, T CEp(X) ⊆ VaR1−(1−p) a∗(1,...,1)

b∗(1,...,1) (||X||max)

• (S1(b∗, α), . . . , Sd(b∗, α)) + Rd

+

• .
• 3. For sufficiently small 1 − p,

VaRp(||X||max)

• (s1(b∗, α), . . . , sd(b∗, α)) + Rd

+

• ⊆ T CEp(X)

where, for 1 ≤ j ≤ d, sj(b∗, α) := α α − 1 1 b∗(1, . . . , 1) +

• ∅=S⊆{i:i=j}

(−1)|S| 1

0 wjd 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 multivari- ate margin C{j}∪S evaluated with the j-th argument being w−α

j

and others being one. In the proof of Theorem 1.4 (1) the Karamata’s theorem (see [21]) was again used to control the convergence of tail integrals so that the limiting process can be passed through integration. In multivariate risk management, the upper (subset) bound presented in Theorem 1.4 (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.

2 Illustrative Examples of Bounds for Tail Risks

We have some examples to examine the quality of the results in Theorem 1.4 when used as approx-

• imations. 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 − uwj, 1 ≤ j ≤ d) ≈ u b∗(w) + u1+ζ b∗

2(w),

u → 0. (2.1) It is intuitive that if ζ is larger (especially if ζ ≥ 1), then the second order term is less important. Note that for the Fr´ echet upper bound copula, CU(1 − uw) = u min{w1, . . . , wd}, and there is no second order term. 6

Example 2.1. The MTCJ copula (or Mardia-Takahasi-Cook-Johnson copula) in dimension d, with dependence increasing in δ, is: C(u; δ) =

• u−δ

1

+ · · · + u−δ

d

− (d − 1) −1/δ, δ > 0. (2.2) Let wj > 0 for j = 1, . . . , d, and let W := w−δ

1

+ · · · + w−δ

d . Then

C(uw; δ) = u[w−δ

1

+ · · · + w−δ

d

− (d − 1)uδ]−1/δ = uW−1/δ[1 − (d − 1)uδ/W]−1/δ ≈ uW−1/δ 1 + (d − 1)δ−1uδ/W] = ub∗(w; δ) + u1+δb∗

2(w; δ), as u → 0,

where b∗(w; δ) = W−1/δ = (w−δ

1

+ · · · + w−δ

d )−1/δ, b∗ 2(w; δ) = (d − 1)δ−1(w−δ 1

+ · · · + w−δ

d )−1/δ−1.

The second order term of C(uw; δ) is O(u1+ζ), where ζ = δ increases with more dependence. Suppose (X1, . . . , Xd) has the multivariate Pareto distribution with univariate survival function x−α for x > 1 for all d margins and the survival copula given in (2.2). That is, F(x) = C(x−α

1 , . . . , x−α d ; δ) =

• xδα

1 + · · · + xδα d − (d − 1)

−1/δ , xj > 1, j = 1, . . . , d. (2.3) An expression for the conditional expectation (given for the first component only because of sym- metry) is: E [X1|X1 > x1, . . . , Xd > xd] = x1 + ∞

0 F(x1 + z1, x2, . . . , xd) dz1

F(x1, . . . , xd) , leading to TCE r−1E [X1 | X1 > rx1, . . . , Xd > rxd] = x1 + ∞

0 F(rx1 + rw1, rx2, . . . , rxd) dw1

F(rx) . (2.4) The above expectations exist for α > 1. Using (2.4), the exact calculation, first and second order approximations of the last summand in (2.4) can be calculated via one dimensional numerical

• integrations. Table 1 has some (representative) results to show how the approximations compare;

we take r = (1 − p)−1/α, d = 2, x1 = x2 = 1, p = 0.999, α = 2 and 5, and δ ∈ [0.1, 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. Example 2.2. We show the quality of the approximations in parts (2) and (3) of Theorem 1.4 for (2.3) with survival copula (2.2). Since b∗(w) = (w−δ

1

+ · · · + w−δ

d )−1/δ, the margins are given by

b∗

S(wj : j ∈ S) = ( j∈S w−δ j )−1/δ, and these can be used to compute sj(b∗, α) and Sj(b∗, α) via

numerical integrations. The exponent function a∗ is in (1.7). If (X1, . . . , Xd) has the distribution in (2.3), the distribution of Xmax = max{X1, . . . , Xd} is FXmax(x) = F(x, . . . , x) = 1 +

