MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES WITH LÉVY COPULAS BY BENJAMIN AVANZI, LUKE C. CASSAR AND BERNARD WONG ABSTRACT In this paper we investigate the potential of Lévy copulas as a tool for moelling epenence between compoun Poisson processes an their applications in insurance. We analyse characteristics regaring the epenence in frequency an epenence in severity allowe by various Lévy copula moels. Through the introuction of new Lévy copulas an comparison with the Clayton Lévy copula, we show that Lévy copulas allow for a great range of epenence structures. Proceures for analysing the fit of Lévy copula moels are illustrate by fi tting a number of Lévy copulas to a set of real ata from Swiss workers compensation insurance. How to assess the fit of these moels with respect to the epenence structure exhibite by the ataset is also iscusse. Finally, we provie a ecomposition of the trivariate compoun Poisson process an iscuss how trivariate Lévy copulas moel epenence in this multivariate setting. KEYWORDS Lévy copula, Depenence, Compoun Poisson process, Insurance, Real ata.. INTRODUCTION In a non-life insurance company, an event may give rise to claims of ifferent types. Such events range from a work-relate accient resulting in claims for meical costs an allowance costs, to a natural peril causing losses in motor an home classes of business. Furthermore, epenence in claims processes can have an impact on both frequency (claim counts) an severity (claim amounts). This has irect implications on pricing, reserving an capital allocation of an insurance company (Embrechts et al., 00; Denuit et al., 005; McNeil et al., 005). It is also highly relevant for solvency purposes an in risk base capital regulatory systems such as Solvency II. A natural an stanar choice for moelling insurance claims processes is the compoun Poisson process (e.g., Bowers et al., 997; Mikosch, 009; Astin Bulletin 4(), 575-609. oi: 0.43/AST.4..36989 0 by Astin Bulletin. All rights reserve.
576 B. AVANZI, L.C. CASSAR AND B. WONG Asmussen an Albrecher, 00). In a multivariate setting, epenence between multiple compoun Poisson processes (loosely interprete as classes of business) can be intuitively represente in a common shock representation (Linskog an McNeil (003); see also Yuen an Wang (00)). In such a representation, classes of business (potentially) share common shocks claims occurring at the same time in two or more ifferent classes accoring to an ientical arrival process. Furthermore, epenence between the sizes of the claims occurring simultaneously can be moelle with istributional copulas. This approach has a number of avantages. Firstly, the common shock moel allows for etaile an separate specification of epenence in frequency an epenence in severity. In aition, as the moel is specifie upon a continuous time (Markov) stochastic process, the moel also allows for the consieration of epenence over alternative time horizons in an internally consistent manner. Unfortunately, ue to its flexibility, the common shock moel becomes increasingly parameter intensive as the number of imensions increases. For example, the case of four classes of business can require the specification of up to fifteen inepenent Poisson arrival processes (because of jumps that can be common to two, three, or four classes), six bivariate istributional copulas, four trivariate istributional copulas, one quavariate istributional copula an twenty-four jump size istributions. An alternative approach is to apply a istributional copula irectly to the aggregate claims of each class at a chosen time horizon, creating a multivariate istribution of aggregate claim amounts (see, for example, McNeil et al., 005; Bargès et al., 009). Similarly, a istributional copula may be applie to the aggregate number of claims over a chosen time horizon, (see, for example, Bäuerle an Grübel, 005; Genest an Neslehová, 007). The moel is then reuce to moelling epenence between ranom variables for a given time horizon. This approach possesses a number of benefits, incluing relative parsimony in moel specification, an in particular with the facilitation of a bottom-up approach to multivariate moel builing whereby moels are built by combining the information of a class of business (i.e. the marginals), with that of the epenence structure across classes (McNeil et al., 005, p. 85). This is in contrast to a common shock base approach where moels are built from common shock events. The focus here is on the classes of business, rather than the common shock events. Unfortunately, as the istributional copula for aggregate claims will epen on the chosen time horizon, in general it is not possible to infer the copula for a ifferent time horizon (to consier the risks face by an insurance company over, or 5 years, for instance, Fosker et al., 00, p. 8). This approach also requires sufficient ata for the aggregate claim amounts in each class of business for the chosen time horizon. For example, if only a single year of ata is available, then using a time horizon of one year woul allow for only ata point for fitting a istributional copula. This also results in an inefficient use of ata where iniviual accient information is known. In contrast to the two methos iscusse above, Lévy copulas provie a new metho which briges the benefits of the common shock an istributional
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 577 copula approaches. Uner this approach, epenence is introuce via a multi variate function (the Lévy copula) which couples the marginal tail integrals of the compoun Poisson processes for each class of business into a multivariate tail integral which completely specifies the esire multivariate (epenent) compoun Poisson processes moel. In a nutshell, the tail integral of a compoun Poisson process (relate to its Lévy measure) represents the expecte number of losses over a threshol (the argument of the function) over one unit of time; refer to the following section for a formal efinition. Such a representation combines the avantages of the common shock moel an the istributional copula approach, by being parsimonious, facilitating a bottom-up approach, allowing changes of time horizon (time consistency), an by being efficient in the way it uses available ata. Lévy copulas were introuce in a series of publications by Tankov (003), Cont an Tankov (004) an Kallsen an Tankov (006). In applications, the Clayton Lévy copula have been use to moel the epenence between compoun Poisson processes firstly to estimate ruin probabilities for an insurance company with multiple classes of business (Bregman an Klüppelberg, 005). Optimal investment an reinsurance problems for a multiline insurer uner a Lévy copula framework was stuie in Bäuerle an Blatter (0). In the closely relate area of operational risk moelling, applications of Lévy copulas between operational loss cells is iscusse in Böcker an Klüppelberg (008), Biagini an Ulmer (009) an Böcker an Klüppelberg (00). On the statistical front, a maximum likelihoo scheme for fitting Lévy copulas to ata is provie in Esmaeili an Klüppelberg (00a), who focus in particular on fitting a Clayton Lévy copula. Aitional theoretical evelopments in more