This communication gives some extensions of the original BÃ¼hlmann model. The paper is devoted to semi-linear credibility, where one examines functions of the random variables representing claim amounts, rather than the claim amounts themselves.contracts, unbiased estimators, structure parameters, several approximating functions, semi-linear credibility theory, unique optimal function, parameter estimation, hierarchical semi-linear credibility theory.

Virginia ATANASIU

English

Research Papers in Economics

Informatica Economică, nr. 2 (42)/2007 
 
126 
More General Credibility Models 
 
Virginia ATANASIU 
Department of Mathematics, Academy of Economic Studies 
e-mail: virginia_atanasiu@yahoo.com 
 
 
This communication gives some extensions of the original Bühlmann model. The paper 
is devoted to semi-linear credibility, where one examines functions of the random variables 
representing claim amounts, rather than the claim amounts themselves. The main purpose of 
semi-linear credibility theory is the estimation of  ( ) ( ) [ ] θ θ μ 1 0 0 + = t X f E  (the net premium for 
a contract with risk parameter: θ ) by a linear combination of given functions of the observ-
able variables:  ( t X X X X ,..., , 2 1
' = ) . So the estimators mainly considered here are linear 
functions of several functions   of the observable random variables. The approxi-
mation to 
n f f f ,..., , 2 1
() θ μ0  based on prescribed approximating functions   leads to the opti-
mal non-homogeneous linearized estimator for the semi-linear credibility model. Also we dis-
cuss the case when taking   for all: 
n f f f ,..., , 2 1
f f p = p , try to find the optimal function  . It should be 
noted that the approximation to 
f
() θ μ0  based on a unique optimal approximating function   
is always better than the one furnished in the semi-linear credibility model based on pre-
scribed approximating functions:  . The usefulness of the latter approximation is 
that it is easy to apply, since it is sufficient to know estimates for the structural parameters 
appearing in the credibility factors. From this reason we give some unbiased estimators for 
the structure parameters. For this purpose we embed the contract in a collective of contracts, 
all providing independent information on the structure distribution. We close this paper by 
giving the semi-linear hierarchical model used in the applications chapter. 
f
n f f f ,..., , 2 1
Mathematics Subject Classification: 62P05. 
Keywords: contracts, unbiased estimators, structure parameters, several approximating func-
tions, semi-linear credibility theory, unique optimal function, parameter estimation, hierar-
chical semi-linear credibility theory. 
 
ntroduction 
In
credi
 this article we first give the semi-linear 
bility model (see Section 1), which in-
volves only one isolated contract. Our prob-
lem (from Section1) is the estimation of 
() ( ) [] θ θ μ 1 0 0 + = t X f E  (the net premium for a 
contract with risk parameter: 
  ( ) t X X X X ,..., , 2 1
' = . So our problem (from 
Section 1) is the determination of the linear 
combination of 1 and the random variables: 
( ) r p X f ,  n p , 1 = ,  t r , 1 =  closest to 
( ) ( ) [ ] θ θ μ 1 0 0 + = t X f E  in the least squares 
sense, where θ  is the structure variable. The 
solution of this problem: 
θ ) by a linear 
combination of given functions   
of the observable variables: 
n f f f ,..., , 2 1
() ( )
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
∑∑
==
2
11
0 0
, 0
n
p
t
r
r p pr X f E Min α θ μ
α α
− −α , where:  ( )
r p pr , α α = , 
is the optimal non-homogeneous linearized 
estimator (namely the semi-linear credibility 
result). In Section 2 we discuss the case 
when taking   for all:  f f p = p , try to find 
the unique optimal function  . It should be 
noted that the approximation to 
I 
f
() μ θ 0
f
 based 
on a unique optimal approximating function 
 is always better than the one furnished in Informatica Economică, nr. 2 (42)/2007 
 
127
the semi-linear credibility model based on 
prescribed approximating functions: 
. The usefulness of the latter ap-
proximation is that it is easy to apply, since it 
is sufficient to know estimates for the struc-
tural parameters:  ,   (with 
n f f f ,..., , 2 1
pq a pq b p , 
q n , 0 = ) appearing in the credibility factors 
 (where  p z n p , 1 = ). To obtain estimates for 
these structure parameters from the semi-
linear credibility model, in Section 3 we em-
bed the contract in a collective of contracts, 
all providing independent information on the 
structure distribution. We close this paper by 
giving the semi-linear hierarchical model 
used in the applications chapter (see Section 
4). 
 
