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

Y:/Taylor & Francis/SACT/articles/SACT188887/SACT188887.3d[x] Tuesday, 11th July

UNCORRECTED PROOF

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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Mot

or c

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Figure 1. Storm damages (both variables are on a logarithmic scale).

2 L. Lescourret and C. Y. Robert


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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.

3Extreme dependence of multivariate catastrophic losses


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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.



UNCORRECTED PROOF

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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UNCORRECTED PROOF

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.



UNCORRECTED PROOF

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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UNCORRECTED PROOF

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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UNCORRECTED PROOF

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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UNCORRECTED PROOF

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.



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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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 (- - -).



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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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.



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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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.



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.

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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.



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.

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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.



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.

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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



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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425

Table 2. Ratios of dependence premiums to premiums without trigger (r2�/10.000 and l2�/40.000).

g \r1 (%) 50.000 (%) 100.000 (%) 150.000 (%) 200.000 (%) 250.000 (%)

30 88 82 77 74 71

40 90 84 80 77 75

50 92 87 84 81 79

60 93 90 87 85 83

70 95 92 90 89 87



UNCORRECTED PROOF

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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



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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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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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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Extreme dependence of multivariate catastrophic lossesAs it will be illustrated in our example,...

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