fragility index of block tailed vectors

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Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 142 (2012) 1837–1848

0378-37

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jspi

Fragility index of block tailed vectors

Helena Ferreira a, Marta Ferreira b,n

a Department of Mathematics University of Beira Interior, Covilh ~a, Portugalb Department of Mathematics – Center of Mathematics, University of Minho, Braga, Portugal

a r t i c l e i n f o

Article history:

Received 23 December 2011

Accepted 30 January 2012Available online 7 February 2012

Keywords:

Multivariate extreme value theory

Tail dependence

Fragility index

Extremal coefficients

58/$ - see front matter & 2012 Elsevier B.V. A

016/j.jspi.2012.01.021

esponding author.

ail address: [email protected] (M. F

a b s t r a c t

Financial crises are a recurrent phenomenon with important effects on the real

economy. The financial system is inherently fragile and it is therefore of great

importance to be able to measure and characterize its systemic stability. Multivariate

extreme value theory provide us such a framework through the fragility index (Geluk

et al., 2007; Falk and Tichy, to appear-a, to appear-b). Here we generalize this concept

and contribute to the modeling of the stability of a stochastic system divided into

blocks. We will find several relations with well-known tail dependence measures in the

literature, which will provide us immediate estimators. We end with an application to

financial data.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

In the last decade, dependencies between financial asset returns have increased, mostly as a consequence ofglobalization effects and relaxed market regulation. Therefore, the concept of tail dependence has been discussed infinancial applications related to market or credit risk, e.g., Hauksson et al. (2001), Ane and Kharoubi (2003), Junker andMay (2005), Straetmans et al. (2008) and Embrechts and Puccetti (2006). The natural framework to model extremaldependence turns out to be the multivariate extreme value theory. The study of systemic stability is an important issuewithin this context of extreme risk dependence. The fragility of a system has been addressed to the fragility index (FI)introduced in Geluk et al. (2007). More precisely, consider a random vector X¼ ðX1, . . . ,XdÞ and Nx :¼

Pdi ¼ 1 1fXi 4xg the

number of exceedances among X1, . . . ,Xd above a threshold x. The FI corresponding to X is the asymptotic conditionalexpected number of exceedances, given that there is at least one exceedance, i.e., FI¼ limx-1 EðNx9Nx40Þ. The stochasticsystem fX1, . . . ,Xdg is called fragile whenever FI41. Theoretical developments, namely, the asymptotic distribution of Nx

conditional to Nx40, can be seen in Falk and Tichy (to appear-a, to appear-b).In this work we generalize some properties of the FI presented in the references above, contributing to the modeling of

the stability of a stochastic system divided into blocks.We shall state some notation that will be used throughout the paper.Consider D¼ fI1, . . . ,Isg a partition of D¼ f1, . . . ,dg. For the random vector X¼ ðX1, . . . ,XdÞ, let XIj

be a sub-vector of Xwhose components have indexes in Ij, with j¼ 1, . . . ,s. If F denotes the d.f. of X, then FIj

denotes the d.f. of sub-vector XIj,

j¼ 1, . . . ,s, and Fi the marginal d.f., i¼ 1, . . . ,d. Let xIjbe a vector of length 9Ij9 with components equal to x 2 R. We will

say that XIjis the jth block of random vector X and denote by Nx the number of blocks where it occurs at least one

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H. Ferreira, M. Ferreira / Journal of Statistical Planning and Inference 142 (2012) 1837–18481838

exceedance of x, i.e.

Nx ¼Xs

j ¼ 1

1fXIj� xIjg:

All operations and inequalities on vectors are meant componentwise.

Definition 1.1. The fragility index (FI) of a random vector X¼ ðX1, . . . ,XdÞ relative to partition D is

limx-1

EðNx9Nx40Þ ð1Þ

whenever the limit exists and is denoted FIðX,DÞ.

If we consider Ij ¼ fjg, j¼ 1, . . . ,d, we find the FI introduced in Geluk et al. (2007) and latter study by Falk and Tichy(to appear-a, to appear-b). This partition will be denoted as Dn, i.e., Dn ¼ fIj ¼ fjg : j¼ 1, . . . ,dg.

We give particular emphasis to random vectors in the domain of attraction of a multivariate extreme value distribution(MEV) and consider either the case of identically distributed margins or tail equivalent margins in the sense considered inFalk and Tichy (to appear-b). In Section 2 we present some asymptotic properties of the distribution of Nx conditionalto Nx40, and find generalizations of results in Falk and Tichy (to appear-b). We prove that FIðX,DÞ exists and relates withthe extremal coefficients E of Tiago de Oliveira (1962) and Smith (1990) in the case of identically distributed margins(Section 3). We define generalized versions of the multivariate tail dependence coefficients of Li (2009), and extend someof its results (Section 4). In Section 5 we relate this latter coefficients with FIðX,DÞ.

For independent margins we have an unit FI. However, the stability of a stochastic system at higher levels can also becharacterized by asymptotic independence (Geluk et al., 2007). Asymptotic independence means that the dependencywhen present vanishes at extreme quantiles and the system is said to be weakly fragile, albeit possibly correlated (e.g.,Gaussian vectors). We extend the concept of asymptotic independent FI in Geluk et al. (2007) for blocks. A second measureis also presented by extending the 2-blocks asymptotic independent coefficient in Ferreira and Ferreira (2011) to the caseof s-blocks, with s42. This issue is considered in Section 6.

Our results relating FIðX,DÞ with well-known tail dependence measures, for which estimators and respective propertieshave already been study in the literature, will provide us immediate estimators (Section 7). We end with an application tofinancial data.

