Electronic copy available at: http://ssrn.com/abstract=1650299

MULTIVARIATE TWEEDIE DISTRIBUTIONS AND SOME RELATED

CAPITAL-AT-RISK ANALYSIS1

Edward Furman2

Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3,

Canada. E-mail: efurman@mathstat.yorku.ca

Zinoviy Landsman

Department of Statistics and Actuarial Research Center, University of Haifa, Haifa 31905,

Israel. E-mail: landsman@stat.haifa.ac.il

Abstract. We study a multivariate extension of the univariate exponential

dispersion Tweedie family of distributions. The class, referred to as the multi-

variate Tweedie family (MTwF), on the one hand includes multivariate Pois-

son, gamma, inverse Gaussian, stable and compound Poisson distributions

and on the other introduces a high variety of new dependent probabilistic

models unstudied so far. We investigate various properties of MTwF and

discuss its possible applications to financial risk management.

Keywords and phrases : Exponential dispersion models, multivariate Tweedie fam-

ily, Cauchy’s functional equations, risk capital allocations, the tail conditional ex-

pectation risk measure.

1We are grateful to a referee for comments that has improved the readability of the paper. Also, the

first author gratefully acknowledges the support of his research by the Natural Sciences and Engineering

Research Council (NSERC) of Canada.2Corresponding author, TEL:+1-416-736-2100 (Ext 33768), FAX:+1-416-736-5757.

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Electronic copy available at: http://ssrn.com/abstract=1650299

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Contents

1. Introduction and motivation 2

2. Univariate Tweedie EDM’s 6

3. The multivariate reduction method 8

4. The multivariate Tweedie family of distributions 9

5. Moments, higher order moments and Chebyshev’s type inequalities 13

6. Multivariate densities 15

7. Asymptotic results 18

8. Capital-at-risk analysis: an example 21

9. Conclusions 25

References 26

1. Introduction and motivation

Exponential dispersion models (EDM’s) play a prominent role in actuarial science and fi-

nance. This can be partly explained by the extent of generalization, as well as by the appeal

of multitude unifications, the EDM’s enable for such widely popular probability distribu-

tion functions (pdf’s) as normal, gamma, inverse Gaussian, stable and many others. The

specificity characterizing probabilistic modeling of actuarial objects is that the underlying

distributions mostly have non-negative supports, and many EDM’s possess this important

phenomenon. As a result, a significant number of scientific papers have been dedicated to

exploring various members of the EDM class in diverse fields of actuarial science (cf., e.g.,

Kaas, Dannenburg & Goovaerts, 1997; Nelder & Verrall, 1997; Landsman & Makov, 1998,

2003; Landsman, 2002; Landsman & Valdez, 2005).

Although univariate EDM’s are considerably rich, analytically tractable and widely ap-

plied, in the multivariate context the case is somewhat different. Specifically, Bildikar and

Patil (1968) studied a multivariate EDM induced by the following probability measure

dPθ,λ = exp(xT θ − λκ(θ))dνλ(x), x = (x1, x2, ..., xn)T ∈ Rn. (1.1)

Here Rn denotes the n-dimensional Euclidean space, θ ∈ (a,b) ⊂ Rn for some finite or

infinite interval (a,b), λ ∈ R+, the measure νλ (x) is independent of θ, and κ (θ) is an

analytic function in θ, independent of x. The family is however not as rich as the univariate

3

EDM, which it aims to generalize. In particular, (1.1) does not include important, especially

in insurance and finance multivariate distributions, having e.g., inverse Gaussian, gamma

or compound Poisson univariate margins. Moreover, the only continuous member of (1.1),

having independent of θ support, is the multivariate normal distribution (cf. loc. cite, for

details).

In addition, Jorgensen (1987) and Jorgensen & Lauritzen (2000) introduced and inves-

tigated certain generalizations of univariate EDM’s. It should be noted however, that the

former generalization has only a single parameter describing its dependence 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 practical

importance in actuarial science.

Seeking to resolve the aforementioned limitation of the multivariate EDM constructed in

Jorgensen & Lauritzen (2000), Song (2000) applied the copula based approach to generate a

multivariate distribution with predefined univariate EDM margins. Song’s model is however

not very tractable analytically. In particular, the joint moments are generally not available

in closed forms and are to be computed via some Monte Carlo techniques. We refer to

e.g., Frees and Valdez (1998) and Cherubini et al. (2004), in which actuarial and financial

applications of copulas are considered.

Broadly speaking, there exist a considerable number of methods of constructing multi-

variate dependent probabilistic structures (cf., e.g., Mardia, 1970; Joe, 1997; Kotz et al.,

2000). However, the insurance and finance industries dictate specific laws that must be

obeyed. Namely, mostly only multivariate models defined on Rn+, preserving unimodality

and positive skewness, can serve as appropriate candidates for model insurance risks. These

peculiarities discard, for instance, such generally important finance models as multivariate

normal, Student-t and Cauchy (cf., e.g., Tobin, 1958; Bawa, 1975; Ross, 1978; Owen & Ra-

binovitch, 1983; Furman & Landsman, 2006). Also, there is always a trade-off between, on

the one hand, the approximation level provided by the model and on the other, its analytic

complexity. Consequently, one has to impose an additional restriction of tractability on the

choice of the multivariate cumulative distribution function (cdf) and its dependence struc-

ture and, thus, to reject even more models, although they might have described insurance

risks well.

4

It must be the above-mentioned difficulties that explain a ‘convention’ to use univariate

distributions to model risks in actuarial science. However, in the last years the concept of

dependence has received its merited attention, which has resulted in numerous papers (cf.,

e.g., Pfeifer & Neslehova, 2004; Bauerle & Grubel, 2005; Alink et al., 2005; Roger et al.,

2005; Centeno, 2005; Kijima, 2006). Also, some new multivariate distributions have been

proposed (cf., e.g., Vernic, 1997, 2000, Furman and Landsman, 2008).

