Extreme dependence of multivariate catastrophic lossesAs it will be illustrated in our example,...
Transcript of Extreme dependence of multivariate catastrophic lossesAs it will be illustrated in our example,...
UNCORRECTED PROOF
Original Article
Extreme dependence of multivariate catastrophic losses
LAURENCE LESCOURRET* and CHRISTIAN Y. ROBERT$
*ESSEC Business School, Avenue Bernard Hirsch, 95021 Cergy-Pontoise, France
$CNAM, 292 rue Saint-Martin, 75003 Paris, France
(Accepted 6 July 2006)
Natural catastrophes cause insurance losses in several different lines of business. An approach to
modelling the dependence in loss severities is to assume that they are related to the intensity of the
natural disaster. In this paper we introduce a factor model and investigate the extreme dependence. We
derive a specific extreme dependence structure when considering an heavy-tailed intensity. Estimation
procedures are presented and their moderate sample properties are compared in a simulation study.
We also motivate our approach by an illustrative example from storm insurance.
Keywords: Catastrophic losses; Multivariate extreme value distributions; Heavy-tailed distributions;
Probability of catastrophic events
1. Introduction
Each year considerable property damage, economic losses, human suffering and deaths
are caused by natural catastrophes such as earthquakes, windstorms and floods. Insurers
have not shown a real willingness to cover such risks in the past although demand for
insurance against natural hazards has steadily increased. One of the reasons is that a
correct assessment of potential losses is still today a difficult task, even though new
methods have been recently developed for modelling the intensity and the frequency of
natural disasters, and for evaluating insurance and reinsurance premiums or capital needs.
Consequently, one of the main concerns for actuaries is to understand and to quantify the
impact of large events which play a crucial role in the risk management and pricing of
catastrophic business.
In this paper, we assume that the intensity of a natural disaster is a common factor of
losses occuring in several different lines of business and we focus on the extreme
dependence in loss severities. This research is motivated by the analysis of an example
from storm insurance. Figure 1 plots aggregate claims of motor policies and aggregate
claims of household policies from a French insurance portfolio for 736 storm events. The
two outliers correspond to the windstorms Lothar and Martin passed over France in
December of 1999 causing the largest insured catastrophe losses in Europe’s history.
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Corresponding author. E-mail: [email protected]
Scandinavian Actuarial Journal, Vol. 00, 2006, 0�00
Scandinavian Actuarial Journal
ISSN 0346-1238 print/ISSN 1651-2030 online # 2006 Taylor & Francis
http://www.tandf.no/saj
DOI: 10.1080/03461230600889645
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During such events, trees are often agents of damage to both residential building and
motor vehicles, even in city centers. The data demonstrate an apparent tendency for large
claim amounts at both lines of business and suggest that the intensity of the storm events
induces dependence in the series across business lines. This natural idea has already been
used in the the context of insurance loss modelling by Lindskog & McNeil (2003) who
consider a common Poisson shock process to model the number of windstorm losses in
France and in Germany, and by Cossette et al. (2003) who propose a model that allows
damage ratios to be functions of the catastrophe intensity.
Understanding the differences between the analysis of individual lines of business or
portfolio lines of business is of much practical interest, especially to make predictions
about the potential losses in stress situations. Among the most important problems are
the description and inference of the probability p associated with the dashed area (see
Figure 1), i.e. the probability that both claim amounts of the same storm event exceed
high thresholds. Since none of the sample points falls into the area, we can not use the
empirical distribution function to estimate p. In a multidimensional space as in a one-
dimensional space, if one has to do inference in the tail of a distribution outside the range
of the observations, a way to proceed is to use extreme value theory and to model the tail
asymptotically as a tail of an extreme value distribution.
To the best of our knowledge, this paper is the first to propose a factor model to
construct specific extreme dependence structures of multivariate losses and to introduce
statistical tools to evaluate the intensity of the dependence. We also provide a detailed
analysis on storm insurance data where such a structure is exhibited.
In Section 2 we introduce the heavy-tailed factor model. Then we present bivariate
extreme value theory and derive from the factor model a class of bivariate extreme value
distributions which takes into account the dependence structure of catastrophic losses.
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Household claims
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or c
laim
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Figure 1. Storm damages (both variables are on a logarithmic scale).
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Several examples are also given. In Section 3 we discuss two common approaches of
estimating bivariate extreme value distribution and introduce an estimator adapted to our
factor model. We also inspect the finite sample behavior of the estimator on simulated
data and compare its performance with those of standard estimators. In the last section
we present the application to storm insurance data. Proofs are gathered in Appendix A
and asymptotic properties of the new estimator are given in Appendix B.
2. Extreme dependence of catastrophic losses
In this section we introduce a common factor for modelling extreme dependence. In order
to compare the subclass of multivariate extreme value distributions of our model with
general extreme value distributions, we only consider the bivariate case. Note however
that the multivariate extreme value distributions are easily derived from this particular
case.
As it will be illustrated in our example, Pareto distributions are often observed on
natural catastrophe insurance data.1 Consequently we will concentrate on models of
multivariate distributions where the marginals are Pareto-type.
2.1. The heavy-tailed factor model
The basic idea of the factor approach is to use a single intensity variable to describe
(aggregate) amounts of losses across different lines of business. Let us denote by Xi,j the
amount of losses of the j-th line of business for the i-th natural disaster. We consider the
following model:
Xi;j �Tj(Yihi;j); j�1; 2; (1)
where Yi is the intensity of the i-th natural disaster and is a common latent factor, the hi,j
are so-called multiplicative disturbances which are independent of Yi, and the Tj are
transformation functions. We assume that:
� Yi has a Pareto-type distribution FY, i.e. there exists a positive constant a for which
FY (y)�1�FY (y)�y�alY (y);
and lY is a slowly varying function at infinity satisfying limy0�lY(ty)/lY(y)�/1, for all
t�/0.
� hi�/(hi,1, hi,2) is a vector of positive random variables which are independent of Yi.
