The discrete-time risk model with correlated classes of business
Hélène Cossette, Etienne Marceau
∗École d’Actuariat, Pavillon Alexandre-Vachon (Local 1620), Université Laval, Sainte-Foy, Québec, Canada, G1K 7P4
Received 1 February 1999; received in revised form 1 October 1999; accepted 24 November 1999
Abstract
The discrete-time risk model with correlated classes of business is examined. Two different relations of dependence are considered. The impact of the dependence relation on the finite-time ruin probabilities and on the adjustment coefficient is also studied. Numerical examples are presented. © 2000 Elsevier Science B.V. All rights reserved.
MSC:IM11; IM12; IM13; IM30
Keywords:Correlated aggregate claims; Shock models; Ruin probability; Adjustment coefficient
1. Introduction
In most actuarial literature related to risk theory, the assumption of independence between classes of business in an insurance book of business is made. In practice, however, there are situations in which this assumption is not verified. In the case of a catastrophe such as an earthquake for example, the damages covered by homeowners and private passenger automobile insurance cannot be considered independent. Papers that treat of a relation of dependence between classes of business include Ambagaspitiya (1998), Cummins and Wiltbank (1983) and Wang (1998).
In the present paper, we study the probability of ruin in the discrete-time risk model proposed by Bühlmann (1970) and also presented in Bowers et al. (1997), Klugman et al. (1998) and Rolski et al. (1999). We first give a brief description of the discrete-time model and we define the probability of ruin overfiniteandinfinite-time within this model. Then, we use a Poisson common shock model and a negative binomial component model, proposed by Wang (1998), to introduce a relation of dependence between classes of business. We present numerical examples to illustrate the impact of the introduction of a relation of dependence on the probability of ruin. We also examine its influence on the adjustment coefficient.
∗Corresponding author. Tel.:+1-418-656-3639; fax:+1-418-656-7790.
E-mail addresses:[email protected] (H. Cossette), [email protected] (E. Marceau) 0167-6687/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.
2. The discrete time model
Assume the discrete-time process{Un, n=0,1,2, . . .}whereUnis the surplus for a book of business of an
insurer at timen (n=0,1,2, . . . )which is defined as
Un =u+cn−Sn, (1)
whereuis the initial surplus,cthe premium income received during each period andSn the total claim amounts
over the firstnperiods. It is also assumed that
Sn=W1+W2+ · · · +Wn, (2)
whereWi represents the total claim amounts for the book of business in the periodiand{Wi, i = 1,2, . . .}is
a sequence of independent and identically distributed random variables withE[Wi] =µW < c. The probability
distribution and density function ofWi (i=1,2, . . . )are denoted byFW(w)andfW(w), respectively.
Given (2), we can rewrite (1) as follows:
Un =u+(c−W1)+(c−W2)+ · · · +(c−Wn). (3)
LetT be the time of ruin defined as
T =inf(n, Un<0)
assuming thatT = ∞ifUn≥0 for alln=1,2, . . . .
Letψ(u,1, n)be the finite-time ruin probability over the periods 1 ton
ψ(u,1, n)=P (T ≤n).
Whenn→ ∞inψ(u,1, n), we have
ψ(u)=P (T <∞)
which is the infinite-time ruin probability. Also,
ϕ(u)=1−ψ(u), ϕ(u,1, n)=1−ψ(u,1, n)
are, respectively, the infinite-and finite-time horizon non-ruin probabilities. Given (3), we have
ϕ(u,1, n)=P (U1≥0, U2≥0, . . . , Un ≥0)
=P (W1≤u+c, W1+W2≤u+2c, . . . , W1+W2+ · · · +Wn≤u+cn).
The analytical expression ofϕ(u,1, n)is given in the following theorem.
