A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
Anna Szymańska*
A PPL IC A T IO N OF SELECTED STATISTICAL M E T H O D S IN A SSE SSIN G H O M O G ENEITY OF INSU R A N C E PO RTFO LIO
Abstract
T h e fo u n d a tio n o f in su ran ce co m p an y activity is p ro p er a d ju stm e n t o f p rem iu m level to the risk level o f the insured. T h e insurer usually g ro u p s policies in p o rtfo lio s characterized w ith sim ilar risk.
H ow ever, th ere exist risk fa cto rs n o t observable directly, h a v in g im p act o n the claim size and frequency. A n im p o rta n t issue, therefore is the assessm ent o f p o rtfo lio hom ogeneity.
T h e p u rp o se o f this w o rk is the assessm ent o f selected m eth o d s o f testin g p o rtfo lio ho m o g en eity illu strated w ith an exam ple o f m o to r insurance.
Key words: hom o g en eity , p o rtfo lio , risk factors.
I. IN T R O D U C T IO N
A set o f insurance policies in p articu lar kind o f insurance is called a p ortfolio. T h e policies o f certain insurance p o rtfo lio are grouped into sets called ta riff classes. A kind o f risk represented by p artic u la r policy is a criterion o f th a t division into classes and is un d ersto o d as an expected loss o f an insurer.
Basic assum ptions o f portfolio con stru ctio n are:
1. A n insurance policy is located in p articu lar ta riff class (which is called a sub -p o rtfo lio o r a group) on the basis o f kn o w n risk factors.
2. T h e classes should be characterized by sim ilar level o f risk an d greater hom ogeneity th a n the w hole portfolio.
3. W ithin a p artic u la r class, sim ilar n u m b er and size o f losses for individual policies are expected. T his implies sim ilar insurance rate.
4. W ithin a p artic u la r class, the policies can be grouped into sub-classes, depending on the n um ber and size o f losses in previous years (bonus-m alus systems). D epending on the system, there arc different m odels o f tran sition from one class to another.
In case o f drivers civil responsibility insurances (О С ), an in surer can observe only part o f factors that decide about the level o f risk, i.e.: production year and type o f car, the aim o f ca r usage, engine capacity, d riv er’s age and gender. H ow ever, there are also factors th a t ca n n o t be directly observed but which considerably influence the risk level o f particu lar driver. T herefore the issue o f su b-portfolio hom ogeneity assessm ent is essential. T h e m ajority o f the m eth o d s o f insurance rates assessm ent need the assum ptio n of hom ogeneity in p o rtfo lio classes as well.
II. S E L E C T E D M E T H O D S
T h is p aper is an attem p t o f indicating m ethods, which m ay be used to assess the hom ogeneity in insurance portfolio, and directs special attention to statistical tests.
Let us assum e th a t insurer registers only occurrence o f a loss o r lack o f it (assum ing the occurrence o f one loss once a year only). T h en the ra n d o m variable (i.e. the num ber o f losses in p ortfo lio ) follow s the binom ial distribution. F ro m portfolio we random ly pick p policies. T h e rand om variable
Yj, j — 1 ,2 ..., p, is the num ber o f losses for y-th policy in n} o f years. T h en the probability estim ato r o f loss occurrence fo r the pooled sam ple (i.e. for w hole portfolio) has the follow ing form (N icm iro, 1997):
p
l Y j
9 = * T - 0 )
I " ;
i= i
F o r testing the hypothesis th at all o f random variables, Y;, follow the same d istrib u tio n one m ay apply the chi-square goodness-of-fit test (D om ański, 2000). T hen the null hypothesis has the form : H 0: = ... = 0j = ... = 0p w here 0j is a stru ctu ra l p aram eter o f the d istrib u tio n o f th e n um b er of losses. We reject H 0, when X2 ^ xl
-In case o f rejecting the null hypothesis, the m ethod presented above m ay be considered effective. A portfo lio m ay be treated as heterogeneous with regard to the n u m b er o f losses. If th ere is no ground for rejecting the null hypothesis one shall look for other m ethods o f portfolio hom ogeneity assessm ent.
In au to m o b ile insurances it is assum ed th a t n u m b er o f losses, X , in hom ogeneous portfolio is a random variable following the Poisson distribution and w ith the p aram eter o f loss intensity A:
P ( X = jc) = e ~ 1- , (x = 0 ,1 ,2 ,...) . (2) x!
T h en the q uestio n o f the exam ination o f p o rtfo lio o r p o rtfo lio classes hom ogeneities is reduced to the verification o f fit between the nu m b er o f losses and the P oisson distribution w ith test (D om ańsk i, 2001).