Moreover there exists a constant d�/a such that Ehdi;j B�; j�/1,2.
� Tj, j�/1,2, are increasing functions such that the inverse functions T1j (x)�
inffy : Tj(y)]xg are regularly varying functions at infinity with index 1/gj�/0, i.e.
T1j (x)�x1=gj lT1
j(x) with lT1
jslowly varying.
� (Yi, hi) are independent and identically distributed (iid) vector random variables (rvs).
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1 See also Benchert & Jung (1974), McNeil (1997), Matthys et al. (2004) for some examples on property
insurance data.
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This construction implies that the amounts of losses Xi,j, i]/1, are iid rvs and have a
Pareto-type distribution.2
Proposition 1 Suppose that Xi,j is specified by Eq. (1). Then
FXj(x)�P(Xi;j �x)�x�bj lXj
(x); i]1; j�1; 2;
where bj�/a/gj and lXjis a slowly varying function.
In such a model Yi induces dependence between the components of the vector
Xi�/{Xi,1, Xi,2}. Let us now focus more specifically on the extreme dependence structure
and study the asymptotic bivariate extreme value distribution of the factor model.
2.2. Bivariate extreme value distributions
i) Definitions
Let {(Yi,1, Yi,2)} for i�/1,. . ., n be iid vector rvs with bivariate distribution F. For j�/1,2
we define Mj,n�/max(Y1,j,. . ., Yn,j). Bivariate extreme value distributions arise as the
limiting joint distribution of normalized componentwise maxima. Assume that there exist
sequences of normalizing functions f1,n, f2,n such that
limn0�
P(M1;n5f1;n(y1);M2;n5f2;n(y2))�G(y1; y2);
where G is a proper multivariate distribution function, nondegenerate in each margin.
Any possible limit distribution G is called a bivariate extreme value distribution. It
follows that the marginal distributions Gj must be one of the univariate extreme
value distributions3: the Weibull distribution Ca(y)�/exp{�/(�/min(y,0))a}, for a�/0,
the Gumbel distribution L(y)�/exp{�/e�y}, or the Frechet distribution Fa(y)�/
exp{�/(max(y,0))�a}, for a�/0.
Note that, in contrast to the univariate case, no natural parametric family exists for
multivariate extreme value distribution G. However G can be described in different forms.
For example, assume that the marginal distributions of G are standard Frechet
distributions F1(y)�/exp{�/y�1}, y�/0. A representation of G is given by
G(y1; y2)�exp
��g A
max(a1y�11 ; a2y�1
2 )S(da)
�; (2)
where a�/(a1, a2) and S is a non-negative, finite measure on A�faj ]0; j�1; 2 : a1�a2�1g satisfying fA ajS(da)�1; j�1; 2 (see e.g. Resnick (1987, Section 5.4)). An
equivalent form is introduced in Pickands (1981)
G(y1; y2)�exp
���
1
y1
�1
y2
�A
�y1
y1 � y2
��; (3)
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2 A related well-known result is Breiman (1965).3 A comprehensive textbook treatment of univariate extremes in insurance and finance is Embrechts et al.
(1997), in which a very extensive list of references may also be found.
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where
A(w)�g1
0
max(a(1�w); (1�a)w)S(da):
A is a convex function bounded above by 1 and below by max(w, 1�/w) and therefore
satisfies A(0)�/A(1)�/1. It is called the dependence function. See also Galambos (1987),
Joe (1997), Hsing, Kluppelberg & Kuhn (2005) for other representations.
ii) Multivariate extreme value distributions for the heavy-tailed factor model
Let us now consider the case of the heavy-tailed factor model. Since the Xi,j have
Pareto-type distribution, the marginal distributions of G are Frechet distributions. It is
well-known (see e.g. Embrechts et al. (1997)) that the choice of the normalizing functions
fj,n(x)�/x1/bjUXj
(n) where UXj(y)�/inf{x : FXj
(x)]/1�/y�1} leads to standard Frechet
distributions for the limiting marginal distributions, i.e.
limn0�
P(Mj;n5fj;n(xj))�F1(xj); j�1; 2:
Before discussing the dependence structure of G, we establish the asymptotic form of the
probability that both amounts of losses Xi,j, j�/1,2 exceed the increasing thresholds
fj,n(xj). This defines a notion of dependence for multivariate extremes which is by no
means the only one available, but which is often of much practical interest. Think for
instance about a multi-line catastrophe Excess-of-loss reinsurance contract where all
losses have to be larger than triggers for the reinsurer beginning to pay.
Proposition 2 Consider the heavy-tailed factor model (1). For x1�/0 and x2�/0,
limn0�
nP(Xi;1�f1;n(x1);Xi;2�f2;n(x2))�P(x1; x2);
where
P(x1; x2)�E min
�x�1
1
hai;1
Ehai;1
; x�12
hai;2
Ehai;2
�: (4)
The asymptotic multivariate extreme value distribution of the heavy-tailed factor model
is derived from P.
Corollary 1 Consider the heavy-tailed factor model (1). For x1�/0 and x2�/0,
limn0�
P(M1;n5f1;n(x1);M2;n5f2;n(x2))�exp(�L(x1; x2));
where
L(x1; x2)�E max
�x�1
1
hai;1
Ehai;1
; x�12
hai;2
Ehai;2
��
1
x1
�1
x2
�P(x1; x2): (5)
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Remark 1 Since L(x1; x2)"x�11 �x�1
2 ; the componentwise maxima are asymptotically
dependent. This dependence plays an important role in many applications such as the
prediction of rare joint events. It must be however stressed that the case of asymptotic
independent marginals is quite observed even when the components are not independent.4 An
important example is the multivariate normal distribution with correlation coefficients
strictly smaller than one.
The strength of dependence is often measured by the tail dependence coefficient (see e.g.