general Lévy process settings can also be foun in Barnorff-Nielsen an Linner (007), Bäuerle et al. (008) an Eer an Klüppelberg (009). In this paper, we first focus on a careful review of the concept of Lévy copula an she some light on how this function is generating epenence between compoun Poisson processes. To ate, there has been limite consieration of the properties enable by specific Lévy copula moels in applications, with the notable exception being the Clayon Lévy copula. Section 3 evelops new Lévy copula moels an illustrates how their epenence structures can be compare. It is illustrate how many of the special properties of the Clayton Lévy copula may not hol in general. Furthermore, it is also important from a practical point of view to consier alternative Lévy copula moels so as to provie aitional flexibility in the type of epenence available to the moeller. This allows for a better unerstaning of the actuarial applications of Lévy copulas an illustrates the range of epenence structures enable by them, in particular with respect to the impact of ifferent moels on the epenence in frequency an/or severity. Section 4 provies a moelling example using a set of worker s compensation claims an the newly evelope Lévy copulas. The issue of moel selection is also iscusse, as the fit of ifferent Lévy copulas to the ata is compare. Finally, as insurance companies normally run more than two (possibly epenent) classes of business, epenence beyon a
578 B. AVANZI, L.C. CASSAR AND B. WONG bivariate setting is of particular relevance. This is investigate in Section 5, where the similarities an ifferences between the bivariate an trivariate cases are highlighte. Such a evelopment is of interest as common jumps can then occur between any sub-set of the consiere processes.. DEPENDENCE BETWEEN COMPOUND POISSON PROCESSES This section provies an introuction to Lévy copulas an their implications on epenence between compoun Poisson processes. Note that whilst compoun Poisson processes in general can have jumps in both positive an negative irections, compoun Poisson processes with only positive jumps are consiere for the purpose of insurance claims moelling. Hence, only positive Lévy copulas are aresse in this paper... Lévy copulas an compoun Poisson processes Consier a bivariate compoun Poisson process {S (t), S (t)}, for instance, to moel two epenent classes of business; see also Sato (999, Theorem 4.3) for a comprehensive efinition of a multivariate compoun Poisson process. It is known that {S (t), S (t)} can be ecompose into unique (superscript =) an common (superscript ;) jumps, so that 9 S() t = S () t + S() t * (.) 9 S () t = S () t + S (), t where S = (t) an S = (t) are inepenent compoun Poisson processes an where S ; (t) an S ; (t) are epenent compoun Poisson processes whose jumps (the common shocks ) occur at the same time (Linskog an McNeil, 003; Esmaeili an Klüppelberg, 00a). In general, the jump size istributions of S i = (t) an S i ; (t) are not ientical. However, the jump size istribution of S i (t) will be a mixture of the jump size istributions for S i = (t) an S i ; (t) (see, for example, Mikosch, 009, Proposition 3.3.4). Let us now introuce more formally the concept of tail integral. The tail integral of a Lévy process measures its expecte number of jumps (above a certain threshol) per unit of time. In the (less general) case of a compoun Poisson process S i (t), i =,, the tail integral boils own to U ( x) = i li Fi ( x), x! ( 0, 3) ) (.) 3 x = 0. where F i (x) is the survival function for the jump size of S i (t). Furthermore, the joint tail integral of a bivariate compoun Poisson process {S (t), S (t)} is given by
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 579 U( x, x ) = Z ] l ] U [ ] U ] 3 \ F( x, x ), ( x, x )! (0, 3) ( x ) x! (0, 3), x = 0 ( x ) x = 0, x! (0, 3) ( x, x ) = (0,0). (.3) where l ; is the Poisson parameter for the (common) jumps in S ; (t) an S ; (t) an F(x, x ) is the joint survival function for the sizes of the common jumps. A formal efinition of the tail integral for a Lévy process with positive jumps is given in Appenix A.. A Lévy copula C couples the marginal tail integrals to the joint tail integral so that C ( U ( x ), U ( x )) = U( x, x ); (.4) see Appenix A. for a formal efinition of positive Lévy copulas. The mechanism is strikingly similar to the one with which istributional copulas couple the marginal istribution functions to the multivariate istribution function an is formalise in what is escribe in Cont an Tankov (004) as a reformulation of Sklar s theorem for tail integrals an Lévy copulas. Theorem.. (Sklar s theorem for Lévy copulas, Tankov, 003) If U is a tail integral with margins U ( ),, U ( ), then there exists a Lévy copula C such that U(x,, x ) = C (U (x ),, U (x )). (.5) If U ( ),, U ( ) are continuous on [0, 3] then this Lévy copula is unique. Otherwise, it is unique on the prouct of the ranges of the marginal tail integrals. The converse is also true. If C is a Lévy copula an U ( ),, U ( ) are marginal tail integrals, then (.5) efines a multiimensional tail integral. On one han, a common shock approach woul require the separate moelling of the Poisson parameters an jump size istributions of S = (t), S = (t), S ; (t) an S ; (t), as well as the epenence structure of the jump sizes of S ; (t) an S ; (t). On the other han, if the Lévy copula is known, only the Poisson parameters an jump size istributions for S (t) an S (t) (which are irectly observable) nee to be specifie. This is because the ecomposition of S (t) an S (t) into unique an common components as shown in (.) stems irectly from the Lévy copula (Böcker an Klüppelberg, 008), as summarise in the following lemma. Lemma.. Common jumps in S ; (t) an S ; (t) arrive at a rate l = C( l, l ), (.6)
580 B. AVANZI, L.C. CASSAR AND B. WONG whereas the sizes of these common jumps have joint survival function F ( x, x ) = C( lf( x), lf ( x)), (.7) l an marginal survival functions Z F ( x) = C( lf( x), l), ] l [ F ( ) = ] x C( l, lf( x)). \ l Unique jumps in S = i (t), i =,, arrive at rates an (.8) 9 i i, l = l - l i =,, (.9) whereas their sizes are istribute with survival functions 9 Fi ( x) = 9 `lifi( x) - l Fi ( x) j, i =,. (.0) l i In general, the istributions of the sizes of common jumps an unique jumps in each compoun Poisson process istributions will not be ientical. However, Lemma.3 provies conitions which must be satisfie by a bivariate Lévy copula to allow for ientically istribute unique an common jump sizes in each compoun Poisson process. Lemma.3. (Ientically istribute unique an common jump sizes) A bivariate compoun Poisson process with Lévy copula C satisfying F( x) C( l, l) = C( lf( x), l), * (.) F ( x) C( l, l ) = C( l, l F ( x)), has unique an common jump sizes in a given compoun Poisson process which are ientically istribute (an are ientical to the marginal jump size istribution of the process). Proof. If the jump size istributions of the common jumps are equivalent to the marginal jump size istributions of the process, then (.) follows from a rearrangement of (.8). In aition, (.0) further implies that as require. 9 Fi ( x) = ( l i Fi ( x) i ( x l l -l F )) - i = F ( x), for i =,, i (.)