Theorem 1. Let{Wi, i =1,2, . . .}be a sequence of i.i.d. random variables and c the annual premium income
constant over each period. Then,
ϕ(u,1, n)=
Z u+c
0
ϕ(u+c−w,1, n−1)dFW(w). (4)
Proof. By the theorem of total probabilities, we have
ϕ(u,1, n)=
Z u+c
0
ϕ(u+c−w,2, n)dFW(w),
where
SinceW1, . . . , Wnare i.i.d., we have
ϕ(y,2, n)=ϕ(y,1, n−1)
from which the result follows.
The calculation of exact values ofϕ(u,1, n) by the direct application of (4) is rarely possible. Therefore, an algorithm which approximatesϕ(u,1, n)is presented in Theorem 2. The application of the algorithm requires the discretization of the distribution functionFW. Methods of discretization are proposed in Klugman et al. (1998) and
Panjer and Willmot (1992). Assume thatFW˜ is the discrete distribution function obtained by one of these methods
andW˜ the corresponding discrete random variable. IfP (W˜ =k)=fk (k=0,1, . . . , M), thenFW˜(k)=P (W˜ ≤
k)=Pk
j=0fj wherefkare the mass probabilities.
We suppose that the premium income is a constant integerpand that the surplus process takes only integer values. We denote byψk,1,nandϕk,1,nthe finite-time ruin and non-ruin probabilities calculated withFW˜ over the periods
1 tonwith an initial surplusk(an integer).
Theorem 2. Let k, p, j be integers. Then,
ϕk,1,n=
min(k+p,M)
X
j=0
ϕk+p−j,1,n−1fj (n=2,3, . . . ), (5)
where
ϕk,1,1=Fmin(k+p,M)=
min(k+p,M)
X
j=0
fj. (6)
Proof. The result follows from Theorem 1.
The non-ruin probability can therefore be recursively calculated by first applying (6) and then (5). This method is used in De Vylder and Goovaerts (1988) and Dickson and Waters (1991).
In the application of the algorithm, the support ofFWis discretized by intervals of lengthr. Setting the parameters
kandpsuch that
k≤ u
r < k+1, p≤ c
r < p+1,
the following inequalities are obtained
ϕk,1,n(p)≤ϕ(u,1, n) < ϕk+1,1,n(p+1).
Therefore, we have
ϕ(u,1, n)≃ 12{ϕk,1,n(p)+ϕk+1,1,n(p+1)}.
In Section 5, special attention is given to the infinite-time ruin probabilityψ(u).
3. Aggregation of dependent classes of business
3.1. Introduction
of these two models is similar to the presentation given in Wang (1998), which proposed different methods of introducing a relation of dependence between risks in the context of individual and collective risk theory. See Ambagaspitiya (1998) for a general method of constructing a vector of dependent random variables from a vector of independent random variables.
In both models, it is assumed that the book of business of the insurer is constituted ofmdependentclasses of business and that the total claim amounts for the book of business in periodiis given by
Wi =Wi,1+Wi,2+ · · · +Wi,m (i=1,2, . . . ),
whereWi,j represents the total claim amounts for thejth class of business in the periodi. Fori6=i′, WiandWi′
are supposed independent and identically distributed. We denote byFW(w)the common probability distribution
function of the random variablesWi (i=1,2, . . . )and letWbe a random variable with this probability distribution
function. For a fixed periodi (i=1,2, . . . ), we assume the different classes of business to be dependent. For the class of businessj (j =1, . . . , m)in the periodi (i=1,2, . . . )we denote byXi,j,kthekth individual
claim and byNi,j the number of claims. Then,
Wi,j =
variable with their common distribution function. Similarly for the random variablesWi,j (i=1,2, . . . ), they are
supposed identically distributed. We denote byW(j )a random variable with their common distribution function. We make the usual assumption thatN(j )andX(j )are independent.
For the class of businessj and for any periodi(i=1,2, . . . ), the premium income is
cj =E[W(j )](1+ηj)=µ(j )E[N(j )](1+ηj) (j =1, . . . , m),
whereηj is the positive risk margin for thejth class of business. The premium income for the book of business in
the periodi (i=1,2, . . . )isc=c1+ · · · +cm.