Schmidt & Stadtmuller (2005))
l� limn0�
P(Xi;1�f1;n(1)½Xi;2�f2;n(1))�P(1; 1)�2�L(1; 1):
Remark 2 It follows from Eqns (3) and (5) that
A(w)�E max
�(1�w)
hai;1
Ehai;1
;wha
i;2
Ehai;2
�: (6)
Note also that if the vector rv (Z1, Z2) has the distribution function G�/exp(�/L), then
A(w)�(E max((1�w)Z1;wZ2))�1 (7)
(see Pickands (1981)). Eq. (6) gives a characterization of A in terms of the distribution of
the unobserved multiplicative disturbances, whereas Eq. (7) gives a characterization of A in
terms of the unknown asymptotic extreme value distribution.
If the vector (hai;1=Eh
ai;1; h
ai;2=Eh
ai;2) has an absolutely continuous distribution function with
density function g, we can characterize S. Straightforward calculations yield
S(da)
da�
1
(a2 � (1 � a)2)3=2 g�
0
u2g
�au
(a2 � (1 � a)2)1=2;
(1 � a)u
(a2 � (1 � a)2)1=2
�du:
Remark 3 Note that the extreme dependence is created by the vector Yhi�/(Yihi,1, Yihi,2).
Its distribution function FYh is a multivariate regularly varying function with a unique index
a�/0 (see Resnick (1987, Section 5.4.2)) and satisfies:
limt0�
1 � FYh(tx1; tx2)
1 � FYh(t; t)�
E max(x�a1 ha
i;1; x�a2 ha
i;2)
E max(hai;1; h
ai;2)
:
Then, by using Corollary 5.18 in Resnick (1987), there exists a sequence an such that5
limn0�
P(max(Y1h1;j ; . . . ;Ynhn;j)5xjan; j�1; 2)�E max(x�a
1 hai;1; x
�a2 ha
i;2)
E max(hai;1; h
ai;2)
:
Our model is much more general since the additional transformations Tj give more flexibility
and allow to have different index bj for each margin.
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4 Besides, it is also possible to define moderate dependence in the class of asymptotically independent
distributions (see Ledford & Tawn (1996, 1997)).5 A related result can be found in Gomes et al. (2004) where Yhi is a vector of rvs satisfying a stochastic
difference equation.
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Remark 4 P and L are two homogeneous functions of order �/1. They are completely
determined on A and are equivalent to A when characterizing the extreme dependence.
If hi,j are iid rvs, Eq. (5) defines a symmetric bivariate extreme value distribution. Let us
denote by H their distribution and by h a generic rv with the same distribution. Then it is
easily seen that
P(x1; x2)�x�11 E
�ha
EhaH
��x2
x1
�1=a
h
���x�1
2 E
�ha
EhaH
��x1
x2
�1=a
h
��
L(x1; x2)�x�11 E
�ha
EhaH
��x2
x1
�1=a
h
���x�1
2 E
�ha
EhaH
��x1
x2
�1=a
h
��;
where H�1�H is the survival distribution function of h. Both characterizations are
useful to compute analytically P and L. A large number of best known symmetric
bivariate extreme value distributions belong to the family defined in Eq. (5). This is
illustrated by the following examples when considering different distributions H.
. The Bernoulli distribution
Assume that H is a Bernoulli distribution with parameter 0B/pB/1. The distribution
of ha is also a Bernoulli distribution with the same parameter and
L(x1; x2)�p max(x�11 ; x�1
2 ):
G is the Marshall-Olkin (1967) distribution.
. The Log-normal distribution
Assume that h�/eX where X has a Gaussian distribution N(m,s2). Then ha has a
Lognormal distribution with parameter am and a2s2 and
L(x1; x2)�X2
l�1
x�1l F((u�1�(u log(xjx
�1l ))=2); j" l); (8)
where u�ffiffiffi2
p=(as): G is the Husler-Reiss (1989) distribution.
. The Weibull distribution
Assume that H is a Weibull distribution (Wei(c,t)), c�/0 and t�/0, i.e. H(x)�exp(�cxt): Then ha has a Weibull distribution Wei(c,t/a) and
L(x1; x2)�x�11 �x�1
2 �(xu1�xu
2)�1=u; (9)
where u�/t/a. G is the Galambos (1975) distribution.
. The Frechet distribution
Assume that H is a Frechet distribution (Fre(c,t)), c�/0 and t�/0, i.e. H(x)�/
exp(�/cx�t). If t/a�/1 then ha has a Frechet distribution Fre(c,t/a) with a finite
mean and
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L(x1; x2)�(x�u1 �x�u
2 )1=u; (10)
where u�/t/a. For m�/2, G is the Logistic or the Gumbel (1960) distribution.
This section has introduced a particular approach to modelling extreme dependence
when there exists an underlying heavy-tailed factor. The main advantage of this approach
is that it yields a specific characterization of the dependence function A which can be used
to construct estimators of the extreme value distribution which should be more accurate
than the standard estimators.
3. Statistical evaluation of bivariate extreme value distributions
Two main approaches have been developed in the literature to estimate a bivariate extreme
value distribution from data on a distribution in its domain of attraction.
The first one is known as the block maxima approach and consists in forming a
sequence of componentwise block maxima and in assuming that their distribution is the
true bivariate extreme value distribution. The distribution may be estimated completely
parametrically by using appropriate estimators: maximum likelihood estimators (Tawn
(1990), Coles & Tawn (1991, 1994)), or specific estimators for nondifferentiable models
(Tawn (1988)). But it may be also estimated by a mixture of parametric and
nonparametric techniques. These methods use a transformation which involves parameter
estimation on the marginals and then consider non-parametric estimators for the
dependence function A (see for this last step Pickands (1981), Deheuvels (1991),
Smith, Tawn & Yuen (1990), Caperaa, Fougeres & Genest (1997) and Hall & Tajvidi
(2000, 2004)).
