Tail Risk of Multivariate Regular Variation

Harry Joe∗ Haijun Li†

June 2009

Abstract

Tail risk refers to the risk associated with extreme values and is often affected by extremaldependence among multivariate extremes. Multivariate tail risk, as measured by a coherent riskmeasure of tail conditional expectation, is analyzed for multivariate regularly varying distribu-tions. Asymptotic expressions for tail risk are established in terms of the intensity measure thatcharacterizes multivariate regular variation. Tractable bounds for tail risk are derived in termsof the tail dependence function that describes extremal dependence. Various examples involvingArchimedean copulas are presented to illustrate the results and quality of the bounds.

Key words and phrases: Coherent risk, tail conditional expectation, regularly varying, cop-ula, tail dependence.

MSC2000 classification: 62H20, 91B30.

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 % is defined as a measurable mapping,

with some coherency principles, from the space of all the performance variables into R [21], and

these coherency principles provide a set of operational axioms that % should satisfy in order to

accurately characterize risky behaviors of portfolios. The coherent risk measure, introduced in [1]

for analyzing economic risk of financial portfolios, is an example of such an axiomatic approach.

Let L be the convex cone1 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∗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 0825960.1A subset L of a linear space is a convex cone if x1 ∈ L and x2 ∈ L imply that λ1x1 + λ2x2 ∈ L for any λ1 > 0

and λ2 > 0. A convex cone is called salient if it does not contain both x and −x for any non-zero vector x.

1

worth of a financial position. A mapping % : L → R is called a coherent risk measure if % satisfies

the following four economically coherent axioms:

1. (monotonicity) For X1, X2 ∈ L with X1 ≤ X2 almost surely, %(X1) ≤ %(X2).

2. (subadditivity) For all X1, X2 ∈ L, %(X1 +X2) ≤ %(X1) + %(X2).

3. (positive homogeneity) For all X ∈ L and every λ > 0, %(λX) = λ%(X).

4. (translation invariance) For all X ∈ L and every l ∈ R, %(X + l) = %(X) + l.

The interpretations of these axioms have been well documented in the literature (see, e.g., [21] for

details), and risk %(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 %(X)−X is acceptable to

regulators/supervisors. The general theory of coherent risk measures was developed for arbitrary

real random variables in [9], and more general convex measures that combine subadditivity and

positive homogeneity into the convexity property were extended to cadlag processes in [7], and to

abstract spaces in [11] that include deterministic, stochastic, single or multi-period cash-stream

structures.

It follows from the duality theory that any coherent risk measure %(X) arises as the supremum

of expected values of X, taken over with respect to a convex set of probability measures on envi-

ronmental 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, 0 < p < 1, then the corresponding coherent

risk measure is known as the worst conditional expectation WCEp(X), which, in the case that loss

variable X is continuous, equals to the tail conditional expectation (TCE) defined as follows,

TCEp(X) := E(X | X > VaRp(X)), (1.1)

where VaRp(X) := infx ∈ R : PrX > 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 [21]. 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., TCEp(X) ≥ VaRp(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.

2

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 distribution2. The explicit relation between TCE and VaR for the phase-type loss

distributions was obtained in [6], from which asymptotic equivalence of TCE and VaR as p → 1

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., [5] for detail),

F (r) := 1− F (r) = r−αL(r), r > 0, α > 0, (1.2)

where L is a slowly varying function; that is, L is a positive function on (0,∞) with property

limr→∞

L(cr)L(r)

= 1, for every c > 0. (1.3)

For example, the Pareto distribution with survival function F (r) = (1+r)−α, r ≥ 0, has a regularly

varying tail. It can be easily verified that if α > 1 for Pareto loss variable X, then

TCEp(X) ≈ α

α− 1VaRp(X), as p→ 1. (1.4)

In fact, (1.4) holds for any loss distribution (1.2) that is regularly varying with heavy-tail index

α > 1. Observe that

TCEp(X) =E(XIX > VaRp(X))

PrX > VaRp(X)

=1

PrX > VaRp(X)

(VaRp(X) PrX > VaRp(X)+

∫ ∞VaRp(X)

PrX > xdx

), (1.5)

where I(A) hereafter denotes the indicator function of set A. By the Karamata theorem (see, e.g.,

[25]), we have∫ ∞VaRp(X)

PrX > xdx ≈ 1α− 1

VaRp(X) PrX > VaRp(X), as p→ 1. (1.6)

Plug this estimate into (1.5), we obtain (1.4) for any regularly varying distribution with heavy-tail

index 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 = (X1, . . . , Xd) from a multi-assets portfolio at the end of a given period, where the i-th

component Xi corresponds to the loss of the financial position on the i-th market. A risk measure2That is, the hitting time distribution of a finite-state Markov chain.

3

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. The coherency

principles that are similar to the univariate case were formulated in [15] for multivariate risk measure

R(X), and it was further shown in [4] that for continuous loss vectors, multivariate TCE’s are

coherent in the sense of [15]. Note, however, that multivariate TCE’s, to be formally defined in

Section 2, are subsets of Rd, which lack tractable expressions even for some widely used multivariate

distributions, such as multivariate normals. The effect of dependence among losses X1, . . . , Xd in

different assets on the multivariate TCE also remains difficult to understand. In this paper, we

study asymptotic behaviors of multivariate TCE’s for multivariate regularly varying distributions.

Our method, based on tail dependence functions developed in [23, 14], not only yields explicit

asymptotic expressions of multivariate TCE’s 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 introduced in [15] and obtain the asymptotic expressions of multivariate

TCE’s for multivariate regularly varying distributions in terms of their intensity measures. In

Section 3, we utilize tail dependence functions to obtain asymptotic bounds for multivariate TCE’s.

Section 4 has some concluding remarks. Throughout this paper, measureability 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 [15]. Let K be a

closed, salient convex cone1 of Rd such that Rd+ ⊆ K. The convex cone K induces a partial order

on Rd: x ≤K y if and only if y ∈ x + K. Note that a convex cone K must be an upper set3

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 = Rd+, 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.. See [15] for details.3A 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

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) ⊂ Rd is closed for

any loss random vector X and 0 ∈ R(0) 6= Rd, 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.