Motivated by the already mentioned limitations of the existing multivariate EDM’s, in

Sections 3 and 4 of the present paper we reintroduce (cf. loc. cite) 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., e.g., Boucher et al., 2008) and background

economy (cf., e.g., Tsanakas, 2008) models. We consider the MTwF class a multivariate

extension of the exponential dispersion Tweedie family since its univariate marginal distri-

butions correspond to the univariate Tweedie ones. Moreover, the proposed family possesses

a dependence structure, which is reflected in its variance/covariance structure and allows

for efficient modeling of multivariate portfolios of dependent risks. As special cases, among

others, MTwF contains the multivariate inverse Gaussian, multivariate gamma, multivariate

stable, multivariate Poisson and multivariate compound Poisson distributions in the sense

that their univariate margins are inverse Gaussian, gamma, positive stable, Poisson and

compound Poisson, respectively. In the light of this, on the one hand, MTwF provides

a general framework to deal with, for instance, the multivariate Poisson of Kocherlatoka

& Kocherlatoka (1992), the multivariate inverse Gaussian of Chhikara & Folks (1989) and

the multivariate gamma of Mathai & Moschopoulos (1991), and on the other, it introduces

some new multivariate dependent models which can be applied to describe insurance risks’

behavior usefully.

We study various seemingly important properties of MTwF. Namely, Section 5 calculates

some higher order moments and applies these moments to produce useful Chebyshev’s type

inequities, Section 6 derives the pdf’s of the random vectors distributed MTwF, and Section 7

develops related asymptotic results. We note in this regard, that the present paper contains

a thorough analysis of various properties of MTwF, and it therefore differs greatly from

Furman & Landsman (2008), in which the model has been passingly introduced with the

main accent made on the multivariate Tweedie compound Poisson special case.

5

To demonstrate one possible application of MTwF, we carry out an investigation of both

the univariate and multivariate capital-at-risk analysis in the framework of the multivariate

Poisson distribution. This distribution is a particular member of MTwF, and it is commonly

applied 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 expectation

(TCE) risk measure and the capital-at-risk allocation based on it. More specifically, let

X ∈ R+ be a risk random variable (rv) with cdf F and decumulative distribution function

(ddf) F = 1− F . Then, for every q ∈ (0, 1), TCE is defined as

TCEq[X] = E [X|X > V aRq[X]] =1

F (V aRq[X])

∫ ∞

V aRq [X]

xdF (x), (1.2)

subject to F (V aRq[X]) > 0 and

V aRq[X] = infx : F (x) ≥ q. (1.3)

Risk measure (1.2) coincides with the expected shortfall (ES) and the conditional Value-at-

Risk (CVaR) under the assumption of continuous distributions (cf., e.g., Hurlimann, 2003;

McNeil, Frei, Embrechts, 2005, Lemma 2.16). It should also be emphasized that TCE is a

coherent risk measure (cf. Artzner et al., 1999; Acerbi & Tasche, 2002; Tasche, 2002), it is a

particular member of the distorted risk functionals (cf. Denneberg, 1994; Wang, 1995, 1996;

Wang et al., 1997), and it can be considered a weighted premium calculation principle (pcp)

(cf. Furman & Zitikis, 2008a).

Let X = (X1, X2, . . . , Xn)T and S = X1 + X2 + · · · + Xn represent the risk inherent in

a financial conglomerate and the total risk of that conglomerate, respectively. Then TCE

allows for a natural allocation of the total risk to its constituents, e.g., business lines Xk,

k = 1, 2, . . . , n (cf., e.g., Denault, 2001; Panjer, 2002; Wang, 2002). Indeed, by the additivity

property of the expectation operator, one obtains that the ‘risk contribution’ of the k−th

business line to the total risk of the conglomerate is

TCEq[Xk|S] = E [Xk|S > V aRq[S]] . (1.4)

Panjer (2002) derived both (1.2) and (1.4) for the n−variate normal random vector X vNn(µ, Σ). He showed that 1.) TCEq[S] can be formulated in terms of the well known

variance premium V P [S] = E[S]+αVar[S] (cf., e.g., Buhlhmann, 1970) with α equal to the

hazard function evaluated at V aRq[S], and 2.) the contribution of Xk to S is stipulated by

E[Xk] and the covariation between them. Landsman and Valdez (2003) extended Panjer’s

6

results for the broader class of elliptical distributions (cf. Furman & Zitikis, 2008b, for

additional generalizations).

Hurlimann (2001) seems to have been the first to consider TCE for positive risks. Lands-

man & Valdez (2005) derived analytic closed form expressions for this risk measure in the

context of EDM’s. However, it should be noted, that both papers cited above deal with the

functionals of form (1.2) and do not touch on the more general problem of allocating risk

capitals (1.4). In this respect, Furman & Landsman (2005) derived analytic expressions for

both (1.2) and (1.4) in the framework of multivariate dependent gamma portfolios. Cai &

Li (2005) and Chiragiev & Landsman (2007) solved a similar problem in the context of the

multivariate phase type and Pareto distributions, respectively. Vernic (2006) considered the

multivariate skew-normal case, see also Dhaene et al. (2008).

In Section 8 we derive analytic expressions for (1.2) and (1.4) in the framework of the

multivariate Poisson distribution, which is an important member of MTwF. Despite of the

quite expository purpose of the section, the appealing simplicity of the obtained allocation

formula seems to be worthy of note.

Next two sections provide the necessary groundwork for introducing the multivariate

Tweedie family.

2. Univariate Tweedie EDM’s

Let ν be a positive non-degenerate measure on R, and define the cumulant transform

κ (θ) = log∫R

eθxν (dx) with the domain Θ =θ ∈ R :

∫R

eθxν (dx) < ∞ 6= φ. For the

canonical parameter θ ∈ Θ, the probability measure

dPθ = exp (θx− κ (θ)) ν (dx) (2.1)

represents a natural exponential family (NEF). Further, let µ = τ (θ) = κ′ (θ) denote the

mean value mapping and Ω = τ (int Θ) be the mean domain. Also, provided (2.1), define the

set of index parameters Λ = λ ∈ R+ such that λκ (θ) = log∫

eθxνλ (dx) for some measure

νλ on R. The natural exponential family generated by νλ is then given by

dPθ,λ = exp (θx− λκ (θ)) νλ (dx) . (2.2)

The probability measure above is referred to as the additive exponential dispersion model

and is denoted by ED (θ, λ) (cf. Jorgensen, 1997). Note that the mgf corresponding to

7

probability measure (2.2) is readily obtained as

M(t) = exp (λ[κ(θ + t)− κ(θ)]) . (2.3)

For regular (or steep) EDM’s, τ ′ (µ) = κ′′ (θ) > 0, 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)

).