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 58.. Changes of time horizon If the Lévy copula for a time horizon of one unit of time is given by C, then the Lévy copula for a time horizon of length T is expresse as u u C T ( u, f, u) = TCc, f, m. (.3) T T This result is ue to the properties of the tail integral an Lévy processes (Barnorff-Nielsen an Linner, 007, Equation 3). This shows how a Lévy copula approach allows for easy changes of time horizon, in contrast to the istributional copula approach. Interestingly, Lévy copulas an the istributional copula of the aggregate claim amounts at time T are relate via the following asymptotic relation, C ( u,, u ) lim f = C ( T,, T + T u f u ), (.4) T " 0 T where C T (,, ) is the time epenent istributional copula for an increment of time length T (Kallsen an Tankov, 006). Whilst the Lévy copula of the process can be interprete as the istributional copula of the aggregate claims amount for small T, in general C T (,, ) cannot be inferre from C..3. Constructing positive Lévy copulas In this section we present two methos for constructing Lévy copulas, ue to Tankov (003), Cont an Tankov (004) an Kallsen an Tankov (006) (see also Bäuerle an Blatter, 0). In Metho., Lévy copulas are erive from a multivariate Lévy process using Sklar s theorem for Lévy copulas. Note, however, that there are only a limite number of multivariate Lévy processes from which Lévy copulas can be erive. As an alternative, Metho. allows for the construction of Archimeean families of Lévy copulas. Metho.. Consier a -imensional spectrally positive Lévy process with continuous marginal tail integrals. A positive Lévy copula C can be constructe as - C( u, f, u ) = U`U ( u ), f, U - ( u ) j, (.5) where U(,, ) is the multivariate tail integral of the multivariate Lévy process an U ( ),, U ( ) are the marginal tail integrals. Remark.. If the marginal tail integrals are not continuous then a Lévy copula can still be constructe from (.5) by an extension proceure, see Tankov (003) an Kallsen an Tankov (006).
58 B. AVANZI, L.C. CASSAR AND B. WONG Metho.. For a function f : [0, 3] " [0, 3] with f(0) = 3 an f(3) = 0 an a efine inverse f ( ), C( u, f, u ) = f _ f( u ) + f+ f( u ) i, (.6) - where the inverse must satisfy k - ( k) (- ) ( f ) ( z) > 0, for z > 0, k =, f,, (.7) an (f ) (k) (z) enotes the k-th erivative of the inverse of f( ) with respect to z. Remark.. When constructing Archimeean istributional copulas, special care is neee in efining the inverse of the generator. The case of Lévy copulas is easier. Archimeean generators of Lévy copulas have a omain of [0, 3] an a range of [0, 3], so there is no nee for a pseuo-inverse (Nelsen, 999)..4. Funamental Lévy copulas For an inepenent multivariate compoun Poisson process, the tail integral of the multivariate process is expresse as Ux f x U x I{ x= f = x = 0} + f+ x I{x= f = x-= 0} (,, ) = ( ) U ( ) ; (.8) see Bregman an Klüppelberg (005). This means that the tail integral of an inepenent -imensional Lévy process is equal to 0 except for the cases where it is equal to the marginal tail integral. As a consequence, the inepenence Lévy copula is given by C ( u, f, u ) = u I + f+ u I, (.9) 9 { u= f= u= 3} { u= f= u-= 3} where the inicator functions are now change since a marginal tail integral evaluate at 0 is equal to 3 by efinition. The comonotonic Lévy copula is erive in a multivariate setting as C ( u, f, u ) = min( u, f, u ); (.0) see Cont an Tankov (004). This implies that the tail integral of a completely epenent -imensional Lévy process is given by the smallest of the marginal tail integrals. Remark.3. Comonotonicity in a multivariate compoun Poisson process means that all jumps in one process are functions of the jumps in the other. However, unless
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 583 the rates of jumps in the marginal compoun Poisson processes are equal, there will always exist unique jumps in the multivariate compoun Poisson process with a comonotonic Lévy copula, so that all arrival processes are not necessarily ientical. This stems from the iscontinuity at 0 of the tail integral of a compoun Poisson process. 3. COMPARISONS AND ILLUSTRATIONS OF BIVARIATE LÉVY COPULAS The current applie literature on Lévy copulas places consierable emphasis on the properties an application of the Clayton Lévy copula. The purpose of this section is to illustrate that Lévy copulas allow for a richer range of epenence structures by eveloping new moels an by comparing their main features. After introucing the pure common shock Lévy copula, an analysis of the Clayton Lévy copula (Tankov 003) is inclue for comparison purposes. Two other new Lévy copulas are also introuce, one that fits well the ata set that is consiere in this paper (see Section 4), an another one that allows for negative epenence in severity. Throughout this section, the epenence structures inuce by the ifferent moels will be compare by examination of their Lévy copula ensity, c ( u, u ) = C( u, u), (3.) u u where u i = U i (x i ), i =,. The volume uner the ensity on [ 0, l ] [ 0, l ] is the expecte number of common jumps per unit time, l # # ( u, u ) u u = l. (3.) l 0 0 c Since this is constant (for given