3.2. Preliminary results
Let us denote by PX(t ), MX(t ), φX(t ), respectively, the probability generating function (pgf), the moment
generating function (mgf) and the characteristic function (chf) of a random variableX.
For a set of random variables(X1, . . . , Xk), the joint probability generating function, the joint moment generating
function and the joint characteristic function are defined as follows:
which is easily demonstrated as follows:
PS(t ) =EtS=EtX1+···+Xk=EtX1. . . tXk=PX1,... ,Xk(t, . . . , t ),
MS(t ) =Eet S=Eet (X1+···+Xk)=Eet X1. . .et Xk=MX1,... ,Xk(t, . . . , t ),
φS(t ) =E
eit S
=Eeit (X1+···+Xk)=Eeit X1. . .eit Xk=φ
X1,... ,Xk(t, . . . , t ).
The probability distribution function of S can be obtained by inverting thechf φS(t ) with the Fast Fourier
Transform (FFT) method. Details on the application of the FFT method in an actuarial context are given in the papers of Bühlmann, (1984), Heckman and Meyers (1983) and Robertson (1992). This method provides an approximation of the probability distribution functionFW ofW which results in the discrete probability distribution functionFW˜
mentioned in Section 2. The resultingFW˜ is used in the algorithm given in (5) and (6) for the estimation of the
non-ruin probabilityϕ(u,1, n).
LetS=X1+ · · · +XNbe a random sum ofNindependent and identically distributed random variables. Then,
φS(t )=PN(φX(t )). (9)
This result, which will prove useful in the following sections, is derived as
φS(t )=E[eit S]=E[E(eit (X1+···+XN)|N]=E
h
(φX(t ))N
i
and can be extended to multivariate random variables. LetS=S(1)+ · · · +S(m) (j =1,2, . . . , m)andS(j )be a random sum ofN(j )independent and identically distributed random variablesXk(j ) (k=1,2, . . . )
S(j )=
N(j )
X
k=1
Xk(j ).
Assume thatN(j ) (j=1,2, . . . , m)are dependent. Then,
φS(1),... ,S(m)(t, . . . , t )=PN(1),... ,N(m)(φX(1)(t1), . . . , φX(m)(tm)), (10)
φS(t )=φS(1),... ,S(m)(t, . . . , t ). (11)
3.3. Poisson model with common shock
The common shock model is presented in Marshall and Olkin (1967, 1988) and in Kocherlakota and Kocherlakota (1992). In Hesselager (1996), bivariate counting distributions and their corresponding compound distributions are considered in a common shock setting.
We consider the case of a book of business divided in three (m = 3)dependent classes of business. The generalization to any numbermof dependent classes of business is easily obtained. It is assumed that a common shock affects the three classes of business and that another common shock has an impact on each couple of classes. Given the assumption made in Section 4 of identical distribution of the random variablesNi,1, Ni,2, Ni,3for any fixed periodi=1,2, . . ., we defineN(j )(j =1,2,3)as follows:
N(1) =N(11)+N(12)+N(13)+N(123), N(2) =N(22)+N(12)+N(23)+N(123), N(3) =N(33)+N(13)+N(23)+N(123),
where
Since the distribution of the sum ofnindependent Poisson random variablesXiwith parameterλi is Poisson with
correlated compound Poisson random variables, has a compound Poisson distribution, as in the independent case presented in Appendix A, but with different parameterλand different claimsize characteristic functionφX(t )to
3.4. Negative Binomial model with common component
Modeling the number of claimsNby a Poisson random variable means that the variance Var[N] is equal to the expectationE[N]. In practice, it may occur that Var[N]> E[N] (see Panjer and Willmot, 1992). The Negative Binomial is often used to model claim numbers in such situations. The probability function of a Negative Binomial random variableNis
The construction presented in the previous section is adapted to a Negative Binomial model. Again, we consider the special case of a book of business subdivided in three dependent classes of business. It is assumed that the number of claims in thejth(j =1,2,3)class of business is the sum of two random variables. The first random variable, denoted byN(jj ),is specific to each class and is independent of the specific random variables of the other classes. The second random variable is denoted byN(j0) (j =1,2,3). A dependence relation is assumed between the second random variables of the different classes. Fori=1,2, . . ., the random variablesN(j ) (j =1,2,3)are defined by
N(j )=N(jj )+N(j0), (15)
where
N(jj )∼NB(αjj, βj) (j =1,2,3), N(j0)∼NB(α0, βj) (j =1,2,3).
