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170 2014, XVII, 2 DOI: 10.15240/ ul/001/2014-2-013
In oduc ion
Ahuge expansion o a compe i i e insu ance
ma ke o ms pa o he ansi ion p ocess in
he o me Socialis coun ies. New in e na ional
ends in secu ing i s inancial s abili y and new
ypes o isk demand a high-powe ed de elopmen
o ac ua ial science and i s applica ion in
insu ance p ac ice. The on-going Eu opean
Union Sol ency II p ojec , aimed a imp o ing
he quali y o assessmen o he sol ency o an
insu e , places emphasis on he modelling o
isk and on in e nal models o managing an
insu e 's isk. This una oidably leads o a mo e
in dep h use o he heo e ical esul s o
ac ua ial science and o hei u he heo e ical
de elopmen .
Imp o ing he quali y o p emium calcula ion
me hods is an e ec i e ac o in educing he
insu ance echnical isk o an insu e . In doing
so i akes in o accoun mo e and mo e a ing
ac o s, hanks o which mo e homogeneous
a ing a i s eme ge allowing he possibili y o
de e mining mo e equi able p emiums. On he
o he hand when he insu e de e mines he
p emium i has o con end wi h a smalle
amoun o in o ma ion conce ning he isk, o
which he gi en a i class is in ended. I he
insu e has inadequa e pas da a ela ing o he
a i class, i can make use o da a om o he
isk classes which a e o a gi en ex en simila .
The insu ance isk, o which he e is
agi en a i class, is called he indi idual isk.
Each a i class o ms pa o he whole
po olio and hus each indi idual isk is pa o
he collec i e, i.e. po olio, isk. On he one
hand he insu e in gene al has a i s disposal
a ela i ely la ge amoun o s a is ical in o ma ion
conce ning he collec i e isk. This in o ma ion
is, howe e , mo e o less limi ed o p icing
indi idual isks, due o he wide a ie y o a i
classes. An insu e usually has li le s a is ical
in o ma ion conce ning indi idual isks and
he e o e should use bo h sou ces o da a. Tha
is he undamen al idea behind c edibili y
heo y.
Back in 1918 Albe W. Whi ney p oposed
ha he c edible p emium Pc ed should be
alinea combina ion o he indi idual p emium
Pind and he collec i e p emium Pcol
Pc ed = z . Pind + (1 – z) . Pcol (1)
whe e Pind is he a e age le el o claim pe
con ac in he gi en a i class and Pcol is he
espec i e mean o he whole insu ance
po olio. F om a s a is ical poin o iew he
c edible p emium is he weigh ed a i hme ical
a e age o he indi idual and collec i e
p emiums, whe e zis he ela i e weigh o he
indi idual p emium and (1 – z) ha o he
collec i e p emium. The weigh zis e e ed o
as he c edibili y ac o . I akes alues in he
in e al 〈0;1〉and exp esses he le el o
con idence in he da a o he gi en insu ance
class when de e mining he p emium o ha
class. A con idence le el (weigh ) o 0 is
assigned o indi idual isk da a which ha e no
use o de e mining he p emium and a le el o
1 is gi en o hose indi idual isk da a which
comple ely su ice in o de o de e mine he
p emium o ha isk class. This means ha , i
z= 0, we mus comple ely ely on he da a om
compa able isks when de e mining he
p emium o he gi en class. I z= 1, hen we
can comple ely ely on he da a o he gi en
indi idual isk.
Bühlmann es ablished he heo e ical
ounda ion o mode n c edibili y heo y, p esen ed
as dis ibu ion ee c edibili y es ima ion.