2. Asymptotic properties of Nx

In this section we present some asymptotic properties of the distribution of Nx conditional to Nx40. We start to relatethis latter with EG

A , the extremal coefficients (Tiago de Oliveira, 1962; Smith, 1990) of the sub-distribution functions of theMEV G corresponding to margins in A, i.e., by assuming G has unit Frechet margins

EGA ¼�log Gð1�1

A ð1Þ, . . . ,1�1A ðdÞÞ, ð2Þ

where, for all A � D, x 2 R

1�1A ðxÞ ¼

1, x 2 A,

1, x =2 A:

(

They may be written through the stable tail dependence function (Huang, 1992)

lGðx�11 , . . . ,x�1

d Þ ¼�log Gðx1, . . . ,xdÞ ¼�log CGðe�1=x1 , . . . ,e�1=xd Þ, ð3Þ

where CG is the copula of G, i.e.

Gðx1, . . . ,xdÞ ¼ CGðG1ðx1Þ, . . . ,GdðxdÞÞ, ðx1, . . . ,xdÞ 2 Rd: ð4Þ

In the sequel we will use notation I ðAÞ ¼ [j2AIj.

Proposition 2.1. If X has d.f. F with identically distributed continuous margins and belongs to the domain of attraction of a MEV

G with unit Frechet margins then, for each k 2 f1, . . . ,sg, we have

limx-1

PðNx ¼ k9Nx40Þ ¼1

EGD

XS�f1,...,sg;9S9 ¼ k

XT�S

ð�1Þ9T9þ1EGI ðT[SC

Þ:

Proof. We have, successively


1�PðNx ¼ 0Þ

XS�f1,...,sg;9S9 ¼ k

P \j2S

XIj�xIj

,\j=2S

XIjrxIj

� �

¼1

1�PðNx ¼ 0Þ

XS�f1,...,sg;9S9 ¼ k

XT�S

ð�1Þ9T9P \j2T[SC

XIjrxIj

!

H. Ferreira, M. Ferreira / Journal of Statistical Planning and Inference 142 (2012) 1837–1848 1839

¼1

1�FðxÞ

XS�f1,...,sg;9S9 ¼ k

XT�S

ð�1Þ9T9þ1ð1�Fðx1�1

I ðT[ScÞð1Þ, . . . ,x1

�1I ðT[Sc

ÞðdÞÞÞ ð5Þ

sinceP

T�Sð�1Þ9T9þ1¼ 0. Assuming w.l.o.g. that F has unit Pareto marginals, we obtained


1�CF 1�1

x1

� � XS�f1,...,sg;9S9 ¼ k

XT�S

ð�1Þ9T9þ1 1�CF 1�1

x1�1I ðT[Sc

Þð1Þ, . . . ,1�1

x1�1I ðT[Sc

ÞðdÞÞ

� �:

�ð6Þ

By hypothesis, F belongs to the domain of attraction of a MEV G, which is equivalent to de Haan and de Ronde (1998):

limt-1

1�CF ð1�y1=x, . . . ,1�yd=xÞ

1=x¼�log CGðe

�y1 , . . . ,e�yd Þ, ðy1, . . . ,ydÞZ0: ð7Þ

Taking limits in (5) and dividing both members by 1=x, conditions (6) and (7) lead us to

limx-1


�log CGðe�1, . . . ,e�1Þ

XS�f1,...,sg;9S9 ¼ k

XT�S

ð�1Þ9T9þ1ð�log CGðe

�1I ðT[Sc Þð1Þ, . . . ,e�1I ðT[Sc ÞðdÞÞÞ

¼1

EGD

XS�f1,...,sg;9S9 ¼ k

XT�S

ð�1Þ9T9þ1EGI ðT[SC

Þ: &

The previous result can be generalized to random vectors X with equivalent marginal distributions, in the sense thatthere exists a d.f. H such that

limx-wðHÞ

1�FiðxÞ

1�HðxÞ¼ gi 2 ð0,1Þ, i¼ 1, . . . ,d, ð8Þ

where w(H) is the right-end-point of H. In this case it is no longer possible an interpretation based on extremal coefficients,as can be seen in the following result.

Proposition 2.2. If X has d.f. F with equivalent marginal distributions in the sense of (8), and belongs to the domain of

attraction of a MEV G with unit Frechet margins then, for each k 2 f1, . . . ,sg, we have

limx-wðHÞ


log CGðe�g1 , . . . ,e�gd Þ

XS�f1,...,sg;9S9 ¼ k

XT�S

ð�1Þ9T9þ1log CGðe�g11I ðT[Sc Þð1Þ, . . . ,e�gd1I ðT[Sc ÞðdÞÞ:

Proof. Observe that

1�Fðx1�1I ðT[Sc

Þð1Þ, . . . ,x1�1I ðT[Sc

ÞðdÞÞ ¼ 1�CF ð1�ð1�F1ðx1�1I ðT[Sc

Þð1ÞÞÞ, . . . ,1�ð1�F1ðx1�1I ðT[Sc

ÞðdÞÞÞÞ

¼ 1�CF ð1�tI ðT[ScÞ

1 ðxÞ, . . . ,1�tI ðT[ScÞ

d ðxÞÞ, ð9Þ

where

tAi ðxÞ ¼ 1AðiÞ

1�FiðxÞ

1�HðxÞð1�HðxÞÞ, i¼ 1, . . . d:

Applying (8), we have

limx-wðHÞ

1�CF ð1�tI ðT[ScÞ

1 ðxÞ, . . . ,1�tI ðT[ScÞ

d ðxÞÞ

1�HðxÞ¼ �log CGðe

�g11I ðT[Sc Þð1Þ, . . . ,e�gd1I ðT[Sc ÞðdÞÞ: ð10Þ

The result follows by retaking expression in (5) and considering (9) and (10). &

If in particular we consider Ij ¼ fjg, j¼ 1, . . . ,d, we find the result of Falk and Tichy (to appear-b).The results above can also be obtained through the relation between CG and D-norms presented in Aulbach et al.

