Insurance: Mathematics and Economics. Multivariate Tweedie distributions and some related capital-at-risk analyses

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1 Insurance: Mathematics and Economics Contents lists available at ScienceDirect Insurance: Mathematics and Economics ournal homeage: wwwelseviercom/locate/ime Multivariate Tweedie distributions and some related caital-at-risk analyses Edward Furman a,, Zinoviy Landsman b a Deartment of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada b Deartment of Statistics and Actuarial Research Center, University of Haifa, Haifa 3195, Israel a r t i c l e i n f o a b s t r a c t Article history: Received Setember 28 Received in revised form December 28 Acceted 8 December 29 Keywords: Exonential disersion models Multivariate Tweedie family Cauchy s functional equations Risk caital allocations The tail conditional exectation risk measure We study a multivariate extension of the univariate exonential disersion Tweedie family of distributions The class, referred to as the multivariate Tweedie family MTwF, on the one hand includes multivariate Poisson, gamma, inverse Gaussian, stable and comound Poisson distributions and on the other hand introduces a high variety of new deendent robabilistic models unstudied so far We investigate various roerties of MTwF and discuss its ossible alications to financial risk management 21 Elsevier BV All rights reserved 1 Introduction and motivation Exonential disersion models EDMs lay a rominent role in actuarial science and finance This can be artly exlained by the extent of generalization, as well as by the aeal of multitude unifications, that the EDMs enable for such widely oular robability distribution functions df s as normal, gamma, inverse Gaussian, stable and many others The secificity characterizing robabilistic modeling of actuarial obects is that the underlying distributions mostly have non-negative suorts, and many EDMs ossess this imortant henomenon As a result, a significant number of scientific aers have been dedicated to exloring various members of the EDM class in diverse fields of actuarial science cf, eg, Kaas et al, 1997; Nelder and Verrall, 1997; Landsman and Makov, 1998, 23; Landsman, 22; Landsman and Valdez, 25 Although univariate EDMs are considerably rich, analytically tractable and widely alied, in the multivariate context the case is somewhat different Secifically, Bildikar and Patil 1968 studied a multivariate EDM induced by the following robability measure dp θ,λ exx T θ λκθdν λ x, x x 1, x 2,, x n T R n 11 Here R n denotes the n-dimensional Euclidean sace, θ a, b R n for some finite or infinite interval a, b, λ R, the measure Corresonding author Tel: x33768; fax: addresses: efurman@mathstatyorkuca E Furman, landsman@stathaifaacil Z Landsman ν λ x is indeendent of θ, and κ θ is an analytic function in θ, indeendent of x The family is however not as rich as the univariate EDM, which it aims to generalize In articular, 11 does not include imortant, esecially in insurance and finance, multivariate distributions, having eg, inverse Gaussian, gamma or comound Poisson univariate margins Moreover, the only continuous member of 11, having indeendent of θ suort, is the multivariate normal distribution cf Bildikar and Patil, 1968, for details In addition, Jorgensen 1987 and Jorgensen and Lauritzen 2 introduced and investigated certain generalizations of univariate EDMs It should be noted however, that the former generalization has only a single arameter describing its deendence structure, and the latter is not marginally closed in the sense that its marginal distributions do not generally belong to the given distribution class The models are therefore of a questionable ractical imortance in actuarial science Seeking to resolve the aforementioned limitation of the multivariate EDM constructed in Jorgensen and Lauritzen 2, Song 2 alied the coula based aroach to generate a multivariate distribution with redefined univariate EDM margins Song s model is however not very tractable analytically In articular, the oint moments are generally not available in closed forms and are to be comuted via some Monte Carlo techniques We refer to eg, Frees and Valdez 1998 and Cherubini et al 24, in which actuarial and financial alications of coulas are considered Broadly seaking, there exist a considerable number of methods for constructing multivariate deendent robabilistic structures cf, eg Mardia, 197; Joe, 1997; Kotz et al, 2 However, the insurance and finance industries dictate secific laws that must /$ see front matter 21 Elsevier BV All rights reserved doi:1116/insmatheco29121