Section 1 (The approximation to  ( ) θ μ0  
based on prescribed approximating func-
tions:  )  n f f f ,..., , 2 1
In this section, we consider one contract with 
unknown and fixed risk parameter: θ , during 
a period of t years. The yearly claim 
amounts are denoted by:  . The risk 
parameter 
t X X ,..., 1
θ  is supposed to be drawn from 
some structure distribution function:  () ⋅ U . It 
is assumed that, for given: θ , the claims are 
conditionally independent and identically dis-
tributed (conditionally i.i.d.) with known 
common distribution function  () θ θ , x FX . The 
random variables   are observable, 
and the random variable   is considered 
as being not (yet) observable. We assume 
that: 
t X X ,..., 1
1 + t X
( ) r p X f ,  n p , 0 = ,  1 , 1 + = t r  have finite 
variance. For:  , we take the function of 
 we want to forecast.  
0 f
1 + t X
We use the notation: 
() ( ) [ ] θ θ μ | r p p X f E =    (1.1) 
( ) 1 , 1 ; , 0 + = = t r n p  
This expression does not depend on r. 
We define the following structure parameters: 
() [ ] ( ) [ ] { } ( ) [ ] r p r p p p X f E X f E E E m = = = θ θ μ |    (1.2), 
()( ) [ ] { } θ | , r q r p pq X f X f Cov E a =    (1.3), 
() () [ ] θ μ θ μ q p pq Cov b , =    (1.4), 
()() [ ] r q r p pq X f X f Cov c , =    (1.5), 
()( ) [ ] θ μq r p pq X f Cov d , =    (1.6), 
for:  ,  p n q , 0 =     ∧ 1 , 1 + = t r . These expres-
sions do not depend on:  1 , 1 + = t r . The 
structure parameters are connected by the fol-
lowing relations: 
pq pq pq b a c + =    (1.7), 
pq pq b d =    (1.8), 
for:  n q p , 0 , = . This follows from the covari-
ance relations obtained in the probability the-
ory where they are very well-known. Just as 
in the case of considering linear combina-
tions of the observable variables themselves, 
we can also obtain non-homogeneous credi-
bility estimates, taking as estimators the class 
of linear combinations of given functions of 
the observable variables, as shown in the fol-
lowing theorem: 
Theorem 1.1 (Optimal non-homogeneous lin-
earized estimators) 
The linear combination of 1 and the random 
variables  ( ) t r n p X f r p , 1 ; , 1 , = =  closest to 
( ) ( ) [ ] θ θ μ | 1 0 0 + = t X f E  and to  ( ) 1 0 + t X f  in 
the least squares sense equals: 
() ∑ ∑∑
= ==
− + =
n
p
p p
n
p
t
r
r p p m z m X f
t
z M
1
0
11
1
   (1.9), Informatica Economică, nr. 2 (42)/2007 
 