Bühlmann and S aub (see [3]) gene alized
Bühlmann’s classical c edibili y model. Wi hou
MULTIDIMENSIONAL CREDIBILITY MODEL
AND ITS APPLICATION
Vie a Pacáko á, E ik ·ol és, Bohdan Linda
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doub he Bühlmann-S aub model is he mos
impo an model in c edibili y heo y. O iginally
i was aimed a he de e mina ion o he c edible
p emium, bu a e ce ain modi ica ions i can
also be used o de e mine a eliable es ima e o
claims equency o o he a e age amoun o
claim. I s impo ance lies in he possibili y o
using i in a wide ange o applica ions in he
ac ua ial p ac ice o non-li e insu ance, bu i
can also be used in li e insu ance and in
einsu ance. Besides ha he Bühlmann-S aub
model ep esen s a base o u he mo e speci ic
models, such as he hie a chical, mul idimen-
sional o eg ession c edibili y models. In his
pape we will conside a gene alisa ion o he
one-dimensional Bühlmann-S aub c edibili y
model o he mul idimensional c edibili y model.
1. Gene alisa ion o he One-
Dimensional Bühlmann-S aub
Model o he Mul idimensional
C edibili y Model
The aim o his pa o a icle is o b ie ly
desc ibe he ma hema ical and s a is ical basis
o one-dimensional Bühlmann-S aub model
( o de ails see [1] o [7]) and poin ou he
analogy o mul idimensional Bühlmann-S aub
model wi h he o iginal Bühlmann-S aub
model. This pa o he a icle also shows he
p ac ical use o mul i-dimensional Bühlmann-
S aub model in insu ance p ac ice.
Le us suppose ha he po olio o
con ac s is di ided in o I isk classes, o which
we wan o de e mine he ne p emium. Le he
i h isk class (i= 1, 2, ..., I) be cha ac e ised by
he indi idual isk p o ile
ϑ
i, which is he
ou come o a andom a iable
θ
i. To calcula e
he ne p emium o each indi idual isk class
we will make use o he quan i ies:
Sij – he agg ega e claim amoun o isk iin
yea j(j= 1, 2, ..., n),
wij – he numbe o con ac s o isk iin
yea j,
Sij
Xij = –––– – he a e age annual le el o
wij
claim o isk iin yea j,
which a e in gene al known.
The one dimensional Bühlmann-S aub
c edibili y model s ems om he ollowing
assump ions Bühlmann, S aub [3], He zog [6],
Pacáko á [8].
BS1: Condi ional on he alue o
θ
i he Xij o
j = 1, 2, ..., na e independen wi h pa ame e s:
E[Xij|
θ
i] =
µ
(
θ
i)(2)
σ
2
(
θ
i)
Va [Xij|
θ
i] = ––––– (3)
wij
BS2:The pai s (
θ
1, X1), (
θ
2, X2) ... a e
independen and
θ
1,
θ
2a e independen and
iden ically dis ibu ed.
I he assump ions BS1aBS2a e me , he
c edible es ima e o he ne p emium is
(4)
whe e he a e age annual le el o claim pe
con ac om he i h isk class
(5)
is he es ima e o he indi idual p emium, µ0is
he collec i e p emium and he c edibili y ac o
has he o m
(6)
The es ima ion o he so-called s uc u al
pa ame e s µ0, σ2aτ2plays a undamen al ole
in c edibili y heo y. De ini ions o he s uc u al
pa ame e s and hei in e p e a ion a e as
ollows:
S uc u al pa ame e In e p e a ion
collec i e p emium
he a iance o he
a e age annual claim
amoun s wi hin a g oup
he in e -g oup a iance
o he a e age annual
claim amoun s
The c edibili y es ima e (4), in which we
eplace he unknown collec i e p emium µ0by
he es ima e
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(7)
is he so-called homogeneous c edibili y
es ima e wi h he o m (see [4])
(8)
The es ima e (7) o he collec i e p emiumis
he weigh ed a i hme ical a e age o he
indi idual p emiums X
—
i, whe e he weigh s a e
he c edibili y ac o s zi.
