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
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 probability ψu, 1, 20 obtained for different relations of dependence between the
classes of business. The differences result from the choice of correlation coefficient between N
1
and N
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
∼ Weibull0.5, 1.125
N
1
∼ Poisson4
Book of business 2 : X
2
∼ Exponential2.25
N
2
∼ Poisson4.
The moments of X
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 in ψu, 1, 20
indicates the correlation coefficient between N
1
and N
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
∼ Weibull0.5, 1.125
N
1
∼ Negative Binomial1, 4
Book of business 2 : X
2
∼ Exponential2.25
N
2
∼ Negative Binomial1, 4.
In Tables 4 and 5, we give the moments of X
i
, N
i
, W
i
and the correlation parameters. The numerical results for the ruin probability are summarized in Table 6, where the last number added in ψu, 1, 20 again indicates the
correlation coefficient between N
1
and N
2
.
142 H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149
Table 3 Ruin probabilities ψu, 1, 20 in the Poisson model
u ψu,
1, 20, 0 ψu,
1, 20, 0.25 ψu,
1, 20, 0.75 ψu,
1, 20, 0.25 ψu,
1, 20, 0 −
1 ψu,
1, 20, 0.75 ψu,
1, 20, 0 −
1 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 of X
i
, N
i
and W
i
Class1 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 ρN
1
, N
2
= ρN
1
, N
2
= 0.25
ρN
1
, N
2
= 0.75
α 0.3125
0.9375 CovN
1
, N
2
5.0000 15.0000
CovW
1
, W
2
6.3281 18.9844
ρW
1
, W
2
0.1716 0.5147
We observe that the increase in the ruin probability ψu, 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
ψu, 1, 10 and ψu, 1, 30.
In both examples, we observe that the ruin probabilities increase as the coefficient of correlation ρN
1
, N
2
increases. It is worth mentioning that with X
1
exponentially distributed and X
2
having a Weibull distribution in both the Poisson and Negative Binomial model, the correlation coefficient ρN
1
, N
2
of 0.75 in the Poisson model produces a correlation coefficient ρW
1
, W
2
of 0.2165 while it leads to a correlation coefficient ρW
1
, W
2
equal to 0.5147 in the Negative Binomial model. This might explain the smaller impact on the ruin probabilities of the introduction of a relation of dependence in the Poisson model than in the Negative Binomial model.
H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149 143
Table 6 Ruin probabilities ψu, 1, 20 in negative binomial model
u ψu,
1, 20, 0 ψu,
1, 20, 0.25 ψu,
1, 20, 0.75 ψu,
1, 20, 0.25 ψu,
1, 20, 0 −
1 ψu,
1, 20, 0.75 ψu,
1, 20, 0 −
1 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