When d = 1, %(X) := infr : r ∈ R(X) is a univariate coherent risk measure satisfying the

four axioms discussed in Section 1, and thus R(X) = [%(X),∞). It was shown in [15] that the

worst conditional expectation for random vector X, defined as

WCEp(X) := x ∈ Rd : E(x−X | B) ≥K 0, ∀B ∈ F with P(B) ≥ 1− p, 0 < p < 1,

is a vector-valued coherent risk measure. Since WCEp(X) = ∩B∈F with P(B)≥1−p(E(X | B) +K)

and K is an upper set, WCEp(X) is also an upper set. For any continuous random vector X,

WCEp(X) equals the tail conditional expectation (TCE) for X, defined as in [4] by,

TCEp(X) := x ∈ Rd : E(x−X |X ∈ A) ≥K 0, ∀A ∈ Qp(X)

=⋂

A∈Qp(X)

(E(X |X ∈ A) +K), 0 < p < 1, (2.1)

where Qp(X) = A ⊆ Rd : A is Borel-measurable and A+K = A,PrX ∈ 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 TCEp(X) is a convex and upper set that consists of all the deterministic 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 [15, 4] 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 Rd = [−∞,∞]d. This can be done using the idea in [9] 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 [9] for details). As a matter of fact,

it can be easily verified that TCEp(X) is coherent in the sense of Definition 2.1 if X, which may

not be bounded, has a continuous density function.

5

The extreme value analysis of TCE TCEp(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) distribution F if there exists a Radon measure µ (i.e., finite on compact

sets), called the intensity measure, on Rd+\04 and a common normalization sequence bn with

bn →∞ such that

nPr

X

bn∈ B

→ µ(B), (2.2)

for all relatively compacts B ⊂ Rd\0 with µ(∂B) = 0. The following equivalent characterizations

of MRV distributions can be found in [24, 25, 2, 3].

Theorem 2.2. (Multivariate Regular Variation) The following statements are equivalent:

1. Random vector X has an MRV df F .

2. There exists a Radon measure µ on Rd\0 such that

limr→∞

1− F (rx)1− F (r1)

= limr→∞

PrX/r ∈ [0, x]cPrX/r ∈ [0, 1]c

= µ([0, x]c), (2.3)

for all continuous points x of µ, where µ([0, x]c) = c∫

Sd−1+

max1≤j≤d (uj/xj)α S(du) for some

positive constants c, α and a probability measure S on Sd−1+ := x ∈ Rd

+ : ||x|| = 1, where

|| · || denote a norm on Rd.

3. There exists a Radon measure µ on Rd\0 such that for every Borel set B ⊂ Rd\0 bounded

away from the origin satisfying that µ(∂B) = 0,

limr→∞

PrX ∈ rBPr||X|| > r

= µ(B), (2.4)

with the homogeneity condition µ(rB) = r−αµ(B).

Observe from (2.3) that the margins Fj , 1 ≤ j ≤ d, of an MRV df F are regularly varying in

the sense of (1.2). Since F1, . . . , Fd are usually assumed to be tail equivalent [25], we have that

F j(x) = Lj(x)/xα, 1 ≤ j ≤ d, where Li(x)/Lj(x) → cij as x → ∞, 0 < cij < ∞. We assume

hereafter that cij = 1 for notational convenience. If cij 6= 1 for some i 6= 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. It is well-known from [24] that a random vector X has an

MRV distribution F if and only if X is in the maximum domain of attraction of a multivariate

extreme value (MEV) distribution with identical Frechet margins H(x; 1/α) := exp−x−α for

4Here Rd+ = [0,∞]d is compact and the punctured version Rd+\0 is modified via the one-point uncompactification

(see, e.g., [25]).

6

x > 0 and α > 0. That is, properly normalized component-wise maximums of random samples

from df F converge weakly, as the sample size goes to infinity, to an MEV distribution with identical

Frechet margins. In general, the margins of an MEV distribution can be expressed in terms of the

generalized extreme value family,

H(x; ξ) := exp−(max1 + ξx, 0)−1/ξ, x ∈ R, ξ ∈ R,

and in particular, with Frechet margins, the extremal dependence structure can be characterized

by intensity measure µ. Note, however, that the parametric feature, enjoyed by the univariate EV

distributions, is lost in the multivariate context.

Theorem 2.3. 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) =∫∞

0µ(Aj(w)∩B)

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

2. As p→ 1,

TCEp(X) ≈⋂

B∈Q||·||

VaR1−(1−p)/µ(B)(||X||) ((u1(B;µ), . . . , ud(B;µ)) +K)

whereQ||·|| := B ⊆ Rd : B+K = B,B∩Sd−1+ 6= ∅, B ⊆ (Bd)c, and Bd := x ∈ Rd : ||x|| ≤ 1

denotes the unit ball in Rd with respect to the norm || · ||.

Proof. To estimate E(X | X ∈ rB) for any upper set B bounded away from 0, consider, for any

1 ≤ j ≤ d,

E(Xj |X ∈ rB) =∫ ∞

0PrXj > x |X ∈ rBdx = r

∫ ∞0

PrXj > rw,X ∈ rBPrX ∈ rB

dw. (2.5)

We first argue that we can pass the limit through the integration. Since upper set B 6= ∅, we have

that the complement Bc 6= Rd is a lower set3 with respect to the ≤K-order. Since K ⊇ Rd+, there

exists a vector w = (w1, . . . , wd) such that the complement Bc ⊇ w − K ⊇ w − Rd+, and thus

B ⊆(∏d

i=1(−∞, wi])c

. Therefore,

PrXj > rw,X ∈ rB ≤ Pr

Xj > rw,X ∈

( d∏i=1

(−∞, rwi])c

≤ Pr Xj > rw,Xj > rwj+ Pr rwj ≥ Xj > rw . (2.6)

Observe from (2.4) and the generalized dominated convergence theorem that, as r →∞,∫ ∞0

Pr rwj ≥ Xj > rwPrX ∈ rB

dw =∫ wj

0

Pr rwj ≥ Xj > rwPrX ∈ rB

dw →∫ wj

0

µ(Aj(w)\Aj(wj))µ(B)

dw,

7

where Aj(w) := (x1, . . . , xd) ∈ Rd : xj > w. For the first summand of (2.6), we have∫ ∞0

Pr Xj > rw,Xj > rwjPrX ∈ rB

dw = wjPr Xj > rwjPrX ∈ rB

+∫ ∞wj

Pr Xj > rwPrX ∈ rB

dw.