EDM’s are classified by their unit variance functions. Below we outline one of the impor-

tant, especially in the insurance context, classes of EDM’s, Tweedie models, which possess

power unit variance functions Vp(µ) = µp, µ ∈ Ω and p ∈ (−∞, 0] ∩ [1,∞). (cf. Tweedie,

1984; Bar-Lev & Enis, 1986; Jorgensen, 1997). In what follows, we add a subscript p to em-

phasize that Tweedie EDM’s are discussed. Choosing an appropriate p parameter, we arrive

at e.g. normal (p = 0, Ωp = R), Poisson (p = 1, Ωp = R+), gamma (p = 2, Ωp = R+),

inverse Gaussian (p = 3, Ωp = R+) and compound Poisson (p ∈ (1, 2), Ωp = R+) distribu-

tions. Other valuable members of Tweedie EDM’s are enumerated in Table 4.1 of Jorgensen

(1997).

It must be emphasized that even more flexibility in modeling insurance losses can be

achieved by considering Tweedie distributions corresponding to non-integer p parameters

(cf., e.g., Dunn & Smyth, 2005).

Although EDM’s are determined by their unit variance functions, in order to construct

the multivariate extension of the univariate Tweedie class, an explicit form of the cumulant

κp(θ) function corresponding to the specific Vp(µ) is required. The former is obtained by

solving two following differential equations

dτ−1p (µ)

dµ=

1

µpand κ′p (θ) = τp (θ) ,

which lead to

κp (θ) =

eθ, p = 1

− log (−θ) , p = 2

(α−1)α

(θ

α−1

)α, p 6= 1, 2

(2.4)

(cf. Jorgensen, 1997, Section 4.1.2). We make extensive use of the above form of κp(θ) in

Section 4.

8

To conclude this section, we note that Tweedie models share with normal and gamma

EDM’s the property that they have both an additive and a reproductive form. Therefore

later on in this paper we discuss additive Tweedie EDM’s only, however this does not imply

any restrictions on the derived results.

3. The multivariate reduction method

We have already mentioned a number of existing generalizations of univariate EDM’s to

the multivariate framework along with the corresponding shortcomings. In this section we

describe an alternative method of constructing multivariate probabilistic models with pre-

defined univariate margins. The resulting multivariate Tweedie family of distributions then

seems to provide an intuitive solution to the situations when observable risks are contami-

nated by a common ‘background’ risk, which makes the whole structure dependent.

It should be noted that the methodology discussed herein is well-known as the ‘common-

shock’ model (cf., e.g., Boucher et al., 2008) in actuarial science and also as the ‘background

economy’ model (cf., e.g., Tsanakas, 2008) in economics. This fact therefore provides addi-

tional interpretations and motivations for MTwF.

Let Y = (Y0, Y1, . . . , Yn)T be an (n+1)-variate random vector with mutually independent

corresponding cdf’s Fi(y; ξi), i = 0, 1, . . . , n, and let X = (X1, X2, . . . , Xn)T be another, say,

resulting, random vector. Denote by T a functional mapping from Rn+1 to Rn, such that

X = T (Y). (3.1)

Definition 3.1. The random vector X ∈ Rn is said to possess the cdf F (x; ξ∗) parameterized

by the vector ξ∗ = (ξ1, ξ2, . . . , ξn)T , such that ξj = ηj(ξ0, ξ1, . . . , ξn) for specific functions

ηj, j = 1, 2, . . . , n.

Some useful examples of mapping (3.1) are X = min (Y), X = max (Y) and X = eTY,

for the unit vector e = (1, 1, . . . , 1)T . In this paper we shall consider linear forms of mapping

(3.1) only. Then it can be rewritten as

X = AY, (3.2)

where A ∈ Matn×(n+1) is an n × (n + 1) matrix. Note that A determines the form of

the multivariate distribution F (x; ξ∗) obtained by the method. Taking, for instance, Yi v

9

Ga (γi, α) and

A =

1 1 0 0 · · · 0

1 0 1 0 · · · 0

1 0 0 1 · · · 0...

......

.... . .

...

1 0 0 0 · · · 1

(3.3)

leads to the structure of Cheriyan (1941) and Ramabhadran (1951). The same matrix A and

the assumption that Yi v IG(ciµ, c2i λ), ci ∈ R+ results in an extension of the bivariate inverse

Gaussian distribution of Chhikara & Folks (1989). Also, by considering again Yi v Ga(γi, α)

and

A =

1 1 0 0 · · · 0

1 1 1 0 · · · 0

1 1 1 1 · · · 0...

......

.... . .

...

1 1 1 1 · · · 1

(3.4)

we arrive at the multivariate gamma model of Mathai & Moschopoulos (1992), see also a

generalized counterpart of Furman (2008).

We note that matrices (3.3) and (3.4) imply specific restrictions on the parameters of

elements of X, i.e., common rate parameters α in the models of Cheriyan (1941), Ram-

abhadran (1951) and Mathai & Moschopoulos (1992) and the relations c0+ck

c0+cland ( c0+ck

c0+cl)2,

where 0 ≤ k, l ≤ n, preserved between the parameters of Yk and Yl in the model of Chhikara

& Folks (1989). We loosen these restrictions in the next section.

4. The multivariate Tweedie family of distributions

Evolving the discussion of two previous sections, let us consider an (n + 1)-variate random

vector Y with mutually independent univariate margins Yi v ED(θi, λi) (recall that without

loss of generality we consider additive EDM’s only). Then, according e.g., to the notions

in the last paragraph of Section 3, it is clear that nor (3.3) neither (3.4) can be applied to

construct a multivariate model with arbitrarily parameterized univariate Tweedie marginal

distributions.

10

Instead, let

A =

β1 1 0 0 · · · 0

β2 0 1 0 · · · 0

β3 0 0 1 · · · 0...

......

.... . .

...

