Is an original paper, which describes techniques for estimating premiums for risks, containing a fraction (a part) of the variance of the risk as a loading on the net risk premium.the linearized credibility formula, the best risk premium – in the sense of the minimal weighted mean squared error -, variance premium

Virginia ATANASIU

English

Research Papers in Economics

Revista Informatica Economică nr.2(46)/2008 
 
12
Linearized Credibility Formula for  
Exponentially Weighted Squared Error Loss Function 
 
Virginia ATANASIU 
Department of Mathematics, Academy of Economic Studies 
virginia_atanasiu@yahoo.com 
 
Is an original paper, which describes techniques for estimating premiums for risks, 
containing a fraction (a part) of the variance of the risk as a loading on the net risk premium. 
Mathematics Subject Classification: 62P05. 
Keywords: the linearized credibility formula, the best risk premium – in the sense of the 
minimal weighted mean squared error -, variance premium. 
 
ntroduction 
One approach is to consider the best risk 
premium (in the sense of the minimal 
weighted mean squared error). In the first 
section it is shown that it can be used as an 
approximation to the variance loaded 
premium, by truncating a series expansion. 
In the second section this premium is 
derived as an optimal estimator minimizing 
a suitable loss function. In the credibility 
theory so far the credibility results described 
are intended to estimate pure net risk 
premiums. An important question arises if 
one is interested in estimating the variance 
loaded premiums. In a top-down approach, 
the collective premium can be distributed 
proportionally to this loaded risk premium. 
Therefore we also consider credibility 
estimates for loaded risk premiums. 
Theory 
Replacing the loss function ()
2 x y −  when 
y  is estimated instead of  x by a slightly 
more general weighted loss function 
( )
hy e x y
2 − , one gets the following best 
function of  X  to estimate a random variable 
Y . 
Minimizing weighted mean squared error 
When  X  and Y  are two random variables, 
and  Y  must be estimated using a function 
( ) X g  of  X , the choice yielding the 
minimal weighted mean squared error 
( ) [ ]
hY e X g Y E
2 ) ( −  is the best risk premium 
(in the sense of the minimal weighted mean 
squared error) for Y , given  X : 
[ ] [ ] [ ] X e E X Ye E X Y H
hY hY | / | | =       (1.1) 
Proof. Write: 
 
() [ ] () [ ] { }= − = − X e X g Y E E e X g Y E
hY hY | ) ( ) (
2 2 ( ) [ ] ∫ = − x X e x g Y E
hY | ) (
2 () x dFX ⋅  
 
For fixed x, the integrand can be written as: 
() [ ]
hZ e p Z E
2 − , with  () x g p =  and Z  
distributed as Y , given  x X =  
( () [] x X Y Z
P
= ≡ |
) (
). This quadratic form in 
p  is minimized taking  p = [ ] [ ]
hZ hZ e E Ze E / 
or what is the same 
( ) x g = [ ] [ ] x X e E x X Ye E
hY hY = = | / |.  
 
Indeed:  () ( ) [] ( ) ( ) ( )
hZ hZ hZ hZ
not
Ze pE e E p e Z E e p Z E p 2
2 2 2
.
− + = − = ϕ . 
 
We have to solve the following 
minimization problem: 
() p Min
p
ϕ           (1.2) 
Since (1.2) is the minimum of a positive 
definite quadratic form, it is suffices to find 
a solution with the first derivative equal to 
zero. Taking the first derivative with respect 
to p , we get the equation: 
( ) ( ) 0 2 2 = −
hZ hZ Ze E e pE .  
So:  ( ) ( )
hZ hZ e E Ze E p / = , because: 
IRevista Informatica Economică nr.2(46)/2008  13
() ( ) 0 2
' ' > =
hZ e E p ϕ . 
If the integrand is chosen minimal for each 
x, the integral over all x is minimized, too. 
Remark 1.1 (the best risk premium and 
variance premium) 
For small h, the best premium approaches 
the variance principle. This can be seen by 
approximating numerator and denominator 
of  [ ] X Y H  up to the first order in h: 
 