(to appear). However, we have chosen to present self-contained proofs using the usual arguments of multivariate extremevalue theory that are more familiar.

3. The fragility index for blocks

In this section we compute the FI for blocks given in (1), whenever X has equally distributed or tail equivalent marginsin the sense of (8), belonging to the domain of attraction of a MEV G with unit Frechet margins.


Proposition 3.1. If X has d.f. F with identically distributed continuous margins and belongs to the domain of attraction of a MEV

G with unit Frechet margins, we have

FIðX,DÞ ¼Ps

j ¼ 1 EGIj

EGD

:

Proof. Observe that

FIðX,DÞ ¼Xs

j ¼ 1

limx-1

PðXIj�xIj

9Nx40Þ ¼Xs

j ¼ 1

limx-1

1�Fðx1�1Ijð1Þ, . . . ,x1�1

IjðdÞÞ

1�Fðx, . . . ,xÞ

¼Xs

j ¼ 1

limx-1

1�CF 1�1

x1Ijð1Þ, . . . ,1�

1

x1IjðdÞ

� �=ð1=xÞ

1�CF 1�1

x, . . . ,1�

1

x

� �=ð1=xÞ

¼1

EGD

Xs

j ¼ 1

EGIj: &

In particular, for partitions corresponding to the family of all margins we obtain the known result FIðX,DnÞ ¼ d=EGD.

Hence, the FI of the system divided into blocks is smaller than the system itself, i.e., FIðX,DÞrFIðX,DnÞ.

Remark 3.1. We can relate FIðX,DÞ with the fragility indexes of the whole system and of each block. More precisely,FIðX,DÞ is a convex linear combination of the ratios FIðX,DnÞ=FIðXIj

,Dn

IjÞ, since we can write

FIðX,DÞ ¼Xs

j ¼ 1

9Ij9d

FIðX,DnÞ

FIðXIj,DnÞ

:

Furthermore, it is also a weighted mean of those ratios:

FIðX,DÞ ¼ 1

s

Xs

j ¼ 1

9Ij9sd

FIðX,DnÞ

FIðXIj,DnÞ

:

Proposition 3.2 (inter-blocks dependence). Under the conditions of Proposition 3.1, we have

(i)

1rFIðX,DÞrPs

j ¼ 1 EGIjWs

j ¼ 1 EGIj

:

(ii)
FIðX,DÞ ¼ 1 if and only if XIj, j¼ 1, . . . s are independent random vectors.P W
(iii)
FIðX,DÞ ¼ sj ¼ 1 EG
Ij= s

j ¼ 1 EGIj

if and only if XIj, j¼ 1, . . . s are totally dependent random vectors in the sense

GðxÞ ¼Vs

j ¼ 1 GIjðxIjÞ.

Proof. By the inequalitiesYs

j ¼ 1

GIjðxIjÞrGðxÞr

s

j ¼ 1

GIjðxIjÞ

we have, successively

ðGf1gðxÞÞ

Ps

j ¼ 1EG

Ij rGEG

D

f1gðxÞr ðGf1gðxÞÞWs

j ¼ 1EG

Ij and_s

j ¼ 1

EGIjrEG

DrXs

j ¼ 1

EGIj: &

Proposition 3.3 (intra-blocks dependence). Under the conditions of Proposition 3.1, we have

(i)
s
EGD

rFIðX,DÞr d

EGD

:

(ii)
FIðX,DÞ ¼ d=EGD if and only if the sub-vectors XIj
, j¼ 1, . . . s, only have independent r.v.’s.
(iii) FIðX,DÞ ¼ s=EG
D if and only if the sub-vectors XIj, j¼ 1, . . . s, only have totally dependent random r.v.’s.

Proof. Just observe that 1rEIjr9Ij9, with the lower and upper bounds corresponding to, respectively, complete

dependence and independence of r.v.’s within sub-vectors XIj, j¼ 1, . . . s. &

Next result presents an extremal coefficient for the amount of dependence between YIj, j¼ 1, . . . ,s, Y� G, through the FI of

X� F in the domain of attraction of the MEV G.


Proposition 3.4. Under the conditions of Proposition 3.1, we have

GðxÞ ¼ ðGI1ðxI1Þ, . . . ,GIs

ðxIsÞÞ

1=FIðX,DÞ:

Proof. Observe that

GðxÞ ¼ G1ðxÞEG

D ¼ ðG1ðxÞ

Ps

j ¼ 1EG

Ij Þ1=FIðX,DÞ

¼ ðG1ðxÞEG

I1 � � �G1ðxÞEG

Is Þ1=FIðX,DÞ

¼ ðGI1ðxI1Þ, . . . ,GIs

ðxIsÞÞ

1=FIðX,DÞ: &

If we consider partition Dn in the previous result, we obtain the known relation (Smith, 1990)

GðxÞ ¼ ðG1ðxÞdÞ1=FIðX,DnÞ

¼ ðG1ðxÞdÞEG=d¼ G1ðxÞ

EG

:

Observe also that we can write FIðY,DÞ instead of FIðX,DÞ with X in the domain of attraction of the MEV distribution of Ysince Y belongs to the same domain of attraction. We finish this section with a generalization of Proposition 3.1 to the caseof equivalent margins and two illustrative examples.

