H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149 135
Since W
1
, . . . , W
n
are i.i.d., we have ϕy,
2, n = ϕy, 1, n − 1 from which the result follows.
The calculation of exact values of ϕu, 1, n by the direct application of 4 is rarely possible. Therefore, an algorithm which approximates ϕu, 1, n is presented in Theorem 2. The application of the algorithm requires the
discretization of the distribution function F
W
. Methods of discretization are proposed in Klugman et al. 1998 and Panjer and Willmot 1992. Assume that F
˜ W
is the discrete distribution function obtained by one of these methods and ˜
W the corresponding discrete random variable. If P ˜
W = k = f
k
k = 0, 1, . . . , M, then F
˜ W
k = P ˜ W ≤
k = P
k j =
f
j
where f
k
are the mass probabilities. We suppose that the premium income is a constant integer p and that the surplus process takes only integer values.
We denote by ψ
k, 1,n
and ϕ
k, 1,n
the finite-time ruin and non-ruin probabilities calculated with F
˜ W
over the periods 1 to n with an initial surplus k an integer.
Theorem 2.
Let k, p, j be integers. Then, ϕ
k, 1,n
=
mink+p,M
X
j =
ϕ
k+p−j, 1,n−1
f
j
n = 2, 3, . . . ,
5 where
ϕ
k, 1,1
= F
mink+p,M
=
mink+p,M
X
j =
f
j
. 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 of F
W
is discretized by intervals of length r. Setting the parameters k
and p such that k ≤
u r
k + 1,
p ≤ c
r p +
1, the following inequalities are obtained
ϕ
k, 1,n
p ≤ ϕu, 1, n ϕ
k+ 1,1,n
p + 1.
Therefore, we have ϕu,
1, n ≃
1 2
{ϕ
k, 1,n
p + ϕ
k+ 1,1,n
p + 1}.
In Section 5, special attention is given to the infinite-time ruin probability ψu.
3. Aggregation of dependent classes of business
3.1. Introduction We present a Poisson model with common shock and a Negative Binomial model with common component and
then study the impact of the relation of dependence in each model on the probability of ruin. These models are based on methods of construction of dependence relations suggested in Marshall and Olkin 1967,1988. Our description
136 H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149
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 of m dependent classes of business and that the total claim amounts for the book of business in period i is given by
W
i
= W
i, 1
+ W
i, 2
+ · · · + W
i,m
i = 1, 2, . . . ,
where W
i,j
represents the total claim amounts for the j th class of business in the period i. For i 6= i
′
, W
i
and W
i
′
are supposed independent and identically distributed. We denote by F
W
w the common probability distribution
function of the random variables W
i
i = 1, 2, . . . and let W be a random variable with this probability distribution
function. For a fixed period i i = 1, 2, . . . , we assume the different classes of business to be dependent. For the class of business j j = 1, . . . , m in the period i i = 1, 2, . . . we denote by X
i,j,k
the kth individual claim and by N
i,j
the number of claims. Then, W
i,j
=
N
i,j
X
k= 1
X
i,j,k
. For j fixed, F
X
j
x , with F
X
j
0 = 0, denotes the common distribution function of the independent and identically distributed random variables X
i,j,k
i = 1, 2, . . . ; k = 1, 2, . . . , N
i,j
. Let X
j
be a random variable with this distribution function. The nth moment of F
X
j
is denoted by µ
j n
with µ
j 1
= µ
j
. For j fixed, N
i,j
i = 1, 2, . . . are identically distributed random variables. We denote by N
j
a random variable with their common distribution function. Similarly for the random variables W
i,j
i = 1, 2, . . . , they are
supposed identically distributed. We denote by W
j
