Compound Poisson approximations for individual models with dependent risks

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1 Insurance: Mathematics and Economics 32 ( Compound Poisson approximations for individual models with dependent riss Christian Genest a,, Étienne Marceau a, Mhamed Mesfioui b a Département de mathématiques et de statistique, Université Laval, Québec, Canada GK 7P4 b Département de mathématiques et d informatique, Université du Québec à Trois-Rivières, C.P. 5, Trois Rivières (Québec, Canada G9A 5H7 Received June 22; received in revised form September 22; accepted 6 November 22 Abstract This paper shows how compound Poisson distributions can be used to approximate the distribution of the total claim amount in the context of single- or multi-class individual ris models where dependence between the contracts arises through mixtures. Some of these models are generated by Archimedean copulas, and others are seen to fall under the purview of a general multi-class shoc model whose structure is both intuitive and easily tractable. A numerical study is used to illustrate the quality of the approximation as a function of the heterogeneity and the dependence in the portfolio. A theoretical result is also provided which helps to explain the effect of dependence on the total claim amount when the contracts are lined through an Archimedean copula model. 22 Elsevier Science B.V. All rights reserved. MSC: IM; IM2; IM3 Keywords: Compound Poisson approximation; Copula; Dependent riss; Individual ris model; Collective ris model. Introduction In the classical individual ris model consisting of n insurance policies over a given period, the ris X l corresponding to the lth contract is usually expressed in the form { Bl if I l =, X l = ( if I l =, where I l is a Bernoulli random variable taing the value when at least one claim is filed while the contract is valid, and B l is a strictly positive random variable representing the sum of all amounts claimed in that period. The B l s are typically assumed to be independent of each other and of all the I l s. Of particular interest is the cumulative distribution function F S of the aggregate claim amount S = X l Corresponding author. Tel.: ; fax: address: genest@mat.ulaval.ca (C. Genest /2/$ see front matter 22 Elsevier Science B.V. All rights reserved. PII: S (225-6

2 74 C. Genest et al. / Insurance: Mathematics and Economics 32 ( and functions thereof such as the stop-loss premium π S (d = E{max(S d,} for d, the value-at-ris VaR α = FS (α for some α (,, or the expected shortfall E(S S >VaR α, also nown as the conditional VaR. As it is generally difficult to compute these quantities for large portfolios, F S often needs to be approximated. When the X l s are stochastically independent, a useful strategy is to approach S by { N Z l if N>, T = if N =, where N follows a Poisson distribution with mean λ and Z,Z 2,... is a sequence of independent and identically distributed random variables with common distribution function F Z. The strategy for selecting the appropriate parameters in T s compound Poisson distribution F T (λ, F Z varies according to the ris measure of interest. Thus if the stop-loss premium or the expected shortfall is of primary concern, then λ and F Z should be chosen in such a way that T dominates S in the convex order, while if the focus is on value-at-ris, stochastic dominance of T is needed to get conservative compound Poisson approximations. To measure the quality of the approximation of S by T, three different distances between their distribution functions are typically used: the Kolmogorov metric, viz. d K (S, T = sup F S (x F T (x, x R the total-variation distance, viz. d TV (S, T = sup P(S A P(T A A B(R with B(R denoting the Borel sets on R, and the stop-loss distance, viz. d SL (S, T = sup π S (x π T (x. x In practice, however, the assumption that riss are independent is not always legitimate. For this reason, various individual ris models incorporating dependence have recently been proposed in the actuarial literature; see, among others, Bäuerle and Müller (998, Albers (999, Cossette et al. (in press or Cossette et al. (22. This, in turn, poses the problem of devising appropriate compound Poisson approximations for such models. When the dependence between the components of a ris portfolio is local or wea, Goovaerts and Dhaene (996 show that the total claim amount S can be approximated to a reasonable degree by a single compound Poisson variable T, which is constructed as in the independent case. For mixture models, however, in which riss are conditionally independent given a (possibly multidimensional mixing parameter, it is seen here that mixtures of compound Poisson approximations provide a simple and efficient alternative. Two classes of mixture models for individual riss are described in Section 2, and corresponding approximations are proposed in Section 3. Since the simplicity of the models considered allows for a clear understanding of the effect of dependence and heterogeneity among riss, the impact of both factors on the quality of the approximation is explored numerically in Section Individual models with dependence among riss A number of strategies have been proposed recently to account for dependence among riss in the individual model. They can be broadly classified in two groups: those in which the dependence is described in terms of random variables, and those where the relation between riss is expressed in terms of distribution functions. Examples of mixture models of each ind are described in turn.