2 352 E Furman, Z Landsman / Insurance: Mathematics and Economics be obeyed Namely, mostly only multivariate models defined on R n, reserving unimodality and ositive skewness, can serve as aroriate candidates for model insurance risks These eculiarities discard, for instance, such generally imortant finance models as multivariate normal, Student-t and Cauchy cf, eg Tobin, 1958; Bawa, 1975; Ross, 1978; Owen and Rabinovitch, 1983; Furman and Landsman, 26 Also, there is always a trade-off between, on the one hand, the aroximation level rovided by the model and on the other, its analytic comlexity Consequently, one has to imose an additional restriction of tractability on the choice of the multivariate cumulative distribution function cdf and its deendence structure and, thus, to reect even more models, although they might have described insurance risks well It must be the above-mentioned difficulties that exlain a convention to use univariate distributions to model risks in actuarial science However, in the last years the concet of deendence has received its merited attention, which has resulted in numerous aers cf, eg, Pfeifer and Neslehova, 24; Bauerle and Grubel, 25; Alink et al, 25; Laeven et al, 25; Centeno, 25; Kiima, 26 Also, some new multivariate distributions have been roosed cf, eg Vernic, 1997, 2; Furman and Landsman, 28 Motivated by the already mentioned limitations of the existing multivariate EDMs, in Sections 3 and 4 of the resent aer we reintroduce cf, Furman and Landsman, 28 a new multivariate class of distributions, which we refer to as the multivariate Tweedie family MTwF, and we relate the class to the common shock cf, eg Boucher et al, 28 and background economy cf, eg Tsanakas, 28 models We consider the MTwF class a multivariate extension of the exonential disersion Tweedie family since its univariate marginal distributions corresond to the univariate Tweedie ones Moreover, the roosed family ossesses a deendence structure, which is reflected in its variance/covariance structure and allows for efficient modeling of multivariate ortfolios of deendent risks As secial cases, among others, MTwF contains the multivariate inverse Gaussian, multivariate gamma, multivariate stable, multivariate Poisson and multivariate comound Poisson distributions in the sense that their univariate margins are inverse Gaussian, gamma, ositive stable, Poisson and comound Poisson, resectively In the light of this, on the one hand, MTwF rovides a general framework to deal with, for instance, the multivariate Poisson of Kocherlatoka and Kocherlatoka 1992, the multivariate inverse Gaussian of Chhikara and Folks 1989 and the multivariate gamma of Mathai and Moschooulos 1991, and on the other, it introduces some new multivariate deendent models which can be alied to describe insurance risks behavior usefully We study various seemingly imortant roerties of MTwF Namely, Section 5 calculates some higher order moments and alies these moments to roduce useful Chebyshev s tye inequalities, Section 6 derives the df s of the random vectors distributed MTwF, and Section 7 develos related asymtotic results We note in this regard, that the resent aer contains a thorough analysis of various roerties of MTwF, and it therefore differs greatly from Furman and Landsman 28, in which the model has been assingly introduced with the main accent made on the multivariate Tweedie comound Poisson secial case To demonstrate one ossible alication of MTwF, we carry out an investigation of both the univariate and multivariate caitalat-risk analyses in the framework of the multivariate Poisson distribution This distribution is a articular member of MTwF, and it is commonly alied for modeling the incidence of insurance risks, a role which it fills fairly adequately To achieve the aforementioned goal, we utilize the well-known tail conditional exectation TCE risk measure and the caitalat-risk allocation based on it More secifically, let X R be a risk random variable rv with cdf F and decumulative distribution function ddf F 1 F Then, for every q, 1, TCE is defined as TCE q X] E X X > VaR q X] ] 1 FVaR q X] subect to FVaR q X] > and xdfx, 12 VaR q X] VaR q X] inf{x : Fx q} 13 Risk measure 12 coincides with the exected shortfall ES and the conditional Value-at-Risk CVaR under the assumtion of continuous distributions cf, eg Hürlimann, 23; McNeil et al, 25, Lemma 216 It should also be emhasized that TCE is a coherent risk measure cf Artzner et al, 1999; Acerbi and Tasche, 22; Tasche, 22, it is a articular member of the distorted risk functionals cf Denneberg, 1994; Wang, 1995, 1996; Wang et al, 1997, and it can be considered a weighted remium calculation rincile c cf Furman and Zitikis, 28a Let X X 1, X 2,, X n T and S X 1 X 2 X n reresent the risk inherent in a financial conglomerate and the total risk of that conglomerate, resectively Then TCE allows for a natural allocation of the total risk to its constituents, eg, business lines X