128 
where   is a solution to the linear system of equations:  n z z z ,..., , 2 1
() [] q p
n
p
pq pq td z d t c 0
1
1 = − + ∑
=
( n q , 1 = )   (1.10) 
or to the equivalent linear system of equations: 
() q p
n
p
pq pq tb z tb a 0
1
= + ∑
=
( n q , 1 = )   (1.11) 
Proof: we have to examine the solution of the problem: 
() ( )
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
− − ∑∑
==
2
11
0 0
, 0
n
p
t
r
r p pr X f E Min α α θ μ
α α
   (1.12) 
Taking the derivative with respect to  0 α  gives: 
() [] () [] 0
11
0 α α θ μ = −∑∑
==
r p
n
p
t
r
pr X f E E , or:  . Inserting this expression for  ∑∑
==
− =
n
p
t
r
p prm m
11
0 0 α α
0 α  into (1.12) leads to the following problem: 
() ( ) (
⎪ ⎭
⎪
⎬
⎫
⎪ ⎩
⎪
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
− − − ∑∑
==
2
11
0 0
n
p
t
r
p r p pr m X f m E Min α θ μ
α )    (1.13) 
On putting the derivatives with respect to  ' qr α  equal to zero, we get the following system of 
equations ( n q , 1 = ;  t r , 1 '= ): 
() ( ) [] ()() [ ∑∑
==
=
n
p
t
r
r q r p pr r q X f X f Cov X f Cov
11
' ' 0 , , α θ μ ]    (1.14) 
Because of the symmetry in time clearly: 
p pt p p α α α α = = = = ... 2 1 , so using the co-
variance results, for  n q , 1 =  this system of 
equations can be written as: 
() [ ∑
=
− + =
n
p
pq pq p q d t c b
1
0 1 α ]    (1.15) 
Now (1.15) and (1.13) lead to (1.9) with: 
t
z p
p = α ,  n p , 1 = . 
 
Section 2 (The approximation to  () θ μ0  
based on a unique optimal approximating 
function:  )  f
The estimator M  for  ( ) θ μ0  of Theorem 1.1 
can be displayed as: 
() ( t X f X f M + + = ... 1 )     
     (2.1), 
where: 
() () ∑ ∑
= =
− + =
n
p
p p
n
p
p p m z
t
m
t
x f z
t
x f
1
0
1
1 1 1
. 
Let us forget now about this structure of   
and look for any function   such that (2.1) 
is closest to:
f
f
( ) θ μ0 . If are considered only 
functions   such that  f ( ) 1 X f  has finite vari-
ance, then the optimal approximating func-
tion   results from the following theorem:  f
Theorem 2.1 ( Optimal approximating func-
tion) 
( ) ( ) t X f X f + +... 1  is closest to  () θ μ0  and to 
( ) 1 0 + t X f  in the least squares sense, if and 
only if   is a solution of the equation:  f
( ) ( ) ( ) [ ] () [] 0 1 1 2 0 1 2 1 ≡ − − + X X f E X X f E t X f
     (2.2) 
Proof: we have to solve the following mini-
mization problem: 
( ) ( )( ) [ ] { }
2
1 1 0 ... t t g X g X g X f E Min − − − +   (2.3) 
Suppose that   denotes the solution to this 
problem, then we consider: 
f
( ) ( ) ( ) X h X f X g α + = , with   arbitrary, 
like in variational calculus. Let: 
() ⋅ h
 Informatica Economică, nr. 2 (42)/2007 
 
129
() ( ) ( ) ( ) ( ) ( ) [] { }
2
1 1 1 0 ... ... t t t X h X h X f X f X f E α α α ϕ − − − − − − = +     (2.4) 
Clearly for   to be optimal,  f () 0 0 ' = ϕ , so for every choice of h: 
()( ) ( ) [] ( ) ( ) [ {} 0 ... ... 1 1 1 0 ] = + + − − − + t t t X h X h X f X f X f E      (2.5), 
must hold. This can be rewritten as: 
() () ( )( ) ( )( ) ( ) [] 0 1 1 2 1 1 1 2 0 = − − − X h X f t t X h X tf X h X tf E      (2.6), 
or: 
() () ( )( ) [] ( ) [ ] {} [] 0 1 1 2 0 1 2 1 1 = + − − − X X f E X X f E t X f X h E      (2.7) 
Because this equation has to be satisfied for 
every choice of the function   one obtains, 
the expression in brackets in (2.7) must be 
identical to zero, which proves (2.2). 
h
An application of Theorem 2.1: 
If   can only take the values 
 and 
1 1,..., + t X X
n ,..., 1 , 0 [] r X q X P pqr = = = 2 1 ,  for: q, 
n r , 0 = , then  ( )( t X f X f + +... 1 )  is closest 
to  ( ) θ μ0  and to  ( ) 1 0 + t X f  in the least squares 
sense, if and only if for  n q , 0 = ,  ( ) q f  is a 
solution of the linear system: 
() ( ) () ()qr
n
r
n
r
qr
n
r
qr p r f p r f t p q f ∑ ∑ ∑
= = =
= − +
0
0
0 0
1        (2.8) 
Indeed:  () ()
()
( )
⎟ ⎟
⎟
⎠
⎞
⎜ ⎜
⎜
⎝
⎛
= ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
= ∑
=
n
r
qr p
q f
q X P
q f
X f
0
1
1 : ,  n q , 0 = ;  () [] ()( = = =∑
=
1 2
0
1 2 X r X P r f X X f E
n
r
 