[] [][ ]
[] () [] ( [] () ) ⋅ + + =
+ +
+ +
=
2 2
2
2 2
| |
| 1
) ( | |
| h O X Y hE X Y E
h O X Y hE
h O X Y hE X Y E
X Y H  
[] ( ) ( )= + − ⋅
2 | 1 h O X Y hE [ ] [ ] ( )
2 | | h O X Y hVar X Y E + + .  
Indeed, we have: 
()
2
2
2 1 ...
!
...
! 2 ! 1
1 h O hY
n
Y
h
Y
h
Y
h e
n
n hY + + ≈ + + + + + = ,  
and so:  ( )
2 2 h O hY Y Ye
hY + + ≅ . Therefore, from (1.1) we get: 
 
[] X Y H | [ ]
[]
[ ( ) ]
() []
[ ] [ ] ()
[] ()
2
2 2
2
2 2
| 1
| |
| 1
|
|
|
h O X Y hE
h O X Y hE X Y E
X h O hY E
X h O hY Y E
X e E
X Ye E
hY
hY
+ +
+ +
=
+ +
+ +
= = .  
Also: () ( ) () ( ) ( )( ) ( )
...
!
1 1 ... 1 1 1
...
! 2
1 1 1
! 1
1
1 1
2 1 +
+ − − − − −
+ +
− − −
+
−
+ = +
− n Z
n
n
Z Z Z , 
 if:  1 < Z . On taking  [] ( )
2 | h O X Y hE Z + =  
one obtains: 
[] () ()
1 2 | 1
−
+ + h O X Y hE ⋅ − ≅ h 1
[] ( )
2 | h O X Y E + , and thus: 
[] [] [] ( )
2 | | | h O X Y hVar X Y E X Y H + + =  (1.3) 
In fact any loss function () ( ) y w x y
2 −  with 
small h= () ) 0 ( / 0 ' w w  leads to the expression 
(1.3). From this remark we may conclude 
that to derive credibility estimates for 
premiums loaded with a fraction of the 
variance, as well as for the best risk 
premium, one may consider a weighted loss 
function. To be able to compute the loaded 
credibility estimates, we will need (co-) 
variances of squares of the observations. 
 
1.  Credibility for the best risk premium 
Consider the original Bühlmann model. 
Applying (1.1) - of the previous section - to 
Y = 1 + t X  and  = X ()
'
1,..., t X X X = , we see 
that the best risk premium - in the sense of 
weighted mean squared error- to charge for 
period ( ) 1 + t  is: 
 
[] [ ] [ ] X e E X e X E X X H
t t hX hX
t t | / | |
1 1
1 1
+ +
+ + =    (2.1) 
For small h, the expansion (1.3) for (2.1) can be rewritten as follows: 
[] [ ] [ ] ( ) [ ] [] X X hVar X X E h O X X hVar X X E X X H t t t t t | | | | | 1 1
2
1 1 1 + + + + + + ≅ + + = =E  
() [] ( ) [ ] ( ) [ ] { } X Var X E h X | | |
2 θ μ θ σ θ μ + +    (2.2) 
Indeed, we have: 
() [] () [] ( ) [ ] ( ) X X E X X X E E X X E E X E t t t | | , | | | | 1 1 1 + + + = = = θ θ θ μ    (2.3), 
and also: 
Var() ( ) [ ] X X E X X E X X t t t | | | 1
2 2
1 1 + + + − =    (2.4) 
But:  () [ ] () [] ( ) ( ) [ ] } { = − = = + + + X X E X E E X X Var E X E t t t | | | | | | 1
2 2
1 1
2 θ θ θ θ σ  
( ) [ ] ( ) [ ] ( ) [ ]− = − = + + + X X X E E X X E E X X E E t t t | , | | | | |
2
1 1
2 2
1 θ θ θ  
 
() [ ] ( ) ( ) [ ] X X E E X X E X X E E t t t | | | | | 1
2 2
1 1
2 θ θ + + + − = −    (2.5), Revista Informatica Economică nr.2(46)/2008 
 