Proposition 3.5. If X has d.f. F with equivalent marginal distributions in the sense of (8), and belongs to the domain of

attraction of a MEV G with unit Frechet margins then, for each k 2 f1, . . . ,sg, we have

FIðX,DÞ ¼Ps

j ¼ 1 log CGðe�g11Ij

ð1Þ, . . . ,e

�gd1Ij ðdÞÞ

log CGðe�g1 , . . . ,e�gd Þ:

Example 3.1. We consider a random vector X of Example 3.2 in Falk and Tichy (to appear-a), i.e., having componentsXi ¼

Pmk ¼ 1 likYk, where Y1, . . . ,Ym are independent r.v.’s with Pareto(a) distribution, a40, and lijZ0 such thatPm

k ¼ 1 laik ¼ 1, i¼ 1, . . . ,d. Taking for H any of the distributions of the margins of X, the equivalence condition (8) holds

with gi ¼ 1, i¼ 1, . . . ,d. The distribution of X belongs to the domain of attraction of

GðxÞ ¼ exp �Xm

k ¼ 1

3di ¼ 1

lik

xi

� �a !

, x40

with Frechet margins, GiðxÞ ¼ expð�x�aÞ. Hence, for u¼ ðu1, . . . ,udÞ 2 ð0,1Þ,

CGðuÞ ¼ exp �Xmk ¼ 1

3di ¼ 1l

aikð�log uiÞ

!:

By Proposition 3.5, we have

FIðX,DnÞ ¼

Pdj ¼ 1

Pmk ¼ 1 3

di ¼ 1l

aik1fjgðiÞPm

k ¼ 1 3di ¼ 1l

aik

¼

Pdj ¼ 1

Pmk ¼ 1 l

ajkPm

k ¼ 1 3di ¼ 1l

aik

¼dPm

k ¼ 1 3di ¼ 1l

aik

as obtained in Falk and Tichy (to appear-a). For any partition D, we have

FIðX,DÞ ¼Ps

j ¼ 1

Pmk ¼ 1 3

di ¼ 1l

aik1IjðiÞPm

k ¼ 1 3di ¼ 1l

aik

:

To illustrate, consider d¼ 3¼m, a¼ 1 and weights

l11 ¼ 4=8, l12 ¼ 2=8, l13 ¼ 2=8,

l21 ¼ 1=8, l22 ¼ 1=8, l23 ¼ 6=8,

l31 ¼ 3=8, l32 ¼ 2=8, l33 ¼ 3=8:

We have

FIðX,DnÞ ¼3

4=8þ2=8þ6=8¼

24

12¼ 2

and, for D¼ ff1;2g,f3gg

FIðX,DÞ ¼ ð4=8þ2=8þ6=8Þþð3=8þ2=8þ3=8Þ

4=8þ2=8þ6=8¼

20

12o2:

Example 3.2. If G has copula

CGðu1, . . . ,udÞ ¼ exp �Xd

i ¼ 1

ð�log uiÞ1=a

!a !, 0oar1


(symmetric logistic model) then, for any partition D, we have

FIðX,DÞ ¼Ps

j ¼ 1 9Ij9a

da¼Xs

j ¼ 1

9Ij9d

� �a

and FIðX,DnÞ ¼ d1�a as already stated in Geluk et al. (2007). In the symmetric model the FI is only a function of the blockssize. If we consider the more general asymmetric logistic model, whose copula is given by

CGðu1, . . . ,udÞ ¼ exp �Xq

k ¼ 1

Xd

i ¼ 1

ð�bki log uiÞ1=ak

!ak( )

,

where bki are non-negative constants such thatPq

k ¼ 1 bki ¼ 1, i¼ 1, . . . ,d, 0oakr1, k¼ 1, . . . ,q, we obtain

FIðX,DÞ ¼Ps

j ¼ 1

Pqk ¼ 1ð

Pi2Ij

b1=ak

ki ÞakPq

k ¼ 1ðPd

i ¼ 1 b1=ak

ki Þak

:

4. Tail dependence for blocks

In the following we always consider that X has continuous marginal d.f.’s. Consider notation MðIjÞ ¼W

i2IjFiðXiÞ, j 2 D.

Definition 4.1. The upper-tail dependence coefficients of X corresponding to partition D of D are defined by, for eachSD! f1, . . . ,sg

tFS ¼ lim

um1P

\j =2 S

MðIjÞ4u9\j2S

MðIjÞ4u

0@ 1A ð11Þ

when the limit exists.

If we consider partition Dn, then SD! f1, . . . ,dg and we find the definition of Li (2009). Further, the case s¼2 lead us todefinition of Ferreira and Ferreira (2011).

Consider

lFS :¼ lim

um1

PT

j2SMðIjÞ4u� �

1�uð12Þ

for each SD! f1, . . . ,sg. Hence, we can write

tFS ¼

lFf1,...,sg

lFS

: ð13Þ

Observe that lFS corresponds to the multivariate upper-tail dependence coefficient LUð1SÞ in Schmidt and Stadtmuller

(2006), where 1S denotes the unit vector with dimension 9S9. In particular, for partition Dn, lFfi,jg ¼LUð1;1Þ corresponds to

the well-known bivariate tail dependence concept (Sibuya, 1960; Joe, 1997).Before we relate the FI with the tail dependence coefficients corresponding to a partition, we present in this section

some extensions of the results in Li (2009).

