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Informatica Economic, nr. 2 (42)/2007126

More General Credibility Models

Virginia ATANASIUDepartment of Mathematics, Academy of Economic Studies

e-mail: [email protected]

This communication gives some extensions of the original Bhlmann 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 ( ) ( )[ ] 100 += tXfE (the net premium fora contract with risk parameter: ) by a linear combination of given functions of the observ-

able variables: ( tXXXX ,...,, 21'

= ). So the estimators mainly considered here are linearfunctions of several functions of the observable random variables. The approxi-

mation to

nfff ,...,, 21( )0 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:

nfff ,...,, 21

ffp = p, try to find the optimal function . It should be

noted that the approximation to

f

( )0 based on a unique optimal approximating functionis 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 bygiving the semi-linear hierarchical model used in the applications chapter.

f

nfff ,...,, 21

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

Incredi

this article we first give the semi-linearbility model (see Section 1), which in-

volves only one isolated contract. Our prob-lem (from Section1) is the estimation of

( ) ( )[ ] 100 += tXfE (the net premium for acontract with risk parameter:

( )tXXXX ,...,, 21'

= . So our problem (from

Section 1) is the determination of the linearcombination of 1 and the random variables:

( )rp Xf , np ,1= , tr ,1= closest to( ) ( )[ ] 100 += tXfE in the least squares

sense, where is the structure variable. Thesolution of this problem:

) by a linearcombination of given functions

of the observable variables:nfff ,...,, 21

( ) ( )

= =

2

1 100

,0

n

p

t

r

rppr XfEMin

, where:rppr ,

= ,

is the optimal non-homogeneous linearized

estimator (namely the semi-linear credibilityresult). In Section 2 we discuss the casewhen taking for all:ffp = p , try to find

the unique optimal function . It should be

noted that the approximation to

I

f

( )0

f

basedon a unique optimal approximating function

is always better than the one furnished in

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Informatica Economic, nr. 2 (42)/2007 127

the semi-linear credibility model based onprescribed 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

nfff ,...,, 21

pqa pqb p ,

q n,0= ) appearing in the credibility factors

(wherepz np ,1= ). To obtain estimates for

these structure parameters from the semi-linear credibility model, in Section 3we em-

bed the contract in a collective of contracts,all providing independent information on thestructure distribution. We close this paper by

giving the semi-linear hierarchical modelused in the applications chapter (see Section4).

Section 1 (The approximation to ( )0 based on prescribed approximating func-

tions: )nfff ,...,, 21

In this section, we consider one contract withunknown and fixed risk parameter: , duringa period of t years. The yearly claim

amounts are denoted by: . The risk

parameter

tXX ,...,1 is supposed to be drawn from

some structure distribution function: ( )U . Itis assumed that, for given: , the claims areconditionally independent and identically dis-tributed (conditionally i.i.d.) with known

common distribution function ( )

,xFX

. The

random variables are observable,

and the random variable is considered

as being not (yet) observable. We assume

that:

tXX ,...,1

1+tX

( )rp Xf , np ,0= , 1,1 += tr have finitevariance. For: , we take the function of

we want to forecast.0f

1+tX

We use the notation:( ) ( ) |rpp XfE= (1.1)

1,1;,0 +== trnp

This expression does not depend on r.We define the following structure parameters:

( ) ( ) ( )rprppp XfEXfEEEm === | (1.2),

( ) ( ) |, rqrppq XfXfCovEa = (1.3),

( ) ( ) qppq Covb ,= (1.4),

( ) ( )rqrppq XfXfCovc ,= (1.5),

( ) ( )qrppq XfCovd ,= (1.6),

for: ,p nq ,0= 1,1 += tr . These expres-

sions do not depend on: 1,1 += tr . Thestructure parameters are connected by the fol-lowing relations:

pqpqpq bac += (1.7),

pqpq bd = (1.8),

for: nqp ,0, = . This follows from the covari-

ance relations obtained in the probability the-ory where they are very well-known. Just asin 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 ofthe 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 ( ) trnpXf rp ,1;,1, == closest to