14
and: 
() []() [ ] ( ) [ ] ( ) [ ]− = − = + X X E E X E X E X Var t | | | | | 1
2 2 2 θ θ μ θ μ θ μ  
( ) X X E t | 1
2
+ −    (2.6), 
because from (2.3) we have:  () [] ( ) X X E X E t | | 1 + = θ μ  and so:  ( ) [ ] () X X E X E t | | 1
2 2
+ = θ μ . 
Furthermore:  
() [ ] () [] ( ) ( ) [ ]+ − = + + + X X E E X X E X Var X E t t | | | | | 1
2 2
1
2 θ θ μ θ σ  
() [ ] ( ) ( ) ( ) X X E X X E X X E X X E E t t t t | | | | | 1
2 2
1 1
2
1
2
+ + + + − = − + θ = 
= () X X Var t | 1 +  (see (2.4)). 
Therefore, 
[] () [] () [ ] ( ) [ ] { } X Var X E h X E X X H t | | | |
2
1 θ μ θ σ θ μ + + = +
. Thus we have (2.2). 
Remark 2.1 The first term in (2.2) denotes 
the expected value part, the second term 
gives the variance part, the last term the 
fluctuation part. 
Remark 2.2 Apart from the optimal 
credibility result (2.1) for this situation, we 
are interested in obtaining a linearized 
credibility formula for estimating Xt+1. 
Therefore, we prove the following 
application of the formula (1.1). 
 