The second approach considers the vectors of observations whose at least one of the
components exceeds a high threshold. Three methods have been introduced. The first
one assumes that a parametric form for the distribution of the excedances is known and
the parameters are estimated by maximum likelihood. It is called the bivariate Peaks
Over Thresholds method (Joe, Smith & Weissman (1992)). The second method
considers non-parametric estimators for the measure �/logG after a transformation
which involves parameter estimation on the marginals. Such a method is semi-
parametric in nature (see for example Einmahl, de Haan & Huang (1993), de Haan
& Resnick (1993), Einmahl, de Haan & Sinha (1997), de Haan & Sinha (1999)). A third
alternative method introduced by Einmahl, de Haan & Piterbarg (2001) is only based
on the ranks of the observations and is completely non-parametric (see also Hsing,
Kluppelberg & Kuhn (2005)).
In this section we propose a new semi-parametric estimator of the dependence function
A which is specific to our heavy-tailed factor model. In order to avoid non-essential
technicalities, we discuss the asymptotic properties of the estimators in Appendix B. Then
we investigate its performance on moderate samples and compare it with some standard
estimators.
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3.1. Estimator of the dependence function
We first provide another characterization of A which will be used for its empirical
evaluation.
Proposition 3 Let
(X i;1; X i;2)���
Xi;1
f1;n(1)
�b1
;
�Xi;2
f2;n(1)
�b2�
If\2j�1
(Xi;j�fj;n(1))g:
Then,
A(w)� limo00
limn0�
E(max((1 � w)X1=(1�o)i;1 ;wX
1=(1�o)i;2 ))
E((1 � w)X1=(1�o)i;1 � wX
1=(1�o)i;2 )
: (11)
This characterization is inspired by Eq. (6). The latent multiplicative disturbances
/haj;i have been replaced by (Xi,j/fj,n(1))bj
/(1�o). Actually it can be shown that, given that
Xi,j�/fj,n(1), (Xi,j/fj,n(1))bj can be approximated for large n by
hai;j
Ehai;j
�Yi
UY (n)
�a
:
Because the intensity Yi is the common multiplicative factor, it may be factorized when
considering linear functions of different Xi,j. Since hi,j and Yi are independent, using a
ratio of two expectations of such functions leads to eliminate E(Yi=UY (n))a: But a critical
issue is that E(Yi=UY (n))a may be not finite depending on lY . Therefore, we first have to
consider a moment of order bj/(1�/o) instead of bj for (Xi,j/fj,n(1)), and then let o tend to
zero.
Characterization (11) motivates our semi-parametric estimator
Aon;k(w)�min
�1;max
�Pn
i�1 (max(1 � w)(X(k)i;1 )1=(1�o);w(X
(k)i;2 )1=(1�o))Pn
i�1 (1 � w)X1=(1�o)i;1 � wX
1=(1�o)i;2
;w; 1�w
��; (12)
where o]/0,
(X(k)i;1 ; X
(k)i;2 )�
��Xi;1
X(k);1
�b1;k
;
�Xi;2
X(k);2
�b2;k�
If\2j�1
(Xi;j�X(k);j )g;
for j�/1,2, X(n),j5/. . .5/X(1),j are the order statistics and bj;k are the Hill estimators, i.e.
b�1j;k �
1
k
Xk
i�1
(logX(i);j�logX(k�1);j):
/Aon;k is not necessarily convex, but it does not violate the other properties required for A.
The asymptotic statistical properties of this estimator are derived when considering a
sequence of integers k�/k(n) that has to be chosen in such a way that typically k is large
(k(n)0/�) but also k is small in comparison to n(k(n)/n0/0). Although the estimator is
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asymptotically biased due to the parameter6 o, it is asymptotically Gaussian (see
Appendix B).
3.2. Simulation study
In this section a simulation study is conducted to compare the performance on moderate
samples of our estimator to six standard estimators:
. four semi-parametric estimators (first approach): for each simulated sample, we
divide [1,. . .,n] into k blocks of length r where r is the integer part of n/k, we compute
componentwise maxima for each block, we fit a generalized extreme-value
distribution to each margin and we transform them to unit exponential margins.
The non-parametric estimators of A considered are those proposed by Caperaa,
Fougeres & Genest (1997), Pickands (1981), Deheuvels (1991) and Hall & Tajvidi
(2000).
. two parametric estimators (second approach):
� the POT estimator (Joe, Smith & Weissman (1992)) with thresholds u1�/X(k),1 and
u2�/X(k),2;
� a moment estimator: u is estimated by equating the empirical tail dependence
coefficient
2�1
k
Xn
i�1
If\2j�1
(Xi;j�X(k);j )g
to the theoritical one (see Schmidt & Stadtmuller (2005));
Data are simula ted from a bivariate factor model where Yi has a Pareto distribution
with index a�/1, hi,1 and hi,2 are iid rvs, T1(x)�/10�/x1/2 and T2(x)�/x. We consider three
different distribution functions H for h (see the examples in Section 2 ii)):
(a) the Log-normal distribution with m�/0 and s2�/1/2 (u�/2),
(b) the Weibull distribution with c�/1 and t�/2 (u�/2),
(c) the Frechet distribution with c�/1 and t�/2 (u�/2).
400 sequences of length n�/750 were simulated from these models. For each estimator
A; we computed a Monte-Carlo approximation to the mean integrated square error
(MISE) Eðf1
0(A(w)�A(w))2dwÞ:
Figure 2 gives MISE for models (a), (b) and (c) in case k�/50, 75, 100, 125, 150. Of
course the parametric estimators perform better than the semi-parametric estimators for
all models and uniformly in k. Note however that, while the parametric approach is
efficient when the model is correct, the previous conclusion can be grossly misleading if
the model is incorrect. Among the semi-parametric estimators, the Hall and Tajvidi
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6 A sequence (on ) tending to zero at an appropriate rate could also be considered, which would suppress the
choice of o and would eliminate the bias.
10 L. Lescourret and C. Y. Robert
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UNCORRECTED PROOF
estimator has the greatest accuracy in terms of MISE, except for the case of model (b) and
k�/100, 125, 150 where Ao�0n;k performs better.