d

• j=1

(−1)j d j

• (jxαδ − j + 1)−1/δ,

x > 0. 7

Table 1: Values of exact TCE minus x1, together with first/second order approximations for the bivariate MTCJ copula with Pareto survival margins; r = (1 − p)−1/α, x1 = x2 = 1, p = 0.999. α = 2 α = 5 δ exact appr1 appr2 exact appr1 appr2 0.1 2.114 4.063 3.349 0.3955 0.5556 0.5079 0.3 2.257 2.464 2.290 0.4382 0.4639 0.4428 0.5 1.968 2.000 1.969 0.4133 0.4180 0.4134 0.7 1.761 1.766 1.761 0.3883 0.3892 0.3883 0.9 1.622 1.624 1.622 0.3690 0.3692 0.3690 1.1 1.526 1.526 1.526 0.3543 0.3543 0.3543 1.3 1.456 1.456 1.456 0.3429 0.3429 0.3429 1.5 1.402 1.402 1.402 0.3338 0.3338 0.3338 1.7 1.360 1.360 1.360 0.3263 0.3263 0.3263 1.9 1.326 1.326 1.326 0.3200 0.3200 0.3200 Based on this distribution, expressions of the form VaRg(p)(||X||max) can be computed numerically. Because of exchangeability, parts (2) and (3) have the form UBd[1d, ∞] ⊆ T CEp(X) ⊆ LBd[1d, ∞]. Table 2 lists the values of LBd and UBd for d = 2, 3 with α = 2 and 5. As might be expected, the ratio UBd/LBd decreases as δ and α increase, and increases as d increases. Example 2.3. We consider general Archimedean copulas which satisfy a regular variation condi-

• tion. Consider a loss vector (X1, . . . , Xd) that has regularly varying margins with heavy-tail index

α > 1, and the Archimedean survival copula C(u; φ) = φ(d

i=1 φ−1(ui)) where the Laplace trans-

form φ is regularly varying at ∞ with tail index β > 0. It follows from Proposition 2.8 of [12] that b∗(w1, . . . , wd) = (w−1/β

1

+ · · · + w−1/β

d

)−β. Observe that (X1, . . . , Xd) is more tail dependent as β

• decreases. Thus, for 1 ≤ j ≤ d,

Sj(b∗, α) = 1 + dβ ∞

1

• wα/β + d − 1

−β dw. sj(b∗, α) = α α − 1dβ + dβ

• ∅=S⊆{i:i=j}

(−1)|S|[(|S| + 1)−β − 1 (wα/β + |S|)−βdw]. It follows from Theorem 1.4 that computable asymptotic bounds are given by (S1(b∗, α), . . . , Sd(b∗, α)) + Rd

+ ⊇ lim p→1

T CEp(X) VaRp(||X||max) ⊇ (s1(b∗, α), . . . , sd(b∗, α)) + Rd

+.

8

Table 2: Bounds for parts (2) and (3) of Theorem 1.4 for the MTCJ copula, with Pareto survival margins; p = 0.999, (1−p)−1/αα/(α−1) = 63.25 and 4.98 provides an intermediate value for α = 2 and 5 respectively. α = 2 α = 5 δ LB2 UB2 LB3 UB3 LB2 UB2 LB3 UB3 0.2 21.46 2908. 11.53 31340. 2.954 211.4 2.133 2208. 0.5 47.21 375.8 41.61 1175. 4.270 30.05 3.967 105.2 0.8 55.07 216.3 51.97 488.9 4.613 17.33 4.456 43.01 1.0 57.48 177.0 55.23 353.8 4.718 14.16 4.605 30.76 1.5 60.29 132.4 59.10 220.2 4.841 10.52 4.782 18.74 2.0 61.45 112.9 60.72 169.2 4.893 8.944 4.857 14.21 3.0 62.38 94.96 62.02 126.8 4.935 7.500 4.918 10.47 4.0 62.74 86.54 62.53 108.4 4.952 6.826 4.942 8.872 5.0 62.91 81.66 62.77 98.22 4.960 6.435 4.953 7.988 8.0 63.11 74.54 63.05 84.07 4.970 5.869 4.967 6.764 Since lim

β→0

∞

1

• wα/β + d − 1

−β dw = ∞

1

w−αdw = 1 α − 1, and lim

β→0

1 (wα/β + |S|)−βdw = 1, we obtain that for fixed α > 1, limβ→0 sj(b∗, α)/Sj(b∗, α) = 1, for 1 ≤ j ≤ d. That is, asymptotic subset and superset bounds are approximately identical for small β.

3 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. The multivariate tail conditional expectation T CEp(X) as p → 1 is essentially linearly related to the value-at-risk of an aggregated norm of X, and the asymptotic proportionality constants 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 approx- imately 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, non-

• verlapping 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 depen- dence; see [16]. When the lower and upper bounds are far apart, reducing the class of relevant upper sets is suggested. 9

The quality of the bounds presented in Theorem 1.4 might be poor for the distributions with weaker tail dependence. In this situation, one may aggregate loss variables with weak tail depen- dence, which also corresponds to choosing some reduced class of specific upper sets B in Theorem 1.2, so that better bounds can be obtained. One can also use the higher order expansions such as (2.1) 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.

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