Assuming ha he e is he same numbe o
con ac s in each isk class in each yea , he
c edible p emium de ined by o mula (8) should
co e he claims ou go o he whole po olio
and o he whole pe iod o n p e ious yea s
conside ed. This esul is exp essed in [1] by
he balance p ope y:
(9)
Fo some classes o business (liabili y,
comme cial i e) a high pe cen age o he o al
claim amoun comes om a small numbe o
e y la ge claims. In hese cases a ela i ely
small numbe o claims, which a e “la ge”
compa ed wi h he “s anda d” claim, ha e a la ge
weigh when he p emium a e is de e mined. I
he e o e makes sense o es ima e he a e age
claim amoun sepa a ely o “la ge” and o
“s anda d” claims. I in ac we we e o calcula e
he c edible p emium using he one-dimensional
Bühlmann-S aub model sepa a ely o “la ge”
and “s anda d” claims, we would no ake in o
accoun any dependency be ween hem.
The e o e i is mo e app op ia e o es ima e he
ec o o ne p emiums, which in his case is
he wo-dimensional ec o
The whole ec o o ne p emiums o indi idual
isk i can be es ima ed by he mul idimensional
c edibili y model, which is a gene alisa ion o
he one-dimensional Bühlmann-S aub model.
Ano he example o he use o he mul idi-
mensional model would be he calcula ion o
he c edible p emium o he sepa a e isk
classes o a whole po olio and he simila isk
classes o o he insu e s. I , he e o e, an
insu e disposes o da a, e en om ano he
insu e , ei he b oken-down o in o al, i can
use he mul idimensional c edibili y model.
Thus, hanks o i , we can calcula e he
p emium o a gi en isk class using ei he pas
da a o ha class o pas da a o he whole o
he po olio o e en his o ical da a o he gi en
isk class o o he whole po olio o simila
isks o ano he insu e .
Whe eas in he one-dimensional
Bühlmann-S aub c edibili y model we es ima e
he ne p emium µ(θi) o he isk i (i= 1, 2, ..., I),
in he mul idimensional Bühlmann-S aub
c edibili y model we es ima e o isk i he whole
ec o o ne p emiums
(10)
In he one-dimensional Bühlmann-S aub
c edibili y model we obse e he quan i ies Xij
( he a e age annual claim amoun o isk iin
yea j) wi h weigh s wij ( he numbe o con ac s
o isk iin yea j). In he mul idimensional
Bühlmann-S aub c edibili y model we obse e
he ec o o a e age annual claim amoun s o
isk iin yea j
(11)
and he weigh s a e he ec o o he numbe o
con ac s o isk iin yea j
(12)
The mul idimensional Bühlmann-S aub
c edibili y model s ems om he ollowing
assump ions ( o de ails see [2]):
MBS1: Condi ional on he alue o θ1 he
alues Xij o j= 1, 2, ..., na e independen wi h
ap-dimensional ec o o mean alues:
(13)
and wi h a co a iance ma ix o ype p
×
p
.
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(14)
MBS2: The pai s ,
whe e he a e indepen-
den and he θ1, θ2, ... a e independen and
iden ically dis ibu ed.
The co a iance ma ix (14) is diagonal, hus
i is assumed ha he componen s o ec o Xij
a e independen . In some si ua ions indepen-
dence is no assu ed. Fo example, i we wan
o use he mul idimensional c edibili y model o
es ima e he claim equencies o la ge claims
and he claim equencies o s anda d claims,
he assump ion o hei independence is no
me . The claim equencies o la ge claims and
o s anda d claims ela e o he same numbe o
con ac s and he e o e he weigh s
a e equal. In he case o equali y o he weigh s
we can eplace he so-
called s anda d assump ion by (see [1]) he so-
called al e na i e assump ion:
In assump ion MBS1in place o he
co a iance ma ix (14) we can conside he
co a iance ma ix
(15)
whe e he weigh s o all he componen s o he
ec o Xij a e equal, i.e.
The s anda d and al e na i e assump ions
do no in ac co e all possible si ua ions, bu
hey do include he majo i y o cases which
a ise in p ac ice. Subjec o hese condi ions we
can gene alise o mulae (4) o (6), ela ing o
he one-dimensional c edibili y es ima e, using
he mul idimensional c edibili y es ima e
de ined la e by o mulae (16) o (21).