3 2.. A general multi-class shoc model C. Genest et al. / Insurance: Mathematics and Economics 32 ( One fruitful way to introduce dependence in a single- or multi-class ris model is in terms of shocs or catastrophes that could affect the entire portfolio or subclasses thereof. A general model of this sort is described below which includes recent proposals by Albers (999 and Cossette et al. (22 as special cases. Consider a portfolio that is divided into m classes comprising n,...,n m contracts, and let X l represent the ris associated with the lth contract in the th class, so that the aggregate claim amount is given by S = m n X l. = Assume that the entire portfolio can be affected by a global shoc, represented by an indicator random variable J, and that all riss in class are subject to a further shoc whose probability of occurrence may differ according to whether the overall catastrophe has taen place (J = or not (J =. Let J (α indicate the presence or absence of a shoc specific to class, givenj = α. For fixed values of α and β = J (α {, }, the ris associated with the lth contract in the th class may then be modeled as in ( by X (αβ l = J (αβ l B (αβ l, where J (αβ l is a Bernoulli random variable indicating whether at least one claim was filed or not, and B (αβ l is a strictly positive random variable representing the total amount claimed. Denote I = I for any probability or indicator function I. The ris X l may then be represented in the form X l = J(J ( X ( l + J ( X ( l + J(J ( X ( l + J ( X ( l. (2 It seems reasonable to suppose that (a J, the J (α s and the J (αβ l s are mutually independent; and that (b the B (αβ l s are independent of each other and of all indicator random variables. These conditions, which will be assumed to hold throughout the paper, are satisfied in the following special cases. Example 2.. Eq. 2. of Albers (999 defines a multi-class dependence model which may be written directly as X l = J ( X ( l + J ( X ( l = J ( J ( l B ( l + J ( J ( l B ( l with the above notation. This is obviously a specialization of (2 in which J =, so that J (, J ( l, J ( l, B ( l and B ( l need not be defined. While there is no provision in this model for a shoc that would be common to all classes, Albers notes that when all riss are combined into a single class, J ( can be assimilated to a global shoc. This is what happens in the second example. Example 2.2. To introduce dependence between the n = n riss of the single-class model (, Cossette et al. (22 suppose that I l may be written as I l = min(j l + J,, (3 in terms of mutually independent Bernoulli random variables J,...,J n with means r,...,r n, respectively, chosen in such a way that r min(q,...,q n with q = P(I l = = P(J = J l = = r r l, l n.

4 76 C. Genest et al. / Insurance: Mathematics and Economics 32 ( This construction, which reduces to independence when r =, is seen to fall under the purview of the above general model if one sets J =, J ( = J and J ( l =, J ( l = J l, B ( l = B ( l = B l in formula (2. For, one then finds X l = J B l + J J l B l, which is equivalent to the representation X l = I l B l with I l defined as in (3. Note that J (, J (, J (, B ( l and B ( l need not be defined in this case, since J =. Example 2.3. When several classes are involved, Cossette et al. (22 model the ris associated with the lth contract in the th class as X l = I l B l with I l = min(j l + J + J,, (4 where J, the J s and the J l s are mutually independent Bernoulli random variables with mean r, r and r l, respectively. Here, r,...,r m are parameters which, in broad terms, increase the dependence among the riss as they get larger. The I l s are independent if and only if r = =r m =. Note that in view of relation (4 and the independence hypothesis on J, the J and the J l s, one has q l = P(I l = = P(J l = J = J = = r l r r. This multi-class model is subsumed by the general shoc model (2 when and J = J, J ( =, J ( = J, J ( l =, J ( l =, J ( l = J l B ( l = B ( l = B ( l = B l. Of course, there are again many variables that need not be defined, such as J ( l, for instance. In the multi-shoc model (2, the distribution function F Xl of X l is easily derived from those of the X (αβ and of the B (αβ l s. Indeed, if E(J = r, E(J (α = r (α and E(J (αβ l = r (αβ l for every choice of l n and m, a straightforward conditioning argument exploiting conditions (a and (b above implies that F Xl (x = r{r ( F ( X l (x + r ( = r[r ( { r ( l + r ( F ( X l l F ( B l + r[r ( { r ( l + r ( l F ( B l (x}+ r{r ( and a simple rearrangement of the terms then yields where F Xl (x = q l + rr ( r( l F ( B l F ( X l (x}+ r ( { r ( l + r ( (x + r ( l F ( B l (x}+ r ( { r ( l + r ( (x + r r ( r ( l F ( B l F ( X l (x}] l F ( B l (x + rr ( r ( (x}] (x} l F ( B l (x + r r ( r ( l F ( B l l s (x, x, q l = rr ( r( l + r r ( r ( l + rr ( r ( l + r r ( r ( l = P(X l. (5