l an l ), the relative repartition of the ensity on [ 0, l ] [ 0, l ] is informative of the epenence structure. First, note that small u an u inicate larger jump sizes (an vice versa), because the expecte number of jumps will be higher as the argument of the tail integral is lower. Thus, a relative higher ensity at small u an u will inicate a propensity for common jumps of large sizes in both components (an vice versa). Similarly, if more ensity is present at small u an large u, common jumps of large sizes in the first component will have a higher propensity to occur with common jumps of small sizes in the secon component, an vice versa. We consier in the rest of this section the following illustration scheme. Assuming l = 00 an l = 00, Lévy copula ensities an the istributional copula of common jump sizes are compare uner three possible values for the expecte number of common jumps l ; = 30, 60, 90. The purpose of this exercise is to emonstrate the range of epenence structures available
584 B. AVANZI, L.C. CASSAR AND B. WONG by using ifferent Lévy copulas, while holing the epenence in frequency constant. 3.. Pure common shock Lévy copula Lemma. showe how the Lévy copula affects both epenence in frequency an epenence in severity in a bivariate compoun Poisson process. However, it is sometimes esirable to assume inepenence between common jump sizes, an use a moel which allows for epenence in the frequency only. We refer to such a epenence structure as a pure common shock moel (not to be confuse with a process consisting of only common jumps; see (.0)). The corresponing Lévy copula representation is given in Definition 3.. Definition 3.. Pure common shock Lévy copula) The pure common shock Lévy copula is given by C ( u, u ) = ( u / l )( u / l ) + [ u - l ( u / l )] I for 0 # # min, c. l l m + [ u - l ( u / l )] I, { u = 3} { u = 3} (3.3) where l an l are the Poisson parameters for the bivariate compoun Poisson process, an where is a parameter which will etermine the intensity of the common jumps, since l = C ( l, l ) = l l. (3.4) Lemma 3.. The pure common shock Lévy copula (3.3) satisfies the necessary conitions of a positive Lévy copula (see Appenix A.). Proof. The positive Lévy copula is clearly increasing in each component u an u, satisfies C (0, u ) = C (u, 0) = 0 an has margins C (3, u ) = u an C (u, 3) = u. For all (a, a ), (b, b )! [0, 3), an with a # b an a # b, C ( b, b )-C ( a, b )-C ( b, a ) + C ( a, a ) = 8_ b / l i-_ a / l ib8_ b / l i-_ a / l ib $ 0, (3.5) an for the case b = 3, b! [0, 3) an (a, a )! [0, 3), C ( b, b )-C ( a, b )-C ( b, a ) + C ( a, a ) = b - a + ( a / l ) 6 ^a / l h-^b / l h@ $ 0, (3.6)
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 585 since (a / l ) # ue to the restriction on. All other cases are proven in a similar way. Note that the upper boun on the Lévy copula parameter in (3.3) is necessary as a result of l i = $ 0 an (.9), so that l # min( l, l ). (3.7) The case of = 0 leas to the inepenence Lévy copula (.9). Lemma 3.. A bivariate compoun Poisson process with epenence specifie by the pure common shock Lévy copula (3.3) with non-zero has inepenent an ientically istribute common an inepenent jump sizes within one process, an inepenent common jump sizes in both processes. Proof. This Lévy copula satisfies C ( l F ( x), l ) = l F ( x) l = F ( x) C ( l, l ), (3.8) an similarly for the secon argument. It follows by Lemma.3 that the resulting common an inepenent jump sizes within one process are inepenent an ientically istribute. Finally, application of (.7) gives F ( x, x ) = C ( lf( x ), lf ( x)), (3.9) ll = F ( x ) F ( x ), (3.0) inicating inepenence. For (u, u )! [0, l ] [0, l ] the Lévy copula ensity for the pure common shock Lévy copula is simply given by the parameter. A plot of this ensity woul then isplay a flat plane at that level, which inicates no prevalence of certain jump sizes over others for given jump sizes in other processes. As the epenence in frequency increases, the height of the plane above 0 also increases. 3.. Clayton Lévy copula The bivariate Clayton (positive) Lévy copula, introuce in Cont an Tankov (004), is given by - - - + = C ( u, u ) `u u j, for > 0. (3.)
586 B. AVANZI, L.C. CASSAR AND B. WONG As " 0, the Clayton Lévy copula (3.) tens to the inepenence Lévy copula, while as " 3, (3.) tens to the comonotonic Lévy copula. A particular property of the Clayton Lévy copula is that the survival copula of the sizes of common jumps is the Clayton istributional copula, that is, - - C( a, a ) = `a + a - j - ; (3.) see Bregman an Klüppelberg (005). The Lévy copula ensities for the three scenarios of l ; are shown in Figure. In contrast to the case of the pure common shock Lévy copula, the Clayton Lévy copula ensity is not a flat plane, reflecting epenence in the sizes of the common jumps. Aitionally, as the epenence in frequency is increase, the intensity of common jumps is more prevalent at larger sizes, since the ensity is increasingly concentrate at small values of u an u. The Clayton Lévy copula is a homogeneous function of orer one, that is, C ( a, au ) = ac ( u u ). (3.3) u, (a) l ; = 30 (b) l ; = 60 (c) l ; = 90 FIGURE : Clayton Lévy copula ensities for l = 00 an l = 00.
