This article was downloaded by: [Universite Laval] On: 02 December 2013, At: 06:57 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Scandinavian Actuarial Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/sact20 Analysis of ruin measures for the classical compound Poisson risk model with dependence Héléne Cossette a , Etienne Marceau a & Fouad Marri a a École d'Actuariat , Université Laval , Québec, G1V OA6, Canada Published online: 26 Aug 2010. To cite this article: Héléne Cossette , Etienne Marceau & Fouad Marri (2010) Analysis of ruin measures for the classical compound Poisson risk model with dependence, Scandinavian Actuarial Journal, 2010:3, 221-245, DOI: 10.1080/03461230903211992 To link to this article: http://dx.doi.org/10.1080/03461230903211992 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms- and-conditions
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with dependence classical compound Poisson risk model ......of light-tailed claim sizes. Boudreault et al. (2006) propose an extension to the classical compound Poisson risk model
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This article was downloaded by: [Universite Laval]On: 02 December 2013, At: 06:57Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Scandinavian Actuarial JournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/sact20
Analysis of ruin measures for theclassical compound Poisson risk modelwith dependenceHéléne Cossette a , Etienne Marceau a & Fouad Marri aa École d'Actuariat , Université Laval , Québec, G1V OA6, CanadaPublished online: 26 Aug 2010.
To cite this article: Héléne Cossette , Etienne Marceau & Fouad Marri (2010) Analysis of ruinmeasures for the classical compound Poisson risk model with dependence, Scandinavian ActuarialJournal, 2010:3, 221-245, DOI: 10.1080/03461230903211992
To link to this article: http://dx.doi.org/10.1080/03461230903211992
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
ISSN 0346-1238 print/ISSN 1651-2030 online # 2010 Taylor & Francis
http://www.tandf.no/saj
DOI: 10.1080/03461230903211992
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f +W (s)�E
�e�sW
��
ll� s
: (3)
The claim amount r.v.’s {Xj, j �N�}, where Xj corresponds to the amount of the jth claim,
are assumed to be a sequence of strictly positive, independent, and identically distributed
(i.i.d.) r.v. with p.d.f. fX, c.d.f. FX, and LT f +X : Throughout the paper, we use the exponent
‘*’ to designate the LT of a function.
In ruin theory, the classical compound Poisson risk model is based on the assumption
of independence between the claim amount random variable Xj and the interclaim time
r.v. Wj. This assumption is appropriate in certain practical circumstances and has the
advantage of simplifying the computation of ruin quantities of interest. However, such a
hypothesis can be restrictive in other practical contexts. For example, in modeling natural
catastrophic events, we can expect that, on the occurrence of a catastrophe, the total claim
amount (or the intensity of the catastrophe) and the time elapsed since the previous
catastrophe are dependent. See e.g. Boudreault (2003) and Nikoloulopoulos & Karlis
(2008) for an application of this type of dependence structure in an earthquake context.
In our paper, we assume that {(Xj,Wj), j �N�} form a sequence of i.i.d. random vectors
distributed as the canonical r.v. (X,W ), in which the components may be dependent. The
joint p.d.f. of (X,W ) is denoted by fX,W (x,t) with t �R� and x �R�. When X and W are
continuous, the associated LT is given by
f +X ;W (s1; s2)�E
�e�s1X e�s2W
��g
�
0g
�
0
e�s1xe�s2tfX ;W (x; t) dxdt: (4)
In this paper, the joint distribution of (X,W) is defined with a Farlie�Gumbel�Morgenstern (FGM) copula.
Recently, some papers consider extensions to the classical risk model considering
dependence models between the claim amount r.v., X, and the interclaim time r.v., W.