The main results of this paper 
In the following we present the main results 
leaving the detailed computations to the 
reader. 
An application  of the formula  (1.1) - 
Linearized credibility formula for 
exponentially weighted squared error loss 
function - 
The solution to the following problem: 
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
− −
+ ∑
=
+
1
1 0
2
1
0 1
,..., ,
t
t
hX
t
r
r r t
c c c e X c c X E Min    (2.7), 
gives:  ( ) [ ]
() []
() [] θ μ
θ μ
θ
θ
θ
θ * *
*
*
* | *
E z
E
X H E
X z M
a
⎭
⎬
⎫
⎩
⎨
⎧
− + =    (2.8) 
Here  [] θ | X H  is the best risk premium for the conditional distribution of X given θ , and: 
[] () [ ] ( ) [ ] [ ] [ ] { } θ θ μ θ μ θ θ | / , |
* * * * X Var E tVar X H tCov z + =    (2.9) 
The asterisks denote expectations taken over 
a weighted structure function 
* U  satisfying: 
( ) ( ) ( ) h h m dU m dU /
* θ θ θ =    (2.10), with: 
( ) [ ] θ θ |
hX
h e E m =    (2.11), and  
() [] ( ) ( ) θ θ θ dU m m E m h h h ∫ = =    (2.12) 
Proof. To have a minimum in (2.7), the derivative with respect to  0 c  must equal to zero, so: 
[] [ ][ ]
1 1 1
1
0 1
+ + + ∑
=
+ + =
t t t hX
r
t
r
r
hX hX
t e X E c e E c e X E , or:  []
[]
[ ]
[]
1
1
1
1
1 1
0 +
+
+
+ ∑
= + − =
t
t
t
t
hX
t
r
hX
r r
hX
hX
t
e E
e X E c
e E
e X E
c , that is: 
0 c [] () [] () () [] h h
t
r
r h h m m E c m m X H E / / |
1
θ θ μ θ θ θ θ ∑
=
− =    (2.13), 
where  () θ h m  and  h m  are as in (2.11) and (2.12), because: 
[] [ ] {} ( )
() ()
⎥
⎥
⎦
⎤
⎢
⎣
⎡
= =
+
+
=
+ + +
+ + θ
θ
θ
θ |
|
|
|
1
1
1
1 1 1
1 1
t
t
t
t t hX
hX
hX
t hX
t
hX
t e E
e E
e X E
E e X E E e X E = 
= () ( ) [] ( ) ( ) [ ] θ θ θ θ θ h h m X H E m X H E | | =  (see (1.1)). 
[ ] ( ) ( ) [ ] ( ) [ ] θ θ h
hX hX hX m E e E E e E e E
t = = =
+ |
1
h m =  
[ ] ( ) [ ] ( ) ( ) [ ]= = =
+ + + θ θ θ | | |
1 1 1 t t t hX
r
hX
r
hX
r e E X E E e X E E e X E  Revista Informatica Economică nr.2(46)/2008  15
() ( ) [ ] ( ) ( ) [ ] ( )( ) [ ] θ θ μ θ θ μ θ θ θ h h
hX m E m E e E X E E = = = | |.  
Therefore, the verification of formula (2.13) is readily performed. Using the notation: 
() [] () () () ( ) () () () [] h h
h
h m m f E dU
m
m
f dU f f E /
* * θ θ θ
θ
θ θ θ θ θ θ = = = ∫ ∫  
(2.13) can be written as:  () [] () [] θ μ θ θ θ
*
1
*
0 | E c X H E c
t
r
r ∑
=
− =    (2.14) 
Inserting (2.14) into (2.7) the problem is reduced to: 
() [] () [] ()
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎭
⎬
⎫
⎩
⎨
⎧
− − −
+ ∑
=
+
1
2
1
* *
1 |
t hX
t
r
r r t c e E X c X H E X E Min θ μ θ θ θ    (2.15), 
where:  ()
'
1,..., t c c c = . On taking the derivative of (2.15) with respect to  t q cq , 1 , =  and 
putting the result equal to zero, one obtains: 
( ) [ ] ( ) ( ) [ ] ( ) { }= − −
+
+
1 * *
1 |
t hX
q t e E X X H E X E θ μ θ θ θ  
() [] () () [] () {}
1 * *
1
+ − − =∑
=
t hX
q r
t
r
r e E X E X c θ μ θ μ θ θ    (2.16) 
Using conditional expectations over θ  an dividing by  h m , this equation can be written as: 
() ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) { }= − − h h m m E X H E X H E / | |
* * θ θ μ θ μ θ θ θ θ θ  
() () [] () () () [] () () {} + − − =∑
≠
=
t
q r
r
h h r m m E E E c
1
* * / θ θ μ θ μ θ μ θ μ θ θ θ  
[ ] ( ) ( ) h h q m m X Var E c / | θ θ θ +    (2.17), 
since:  [ ] [ ] { } ( ) ( ) [ ]= = =
+ + +
+ + + θ θ θ | | |
1 1 1
1 1 1 q
hX
t
hX
q t
hX
q t X E e X E E e X X E E e X X E
t t t  
() ( ) ( ) [] ( ) ( ) ( ) [ ] θ μ θ θ θ μ θ θ θ h h m X H E m X H E | | = = , from which: 
[ ]=
+
+ h
hX
q t m e X X E
t /
1
1 ( ) [ ( ) ( ) ] h h m m X H E / | θ μ θ θ θ ; 
() [] { } ( ) [ ] { } ( ) ( ) [ ] { ( )} θ θ μ θ θ μ θ μ θ θ θ | |
1 1 1
1
* *
1
*
1
+ + +
+ + + = =
t t t hX
t
hX
t
hX
t e X E E E e E X E E e E X E = 
() [] ( ) [ ] ( ) [ ] ( ) ( ) [ ] () [] { θ μ θ θ θ μ θ θ μ θ θ θ θ θ
* *
1
* | |
1 E E m X H E E e X E E E h
hX
t
t = =
+
+ · 
· () ( )} θ θ h m X H | , from which:  ( ) [ ] { }=
+
+ h
hX
t m e E X E
t /
1 *
1 θ μ θ () [] { () θ θ μ θ θ |
* X H E E · 
· ( )} h h m m / θ ; 
() [] { } ( ) [ ] ( ) () [ ] () [ ⋅ = = =
+ + θ μ θ θ θ θ θ θ θ E X H E e X E X H E e X X H E E
t t hX
q
hX
q | ... | |
* * * 1 1  
() ] () [] ( ) ( ) { } θ θ μ θ θ θ θ h h m X H E E m |
* = ⋅ , from which:  ( ) [ ] { / |
1 * + t hX
qe X X H E E θ θ  
} ( ) [ ] ( ) ( ) { } h h h m m X H E E m / | /
* θ θ μ θ θ θ = ; 
() [] () [] { } ( ) [ ] ( ) [ ] ( ) () [] θ θ μ θ θ μ θ θ θ θ θ θ | ... | |
* * * * * 1 1 X H E e E E X H E e E X H E E
t t hX hX = = =
+ + · 
· () [] () [] θ θ μ θ h m E E
* = () [] ( ) [ ] ( ) [ ] () [ ] { θ θ θ μ θ θ θ θ θ θ | |
* * * X H E E m E E X H E h = () [] θ μ θ
* E · 
· () } θ h m , from which:  ( ) [ ] ( ) [ ] { } { () [] θ θ μ θ θ θ θ θ | / |
* * * 1 X H E E m e E X H E E h
hXt =
+ · 
· () [] ( ) } h h m m E /
* θ θ μ θ ; if  q r ≠  then we obtain: 
( ) ( ) [ ] ( ) [ ( ) ( )]= = =
+ + + θ θ θ θ | | | |
1 1 1 t t t hX
q r
hX
q r
hX
q r e E X E X E E e X X E E e X X E  
= ()() () [] ( ) ( ) ( ) [] θ θ μ θ μ θ θ μ θ μ θ h h m E m E = , from which:  ( )=
+
h
hX
q r m e X X E
t /
1  
= ()() ( ) [] h h m m E / θ θ μ θ μ θ ; if  q r =  then we obtain: 
( ) ( ) [ ] ( ) ( ) [ ] ( ) () [ ] θ θ θ θ θ h
hX
q
hX
q
hX
q m X E E e E X E E e X E E e X E
t t t | | | |
2 2 2 2 1 1 1 = = =
+ + + = Revista Informatica Economică nr.2(46)/2008 
 