5 C. Genest et al. / Insurance: Mathematics and Economics 32 ( Accordingly, each ris X l may be expressed in the form ( as { Bl if I l =, X l = if I l =, in terms of an indicator random variable I l with P(I l = = q l defined as in Eq. (5, and a strictly positive random variable B l with mixture distribution F Bl = rr( r( l F ( q B l l + r r( r ( l F ( q B l l + rr( r ( l r r( r ( l F ( q B + l l F ( q B l l In this model, however, the X l s are generally far from being independent, which maes it hard to evaluate directly the distribution F S of the total claim amount S of the portfolio. As will be seen in Section 3, a handy way of circumventing this problem lies in the fact that under conditions (a and (b, relation (2 induces a mixture structure for S. In other words, there exists a latent random vector Θ with distribution M for which F S may be expressed in the form F S (x = F S (θ(x dm(θ, (6 where S (θ is distributed as S given Θ = θ. To be specific, let Θ = (Θ,...,Θ m tae values in {, } m+ in such a way that and P(Θ = α = P(J = α = r α r α P(Θ = β,...,θ m = β m Θ = α = P(J (α = β,...,j (α m = β m = for all α, β,...,β m {, }.ForΘ = θ, write X (θ l = X(αβ l and let S (θ = n X (θ l and S (θ = Then in view of (2, one has ( m n F S (x = P X l x = = α= β = = θ {,} m+ P m = ( m n P β m = ( m = S (θ = m. (r (α = β ( r (α β S (θ. (7 X (αβ l x P(Θ = θ = x P {Θ = (α, β,...,β m } θ {,} m+ F S (θ(xp (Θ = θ, which is the same as (6 for Θ discrete. The advantage of this representation is that for given θ, the S (θ s are mutually independent and each of them is a finite sum of mutually independent random variables.

6 78 C. Genest et al. / Insurance: Mathematics and Economics 32 ( Single-class ris models based on copulas Eq. (2, and relations (3 and (4 which stem from it in special cases, introduce dependence between riss through shocs represented by indicator variables. In the single-class portfolio problem, where each ris may be written as in ( in the form X l = I l B l with Bernoulli random variables I l with expectation q l, another way of modeling dependence among the components of the vector I = (I,...,I n is to write their joint cumulative distribution function as F I (i,...,i n = C{F I (i,...,f In (i n }, (8 where F Il (t = P(I l t and C :[, ] n [, ] is a copula, that is, the restriction to [, ] n of a cumulative distribution function with uniform marginals on the unit interval. The advantage of this approach, considered by Cossette et al. (22, is that the distributions F Il of the occurrence variables I l and their joint dependence structure can then be modeled separately. Writing F I in the form (8 is not restrictive per se, since it was shown by Slar (959 that every multivariate distribution function may be written this way for some copula C. What is an assumption, however, is to suppose that C is of a specific form or that it belongs to a given class C of copulas, such as C γ (u,...,u n = ( n + u γ + +u γ n /γ, γ > nown as the model of Clayton (978. The reader may refer to the review article by Frees and Valdez (998 or to the boos by Hutchinson and Lai (99, Joe (997, Nelsen (999 and Drouet-Marie and Kotz (2 for a large selection of models of this sort. Of particular interest here are copula-based dependence models that induce a mixing structure for the total claim distribution F S. Two classes of this type are described below: the elementary Fréchet mixture model, and the much richer structure associated with the so-called Archimedean copulas (Genest and MacKay, 986. By definition, the latter are expressible in the form C(u,...,u n = φ {φ(u + +φ(u n } (9 for some function φ : (, ] [, such that φ( = and ( l dl dt l φ (t, l n. ( Popular choices for actuarial applications include (a φ γ (t = (t γ /γ with γ>, which yields the Clayton model defined above; (b φ γ (t ={log(/t} γ with γ, which generates Gumbel s family of (extreme value copulas; (c φ γ (t = log{(e γ /(e γt } with <γ <, which corresponds to the positively associated members of Fran s family. See Genest et al. (998 for a three-parameter family of Archimedean copulas that encompasses all these models. Example 2.4. Suppose that the underlying copula of the vector I is of the form C γ (u,...,u n = ( γu u n + γ min(u,...,u n for some γ, so that I,...,I n are mutually independent with probability γ, but comonotonic in the sense of Wang and Dhaene (997 with probability γ. Further assume without loss of generality that the I l s are labeled in such a way that q l = P(I l = P(I l = = q l when l l. Then I l = F I l (V, l n