βn 0 0 0 · · · 1

, (4.1)

subject to βj = θ0/θj, j = 1, 2, . . . , n. Further, to acquire an intuition, note that the mgf of

the j-th univariate marginal component of the mapping X = AY, or in other words the mgf

of Xj = βjY0 + Yj is

Mj (t) = exp (λj [κ (θj + t)− κ (θj)] + λ0 [κ (θ0 + βjt)− κ (θ0)])

= exp (λj [κ (θj + t)− κ (θj)] + λ0 [κ ((θj + t)βj)− κ (βjθj)]) ,

and it must be of form (2.3) for Xj to be an additive EDM. Let x = θj + t and y = βj for

the sake of notational convenience, then the mgf of Xj is formulated as

Mj (t) = exp (λjκ (x)− λjκ (θj) + λ0κ (xy)− λ0κ (θjy)) , (4.2)

which (for θj 6= θ0) 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) . (4.3)

The equation is known to possess the following most general non-constant and continuous

non-trivial solutions (cf., e.g., Castillo & Ruiz-Cobo, 1992)

κ (x) = a log (x) + b ; f (x) = 1 ; g (x) = a log (x)

κ (x) = axc + b ; f (x) = xc ; g (x) = b (1− xc)

where a, b and c are constants such that a 6= 0 and c 6= 0 but otherwise arbitrary.

Straightforward comparison of the aforementioned non-trivial solutions of equation (4.3)

to cumulant function (2.4) implies that MTwF is the only possible multivariate extension of

EDM’s, given Definition 3.1 and matrix (4.1). Indeed, for θj 6= θ0, the solutions imply that

for Xj to be an EDM, the corresponding cumulant function must be of either logarithmic

or power form, which readily yields the cumulant function of the Tweedie class. Theorem

4.1 ensures that the proposed multivariate probabilistic model possesses univariate mar-

gins belonging to univariate Tweedie EDM’s, and it establishes these marginal distributions

explicitly.

11

We note in passing that the case when θj ≡ θ0 implies y ≡ 1 in mgf (4.2), and this is nothing

else but the well-known closure under convolutions property of the additive EDM’s. In our

context, equal canonical parameters correspond to the multivariate Poisson distribution,

which is certainly a member of MTwF. In such a case we omit indexation (cf. Theorem 4.1,

p = 1).

Theorem 4.1. Let X = AY and Y be an (n + 1)-variate random vector with mutually

independent additive Tweedie margins Yi v Twp (θi, λi) , i = 0, 1, . . . , n. Let βj = θ0/θj in

(4.1). Then the j-th univariate margin of X is

Xj v

Tw1 (θ, λ0 + λj) , p = 1

Tw2 (θj, λ0 + λj) , p = 2

Twp

(θj, λ0

(θ0

θj

)α

+ λj

), p 6= 1, 2

,

where α = (p− 2)/(p− 1).

Proof. We prove only the case p 6= 1, 2. Hence,

log Mj (t)

= λ0(α− 1)1−α

α

((θ0 + t

θ0

θj

)α

− (θ0)α

)+ λj

(α− 1)1−α

α((θj + t)α − (θj)

α)

=(α− 1)1−α

α(θj + t)α

(λ0

(θ0

θj

)α

+ λj

)− (α− 1)1−α

α(θj)

α

(λ0

(θ0

θj

)α

+ λj

)

=

(λ0

(θ0

θj

)α

+ λj

)(α− 1)1−α

α((θj + t)α − (θj)

α)

= λj [κp(θj + t)− κp(θj)] ,

where λj = λ0

(θ0

θj

)α

+ λj, which completes the proof. ¤

We now naturally arrive at the following definition of additive MTwF.

Definition 4.1. Let Y be an (n + 1)-variate random vector with mutually independent ad-

ditive Tweedie margins Yi v Twp (θi, λi) , i = 0, 1, . . . , n, and let βj = θ0/θj in (4.1). Also,

let θ ∈ Θ ⊂ Rn be the n-variate vector of canonical parameters. Here, Θ is the Carte-

sian product of the domains Θ ⊂ R. Then the joint distribution of X = AY, denoted by

X v Twn,p

(θ, λ

), is the n-variate additive Tweedie distribution.

Note 4.1. To obtain the reproductive form of MTwF, we take βj = λ0θ0

λjθjin Definition 4.1.

12

It is well known that univariate Tweedie EDM’s are closed under convolutions for common

canonical parameters. We further introduce the multivariate counterpart of such a closure

in the framework of MTwF. To achieve this goal, we make extensive use of the multivariate

mgf of X v Twn,p

(θ, λ

), which is readily available and can be formulated as

MX (t)

= E[eXT t

]= E

[exp

(n∑

j=1

tj

(θ0

θj

Y0 + Yj

))]

= MY0

(n∑

j=1

θ0

θj

tj

)n∏

j=1

MYj(tj) , (4.4)

with Y0, Y1, . . . , Yn given in Definition 4.1.

Theorem 4.2. Let X1 v Twn,p(θ, λ1) and X2 v Twn,p(θ, λ2) be two independent multi-

variate Tweedie random vectors. Then S = X1 + X2 v Twn,p(θ, λ1 + λ2).

Proof. Due to the independence of X1 and X2 and evolving equation (4.4), the mgf of S is

written, for β = ( θ0

θ1, θ0

θ2, . . . , θ0

θn)T , X1,j = βjY1,0 + Y1,j and X2,j = βjY2,0 + Y2,j, j = 1, . . . , n,

as

MS(t) = MY1,0

(βT t

) n∏j=1

MY1,j(tj)×MY2,0

(βT t

) n∏j=1

MY2,j(tj) .

Therefore,

log MS(t)

=((λ1,0 + λ2,0)

[κp

(θ0 + βT t

)− κp (θ0)]) n∑

j=1

(λ1,j + λ2,j) (κp (θj + tj)− κp (θj)) ,

which necessarily means that the mgf of S is of form (4.4) with λS = (λ1,1 + λ2,1, λ1,2 +

λ2,2, . . . , λ1,n + λ2,n) and hence completes the proof. ¤

Literally speaking, Theorem 4.2 implies that combining independent portfolios of risks

Twn,p(θ, λ1) and Twn,p(θ, λ2), produces another portfolio which is also in MTwF. Such a

multivariate additivity property allows for a fairly tractable treatment of the MTwF random

vectors and therefore portfolios of risks described by this multivariate probabilistic model.

13

5. Moments, higher order moments and Chebyshev’s type inequalities

Many problems in actuarial science require to evaluate moments of order-m of the risk rv

X. This section is dedicated to deriving these useful expressions in the general context of

MTwF.

According to Definition 4.1, the evaluation of, say E[Xmj ], j = 1, 2, . . . , n requires the

moments E [Y mi ] , i = 0, 1, . . . , n, which are readily available. Indeed, as the corresponding

cumulants are

km =∂m log MYi

(t)

∂tm|t=0 = λiκ

(m)p (θi) ,

and the mgf of Yi is of exponential form, we readily obtain that

E [Y mi ] =

∂mMYi(t)

∂tm|t=0

=∑ m!

h!j!k! · · · l!(

k1

1!