Among them, Albrecher & Teugels (2006) consider a dependence structure for (X,W )
based on a copula. By employing the underlying random walk structure of the risk model,
they derive exponential estimates for finite and infinite-time ruin probabilities in the case
of light-tailed claim sizes. Boudreault et al. (2006) propose an extension to the classical
compound Poisson risk model assuming a dependence structure for (X,W ), in which the
distribution of the next claim amount is defined in terms of the time elapsed since the last
claim. They derive the defective renewal equation satisfied by the expected discounted
penalty function. They also obtain an explicit expression for the LT of the time of ruin
assuming that the claim amount belongs to a large class of distributions.
The present paper is organized as follows. In Section 2, we briefly recall basic notions
on copulas and present properties of the FGM copula. Basic definitions for ruin measures
are given in Section 3. We derive the generalized Lundberg equation and analyze its
properties in Section 4. In Section 5, we obtain an integro-differential equation for the
expected discounted penalty function and, in the following section, we derive the LT of the
expected discounted penalty function. In Section 7, we derive the defective renewal
equation for the expected discounted penalty function. An explicit expression for the LT
of the discounted value of a general function of the deficit at ruin is obtained for claim
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amounts having an exponential distribution in Section 8. Numerical examples are also
provided in Section 8.
2. Dependence structure based on Farlie�Gumbel�Morgenstern (FGM) copula
A bivariate copula C is a joint distribution function on [0,1]�[0,1] with standard uniform
marginal distributions. By the theorem of Sklar (see e.g. Nelsen (2006)), any bivariate
distribution function F with marginals F1 and F2 can be written as F(x1,x2)�C(F1(x1),F2(x2)), for some copula C. This copula is unique if F is continuous. Otherwise
it is uniquely defined on the range of the marginals. We refer the reader to Joe (1997) or
Nelsen (2006) for further details on copulas. Modeling the dependence structure between
r.v. using copulas has become popular in actuarial science and financial risk management.
The reader may consult, e.g. Frees & Valdez (1998), Wang (1998), Bouye et al. (2000),
Denuit et al. (2005), and McNeil et al. (2005) for applications of copulas in actuarial
science and financial risk management. As in the present paper, Albrecher & Teugels
(2006) use copulas to define the joint distribution for the interclaim time r.v. and the claim
amount r.v.
Assume a bivariate random vector (U,V ) with continuous uniform marginals and with
a dependence structure defined by a copula FU,V (u,v)�C(u,v) with (u,v) � [0,1]�[0,1].
Important copulas are the independence copula with C�(u,v)�uv; the comonotonic
copula with C�(u,v)�min(u,v); the countermonotonic copula with C�(u,v)�max(u�v � 1;0). It is important to mention that all copulas satisfy the inequalities C�(u,v)5
C(u,v)5C�(u,v), for (u,v) � [0,1]�[0,1].
The joint p.d.f. associated to a copula C is defined by
c(u1; u2)�@2
@u1@u2
C(u1; u2): (5)
Let the bivariate distribution function FX,W of (X,W ) with marginals FX and FW be
defined as FX,W (x,t)�C(FX (x),FW (t)) for (x,t) � R��R�. The joint p.d.f. of (X,W ) is
given by
fX ;W (x; t)�c(FX (x);FW (t)) fX (x) fW (t); (6)
for (x,t) �R��R�.
The FGM copula is given by
CFGMu (u1; u2)�u1u2�uu1u2(1�u1)(1�u2);
�15u51, where CFGM0 �C�: The FGM copula allows negative and positive depen-
dence, includes the independence copula (u�0), but does not include the comonotonic
and the countermonotonic copulas as limit cases. In addition, the FGM copula is a
perturbation of the independence copula and it is not Archimedian. The FGM copula is a
first order approximation of the Plackett copula (Nelsen 2006, p. 100) and of the Frank
copula (p. 133). This copula is attractive due to its simplicity and its tractability. However,
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the FGM copula is restrictive because it is only useful when dependence between the two
marginals is modest in magnitude. For the FGM copula, Kendall’s tau and Spearman’s
rho are t�2u9
and r�u3; respectively. It means that �
2
95t5
2
9and �
1
35r5
1
3: It can
also be shown that the Pearson’s correlation coefficient goes from�1
3to
1
3: Even it seems
a bit disappointing at first sight, one can see in the numerical example of Section 8 that
the dependence parameter may have a significant impact on the ruin measures. To our
knowledge, no copula, which includes the lower and the upper Frechet bounds as special
cases, has this property of tractability.