7 C. Genest et al. / Insurance: Mathematics and Economics 32 ( for some random variable V with uniform distribution on [, ], so that I = =I l = and I l = =I n = if and only if (up to a set of measure zero q l V q l for l n (with q =. Furthermore, I = =I n = when V q n. Conditionally on the I l s being comonotonic, therefore, the distribution of S is then given by F com (s = (q l q l F B(l (s + ( q n [, (s with B (l = B l + +B n, l n. Unconditioning, one can then write F S (s = ( γf S (s + γf com (s where S = X + +X n is the sum of mutually independent random variables X l whose univariate distributions are the same as those of the X l s. The above model can be viewed as a special case of the previous shoc model with a single class in which the indicator random variables J (α l, l n, are either comonotonic or mutually independent according as a global shoc J with P(J = = γ occurs (α = or not (α =. The following example, however, is of a different type. Example 2.5. Suppose that the joint distribution of the vector I is of the form (8 with Archimedean copula C generated by φ. Further assume that condition ( is verified for all integers n, so that φ is completely monotone and hence the Laplace transform of a distribution M whose support is included in [,. Following Marshall and Olin (988, F I (i,...,i n may then be viewed as a mixture of powers, that is, it can be written in the form n F I (i,...,i n = {FI l (i l } θ dm(θ, ( where F I l (t = exp[ φ{f Il (t}] is the cumulative distribution function of a Bernoulli random variable with expectation exp{ φ( q l }, l n. In other words, Θ is a latent variable conditional upon which the I l s are mutually independent Bernoulli random variables with mean E(I l Θ = θ = exp{ θφ( q l } q lθ, l n. (2 For given θ >, let I θ,...,i nθ be mutually independent indicator random variables such that E(I lθ = q lθ. Assuming, as in the standard single-class model, that the B l s are independent of each other and of all the indicators, the components X (θ l = { Bl if I lθ =, if I lθ =, of the sum S (θ = X (θ + +X n (θ are then mutually independent, so that F S admits the mixture representation (6. For an example of application of model (, see Cossette et al. (in press.

8 8 C. Genest et al. / Insurance: Mathematics and Economics 32 ( Multi-class ris models based on copulas To define a multi-class extension of the Archimedean model of Section 2.2, suppose that the joint distribution of the (n + +n m -dimensional vector I = (I,...,I m of occurrence random variables is given by m n F I (i,...,i m = {FI l (i l } θ dm(θ,...,θ m, (3 = where M is the m-variate distribution function of the latent vector Θ = (Θ,...,Θ m [, m and F I l (t = exp[ φ {F Il (t}] is the cumulative distribution function of a Bernoulli random variable with expectation exp{ φ ( q l } with l n and m. Conditional on the value of Θ, the I l s are then mutually independent Bernoulli random variables with E(I l Θ = θ = E(I l Θ = θ = exp{ θ φ ( q l }=q lθ. For all m and l n, let I (θ l be mutually independent indicator random variables with mean q lθ, and assume as before that the B l s are independent among themselves and from all indicators. Conditional on Θ = θ, the sums S (θ and S (θ defined as in (7, but with X (θ l = X(θ l = I (θ l B l are then composed of mutually independent terms. Consequently, the total claim distribution F S may again be represented in the form (6. 3. Approximation strategy and applications In the individual and collective ris models described in Section 2, dependence is introduced via a (possibly multivariate mixing variable Θ which is either latent, as in Archimedean copulas, or explicitly represented by the indicators of the common and class shocs in the general construction of Section 2.. In all cases, it was seen that the total claim distribution could be expressed in the form F S (x = F S (θ(x dm(θ involving the distribution function of a double sum m m S (θ = S (θ n = = = X (θ l of random variables X (θ l that are mutually independent given Θ = θ. Furthermore, the latter may all be expressed in the form X (θ l = I (θ l B(θ l as in (. To evaluate F S, therefore, one could see (if necessary a suitable discretization of the B (θ l s and use, say, the algorithms of DePril that can be found in the boo by Rolsi et al. (999, Chapter 4. As a natural alternative often considered in actuarial science, compound Poisson approximations could be developed for F S by exploiting standard results to approach each sum S (θ of independent riss given Θ = θ by a Poisson variable T (θ. See the authoritative text by Barbour et al. (992 for a survey of the topic, and Rolsi et al. (999, Chapter 4, among other, for actuarial applications.