Proposition 4.1. If X has MEV distribution G with standard Frechet margins and spectral measure� W defined on the

d-dimensional unit sphere Sd then, for each SD! f1, . . . ,sg, we have

tGS ¼

RSd

Vsj ¼ 1

Wi2Ij

wi dWðwÞRSd

Vj2s

Wi2Ij

wi dWðwÞ: ð14Þ

Proof. From the spectral representation of G we obtain, for u sufficiently close to 1

P\s

j ¼ 1

MðIjÞ4u

0@ 1A¼ 1�X

|aS�f1,...,sg

ð�1Þ9S9þ1G �1�1I ðSÞð1Þ

log u, . . . ,�

1�1I ðSÞðdÞ

log u

!

� 1�X

|aS�f1,...,sg

ð�1Þ9S9þ1 1þ log u

ZSd

_i2I ðSÞ

wi dWðwÞ

0@ 1A¼ 1�

X|aS�f1,...,sg

ð�1Þ9S9þ1 1�ð�log uÞ

ZSd

_j2S

_i2Ij

wi dWðwÞ

0@ 1A: ð15Þ


Since X|aS�f1,...,sg

ð�1Þ9S9þ1¼ 1

and X|aS�f1,...,sg

ð�1Þ9S9þ1_j2S

aj ¼^

j2f1,...,sg

aj

expression in (15) becomes

P\s

j ¼ 1

MðIjÞ4u

0@ 1A� ð�log uÞ

ZSd

s

j ¼ 1

_i2Ij

wi dWðwÞ

0@ 1A:Analogously, we obtain

P\j2S

MðIjÞ4u

0@ 1A� ð�log uÞ

ZSd

^j2S

_i2Ij

wi dWðwÞ

0@ 1A: &

For the particular case Dn, the previous result is the one found in Li (2009). Note also that the numerator of (14) can beexpressed through extremal coefficients as follows:X

|aS�f1,...,sg

ð�1Þ9S9þ1Z

Sd

_j2S

_i2Ij

wi dWðwÞ ¼X

|aS�f1,...,sg

ð�1Þ9S9þ1EGIðSÞ ¼ E

GI1þ � � � þEG

Is�ðEG

I1[I2þEG

I1[I3þ � � �Þ� � � � þð�1Þ9S9þ1EG

I1[��[Is

a generalization of result (15) in Ferreira and Ferreira (2011) where s¼2.The next result highlights the connections between tail dependence and extremal coefficients.

Corollary 4.1. Under the conditions of Proposition 4.1, we have

(i)
lG
S ¼X

|aT�S

ð�1Þ9T9þ1EGIðTÞ,

(ii)

tGS ¼

P|aT�f1,...,sgð�1Þ9T9þ1EG

I ðTÞP|aT�Sð�1Þ9T9þ1EG

I ðTÞ:

We end this section with a generalization of Theorem 2.6 in Li (2009), by adapting the arguments to subsets of D thatcorrespond to unions of blocks in D.

Proposition 4.2. If F belongs to the domain of attraction of a MEV G with unit Frechet margins then, for any partition D and

|aS � f1, . . . ,sg, the non-null upper-tail dependence coefficients tFS are the same as the corresponding ones of G.

5. Fragility index and tail dependence for blocks

In this section we shall see that the asymptotic d.f.’s of the conditional probability of k exceedances between blocks,I1, . . . ,Is, can be derived through the tail dependence coefficients given in (12). More precisely, if the d.f. F of X has taildependence coefficients lF

S corresponding to partition D, we can obtain limx-1 PðNx ¼ k9Nx40Þ from this latter. Inparticular, for F in the domain of attraction of a MEV G, besides the representations presented in the previous results, wecan write those limiting probabilities through tail dependence coefficients lG

S .

Proposition 5.1. Let X be a random vector with d.f. F with continuous and identically distributed margins. Let D be a partition

of D for which the tail dependence coefficients lFS corresponding to F exist for each S � f1, . . . ,sg. Then

(i)
For each k 2 f1, . . . ,sg
limx-1

PðNx ¼ k9Nx40Þ ¼

PS�f1,...,sg,9S9 ¼ k

PT�SC ð�1Þ9T9lF

T[SPsk ¼ 1

PS�f1,...,sg,9S9 ¼ k


T[S

as long as the numerator is non-null.


(ii)
If F belongs to the domain of attraction of a MEV G with unit Frechet marginals, the limits in (i) exist and coincide for both
distributions, F and G.

Proof. Just observe that

limx-1

PðNx ¼ k9Nx40Þ ¼ limum1

PS�f1,...,sg,9S9 ¼ k

PT�SC ð�1Þ9T9Pð

TT[SMðIjÞ4uÞPs

k ¼ 1

PS�f1,...,sg,9S9 ¼ k

PT�SC ð�1Þ9T9Pð

TT[SMðIjÞ4uÞ

:

Now divide both terms of the ratio by 1�u. &

Analogously, we obtain the FI through lFS or lG

S .

Proposition 5.2. Under the conditions of Proposition 5.1, we have

(i)

FIðX,DÞ ¼Ps

j ¼ 1 lFfjgPs

k ¼ 1

PS�f1,...,sg,9S9 ¼ k


T[S

as long as the numerator is non-null.
(ii) If F belongs to the domain of attraction of a MEV G with unit Frechet marginals, FIðX,DÞ exists and the the expression in (i)
coincides for both distributions, F and G.

The statements in (ii) of the two propositions are consequences of Proposition 4.2.