a random variable with their common distribution function. We make the usual assumption that N
j
and X
j
are independent. For the class of business j and for any period ii = 1, 2, . . . , the premium income is
c
j
= E [W
j
]1 + η
j
= µ
j
E [N
j
]1 + η
j
j = 1, . . . , m,
where η
j
is the positive risk margin for the j th class of business. The premium income for the book of business in the period i i = 1, 2, . . . is c = c
1
+ · · · + c
m
. 3.2. Preliminary results
Let us denote by P
X
t , M
X
t , φ
X
t , respectively, the probability generating function pgf, the moment
generating function mgf and the characteristic function chf of a random variable X. For a set of random variables X
1
, . . . , X
k
, the joint probability generating function, the joint moment generating function and the joint characteristic function are defined as follows:
P
X
1
,... ,X
k
t
1
, . . . , t
k
= E h
t
X
1
1
. . . t
X
k
k
i M
X
1
,... ,X
k
t
1
, . . . , t
k
= E e
t
1
X
1
+···+t
k
X
k
= P
X
1
,... ,X
k
e
t
1
, . . . , e
t
k
φ
X
1
,... ,X
k
t
1
, . . . , t
k
= E e
it
1
X
1
+···+t
k
X
k
= P
X
1
,... ,X
k
e
it
1
, . . . , e
it
k
. 7
For any k correlated random variables X
1
, . . . , X
k
with joint pgf P
X
1
,... ,X
k
, joint mgf M
X
1
,... ,X
k
and joint chf φ
X
1
,... ,X
k
, the pgf, mgf and chf of the sum S = X
1
+ · · · + X
k
are, respectively, P
S
t = P
X
1
,... ,X
k
t, . . . , t , M
S
t = M
X
1
,... ,X
k
t, . . . , t , φ
S
t = φ
X
1
,... ,X
k
t, . . . , t , 8
H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149 137
which is easily demonstrated as follows: P
S
t = E
t
S
= E t
X
1
+···+X
k
= E t
X
1
. . . t
X
k
= P
X
1
,... ,X
k
t, . . . , t , M
S
t = E
e
t S
= E e
t X
1
+···+X
k
= E e
t X
1
. . . e
t X
k
= M
X
1
,... ,X
k
t, . . . , t , φ
S
t = E
e
itS
= E e
it X
1
+···+X
k
= E e
itX
1
. . . e
itX
k
= φ
X
1
,... ,X
k
t, . . . , t . The probability distribution function of S can be obtained by inverting the chf φ
S
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 function F
W
of W which results in the discrete probability distribution function F
˜ W
mentioned in Section 2. The resulting F
˜ W
is used in the algorithm given in 5 and 6 for the estimation of the non-ruin probability ϕu, 1, n.
Let S = X
1
+ · · · + X
N
be a random sum of N independent and identically distributed random variables. Then, φ
S
t = P
N
φ
X
t . 9
This result, which will prove useful in the following sections, is derived as φ
S
t = E [e
itS
] = E[Ee
it X
1
+···+X
N
|N ] = E
h φ
X
t
N
i and can be extended to multivariate random variables. Let S = S
1
+ · · · + S
m
j = 1, 2, . . . , m and S
j
be a random sum of N
j
independent and identically distributed random variables X
j k
k = 1, 2, . . .
S
j
=
N
j
X
k= 1
X
j k
. Assume that N
j
j = 1, 2, . . . , m are dependent. Then,
φ
S
1
,... ,S
m
t, . . . , t = P
N
1
,... ,N
m
φ
X
1
t
1
, . . . , φ
X
m
t
m
, 10
φ
S
t = φ
S
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 number m of 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 variables N
i, 1
, N
i, 2
, N
i, 3
for any fixed period i = 1, 2, . . . , we define N
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
N
uv
∼ Poissonλ
uv
u, v = 1, 2, 3,
N
123
∼ Poissonλ
123
.
138 H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149
Since the distribution of the sum of n independent Poisson random variables X
i
with parameter λ
i
is Poisson with parameter
P
n i=
1
λ
i
, we obtain N
r
∼ Poissonλ
r
r = 1, 2, 3,
with λ
1
= λ
11
+ λ
12
+ λ
13
+ λ
123
, λ
2
= λ
22
+ λ
12
+ λ
23
+ λ
123
, λ
3
= λ
33
+ λ
13
+ λ
23
+ λ
123
. Also,
Cov[N
u
, N
v
] = Var[N
uv
] + Var[N
123
] u 6= v.