k, k 1, 2,, n cf, eg Denault, 21; Paner, 22; Wang, 22 Indeed, by the additivity roerty of the exectation oerator, one obtains that the risk contribution of the kth business line to the total risk of the conglomerate is TCE q X k S] E X k S > VaR q S] ] 14 Paner 22 derived both 12 and 14 for the n-variate normal random vector X N n µ, Σ He showed that 1 TCE q S] can be formulated in terms of the well-known variance remium VPS] ES] αvars] cf, eg Bühlmann, 197 with α equal to the hazard function evaluated at VaR q S], and 2 the contribution of X k to S is stiulated by EX k ] and the covariation between them Landsman and Valdez 23 extended Paner s results for the broader class of ellitical distributions cf Furman and Zitikis, 28b, for additional generalizations Hürlimann 21 seems to have been the first to consider TCE for ositive risks Landsman and Valdez 25 derived analytic closed form exressions for this risk measure in the context of EDMs However, it should be noted, that both aers cited above deal with the functionals of form 12 and do not touch on the more general roblem of allocating risk caitals 14 In this resect, Furman and Landsman 25 derived analytic exressions for both 12 and 14 in the framework of multivariate deendent gamma ortfolios Cai and Li 25 and Chiragiev and Landsman 27 solved a similar roblem in the context of the multivariate hase tye and Pareto distributions, resectively Vernic 26 considered the multivariate skew-normal case, see also Dhaene et al 28 In Section 8 we derive analytic exressions for 12 and 14 in the framework of the multivariate Poisson distribution, which is an imortant member of MTwF Desite of the quite exository urose of the section, the aealing simlicity of the obtained allocation formula seems to be worthy of note Next two sections rovide the necessary groundwork for introducing the multivariate Tweedie family 2 Univariate Tweedie EDMs Let ν be a ositive non-degenerate measure on R, and define the cumulant transform κ θ log R eθx ν dx with the domain Θ { θ R : R eθx ν dx < } For the canonical arameter θ Θ, the robability measure dp θ ex θx κ θ ν dx 21

3 E Furman, Z Landsman / Insurance: Mathematics and Economics reresents a natural exonential family NEF Further, let µ τ θ κ θ denote the mean value maing and Ω τ int Θ be the mean domain Also, rovided 21, define the set of index arameters Λ {λ R } such that λκ θ log e θx ν λ dx for some measure ν λ on R The natural exonential family generated by ν λ is then given by dp θ,λ ex θx λκ θ ν λ dx 22 The robability measure above is referred to as the additive exonential disersion model and is denoted by ED θ, λ cf Jorgensen, 1997 Note that the mgf corresonding to robability measure 22 is readily obtained as Mt ex λκθ t κθ] 23 For regular or stee EDMs, τ µ κ θ >, and hence τ µ is a strictly increasing function Then one can define the unit variance function V : Ω R by V µ τ τ 1 µ EDMs are classified by their unit variance functions Below we outline one of the imortant, esecially in the insurance context, classes of EDMs, Tweedie models, which ossess ower unit variance functions V µ µ, µ Ω and, ] 1, cf Tweedie, 1984; Bar-Lev and Enis, 1986; Jorgensen, 1997 In what follows, we add a subscrit to emhasize that Tweedie EDMs are discussed Choosing an aroriate arameter, we arrive at eg normal, Ω R, Poisson 1, Ω R, gamma 2, Ω R, inverse Gaussian 3, Ω R and comound Poisson 1, 2, Ω R distributions Other valuable members of Tweedie EDMs are enumerated in Table 41 of Jorgensen 1997 It must be emhasized that even more flexibility in modeling insurance losses can be achieved by considering Tweedie distributions corresonding to non-integer arameters cf, eg, Dunn and Smyth, 25 Although EDMs are determined by their unit variance functions, in order to construct the multivariate extension of the univariate Tweedie class, an exlicit form of the cumulant κ θ function corresonding to the secific V µ is required The former is obtained by solving two following differential equations dτ 1 µ 1 and κ dµ µ θ τ θ, which lead to e θ, 1 log θ κ θ, 2 α 1 θ α, 1, 2 α α 1 24 cf Jorgensen, 1997, Section 412 We make extensive use of the above form of κ θ in Section 4 To conclude this section, we note that Tweedie models share with normal and gamma EDMs the roerty that they have both an additive and a reroductive form Therefore later on in this aer we discuss additive Tweedie EDMs only, however this does not imly any restrictions on the derived results 3 The multivariate reduction method We have already mentioned a number of existing generalizations of univariate EDMs to the multivariate framework along with the corresonding shortcomings In this section we describe an alternative method of constructing multivariate robabilistic models with redefined univariate margins The resulting multivariate