9 Upon unconditioning, the fact that d(s,t C. Genest et al. / Insurance: Mathematics and Economics 32 ( d(s (θ,t (θ dm(θ (4 can then be used to derive bounds on the overall quality of the approximation for specific choices of the distance d. Witte (99 gives a series of such bounds for d = d K, d TV or d SL, and the conditions under which they apply. Other ey references concerning compound Poisson approximations include Serfozo (986, 988 and Vellaisamy and Chaudhuri (999. Detailed descriptions of this Poisson approximation strategy are given below, both for the general multi-class model and the mixture models based on Archimedean copulas. 3.. Poisson approximation for the multi-class shoc model Suppose that the ris X l associated with the lth contract in the th class of a portfolio can be affected as in (2 by a random global catastrophe J and a class-specific shoc J (α whose probability of occurrence depends on the value of the indicator J = α {, }. It was seen in Section 2. that given a vector Θ = θ = (α, β,...,β m whose components indicate whether the global catastrophe and the m class-specific shocs occurred or not, the random sums S (θ and S (θ,...,s(θ m defined in (7 consist of mutually independent terms. In particular, for each m, S (θ = S (αβ may thus be approximated by a Poisson variable T (αβ chosen in such a way that either or E(T (αβ = E(S (αβ P(T (αβ = = P(S (αβ =, so that T (αβ dominates S (αβ in the convex or in the stochastic dominance order, respectively. A Poisson approximation of S is then given by T = JT ( + JT ( with m m T (α = J (α T (α + J (α T (α, α =,. (7 = = An evaluation of the right-hand side of (4 now yields α= β = { m r α r α β m = (r (α = β ( r (α β } ( m d S (αβ, = m = T (αβ which provides an upper bound on the quality of the above approximation, whether d stands for the Kolmogorov, the total-variation or the stop-loss distance. A looser, but possibly easier bound to compute can be derived from the inequality d ( m = S (αβ, m = T (αβ m = ( d S (αβ,t (αβ, which obtains because for fixed α, the S (αβ s and the T (αβ s form a collection of 2m mutually independent random variables., (5 (6

10 82 C. Genest et al. / Insurance: Mathematics and Economics 32 ( The following proposition summarizes the above developments. Proposition 3.. In the general multi-class model (2, a Poisson approximation to the total claim amount S is provided by T = JT ( + JT ( with T (α defined as in (7 in terms of Poisson random variables T (αβ chosen as per relations (5 or (6. An upper bound on the quality of this approximation is given by { m } r α r α (r (α β ( r (α m β d(s (αβ,t (αβ, α= β = β m = = where d represents either the Kolmogorov, total-variation or stop-loss distance. = Example 3.. In the single-class model of Albers (999 described in Example 2., one has J = and hence r =. An upper bound on the quality of the approximation T = T ( = J ( T ( + J ( T ( of S is thus given by r ( d(s(,t ( + r ( d(s(,t (. (8 Here, T (β is a Poisson random variable whose mean or whose probability of being equal to zero matches that of S (β = X (β + +X (β n, that is, the total claim amount when the common shoc J occurs (β = or not (β =. To illustrate this result concretely, consider for example the result of Gerber (984 reported as Theorem 4.6. of Rolsi et al. (999, which states that for β =,, one has and d TV (S (β,t (β d SL (S (β,t (β n (r (β l 2 n (r (β l 2 E(B (β l. Using these inequalities in bound (8 yields and d TV (S, T r ( d SL (S, T r ( n (r ( n (r ( l 2 + r ( l 2 E(B ( l n (r ( l 2 n + r( (r ( l 2 E(B ( as upper bounds for the quality of the proposed Poisson approximation to the total claim amount in the model of Albers (999. Example 3.2. In the single-class model of Cossette et al. (22, it has already been seen that J = and r =. Denoting J ( = J, J ( l = J l and B ( l = B ( l = B l as in Example 2.2, and recalling that J ( l = for all l n, one has S = J S ( + J S ( = J B l + J J l B l, l