Among the recent applications of the FGM copula, we mention Prieger (2002) who
uses it in the modeling selection into health insurance plans. The FGM copula (its
multivariate version) is also applied in the context of sums of dependent r.v. by Geluk &
Tang (2008) and in the analysis of the behavior of discrete-time risk models with
dependent financial risks by Tang & Vernic (2007). Gebizlioglu & Yagci (2008) apply the
FGM copula to establish tolerance intervals for quantiles of bivariate risks in the context
of risk measurement. The FGM copula is also used in a stereological context by Benes
et al. (2003). Smith (2008) uses it in the context of stochastic frontiers.
Due to its simplicity, several extensions were considered by various authors (see e.g.
Drouet-Mari & Kotz (2001) for a review of the FGM copula and some its extensions).
For the FGM copula, the expression for Eq. (5) is given by
cFGMu (u1; u2)�1�u(1�2u1)(1�2u2): (7)
The bivariate distribution function FX,W of (X,W ) with marginals FX and FW and defined
with the FGM copula is given by
FX ;W (x; t)�FX (x)FW (t)�uFX (x)FW (t)(1�FX (x))(1�FW (t));
for (t,x) � R��R�. Combining Eqs. (6) and (7), we obtain the expression for the joint
p.d.f. of (X,W)
fX ;W (x; t)�fX (x) fW (t)�u fX (x) fW (t)(1�2FX (x))(1�2FW (t)); (8)
and, given from Eq. (1), we have
fX ;W (x; t)�fX (x)le�lt�ufX (x)le�lt(1�2FX (x))(2e�lt�1): (9)
Defining hX (x)�(1�2FX (x))fX (x) and denoting by h+X (s) its associated LT, Eq. (9) can be
written as
fX ;W (x; t)�fX (x)le�lt�uhX (x)(2le�2lt�le�lt): (10)
In Figure 1, we provide an illustration of the dependence relation between the claim
amount r.v. X and the interclaim r.v. W. We assume that the claim amount r.v. X and the
interclaim r.v. W both follow an exponential distribution with means 9 and 12,
respectively.
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We conclude this section with the following result shown in Rolski et al. (1999,
Theorem 6.1.12):
limt0�
S(t)
t� lim
t0�
E[S(t)]
t�
E[X ]
E[W ](with probability 1);
which holds whatever the dependence structure between the r.v. X and W. However, the
combined effect of the timing of a claim and its amount can have a significant impact on
the surplus associated to the insurance portfolio (as illustrated in Figure 1) and
consequently, on the behavior of ruin measures. This aspect is clearly illustrated in the
numerical example provided in Section 8.
3. Ruin measures
We define the time of ruin as the r.v. T where T� inf t]0ft;U(t)B0g with T�� if U(t)]
0 for all t]0 (i.e. ruin does not occur). To ensure that ruin will not occur almost surely,
the premium rate p is such that
E[pWi�Xi]�0; i�1; 2; . . . ; (11)
providing a positive safety loading. The deficit at ruin and the surplus just prior to ruin
are, respectively, denoted by jU(T)j and U(T�). In the recent years, a fair amount
of research in ruin theory has been devoted to the analysis of the expected value of
Claim amounts with FGM copula (θ = –1 and 1)
0
5
10
15
20
25
30
35
3,24 43,75 69,45 72,48 73,79 76,08
Times of occurence
Cla
im a
mou
nts
Xi (negative dependence)
Xi (positive dependence)
Figure 1. Simulation of claim occurrences and claim amounts with FGM copula with u��1 and u�1.
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the discounted penalty function. Introduced by Gerber & Shiu (1998), this function is