The distribution of the sum ofnindependent Negative Binomial random variablesYi with parameters(αi, β)is
Negative Binomial with parameters(Pn
i=1αi, β). Hence, we have
N(j )∼NB(αj, βj) (j =1,2,3),
whereαj =αjj+α0. For a fixedi, we assume that the random variablesN(jj ) (j =1,2,3)are independent. The random variablesN(j0) (j=1,2,3)are dependent and they are modeled by a common Poisson–Gamma mixture with
(1) N(j0)|2=θ∼Poisson(θβj) (j =1,2,3),
(2) 2∼Gamma(α0,1),
(3) N(j0)|2=θ are independent (j =1,2,3).
(16)
The construction (16) of a vector of dependent Negative Binomial random variables by a common mixture may be found in Kocherlakota and Kocherlakota (1992). The representation of the vector(N(1), N(2), N(3))as a sum of two independent vectors as in (15) and where one of the vectors has dependent components modeled by a common Poisson–Gamma mixture is given in Wang (1998). The negative binomial model with common component differs from the common shock Poisson model presented in the previous section in that we assume thatN(j )has a marginal Negative Binomial distribution with parameters(αj, βj).
PN(1),N(2),N(3)(t1, t2, t3)=E
The probability distributionFW ofWis approximated by taking the inverse of (17) using the FFT method. This
procedure is explained in the next section.
3.5. Approximation of FW
Within both models, the non-ruin probability can be approximated by usingFW˜ which is obtained from the
Table 1
Moments ofX(i), N(i)andW(i)
Class (1) Class (2)
E[X(i)] 1.1250 1.1250
E[X(i)2] 2.5313 2.5313
E[N(i)] 4.0000 4.0000
Var[N(i)] 4.0000 4.0000
E[W(i)] 4.5000 4.5000
Var[W(i)] 10.1250 30.375
Table 2
Correlation parameters
ρ(N(1), N(2))=0 ρ(N(1), N(2))=0.25 ρ(N(1), N(2))=0.75
λ0 0 1.0000 3.0000
Cov(N(1), N(2)) 0 1.0000 3.0000
Cov(W(1), W(2)) 0 1.2656 3.7969
ρ(W(1), W(2)) 0 0.0727 0.2165
4. Numerical examples
We study the impact on the probability of ruin of a relation of dependence between two classes of business of an insurance book of business. In a first example, we consider the aggregation of the classes of business via a common shock model and in a second one, the aggregation is made via the Negative Binomial model with common component. We compare the ruin probabilityψ(u,1,20)obtained for different relations of dependence between the classes of business. The differences result from the choice of correlation coefficient betweenN(1)andN(2)which are either 0, 0.25 or 0.75. The case with a correlation coefficient of zero corresponds to the case of independent classes of business.
Example 1(Poisson model with common shock). The characteristics of the two books of business are the following:
Book of business #1 : X(1)∼Weibull(0.5,1.125) N(1)∼Poisson(4)
Book of business #2 : X(2)∼Exponential(2.25) N(2)∼Poisson(4).