Tweedie family of distributions then seems to rovide an intuitive solution to the situations when observable risks are contaminated by a common background risk, which makes the whole structure deendent It should be noted that the methodology discussed herein is well-known as the common shock model cf, eg Boucher et al, 28 in actuarial science and also as the background economy model cf, eg Tsanakas, 28 in economics This fact therefore rovides additional interretations and motivations for MTwF Let Y Y, Y 1,, Y n T be an n 1-variate random vector with mutually indeendent corresonding cdf s F i y; ξ i, i, 1,, n, and let X X 1, X 2,, X n T be another, say, resulting, random vector Denote by T a functional maing from R n1 to R n, such that X TY 31 Definition 31 The random vector X R n is said to ossess the cdf Fx; ξ arameterized by the vector ξ ξ 1, ξ 2,, ξ n T, such that ξ η ξ, ξ 1,, ξ n for secific functions η, 1, 2,, n Some useful examles of maing 31 are X min Y, X max Y and X e T Y, for the unit vector e 1, 1,, 1 T In this aer we shall consider linear forms of maing 31 only Then it can be rewritten as X AY, 32 where A Mat n n1 is an n n 1 matrix Note that A determines the form of the multivariate distribution Fx; ξ obtained by the method Taking, for instance, Y i Ga γ i, α and A leads to the structure of Cheriyan 1941 and Ramabhadran 1951 The same matrix A and the assumtion that Y i IGc i µ, c 2λ, i c i R results in an extension of the bivariate inverse Gaussian distribution of Chhikara and Folks 1989 Also, by considering again Y i Gaγ i, α and A we arrive at the multivariate gamma model of Mathai and Moschooulos 1992, see also a generalized counterart of Furman 28 For a non-linear form of maing 31, see eg Asimit et al, forthcoming We note that matrices 33 and 34 imly secific restrictions on the arameters of elements of X, ie, common rate arameters α in the models of Cheriyan 1941, Ramabhadran 1951 and Mathai and Moschooulos 1992 and the relations c c k c c l and c c k c c l 2, where k, l n, reserved between the arameters of Y k and Y l in the model of Chhikara and Folks 1989 We loosen these restrictions in the next section 4 The multivariate Tweedie family of distributions Evolving the discussion of two revious sections, let us consider an n 1-variate random vector Y with mutually indeendent

4 354 E Furman, Z Landsman / Insurance: Mathematics and Economics univariate margins Y i EDθ i, λ i recall that without loss of generality we consider additive EDMs only Then, according eg, to the notions in the last aragrah of Section 3, it is clear that nor 33 neither 34 can be alied to construct a multivariate model with arbitrarily arameterized univariate Tweedie marginal distributions Instead, let β 1 1 β 2 1 A β 3 1, 41 β n 1 subect to β θ /θ, 1, 2,, n Further, to acquire an intuition, note that the mgf of the th univariate marginal comonent of the maing X AY, or in other words the mgf of X β Y Y is M t ex λ κ θ t κ θ ] λ κ θ β t κ θ ] ex λ κ θ t κ θ ] λ κ θ tβ κ β θ ], and it must be of form 23 for X to be an additive EDM Let x θ t and y β for the sake of notational convenience, then the mgf of X is formulated as M t ex λ κ x λ κ θ λ κ xy λ κ θ y, 42 which for θ θ is the mgf of a member of EDM if and only if the function κ x is the solution of the generalized Cauchy s functional equation κ xy κ x f y g y 43 The equation is known to ossess the following most general nonconstant and continuous non-trivial solutions cf, eg, Castillo and Ruiz-Cobo, 1992 κ x a log x b; f x 1; g x a log x κ x ax c b; f x x c ; g x b 1 x c where a, b and c are constants such that a and c but otherwise arbitrary Straightforward comarison of the aforementioned non-trivial solutions of Eq 43 to cumulant function 24 imlies that MTwF is the only ossible multivariate extension of EDMs, given Definition 31 and matrix 41 Indeed, for θ θ, the solutions imly that for X to be an EDM, the corresonding cumulant function must be of either logarithmic or ower form, which readily yields the cumulant function of the Tweedie class Theorem 41 ensures that the roosed multivariate robabilistic model ossesses univariate margins belonging to univariate Tweedie EDMs, and it establishes these marginal distributions exlicitly We note in assing that the case when θ θ imlies y 1 in mgf 42, and this is nothing else but the well-known closure under convolutions roerty of the additive EDMs In our context, equal canonical arameters corresond to the multivariate Poisson distribution, which is certainly a member of MTwF In such a case we omit indexation cf Theorem 41, 1 Theorem 41 Let X AY and Y be an n 1-variate random vector with mutually indeendent additive Tweedie margins Y i Tw θ i, λ i, i, 1,, n Let β θ /θ in 41 Then the th univariate margin of X is Tw 1 θ, λ λ, 1 Tw 2 θ, λ λ X, 2 