6. Asymptotic independence

If X has independent margins Xi, i¼ 1, . . . ,d, we have an unit FI. As Geluk et al. (2007) observed, in this case we might havean asymptotic independence characterized by a dependency that vanishes at extreme quantiles. Asymptotic independencemeans that the system is weakly fragile, albeit possibly correlated (e.g., Gaussian vectors). Geluk et al. (2007) have defineda fragility index for asymptotic independence (AIFI) given by

ZD ¼1

dlimx-1

Pdi ¼ 1 log PðXi4xÞ

log PðX14x, . . . ,Xd4xÞ: ð16Þ

In case d¼2 we find the Ledford and Tawn coefficient of asymptotic independence (Ledford and Tawn, 1996, 1997) and, ifd42, a multivariate extension of this latter (Ferreira and Ferreira, 2012).

Here we consider an extension of the AIFI in (16) for blocks, in the same spirit of the FI in Proposition 3.1, i.e., by relatingthe AIFI within blocks with the AIFI of the whole vector.

Let ZA be the AIFI of sub-vector XA of X, with A � D, i.e.

ZA ¼1

9A9limx-1

Pi2Alog PðXi4xÞ

log PðXA4xAÞ: ð17Þ

Definition 6.1. Let X¼ ðX1, . . . ,XdÞ be a random vector with FIðX,DÞ ¼ 1. Then the AIFI of X¼ ðX1, . . . ,XdÞ relative topartition D of D is

1

slimx-1

Psj ¼ 1 log PðXIj

4xIjÞ

log PðX14x, . . . ,Xd4xÞð18Þ

whenever the limit exists and is denoted ZðX,DÞ.

Proposition 6.1. Let X¼ ðX1, . . . ,XdÞ be a random vector with FIðX,DÞ ¼ 1. Assume that (16) holds and that (17) holds for all

Ij 2 D, j¼ 1, . . . ,s, with limit given by ZIj, respectively. If X has identically distributed or equivalent margins in the sense of (8),

then

ZðX,DÞ ¼ ZD

1

s

Xs

j ¼ 1

1

ZIj

: ð19Þ

Proof. Observe that by (16) and (17), then

ZðX,DÞ ¼ limx-1

ZD

1

s

Psj ¼ 1

1

ZIj9Ij9

Pi2Ij

log PðXi4xÞ

1

d

Pdi ¼ 1 log PðXi4xÞ


and by (8), we have

ZðX,DÞ ¼ limx-1

ZD

1

s

Psj ¼ 1

1

ZIj

1

9Ij9

Pi2Ij

log giþ logð1�HðxÞÞ

!1

d

Pdi ¼ 1 log giþ logð1�HðxÞÞ

: &

In particular, we have ZðX,DnÞ ¼ ZD and hence, the AIFI of the system divided into blocks is larger than the AIFI of thesystem itself, i.e., ZðX,DÞZZðX,DnÞ.

Remark 6.1. In the particular case of FIðX,DnÞ41 we have ZðX,DÞ ¼ 1.Observe also that, by Proposition 6.1, we can relate ZðX,DÞ with the fragility indexes of the whole system and of each

block. More precisely, ZðX,DÞ is the arithmetic mean of the ratios ZðX,DnÞ=ZðXIj,DnÞ.

Example 6.1. Consider X¼ ðX1, . . . ,XdÞ a standard d-variate Gaussian random vector with d.f. Fdð�; ðSi,jÞi,j2DÞ having positivedefinite correlation matrix ðSi,jÞi,j2D. We have, for all A � D, Z�1

A ¼ 1AðSi,jÞ�1i,j2A1T

A, the sum of all elements of the sub-matrixðSi,jÞ

�1i,j2A (Geluk et al., 2007; Hua and Joe, 2011). For illustration, consider dimension d¼4, constant correlation r, and take

s¼3 with I1 ¼ f1;2g, I2 ¼ f3g and I3 ¼ f4g. We have ZD ¼ ð3rþ1Þ=4, ZI1¼ ðrþ1Þ=2 and ZI2

¼ ZI3¼ 1. Hence

ZðX,DÞ ¼ 1

3

3rþ1

4

2

rþ1þ1þ1

� �¼ð3rþ1Þðrþ2Þ

6ðrþ1Þ:

In the sequel we consider positive/negative association of a random vector in the sense of Ledford and Tawn (1996,1997).

Proposition 6.2 (inter-blocks asymptotic independence). Under the conditions of Proposition 6.1, we have

(i)
ZðX,DÞr1=s in case of positive association between sub-vectors XIj, j¼ 1, . . . s, and ZðX,DÞZ1=s in case of negative
association.
(ii)
ZðX,DÞZVs

j ¼ 1 ZIj

s

Xs

j ¼ 1

1

ZIj

:

(iii)
ZðX,DÞ ¼ 1=s if and only if the sub-vectors XIj, j¼ 1, . . . s, are independent.V P
(iv)
ZðX,DÞ ¼ ð sj ¼ 1 ZIj
=sÞ sj ¼ 1ð1=ZIj

Þ if and only if the sub-vectors XIj, j¼ 1, . . . s, are totally dependent.