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 587 (a) l ; = 30 (b) l ; = 60 (c) l ; = 90 FIGURE : Simulations from the istributional copula of common jump sizes uner the Clayton Lévy copula for l = 00 an l = 00. An important consequence of (3.3) is that a Clayton Lévy copula for a new time horizon of length T is unchange from the original so that C = C,. (3.4),T ( u, u ) ( u, u ) for T > 0 To stuy epenence in severity, Figure shows scatterplots of 000 simulations from the istributional copula of the sizes of common jumps. Uner the Clayton Lévy copula, the survival copula of the common jumps sizes is a Clayton istributional copula. The istributional copula of the sizes of common jumps, C(, ) is then erive from the survival copula C(, ) using the relationship Ca (, a ) = a+ a - + C( -a, -a) ; (3.5) see Nelsen (999). As the epenence in frequency increases, the epenence in the sizes of common jumps is increasingly evient in the right-tail. That is, the prevalence of common jumps of relatively large sizes in both component increases, which further confirms our euctions from the Clayton Lévy ensities in Figure. 3.3. Archimeean moel I In this section we introuce Archimeean moel I, constructe using Metho.. Archimeean moel I is an extension of a Lévy copula introuce in Chapter 5 of Cont an Tankov (004). Definition 3.. (Archimeean moel I) -^u+ uh C ( u, ) ln e u = - e, for > 0, e -u e -^u+ uh e -u o - + (3.6)
588 B. AVANZI, L.C. CASSAR AND B. WONG is a bivariate Archimeean positive Lévy copula with Archimeean generator -z ( z) = e -z f. - e (3.7) In contrast to the Clayton Lévy copula, Archimeean moel I oes not ten to the inepenence Lévy copula as " 0. Instea, the egree of epenence enable uner Archimeean moel I is restricte as uu lim C ( u, ). 0 u = (3.8) " u + u This means that as " 0, Archimeean moel I tens to a Clayton Lévy copula with a parameter of. Archimeean moel I tens to the comonotonic Lévy copula as " 3. Even though Archimeean moel I is not a homogeneous function of orer one, the time scale Lévy copula is erive by a simple ajustment of the parameter. If C is an Archimeean moel I Lévy copula efine for a time horizon equal to one unit of time, then the equivalent Lévy copula for a time horizon of length T is given by C, T (, u) = C (, u). T u u (3.9) This is a very convenient result as it means that epenence in multivariate compoun Poisson processes may be time scale with a simple change of parameter for the Lévy copula. This property is a result of the Archimeean generator f( ) being a function of z. The survival copula of the sizes of common jumps is then erive as C( a, a ) = - h( a, l, l, ) h( a, l, l, ) lne o h( a, l, l, ) + h( a, l, l, ) - h( a, l, l, ) h( a, l, l, ), - ( l+ l) ln - e e -l -l - ( l+ l) o e + e -e (3.0) where - m - e e ^m e + mh -m e - - + h ( z, l, l, ) =. -l zlne - m ^ - e + mh o- l -^m - e + mh ^m e + mh - - -m -m -m e - m - e + m + e -m e - m - e + m + zln e o- l zln e o ^ h ^ h e e - e + e (3.) Notice that in contrast to the case of the Clayton Lévy copula, the survival copula of the sizes of common jumps is epenent on the values of l an l.
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 589 N/A (a) l ; = 30 (b) l ; = 60 (c) l ; = 90 FIGURE 3: Archimeean moel I ensities for l = 00 an l = 00. Aitionally, it oes not bear any resemblance to any commonly known bivariate Archimeean istributional copulas (see, for instance, Nelsen, 999). Due to the restriction inicate by (3.8), a epenence in frequency of l ; = 30 cannot be prouce by this Lévy copula, which explains why there is no ensity for that case in Figure 3. For the cases of l ; = 60 an l ; = 90, Archimeean moel I moels positive jump epenence as the Lévy copula ensity is concentrate at those values where u = u ; not issimilar to the Clayton Lévy copula. However, there is a notable ifference in the way that the Lévy copula ensity changes with changes in l ; compare to the Clayton Lévy copula. As is observe in the Lévy copula ensities in Figure 3, there is a lack of significant change in the istributional copula of common jump sizes, shown in Figure 4. However, as epenence in frequency increases, the epenence in sizes of common jumps becomes stronger an is also positive an preominantly in the right tail.
590 B. AVANZI, L.C. CASSAR AND B. WONG N/A (a) l ; = 30 (b) l ; = 60 (c) l ; = 90 FIGURE 4: Simulations from the istributional copula of common jump sizes uner Archimeean moel I for l = 00 an l = 00. 3.4. Archimeean moel II Both previous Lévy copulas moel positive epenence in both frequency an severity. Although only positive epenence in frequency is possible uner a Lévy copula moel (since l ; $ 0), we present here a Lévy copula which allows for both negative epenence in severity, as well as epenence in the left-tail. Definition 3.3. (Archimeean moel II) u u C ( u, u ) = lna_ e -i + _ e - i k + n, for > 0, (3.) - - - is a bivariate positive Lévy copula with Archimeean generator z - f ( z) = ^e -h. (3.3) Similar to the Clayton case, as " 0, Archimeean moel II tens to the inepenence Lévy copula. As " 3, Archimeean moel II tens to the comonotonic Lévy copula. The Lévy copula for a time horizon of length T, expresse in terms of an Archimeean moel II Lévy copula efine for a time horizon of length one, is erive as C ( u, u ) = TlnfbeT -l + be T - l n + p. (3.4),T u u - - - Clearly, Archimeean moel II is not a homogeneous function of orer one, nor oes it exhibit the same time scaling property as Archimeean moel I.