α θ Tw θ, λ λ, 1, 2, θ where α 2/ 1 Proof We rove only the case 1, 2 Hence, α 1 1 α log M t λ θ t θ α θ α α θ α 1 1 α λ θ α α t θ α α 11 α θ α α θ t λ λ α θ α 11 α α α θ θ λ λ α θ α θ α 1 1 α λ λ θ α α t θ θ α λ κ θ t κ θ ], where λ λ θ θ α λ, which comletes the roof We now naturally arrive at the following definition of additive MTwF Definition 41 Let Y be an n 1-variate random vector with mutually indeendent additive Tweedie margins Y i Tw θ i, λ i, i, 1,, n, and let β θ /θ in 41 Also, let θ R n be the n-variate vector of canonical arameters Here, is the Cartesian roduct of the domains Θ R Then the oint distribution of X AY, denoted by X Tw n, θ, λ, is the n- variate additive Tweedie distribution Note 41 To obtain the reroductive form of MTwF, we take β λ θ λ θ in Definition 41 It is well-known that univariate Tweedie EDMs are closed under convolutions for common canonical arameters We further introduce the multivariate counterart of such a closure in the framework of MTwF To achieve this goal, we make extensive use of the multivariate mgf of X Tw n, θ, λ, which is readily available and can be formulated as M X t E e XT t ] E ex ] θ t Y Y θ θ n M Y t M Y t, 44 θ with Y, Y 1,, Y n given in Definition 41 Theorem 42 Let X 1 Tw n, θ, λ 1 and X 2 Tw n, θ, λ 2 be two indeendent multivariate Tweedie random vectors Then S X 1 X 2 Tw n, θ, λ 1 λ 2 Proof Due to the indeendence of X 1 and X 2 and evolving Eq 44, the mgf of S is written, for β θ θ 1, θ θ 2,, θ θ n T, X 1, β Y 1, Y 1, and X 2, β Y 2, Y 2,, 1,, n, as M S t n n M Y1, β T t M Y1, t MY2, β T t M Y2, t Therefore, log M S t λ 1, λ 2, κ θ β T t κ θ ] λ 1, λ 2, κ t κ, which necessarily means that the mgf of S is of form 44 with λ S λ 1,1 λ 2,1, λ 1,2 λ 2,2,, λ 1,n λ 2,n and hence comletes the roof

5 E Furman, Z Landsman / Insurance: Mathematics and Economics Literally seaking, Theorem 42 imlies that combining indeendent ortfolios of risks Tw n, θ, λ 1 and Tw n, θ, λ 2, roduces another ortfolio which is also in MTwF Such a multivariate additivity roerty allows for a fairly tractable treatment of the MTwF random vectors and therefore ortfolios of risks described by this multivariate robabilistic model 5 Moments, higher order moments and Chebyshev s tye inequalities Many roblems in actuarial science require to evaluate moments of order-m of the risk rv X This section is dedicated to deriving these useful exressions in the general context of MTwF According to Definition 41, the evaluation of, say EX m ], 1, 2,, ] n requires the moments E Y m i, i, 1,, n, which are readily available Indeed, as the corresonding cumulants are k m m log M Yi t t m t λ i κ m θ i, and the mgf of Y i is of exonential form, we readily obtain that ] m E Y m M Yi t i t m t m! h k l k1 k2 k3 kl, 51 h!!k! l! 1! 2! 3! L! where the summation is over all solutions in non-negative integers of the equation h 2 3k Ll m Eq 51 imlies, for instance, that E Y i ] k 1 λ i κ θ i, ] E Y 2 i k 2 1 k 2 λ i κ θ i 2 λ i κ 2 θ i and Var Y i ] λ i κ 2 θ i, ] E Y 3 i k3 3k 1 k 2 k 3 1 and Skew Y i ] λ iκ 3 θ i 3/2 λ i κ 2 θ i Further, alying again Definition 41, we formulate higher order moments of X as ] m ] E X m θ E Y Y θ m m θ r r θ r E Y r ] ] E Y m r Last equations yield, for instance, that the exectation of X is ] θ E X λ κ θ θ λ κ, the variance of X is ] 2 θ Var X λ κ 2 θ λ κ 2, θ and the skewness of X is Skew X ] 3 θ λ θ κ 3 θ λ κ 3 2 θ λ θ κ 2 θ λ κ 2 3/2 Similar rocedure allows us to evaluate the following roduct moments ] n m ] E X n i X m θ θ E Y Y i Y Y θ i θ n θ r m m ] θ s E Y Y n r i Y Y m s r θ r i s θ s m n m θ r s θ ] E Y rs r s θ r s i θ ] ] E Y n r i E Y m s, ] where E Y rs, E Y n r i ] ] and E Y m s are available from Eq 51 To comute the cumulants of X we need the logarithm of its mgf According to, say 44, we have that log M X t λ κ θ θ ] t κ θ θ λ κ θ t κ θ ] Consequently, the mth cumulant of X is K m m log M X t m θ λ t m κ m t θ θ λ κ m Also, note that for β θ θ 1, θ θ 2,, θ θ n T, we have that log M X t λ κ θ β T t κ θ ] ] λ κ t κ, and therefore the oint cumulants of X i and X are K mi,m m im log M X t im θm t m i i t m t θ m i i θ m for i Hence, the covariance of X i and X is λ κ m im θ, Cov X i, X ] θ 2 θ i θ λ κ 2 θ θ 2 θ i θ Var Y, for i 51 Chebyshev s tye inequalities As an illustration of the revious results, we further derive some useful Chebyshev s tye inequalities in the case of the multivariate Tweedie distributions Theorem 51 Let ε and X, 1, 2,, n be some arbitrary ositive constants and the comonents of the multivariate additive Tweedie family, corresondingly Then P X 1 ε 1,, X n ε n θ 1 λ κ θ θ λ κ / ε 52 and P X 1 ε 1,, X n ε n θ λ κ θ θ λ κ / ε 53