If the dependence structure can be identified, the parametric estimators should be
chosen. Otherwise our specific estimator and the Hall and Tajvidi estimator are good
candidates.
4. Data example
The model is now applied to the 736 pairs of storm damages (motor claim amounts and
household claim amounts) presented in Introduction. The claim amounts were
constructed from 150,000 individual weather-related domestic insurance claims for the
11-years period, 1 January 1990 to the 31 December 2000. These claims are related to
damage caused by high winds, tornadoes, lightnings, hails, torrential rains and weights of
snow. They are referred to by the insurer as ‘storm damage’. There is no uniformly
accepted definition of what constitutes a storm, and insurers use different thresholds of
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325
60 80 100 120 140
0.00
000.
0005
0.00
100.
0015
0.00
200.
0025
0.00
30
k
40
.0.
k
..
k
.
k
1
.
k
.
k
.
(a) (b) (c)
k
60 80 100 120 140
kkkkkkkk kkkkkkkk
60 80 100 120 140
0.00
000.
0005
0.00
100.
0015
0.00
200.
0025
0.00
30.0
.......
0.00
000.
0005
0.00
100.
0015
0.00
200.
0025
0.00
30.0
.......
Figure 2. Monte Carlo approximations to MISE of different estimators of the dependence function for models
(a ), (b ) and (c ). Results are for k�/50, 75, 100, 125, 150. The different estimators are: Ao�0n;k (/� � �); Caperaa et al.
estimator (- �/- �/), Pickands’ estimator (� � �), Deheuvels’ estimator (� �/� �/), Hall and Tajvidi estimator (*(*),
the POT estimator (**), the moment estimator (- - -).
11Extreme dependence of multivariate catastrophic losses
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UNCORRECTED PROOF
maximum wind gust and temporal scale in order to accept a claim as valid. Most current
treaties between reinsurers and insurers define occurrences using a 72 hours scale, which
was also chosen here to obtain 736 storm damages. The claims were adjusted for changes
in price7 and exposure levels.
4.1. Estimation of the factor model
We first consider univariate distributions. Pareto-type distributions are graphically
detected through Pareto-quantile plots (�/log(1�/k/(n�/1)), log(X(k),l))l�1,. . .,n (see
Beirlant et al., 1996). Specifically, when the data are generated from a random sample
of a Pareto-type distribution, the plot should look roughly linear at right. Our data
illustrate this feature (see Figure 3).
In Figure 4, Hill estimators are plotted with respect to the k upper order statistics. For a
large number of values of k, the estimated tail indexes are below one. If the true
parameters are smaller than one, claims size distributions have infinite means. Thus these
catastrophic risks should be uninsurable. In practice, there always exists a maximum
insured value and the loss amount can not exceed this value. This remark raises however
the question whether the data have been censored (see Matthys et al. (2004) for an
approach to estimating the index bj in the presence of right censoring).
We now consider the estimation of the dependence structure. The choice of the number
k of order statistics to be used in the estimation is not an easy task. It is well-known that,
when k increases the variance of extreme value estimators decreases but their bias gets
larger. Therefore a trade-off between bias and variance has to be made. For example k
should be selected such that it mimimizes the MISE. The simulation study shows that a k
greater or equal to 100 leads to more accurate estimates. Nevertheless we first consider
several values for k�/75, 100, 125, 150 and then choose a value for this parameter.
Graphs on Figure 5 show the biplots of the logarithms of the claim amounts after
transformation (log(X(k)i;1 ); log(X
(k)i;2 )): They are symmetric with respect to the main
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350
351
0.0 0.5 1.0 1.5 2.0 2.5 3.0Exponential Quantiles
0.0 0.5 1.0 1.5 2.0 2.5 3.0Exponential Quantiles
0
1
2
3
4
5
6
7lo
g (H
ouse
hold
cla
ims)
0.0
0.8
1.6
2.4
3.2
4.0
4.8
log
(Mot
or c
laim
s)
Figure 3. Pareto quantile plots for Motor claim amounts and Household claim.
7 Changes in construction costs and other factors that affect the prices of property exposed to loss.
12 L. Lescourret and C. Y. Robert
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UNCORRECTED PROOF
diagonal, which suggests to consider a symmetric dependence structure. Figure 6 displays
the Quantile-Quantile plot of log(X(k)i;1 =X
(k)i;2 ) with respect to the Gaussian distribution and
illustrates the distribution of a log(hi,1/hi,2). Plots are quite linear. Therefore a factor
model with Log-normal disturbances seems to be a good choice to fit the data and we
select the parametric dependence function of the Husler and Reiss model.
Figure 7 shows the parametric dependence function estimates for the Husler and Reiss
model and the semi-parametric dependence function estimates. Note that we have
constrained the semi-parametric estimators to be symmetric in the following way
As(w)�A(w) � A(1 � w)
2:
The choice of k�/150 seems to be the most appropriate one because the parametric
estimators and our estimator and the Hall and Tajvidi estimator are quite close.
4.2. Actuarial applications
This section describes two applications using an estimated extreme dependence function:
i) the estimation of the probability of a catastrophic event, ii) the evaluation of
reinsurance premiums. The numerical values are computed by using the moment
estimator with k�/150.
i) We first discuss how to make inference of claim frequencies in an area of the sample
where there is a very small amount of data or even no observation at all (as in the dashed
area in Figure 1). Suppose that we want to estimate the probability p of an extreme set of
the form f\2j�1(Xi;j �tj)g; i.e. the probability that all thresholds tj are exceeded. Using
Proposition 2.2, natural estimators of the probability are given by
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365
366
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369
370
371
372
0 50 100 150 200k
0.0
0.5
1.0
1.5
2.0 Household claimsMotor claims
Figure 4. Hill estimators for Household claim amounts and Motor claim amounts.