I assump ions MBS1aMBS2a e me , we
can exp ess he c edibili y es ima e o he ne
p emium as ollows
(16)
whe e Biis he indi idual es ima e and he
collec i e es ima e o he ec o
µ
(θi). Ziis he
c edibili y ma ix and i can be shown ha , a e
gene alising o mula (6) o he c edibili y ac o ,
he ollowing applies
(17)
I is clea ha he ec o o a e age annual
claim amoun s o he i h isk class ep esen s
he indi idual es ima e o he ec o
µ
(θi) and
he e o e
(18)
he e, o k= 1, 2, ..., p, he k h elemen o he
ec o Biis
(19)
We can he e o e exp ess he ec o Biby
he o mula
(20)
whe e he ma ix o weigh s has he elemen s
(21)
I s anda d assump ion (14) is me , he ma ix
has elemen s
(22)
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I he co a iance ma ix o he ec o
mee s he al e na i e assump ion (15), he
ollowing applies
(23)
The mul idimensional c edibili y model has
h ee s uc u al pa ame e ec o s , Sand T
which a e analogous o he s uc u al pa a-
me e s
µ
0,
σ
2and
τ
2o he one-dimensional
Bühlmann-S aub c edibili y model. Thei
de ini ions a e as ollows:
The ma ices o s uc u al pa ame e s
is he ec o o collec i e p emiums
is he co a iance ma ix o he ec o
is he co a iance ma ix o he
collec i e p emium ec o
We ob ain he homogeneous c edibili y es ima e
i in o mula (16) we eplace he
unknown ec o µo he collec i e p emiums by
i s es ima e
(24)
Simila o he one-dimensional Bühlmann-
S aub c edibili y model, also in he mul idi-
mensional model he homogeneous es ima e
and he indi idual elemen s o he ec o
sa is y he balance p ope y
(25)
To compa e he accu acy o c edibili y
es ima es, collec i e es ima e and indi idual
es ima es, he e a e used quad a ic losses. The
quad a ic loss o he c edible p emium is
zi-mul iple o he quad a ic loss o he indi idual
p emium and (1 – zi)-mul iple o he quad a ic
loss o he collec i e p emium (see [9]). Since zi
and (1 – zi) ake alues om in e al 〈0;1〉, so
he quad a ic loss o he c edibili y p emium is
less o a he mos equal o quad a ic losses o
he collec i e and he indi idual p emium.
2. Es ima ion o he S uc u al
Pa ame e s o he One-
Dimensional Bühlmann-S aub
and he Mul idimensional
C edibili y Models
In he one-dimensional Bühlmann-S aub
c edibili y model i is necessa y o es ima e, no
only he collec i e p emium as shown in
o mula (8), bu also he s uc u al pa ame e s
o a iabili y σ2and τ2. The s uc u al pa ame e
σ2is he mean o he wi hin g oup a iances
σ2(θi). As he es ima e o he a iance σ2(θi)in
he i h isk class we can ake he sample a iance
(26)
I is possible o es ima e he mean o hese
a iances as he a i hme ic mean. As epo ed
by Bühlmann and Gisle [1] he op imal weigh s
should be p opo ional o .
This exp ession depends on he momen s up o
he ou h o de . These momen s a e unknown
and hei es ima ion equi es u he s uc u al
pa ame e s, he e o e such a p ocedu e is no
app op ia e. In he special case, whe e Xij a e
condi ionally on θino mally dis ibu ed, he
abo e men ioned exp ession does no depend
on i. The e o e i is op imal o use he equal
weigh s o all pa ial a iances. On he one hand,
he mos o insu ance da a a e no no mally
dis ibu ed, on he o he hand, weigh s, which
would be be e , a e no known. Fo his eason,
Bühlmann and S aub [3] sugges ed o
es ima e he s uc u al pa ame e σ2using he
simple a i hme ic mean o he indi idual
a iances :
(27)
The cha ac e is ic is an unbiased and
consis en es ima e o he s uc u al pa ame e
σ2. A eade can ind p oo s o hese p ope ies
in [1] on he page 94.