Using the Karamata theorem (1.6), we have, as r →∞,∫ ∞wj

Pr Xj > rwPrX ∈ rB

dw =∫ ∞rwj

Pr Xj > xrPrX ∈ rB

dx ≈ 1α− 1

rwj Pr Xj > rwjrPrX ∈ rB

.

Thus, we have, via (2.4), as r →∞,∫ ∞0

Pr Xj > rw,Xj > rwjPrX ∈ rB

dw → wjµ(Aj(wj))µ(B)

+1

α− 1wjµ(Aj(wj))

µ(B)=∫ ∞

0

µ(Aj(w ∨ wj))µ(B)

dw,

where the last equality follows from the direct calculation via (2.3). Therefore, we have

limr→∞

∫ ∞0

(Pr rwj ≥ Xj > rw

PrX ∈ rB+

Pr Xj > rw,Xj > rwjPrX ∈ rB

)dw

=∫ ∞

0

(µ(Aj(w)\Aj(wj))

µ(B)+µ(Aj(w ∨ wj))

µ(B)

)dw

=∫ ∞

0limr→∞

(Pr rwj ≥ Xj > rw

PrX ∈ rB+

Pr Xj > rw,Xj > rwjPrX ∈ rB

)dw, (2.7)

where the second equality follows from (2.4). Because of (2.6), (2.7) and the generalized dominated

convergence theorem, we have from (2.5) that

limr→∞

1rE(Xj |X ∈ rB) = lim

r→∞

∫ ∞0

PrXj > rw,X ∈ rBPrX ∈ rB

dw

=∫ ∞

0limr→∞

PrXj > rw,X ∈ rBPrX ∈ rB

dw =∫ ∞

0

µ(Aj(w) ∩B)µ(B)

dw. (2.8)

This concludes the proof of statement (1).

For statement (2), we simplify the asymptotic expression for (2.1). For any upper set A ∈Qp(X), there exists an upper set B with B ∩ Sd−1

+ 6= ∅ and a positive number rB such that

A = rBB. Consider p → 1. Since PrX ∈ rB is decreasing in r, we can find rB,p ≥ rB for any

A = rBB such that

PrX ∈ A ≥ PrX ∈ rB,pB = 1− p, as p→ 1.

It follows from (2.8) that E(Xj |X ∈ rB,pB) is asymptotically increasing for sufficiently small 1−pand goes to +∞ as p → 1, and thus we have E(X | X ∈ A) ≤ E(X | X ∈ rB,pB) for sufficiently

small 1 − p. Since E(X | X ∈ A) + K ⊇ E(X | X ∈ rB,pB) + K for sufficiently small 1 − p, we

have, as p→ 1,

(E(X |X ∈ A) +K) ∩ (E(X |X ∈ rB,pB) +K) = E(X |X ∈ rB,pB) +K.

8

Observe that rB,pB ∈ Qp(X), and we have

limp→1

[( ⋂B∈Q

(E(X |X ∈ rB,pB) +K)

)\ TCEp(X)

]

= limp→1

⋂A∈Qp(X)

(E(X |X ∈ A) +K)

\ TCEp(X)

= ∅,

where Q := B ⊆ Rd : B + K = B,B ∩ Sd−1+ 6= ∅, B is bounded away from 0 and PrX ∈

rB,pB = 1− p. That is, (2.1) can be rewritten as follows, for sufficiently small 1− p,

TCEp(X) ≈⋂B∈Q

(E(X |X ∈ rB,pB) +K). (2.9)

For any B ∈ Q, there exists a real number rB with rB ≥ 1 such that rBB ∈ Q||·|| = B ⊆ Rd :

B + K = B,B ∩ Sd−1+ 6= ∅, B ⊆ (Bd)c. That is, for any B ∈ Q with PrX ∈ rB,pB = 1 − p, we

can find a B′ ∈ Q||·|| and a real number rB′,p (e.g., rB′,p = rB,p/rB) such that rB,pB = rB′,pB′.

Thus (2.9) can be rewritten further as

TCEp(X) ≈⋂

B∈Q||·||,PrX∈rB,pB=1−p

(E(X |X ∈ rB,pB) +K), (2.10)

for sufficiently small 1 − p. Observe that as p → 1, rB,p → ∞, and thus it follows from (2.4) that

for sufficiently small 1− p,µ(B) Pr||X|| > rB,p ≈ 1− p,

which implies that rB,p ≈ VaR1−(1−p)/µ(B)(||X||) as p→ 1. Therefore, (2.8) and (2.10) imply that

TCEp(X) ≈⋂

B∈Q||·||

VaR1−(1−p)/µ(B)(||X||) ((u1(B;µ), . . . , ud(B;µ)) +K)

as p→ 1, where uj(B;µ) =∫∞0

µ(Aj(w)∩B)µ(B) dw, 1 ≤ j ≤ d.

3 Bounds for Tail Risks via Tail Dependence Functions

Theorem 2.3 shows how extremal dependence, as described by the intensity measure, would affect

tail risks, but the asymptotic expression obtained in Theorem 2.3 is intractable for some multivariate

distributions. In this section, we utilize the method of tail dependence functions introduced in

[23, 14] to derive tractable bounds for TCE. For notational convenience, we only consider the case

where K = Rd+.