The moments ofX(i), N(i), W(i)are summarized in Table 1. Also, the correlation parameters are given in Table 2. The numerical results for the ruin probability are shown in Table 3, where the last number added inψ(u,1,20)
indicates the correlation coefficient betweenN(1)andN(2).
Example 2(Negative Binomial model with common component). In this example, the characteristics of the books
of business are
Book of business #1 : X(1)∼Weibull(0.5,1.125) N(1)∼Negative Binomial(1,4)
Book of business #2 : X(2)∼Exponential(2.25) N(2)∼Negative Binomial(1,4).
Table 3
Ruin probabilitiesψ(u,1,20)in the Poisson model
u ψ(u,1,20,0) ψ(u,1,20,0.25) ψ(u,1,20,0.75) ψ(u,1,20,0.25) ψ(u,1,20,0) −1 (%)
ψ(u,1,20,0.75) ψ(u,1,20,0) −1 (%)
0 0.6317 0.6389 0.6518 1.1 3.2
10 0.3207 0.3343 0.3592 4.2 12.0
20 0.1643 0.1752 0.1961 6.6 19.4
30 0.0836 0.0909 0.1055 8.7 26.2
40 0.0420 0.0465 0.0558 10.7 32.9
50 0.0208 0.0234 0.0290 12.5 39.4
60 0.0102 0.0117 0.0149 14.7 46.1
70 0.0050 0.0057 0.0075 14.0 50.0
80 0.0024 0.0028 0.0037 16.7 54.2
90 0.0011 0.0014 0.0018 27.3 63.6
100 0.0005 0.0006 0.0009 20.0 80.0
110 0.0003 0.0003 0.0004 0.0 33.3
120 0.0001 0.0001 0.0002 0.0 100.0
130 0.0001 0.0001 0.0001 0.0 0.0
140 0.0000 0.0000 0.0000 0.0 0.0
150 0.0000 0.0000 0.0000 0.0 0.0
Table 4
Moments ofX(i), N(i)andW(i)
Class(1) Class (2)
E[X(i)] 1.1250 1.1250
E[X(i)2] 2.5313 2.5313
E[N(i)] 4.0000 4.0000
Var[N(i)] 20.0000 20.0000
E[W(i)] 4.5000 4.5000
Var[W(i)] 28.1250 48.3750
Table 5
Correlation parameters
ρ(N(1), N(2))=0 ρ(N(1), N(2))=0.25 ρ(N(1), N(2))=0.75
α0 0 0.3125 0.9375
Cov(N(1), N(2)) 0 5.0000 15.0000
Cov(W(1), W(2)) 0 6.3281 18.9844
ρ(W(1), W(2)) 0 0.1716 0.5147
We observe that the increase in the ruin probabilityψ(u,1,20)with the introduction of a relation of dependence is more important in the Negative Binomial model than in the Poisson model. We have observed similar results for
ψ(u,1,10)andψ(u,1,30).
In both examples, we observe that the ruin probabilities increase as the coefficient of correlationρ(N(1), N(2))
increases. It is worth mentioning that withX(1)exponentially distributed andX(2)having a Weibull distribution in both the Poisson and Negative Binomial model, the correlation coefficientρ(N(1), N(2))of 0.75 in the Poisson model produces a correlation coefficientρ(W(1), W(2))of 0.2165 while it leads to a correlation coefficientρ(W(1), W(2))
Table 6
Ruin probabilitiesψ(u,1,20)in negative binomial model
u ψ(u,1,20,0) ψ(u,1,20,0.25) ψ(u,1,20,0.75) ψ(u,1,20,0.25) ψ(u,1,20,0) −1 (%)
ψ(u,1,20,0.75) ψ(u,1,20,0) −1 (%)
0 0.6867 0.6933 0.6972 1.0 1.5
10 0.4599 0.4971 0.5133 8.1 11.6
20 0.3025 0.3530 0.3743 16.7 23.7
30 0.1956 0.2474 0.2696 26.5 37.8
40 0.1243 0.1711 0.1916 37.7 54.1
50 0.0776 0.1167 0.1345 50.4 73.3
60 0.0476 0.0785 0.0932 64.9 95.8
70 0.0287 0.0522 0.0638 81.9 122.3
80 0.0171 0.0342 0.0432 100.0 152.6
90 0.0100 0.0222 0.0289 122.0 189.0
100 0.0058 0.0142 0.0191 144.8 229.3
110 0.0033 0.0090 0.0126 172.7 281.8
120 0.0019 0.0057 0.0081 200.0 326.3
130 0.0010 0.0035 0.0052 250.0 420.0
140 0.0006 0.0022 0.0033 266.7 450.0
150 0.0003 0.0013 0.0021 333.3 600.0
5. Dependence and the adjustment coefficient
The purpose of this section is to measure the impact of the dependence on the infinite-time ruin probability through the adjustment coefficient.