An unbiased and, p o ided he e is no
dominan isk class in he po olio, also
consis en es ima e o he pa ame e τ2(see
e.g. [7]) is gi en by
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(28)
whe e
(29)
(30)
And
(31)
Rema k:
I he si ua ion a ises, ha is nega i e,
di e ences be ween he isk classes a e no
demons able. In his case we se he
pa ame e τ2 o 0. The c edibili y ac o s (6)
hen equal 1 and he c edibili y p emiums o all
he isk classes a e he same and a e equal o
he a e age annual claim amoun o he whole
po olio (31).
So ha he es ima e o he s uc u al pa a-
me e τ2can also be used in he si ua ion desc ibed
in he ema k, he es ima e o he s uc u al
pa ame e τ2is de i ed om he o mula
(32)
The homogeneous c edibili y es ima e in
he mul idimensional c edibili y model al eady
has inco po a ed in i an es ima e o he ec o
µin acco dance wi h o mula (24). Hence we
will concen a e only on es ima ion o he ma ix
o he s uc u al pa ame e s Sand T.
To es ima e he diagonal elemen s o he
ma ices SaTwe can use analogous
app oaches o hose in he case o he one-
dimensional Bühlmann-S aub model. The
diagonal elemen s o he ma ix Sa e es ima ed
in a simila way o ha o he pa ame e σ2in
he Bühlmann-S aub model, in acco dance
wi h o mula (27) hus:
(33)
The diagonal elemen s o he ma ix T again
can be es ima ed simila ly o he s uc u al
pa ame e τ2in he Bühlmann-S aub model.
Analogously o o mulae (32) and (28) o (31)
we hen ge
(34)
whe e
(35)
and
(36)
(37)
(38)
We will deno e he non-diagonal elemen in
he k h ow and he l h column o he ma ix Tby
τkl. Fi s we es ima e τkl using he weigh s
hus:
(39)
Simila ly we es ima e τkl using he weigh s
:
(40)
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whe e
(41)
The ma ix Tis he co a iance ma ix,
which means ha i s diagonal elemen s a e he
a iances τ2
kand i s non-diagonal elemen s a e
he co a iances τkl. By aking he a io o he
co a iance τkl and he p oduc o he s anda d
de ia ions τk. τlwe ge he co ela ion coe icien ,
which akes alues in he in e al 〈–1; 1〉. On
he basis o his i mus be he case ha |τkl|≤τk. τl.
I his condi ion is me , he simple a i hme ic
mean o he es ima es (39) and (40) is an
es ima e o he co a iance τkl. In gene al he
ollowing es ima e is p oposed (see [1])
(42)
To es ima e he s uc u al pa ame e s o he
one-dimensional and also he mul idimensional
Bühlmann-S aub c edibili y model we can use
s a is ical me hods, o which we can ind
p ocedu es in s a is ical p og amming packages.
We will concen a e only on he mul idimen-
sional c edibili y model. So he k h diagonal
elemen (33) o he ma ix Sis he simple
a i hme ic mean o he sample a iances
(43)
whe e
(44)
in each o he isk g oups, ela ing o he k h
elemen o he es ima ed ec o
Le us modi y he elemen s o ma ix T. We
can ew i e he cha ac e is ics T(k) in he o m
(45)
whe e is he sample a iance o he
a iable B(k) wi h ela i e weigh s :
(46)
We hen know ha in acco dance wi h
o mula (45) we can exp ess he es ima e (35)
o he diagonal elemen τ2
ko he ma ix Tby he
o mula
(47)
In he case o he es ima es (39), (40) o he
non-diagonal elemen s τkl o he ma ix Ti is
again he case ha
(48)
(49)
whe e is he sample co a-
iance be ween a iables B(k), B(l) wi h ela i e
weigh s :
(50)
and is he sample co a ian-
ce be ween a iables B(k), B(l) wi h ela i e
weigh s :
(51)
Fo k= 1 we ge he sample a iance (46)
and he sample co a iance (50) ( o i= 1, 2, ..., p)
om he i s ow o he sample co a iance
ma ix o a iables B(1), B(2), ..., B(p) making use
o he ela i e weigh s . Simila ly o k= 2
.