The joint pgf of N
1
, N
2
, N
3
is P
N
1
,N
2
,N
3
t
1
, t
2
, t
3
= E h
t
N
11
+N
12
+N
13
+N
123
1
. . . t
N
33
+N
13
+N
23
+N
123
3
i =
n Q
3 j =
1
E h
t
N
jj
j
io E
h t
1
t
2 N
12
i E
h t
1
t
3 N
13
i E
h t
2
t
3 N
23
i E
h t
1
t
2
t
3 N
123
i =
exp h
P
3 j =
1
λ
jj
t
j
− 1
i +
exp h
P
j k
t
j
t
k
− 1
i +
exp [λ
123
t
1
t
2
t
3
− 1] .
Given 10, the joint chf of W
1
, W
2
, W
3
is φ
W
1
,W
2
,W
3
t
1
, t
2
, t
3
= P
N
1
,N
2
,N
3
φ
X
1
t
1
, φ
X
2
t
2
, φ
X
3
t
3
= e
A+B+C
12 with
A =
3
X
j = 1
λ
jj
φ
X
j
t
j
− 1
B = X
j k
λ
j k
φ
X
j
t
j
φ
X
k
t
k
− 1
C = λ
123
φ
X
1
t
1
φ
X
2
t
2
φ
X
3
t
3
− 1.
By 9 and 12, we have φ
W
t = φ
W
1
,W
2
,W
3
t, t, t = e
[λφ
X
t − 1]
, 13
where λ = λ
11
+ λ
22
+ λ
33
+ λ
12
+ λ
13
+ λ
23
+ λ
123
and φ
X
t = λ
11
λ φ
X
1
t + λ
22
λ φ
X
2
t + λ
33
λ φ
X
3
t + λ
12
λ φ
X
1
+X
2
t + λ
13
λ φ
X
1
+X
3
t +
λ
23
λ φ
X
2
+X
3
t + λ
123
λ φ
X
1
+X
2
+X
3
t , 14
with φ
X+Y
t = φ
X
t φ
Y
t . The random variable W = W
1
+ W
2
+ W
3
, where W
i
i = 1, 2, 3 are
correlated compound Poisson random variables, has a compound Poisson distribution, as in the independent case presented in Appendix A, but with different parameter λ and different claimsize characteristic function φ
X
t to
which is associated the distribution function F
X
x = λ
11
λ F
X
1
x + λ
22
λ F
X
2
x + λ
33
λ F
X
3
x + λ
12
λ F
X
1
+X
2
x + λ
13
λ F
X
1
+X
3
x +
λ
23
λ F
X
2
+X
3
x + λ
123
λ F
X
1
+X
2
+X
3
x. Also,
µ = λ
11
λ µ
1
+ λ
22
λ µ
2
+ λ
33
λ µ
3
+ λ
12
λ µ
1
+ µ
2
+ λ
13
λ µ
1
+ µ
3
+ λ
23
λ µ
2
+ µ
3
+ λ
123
λ µ
1
+ µ
2
+ µ
3
= λ
1
λ µ
1
+ λ
2
λ µ
2
+ λ
3
λ µ
3
.
H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149 139
3.4. Negative Binomial model with common component Modeling the number of claims N by a Poisson random variable means that the variance Var[N ] is equal to the
expectation E[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 variable N is
P N = n = α + n − 1
α − 1
1 1 + β
α
β 1 + β
n
, α, β
0, n = 0, 1, 2, . . . . The mean and the variance are, respectively, µ = αβ and σ
2
= αβ 1 + β = µ1 + β. The corresponding pgf is
[1 − βt − 1]
−α
. 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 the j th j = 1, 2, 3 class of business is the sum of two random variables. The first random
variable, denoted by N
jj
, is specific to each class and is independent of the specific random variables of the other
classes. The second random variable is denoted by N
j
j = 1, 2, 3. A dependence relation is assumed between
the second random variables of the different classes. For i = 1, 2, . . . , the random variables N
j
j = 1, 2, 3 are
defined by N
j
= N
jj
+ N
j
, 15
where N
jj
∼ NBα
jj
, β
j
j = 1, 2, 3,
N
j
∼ NBα
, β
j
j = 1, 2, 3.