13Extreme dependence of multivariate catastrophic losses
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UNCORRECTED PROOF
pn�k
n
�1
x1;k
�1
x2;k
��1�A
�x1;k
x1;k � x2;k
��(13)
where xj;k�(tj=X(k);j)bj;k and A is the estimate of the extreme dependence function.
For example, suppose that we want to evaluate the probability that damages exceed
twice Lothar’s losses (for the French insurer). This probability is estimated by pn�1:67�10�4; which corresponds to a storm every 90 years.8 This return period is
consistent with the figures provided by Swiss Re experts (Bresch, Bisping & Lemcke
(2000)) who make an univariate analysis of the sum of the claim amounts of several lines
of business.
373
374
375
376
377
378
379
380
8 See Lescourret & Robert (2004) for a confident interval based on a Asymptotic Least Squares method.
0 1 2 3 4 5 6 0 1 2 3 4 5 6
01
23
45
6
k=75~
log(
Xi(k
) )
01
23
45
6
~lo
g(X
i(k) )
01
23
45
6
~lo
g(X
i(k) )
01
23
45
6
~lo
g(X
i(k) )
~log(Xi
(k)) ~log(Xi
(k))
0 1 2 3 4 5 6 0 1 2 3 4 5 6~
log(Xi(k)) ~
log(Xi(k))
k=100
k=125
k=150
Figure 5. Logarithms of the claim amounts after transformation. Results are for k�/75, 100, 125, 150.
14 L. Lescourret and C. Y. Robert
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UNCORRECTED PROOF
We refer to de Haan & Sinha (1999) or Draisma et al. (2004) for a rigorous approach of
the semi and non-parametric estimation of failure sets and to Hall & Tajvidi (2004) for
a-level prediction regions.
ii) We now consider a catastrophe excess of loss reinsurance contract. This contract
indemnifies against a aggregate loss Xi,2 related to motor insurance arising from a
catastrophic event in excess of a specified amount r2 up to a specified limit l2. Moreover if
the aggregate loss Xi,1 related to household insurance is less than a trigger r1 the contract
only pays a proportion g of the indemnity. The payout of the reinsurance is specified as
follows
h(Xi;1;Xi;2)�max(0; min(Xi;2�r2; l2))�(g�(1�g)IfXi;1 � r1g):
Because it is common practice to assume independence, we provide Table 1 which gives
ratios of independence to dependence reinsurance premiums.
381
382
383
384
385
386
387
388
389
390
391
392
–2 –1 0 1 2 -2 -1 0 1 2
–2–1
01
2
k=75
Normal quantiles
Sam
ple
quan
tiles
–2–1
01
2
Sam
ple
quan
tiles
–2–1
01
2
Sam
ple
quan
tiles
–2–1
01
2
Sam
ple
quan
tiles
k=100
k=125
Normal quantiles
–2 –1 0 1 2 –2 –1 0 1 2Normal quantiles Normal quantiles
k=150
Figure 6. QQ-plots of log(X(k)i;1 =X
(k)i;2 ) with respect to the Gaussian distribution. Results are for k�/75, 100, 125,
150.
15Extreme dependence of multivariate catastrophic losses
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UNCORRECTED PROOF
In order to compare with the case without trigger r1, i.e.
h(Xi;2)�max(0; min(Xi;2�r2; l2));
we also provide Table 2 which gives ratios of dependence reinsurance premiums to
reinsurance premiums without trigger.
393
394
395
396
0.0 0.2 0.4 0.6 0.8 1.0
k=750.0 0.2 0.4 0.6 0.8 1.0
k=125
0.0 0.2 0.4 0.6 0.8 1.0
k=1000.0 0.2 0.4 0.6 0.8 1.0
k=150
0.5
0.6
0.7
0.8
0.9
1.0
A(w
)
0.5
0.6
0.7
0.8
0.9
1.0
A(w
)
0.5
0.6
0.7
0.8
0.9
1.0
A(w
)
0.5
0.6
0.7
0.8
0.9
1.0
A(w
)
Figure 7. Estimators of the dependence function for models (a ), (b ) and (c ). The different estimators As are:
Ao�0n;k (/� � �); Caperaa et al. estimator (- �/- �/), Pickands’ estimator (� � �), Deheuvels’ estimator (� �/� �/), Hall and
Tajvidi estimator (*(*), the POT estimator (**), the moment estimator (- - -). Results are for k�/75, 100,
125, 150.
Table 1. Ratios of independence to dependence premiums (r2�/10.000 and l2�/40.000).
g \r1 (%) 50.000 (%) 100.000 (%) 150.000 (%) 200.000 (%) 250.000 (%)
30 34 37 39 41 42
40 44 47 50 52 53
50 55 56 60 62 63
60 64 67 69 71 72
70 74 76 78 79 80
16 L. Lescourret and C. Y. Robert
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UNCORRECTED PROOF
Note that the reinsurance premiums without trigger are equal to the premiums when
P(Xi;1�r1½Xi;2�r2)�1;
which depicts a very strong dependence.
Table 1 shows that it could result in substantial undervaluations if the unrealistic
assumption of independence between both lines of business was made. According to this
table, these undervaluations are larger for lower limits r1. Table 2 shows that the
overvaluations would not be so important if, on the contrary, the assumption of a too
strong dependence was made (except maybe for large triggers r1).
5. Conclusion
In this paper we have focused on the extreme dependence of catastrophe risks across
different lines of business where the intensity of the natural disaster can be considered as a
common underlying factor. Such a model may also be relevant for various problems of
practical interests ranging from environmental impact evaluation to financial risk
management. A first example is problems involving spatial dependence for floods which
may occur at several sites along a coast line, or at various rain-gauges in a national
network. A second example is the wave height and still-water level which are two
important variables for causing floods along a sea cost during storm events. A last
example is the study of extremes of stocks returns where the market return can be
considered as a common underlying factor.
Acknowledgements
The authors would like to thank Charles Levi for providing the data and an anonymous
referee for helpful comments.