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we can ead o he sample a iance (46) and
sample co a iance (50) ( o l= 1, 2, ..., p) om
he second ow o he sample co a iance
ma ix o a iables B(1), B(2), ..., B(p) by making
use o he ela i e weigh s . Analogously
we can ob ain he sample a iance (46) and
sample co a iance (50) o k = 3, 4, ..., p. F om
hese elemen s we can hen de elop he ma ix
(52)
whose k h ow is he k h ow o he sample
co a iance ma ix o a iables B(1), B(2), ..., B(p)
making use o he ela i e weigh s .
I we look a o mula (48) and also o mula
(49) we see ha he es ima es o he non-
diagonal elemen s o he k h ow o he ma ix
Ta e a mul iple o he k h ow o he
ma ix SB. F om o mula (47) i is ob ious ha
he es ima e o he k h diagonal elemen o
he ma ix Tis a mul iple o he di e-
ence be ween he k h diagonal elemen o he
ma ix SBand he k h diagonal elemen o he
ma ix . Gi en ha he ma ix is diagonal
wi h elemen s we can de ine
he ma ix
(53)
whe e ⊗is he Hadama d ma ix p oduc and
ma ix Co ype p
×
phas elemen s
(54)
An es ima e o he co a iance ma ix Tis
hen he ma ix
(55)
I any o i s diagonal elemen s a e nega i e,
hen in acco dance wi h o mula (34) we
eplace hem by he alue 0 and i any o i s
non-diagonal elemen s do no mee he
condi ion hen in acco dance wi h
o mula (42) we eplace hem by he alue
sgn
3. Example o Applica ion
An unnamed insu ance company di ides i s
MTPL po olio in o eigh a i classes. Fo his
ype o insu ance he insu e has a ailable no
only i s own da a, bu also summa y da a om
o he companies. Speci ically o each isk
class i has a ailable he ollowing s a is ics:
X
–
i– he a e age claim amoun (in €), pe
con ac yea , o a pe iod o
nyea s,
– he sample s anda d de ia ion o
he a e age claim amoun s o he
pe iod o n yea s,
wi• – he numbe o con ac s o he pe iod
o nyea s.
These da a a e se ou in Table 1.
The ac ua y has he ask o se ing he ne
p emium o he nex insu ance yea . Gi en ha
one expec s a signi ican dependence be ween
he own company da a and ha o he o he
insu e s, i is desi able o use he
mul idimensional c edibili y model o se he ne
p emium o each a i class. In o de o ge he
c edibili y es ima e o he ne p emium using
he mul idimensional model we need o ca y
ou he calcula ions as shown in Table 2.
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Finance
178 2014, XVII, 2
Tab. 1: Own da a and da a om o he insu e s
Sou ce: own
Tab. 2: Table o calcula ions
Sou ce: own calcula ion
By aking he simple a i hme ic means o
he sample a iances and he simple
a i hme ic means o he sample a iances
we ob ain es ima es o he diagonal elemen s o
he ma ix S
An es ima e o he ma ix Sis he e o e he
ma ix
To a oid he ela i ely leng hy calcula ions
o es ima e ma ix Twe will use he adjus men s
which we made ea lie in he a icle (exp essions
(45) o (52)). To calcula e he elemen s o
ma ix (52) we can use se e al o s a is ical
so wa e applica ions. To es ima e ma ix Tin
his a icle we use co ela ion analysis
p ocedu e in SAS En e p ise Guide applica ion.
Fi s we es ima e he co a iance ma ix o
a iables B(1) aB(2) ( he alues a e shown in
Tables 1 and 2), whe eby we make use o he
ela i e weigh s . Simila ly we es ima e he
co a iance ma ix o a iables B(1) aB(2)
making use o he ela i e weigh s . We hus
ge he co a iance ma ices shown in Table 3.
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