6 356 E Furman, Z Landsman / Insurance: Mathematics and Economics Proof The first art follows straightforwardly from the following well-known inequality P X > ε EX] ε, by recalling that ] θ E X λ κ θ θ λ κ Indeed, n { } P X ε 1 { } P X > ε 1 EX ] ε Inequality 53 follows from the note that P X 1 ε 1,, X n ε n P X i ε i, which comletes the roof 6 Multivariate densities i1 It turns out that Definition 41 allows to derive multivariate df s of a general member of MTwF Recall that we deal with the random vector X Tw n, θ, λ and, for β θ /θ, consider the following transformation i1 X β Y Y, 1, 2,, n and i, 1,, n We further assume that the measure ν λ in Eq 22 ossesses df c ; λ, then, due to the indeendence of its marginal comonents, the oint df of Y Y, Y 1,, Y n T can be formulated as f Y y, y 1,, y n f Yi y i i c y i ; λ i ex θ i y i λ i κ θ i i c y ; λ c y ; λ ex y λ κ θ y λ κ θ Substituting Y X β Y, we obtain the df of X Y, X 1,, X n T which in such a case equals f X y, x 1,, x n c y ; λ c x β y ; λ ex θ x n 1 θ y λ i κ θ i Finally, observing that Y min X 1, X 2,, X n, we can write the df of X as f X x 1, x 2,, x n ex x λ κ λ κ θ xmin c y ; λ i c x β y ; λ ex n 1 θ y dy, 61 where x min min x 1, x 2,, x n Last equation is quite comlicated and cannot be solved analytically Moreover, it imlies that the df of X is of different form for each of n! ermutations of x 1, x 2,, x n In what follows, we consider two examles which show that in some articular cases the df of X can be handled in site of the certain difficulties related to its unleasant form Examle 61 Multivariate gamma distribution Let X Gaγ, α denote a gamma rv with shae and rate arameters equal γ and α, resectively The df of X is f x 1 Γ γ e αx x γ 1 α γ x γ 1 ex αx γ logα, Γ γ which, for θ α, λ γ and κθ log θ, can be reformulated as 22 Moreover, from the form of cumulant function 24, it follows that X is a member of the univariate Tweedie class with 2, or more recisely X Tw 2 α, γ We now can carry out necessary substitutions and find the multivariate df of X Tw n,2 θ, λ Due to Eq 61, the df is given by α γ i xmin i f X x 1, x 2,, x n ex α x y γ 1 Γ γ i i x α γ 1 y e n 1α y dy α The above multivariate df agrees well with that of Mathai and Moschooulos 1991 Note that taking α 1, 1,, 1 T, we arrive at the model of Cheriyan 1941 and Ramabhadran 1951 In such a case, the latter equation simlifies even more, reducing to ex n x f X x 1, x 2,, x n Γ γ i i xmin y γ 1 x y γ 1 e n 1y dy We further consider another examle of a member of MTwF In order to emhasize that the results of this aer can be equally alied to reroductive EDMs, we discuss the multivariate inverse Gaussian distribution, which is MTwF with 3 Examle 62 Multivariate inverse Gaussian distribution Let Y IGµ, λ denote an inverse Gaussian rv with df λ f Y y 2πy ex λy µ2, y > 3 2µ 2 y Slight reformulation of the above df yields f Y y e λ λ 2y 2πy ex λ y 3 2µ 2 µ 1, which, for θ 2µ 2 1 and κθ 2θ can be seen as a reroductive EDM cf Jorgensen, 1997 Then the ower form of κθ imlies that Y is a univariate Tweedie with 3, and the dual transformation X λy leads to the additive Tweedie EDM, or more recisely, to X Tw 3 1, λ, ossessing the following 2µ 2 df f x e λ2 2x λ 2 2πx 3 ex x 2µ 2 λµ 1

7 E Furman, Z Landsman / Insurance: Mathematics and Economics We now can establish the df of the multivariate inverse Gaussian distribution, ie, of X Tw n,3 θ, λ Making use of Eq 61, we obtain that f X x 1, x 2,, x n ex x λ λ 2µ 2 µ µ xmin e λ2 λ 2 2y e 2πy 3 n 1 y ex dy 2µ 2 λ 2 2x µ µ 2 y λ 2 2πx µ µ 2 y 3 with x min min x 1, x 2,, x n From the above df we can obtain eg, the multivariate inverse Gaussian distribution of Chhikara and Folks 1989, which is f X x 1, x 2,, x n ex x 2µ λ λ 2 µ µ xmin e λ2 λ 2 2y e λ2 λ 2 2x y 2πy 3 2πx y 3 n 1 y ex dy 2µ 2 7 Asymtotic results In this section we develo some seemingly useful asymtotic results for the multivariate Tweedie distributions The usefulness of the results derived herein can be ustified, for instance, by the formally unleasant nature of multivariate densities 61 Let X X 1, X 2,, X n T Tw n, θ, λ, and denote by Z the standardized MTwF random vector ossessing univariate margins Z X E X ] Var X ], which, after alying the results of Section 5, yields θ Z X β λ κ θ λ κ β 2λ κ 2 θ λ κ 2 X β 2λ κ 2 θ λ κ 2 β λ κ θ λ κ β 2λ κ 2 θ λ κ 2, for β θ /θ, We are now in a osition to establish the asymtotic distribution of Z Theorem 71 Let λ and λ be some ositive constant Then the standardized univariate Tweedie rv Z is asymtotically standard normal rv N, 1 Proof The roof relies on the mgf of Z, which is as follows M Z t ex β λ κ θ λ κ β 2λ κ 2 θ λ κ 2 t t M X β 2λ κ 2 θ λ κ 2, θ where the mgf of X is given by M X t ex λ κ θ θ ] t κ θ θ λ κ θ t κ θ ] Consequently, log M Z t β λ κ θ λ κ β 2λ κ 2 θ λ κ 2 t t λ κ θ β β 2λ κ 2 θ λ κ 2 κ θ t λ κ θ β 2λ κ 2 θ λ κ 2 κ Exanding into a ower series, we obtain that log M Z t β λ κ θ λ κ β 2λ κ 2 θ λ κ 2 t β λ κ θ β 2λ κ 2 θ λ κ 2 t β 2 λ κ 2 θ β 2λ κ 2 θ λ κ 2 t2 2 λ κ β 2λ κ 2 θ λ κ 2 t λ κ 2 β 2λ κ 2 θ λ κ 2 t o β 2λ κ 2 θ λ κ 2 t2 2 2o 1 β 2λ κ 2 θ λ κ 2 as λ, which comletes the roof, θ In a similar fashion we establish the following theorem for the vector Z Z 1, Z 2,, Z n T Theorem 72 The standardized vector Z is asymtotically multivariate normal with the vector of means µ,,, T and the covariance matrix Σ with Σ i,i 1 and Σ i, given below, for i, 1, 2,, n, 1 If λ is some ositive constant and λ, then Σ i, 2 If both λ and λ go to infinity such that λ /λ k, then β β i κ 2 θ Σ i, β 2 κ2 θ m κ 2 β 2 i κ2 θ m i κ 2 θ i