5.1. Adjustment coefficient and infinite-time ruin probability
The infinite-time ruin probabilityψ(u)depends on the probability distribution ofW. We consider the probability of ruinψ(u)in the case where it is assumed that themgf ofW, MW(r) = E[erW], exists. We define R as the
adjustment coefficientwhich is the strictly positive solution of the equation
MW(r)=ecr. (18)
It is possible to write the ruin probabilityψ(u)for a book of business as a function of the adjustment coefficientR.
Theorem 3. IfMW(r)exists and foru >0,we have
ψ(u)= exp(−Ru)
E[exp(−RUT)|T <∞]
, (19)
where R is the adjustment coefficient and T the time of ruin.
Proof. The proof of this result is given in Gerber (1979) or Bowers et al. (1997).
WhenT < ∞, it is clear thatUT < 0. The denominator in (19) is thus greater than one which leads to the
following well known inequality:
ψ(u) <e−Ru.
Given the adjustment coefficientR, we have an upper bound for the probability of ruin which is useful, for example, in evaluating reinsurance agreements. Usually, the adjustment coefficientR is considered as a crude measure of dangerousness of the risk processUnrepresenting the surplus of the book of business. The smaller is the adjustment
coefficientR, the more dangerous is the processUn. Our objective is to examine the impact onR of assuming a
5.2. Exponential ordering and adjustment coefficient
According to Goovaerts et al. (1990), the riskX precedes the risk Y in the exponential order if, for allr >
0, MX(r)≤ MY(r). The exponential order is denoted byX <e Y. In the following theorem, the ordering of the adjustment coefficient is demonstrated on the basis of the exponential order.
Let Wa andWb be the total claimsize random variables of two distinct books of business. We assume that
E[Wa] =E[Wb]=E[W] and thatca =cb =c=(1+θ )E[W], whereθ >0. The corresponding adjustment coefficientsRa andRb are solutions to MWa(r) = h(r) andMWb(r) = h(r), whereh(r) = e
cr. We have the
following result.
Theorem 4. IfWa<eWb,thenRa> Rb.
Proof. The adjustment coefficients Ra andRb are solutions toMWa(r) = ecr andMWb(r) = ecr, respectively.
Since Wa <e Wb, we have MWa(r) ≤ MWb(r) for all r ≥ 0. Also, MWa(0) = MWb(0) = h(0) = 1 and
M′Wa(0)=M
′
Wb(0)=E[W]< h
′(0)=cby assumption. It follows thath(r)will meetM
Wb(r)beforeMWa(r)
and thereforeRa > Rb.
5.3. Poisson model with common shock and adjustment coefficient
We compare two books of business. The first book of business is the combination of two(m = 2)classes of business assumeddependentwhile the second one is the aggregation of the same two classes of business but assumed independent. The following results can be generalized to any finite number of classes of business.