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

9

copula of F . The copula-based approach for extremal dependence analysis is especially effective in

developing versatile parametric dependence models [13, 22]. Assume that df F of random vector

X = (X1, . . . , Xd) has continuous margins F1, . . . , Fd, and then from [26], the copula C of F can

be uniquely expressed as

C(u1, . . . , ud) = F (F−11 (u1), . . . , F−1

d (ud)), (u1, . . . , un) ∈ [0, 1]d,

where F−1j , 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 lower

or upper tail dependence parameters, for example, are the conditional probabilities that random

vector (U1, . . . , Ud) := (F1(X1), . . . , Fd(Xd)) with standard uniform margins belongs to lower or

upper tail orthants given that a univariate margin takes extreme values (small or large):

λL = limu↓0

PrU1 ≤ u, . . . , Ud ≤ u | Ud ≤ u = limu↓0

C(u, . . . , u)u

λU = limu↓0

PrU1 > 1− u, . . . , Ud > 1− u | Ud > 1− u = limu↓0

C(1− u, . . . , 1− u)u

. (3.1)

where C denotes the survival function of C. Bivariate tail dependence has been widely studied

[13], and various multivariate versions of tail dependence parameters have also been introduced

and studied in [16, 18].

Observe from (3.1) that the tail dependence parameters of copula C are the conditional tail

probabilities that components Ui’s go to extremes at the same rate (same relative scale), and thus

they describe only some aspects of extremal dependence. The tail dependence parameters also

lack operational properties to facilitate the extremal dependence analysis of certain multivariate

distributions, such as vine copulas, that are constructed from basic building blocks of bivariate dis-

tributions. To overcome these deficiencies, the lower and upper tail dependence functions, denoted

by b(·;C) and b∗(·;C) respectively, were introduced in [16, 23, 14] as follows,

b(w;C) := limu↓0

C(uwj , 1 ≤ j ≤ d)u

, ∀w = (w1, . . . , wd) ∈ Rd+;

b∗(w;C) := limu↓0

C(1− uwj , 1 ≤ j ≤ d)u

, ∀w = (w1, . . . , wd) ∈ Rd+. (3.2)

Since b(w; C) = b∗(w;C) where C(u1, . . . , ud) = C(1− u1, . . . , 1− ud) is the survival copula of C,

we focus only on upper tail dependence in this paper and any result about upper tail dependence

can be easily translated into the result for lower tail dependence. The explicit expression of b∗ for

elliptical distributions was obtained in [16]. A theory of tail dependence functions was developed

in [23, 14] based on the Euler’s formula for homogeneous functions:

b∗(w;C) =d∑j=1

wjtj(wi, i 6= j | wj), ∀w = (w1, . . . , wd) ∈ Rd+, (3.3)

10

where the upper conditional tail dependence functions tj(wi, i 6= j | wj) := limu↓0 PrUi > 1 −uwi, ∀i 6= j | Uj = 1 − uwj, 1 ≤ j ≤ d, are homogeneous of order zero. For the copulas with

explicit expressions, the tail dependence functions can be obtained directly with relative ease. For

the copulas without explicit expressions, the tail dependence functions can be obtained via (3.3) by

exploring closure properties of conditional distributions. For example, the tail dependence function

of the t distribution can be obtained by (3.3) (see [23]). If follows from (3.1)–(3.3) that the upper

tail dependence parameter can be expressed as

λU =d∑j=1

limu↓0

PrUi > 1− u, ∀i 6= j | Uj = 1− u,

which extends the well-known formula (see, e.g., [10]) for bivariate tail dependence parameters to

the multivariate case. It was shown in [14] that b∗(w;C) > 0 for all w ∈ Rd+ if and only if λU > 0.

Unlike λU , however, the tail dependence function provides all the extremal dependence information

of copula C as specified by its extreme value copula [23, 14].

Using the inclusion-exclusion principle, we define the upper exponent function of C as follows

a∗(w;C) :=∑

S⊆1,...,d,S 6=∅

(−1)|S|−1b∗S(wi, i ∈ S;CS), (3.4)

where b∗S(wi, i ∈ S;CS) denotes the upper tail dependence function of the margin CS of C with

component indexes in S. Similar to tail dependence functions, the exponent function has the

following homogeneous representation:

a∗(w;C) =d∑j=1

wjtj(wi, i 6= j | wj), ∀w = (w1, . . . , wd) ∈ Rd+, (3.5)

where tj(wi, i 6= j | wj) = limu↓0 PrUi ≤ 1 − uwi,∀i 6= j | Uj = 1 − uwj, 1 ≤ j ≤ d, is

homogeneous of order zero. It was shown in [14] that tail dependence functions b∗S(wi, i ∈ S;CS)and the exponent function a∗(w;C) are uniquely determined from one to another.

It follows from Theorem 2.4 of [18] and (2.3) that

µ([1,∞]d) =b∗(1, . . . , 1;C)a∗(1, . . . , 1;C)

, and µ([0,1]c) = 1. (3.6)

In fact, the detailed relations between the intensity measure µ and tail dependence function b∗ have

been established in [19], and in particular,

µ([w,∞]) =b∗(w−α1 , . . . , w−αd ;C)

a∗(1, . . . , 1;C), ∀w = (w1, . . . , wd) ∈ Rd

+.

The Radon-Nikodym derivative of µ with respect to the Lebesgue measure is then given by

dµ

dv

∣∣∣v=w

=

(αd∏di=1w

−α−1i

a∗((1, . . . , 1);C)

)∂b∗(v1, . . . , vd;C)∂v1, . . . , ∂vd

∣∣∣vi=w−αi di=1

, ∀w = (w1, . . . , wd) ∈ Rd+. (3.7)

11

If the tail dependence function is explicitly known, such as in the case of Archimedean copulas

[14], then the Radon-Nikodym derivative (3.7) of the corresponding intensity measure µ can be

calculated explicitly, and

µ(Aj(w) ∩B) =∫Aj(w)∩B

dµ

dv

∣∣∣v=x

dx, µ(B) =∫B

dµ

dv

∣∣∣v=x

dx.

Therefore, by Theorem 2.3 (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 TCEp(X)

via Theorem 2.3 (2) is still cumbersome because B ∈ Q||·|| can be quite arbitrary. More tractable

bounds for TCEp(X) can be established using the tail dependence and exponent functions, as

shown in the next theorem.