The distribution of the sum of n independent Negative Binomial random variables Y
i
with parameters α
i
, β is
Negative Binomial with parameters P
n i=
1
α
i
, β . Hence, we have
N
j
∼ NBα
j
, β
j
j = 1, 2, 3,
where α
j
= α
jj
+ α . For a fixed i, we assume that the random variables N
jj
j = 1, 2, 3 are independent. The
random variables N
j
j = 1, 2, 3 are dependent and they are modeled by a common Poisson–Gamma mixture
with 1 N
j
|2 = θ ∼ Poissonθβ
j
j = 1, 2, 3,
2 2 ∼ Gammaα ,
1, 3 N
j
|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 that N
j
has a marginal Negative Binomial distribution with parameters α
j
, β
j
. The joint probability distribution function of N
10
, N
20
, N
30
is P
N
10
,N
20
,N
30
t
1
, t
2
, t
3
= E h
E h
t
N
10
1
t
N
20
2
t
N
30
3
|2 ii
= M
2
[β
1
t
1
− 1 + β
2
t
2
− 1 + β
3
t
3
− 1]
= [1 − β
1
t
1
− 1 − β
2
t
2
− 1 − β
3
t
3
− 1]
−α
. Therefore, the joint pgf of N
1
, . . . , N
3
is
140 H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149
P
N
1
,N
2
,N
3
t
1
, t
2
, t
3
= E h
t
N
11
+N
10
1
t
N
22
+N
20
2
t
N
33
+N
30
3
i = E
h t
N
11
1
i E
h t
N
22
2
i E
h t
N
33
3
i E
h t
N
10
1
t
N
20
2
t
N
30
3
i =
3
Y
j = 1
[1 − β
j
t
j
− 1]
−α
jj
P
N
10
,N
20
,N
30
t
1
, t
2
, t
3
=
3
Y
j = 1
[1 − β
j
t
j
− 1]
−α
jj
1 −
3
X
j = 1
β
j
t
j
− 1
−α
. Also, we have
Cov[N
u
, N
v
] = CovN
uu
+ N
u
, N
vv
+ N
v
= Cov[EN
u
|2, EN
v
|2 ] + E[CovN
u
, N
v
|2 ].
Since N
j
|2 = θ j = 1, 2, 3 are independent, we obtain
Cov[N
u
, N
v
] = β
u
β
v
Var[2] = α β
u
β
v
. The marginal distribution of W
j
j = 1, 2, 3 is a compound Negative Binomial with parameters α
j
, λ
j
and F
X
j
. Its chf φ
W
j
t is
φ
W
j
t = [1 − β
j
φ
X
j
t
j
− 1]
−α
jj
[1 − β
j
φ
X
j
t
j
− 1]
−α
= [1 − β
j
φ
X
j
t
j
− 1]
−α
j
. By 10, the joint chf of W
1
, W
2
, W
3
is φ
W
1
,W
2
,W
3
t
1
, t
2
, t
3
= P
N
1
,N
2
,N
3
φ
X
1
t
1
, φ
X
2
t
2
, φ
X
3
t
3
= A × B, where
A =
3
Y
j = 1
[1 − β
j
φ
X
j
t
j
− 1]
−α
jj
B = [1 − β
1
φ
X
1
t
1
− 1 − β
2
φ
X
2
t
2
− 1 − β
3
φ
X
3
t
3
− 1]
−α
. Given 9, the chf of W is
φ
W
t = φ
W
1
,W
2
,W
3
t, t, t . 17
The probability distribution F
W
of W is approximated by taking the inverse of 17 using the FFT method. This procedure is explained in the next section.
3.5. Approximation of F
W
Within both models, the non-ruin probability can be approximated by using F
˜ W
which is obtained from the FFT method applied to φ
W
t . The first step is to discretize F
X
j
j = 1, . . . , m see Klugman et al., 1998 or
Panjer and Willmot, 1992 and take their Fourier transform. These transforms are inserted in either 13 or 17. The resulting transforms are inverted with the FFT method which produces the vector of mass probabilities defining the
probability distribution function F
˜ W
. This approximation of F
W
is used in 5 and 6 in the evaluation of ϕ
k, 1,n
.
H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149 141
Table 1 Moments of X
i
, N
i
and W
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 ρN
1
, N
2
= ρN
1
, N
2
= 0.25
ρN
1
, N
2
= 0.75
λ 1.0000
3.0000 CovN
1
, N
2
1.0000 3.0000
CovW
1
, W
2
1.2656 3.7969
ρW
1
, W
2
0.0727 0.2165
4. Numerical examples