)h (k2

2!

)j (k3

3!

)k

· · ·(

kL

L!

)l

, (5.1)

where the summation is over all solutions in non-negative integers of the equation h + 2j +

3k + · · ·+ Ll = m.

Equation (5.1) implies, for instance, that

E [Yi] = k1 = λiκ′p (θi) ,

E[Y 2

i

]= k2

1 + k2 =(λiκ

′p (θi)

)2+ λiκ

(2)p (θi) and Var [Yi] = λiκ

(2)p (θi) ,

E[Y 3

i

]= k3 + 3k1k2 + k3

1 and Skew [Yi] =λiκ

(3)p (θi)(

λiκ(2)p (θi)

)3/2.

Further, applying again Definition 4.1, we formulate higher order moments of Xj as

E[Xm

j

]= E

[(θ0

θj

Y0 + Yj

)m]=

m∑r=0

(m

r

)(θ0

θj

)r

E [Y r0 ]E

[Y m−r

j

].

Last equations yield, for instance, that the expectation of Xj is

E [Xj] =θ0

θj

λ0κ′p (θ0) + λjκ

′p (θj) ,

the variance of Xj is

Var [Xj] =

(θ0

θj

)2

λ0κ(2)p (θ0) + λjκ

(2)p (θj) ,

14

and the skewness of Xj is

Skew [Xj] =

(θ0

θj

)3

λ0κ(3)p (θ0) + λjκ

(3)p (θj)

((θ0

θj

)2

λ0κ(2)p (θ0) + λjκ

(2)p (θj)

)3/2.

Similar procedure allows us to evaluate the following product moments

E[Xn

i Xmj

]

= E

[(θ0

θi

Y0 + Yi

)n (θ0

θj

Y0 + Yj

)m]

= E

[n∑

r=0

(n

r

)(θ0

θi

Y0

)r

Y n−ri

m∑s=0

(m

s

)(θ0

θj

Y0

)s

Y m−sj

]

=n∑

r=0

m∑s=0

(n

r

)(m

s

) (θ0

θi

)r (θ0

θj

)s

E[Y

(r+s)0

]E

[Y n−r

i

]E

[Y m−s

j

],

where E[Y

(r+s)0

], E

[Y n−r

i

]and E

[Y m−s

j

]are available from equation (5.1).

To compute the cumulants of Xj we need the logarithm of its mgf. According to, say

(4.4), we have that

log MXj(t) = λ0

[κp

(θ0 +

θ0

θj

t

)− κp (θ0)

]+ λj [κp (θj + t)− κp (θj)] .

Consequently, the m-th cumulant of Xj is

Km =∂m log MXj

(t)

∂tm|t=0 =

(θ0

θj

)m

λ0κ(m)p (θ0) + λjκ

(m)p (θj) .

Also, note that for β = ( θ0

θ1, θ0

θ2, . . . , θ0

θn)T , we have that

log MX (t) = λ0

[κp

(θ0 + βT t

)− κp (θ0)]+

n∑j=1

(λj [κp (θj + tj)− κp (θj)]) ,

and therefore the joint cumulants of Xi and Xj are

Kmi,mj=

∂mi+mj log MX (t)

∂tmii ∂t

mj

j

|t=0 =θ

(mi+mj)0

θmii θ

mj

j

λ0κ(mi+mj)p (θ0) ,

for i 6= j. Hence, the covariance of Xi and Xj is

Cov [Xi, Xj] =θ20

θiθj

λ0κ(2)p (θ0) =

θ20

θiθj

V ar (Y0) , for i 6= j.

15

5.1. Chebyshev’s type inequalities. As an illustration of the previous results, we further

derive some useful Chebyshev’s type inequalities in the case of the multivariate Tweedie

distributions.

Theorem 5.1. Let εj and Xj, j = 1, 2, . . . , n be some arbitrary positive constants and the

components of the multivariate additive Tweeide family, correspondingly. Then

P (X1 ≤ ε1, . . . , Xn ≤ εn) ≥ 1−n∑

j=1

(θ0

θj

λ0κ′p (θ0) + λjκ

′p (θj)

)/εj (5.2)

and

P (X1 ≥ ε1, . . . , Xn ≥ εn) ≤n∑

j=1

(θ0

θj

λ0κ′p (θ0) + λjκ

′p (θj)

)/

n∑j=1

εj. (5.3)

Proof. The first part follows straightforwardly from the following well known inequality

P (X > ε) ≤ E[X]

ε,

by recalling that

E [Xj] =θ0

θj

λ0κ′p (θ0) + λjκ

′p (θj) .

Indeed,

P

(n⋂

j=1

Xj ≤ εj)

≥ 1−(

n∑j=1

P (Xj > εj))

≥ 1−n∑

j=1

E[Xj]

εj

.

Inequality (5.3) follows from the note that

P (X1 ≥ ε1, . . . , Xn ≥ εn) ≤ P (n∑

i=1

Xi ≥n∑

i=1

εi),

which completes the proof. ¤

6. Multivariate densities

It turns out that Definition 4.1 allows to derive multivariate pdf’s of a general member of

MTwF. Recall that we deal with the random vector X v Twn,p

(θ, λ

)and, for βj = θ0/θj,

consider the following transformation

Xj = βjY0 + Yj, j = 1, 2, . . . , n and i = 0, 1, . . . , n.

16

We further assume that the measure νλ (·) in equation (2.2) possesses pdf c (·; λ) , then, due

to the independence of its marginal components, the joint pdf of Y = (Y0, Y1, . . . , Yn)T can

be formulated as

fY (y0, y1, . . . , yn)

=n∏

i=0

fYi(yi) =

n∏i=0

c (yi; λi) exp (θiyi − λiκp (θi))

= c (y0; λ0)n∏

j=1

c (yj; λj) exp

(n∑

j=1

(θjyj − λjκp (θj)) + θ0y0 − λ0κp (θ0)

).

Substituting Yj = Xj − βjY0, we obtain the pdf of X∗ = (Y0, X1, . . . , Xn)T which in such

a case equals

fX∗ (y0, x1, . . . , xn)

= c (y0; λ0)n∏

j=1

c (xj − βjy0; λj) exp

(n∑

j=1

θjxj − (n− 1) θ0y0 −n∑

i=0

λiκp (θi)

).