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES (a) l ; = 30 59 (b) l ; = 60 (c) l ; = 90 FIGURE 5: Archimeean moel II ensities for l = 00 an l = 00. (a) l ; = 30 (b) l ; = 60 (c) l ; = 90 FIGURE 6: Simulations from the istributional copula of common jump sizes uner Archimeean moel II for l = 00 an l = 00. 94838_Astin4- Avanzi.in 59 // 08:33
59 B. AVANZI, L.C. CASSAR AND B. WONG As illustrate in Figure 5, the epenence structure enable uner Archimeean moel II is clearly istinct from those enable by the Clayton Lévy copula an Archimeean moel I. The fact that this ensitity has mass at points where u is small an u is large, an vice versa, suggests negative epenence in the sizes of the common jumps. This is confirme in Figure 6. Clearly, Archimeean moel II allows for negative epenence in the sizes of common jumps. In aition to this, as the epenence in frequency increases (l ; increases), the sizes of common jumps ten to positive epenence. On the other han, as l ; ecreases, the epenence in severity becomes increasingly negatively epenent. 4. MODELLING EXAMPLE: APPLICATION TO SWISS WORKERS COMPENSATION CLAIMSC In this section Lévy copulas are use to moel epenence in a real set of ata provie by SUVA ( Schweizerische Unfallversicherungsanstalt ). SUVA is a boy incorporate uner Swiss public law which provies accient an occupational isease compensation insurance to aroun million employe an unemploye people in Switzerlan (almost a thir of Swiss resients). 4.. Data analysis The ataset use in this moelling example is a ranom sample of 5% of the claims from class 4A, relating to the construction sector, from accient year 999. The sample size of the ataset is 36. Each claim is ivie into two claim classes. The first class relates to meical costs whilst the secon correspons to aily allowance costs. Importantly, claims have been subject to 3 years of evelopment. That is, claims ata is at 00 year en. Depenence in frequency between meical claims an aily allowance claims is evient by the existence of 089 accients which resulte in a claim in both classes. That is, there were 089 common claims. Table breaks own the claim numbers in terms of unique an common claims for each class. TABLE NUMBER OF UNIQUE AND COMMON CLAIM PAYMENTS IN EACH CLASS Allowance Claim No claim Total Meical Claim 089 60 49 No claim 0 67 77 Total 099 7 36
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 593 Swiss law requires all worker s compensation accients to be reporte to SUVA even if the accient oes not result in a claim payment. There are a total of 67 reporte accients with a claim size of 0 Swiss francs (CHF) in both meical an aily allowance classes. Our moelling approach is to let S (t) be a compoun Poisson process for meical claims an S (t) be a compoun Poisson process for aily allowance claims. We let C enote the Lévy copula with parameter specifying the epenence between the two processes. The jump sizes for each process are reflecte in positive claim amounts an the moel will assume that a claim payment of 0 oes not reflect a jump in the compoun Poisson process. This means that the jump size istributions of S (t) an S (t) o not have masses at 0. As such, the 67 accients which resulte in no losses in either class will be ignore, as the moel is concerne with those accients that resulte in claims only, leaving a total of 59 accients to which the bivariate compoun Poisson process will be fitte. As a result of the thinning property of the compoun Poisson process (see, for instance, Esmaeili an Klüppelberg, 00a), removing these ata points oes not change the assumption of a bivariate compoun Poisson process. TABLE SUMMARY STATISTICS FOR CLAIM SIZES IN EACH CLASS Statistic Meical claim sizes Allowance claim Mean 49.77 6 760.3 Stanar eviation 5 764.39 7 890. Skewness 8.88 6.35 Kurtosis 05.03 5.03 Minimum 5 6 Meian 49 763 Maximum 97 506 86 850 Table shows summary statistics for the claim sizes in each class, where accients without claims have been remove. Sample kurtoses of 05.03 an 5.03 for meical claim sizes an aily allowance claim sizes respectively, shown in Table, suggest a heavy taile claim size istribution for both classes. Depenence in frequency is evient by the presence of 089 claims common to both classes. Depenence in the severity of these claims is evient in Figure 7, showing scatter plots of the claim sizes, the logarithm of claim sizes an the empirical copula (whole an upper-right quarant), respectively. Figure 7 suggests positive epenence in meical an aily allowance claim sizes for the 089 common claims. Furthermore, this epenence appears to be right-taile. That is, there is stronger positive epenence amongst larger claim sizes as oppose to smaller claim sizes.
594 B. AVANZI, L.C. CASSAR AND B. WONG (a) Claim sizes (b) Logarithm of claim sizes (c) Empirical copula () Upper-right quarant of the empirical copula FIGURE 7: Scatterplots for common meical an aily allowance claims. 4.. Parameter estimation In fitting a bivariate compoun Poisson process, the Poisson parameters, marginal jump size istribution parameters an Lévy copula parameters are estimate simultaneously. The fit will epen on the choice of marginal jump istributions F (x) an F (x) with parameters q an q respectively, an the choice of a Lévy copula C with parameter(s). A maximum likelihoo estimation metho for a bivariate compoun Poisson process requires the following observation scheme (Esmaeili an Klüppelberg, 00a). Let n be the total number of claims (jumps) occurring in a
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 595 time interval of length T. The number of jumps in each class is n an n. The number of claims common to both classes is n ;, an the number of claims unique to each class is expresse as n = an n = respectively. The jump sizes in = the first an secon components are enote by x, f, x = = n = an y, f, y = = n respectively, while the sizes of the observe common jumps in both components are enote by (x ;,y ; ),, ( x n, y n ). Maximising the full likelihoo function can become numerically intensive for large atasets. Furthermore, as the full likelihoo function is not the same uner ifferent Lévy copulas, this maximisation must be performe for each of the Lévy copula caniates in orer to select one. To aress this issue, one can use a metho analogous to the inference functions for margins ( IFM ) metho (Joe, 997, Chapter 0.) in orer to heuristically select a Lévy copula moel; see also Esmaeili an Klüppelberg (00b). This relies on the following representation of the