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Appendix A
Proof of Proposition 1: By using the Dominated Convergence Theorem (DCT) and
bounds of regularly varying functions (Bingham et al. (1987) Theorem 1.5.6), we deduce
that
limy0�
P(Yihi;j � y)
P(Yi � y)� lim
y0�
E[P(Yihi;j � y½hi;j)]
P(Yi � y)�Eha
i;j :
Then for large x
P(Xi;j �x)�P(Yihi;j �T1j (x))�(1�o(1))Eha
i;j(T1j (x))�alY (T1
j (x))
�x�bj (1�o(1))Ehai;j(lT1
j(x))�alY (x1=gj lT1
j(x))�x�bj lXj
(x):
It is easy to see that lXjis a slowly varying function. I
Proof of Proposition 2: First remark that for xj�/0
limn0�
nP(Xi;j �Tj((xjEhai;j)
1=aUY (n)))� limn0�
nP(Yihi;j �(xjEhai;j)
1=aUY (n))
�(1�o(1)) limn0�
nEhai;j((xjEh
ai;j)
1=aUY (n))�alY ((xjEhai;j)
1=aUY (n))
�(1�o(1))x�1j lim
n0�n(UY (n))�alY (UY (n))
�(1�o(1))x�1j ;
and therefore
limn0�
nP(Xi;j �fj;n(xj); j�1; 2)� limn0�
nP(Yihi;j �(xjEhai;j)
1=aUY (n); j�1; 2):
Then we have
limn0�
nP(Xi;j �fj;n(xj); j�1; 2)� limn0�
P(Yihi;j � (xjEhai;j)
1=aUY (n); j � 1; 2)
P(Yi � UY (n))
� limy0�
P(Yiminj�1;2(x�1j ha
i;j=Ehai;j)
1=a� y)
P(Yi � y)
�P(x1; x2);
by using the same arguments as for the proof of Proposition 1. I
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
19Extreme dependence of multivariate catastrophic losses
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UNCORRECTED PROOF
Proof of Corollary 1: Corollary 1 is deduced from the following calculation
limn0�
logP(Mj;n5fj;n(xj); j�1; 2)� limn0�
logFn(Xi;j 5Tj((xjEhai;j)
1=aUY (n)); j�1; 2)
� limn0�
logPn(fYihi;15(x1Ehai;1)1=aUY (n)g \ fYihi;25(x2Eh
ai;2)1=aUY (n)g)
�� limn0�
nP(fYihi;1�(x1Ehai;1)1=aUY (n)g@fYihi;2�(x2Eh
ai;2)1=aUY (n)g)
�x�11 �x�1
2 �P(x1; x2): I
Proof of Proposition 3: Let w1�/1�/w and w2�/w. Recall that
Xi;j �Tj(Yihi;j)�(Yihi;j)gj lTj
(Yihi;j);
fj;n(1)�(1�o(1))Tj((Ehai;j)
1=aUY (n))
�(1�o(1))((Ehai;j)
1=aUY (n))gj lTj((Eha
i;j)1=aUY (n)):
For large n, we can use the following approximation
maxj�1;2
�wj
�Xi;j
fj;n(1)
�bj=(1�o)�If\2
j�1(Xi;j�fj;n(1))g�(1�o(1))
�Yi
UY (n)
�a=(1�o)
maxj�1;2
�wj
��hi;j
(Ehai;j)
1=a
�a=(1�o)�lTj((Yi=UY (n))hi;jUY (n))
lTj((Eha
i;j)1=aUY (n))
�bj=(1�o)��
If Yi
UY (n)� (minl�1;2(ha
i;j=Eha
i;j)1=a)�1g
:
Moreover, given that Yi�/UY(n), (Yi/UY(n)) converges in law to a Pareto distribution with
index a
limn0�
P(Yi�uUY (n)jYi�UY (n))�u�a:
Using the DCT yields
limn0�
E(maxj�1;2(wj(Xi;j=fj;n(1))bj=(1�o))If\2j�1
(Xi;j�fj;n(1))g)
P(Yi � UY (n))
� limn0�
EE(maxj�1;2(wj(Xi;j=fj;n(1))bj=(1�o))If\2j�1
(Xi;j�fj;n(1))gjhi;j ; j � 1; :::;m)
P(Yi � UY (n))
�E
��g
�
(minj�1;2(hai;j=Eha
i;j))�1
ua=(1�o)au�a�1du
�maxj�1;2
�wj
�ha
i;j
Ehai;j
�1=(1�o)��
�1 � o
oE
��minj�1;2
(hai;j=Eh
ai;j)
�o=(1�o)
maxj�1;2
�wj
�ha
i;j
Ehai;j
�1=(1�o)��:
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20 L. Lescourret and C. Y. Robert
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UNCORRECTED PROOF
By applying the same method we get
limn0�
E
�P2
j�1 wj((Xi;j=fj;n(1))bj=(1�o))If\2j�1
(Xi;j � fj;n(1))g
�
P(Yi � UY (n))
�1 � o
oE
��minj�1;2
(hai;j=Eh
ai;j)
�o=(1�o) X2
j�1
wj
�ha
i;j
Ehai;j
�1=(1�o)�:
And by using again the DCT, we have
A(w)� limo00
limn0�
E(maxj�1;2(wj(Xi;j=fj;n(1))bj=(1�o))If\2j�1
(Xi;j�fj;n(1))g)
E
�P2
j�1 wj((Xi;j=fj;n(1))bj=(1�o))If\2j�1
(Xi;j�fj;n(1))g
� :
Appendix B: Asymptotic properties of Aon;k
We first introduce an estimator of P and discuss its asymptotic properties. Then we
derive the asymptotic properties of our estimator.
i) An estimator of PThis paragraph presents a semi-parametric estimator of P and discusses its asymptotic
properties. We will use the following notations: x�/(x1, x2), R2��fx : xl ]0g\f0g;
[x,�[�/[x1,�[�/[x2,�[. The basic idea of the approach is a point process representation
already widely used for the probabilistic characterization of multivariate extremes (see e.g.