Theorem 3.1. 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;C) > 0. Let

|| · ||max denote the maximum norm.

1. For 1 ≤ j ≤ d,

limr→∞

1rE(Xj |X ∈ r(x,∞]) =

∫ ∞0

b∗(x−α1 , . . . , (wj ∨ xj)−α, . . . , x−αd ;C)b∗(x−α1 , . . . , x−αd ;C)

dwj .

2. For sufficiently small 1− p,

TCEp(X) ⊆ VaR1−(1−p)a

∗(1,...,1;C)b∗(1,...,1;C)

(||X||max)(

(S1(b∗, α), . . . , Sd(b∗, α)) + Rd+

)where Sj(b∗, α) =

∫∞0

b∗(1,...,1,(wj∨1)−α,1,...,1;C)b∗(1,...,1;C) dwj , 1 ≤ j ≤ d.

3. For sufficiently small 1− p,

VaRp(||X||max)(

(s1(b∗, α), . . . , sd(b∗, α)) + Rd+

)⊆ TCEp(X)

where, for 1 ≤ j ≤ d,

sj(b∗, α) :=α

α− 11

b∗(1, . . . , 1;C)

+∑

∅6=S⊆i:i 6=j

(−1)|S|b∗j∪S(1, . . . , 1;Cj∪S)−

∫ 10 b∗j∪S(w−αj , 1, . . . , 1;Cj∪S)dwj

b∗(1, . . . , 1;C).

Proof. Let F have margins F1, . . . , Fd that are regularly varying in the sense of (1.2). Since

F1, . . . , Fd are tail equivalent [25], we have that F j(x) = Lj(x)/xα, 1 ≤ j ≤ d, where Li(x)/Lj(x)→1 as x→∞.

12

(1) Without loss of generality, let j = 1. The straightforward calculation shows

E(X1 |X > rx) =∫ ∞

0

PrX1 > x,X1 > rx1, . . . , Xd > rxdPrX1 > rx1, . . . , Xd > rxd

dx

= rx1 +∫ ∞rx1

PrX1 > x,X2 > rx2, . . . , Xd > rxdPrX1 > rx1, . . . , Xd > rxd

dx

= r

(x1 +

∫ ∞x1

PrX1 > rw,X2 > rx2, . . . , Xd > rxdPrX1 > rx1, . . . , Xd > rxd

dw

)= r

(x1 +

∫ ∞x1

PrU1 > F1(rw), U2 > F2(rx2), . . . , Ud > Fd(rxd)PrU1 > F1(rx1), . . . , Ud > Fd(rxd)

dw

).

Applying the Karamata theorem and generalized dominated convergence theorem, we are allowed

to pass the limit through the integral. Since Lj , 1 ≤ j ≤ d, are slowly varying and the margins are

tail equivalent, we have,

limr→∞

1rE(X1 |X > rx)

= x1 + limr→∞

∫ ∞x1

PrU1 > 1− L1(rw)/(rw)α, . . . , Ud > 1− Ld(rxd)/(rxd)αPrU1 > 1− L1(rx1)/(rx1)α, . . . , Ud > 1− Ld(rxd)/(rxd)α

dw

= x1 +∫ ∞x1

limr→∞

PrU1 > 1− w−αL1(r)r−α, U2 > 1− x−α2 L1(r)r−α, . . . , Ud > 1− x−αd L1(r)r−αPrU1 > 1− x−α1 L1(r)r−α, . . . , Ud > 1− x−αd L1(r)r−α

dw

= x1 +∫ ∞x1

limu→0

PrU1 > 1− w−αu, U2 > 1− x−α2 u, . . . , Ud > 1− x−αd uPrU1 > 1− x−α1 u, . . . , Ud > 1− x−αd u

dw

= x1 +∫ ∞x1

b∗(w−α, x−α2 , . . . , x−αd ;C)b∗(x−α1 , x−α2 , . . . , x−αd ;C)

dw =∫ ∞

0

b∗((w1 ∨ x1)−α, x−α2 , . . . , x−αd ;C)b∗(x−α1 , . . . , x−αd ;C)

dw1.

(2) It follows from (2.10) that as p→ 1,

TCEp(X) ⊆⋂

x∈Sd−1+

(E(X |X ∈ rx,p(x,∞]) + Rd+)

where rx,p satisfies PrX ∈ rx,p(x,∞] = 1 − p. Since b∗(1;C) > 0, it follows from Theorem 2.4

of [18] that µ((1,∞]) > 0 (see also (3.6)). Since ||X||max is regularly varying at ∞, we have for

sufficiently small 1− p, there exists r1,p, such that

µ((1,∞]) Pr||X||max > r1,p = 1− p,

which implies that r1,p ≈ VaR1−(1−p)/µ((1,∞])(||X||max) as p → 1. Observe that as p → 1,

r1,p →∞, and thus it follows from (2.4) that for sufficiently small 1− p,

PrX ∈ r1,p(1,∞] ≈ µ((1,∞]) Pr||X||max > r1,p = 1− p.

Therefore, as p→ 1,

TCEp(X) ⊆⋂

x∈Sd−1+

(E(X |X ∈ rx,p(x,∞]) + Rd+) ⊆ E(X |X ∈ r1,p(1,∞]) + Rd

+. (3.8)

13

Observe from (1) and (3.6) that as p→ 1,

E(X |X ∈ r1,p(1,∞]) ≈ VaR1−(1−p)a

∗(1,...,1;C)b∗(1,...,1;C)

(||X||max)(S1(b∗, α), . . . , Sd(b∗, α))

where Sj(b∗, α) = 1 +∫∞1

b∗(1,...,1,w−αj ,1,...,1;C)

b∗(1,...,1;C) dwj , 1 ≤ j ≤ d. Plug this estimate into (3.8), we

obtain (2).