11 C. Genest et al. / Insurance: Mathematics and Economics 32 ( which can then be approximated by T = T ( = J T ( + J T ( with T ( = B + +B n while T ( is chosen to match either the mean of J B + +J n B n or its probability of being equal to zero. In this case, one may tae d(s (,T ( = and r ( = E( J in the bound (8. Proceeding as in the previous example, therefore, upper bounds on the quality of this approximation, based on Gerber s result, would tae the form and d TV (S, T r ( d SL (S, T r ( n (r ( l 2 n (r ( l 2 E(B l. Example 3.3. In the multi-class model of Cossette et al. (22, one has J = J, J ( =, J ( = J, J ( l =, J ( l =, J ( l = J l and B ( l = B ( l = B ( l = B l for all l n and m. The total claim amount may thus be written as m n m ( n n S = J B l + B l + J l B l. = It can be approximated by T = J in which T ( m n B l + = E(T ( = E J = m J = ( J n J J B l + J T ( is chosen in such a way that either ( n J l B l or P(T ( = = P ( n J l B l =. Since in this approximation T ( = S ( and T ( is distributed as S given J =, a bound on the quality of this approximation is then of the form { m } r (r ( β ( r ( m β {β =}d(s (,T (. and β = β = = In view of Gerber s result, one can see that d TV (S ( d SL (S (,T (,T ( n (r ( l 2 n (r ( l 2 E(B l. Consequently, the upper bound of d TV (S, T is given by { m } r (r ( β ( r ( m n β {β =}(r ( l 2, β = β = = = =

12 84 C. Genest et al. / Insurance: Mathematics and Economics 32 ( while the upper bound of d SL (S, T equals { m } r (r ( β ( r ( m n β {β =}(r ( l 2 E(B l. β = β = = = 3.2. Poisson approximation for the single-class Archimedean model Suppose that the riss X l = I l B l in a single-class model are such that the B l s are independent of the I l s, and the latter are Bernoulli random variables with E(I l = q l and copula C(i,...,i n = φ {φ(i + +φ(i n } for some completely monotonic generator φ : (, ] [, whose inverse φ is the Laplace transform of a distribution M whose support is a subset of [,. It was shown in Example 2.5 that conditional on a realization θ of the latent variable Θ with distribution M, the total claim amount S = n I l B l is distributed as S (θ = n I lθ B l, where the I lθ s are mutually independent Bernoulli random variables with mean E(I lθ = q lθ given in (2. Now suppose that an approximation is sought for the stop-loss premium or expected shortfall associated with the portfolio S. For each fixed θ, one could then approach S (θ by a compound Poisson random variable T (θ with parameters q θ = q lθ and F Zθ = q lθ q θ F Bl, so that E(T (θ = E(S (θ. The mixture random variable T with distribution function F T (t = P(T (θ tdm(θ is then an approximation of S satisfying E(T = E(S. Examples of application of inequality (4 are given below for this approximation when d is either the total-variation or stop-loss distance. Similar results could be obtained using other bounds or a different approximation T of S ensuring that P(T = = P(S =. Example 3.4. When d is either the total-variation or the stop-loss distance, the same result of Gerber (984 mentioned above implies that d TV (S (θ,t (θ qlθ 2 and d SL (S (θ,t (θ qlθ 2 E(B l, where q lθ is defined as in (2.Inviewof(4 and the fact that φ (t = E(e tθ, it follows that d TV (S, T = Similarly, one has d SL (S, T d TV (S (θ,t (θ dm(θ q 2 lθ dm(θ { 2e θφ( q l + e 2θφ( q l } dm(θ = [φ {2φ( q l }+2q l ]E(B l. [φ {2φ( q l }+2q l ]. (9

13 C. Genest et al. / Insurance: Mathematics and Economics 32 ( Of course, both these bounds immediately reduce to those initially found by Gerber (984 for independent riss when φ(t = log(/t is the generator of the independence distribution. As a concrete illustration, suppose that the underlying copula of the vector I of occurrence variables is from the Gumbel family with generator φ γ (t ={log(/t} γ and parameter γ. One may then conclude that d TV (S, T {( q l 2/γ + 2q l }, which reduces to Gerber s original bound when γ = (the case of independence and increases with γ until it approaches q + +q n as γ, that is, when the I l s become comonotonic. Example 3.5. When the B l s are independent and identically distributed, Goovaerts and Dhaene (996 show that d TV (S (θ,t (θ e q θ q θ q 2 lθ (see also Barbour and Hall, 984. In this special case, therefore, it also follows from (4 that d TV (S, T qlθ 2 e q θ dm(θ. q θ 3.3. Poisson approximation for the multi-class Archimedean model When the dependence between the multivariate indicator random vectors I,...,I m is modeled through Eq. (3, it was shown in Section 2.3 that conditionally on the value of a latent vector Θ, the total claim amount may be written as S (θ = m n I (θ l B l = in terms of mutually independent Bernoulli random variables I (θ l with mean q lθ. The approximation of each of these S (θ s by a compound Poisson random variable T (θ with parameters q θ = m n q lθ and F Zθ = = m n = q lθ F Bl q θ leads once again to a mixture random variable T with distribution (9 such that E(T = E(S. Proceeding exactly as in the previous section, one can see for instance that for such an approximation and d TV (S, T d SL (S, T m n = m n = [φ [φ {2φ ( q l }+2q l ] {2φ ( q l }+2q l ]E(B l.