)( )
∑
∑
=
=
= =
n
r
qr
qr
n
r p
p
r f q
0
0
;  () [] ()() ()
∑
∑ ∑
=
= =
= = = =
n
r
qr
qr
n
r
n
r p
p
r f q X r X P r f X X f E
0
0
0 1 2
0
0 1 2 0 . 
Inserting these expressions for:  ( ) 1 X f , 
() [] 1 2 X X f E  and  () [ 1 2 0 X X f E ]  into (2.2) 
leads to (2.8). 
 
Section 3 (Parameter estimation) 
It should be noted that the approximation to 
() θ μ0  based on a unique optimal approxi-
mating function   is always better than the 
one furnished in Section 1 based on pre-
scribed approximating functions: 
. The usefulness of the latter ap-
proximation is that it is easy to apply, since it 
is sufficient to know estimates for the struc-
tural parameters  ,   (with 
f
n f f f ,..., , 2 1
pq a pq b p ,  q n , 0 = ) 
appearing in the credibility factors   (where  p z
n p , 1 = ). From this reason we give some un-
biased estimators for the structure parame-
ters. For this purpose we consider   con-
tracts,
k
k j , 1 = , and     independent and 
identically distributed vectors 
k ( 2 ≥ )
( ) ( ) jt j j j j X X X ,..., , , 1
' θ θ = , for  k j , 1 = . The 
contract indexed j is a random vector consist-
ing of a random structure parameter  j θ  and 
observations:  , where  jt j X X ,..., 1 k j , 1 = . For 
every contract  k j , 1 =  and for  j θ  fixed, the 
variables:   are conditionally inde-
pendent and identically distributed.  
jt j X X ,..., 1
Theorem 3.1 (Unbiased estimators for the 
structure parameters) 
Let: 
    ∑∑
==
= =
k
j
jr
t
r
p
p
p X f
kt
X
kt
m
11
..
^
) (
1 1
   ( 3 . 1 )
⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛ −
−
= ∑∑
==
q
j
q
jr
k
j
t
r
p
j
p
jr pq X
t
X X
t
X
t k
a .
11
.
^ 1 1
) 1 (
1
   (3.2) Informatica Economică, nr. 2 (42)/2007 
 
130 
t
a
X
kt
X
t
X
kt
X
t k
b
pq q q
j
k
j
p p
j pq
^
.. .
1
.. .
^ 1 1 1 1
1
1
− ⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛ −
−
= ∑
=
   (3.3), 
, then:  ,  ,  , where:  ,  , 
,  , with 
p p m m E = ⎟
⎠
⎞
⎜
⎝
⎛
^
pq pq a a E = ⎟
⎠
⎞
⎜
⎝
⎛
^
pq pq b b E = ⎟
⎠
⎞
⎜
⎝
⎛
^
∑
=
=
t
r
p
jr
p
j X X
1
. ∑
=
=
t
r
q
jr
q
j X X
1
.
∑∑
==
=
k
j
t
r
p
jr
p X X
11
.. ∑∑
==
=
k
j
t
r
q
jr
q X X
11
.. ( ) jr p
p
jr X f X = ,  k j , 1 ( =  and  t r , 1 = ), 
( ) jr q
q
jr X f X = ,  k j , 1 ( =  and  t r , 1 = ), for  p ,  n q , 0 = , such that  q p < . 
Proof: note that the usual definitions of the structure parameters apply, with  j θ  replacing θ  
and   replacing   so:  jr X r X () [] = = = = ⎟
⎠
⎞
⎜
⎝
⎛ ∑ ∑ p
r j
p
r j
jr p p m
kt
kt
m
kt
X f E
kt
m E
, ,
^ 1 1
p m ;  
() () [ () () () − ⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛ − +
−
= ⎟
⎠
⎞
⎜
⎝
⎛ ∑
q
j
p
jr
q
j
p
jr
q
jr
p
jr
r j
q
jr
p
jr pq X
t
E X E X
t
X Cov X E X E X X Cov
t k
a E . .
,
^ 1 1
, ,
1
1
 