log-likelihoo function for the bivariate compoun Poisson process. l(, l, l, q, q ) = n ln l - l T+ ln f ( x; q ) + n ln l - l T+ ln f ( y; q ) 9 n + ln f - i = u 9 u= lf( xi ; q) 9 n + ln f - + / / i = u 9 u= lf( yi ; q) n / ln u u n n / / i = C ( u, l ) i i i = C ( l, u ) p + C ( l, l ) T C ( u, u ), i = u= lf( xi; q), u= lf( yi; q) p (4.) assuming the existence of C uu (u, u ) for all (u, u )! (0, l ) (0, l ). Using an IFM approach involves the estimation of parameters l, l, q an q firstly without consieration of the epenence structure between the two compoun Poisson processes. That is, these parameters are estimate by maximisation of the log-likelihoo * l ( l, l, q, q ) n n = n lnl - l T+ ln f ( x ; q ) + n ln l - l T+ ln f ( y ; q ), / / i i = i = i (4.) proucing parameter estimates (l, l, q, q ). This is where the choice of the marginal jump size istributions occurs. Then, using the parameters estimate above (of the best marginal moels), ifferent Lévy copulas are fit to the ata by estimating their parameter(s) through maximisation of l (, l, l, q, q ) (keeping the parameters l an q constant). Finally, once the jump size istributions
596 B. AVANZI, L.C. CASSAR AND B. WONG an Lévy copula have been chosen, all parameters can be estimate simultaneously on maximisation of the full likelihoo l (, l, l, q, q ). This metho is less computationally intensive than simultaneous maximisation of all parameters using the full likelihoo an trial an error of ifferent Lévy copula moels an jump size istributions. 4.3. Maximum likelihoo estimation IFM approach In fitting the bivariate compoun Poisson process, we begin with an IFM approach. Firstly, we choose a time unit of one year so that T =. We then erive maximum likelihoo estimates for l an l base on the marginal compoun Poisson processes. The estimates for l an l are erive as l = 49 an l = 099. In fitting the marginal jump size istributions we let X enote the logarithm of meical claim sizes an X enote the logarithm of aily allowance claim sizes. Table 3 shows the maximise log-likelihoo for a number of istributions when fit to the logarithm of both meical claim sizes an aily allowance claim sizes. It can be seen that a Gumbel istribution maximises the log-likelihoo for the logarithm of meical claim sizes while a Gaussian istribution maximises the log-likelihoo for the logarithm of aily allowance claim sizes. The parameter estimates for the Gumbel istribution are a = 5.476 an b =.048, while the parameter estimates for the Normal istribution are m = 7.6305 an s =.4403. TABLE 3 MAXIMISED LOG-LIKELIHOOD VALUES FOR FITTING THE LOGARITHM OF CLAIM SIZE DATA Maximise log-likelihoo Distribution X X Gaussian 3960.3 960.43 Gumbel 3759. 995.93 Weibull 407.53 003.6 Cauchy 4056.66.89 The final step is to maximise the full likelihoo assuming marginal parameter estimates l, l, a, b, m an s constant while eriving an estimate for the Lévy copula parameter uner ifferent Lévy copulas. The maximise log likelihoo, parameter estimates for an the implie value for l ; uner the IFM metho are shown in Table 4. The IFM metho ientifies Archimeean moel I as an appropriate Lévy copula for use in the moel, base on the maximise value of the likelihoo function, while Archimeean moel II performs poorly, which coul have been expecte because of its tenency to moel negative epenence in severity.
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 597 TABLE 4 MAXIMISED LOG-LIKELIHOOD AND LÉVY COPULA PARAMETER ESTIMATES UNDER THE IFM METHOD. Lévy copula Maximise l Implie l ; Pure common shock 7845.03 0.0004406 089.00 Clayton 850.8.459 003.6 Archimeean moel I 864.06 0.004983 079.66 Archimeean moel II 4564.90 0.000 099.00 The implie l ; is calculate as C (l, l ) an inicates the estimate expecte number of common jumps per unit of time. A value of 089 represents the maximum likelihoo estimate for the number of common jumps. The pure common shock Lévy copula reprouces this estimate, since it affects the epenence in frequency only. 4.4. Maximum likelihoo estimation full moel In this section fitting results using the full likelihoo for all Lévy copulas iscusse in this paper are presente (with the exception of Archimeean moel II, which performe significantly poorly in comparison to the other caniate Lévy copulas). Aitional Archimeean Lévy copulas were also teste but exclue from this analysis ue to their relatively poor fit. TABLE 5 MAXIMISED LOG-LIKELIHOOD AND PARAMETER ESTIMATES OF THE BIVARIATE COMPOUND POISSON PROCESS FOR EACH LÉVY COPULA. Lévy copula Maximise l l l Implie l ; Pure common shock 7845.03 0.0004406 49.00 099.00 089.00 (0.0000093) (47.4) (33.5) Clayton 8536.43.63 76.90 066.7 984.6 (0.06886) (46.6) (3.68) Archimeean moel I 863.7 0.005358 39.4 3.3 093.74 (0.0000) (47.0) (3.75) Lévy copula a b m s Pure common shock 5.476.048 7.6305.4403 (0.045) (0.079) (0.0434) (0.0307) Clayton 5.007.404 7.70.5498 (0.053) (0.089) (0.0430) (0.05) Archimeean moel I 5.94.0785 7.679.405 (0.037) (0.070) (0.0400) (0.036)
598 B. AVANZI, L.C. CASSAR AND B. WONG CLAYTON (a) Meical claim sizes (b) Allowance claim sizes ARCHIMEDEAN MODEL I (c) Meical claim sizes () Allowance claim sizes FIGURE 8: Quantile-quantile plots for fitte marginal jump size istributions uner Clayton Lévy copula an Archimeean moel I Table 5 shows the maximise log-likelihoo an corresponing parameter estimates for each Lévy copula. The stanar errors of each parameter estimate are given in parentheses an are calculate as the square roots of the iagonal entries in the inverse Hessian matrix of the log-likelihoo function (Klugman et al., 008, Chapter 5.3). Note that the estimates for, the Lévy copula parameter, are not comparable across ifferent Lévy copulas. As initially suggeste from analysis uner the IFM metho, Archimeean moel I maximises the log-likelihoo function for the bivariate compoun Poisson process. With the exception of the case of the pure common shock Lévy copula, the parameter estimates l, l, a, b, m an s iffer from those