If p o rtfo lio is non-hom ogeno us then the param eter o f loss intensity has usually the gam m a d istrib u tio n w ith p aram eters a. and ß while the num ber o f losses has the negative binom ial d istrib u tio n w ith param eters p and к (Ilo ssa c k et al., 1999). It m eans th a t its prob ab ility fun ctio n has the follow ing form :
V o - P ) * , (x = 0 ,1 ,2 ,...) , (3)
where
к = a and p = /?/(l -f ß). (4) T herefo re, if the d istrib u tio n o f the num ber o f losses fits ( x2 fit-test) the negative binom ial d istrib u tio n , then there is n o g ro u n d for rejecting the hypothesis o f portfolio heterogeneity.
A n o th e r m ethod o f portfolio hom ogeneity assessm ent w ith respect to the n u m b er o f losses m ay be the graphical m eth o d p ro p o sed by H ossack. T his m eth o d it is assum es th a t the num b er o f losses in p o rtfo lio follows the negative binom ial distribution. Therefore we can say th a t non-hom ogeneity o f p o rtfo lio is assum ed. T hen, the param eters o f the g am m a d istribution o f ra n d o m variable o f losses intensity are calculated. T he next step is to draw a grap h o f the density o f the probability d istrib u tio n o f the num ber o f losses. I f the grap h is sim ilar to the graph o f th e p ro b a b ility density of the gam m a d istrib u tio n we infer th a t the p o rtfo lio is heterogeneous.
UI. APPLICATIONS
T h e study co nducted was based on d a ta from one insurance com pany (in the city o f Ł ódź), for О С autom obile insurances fo r the year 2000. F ro m the w hole p o rtfo lio , containing 31734 policies, 15 867 o f them were
draw n independently and grouped according to d riv er’s age. T h e d a ta are presented in T ab le 1.
I his study aim s to assess the portfolio hom ogeneity with th e m ethods presented above. A pplication o f statistical inference m ethods needs one sample. T h a t gives ad e q u ate num bers o f observation in specified (according to the n u m b er o f losses) classes.
T able 1. N u m b e r o f losses in tw o p o rtfo lio g ro u p s and in the w hole p o rtfo lio
G ro u p I 11 W hole
p o rtfo lio A ge o f a d riv er (in years) less th an 25 25 o r m ore
N o. o f claim s 0 2 907 10 221 13 128 1 592 1 843 2 435 2 66 210 276 3 5 18 23 4 0 5 5 N o. o f o b serv atio n s 3 570 12 292 15 867
S ourcc: In su re r’s com pany d ata.
F o r p o rtfo lio groups and for the whole p o rtfo lio (from T ab le 1), the fit betw een the distrib u tio n o f the num ber o f claim s and the Poisson d istrib u tio n and the negative binom ial d istrib u tio n was exam ined using the chi-square test.
Let r be the num ber o f classes, n, em pirical a m o u n t in i-th class, npt theoretical (expected) a m o u n t in i-th class. T he chi-square goodness-of-fit test statistic has the follow ing form:
2 v ’ ( n i - nPi) 2
X2 = I — --- — (5)
i = i " P i
O n the basis o f d ata from Table 1, the param eter o f the Poisson distribution was assessed assum ing
Я = x (6)
and the p aram eters o f the negative binom ial d istrib u tio n were assessed assum ing
where x is the value o f the sam ple m ean from the sam ple, and s2 is the sam ple variance. T h e param eters o f the gam m a d istrib u tio n were calculated according to fo rm u la (4).
T able 2. P aram eters o f the d istrib u tio n o f frequency o f claim s, based on d a ta from T ab le 1 G ro u p A verage n u m b er V ariance o f n u m b er
P aram eters o f negative binom ial d istrib u tio n
P a ra m e te rs o f G a m m a d istrib u tio n o f losses o f losses P к a fi I 0.207 0.209 0.98 16.95 16.95 81.88 II 0.19 0.2 0.95 3.61 3.61 19
W hole p o rtfo lio 0.2 0.21 0.952 3.96 3.96 19.8
Source: O w n research.
B asing on d a ta fro m T able 2 the theoretical am o u n ts o f th e n u m ber of losses were calculated. R esults are presented in T ables 3-5.
T able 3. T h eo retical an d em pirical a m o u n ts fo r group I o f in su ran ce p o rtfo lio
N o . o f claim s E m pirical a m o u n ts T heo retical a m o u n ts o f P oisson d istrib u tio n
T h eo retical a m o u n ts o f negative binom ial d istrib u tio n
0 2 907 2 903 2 962 1 592 600 54 2 66 63 75 3 5 4 9 4 0 0 0 W hole 3 570 3 570 3 570 S ource: O w n calculations.