14 86 C. Genest et al. / Insurance: Mathematics and Economics 32 ( If in addition the B l s are identically distributed, one has also d TV (S, T m n = q 2 lθ e q lθ q θ dm(θ. Similar bounds could obviously be devised for a different approximation T satisfying P(T = = P(S =. In addition, formula (4 would also be applicable to the more sophisticated bounds surveyed by Witte (99, under whatever particular conditions mae the latter valid. 4. Effect of heterogeneity and dependence on the quality of the approximations In order to quantify the effect of dependence on the quality of the Poisson approximation, a fictitious portfolio was considered which consisted of a single class comprising contracts. To introduce heterogeneity, this portfolio was divided into subgroups labeled (, 2 with, 2 {,...,}. It was assumed that each of the riss in subgroup (, 2 could be expressed as a product X (, 2 l = I (, 2 lb (, 2 l, l of two independent random variables, namely a Bernoulli random variable I (, 2 l with P(I (, 2 l = =. +.5q( 2 and a geometric random variable B (, 2 l distributed on the integers {, 2,...} with mean E(B (, 2 l = 2 + b( 2. Here, q and b are arbitrary non-negative parameters ranging in [, 4] and [, 2], respectively. To induce dependence between the contracts, a single-class ris model based on a Fran copula was used. In other words, the dependence between the n = indicator random variables I (, 2 l was assumed to be of the form (8 with Archimedean copula C of the type (9 with generator φ γ (t = log { e γ e γt }, γ >. The statistical properties of this class of multivariate copulas were studied by Nelsen (986 and Genest (987, who show that the parameter γ is related to Kendall s coefficient of concordance between any two components (that is, between any two I (, 2 l s via τ(γ = + 4 γ ( γ γ t e t dt. Three choices of γ were considered that correspond to independence (τ =, γ =, wea dependence (τ = /8, γ.3956, and moderate dependence (τ = /4, γ between the riss. An advantage of woring with Fran s copula with parameter γ > is that its associated mixture distribution M is the discrete logarithmic distribution with probability function P(Θ = θ = ( e γ θ, θ {, 2,...}. θγ In this special case, therefore, it is actually possible to determine F S (x = F S (θ(xp (Θ = θ θ=

15 C. Genest et al. / Insurance: Mathematics and Economics 32 ( Table Exact and compound Poisson-based approximation for the stop-loss premium corresponding to a single-class portfolio comprising heterogeneous contracts whose occurrence random variables are independent (τ = or stochastically related (τ = /8 or /4 through a Fran copula model x τ = τ = /8 τ = /4 π S (x π T (x π S (x π T (x π S (x π T (x exactly by applying a standard fast Fourier transform algorithm to compute F S (θ for all θ {,...,θ } for some sufficiently large value of θ. A similar numerical procedure was used to compute F T as defined in (9 in terms of the conditional Poisson approximations T (θ with E(T (θ = E(S (θ, as detailed in Section 3.2. As an illustration, Table shows the exact and approximate values of the stop-loss premium for this portfolio when q = 3, b = 2, and Kendall s tau is successively equal to, /8 and /4. Two observations are apparent from this table, in which of course π S ( = π T ( = E(S = E(T = and π T (x π S (x for all x. First, the approximation is seen to be quite satisfactory for all values of τ and x; the difference π S (x π T (x is always smaller than, and quite often much smaller than this. Second, it appears that for each fixed value of x, π S (x is a monotone increasing function of γ, so that larger degrees of dependence among the riss translates into a greater stop-loss premium for any given retention amount x. This empirical finding may actually be confirmed using the following extension of Proposition 3 in Denuit et al. (22. Proposition 4.. Let (M,...,M n be a vector of integer-valued random variables. Given sequences (Y,..., (Y n of mutually independent, positive random variables, define the random sum { Ml = W Ml = Y l if M l >, if M l =, for each l {,...,n}. Suppose that the M l s are joined by a multivariate distribution whose underlying copula is of the form (9 with generator φ = φ γ depending on a parameter γ. Let also W = W Ml.