( ) ( )[ ] |100 += tXfE and to ( )10 +tXf inthe least squares sense equals:

( ) == =

+=n

p

pp

n

p

t

r

rpp mzmXft

zM1

01 1

1 (1.9),

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where is a solution to the linear system of equations:nzzz ,...,, 21

( )[ ] qpn

p

pqpq tdzdtc 01

1 =+=

( nq ,1= ) (1.10)

or to the equivalent linear system of equations:( ) qp

n

p

pqpq tbztba 01

=+=

( nq ,1= ) (1.11)

Proof: we have to examine the solution of the problem:

( ) ( )

= =

2

1 100

,0

n

p

t

r

rppr XfEMin

(1.12)

Taking the derivative with respect to 0 gives:

( )[ ] ( )[ ] 01 1

0 = = =

rp

n

p

t

r

pr XfEE , or: . Inserting this expression for= =

=n

p

t

r

pprmm

1 1

00

0 into (1.12) leads to the following problem:

( ) ( )(

= =

2

1 100

n

p

t

r

prppr mXfmEMin

) (1.13)

On putting the derivatives with respect to 'qr equal to zero, we get the following system of

equations ( nq ,1= ; tr ,1'= ):

( ) ( )[ ] ( ) ( )[= =

=n

p

t

r

rqrpprrq XfXfCovXfCov1 1

''0 ,, ] (1.14)

Because of the symmetry in time clearly:pptpp ==== ...21 , so using the co-

variance results, for nq ,1= this system of

equations can be written as:

( )[=

+=n

p

pqpqpq dtcb1

0 1 ] (1.15)

Now (1.15) and (1.13) lead to (1.9) with:

t

zpp = , np ,1= .

Section 2(The approximationto ( )0 based on a unique optimal approximatingfunction: )f

The estimator for ( )0 of Theorem 1.1can be displayed as:

( ) ( tXfXfM ++= ...1 ) (2.1),

where:

( ) ( ) == +=

n

ppp

n

ppp mztmtxfztxf 1

01

111

.

Let us forget now about this structure ofand look for any function such that (2.1)

is closest to:

ff

( )0 . If are considered only

functions such thatf ( )1Xf has finite vari-ance, then the optimal approximating func-tion results from the following theorem:f

Theorem 2.1 (Optimal approximating func-tion)

( ) ( )tXfXf ++ ...1 is closest to ( )0 and to

( )10 +tXf in the least squares sense, if andonly if is a solution of the equation:f( ) ( ) ( )[ ] ( )[ ] 01 120121 + XXfEXXfEtXf

(2.2)Proof: we have to solve the following mini-mization problem:

( ) ( ) ( )[ ]2110 ... ttg

XgXgXfEMin + (2.3)

Suppose that denotes the solution to this

problem, then we consider:

f

( ) ( ) ( )XhXfXg += , with arbitrary,like in variational calculus. Let: ( )h

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( ) ( ) ( ) ( ) ( ) ( )[ ]{ }21110 ...... ttt XhXhXfXfXfE = + (2.4)Clearly for to be optimal,f ( ) 00' = , so for every choice of h :

( ) ( ) ( )[ ] ( ) ( )[{ } 0...... 1110 ] =+++ ttt XhXhXfXfXfE (2.5),

must hold. This can be rewritten as:( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 01 1211120 = XhXfttXhXtfXhXtfE (2.6),

or:

( ) ( ) ( ) ( )[ ] ( ) ]{ }[ ] 01 1201211 =+ XXfEXXfEtXfXhE (2.7)Because this equation has to be satisfied forevery choice of the function one obtains,the expression in brackets in (2.7) must beidentical to zero, which proves (2.2).

h

An application of Theorem 2.1:If can only take the values

and

11 ,..., +tXX

n,...,1,0 [ ]rXqXPpqr === 21 , for: q ,

nr ,0= , then ( ) ( tXfXf ++ ...1 ) is closestto ( )0 and to ( )10 +tXf in the least squares

sense, if and only if for nq ,0= , ( )qf is asolution of the linear system:

( ) ( ) ( ) ( )qr

n

r

n

r

qr

n

r

qr prfprftpqf ===

=+0

000

1 (2.8)

Indeed: ( ) ( )( )

( )

=

= =

n

r

qrp

qf

qXP

qfXf

01

1 : , nq ,0= ; ( )[ ] ( ) ( === =

120

12 XrXPrfXXfEn

r

) ( )

=

=

==n

r

qr

qrn

r p

prfq

0

0

; ( )[ ] ( ) ( ) ( )

=

==

====n

r

qr

qrn

r

n

r p

prfqXrXPrfXXfE

0

0012

00120 .

Inserting these expressions for: ( )1Xf ,( )[ ]12 XXfE and ( )[ 120 XXfE ] into (2.2)

leads to (2.8).

Section 3 (Parameter estimation)It should be noted that the approximation to

( )0 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 itis sufficient to know estimates for the struc-

tural parameters , (with

f

nfff ,...,, 21

pqa pqb , q n,0= )

appearing in the credibility factors (wherepz

np ,1= ). From this reason we give some un-

biased estimators for the structure parame-

ters. For this purpose we consider con-

tracts,

k

kj ,1= , and independent and

identically distributed vectors

k ( 2 )

( )jtjjjj XXX ,...,,, 1'

= , for kj ,1= . The

contract indexed j is a random vector consist-ing of a random structure parameter j and

observations: , wherejtj XX ,...,1 kj ,1= . For

every contract kj ,1= and for j fixed, the

variables: are conditionally inde-

pendent and identically distributed.jtj XX ,...,1

Theorem 3.1 (Unbiased estimators for thestructure parameters)Let:

= =

==k

j

jr

t

r

p

p

p Xfkt

Xkt

m1 1

..

^

)(11

(3.1)

=

= =q

j

q

jr

k

j

t

r

p

j

p

jrpqX

tXX

tX

tka

.1 1 .

^ 11

)1(

1 (3.2)

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t

aX

ktX

tX

ktX

tkb

pqqq

j

k

j

pp

jpq

^

...1

...

^ 1111

1

1

=

=

(3.3),

, then: , , , where: , ,

, , with

pp mmE =

^

pqpq aaE =

^

pqpq bbE =

^

==

t

r

p

jr

p

j XX 1. ==

t

r

q

jr

q

j XX 1.

= =

=k

j

t

r

p

jr

p XX1 1

.. = =

=k

j

t

r

q

jr

q XX1 1

.. ( )jrppjr XfX = , kj ,1( = and tr ,1= ),

( )jrqqjr XfX = , kj ,1( = and tr ,1= ), for p , nq ,0= , such that qp< .Proof: note that the usual definitions of the structure parameters apply, with j replacing

and replacing so:jrX rX ( )[ ] ====

prj

p

rj

jrpp mkt

ktm

ktXfE

ktmE

,,

^ 11pm ;

( ) ( )[ ( ) ( ) ( )

+

=

qjpjrqjpjrqjrpjrrj

q

jr

p

jrpq X

t

EXEX

t

XCovXEXEXXCov

tk

aE ..,

^ 11,,

1

1

( )qjrpjqjrpj XEXt

EXXt

Cov

..

1,

1

( )

=

+

+

1

1111,

1....

tkX

tEX

tEX

tX

tCov qj

p

j

q

j

p

j

( )

+

++

+

+++

rj

qppqpqqppqpqqppqpqqppqpq mmbat

mmbat

mmbat

mmba,

111

=( ) ( )

( )pqpq

rj

pqpqpqpq aat

tkt

tkba

tba

tk=

=

+

1

1

11

1

1

,

;

=

j

pqt

Covk

bE1

1

1^

,1111

,1111

, ............pqp

j

qp

j

q

j

p

j

q

j

p

j X

kt

CovX

kt

EX

t

EX

kt

X

t

CovX

t

EX

t

EX

t

X

+

=

+

+

t

aX

ktEX

ktEX

ktX

ktCovX

tEX

ktEX

t

pqqpqpq

j

pq

j ............