8 358 E Furman, Z Landsman / Insurance: Mathematics and Economics Proof The mgf of Z can be formulated in terms of the mgf s of Y, Y 1,, Y n as M Z t E ex n t X β 2λ κ 2 θ λ κ 2 β λ κ θ λ κ β 2λ κ 2 θ λ κ 2 t β λ κ ex θ λ κ β 2λ κ 2 θ λ κ 2 t E ex n t X β 2λ κ 2 θ λ κ 2 β λ κ ex θ λ κ β 2λ κ 2 θ λ κ 2 t M Y n t β β 2λ κ 2 θ λ κ 2 t M Y β 2λ κ 2 θ λ κ 2 The logarithm of the latter equation is therefore β λ κ log M Z t θ λ κ β 2λ κ 2 θ λ κ 2 t λ t κ θ β β 2λ κ 2 θ λ κ 2 κ θ t λ κ θ β 2λ κ 2 θ λ κ 2 κ, which after exanding into a ower series yields 1 log M Z t λ 2 κ2 θ n λ κ2 θ 2 β β 2 λ κ 2 t β 2λ κ 2 θ λ κ 2 2 t θ λ κ 2 Hence, if λ is some fixed ositive constant and λ, then log M Z t 1 2 λ κ 2 t 2 β 2λ κ 2 θ λ κ 2 θ t 2 2 Alternatively, when both λ and λ go to infinity such that λ /λ k, we have that 2 log M Z t 1 2 i t 2 β β i κ 2 θ t t i β 2 κ2 θ m κ 2, β 2 i κ2 θ m i κ 2 θ i which comletes the roof We note in assing that Theorem 72 is intuitively clear, bearing in mind that λ has a crucial imortance for the deendence structure of Tw n, θ, λ In the light of this, the first statement of Theorem 72 addresses the case when the deendence is asymtotically vanishing, while the second statement relates to the contrary case 8 Caital-at-risk analysis: An examle In the modern concetion of risk management, where risks are mostly reresented by non-negatively valued rv s cf, eg Artzner et al, 1999, the following three substantial subects have to be addressed 1 An aroriate multivariate deendent robabilistic model Fx 1, x 2,, x n that rovides a satisfactory fit for a real life multi-line business, 2 A risk measure H : X, ], which actually measures the degree of riskiness that each X X imlies, 3 Analytic exressions for the chosen risk measure H, given the multivariate cdf F We have so far introduced and studied the multivariate Tweedie family of distributions and therefore addressed oint 1 above In what follows, we consider risk functional 12, and we derive formulas for S e T X, where, for e 1, 1,, 1 T, X X 1, X 2,, X n T Tw n,1 θ, λ is a multivariate Poisson distribution There is a considerable amount of literature in actuarial science concerning the distribution of the aggregate loss S when the univariate marginal elements of X are indeendent Much less is known of the case when some stochastical deendence is assumed, and yet this is a common situation in ractice cf, eg Denuit et al, 25 In our context, due to Definition 41 and keeing the Poisson case 1 in mind, we arrive at the following reresentation of S, for Y i Tw 1 λ i e θ, i, 1,, n, S X ny Y 81 Then the distribution of S can be seen as a comound Poisson Indeed, for any ositive constants c i and Poisson rv s Y i, the roduct c i Y i can be interreted as a comound Poisson rv with the corresonding Poisson arameter λ i e θ and a degenerate claim amount distribution at c i Consequently, the sum S i c iy i is distributed comound Poisson with Λ e θ λ i i and the following corresonding robability mass function mf of claim amount { λi e θ c c c i Λ otherwise We further state some auxiliary results as lemmas The former lemma is quite general and it concerns arbitrarily distributed indeendent risks with non-negative suorts cf Furman and Landsman, 25 The latter, reveals how the size-biased rv associated with a Poisson rv, can be handled

9 E Furman, Z Landsman / Insurance: Mathematics and Economics Lemma 81 Let X reresent a multivariate ortfolio of arbitrary indeendent non-negative risks X 1, X 2,, X n with df s f x, 1, 2,, n and finite exectations Then VaRq S] E X S > VaR q S] ] E X ] F S X X F S VaRq S], 82 where X is the size-biased rv associated with X cf, eg, Patil and Rao, 1978 Lemma 82 Let X Tw 1 θ, λ be a Poisson rv Then the size-biased rv associated with it, is also Poisson with a translated to the right suort Proof The roof relies on the following ddf of X F X x E X1X > x] lx E X] 1 λe θ tx1 t λt t! ex θt λe θ λ l l! ex θl λe θ F X x 1, x 1, where 1A is the indicator function of the set A Then X d X 1, which comletes the roof Here, d stands for the equality in distribution The next theorem derives risk measure 12 for S We note that in what follows S, S 1 and S n are all Poisson rv s having differently