LetWD andWI represent the total claim amounts for the dependent and the independent books of business, respectively. The random variable WD is defined within the Poisson model with common shock. The random variableWIis defined within the classical Poisson model assuming independence between the classes of business (see Appendix A). The marginal distribution of total claims for the class of businessj (j =1,2)in the dependent case is identical to the marginal distribution of the total claims for thejth class of business in the independent case. The number of claims in thejth class of businessN(j ) (j =1,2)is a Poisson random variable with parameterλj
in both cases. In the particular case ofWD, the parametersλj are the sum of two parameters
λj =λjj+λ12, j =1,2,
sinceN(j )=N(jj )+N(12). ThemgfMWD(r)andMWI(r)are, respectively,
MWD(r)=exp(λ11MX(1)(r)+λ22MX(2)(r)+λ12MX(1)(r)MX(2)(r)−λ11−λ22−λ12), withλ11 =λ1−λ12, λ22 =λ2−λ12and
MWI(r)=exp(λ1MX(1)(r)+λ2MX(2)(r)−λ1−λ2).
Corollary 5. DefineRDandRIas the adjustment coefficients which are solutions toMWD(r)=ecrandMWI(r)=
ecr,respectively.
ThenRD< RI.
Proof. The inequalityMWD(r)≥MWI(r), which must be verified forr ≥0, is equivalent to
λ11MX(1)(r)+λ22MX(2)(r)+λ12MX(1)(r)MX(2)(r)−λ11−λ22−λ12 ≥λ1MX(1)(r)+λ2MX(2)(r)−λ1−λ2 and also to
λ11MX(1)(r)+λ22MX(2)(r)+λ12MX(1)(r)MX(2)(r)−λ11−λ22−λ12≥(λ11+λ12)MX(1)(r)
The inequality (20) reduces to
MX(1)(r)+MX(2)(r)≤MX(1)(r)MX(2)(r)+1
which is true forr≥0. The result follows from Theorem 4.
From Corollary 5, we can say that the risk process for the book of business under the dependence assumption is more dangerous than the book of business under the independence assumption.
We now compare two books of business that are both modelized with the Poisson model with common shock. They differ however in the strength of the dependence relation between the classes of business. The marginal distributions ofN(1)andN(2)are identical but the common shock parameterλ12 in their marginal distribution is different. We have the following result.
Corollary 6. DefineRDandRD′as the adjustment coefficients which are solution toMWD(r)=ecrandMWD′(r)=
ecr.
Ifλ12> λ′12≥0,thenRD< RD′.
Proof. The inequalityMWD(r)≥MWD′(r)must be verified forr≥0. This is equivalent to verifying
(λ1−λ12)MX(1)(r)+(λ2−λ12)MX(2)(r)+λ12MX(1)(r)MX(2)(r)−λ1−λ2+λ12≥(λ1−λ′12)MX(1)(r) +(λ2−λ′12)MX(2)(r)+λ′12MX(1)(r)MX(2)(r)−λ1−λ2+λ′12.
Letλ12=λ′12+γ withγ >0. Then, it follows that
γ MX(1)(r)+γ MX(2)(r)≤γ MX(1)(r)MX(2)(r)+γ
which is true forr≥0. Again, the result follows from Theorem 4. Both Corollaries 5 and 6 can be generalized to any finite number of books of business.
5.4. Negative Binomial with common component
As in the previous section, we consider two books of business where the first book of business is the combination of two dependent classes of business and the second one is the combination of two independent classes of business. In the present section, we define the random variable WD within the Negative Binomial model with common component and the random variable WI within the classical Negative Binomial model assuming independence between the classes of business (see Appendix A). ThemgfMWD(r)is
MWD(t )= 2
Y
j=1
[1−βj(MX(j )(t )−1)]−αjj
1−
2
X
j=1
βj(MX(j )(tj)−1)
−α0
,
whereαjj =αj−α0 (j =1,2), and themgfMWI(r)is
MWI(r)= 2
Y
j=1
[1−βj(MX(j )(t )−1)]−αjj−α0.