16
= ( ) () [ ] θ θ θ h m X E E |
2 , from which:  ( ) ( ) [ ( ) ] h h h
hX
q m m X E E m e X E
t / | /
2 2 1 θ θ θ =
+ ; also, we have 
() [] { } ( ) [ ] { } ( ) ( ) [ ] ( ) { ( | | |
1 1 1 * * * + + + = =
t t t hX
r
hX
r
hX
r e E X E E E e E X E E e E X E θ θ μ θ θ μ θ μ θ θ θ  
| )} ( ) [] () () { } ( ) [ ] ( ) ( ) { } θ θ μ θ μ θ θ μ θ μ θ θ θ θ h h m E E m E E
* * = = , from which: E { r X · 
· () [] } () [] ( ) ( ) { } h h h
hX m m E E m e E
t / /
* * 1 θ θ μ θ μ θ μ θ θ θ =
+ , where  t r , 1 = ; similarly, we have: 
() [] { } ( ) [ ] ( ) ( ) { } h h h
hX
q m m E E m e X E E
t / /
* * 1 θ θ μ θ μ θ μ θ θ θ =
+ , where  t q , 1 = ; we write: 
() [] { } () [] ( ) ( ) [ ] ( ) [ ]= = =
+ + θ θ μ θ μ θ μ θ θ θ h
hX hX m E E e E E e E E
t t 2 * 2 * 2 * 1 1 { () [] θ μ θ
2 * E E · 
() } () [] () { } θ θ μ θ θ θ h h m E E m
2 * = , from which:  ( ) [ ] { } { () [] θ μ θ μ θ θ θ
2 * 2 * /
1 E E m e E E h
hXt =
+ · 
· } h
hX m e
t /
1 + ; finally, we observe that: 
() [] () { } ( ) [ { () [ ] () [ ] () ] θ θ μ θ μ θ θ μ θ θ θ θ | 2 |
* 2 * 2 2 * 1 X E E E X E E e E X E
t hX
q − + = −
+ · 
· () } θ h m = ( ) () ( ) [ ] ( ) { } ( ) [ { θ θ θ θ θ θ θ | | 2 | |
2 2 2 2 X E E m X E X E X E E h = − + - 
- () ] () } () ( ) [] θ θ θ θ θ h h m X Var E m X E | |
2 = , from which:  ( { () [] )
2 * θ μ θ E X E q − · 
· } () ( ) [ ] h h h
hX m m X Var E m e
t / | /
1 θ θ θ =
+ , where:  ( ) ( [
1 1 2 2 + + =
t t hX
q
hX
q e X E E e X E | 
| )] ( ) ( ) [ ] ( ) ( ) [ ] ( ) [ θ θ θ θ θ θ θ | | | |
2 2 2 1 X E E m X E E e E X E E h
hX
q
t = = =
+ () ] θ h m , 
() [] { } () [ ] ( ) ( ) [ ] ( ) [ ] ( ) [ ] { ⋅ = = = =
+ + θ μ θ θ μ θ μ θ μ θ θ θ θ
2 * 2 * 2 * 2 * ...
1 1 E E m E E e E E e E E h
hX hX t t () } θ θ E mh =  
() [] () { } θ θ μ θ h m E
2 * ,  ( ) [ ] { } ( ) [ ] ( )
1 1 * * + + =
t t hX
q
hX
q e X E E e E X E θ μ θ μ θ θ = 
=…= () [] () () [] ( ) [ ] ( ) ( ) { } θ θ μ θ μ θ θ μ θ μ θ θ θ θ h h m E E m E E
* * = = { ( ) [ ] () θ θ μ θ θ |
* X E E E · 
· ( )} θ h m , with  t q , 1 = . 
Defining 
* Cov and 
* Var  by using 
*
θ E  instead of  θ E  (2.17) becomes: 
() ( ) [] ( ( ) ( [ ( ) [ ]) ( )( ) [ ] ( )]) θ μ θ μ θ θ θ μ θ θ θ θ
* * * * | | , | E X H E X H E X H Cov − − = = 
= () () [] () [] () [] ( [ | | ,
1
* * *
1
* X Var E c Var c X Var E c Cov c
t
q r
r
q r q
t
q r
r
r ∑ ∑
≠
=
≠
=
+ = + θ θ θ μ θ θ μ θ μ )] θ ,  
where  t q , 1 = . 
Because of the symmetry of this system of equations in the variables:  t c c c ,..., , 2 1  one obtains 
t c c c = = = ... 2 1  and therefore: 
() ( ) [] ( ) [ ] ( ) [ ] { } θ θ μ θ μ θ θ | / , |
* * * X Var E tVar X H Cov c + =    (2.18) 
Inserting (2.18) into (2.14) and taking  ct z =
*  as in (2.9) one obtains: 
[] [] () [] [] [] () [] [] [] θ θ μ θ θ μ θ θ θ θ θ θ | | |
* *
*
*
1
* *
0 X H E tE
t
z
X H E E c X H E c
t
r
= − = − = ∑
=
- 
- () [] θ μ θ
* *E z . Consequently: 
[] [] () [] + = + − = + = ∑ ∑
= =
X z X
t
z
E z X H E X c c M
t
r
r
t
r
r r
a *
1
*
* * *
1
0 |
*
θ μ θ θ θ  
+ [] []
() []
() [] θ μ
θ μ
θ
θ
θ
θ * *
*
* |
E z
E
X H E
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− , as was to be proven. Taking the limit for:  0 → h , the 
original Bühlmann credibility formula results (see (1.1)). 
 