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 599 prouce uner the IFM metho. Recall from Lemma. that the Lévy copula affects the istribution of unique jump sizes as well as the istribution of common jump sizes an their epenence structure. As a result of this, the fitting proceure will estimate parameters base on the fit of unique jump sizes an common jump sizes, resulting in ifferent marginal jump size parameters. Also, as the epenence in frequency (via the expecte number of common jumps) an the epenence in the severity (of those common jumps) are fit simultaneously, they compete with each other an in oing so, yiel ifferent parameter estimates for the marginal Poisson parameters an jump size istribution parameters. Figure 8 shows quantile-quantile plots for the marginal jump sizes using the fitte parameters from the Clayton Lévy copula an from Archimeean moel I. While the parameter estimates for the marginal jump size istributions iffer uner the IFM metho, the quality of fit is still reasonable. Further tests for the gooness-of-fit for the marginal jump size istributions can also be employe (Klugman an Rioux, 006). Even though the pure common shock Lévy copula prouces the same marginal parameters estimates as uner the IFM metho, it also prouces the lowest value for the maximise log-likelihoo. This is because the pure common shock Lévy copula assumes inepenent jump sizes, which is an invali assumption as suggeste by Figure 7. In aition to this, we will see in the following section that the assumption of ientically istribute unique an common jump sizes in each class is incorrect. 4.5. Depenence gooness-of-fit An initial assessment of the Lévy copula fit woul be to compare the fitte epenence in frequency as measure by the implie l ;. In Table 5 we see that the pure common shock Lévy copula an Archimeean moel I prouce a goo fit for epenence in frequency, as l ; is relatively close to the observe number 089 (in the case of the pure common shock Lévy copula it is equal). However, the epenence in frequency is merely one aspect of the fitte bivariate compoun Poisson process than can be assesse. In orer to iscuss the fit of the moel in terms of epenence in severity (the sizes of common jumps), Figure 9 plots 089 simulations from the istributional copula of the sizes of common jumps erive uner the Clayton Lévy copula an Archimeean moel I. The pure common shock Lévy copula only allows for inepenence in common jump sizes an was consequently omitte from this analysis. On comparison of Figure 9 with the empirical copula of the common claims in Figure 7, the istributional copula uner the Clayton Lévy copula appears to offer a better fit for the epenence in the sizes of common jumps. While Archimeean moel I offers right-tail positive epenence, its upper-right quarant oes not fit as well as the one of the Clayton Lévy copula. Note that more sophisticate methos can be use in testing the gooness-of-fit of istributional copulas (cf. Genest et al., 009). While these traitional gooness-of-fit approaches are inconclusive, it is also possible to plot the theoretical tail integrals against the empirical tail
600 B. AVANZI, L.C. CASSAR AND B. WONG CLAYTON (a) Empirical copula (b) Upper-quarant of the empirical copula ARCHIMEDEAN MODEL I (c) Empirical copula () Upper-quarant of the empirical copula FIGURE 9: Simulations from the istributional copula of common jump sizes uner caniate Lévy copula moels. integrals for unique jumps an common jumps in each component, an for each fitte moel. The avantage of this approach is that it assesses both fit of the epenence in frequency an severity at the same time. For the common components (S i ; (t), i =, ) the empirical tail integrals are efine as i;n ( number of common jumps in component i of size > x U x) =,for i =,, T (4.3)
MODELLING DEPENDENCE IN INSURANCE CLAIMS PROCESSES 60 while the empirical tail integrals for unique jumps are efine as 9 i; n ( number of unique jumps in component i of size > x U 9 x) =,for i =,. i T (4.4) Figure 0 shows plots of empirical tail integrals against theoretical fitte tail integrals of unique jumps an common jumps in each component for the Clayton, pure common shock, an the Archimeean moel I Lévy copulas (where the tail integral comparisons for unique allowance claims were not plotte ue to a small number of ata for these types of claims). Note that for each Lévy copula, both curves start at the moel an empirical versions CLAYTON PURE COMMON SHOCK ARCHIMEDEAN MODEL I FIGURE 0: Empirical (grey) an theoretical (black) tail integrals for meical (common an unique) an allowance (common) jumps with three caniate Lévy copula moels.
60 B. AVANZI, L.C. CASSAR AND B. WONG of l =, l ; an l ;, respectively, an are then shape accoring to the moel an empirical versions of F = (x), F ; (x) an F ; (x) respectively. Although the Clayton Lévy copula prouce a relatively high maximise loglikelihoo, we see that the fit of the tail integrals uner the Clayton Lévy copula is rather poor. Whilst the pure common shock Lévy copula fits the epenence in frequency perfectly, the jump size istributions for common an unique jumps are not fitte well at all. Finally, it can be seen that Archimeean moel I fits the tail integral components rather well in comparison to the other Lévy copulas. In view of the above, Archimeean moel I seems to be the most appropriate choice of Lévy copula for the epenence structure exhibite in the SUVA ataset. 5. TRIVARIATE COMPOUND POISSON PROCESS In this section, Lemmas 5. an 5. exten Lévy copula results for the bivariate compoun Poisson process to a trivariate compoun Poisson process using a trivariate Lévy copula, aopting a similar approach to the one use in the bivariate case by Esmaeili an Klüppelberg (00a). Care has to be taken in settings beyon the bivariate case, as common jumps can now be common between two out of the three processes, or between all three processes. In particular, Example 5. illustrates the similarities an ifferences between the bivariate an trivariate cases. Results for the trivariate compoun Poisson process can easily be generalise for use with multivariate compoun Poisson processes of any imension. Lemma 5.. (Trivariate compoun Poisson process) The constituents of a trivariate compoun Poisson process {S (t), S (t), S 3 (t)}, can be expresse as Z 9 S() t = S () t + S; () t + S3 ; () t + S;3(), t ] 9 [ S() t = S () t + S;() t + S; 3() t + S;3(), t (5.) ] 9 S3() t = S3 () t + S3;3() t + S3; 3() t + S3;3(). t \ The compoun Poisson processes enote by S ; i; ij (t) feature an arrival process common with compoun Poisson processes S ; j; ij (t), for i, j =,, 3 an i j only. The three compoun Poisson processes enote as S ; i; 3 (t), for i =,, 3 all feature a common arrival process. Proof. Decomposing the tail integral of S (t) in terms of the Lévy measure gives, U ( x ) = n([ x, 3) #{0} #{0}) + lim n([ x, 3) #{0} #[ x, 3)) + 3 x3 " 0 + lim n([ x, 3) #[ x, 3) #{0}) + lim n([ x, 3) #[ x, 3) #[ x, 3)) + + 3 x" 0 x, x3" 0 9 ( ) ; ( x) ; 3( x) ; 3( ), = U x + U + U + U x (5.)