(3) In light of (2.10), consider, for any B ∈ Q||·||maxwith PrX ∈ rB,pB = 1− p,

E(Xj |X ∈ rB,pB) =E(XjIX ∈ rB,pB)

PrX ∈ rB,pB.

Since (1,∞]d ⊆ B ⊆ [0,1]c for any B ∈ Q||·||max, we have

E(Xj |X ∈ rB,pB) ≤E(XjIX ∈ rB,p[0,1]c)

PrX ∈ rB,p(1,∞]d=∫ ∞

0

PrXj > x ∩ X ∈ rB,p[0,1]cPrX ∈ rB,p(1,∞]d

dx.(3.9)

If x > rB,p then

PrXj > x ∩ X ∈ rB,p[0,1]c = PrXj > x.

If x ≤ rB,p then

PrXj > x ∩ X ∈ rB,p[0,1]c = PrXj > x ∩ (∪di=1Xi > rB,p)

= Pr∪di=1(Xj > x ∩ Xi > rB,p) = Pr(∪i 6=jXj > x,Xi > rB,p) ∪ Xj > rB,p

=∑

S⊆i:i 6=j

(−1)|S| PrXj > rB,p, Xi > rB,p, i ∈ S −∑

∅6=S⊆i:i 6=j

(−1)|S| PrXj > x,Xi > rB,p, i ∈ S

= PrXj > rB,p+∑∅6=S⊆i:i 6=j

(−1)|S| (PrXj > rB,p, Xi > rB,p, i ∈ S − PrXj > x,Xi > rB,p, i ∈ S) . (3.10)

Since the margins are tail equivalent and slowly varying, we have, for any 0 ≤ wj ≤ 1, and any

∅ 6= S ⊆ i : i 6= j,

limp→1

PrXj > rB,pwj , Xi > rB,p, i ∈ SPrX ∈ rB,p(1,∞]d

= limp→1

PrUj > 1− w−αj r−αB,pLj(rB,pw), Ui > 1− r−αB,pLi(rB,p), i ∈ SPrUi > 1− r−αB,pLi(rB,p), 1 ≤ i ≤ d

= limrB,p→∞

PrUj > 1− w−αj r−αB,pL1(rB,p), Ui > 1− r−αB,pL1(rB,p), i ∈ SPrUi > 1− r−αB,pL1(rB,p), 1 ≤ i ≤ d

=b∗j∪S(w−αj , 1, . . . , 1;Cj∪S)

b∗(1, . . . , 1;C),

14

where b∗j∪S(w−αj , 1, . . . , 1;Cj∪S) denotes the upper tail dependence function of the multivariate

margin Cj∪S evaluated with the j-th argument being w−αj and others being one. Similarly,

limp→1

PrXj > rB,p, Xi > rB,p, i ∈ SPrX ∈ rB,p(1,∞]d

=b∗j∪S(1, . . . , 1;Cj∪S)

b∗(1, . . . , 1;C),

limp→1

PrXj > rB,pPrX ∈ rB,p(1,∞]d

=1

b∗(1, . . . , 1;C). (3.11)

Using the bounded convergence theorem, we then have, for sufficiently small 1− p,∫ 1

0

∑∅6=S⊆i:i 6=j

(−1)|S|PrXj > rB,p, Xi > rB,p, i ∈ S − PrXj > rB,pwj , Xi > rB,p, i ∈ S

PrX ∈ rB,p(1,∞]ddwj

≈∑

∅6=S⊆i:i 6=j

(−1)|S|b∗j∪S(1, . . . , 1;Cj∪S)−

∫ 10 b∗j∪S(w−αj , 1, . . . , 1;Cj∪S)dwj

b∗(1, . . . , 1;C). (3.12)

Plug (3.11) and (3.12) into (3.10), and we have, for sufficiently small 1− p,∫ rB,p

0

PrXj > x ∩ X ∈ rB,p[0,1]cPrX ∈ rB,p(1,∞]d

dx ≈rB,p

b∗(1, . . . , 1;C)+

rB,p∑

∅6=S⊆i:i 6=j

(−1)|S|b∗j∪S(1, . . . , 1;Cj∪S)−

∫ 10 b∗j∪S(w−αj , 1, . . . , 1;Cj∪S)dwj

b∗(1, . . . , 1;C).(3.13)

On the other hand, using the Karamata theorem (1.6), we have, for sufficiently small 1− p,∫ ∞rB,p

PrXj > x ∩ X ∈ rB,p[0,1]cPrX ∈ rB,p(1,∞]d

dx =∫ ∞rB,p

PrXj > xPrX ∈ rB,p(1,∞]d

dx

≈ rB,p1

α− 1PrXj > rB,p

PrX ∈ rB,p(1,∞]d≈

rB,p(α− 1)b∗(1, . . . , 1;C)

. (3.14)

Combining (3.13) and (3.14) into (3.9), we have, for sufficiently small 1− p,

E(Xj |X ∈ rB,pB) ≤ α

α− 1rB,p

b∗(1, . . . , 1;C)

+ rB,p∑

∅6=S⊆i:i 6=j

(−1)|S|b∗j∪S(1, . . . , 1;Cj∪S)−

∫ 10 b∗j∪S(w−αj , 1, . . . , 1;Cj∪S)dwj

b∗(1, . . . , 1;C).

As p → 1, rB,p ≈ VaR1−(1−p)/µ(B)(||X||max) ≤ VaR1−(1−p)/µ([0,1]c)(||X||max) = VaRp(||X||max).

Thus, for sufficiently small 1− p,

E(Xj |X ∈ rB,pB)VaRp(||X||max)

≤ α

α− 11

b∗(1, . . . , 1;C)

+∑

∅6=S⊆i:i 6=j

(−1)|S|b∗j∪S(1, . . . , 1;Cj∪S)−

∫ 10 b∗j∪S(w−αj , 1, . . . , 1;Cj∪S)dwj

b∗(1, . . . , 1;C)=: sj(b∗, α),

15

for any B ∈ Q||·||max. Therefore,

TCEp(X) ⊇ VaRp(||X||max)(

(s1(b∗, α), . . . , sd(b∗, α)) + Rd+

),

for sufficiently small 1− p.