translated suorts Theorem 81 Let X Tw n,1 θ, λ denote a multivariate Poisson random vector and S be the sum of its univariate marginal elements The tail conditional exectation risk measure for S is then TCE q S] nλ e θ F Sn VaRq S] F S VaRq S] λ e θ F S1 VaRq S] F S VaRq S] 83 Proof First, we note that in a similar fashion to the roof of Lemma 82, it can be shown that ny d ny n Then alying Lemma 81, we obtain that E ny S > VaR q S] ] E ny ] F Sn VaRq S] F S VaRq S] nλ e θ F Sn VaRq S] F S VaRq S] and ] E Y S > VaR q S] which comletes the roof E Y S > VaR q S] ] λ e θ F S1 VaRq S] F S VaRq S], as Corollary 81 Suose λ and therefore Y In such a limit case, the random vector X becomes a vector of n indeendent univariate Poisson rv s, and the formula for TCE of S simlifies to TCE q S] λ e θ F S1 sq 84 F S sq It should be also emhasized that, since for discrete rv s the following relation holds PS s PS > s 1 PS > s, we can reformulate Eq 84 as following TCE q S] λ e θ 1 f SVaR q S] F S VaRq S] ES] h VarS], 85 where h is the hazard function Last exression is equivalent to the formulas for the TCE risk measure of a normally distributed rv cf, eg Paner, 22; Landsman and Valdez, 23 In the light of this, Poisson rv s can be seen as discrete counterarts of normal ones We must outline here, that form 85 does not always hold For instance, Furman and Landsman 25 showed that, for S Tw 2 θ, λ, TCE q S] ES] h VaR q S] 81 The TCE based allocation rule In addition to three basic illars enumerated at the beginning of this section, a significant number of financial institutions have recently adoted the so-called risk caital framework According to Zaik et al 1996, two central elements of such framework are 1 Assessing the risk caital and holding sufficient amount of caital to cover risks, and 2 Allocating the risk caital to each oerating division or deartment In Theorem 81 we addressed oint 1 In the next theorem we treat the second oint, and we show that the contribution of each univariate marginal risk to the shortfall in the deendent multivariate Poisson case is stiulated by its mathematical exectation Theorem 82 The contribution of X to the shortfall, in the case of X Tw n,1 θ, λ is ] TCE q X S λ e θ F Sn VaRq S] F S VaRq S] λ e θ F S1 VaRq S] F S VaRq S] 86 Proof The roof follows from Lemmas 81 and 82 We are often interested in the relative contribution of riskiness of X to S, which, in the deendent multivariate Poisson case, is formulated as ] TCE q X S λ F Sn VaRq S] λ F S1 VaRq S] TCE q S] nλ F Sn VaRq S] n λ F S1 VaRq S] Then, assuming that nor F Sn VaRq S] neither F S1 VaRq S] are zeros, the right-hand side of the last equation can be rewritten in the following way ] TCE q X S λ 1 TCE q S] F λ S1VaR q S] λ F SnVaR q S] nλ λ nλ λ λ,

10 36 E Furman, Z Landsman / Insurance: Mathematics and Economics which, in the limit case λ, reduces to ] TCE q X S λ EX ] TCE q S] ES] 87 λ It therefore turns out that, although in the deendent multivariate Poisson case the relative contribution of riskiness of X to S is quite lengthy, it is surrisingly simle in the indeendent case We note in this regard that although X 1,, X n are indeendent, the air X, S, 1,, n is certainly deendent, and thus Eq 87 can be of some ractical imortance 9 Conclusions In this aer we have thoroughly studied the multivariate Tweedie family of distributions, which we consider a multivariate extension of the well-known exonential disersion Tweedie models To formulate MTwF, we have utilized the multivariate reduction method, which for this urose has been formulated in a quite general form As its very name imlies, the multivariate robabilistic model discussed herein ossesses univariate marginal distributions which corresond to the univariate Tweedie ones Moreover, the deendence structure of MTwF is reflected in its variance/ covariance structure and allows for quite efficient modeling of multivariate ortfolios of deendent insurance losses Excet for the multivariate normal distributions, members of MTwF ossess non-negative suorts, ositive skewness and some of them are relatively tolerant to large risks, which resonds well to the eculiarities of the insurance industry demands Indeed, on the one hand, MTwF contains as secial cases the multivariate Poisson 1, multivariate gamma 2, multivariate inverse Gaussian 3 and multivariate comound Poisson 1 < < 2 distributions, and on the other, it introduces a rich variety of other multivariate models corresonding to non-integer values of outside the, 1 interval Moreover, MTwF rovides a general framework to deal with such well-known existing multivariate distributions as the multivariate inverse Gaussian distribution of Chhikara and Folks 1989, multivariate gamma of Mathai and Moschooulos 1991 and 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