16 88 C. Genest et al. / Insurance: Mathematics and Economics 32 ( If φ γ φ γ is a Laplace transform for some possible parameter values γ and γ, then (M,...,M n sm (M,...,M n (2 in the supermodular ordering, as defined in Shaed and Shanthiumar (997, Bäuerle and Müller (998, or Denuit et al. (22. As a consequence, (W M,...,W Mn sm (W M,...,W M n (2 and hence W cx W (22 in the classical stop-loss or convex ordering defined in the above references. Proof. Relation (2 is a direct consequence of Theorem 3. of Wei and Hu (22. Relation (2 then follows from part (iv of Proposition 2 of Denuit et al. (22, and finally, part (i of Corollary 2 in the latter paper implies that W cx W, as claimed. Corollary 4.. Suppose that in the setting of Proposition 4., the underlying copula of the M i s belongs to the Clayton, Gumbel or Fran families, as defined in Section 2.2. If φ γ represents a generator for any one of these three families, then φ γ φ γ is a Laplace transform whenever γ<γ, and hence relations (2 (22 obtain. Proof. This is a straightforward consequence of results stated in Appendix A. of Joe (997. Corollary 4.2. Consider a single-class model consisting of n riss of the form X l = I l B l, where the B l s are positive random variables that are mutually independent among themselves and of the I l s. Suppose that the latter are Bernoulli random variables whose joint distribution is of the form (8 with Archimedean copula C from the Clayton, Gumbel or Fran families, as parameterized in Section 2.2. The stop-loss premium of the total claim amount S = X + +X n is then a monotone increasing function of the dependence parameter γ, in other words, γ<γ π S (x π S (x, x. Proof. It suffices to set I l = M l and B l = Y l for l n. To investigate in greater depth the impact of heterogeneity and the strength of dependence on the quality of the compound Poisson approximations, the observed Kolmogorov distance d γ (q, b = sup F S (x F T (x x R between F S and its approximation F T was plotted as a function of q and b in Figs. 3 for the three values of γ corresponding to τ = (independence, τ = /8 (small dependence, and τ = /4 (moderate dependence. Three main observations emerge from these pictures: (i In all graphics, the influence of b seems to be much smaller than that of q; that is, greater heterogeneity in the occurrence probabilities is more detrimental to the approximation than heterogeneity in the average amounts of the claims. (ii The approximation tends to improve as γ increases; in other words, the compound Poisson approximation performs better when the occurrence random variables are dependent than when they are not; nevertheless, the approximation remains quite good even in the case of independence, since d γ (q, b.5 on the entire domain.

17 C. Genest et al. / Insurance: Mathematics and Economics 32 ( Fig.. Graph of the Kolmogorov distance d γ (q, b between F S and its compound Poisson-based approximation F T for a single-class portfolio comprising heterogeneous contracts whose occurrence random variables are independent. Fig. 2. Graph of the Kolmogorov distance d γ (q, b between F S and its compound Poisson-based approximation F T for a single-class portfolio comprising heterogeneous contracts whose occurrence random variables are stochastically related through a Fran copula model with τ = /8.

18 9 C. Genest et al. / Insurance: Mathematics and Economics 32 ( Fig. 3. Graph of the Kolmogorov distance d γ (q, b between F S and its compound Poisson-based approximation F T for a single-class portfolio comprising heterogeneous contracts whose occurrence random variables are stochastically related through a Fran copula model with τ = /4. (iii The approximation tends to deteriorate as q increases; in other words, the compound Poisson approximation performs better when the occurrence random variables are homogeneous than when they are not. Indeed, the approximation is best when q =, even in the presence of heterogeneity in the means of the claim amounts B l. The same phenomena, which were verified experimentally using other portfolio compositions and copula dependence structures, are generally in accordance with intuition, except perhaps as concerns item (ii. Theoretical arguments in support of these conclusions would be well worth developing. Acnowledgements Partial funding in support of this wor was provided by the Natural Sciences and Engineering Research Council of Canada, by the Fonds québécois de recherche sur la nature et les technologies, and by the Chaire en assurance L Industrielle Alliance (Université Laval. The authors are grateful to Jean-François Plante for some programming assistance. References Albers, W., 999. Stop-loss premiums under dependence. Insurance: Mathematics and Economics 24, Barbour, A.D., Hall, P., 984. On the rate of Poisson convergence. Mathematical Proceedings of the Cambridge Philosophical Society 95, Barbour, A.D., Holst, L., Janson, S., 992. Poisson Approximation. Oxford University Press, Oxford. Bäuerle, N., Müller, A., 998. Modeling and comparing dependencies in multivariate ris portfolios. ASTIN Bulletin 28,

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