 
 Revista Informatica Economică nr.2(46)/2008  17
Conclusions 
In this paper we have obtained the best risk 
premium - in the sense of weighted mean 
squared error - to charge for period ( ) 1 + t , by 
truncating a series expansion. To be able to 
compute the loaded credibility estimates, we 
demonstrated the relevant (co-) variances of 
squares of the observations. The fact that it is 
based on complicated mathematics, involving 
conditional expectations, needs not bother 
the user more than it does when he applies 
statistical tools like SAS, GLIM, 
discriminant analysis, and scoring models. 
Apart from the optimal credibility result (2.1) 
for this situation, we have presented a 
linearized credibility formula for 
exponentially weighted squared error loss 
function (a linearized credibility formula for 
estimating Xt+1), using the greatest accuracy 
theory. 
 
References 
[1] Daykin, C.D., Pentikäinen, T., Pesonen, 
M.,  Practical Risk Theory for Actuaries, 
Chapman & Hall, 1993. 
[2]  De Vylder, F., Optimal semilinear 
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An Introduction to Non-Life Insurance Mathematics, volume of the “Mannheim Series”,

Credibility for Esscher premiums,

Loss distributions,

Optimal semilinear credibility,

Practical Risk Theory for Actuaries,

Semilinear credibility with several approximating functions,

Linearized Credibility Formula for Exponentially Weighted Squared Error Loss Function

http://revistaie.ase.ro/content/46/Atanasiu.pdf

Linearized Credibility Formula for Exponentially Weighted Squared Error Loss Function

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