Finally, observing that 0 ≤ Y0 ≤ min (X1, X2, . . . , Xn), we can write the pdf of X as

fX (x1, x2, . . . , xn)

= exp

(n∑

j=1

(θjxj − λjκp (θj))− λ0κp (θ0)

)

×∫ xmin

0

c (y0; λ0)n∏

j=1

c (xj − βjy0; λj) exp (− (n− 1) θ0y0) dy0, (6.1)

where xmin = min (x1, x2, . . . , xn).

Last equation is quite complicated and cannot be solved analytically. Moreover, it implies

that the pdf of X is of different form for each of n! permutations of x1.x2, . . . , xn. In what

follows, we consider two examples which show that in some particular cases the pdf of X

can be handled in spite of the certain difficulties related to its unpleasant form.

Example 6.1. Multivariate gamma distribution. Let X v Ga(γ, α) denote a gamma rv with

shape and rate parameters equal γ and α, respectively. The pdf of X is

f(x) =1

Γ(γ)e−αxxγ−1αγ =

xγ−1

Γ(γ)exp (−αx + γ log(α)) ,

which, for θ = −α, λ = γ and κ(θ) = − log(−θ), can be reformulated as (2.2). Moreover,

from the form of cumulant function (2.4), it follows that X is a member of the univariate

Tweedie class with p = 2, or more precisely X v Tw2(−α, γ).

17

We now can carry out necessary substitutions and find the multivariate pdf of X vTwn,2(θ, λ). Due to equation (6.1), the pdf is given by

fX (x1, x2, . . . , xn)

= exp

(n∑

j=1

(−αjxj)

)n∏

i=0

αγi

i

Γ(γi)

∫ xmin

0

yγ0−10

n∏j=1

(xj − α0

αj

y0)γj−1e(n−1)α0y0dy0.

The above multivariate pdf agrees well with that of Mathai & Moschopoulos (1991). Note

that taking α=(1, 1, . . . , 1)T , we arrive at the model of Cheriyan (1941) and Ramabhadran

(1956). In such a case, the latter equation simplifies even more, reducing to

fX (x1, x2, . . . , xn) =exp

(−∑n

j=1 xj

)∏n

i=0 Γ(γi)

∫ xmin

0

yγ0−10

n∏j=1

(xj − y0)γj−1e(n−1)y0dy0.

We further consider another example of a member of MTwF. In order to emphasize that

the results of this paper can be equally applied to reproductive EDM’s, we discuss the

multivariate inverse Gaussian distribution, which is MTwF with p = 3.

Example 6.2. Multivariate inverse Gaussian distribution. Let Y v IG(µ, λ) denote an

inverse Gaussian rv with pdf

fY (y) =

√λ

2πy3exp

(−λ(y − µ)2

2µ2y

), y > 0.

Slight reformulation of the above pdf yields

fY (y) = e−λ2y

√λ

2πy3exp

(λ

(− y

2µ2+ µ−1

)),

which, for θ = (−2µ2)−1 and κ(θ) = −√−2θ can be seen as a reproductive EDM (cf.

Jorgensen, 1997). Then the power form of κ(θ) implies that Y is a univariate Tweedie with

p = 3, and the dual transformation X = λY leads to the additive Tweedie EDM, or more

precisely, to X v Tw3(− 12µ2 , λ), possessing the following pdf

f(x) = e−λ2

2x

√λ2

2πx3exp

(− x

2µ2+ λµ−1

).

18

We now can establish the pdf of the multivariate inverse Gaussian distribution, i.e., of

X v Twn,3(θ, λ). Making use of equation (6.1), we obtain that

fX (x1, x2, . . . , xn)

= exp

(n∑

j=1

(− xj

2µ2j

− λj

µj

)− λ0

µ0

)

×∫ xmin

0

e− λ2

02y0

√λ2

0

2πy30

n∏j=1

e− λ2

j

2(xj−(µjµ0

)2y0)

√λ2

j

2π(xj − (µj

µ0)2y0)3

exp

((n− 1) y0

2µ20

)dy0

with xmin = min (x1, x2, . . . , xn).

From the above pdf we can obtain e.g., the multivariate inverse Gaussian distribution of

Chhikara & Folks (1989), which is

fX (x1, x2, . . . , xn)

= exp

(n∑

j=1

(− xj

2µ2− λj

µ

)− λ0

µ

)

×∫ xmin

0

e− λ2

02y0

√λ2

0

2πy30

n∏j=1

e− λ2

j2(xj−y0)

√λ2

j

2π(xj − y0)3exp

((n− 1) y0

2µ2

)dy0.

7. Asymptotic results

In this section we develop some seemingly useful asymptotic results for the multivariate

Tweedie distributions. The usefulness of the results derived herein can be justified, for

instance, by the formally unpleasant nature of multivariate densities (6.1).

Let X = (X1, X2, . . . , Xn)T vTwn,p

(θ, λ

), and denote by Z the standardized MTwF ran-

dom vector possessing univariate margins

Zj =Xj − E [Xj]√

Var [Xj],

which, after applying the results of Section 5, yields

Zj =Xj − βjλ0κ

′p (θ0)− λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

=Xj√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

− βjλ0κ′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

,

for βj = θ0/θj, We are now in a position to establish the asymptotic distribution of Zj.

19

Theorem 7.1. Let λj → ∞ and λ0 be some positive constant. Then the standardized

univariate Tweedie rv Zj is asymptotically standard normal rv N(0, 1).

Proof. The proof relies on the mgf of Zj, which is as follows

MZj(t) = exp

− βjλ0κ

′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

t

MXj

t√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

,

where the mgf of Xj is given by

MXj(t) = exp

(λ0

[κp

(θ0 +

θ0

θj

t

)− κp (θ0)

]+ λj [κp (θj + t)− κp (θj)]

).

Consequently,

log MZj(t)

= − βjλ0κ′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

t

+λ0

κp

θ0 + βj

t√β2

j λ0κ(2)p (θ0) + λjκ

(2)p (θj)

− κp (θ0)

+λj

κp

θj +

t√β2

j λ0κ(2)p (θ0) + λjκ

(2)p (θj)

− κp (θj)

.