Corollary 7. DefineRDandRIas the adjustment coefficients which are solution toMWD(r)=ecrandMWI(r)=
ecr.
ThenRD< RI.
Proof. We must verify thatMWD(r)≥MWI(r), forr ≥0. This is equivalent to verifying
and then
[1−β1(MX(1)(t1)−1)−β2(MX(2)(t2)−1)]≤[1−β1(MX(1)(t )−1)] [1−β2(MX(2)(t )−1)]. This leads to
[β1(MX(1)(t )−1)] [β2(MX(2)(t )−1)]≥0,
which is true forr≥0. By Theorem 4, we obtain the desired result. As in the Poisson model, we can say that the risk process for the book of business under the dependence assumption is more dangerous than the one for the book of business under the independence assumption.
Again, we compare two books of business which differ in the strength of the dependence relation between the classes of business but here in the case of the Negative Binomial model with common component. The marginal distributions ofN(1)andN(2) are identical but the common shock parameterα0 in their marginal distribution is different.
Corollary 8. Define RD and R′D as the adjustment coefficients which are solutions to MWD(r) = ecr and
MWD′(r)=ecr.
Ifα0> α′0≥0,thenRD< RD′.
Proof. The inequalityMWD(r)≥MWD′(r)must be verified forr≥0. This is equivalent to verifying that
[1−β1(MX(1)(t1)−1)−β2(MX(2)(t2)−1)]−α0[1−β1(MX(1)(t )−1)]α0[1−β2(MX(2)(t )−1)]α0 ≥[1−β1(MX(1)(t1)−1)−β2(MX(2)(t2)−1)]−α
′
0[1−β
1(MX(1)(t )−1)]α
′
0[1−β
2(MX(2)(t )−1)]α
′
0.
Letα0=α′0+γ withγ >0. It follows that
[1−β1(MX(1)(t1)−1)−β2(MX(2)(t2)−1)]−γ[1−β1(MX(1)(t )−1)]α0[1−β2(MX(2)(t )−1)]γ ≥1 then
[1−β1(MX(1)(t1)−1)−β2(MX(2)(t2)−1)]≤[1−β1(MX(1)(t )−1)] [1−β2(MX(2)(t )−1)]. This leads to
[β1(MX(1)(t )−1)] [β2(MX(2)(t )−1)]≥0,
which is true forr≥0. By Theorem 4,RD< RD′.
Acknowledgements
This research was funded by individual operating grants from the Natural Sciences and Engineering Research Council of Canada and by a joint grant from the Chaire en Assurance L’Industrielle-Alliance (Université Laval).
Appendix A
A.1. Poisson model
Consider three classes of business with no common shock between them. The definitions ofW(j ) (j =1,2,3)
andX(j )remain unchanged. We have
N(j )∼Poisson(λj) (j =1,2,3), random variables, has a compound Poisson distribution with parameterλand claimsize characteristic functionφX(t )
to which is associated the following distribution function:
By (10), the jointchfof(W(1), W(2), W(3))is
φW(1),W(2),W(3)(t1, t2, t3)=PN(1),N(2),N(3)(φX(1)(t1), φX(2)(t2), φX(3)(t3))= 3
Y
j=1
[1−βj(φX(j )(tj)−1)]−αjj.
Appendix B. Descriptions of some distributions
• Exponential:
F (x)=1−e−(x/β), x >0.
MX(t )=(1−βt ), β >0.
• Weibull:
F (x)=1−e−(x/β)τ, x >0.
E[Xθ]=βθŴ
1+θ
τ
, τ >0 β >0.
• Poisson:
P (N =n)=e−λ(λ)
n
n! , n=0,1,2, . . . .
PN(t )=eλ(t−1), λ >0.
• Negative Binomial:
P (N =n)=
α+n−1
α−1
1 1+β
α β
1+β
n
, n=0,1,2, . . . .
PN(t )=[1−β(t−1)]−α, α, β >0.
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