Remark 3.2. Observe that if d = 1, then Theorem 3.1 (2) and (3) reduce to (1.4). In multivariate

risk management, the upper (subset) bound presented in Theorem 3.1 (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.

In the remainder of this section, we have some examples to examine the quality of the results

in Theorem 3.1 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− uwj , 1 ≤ j ≤ d) ≈ u b∗(w;C) + u1+ζ b∗2(w;C), u→ 0. (3.15)

It is intuitive that if ζ is larger (especially if ζ ≥ 1), then the second order term is less important.

It remains an open question whether it can be proved in any generality that ζ increases as the

amount of extremal dependence in a parametric family increases. Note that for the Frechet upper

bound copula, CU (1− uw) = uminw1, . . . , wd, and there is no second order term (i.e., ζ =∞).

Example 3.3. (a) Analysis of complete dependence (the Frechet upper bound). Let CU be

the Frechet upper bound copula of dimension d. Then b∗(w;CU ) = minw1, . . . , wd and

b∗(1;CU ) = 1, a∗(1;CU ) = 1. In part (2) of Theorem 3.1, 1− (1−p)a∗/b∗ = p, and for α > 1,

Sj(b∗, α) = 1 +∫∞1 min1, w−αdw = 1 + (α− 1)−1 = α/(α− 1). In part (3) of Theorem 3.1,

for α > 1, sj(b∗, α) = α/(α−1) +∑∅6=S⊆i:i 6=j(−1)|S|0 = 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;C) → 0 and a∗(1;C) → d and 1 − (1 − p)a∗(1;C)/b∗(1;C) > 0

only if p > 1− b∗(1;C)/a∗(1;C) so that for small b∗(1;C), the result in part (2) is non-trivial

only for large p near 1. This is a hint that all of the limiting results of Theorem 3.1 are

worse for weak tail dependence. In this case, one has to use Theorem 2.3 to approximate the

multivariate TCE.

Example 3.4. We show some details for three copula families to illustrate Theorem 3.1. The first

copula is the exchangeable MTCJ copula (or Mardia-Takahasi-Cook-Johnson copula, see [20, 27, 8]),

the second is a mixture of the MTCJ copula and the independence copula, and the third is a

non-exchangeable trivariate copula. Second order expansions of the tail dependence functions are

obtained and the approximation from part (1) of Theorem 3.1 is summarized in Tables for some

special cases.

16

(a) The MTCJ copula in dimension d, with dependence increasing in δ, is:

C(u; δ) =[u−δ1 + · · ·+ u−δd − (d− 1)

]−1/δ, δ > 0. (3.16)

Let wj > 0 for j = 1, . . . , d, and let W = w−δ1 + · · ·+ w−δd . Then

C(uw; δ) = u[w−δ1 + · · ·+ w−δd − (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+δb2(w; δ), as u→ 0,

where

b(w; δ) = b(w;C) = W−1/δ = (w−δ1 + · · ·+ w−δd )−1/δ,

b2(w; δ) = b2(w;C) = (d− 1)δ−1(w−δ1 + · · ·+ w−δd )−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) is multivariate Pareto of the form used in [20]; the univariate survival

function is x−α for x > 1 for all d margins and the copula is given in (3.16). That is,

F (x) = C(x−α1 , . . . , x−αd ; δ) =[xδα1 + · · ·+ xδαd − (d− 1)

]−1/δ, xj > 1, j = 1, . . . , d. (3.17)

An expression for the conditional expectation (given for the first component only because of

symmetry) 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). (3.18)

The above expectations exist for α > 1. Because we are using the copula with survival

functions, we will use the above b, b2 in their upper tail dependence form of b∗, b∗2.

• Exact calculation of the last summand in (3.18):∫∞0 C

((r[x1 + w1])−α, (rx2)−α, . . . , (rxd)−α; δ

)dw1

C((rx1)−α, . . . , (rxd)−α; δ

)=

∫∞0

[(r[x1 + w1])αδ + (rx2)αδ + · · · (rxd)αδ − (d− 1)

]−1/δdw1[

(rx1)αδ + · · ·+ (rxd)αδ − (d− 1)]−1/δ

.

17

• First order approximation of the last summand in (3.18):∫∞0 b∗

((x1 + w1)−α, x−α2 , . . . , x−αd ; δ

)dw1

b∗(x−α1 , . . . , x−αd ; δ

) =

∫∞0

((x1 + w1)αδ + xαδ2 + · · ·+ xαδd

)−1/δdw1(

xαδ1 + · · ·+ xαδd)−1/δ

.

This can be computed via numerical integration. Let the numerator and denominator

of the above be denoted as N1 = N1(x;α, δ) and D1 = D1(x;α, δ).

• Second order approximation of the last summand in (3.18):

r−αN1 + r−α(1+δ)∫∞0 b∗2

((x1 + w1)−α, x−α2 , . . . , x−αd ; δ

)dw1

r−αD1 + r−α(1+δ)b∗2(x−α1 , . . . , x−αd ; δ

)=N1 + (d− 1)r−αδδ−1

∫∞0

((x1 + w1)αδ + xαδ2 + · · ·+ xαδd

)−1/δ−1dw1

D1 + (d− 1)r−αδδ−1(xαδ1 + · · ·+ xαδd

)−1/δ−1.

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.

(b) Mixture model with MTCJ and independence copulas. Now, the second order term is between

O(u) and O(u2), depending on the amount of dependence in the copula. Let

C(u; δ, β) = (1− β)d∏j=1

uj + β[u−δ1 + · · ·+ u−δd − (d− 1)]−1/δ, δ > 0, 0 < β < 1

so that dependence increases as δ and β increase. Let W = w−δ1 + · · ·+ w−δd . Then

C(uw; δ, β) ≈ (1− β)udd∏j=1

wj + βuW−1/δ[1 + (d− 1)δ−1uδ/W

]= u b(w; δ, β) + u1+ζb2(w; δ, β),

where

b(w; δ, β) = βW−1/δ = β(w−δ1 + · · ·+ w−δd )−1/δ,

b2(w; δ, β) =

(d− 1)βδ−1(w−δ1 + · · ·+ w−δd )−1/δ−1 if δ < d− 1,

(1− β)∏dj=1wj + (d− 1)βδ−1(w−δ1 + · · ·+ w−δd )−1/δ−1 if δ = d− 1,

(1− β)∏dj=1wj 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 0 (i.e., weak dependence). Similar to part (a), we list the exact

TCE and the first/second order approximations for the last summand in (3.18).