Expanding into a power series, we obtain that

log MZj(t)

= − βjλ0κ′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

t +βjλ0κ

′p (θ0)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

t

+β2

j λ0κ(2)p (θ0)

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

t2

2+

λjκ′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

t

+λjκ

(2)p (θj)

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

t2

2+ 2o

1√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

=t2

2+ 2o

1√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

, as λj →∞,

which completes the proof. ¤

In a similar fashion we establish the following theorem for the vector Z = (Z1, Z2, . . . , Zn)T .

20

Theorem 7.2. The standardized vector Z is asymptotically multivariate normal with the

vector of means µ = (0, 0, . . . , 0)T and the covariance matrix Σ with Σi,i = 1 and Σi,j given

below, for i, j = 1, 2, . . . , n,

(1) If λ0 is some positive constant and λj →∞, then

Σi,j = 0.

(2) If both λ0 and λj go to infinity such that λj/λ0 → k, then

Σi,j =βjβiκ

(2)p (θ0)√

β2j κ

(2)p (θ0) + mjκ

(2)p (θj)

√β2

i κ(2)p (θ0) + miκ

(2)p (θi)

.

Proof. The mgf of Z can be formulated in terms of the mgf’s of Y0, Y1, . . . , Yn as

MZ (t)

= E

exp

n∑j=1

tjXj√β2

j λ0κ(2)p (θ0) + λjκ

(2)p (θj)

−n∑

j=1

βjλ0κ′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

tj

= exp

−

n∑j=1

βjλ0κ′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

tj

E

exp

n∑j=1

tjXj√β2

j λ0κ(2)p (θ0) + λjκ

(2)p (θj)

= exp

−

n∑j=1

βjλ0κ′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

tj

MY0

n∑j=1

βjtj√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

×n∏

j=1

MYj

tj√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

.

The logarithm of the latter equation is therefore

log MZ (t)

= −n∑

j=1

βjλ0κ′p (θ0) + λjκ

′p (θj)√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

tj

+λ0

κp

θ0 +

n∑j=1

βjtj√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

− κp (θ0)

+n∑

j=1

λj

κp

θj +

tj√β2

j λ0κ(2)p (θ0) + λjκ

(2)p (θj)

− κp (θj)

,

21

which after expanding into a power series yields

log MZ (t)

= λ01

2κ(2)

p (θ0)

n∑j=1

βjtj√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

2

+n∑

j=1

λj

κ

(2)p (θj)

2

tj√

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

2 .

Hence, if λ0 is some fixed positive constant and λj →∞, then

log MZ (t) → 1

2

n∑j=1

λjκ(2)p (θj) t2j

β2j λ0κ

(2)p (θ0) + λjκ

(2)p (θj)

=n∑

j=1

t2j2

.

Alternatively, when both λ0 and λj go to infinity such that λj/λ0 → k, we have that

log MZ (t) → 1

2

n∑j=1

t2j +∑

i6=j

βjβiκ(2)p (θ0) tjti√

β2j κ

(2)p (θ0) + mjκ

(2)p (θj)

√β2

i κ(2)p (θ0) + miκ

(2)p (θi)

,

which completes the proof. ¤

We note in passing that Theorem 7.2 is intuitively clear, bearing in mind that λ0 has

a crucial importance for the dependence structure of Twn,p(θ, λ). In the light of this, the

first statement of Theorem 7.2 addresses the case when the dependence is asymptotically

vanishing, while the second statement relates to the contrary case.

8. Capital-at-risk analysis: an example

In the modern conception of risk management, where risks are mostly represented by non-

negatively valued rv’s (cf., e.g., Artzner et al., 1999), the following three substantial subjects

have to be addressed

1. An appropriate multivariate dependent probabilistic model F (x1, x2, . . . , xn) that

provides a satisfactory fit for a real life multi-line business,

2. A risk measure H : X → [0,∞], which actually measures the degree of riskiness that

each X ∈ X implies,

3. (Analytic) expressions 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 point 1 above. In what follows, we consider risk functional (1.2),

22

and we derive formulas for S = eTX, where, for e = (1, 1, . . . , 1)T , X = (X1, X2, . . . , Xn)T vTwn,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 independent. Much

less is known of the case when some stochastical dependence is assumed, and yet this is a

common situation in practice (cf., e.g., Denuit et al., 2005). In our context, due to Definition

4.1 and keeping the Poisson case (p = 1) in mind, we arrive at the following representation

of S, for Yi v Tw1(λieθ), i = 0, 1, . . . , n,

S =n∑

j=1

Xj = nY0 +n∑

j=1

Yj. (8.1)

Then the distribution of S can be seen as a compound Poisson. Indeed, for any positive

constants ci and Poisson rv’s Yi, the product ciYi can be interpreted as a compound Poisson rv

with the corresponding Poisson parameter λieθ and a degenerate claim amount distribution

at ci. Consequently, the sum S =∑n

i=0 ciYi is distributed compound Poisson with Λ =

eθ∑n

i=0 λi and the following corresponding probability mass function (pmf) of claim amount

p (c) =

λieθ

Λc = ci

0 otherwise.

We further state some auxiliary results as lemmas. The former lemma is rather general

and it concerns arbitrarily distributed independent risks with non-negative supports (cf.

Furman & Landsman, 2005). The latter, reveals how the size-biased rv associated with a

Poisson rv, can be handled.

Lemma 8.1. Let X represent a multivariate portfolio of arbitrary independent non-negative

risks X1, X2, ..., Xn with pdf’s fj (x) , j = 1, 2, ..., n and finite expectations. Then

E [Xj|S > V aRq[S]] = E [Xj]F S−Xj+X∗

j(V aRq[S])

F S (V aRq[S]), (8.2)

where X∗j is the size-biased rv associated with Xj (cf., e.g., Patil & Rao, 1978).

Lemma 8.2. Let X v Tw1(θ, λ) be a Poisson rv. Then the size-biased rv associated with

it, is also Poisson with a translated to the right support.

23

Proof. The proof relies on the following ddf of X∗

FX∗ (x) =E [X1(X > x)]

E [X]=

1

λeθ

∞∑t=x+1

tλt

t!exp

(θt− λeθ

)

=∞∑

l=x

λl

l!exp

(θl − λeθ

)= FX (x− 1) , x ≥ 1,

where 1(A) is the indicator function of the set A.

Then X∗ d= X + 1, which completes the proof. (Here, d stands for the ‘equality in

distribution’). ¤

The next theorem derives risk measure (1.2) for S. We note that in what follows S, S + 1

and S + n are all Poisson rv’s having differently translated supports.