Proof. Observe that PðX14x, . . . ,Xd4xÞrVs

j ¼ 1 PðXIj4xIjÞ, with the upper bound corresponding to total dependence

between sub-vectors XIj, j¼ 1, . . . ,s. Under positive association we have PðX14x, . . . ,Xd4xÞZ

Qsj ¼ 1 PðXIj

4xIjÞ and

negative association otherwise, with the bounds corresponding to independence between sub-vectors XIj, j¼ 1, . . . ,s. &

Proposition 6.3 (intra-blocks asymptotic independence). Under the conditions of Proposition 6.1, we have

(i)
ZðX,DÞrZDd=s in case the sub-vectors XIj, j¼ 1, . . . ,s, only have positively associated r.v.’s and ZðX,DÞZZDd=s in case of
negative association.
(ii) ZðX,DÞZZD.
(iii)
ZðX,DÞ ¼ ZDd=s if and only if the sub-vectors XIj, j¼ 1, . . . ,s, only have independent r.v.’s.
(iv)
ZðX,DÞ ¼ ZD if and only if the sub-vectors XIj, j¼ 1, . . . ,s, only have totally dependent random r.v.’s.
Proof. For each j¼ 1, . . . ,s, under positive association of the r.v.’s in XIjwe have PðXIj

4xIjÞZQ

i2IjPðXi4xÞ, and hence

1=ZIjr9Ij9, with the upper bound corresponding to independence. For negative association we have

PðXIj4xIjÞrQ

i2IjPðXi4xÞ. Observe also that total dependence within each block means PðXIj

4xIjÞ ¼V

i2Ijgið1�HðxÞÞ. &

As already mentioned, the definition of the AIFI in (18) measures the asymptotic independent fragility of a system dividedinto blocks (sub-vectors), by relating the asymptotic independent fragility within the blocks and in the whole system. If in(16) we generalize for blocks the concept of an exceedance of a r.v., Xi4x, through events XIj

� xIj, we obtain another

coefficient for asymptotic tail independence. In this way, we extend the coefficient of asymptotic tail independence that wasconsidered in Ferreira and Ferreira (2011) for the particular case of a partition D¼ fI1,I2g of D¼ f1, . . . ,dg.


Definition 6.2. Let X¼ ðX1, . . . ,XdÞ be a random vector with FIðX,DÞ ¼ 1. The coefficient of asymptotic independence ofX¼ ðX1, . . . ,XdÞ relative to partition D of D is

limx-1

1

s

Psj ¼ 1 log PðXIj

�xIjÞ

log PðTs

j ¼ 1fXIj�xIjgÞ

ð20Þ

whenever the limit exists, and is denoted ZðI1 ,...,IsÞ.

The following result is therefore an immediate extension of Proposition 2.4 in Ferreira and Ferreira (2011).

Proposition 6.4. Let X¼ ðX1, . . . ,XdÞ be a random vector with FIðX,DÞ ¼ 1. Assume that the limit in (20) exists and, for all

|aKj � Ij, j¼ 1, . . . ,s, Eq. (17) holds for [sj ¼ 1Kj. Then

ZðI1 ,...,IsÞ¼maxfZfi1 ,...,isg

: ij 2 Ij, j¼ 1, . . . ,sg: ð21Þ

Similar to ZðX,DÞ, coefficient ZðI1 ,...,IsÞis also based on the coefficient of Ledford and Tawn or multivariate extensions of

this latter.In the example below, one can see that the asymptotic tail independent coefficients, ZðX,DÞ and ZðI1 ,...,IsÞ

, are different.

Example 6.2. Consider fVigiZ1 an i.i.d. sequence of unit Pareto r.v.’s. Let ðX1,X2,X3Þ be a random vector such that,X1 ¼minðV1,V2Þ, X2 ¼minðV2,V3Þ and X3 ¼minðV3,V4Þ. If D¼ fI1,I2g, with I1 ¼ f1;2g and I2 ¼ f3g, we have PðXi4xÞ ¼ x�2,i¼ 1;2,3, PðX14x,X24xÞ ¼ PðX24x,X34xÞ ¼ x�3 and PðX14x,X34xÞ ¼ PðX14x,X24x,X34xÞ ¼ x�4. Therefore

FIðX,DÞ ¼ limx-1¼

PðS2

i ¼ 1fXi4xgÞþPðX34xÞ

PðS3

i ¼ 1fXi4xgÞ¼

2x�2�x�3þx�2

3x�2�2x�3�x�4þx�4¼ 1,

ZðX,DÞ ¼ 1

2limx-1

P2j ¼ 1 log PðXIj

4xIjÞ

log PðX14x,X24x,X34xÞ¼

1

2limx-1

log PðX14x,X24xÞþPðX34xÞ


1

2

�3�2

�4¼

5

8

and

ZðI1 ,I2Þ¼

1

2limx-1

log PðXI1�xI1Þþ log PðXI2

�xI2Þ

log PðXI1�xI1

,XI2�xI2Þ

¼1

2limx-1

log ðPðX14xÞþPðX24xÞ�PðX14x,X24xÞÞþ log PðX34xÞ

log ðPðX14x,X34xÞþPðX24x,X34xÞ�PðX14x,X24x,X34xÞÞ

¼1

2limx-1

log ð2x�2�x�3Þþ log x�2

log ðx�4þx�3�x�4Þ¼ 2=3:

Observe that the same results are obtained if we apply Propositions 6.1 and 6.4, respectively. More precisely, we have

ZD ¼ Zf1;2,3g ¼1

3limx-1

P3i ¼ 1 log PðXi4xÞ


1

3

3 log x�2

log x�4¼

1

2,

Zf1;3g ¼1

2

2 log x�2

log x�4¼

1

2and Zf2;3g ¼

1

2

2 log x�2

log x�3¼

2

3:

Obviously ZI2¼ 1 and

ZI1¼

1

2limx-1

P2i ¼ 1 log PðXi4xÞ

log PðX14x,X24xÞ¼

1

2

2 log x�2

log x�3¼

2

3:

Hence, by Propositions 6.1 and 6.4, respectively

ZðX,DÞ ¼ 12 ð

12 ð3=2þ1ÞÞ ¼ 5

8

and

ZðI1 ,I2Þ¼maxfZf1;3g,Zf2;3gg ¼maxf1=2;2=3g ¼ 2=3:

7. Estimation of the fragility index for blocks

In the previous sections we have related the FI for blocks with well-known tail dependence measures. This will allow to obtainimmediate estimators for our index through the estimators of those coefficients that are already studied in the literature.