Resnick (1987), de Haan & Resnick (1993) and de Haan & Sinha (1999)). The limiting
result
limn0�
nP�
((Xi;1=UX1(n))b1 ; (Xi;2=UX2
(n))b2 ) [x;�[��P(x)
derived from Eq. (4) suggests to define an extremal dependence measure m concentrating
on R2� by m([x,�[)�/P(x) Then it is natural to consider empirical measures as candidates
for an estimator of m. For x R2� and AƒR2
�; let
ex(A)�1 if x A;0 if x Ac;
�
and define the estimator
mn([x;�[)�1
k
Xn
i�1
e� Xi;1
X(k);1
b1;k;
Xi;2
X(k);2
b2;k
�([x;�[); k5n:
Let us assume that P has continuous first order partial derivatives. Since P is
homogeneous of order �/1, we deduce that
X2
j�1
@P(x)
@xj
xj ��P(x): (B:1)
A second order refinement of Eq. (4) is also needed: we assume that there exist a non
constant function c from R2� to R and a positive function c such that
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21Extreme dependence of multivariate catastrophic losses
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UNCORRECTED PROOF
limn0�
nP(\2j�1fXi;j � x
1=bj
j UXj(n)g) �P(x)
c(UX(n))�c(x)B�;
locally uniformly for x R2�; where c(UX(t))�/c(Ux1
(t),Ux2(t)) is regularly varying and
c(UX(t))0/0 as t0/�. This condition allows to control the asymptotic bias of the
estimators. For more details, see de Haan & Stadtmuller (1996) or de Haan & Resnick
(1993). Let D2(R2�) be the space of cadlag functions from R2
� to R� equipped with the
Skohorod topology. We denote by ej the point of R2� such that the j-th element is equal to
one and the other is nul. By using the methodology developed by de Haan & Resnick
(1993), we derive the following proposition.
PROPOSITION B.1: Suppose that k�/k(n) is such that k0/�, k/n0/0, as n0/�, then
mn([x;�[)0PP(x):
Moreover if
limn0�
ffiffiffik
pc(UX(n=k)))�0; (B:2)
then
ffiffiffik
p(mn([x;�[)�m([x;�[))[V (x)“W (x)�
X2
j�1
@P(x)
@xj
xj(W (ej)�bj logxjGj); (B:3)
weakly in D2(R2�), where W is a zero mean Gaussian random field (W (x); x R2
�) with
covariance function
cov(W (x);W (x?))�m([x;�[\[x?;�[);
and
Gj �1
bj
�g
�
1
W (yej)dy
y�W (ej)
�:
ii) Asymptotic properties of Aon;k
Let w1�/1�/w, w2�/w and define
Aon;k(w)�
Pn
i�1 maxj�1;2wj(X(k)i;j )1=(1�o)
Pn
i�1
P2
j�1 wjX1=(1�o)i;j
:
First note that
Aon;k(w)�
g�
1g
�
1
maxj�1;2(wju1=(1�o)j )mn(du)
g�
1g
�
1
�X2
j�1wju
1=(1�o)j
�mn(du)
:
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UNCORRECTED PROOF
By using Proposition B.1, Proposition 3 and technical arguments (see e.g., Resnick &
Starica (1995)), it may be proved that, if k�/k(n) is such that k0/�, k/n0/0, as n0/�,
then
Aon;k(w)0
P g�
1g
�
1
maxj�1;2(wju1=(1�o)j )dP(u)
g�
1g
�
1
�X2
j�1wju
1=(1�o)j
�dP(u)
�E(minj�1;2(ha
i;j=Ehai;j)
o=(1�o)maxj�1;2(wj(hai;j=Eh
ai;j)
1=(1�o)))
E
�minj�1;2(ha
i;j=Ehai;j)
o=(1�o) P2
j�1 (wj(hai;j=Eh
ai;j)
1=(1�o))
��Ao(w):
Let us now focus on the asymptotic normality. If k�/k(n) is such that k0/�, k/n0/0 and
limn0�
ffiffiffik
pc(UX(n=k)))�0; as n0/�, then
limn0�
supkxk�a
kxk1=4fffiffiffik
p(mn([x;�[)�P(x))�V (x)g�0
in probability for all a�/0 (see e.g. Einmahl (1992)). Using this remark, we can prove that
ffiffiffik
p �g
�
1g
�
1
maxj�1;2
(wju1=(1�o)j )mn(du)�g
�
1g
�
1
maxj�1;2
(wju1=(1�o)j )dP(u)
�
[g�
1g
�
1
maxj�1;2
(wju1=(1�o)j )V (du)
and
ffiffiffik
p �g
�
1g
�
1
�X2
j�1
wju1=(1�o)j
�mn(du)�g
�
1g
�
1
�X2
j�1
wju1=(1�o)j
�dP(u)
�
[g�
1g
�
1
�X2
j�1
wju1=(1�o)j
�V (du):
Then we haveffiffiffik
p(Ao
n;k(w)�Ao(w))[V oA(w)�
1
E
�minj�1;2(ha
i;j=Ehai;j)
o=(1�o) P2
j�1 (wj(hi;j=Ehai;j)
a=(1�o))
�g�
1g
�
1
maxj�1;2
(wju1=(1�o)j )V (du)
�Ao(w)
E
�minj�1;2(ha
i;j=Ehai;j)
o=(1�o) P2
j�1 (wj(hi;j=Ehai;j)
a=(1�o))
�g�
1g
�
1
�X2
j�1
wju1=(1�o)j
�V (du)
weakly in D([0,1]) and Aon;k(w) is asymptotically Gaussian. The asymptotic variance can be
performed numerically. Finally, if max(w, 1�/w)B/Ao(w)B/1 for all w // [0,1], we also haveffiffiffik
p(Ao
n;k(w)�Ao(w))[V oA(w):
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23Extreme dependence of multivariate catastrophic losses
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