Theorem 8.1. Let X v Twn,1(θ, λ) denote a multivariate Poisson random vector and S be

the sum of its univariate marginal elements. The tail conditional expectation risk measure

for S is then

TCEq [S] = nλ0eθ F S+n (V aRq[S])

F S (V aRq[S])+

n∑j=1

λjeθ F S+1 (V aRq[S])

F S (V aRq[S]). (8.3)

Proof. First, we note that in a similar fashion to the proof of Lemma 8.2, it can be shown

that (nY0)∗ d

= nY0 + n. Then applying Lemma 8.1, we obtain that

E [nY0|S > V aRq[S]] = E [nY0]F S+n (V aRq[S])

F S (V aRq[S])= nλ0e

θ F S+n (V aRq[S])

F S (V aRq[S])

and

E

[n∑

j=1

Yj|S > V aRq[S]

]=

n∑j=1

E [Yj|S > V aRq[S]] =n∑

j=1

λjeθ F S+1 (V aRq[S])

F S (V aRq[S]),

which completes the proof. ¤

Corollary 8.1. Suppose λ0 → 0 and therefore Y0a.s.→ 0. In such a limit case, the random

vector X becomes a vector of n independent univariate Poisson rv’s, and the formula for

TCE of S simplifies to

TCEq [S] =n∑

j=1

λjeθ F S+1 (sq)

F S (sq). (8.4)

It should be also emphasized that, since for discrete rv’s the following relation holds

P (S = s) = P (S > s− 1)− P (S > s),

24

we can reformulate equation (8.4) as following

TCEq [S] =n∑

j=1

λjeθ

(1 +

fS(V aRq[S])

F S (V aRq[S])

)

= E[S] + h ·Var[S], (8.5)

where h is the hazard function. Last expression is equivalent to the farmulas for the TCE

risk measure of a normally distributed rv (cf., e.g., Panjer, 2002; Landsman & Valdez, 2003).

In the light of this, Poisson rv’s can be seen as discrete counterparts of normal ones.

We must outline here, that form (8.5) does not always hold. For instance, Furman &

Landsman (2005) showed that, for S v Tw2(θ, λ),

TCEq[S] = E[S] + h · V aRq[S].

8.1. The TCE based allocation rule. In addition to three basic pillars enumerated at the

beginning of this section, a significant number of financial institutions have recently adopted

the so-called risk capital framework. According to Zaik et al. (1996), two central elements

of such framework are

1. Assessing the risk capital and holding sufficient amount of capital to cover risks, and

2. Allocating the risk capital to each operating division or department.

In Theorem 8.1 we addressed point 1. In the next theorem we treat the second point,

and we show that the contribution of each univariate marginal risk to the shortfall in the

dependent multivariate Poisson case is stipulated by its mathematical expectation.

Theorem 8.2. The contribution of Xj to the shortfall, in the case of X v Twn,1(θ, λ) is

TCEq [Xj|S] = λ0eθ F S+n (V aRq[S])

F S (V aRq[S])+ λje

θ F S+1 (V aRq[S])

F S (V aRq[S]). (8.6)

Proof. The proof follows from Lemmas 8.1 and 8.2. ¤

We are often interested in the relative contribution of riskiness of Xj to S, which, in the

dependent multivariate Poisson case, is formulated as

TCEq [Xj|S]

TCEq [S]=

λ0F S+n (V aRq[S]) + λjF S+1 (V aRq[S])

nλ0F S+n (V aRq[S]) +∑n

j=1 λjF S+1 (V aRq[S]).

25

Then, assuming that nor F S+n (V aRq[S]) neither F S+1 (V aRq[S]) are zeros, the right-hand

side of the last equation can be rewritten in the following way

TCEq [Xj|S]

TCEq [S]=

λj∑nj=1 λj

+1

F S+1(V aRq [S])

F S+n(V aRq [S])

∑nj=1 λj + nλ0

(λ0 − nλ0

λj∑nj=1 λj

),

which, in the limit case λ0 → 0, reduces to

TCEq [Xj|S]

TCEq [S]=

λj∑nj=1 λj

=E[Xj]

E[S]. (8.7)

It therefore turns out that, although in the dependent multivariate Poisson case the rel-

ative contribution of riskiness of Xj to S is quite lengthy, it is surprisingly simple in the

independent case. We note in this regard that although X1, . . . , Xn are independent, the

pair (Xj, S), j = 1, . . . , n is certainly dependent, and thus equation (8.7) can be of some

practical importance.

9. Conclusions

In this paper we have thoroughly studied the multivariate Tweedie family of distributions,

which we consider a multivariate extension of the well-known exponential dispersion Tweedie

models. To formulate MTwF, we have utilized the multivariate reduction method, which for

this purpose has been formulated in a quite general form.

As its very name implies, the multivariate probabilistic model discussed herein possesses

univariate marginal distributions which correspond to the univariate Tweedie ones. More-

over, the dependence structure of MTwF is reflected in its variance/covariance structure

and allows for quite efficient modeling of multivariate portfolios of dependent insurance

losses. Except for the multivariate normal distributions, members of MTwF possess non-

negative supports, positive skewness and some of them are relatively tolerant to large risks,

which responds well to the peculiarities of the insurance industry demands. Indeed, on the

one hand, MTwF contains as special cases the multivariate Poisson (p = 1), multivariate

gamma (p = 2), multivariate inverse Gaussian (p = 3) and multivariate compound Poisson

(1 < p < 2) distributions, and on the other, it introduces a rich variety of other multivari-

ate models corresponding to non-integer values of p outside the (0, 1) interval. Moreover,

MTwF provides a general framework to deal with such well-known existing multivariate

distributions as the multivariate inverse Gaussian distribution of Chhikara & Folks (1989),

26

multivariate gamma of Mathai & Moschopoulos (1991) and the bivariate and multivariate

Poisson distributions of Vernic (1997, 2000).

We have studied various important properties of MTwF. Namely, we have derived the

multivariate pdf and mgf of its general member, calculated higher order moments and their

products, constructed some useful Chebyshev’s type inequalities and produced related as-

ymptotic results.

Last but not least, we have illustrated a possible application of MTwF to risk management.

More specifically, we have developed analytic expressions for the tail conditional expectation

risk measure and the risk capital allocation based on it in the framework of the dependent

Poisson random vectors.

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