18

• Exact (assuming α > 1 as before): with Px =∏dj=1 x

−αi ,

β∫∞0

(r[x1 + w1])αδ + (rx2)αδ + · · ·+ (rxd)αδ − (d− 1)

−1/δdw1 + (1− β)r−dαPxx1/(α− 1)

β

(rx1)αδ + · · ·+ (rxd)αδ − (d− 1)−1/δ + (1− β)r−dαPx

since∫∞0 (x1 + w)−αdw = 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∫∞0 b∗

((x1 + w1)−α, x−α2 , . . . , x−αd ; δ, β

)dw1 + r−α(d−1)

∫∞0 b∗2

((x1 + w1)−α, x−α2 , . . . , x−αd ; δ, β

)dw1

b∗(x−α1 , . . . , x−αd ; δ, β

)+ r−α(d−1)b∗2

(x−α1 , . . . , x−αd ; δ, β

)Table 2 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, β = 0.25, α = 2 and 5, δ ∈ [0.1, 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. This is because the accuracy is

of order O(ud) = O(u2) rather than order O(u1+δ) (see part (a)) that increases as δ > 1

becomes larger.

(c) Consider a trivariate nested Archimedean copula (Joe 1993) that is non-exchangeable. Let

C(u; δ1, δ2) =[(u−δ21 + u−δ22 − 1

)δ1/δ2 + u−δ13 − 1]−1/δ1

, δ2 ≥ δ1 > 0.

Similar to part (a), one gets C(uw; δ1, δ2) ≈ u b(w; δ1, δ2) + u1+δ1b2(w; δ1, δ2), where

b(w; δ1, δ2) =[(w−δ21 + w−δ22

)δ1/δ2 + w−δ13

]−1/δ1=: W−1/δ1 ,

b2(w; δ1, δ2) = δ−11 W−1/δ1−1.

The second order term has a parameter δ1 which is the weakest dependence parameter of the

bivariate margins.

Example 3.5. We show the quality of the approximations in parts (2) and (3) of Theorem 3.1 for

(3.17) with copula (3.16). Since b∗(w; C) = b(w;C) = (w−δ1 + · · ·+ w−δd )−1/δ, the margins are

b∗S(wj : j ∈ S) =(∑j∈S

w−δj

)−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 (3.4). If (X1, . . . , Xd) has the distribution in (3.17), the distribution of Xmax =

19

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

Table 2: Values of exact TCE minus x1, together with first/second order approximations for the

bivariate mixture of independence and MTCJ copulas, with Pareto survival margins; r = (1 −p)−1/α, x1 = x2 = 1, p = 0.999, β = 0.25.

α = 2 α = 5

δ exact appr1 appr2 exact appr1 appr2

0.1 1.951 4.063 3.349 0.3742 0.5556 0.5079

0.3 2.227 2.464 2.290 0.4338 0.4639 0.4428

0.5 1.957 2.000 1.969 0.4114 0.4180 0.4134

0.7 1.755 1.766 1.761 0.3872 0.3892 0.3883

0.9 1.622 1.624 1.622 0.3683 0.3692 0.3690

1.1 1.523 1.526 1.526 0.3538 0.3544 0.3542

1.3 1.453 1.456 1.455 0.3424 0.3429 0.3428

1.5 1.400 1.402 1.402 0.3334 0.3338 0.3337

1.7 1.358 1.360 1.360 0.3259 0.3263 0.3262

1.9 1.324 1.326 1.325 0.3197 0.3200 0.3199

20

Table 3: Bounds for parts (2) and (3) of Theorem 3.1 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

maxX1, . . . , Xd is

FXmax(x) = F (x, . . . , x) = 1 +d∑j=1

(−1)j(d

j

)(jxαδ − j + 1)−1/δ, x > 0.

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,∞] ⊆ TCEp(X) ⊆ LBd[1d,∞].

Table 3 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 3.6. We consider general Archimedean copulas which satisfy a regular variation con-

dition. Consider a loss vector (X1, . . . , Xd) with copula C and regularly varying margins having

heavy-tail index α > 1. Assume that the survival copula C of C is an Archimedean copula

C(u;φ) = φ(∑d

i=1 φ−1(ui)) where the Laplace transform φ is regularly varying at ∞ in the sense

of (1.2) with tail index β > 0. It follows from Proposition 2.8 of [14] that

b∗(w1, . . . , wd;C) = (w−1/β1 + · · ·+ w

−1/βd )−β.

21

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

)−βdw.

sj(b∗, α) =α

α− 1dβ + dβ

∑∅6=S⊆i:i 6=j

(−1)|S|[(|S|+ 1)−β −∫ 1

0(wα/β + |S|)−βdw].

It follows from Theorem 3.1 that computable asymptotic bounds are given by

(S1(b∗, α), . . . , Sd(b∗, α)) + Rd+ ⊇ lim

p→1

TCEp(X)VaRp(||X||max)

⊇ (s1(b∗, α), . . . , sd(b∗, α)) + Rd+.

Since

limβ→0

∫ ∞1

(wα/β + d− 1

)−βdw =

∫ ∞1

w−αdw =1

α− 1, and lim

β→0

∫ 1

0(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 for multivariate TCE are approximately identical

for small β.

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 TCEp(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 (see (3.7) and Theorem 3.1).

Weak tail dependence can occur at some margins in high-dimensional distributions such as

vine copulas (see [14]), and the quality of the bounds presented in Theorem 3.1 is rather poor

for the distributions with weak tail dependence. In this situation, the higher order expansions

such as (3.15) should be used 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.

22

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