111,

1111,

1

1

k

+

+

+

++

+ pqqppqpqqppqpqqp

j

pqpq akt

mmbk

akt

mmbk

akt

mmbat

111111

=

+

=

+

+

t

ab

ka

ktba

tkt

ammb

k

pq

pqpqpqpq

j

pq

qppq

111

1

11pqb

k

kk

k

1

1

1

+

pq

pqpq

pq

pq

pq bt

a

t

ab

t

aa

kt

kk

k=+=

+

1

1

1.

Section 4(Applications of semi-linearcredibility theory)We close this paper by giving the semi-linear hierarchical modelused in the appli-cations chapter. Like in Jewells hierarchicalmodel we consider a portfolio of contracts,which can be broken up into P sectors eachsector consisting of groups of con-

tracts. Instead of estimating: ,

p pk

1,, +tjpX

jpptjpjppXE ,, 1,, += (the pure net

risk premium of the contract ),( )jp,

( )ptjpp XE 1,, += (the pure net risk pre-

mium of the sector ), we now estimate:p

1,,0 +tjpXf ,

( )jpptjpjpp

XfE ,, 1,,00 += (the pure

net risk premium of the contract ),( )jp,

( ) ( )ptjpp XfE 1,,00 += (the pure net risk

premium of the sector p ), where Pp ,1=

and pkj ,1= . In semi-linear credibility the-

ory the following class of estimators is con-

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sidered: ,

where are functions given in

advance. Let us consider the case of onegiven function in order to approximate

( )qirp

n

p

P

q

k

i

t

r

pqir Xfq

= = = =

+1 1 1 1

0

( ) ( ) nff ,...,1

1f

1,,0 +tjpXf or p0 and jpp ,0 . We

formulate the following theorem:Theorem 4.1 (Hierarchical semi-linearcredibility)Using the same notations as introduced forthe hierarchical model of Jewell and denoting

pjspjs XfX 00 = and pjspjs XfX 1

1 = one ob-

tains the following least squares estimates forthe pure net risk premiums:

( ) ( ) 110^

0 pzwppp Xzmzm += ,

( ) ( ) 110^

0 , pjwpjpjpjp Xzmzm += (3.1)

where: 1

1 .

1pjr

t

r pj

pjr

pjw Xw

wX

=

= ,

1

1 .

1pjw

k

j p

pj

pzw Xz

zX

p

=

= ,

11.1101. 1/ dwcdwz pjpjpj += (the credibility factor on contract level), with:

( )1 '001 , pjrpjr XXCovd = , ( )1 '111 , pjrpjr XXCovd = ,'rr , ( ) ( )11111 , pjrpjrpjr XVarXXCovc == ,

and: 11.1101. 1/ DzCDzz ppp += (the

credibility factor at sector level), with:

( )1 '001 , wpjpjw XXCovD = ,( )1 '111 , wpjpjw XXCovD = , ,'jj =11C

( ) ( )111 , pjwpjwpjw XVarXXCov == .Remark 4.1: the linear combination of 1 and

the random variables (1pjrX Pp ,1= ,

pkj ,1= , tr ,1= ) closest to 1,,0 +tjpXf and

to p0 in the least squares sense equals

( )p^

0 , and the linear combination of 1 and

the random variables (1pjrX Pp ,1= ,

pkj ,1= , tr ,1= ) closest to jpp ,0 in

the least squares sense equals .( )pjp ,^

0

References

[1] Goovaerts, M.J., Kaas, R., VanHerwaarden, A.E., Bauwelinckx,T.:Insurance Series, volume 3, EfectiveActuarial Methods, University of Amster-dam, The Netherlands, 1991.[2] Pentikinen, T., Daykin, C.D., and Pe-sonen, M.:Practical Risk Theory for Actuar-

ies,Universit Pierr et Marie Curie, 1990.[3] Sundt, B.:An Introduction to Non-LifeInsurance Mathematics, Veroffentlichungendes Instituts fr Versicherungswissenschaftder Universitt Mannheim Band 28 , VVWKarlsruhe, 1984.

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