()
q
jr
p
j
q
jr
p
j X E X
t
E X X
t
Cov ⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛ − . .
1
,
1
()
⋅
−
= ⎥
⎦
⎤
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛ + ⎟
⎠
⎞
⎜
⎝
⎛ +
1
1 1 1 1
,
1
. . . . t k
X
t
E X
t
E X
t
X
t
Cov
q
j
p
j
q
j
p
j  
() ∑ ⎥
⎦
⎤
⎢
⎣
⎡
+ ⎟
⎠
⎞
⎜
⎝
⎛ + + − ⎟
⎠
⎞
⎜
⎝
⎛ + − − ⎟
⎠
⎞
⎜
⎝
⎛ + − + + ⋅
r j
q p pq pq q p pq pq q p pq pq q p pq pq m m b a
t
m m b a
t
m m b a
t
m m b a
,
1 1 1
= () ()
( )
pq pq
r j
pq pq pq pq a a
t
t
kt
t k
b a
t
b a
t k
=
−
−
= ⎟
⎠
⎞
⎜
⎝
⎛ − − +
− ∑
1
1
1 1
1
1
,
;  ⋅ ⎢
⎣
⎡
⎜
⎝
⎛
−
= ⎟
⎠
⎞
⎜
⎝
⎛ ∑
j
pq t
Cov
k
b E
1
1
1 ^
 
,
1 1 1 1
,
1 1 1 1
, .. .. . .. . . . . .
p q p
j
q p
j
q
j
p
j
q
j
p
j X
kt
Cov X
kt
E X
t
E X
kt
X
t
Cov X
t
E X
t
E X
t
X ⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛ + ⎟
⎠
⎞ ⋅  
= − ⎥
⎦
⎤
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛ + ⎟
⎠
⎞
⎜
⎝
⎛ + ⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛ − ⎟
⎠
⎞
t
a
X
kt
E X
kt
E X
kt
X
kt
Cov X
t
E X
kt
E X
t
pq q p q p q
j
p q
j .. .. .. .. . .. .
1 1 1
,
1 1 1 1
, ⋅
−1
1
k
 
+ ⎜
⎝
⎛ + − ⎟
⎠
⎞
⎜
⎝
⎛ + − − ⎟
⎠
⎞
⎜
⎝
⎛ + − + ⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛ + ⋅∑ pq q p pq pq q p pq pq q p
j
pq pq a
kt
m m b
k
a
kt
m m b
k
a
kt
m m b a
t
1 1 1 1 1 1
 
= − ⎟
⎠
⎞ − − + ⎜
⎝
⎛
−
= − ⎥
⎦
⎤
+ ⎟
⎠
⎞ + ∑ t
a
b
k
a
kt
b a
t k t
a
m m b
k
pq
pq pq pq pq
j
pq
q p pq
1 1 1
1
1 1
pq b
k
k
k
k
1
1
1 −
−
+  
pq
pq pq
pq
pq
pq b
t
a
t
a
b
t
a
a
kt
k
k
k
= − + = −
−
−
+
1
1
1
. 
 