Proposition 3.1 presents an estimation procedure for the FI based on the extremal coefficients of Tiago de Oliveira(1962) and Smith (1990) given in (2). Observe that they can be expressed through the stable tail dependence function, lG,defined in (3). There are several references in the literature on the estimation of the stable tail dependence function. In aparametric framework, a model for lG must be imposed. Non-parametric estimators are usually based on an arbitrarilychosen parameter, corresponding to the number of top order statistics to use in order to provide the best trade-off


between bias and variance, which is not an easy task. For a survey, we refer Krajina (2010) or Beirlant et al. (2004). Themore recent work in Ferreira and Ferreira (2011) presents a simpler non-parametric estimator based on sample means,which we shall adopt here.

For A � D, denote MðAÞ ¼W

i2AFiðXiÞ. Consider

bEGD ¼

MðDÞ

1�MðDÞand bEG

Ij¼

MðIjÞ

1�MðIjÞ, ð22Þ

where MðAÞ is the sample mean

MðAÞ ¼1

n

Xn

i ¼ 1

_j2A

bF jðXðiÞj Þ ð23Þ

and bF j, j 2 A, is the (modified) empirical d.f. of Fj

bF jðuÞ ¼1

nþ1

Xn

k ¼ 1

1fXðkÞ

jrug

:

The denominator nþ1 instead of n in the ordinary empirical d.f. concerns estimation accuracy and other modifications canbe used (see, for instance, Beirlant et al., 2004). Based on Proposition 3.1, and for a partition D of D, we consider estimator

bFIðX,DÞ ¼Ps

j ¼ 1bEG

IjbEGD

, ð24Þ

which is consistent given the consistency of estimators bEGD and bEG

Ijalready stated in Ferreira and Ferreira (2011).

By Proposition 5.2, we can also estimate FIðX,DÞ based on the tail dependence coefficients in (12). As alreadymentioned, these correspond to multivariate upper-tail dependence coefficients considered in Schmidt and Stadtmuller(2006), for which non-parametric estimators have been studied. We remark that these estimators are also based on asimilar procedure as described above for the stable tail dependence function, i.e., it comprises the choice of a number oftop order statistics to be used, not too large neither too small to avoid, respectively, large bias and large variance.

Remark 7.1. In case of asymptotic independence considered in Section 6, immediate estimators for the asymptoticindependent coefficients, ZðX,DÞ and ZðI1 ,...,IsÞ

, can be derived from the found relations with the Ledford and Tawncoefficient (or multivariate extensions), whose estimation has already been studied in the literature (see, for instance,Draisma et al., 2004 or, for a survey Beirlant et al., 2004). In particular, the Ledford and Tawn coefficient can be estimatedas the extreme value index of r.v. minð1=ð1�F1ðX1ÞÞ,1=ð1�F2ðX2ÞÞÞ and, similarly, its multivariate versions in (17) can beestimated as the extreme value index of r.v. mini2Að1=ð1�FiðXiÞÞÞ.

7.1. An application to financial data

We now illustrate the estimation of the FI for blocks through an application to data analyzed by Ferreira and Ferreira(2011). The data are the series of negative log-returns of the closing values of the stock market indexes, CAC 40 (France),FTSE100 (UK), SMI (Swiss), XDAX (German), Dow Jones (USA), Nasdaq (USA), SP500 (USA), HSI (China), Nikkei (Japan). Theperiod covered is from January 1993 to March 2004. Since we do not have a sample of maximum values, we consider themonthly maximums in each market. We group the indexes in Europe (CAC 40, FTSE100, SMI, XDAX), USA (Dow Jones,Nasdaq) and Far East (HSI, Nikkei). The presence of dependence within these groups was already evidenced in Ferreira andFerreira (2011). We are interested in assessing the fragility within the system of the financial stock markets whenever

grouped in the three big world markets referred: Europe, USA and Far East. To this end, we use estimator bFIðX,DÞ in (24). In

Table 1 are the obtained estimates, as well as, the estimates of the extremal coefficient (bEG) within each group and in the

whole system (we denote the whole system, i.e., the vector of all observations as ‘‘Global’’). The estimates of the FI withineach financial market group and in the whole system are also presented. One can see that the USA is the most fragile

Table 1

Estimates of the extremal coefficient (bEG) within each block (Europe, USA, Far East) and in the whole system (Global)

comprising the three blocks, as well as, estimates of the FI (bFI) within each block, in the whole system and within the

system divided into blocks (bFIðX,DÞ).

Block bEG bFI bFIðX,DÞ

Europe 2.243980009 1.78254707 1.370940479

USA 1.590711176 1.885948905

Far East 1.673156121 1.195345715

Global 4.017568517 2.240160924


financial system with bFI ¼ 1:885948905. Observe also that the FI of the whole system is almost twice the FI of the systemdivided into blocks.

Acknowledgement

Helena Ferreira was partially supported by the research unit ‘‘Centro de Matematica’’ of the University of Beira Interiorand the research project PTDC/MAT/108575/2008 through the Foundation for Science and Technology (FCT) co-financed byFEDER/COMPETE.

Marta Ferreira was partially supported by FEDER funds through ‘‘Programa Operacional Factores de Competitividade -COMPETE’’, by Portuguese funds through FCT - ‘‘Fundac- ~ao para a Ciencia e a Tecnologia’’, within the Project Est-C/MAT/UI0013/2011 and by PTDC/FEDER grants from Portugal.

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fragility index of block tailed vectors

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