Section 4 (Applications of semi-linear 
credibility theory) 
We close this paper by giving the semi-
linear hierarchical model used in the appli-
cations chapter. Like in Jewell’s hierarchical 
model we consider a portfolio of contracts, 
which can be broken up into P  sectors each 
sector   consisting of   groups of con-
tracts. Instead of estimating:  , 
p p k
1 , , + t j p X
( ) [ ]
j p p t j p j p p X E θ θ θ θ μ , , 1 , , + =  (the pure net 
risk premium of the contract  ),  () j p,
( ) [ ] p t j p p X E θ θ ν 1 , , + =  (the pure net risk pre-
mium of the sector  ), we now estimate:  p
( ) 1 , , 0 + t j p X f , 
( ) ( ) [ ]
j p p t j p j p p X f E θ θ θ θ μ , , 1 , , 0 0 + =  (the pure 
net risk premium of the contract  ),  () j p,
( ) ( ) [ ] p t j p p X f E θ θ ν 1 , , 0 0 + =  (the pure net risk 
premium of the sector  p ), where  P p , 1 =  
and  p k j , 1 = . In semi-linear credibility the-
ory the following class of estimators is con-Informatica Economică, nr. 2 (42)/2007 
 
131
sidered: , 
where   are functions given in 
advance. Let us consider the case of one 
given function   in order to approximate 
() qir p
n
p
P
q
k
i
t
r
pqir X f
q
∑∑∑∑
====
+
1111
0 α α
() () ⋅ ⋅ n f f ,..., 1
1 f
( ) 1 , , 0 + t j p X f  or  ( ) p θ ν 0  and ( )
j p p θ θ μ , 0 . We 
formulate the following theorem: 
Theorem 4.1 ( Hierarchical semi-linear 
credibility) 
Using the same notations as introduced for 
the hierarchical model of Jewell and denoting 
( ) pjs pjs X f X 0
0 =  and  ( ) pjs pjs X f X 1
1 =  one ob-
tains the following least squares estimates for 
the pure net risk premiums: 
()( )
1
1 0
^
0 pzw p p p X z m z m + − = θ ν , 
() ( )
1
1 0
^
0 , pjw pj pj pj p X z m z m + − = θ θ μ    (3.1) 
where: 
1
1 .
1
pjr
t
r pj
pjr
pjw X
w
w
X ∑
=
= , 
1
1 .
1
pjw
k
j p
pj
pzw X
z
z
X
p
∑
=
= , 
( ) [ ] 11 . 11 01 . 1 / d w c d w z pj pj pj − + =  
(the credibility factor on contract level), with: 
( )
1
'
0
01 , pjr pjr X X Cov d = ,  ( )
1
'
1
11 , pjr pjr X X Cov d = , 
' r r ≠ ,  ( ) ( )
1 1 1
11 , pjr pjr pjr X Var X X Cov c = = , 
and:  ( ) [ ] 11 . 11 01 . 1 / D z C D z z p p p − + =  (the 
credibility factor at sector level), with: 
( )
1
'
0
01 , w pj pjw X X Cov D = , 
( )
1
'
1
11 , w pj pjw X X Cov D = ,  ,  ' j j ≠ = 11 C  
( ) ( )
1 1 1 , pjw pjw pjw X Var X X Cov = = . 
Remark 4.1: the linear combination of 1 and 
the random variables   (
1
pjr X P p , 1 = , 
p k j , 1 = ,  t r , 1 = ) closest to  ( ) 1 , , 0 + t j p X f  and 
to  ( ) p θ ν 0  in the least squares sense equals 
( ) p θ ν
^
0 , and the linear combination of 1 and 
the random variables   (
1
pjr X P p , 1 = , 
p k j , 1 = ,  t r , 1 = ) closest to  ( )
j p p θ θ μ , 0  in 
the least squares sense equals  .  () pj p θ θ μ ,
^
0
 
References 
[1] Goovaerts, M.J., Kaas, R., Van 
Herwaarden, A.E., Bauwelinckx,T.: 
Insurance Series, volume 3, Efective 
Actuarial Methods, University of Amster-
dam, The Netherlands, 1991. 
[2] Pentikäinen, T., Daykin, C.D., and Pe-
sonen, M.: Practical Risk Theory for Actuar-
ies, Université Pierré et Marie Curie, 1990. 
[3] Sundt, B.:  An Introduction to Non-Life 
Insurance Mathematics, Veroffentlichungen 
des Instituts für Versicherungswissenschaft 
der Universität Mannheim Band 28 , VVW 
Karlsruhe, 1984. 
 

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