T able 4. E m pirical an d theoretical am o u n ts fo r group II o f in su ran ce p o rtfo lio
N o . o f claim s E m p irical a m o u n ts
T heo retical a m o u n ts o f P oisson d istrib u tio n
T h eo retical a m o u n ts o f negative b inom ial d istrib u tio n
0 10 221 10 169 10 217 1 1 843 1 932 1 844 2 210 183 212 3 18 12 19 4 5 1 5 W hole 12 297 12 297 12 297
T abic 5. E m pirical and theoretical am o u n ts for the w hole insurance p o rtfo lio
N o. o f claim s E m pirical am o u n ts T heoretical a m o u n ts o f P oisson d istrib u tio n
T h eo retical a m o u n ts o f negative binom ial d istrib u tio n
0 13 128 13 121 13 184 1 2 435 2 493 2 379 2 276 236 275 3 23 15 25 4 5 2 3 W hole 15 867 15 867 15 867 Source: O w n calculations.
For the group I.
We shall verify the null hypothesis, H 0, th a t thed istrib u tio n o f the nu m b er o f losses in group I is the Poisson d istrib ution against alternative hypothesis, H t , th a t the d istrib u tio n o f n um ber o f losses in g ro u p I is n o t the Poisson distribution. T he value o f statistics x 2 — 0.5049. F o r a = 0.05 there is no ground for rejecting the null hypothesis.
W e shall verify the null hypothesis, H 0, th a t the d istrib u tio n o f the nu m b er o f losses in group I is the negative binom ial d istrib u tio n against alternative hypothesis, H ,, th a t the d istribu tio n o f the n u m ber o f losses in g ro u p I is no t the negative binom ial d istrib ution . T h e value o f statistics X 2 = 11.34. F o r a = 0.05 there is n o ground for rejecting the null hypothesis.
H ence, one m ay assum e th a t the distribu tio n o f th e n u m b er o f losses in g ro u p I is hom ogenous.
For the group II.
We shall verify the null hypothesis, I I 0, th a t the d istrib u tio n o f num ber o f losses in g roup II is the Poisson d istribution against alternative hypothesis, H 1; th a t distrib ution o f n um b er o f losses in g ro u p II is no t the Poisson distribution. T he value o f statistics x 2 — 27.349. F o r a = 0.05 we reject the null hypothesis to the adv antage o f the alternative hypothesis.W e shall verify the null hypothesis H 0, th a t the d istrib u tio n o f losses in group II is the negative binom ial distribution against alternative hypothesis, H j, th a t d istrib u tio n o f n um ber o f losses in g rou p II is n o t th e negative binom ial distribution. T he value o f statistics x 2 = 0.07. F o r a = 0.05 there is no gro u n d for rejecting the null hypothesis.
H ence, one m ay assum e th a t the d istrib u tio n o f n u m b er o f losses in g ro u p II is n o t hom ogenous.
For the whole portfolio.
We shall verify the null hypothesis, H 0, th at the distribution o f the num ber o f losses in portfolio is the Poisson distribution ag ainst alternative hypothesis, H l5 th a t distrib ution o f n u m ber o f losses in p o rtfo lio is no t the Poisson distribution. T he value o f statistics x 2 = 16.899. F o r a = 0.05 there is no grou nd for rejecting the null hypothesis.W c shall verify the null hypothesis, H 0, th a t th e d istrib u tio n o f the nu m b er o f losses in p ortfolio is the negative binom ial d istrib u tio n against alternative hypothesis, H t , th a t distribu tio n o f n um ber o f losses in portfolio is no t the negative binom ial distribution. T h e value o f statistics x 2 — 3.053. F o r a = 0.5 there is no ground fo r rejecting the null hypothesis.
H ence, one m ay assum e th a t the distribution o f the n um b er o f losses in the to tal p ortfolio is no t hom ogenous.
F o r the p urpo se o f results com parison let us apply the graphical m ethod o f insurance p o rtfo lio valu atio n th a t was proposed by H ossack (1999).
X
F igure 1. D ensity fu n ctio n o f the g am m a d istrib u tio n in tw o g ro u p s o f p o rtfo lio and in the w hole p o rtfo lio (see d a ta fro m T ab le 1)
Source: O w n calculations
O n the basis o f the graphs obtained, group 1 is h o m ogenou s (the graph o f density function is high and slender with small stan d ard deviation) and g ro u p 11 and the whole portfolio are n o t hom ogenous. T h e biggest n o n hom ogeneity occurs in group II.
V. S U M M A R Y
T h e results obtained with the graphical m ethod confirm the conclusions draw n from the application o f the chi-square goodness-of-fit test. H ow ever, b o th graphical m ethod and chi-square test for co nform ity, require large a m o u n t o f d a ta to estim ate th e p aram eters o f th e n eg ative binom ial distribution. G ro u p in g the portfolio according to large n u m b er o f factors m ay cause a reductio n o f the n um ber o f observations in p artic u la r groups.
In consequence, it can preclude the application o f these m eth od s. M oreover, practice often shows th a t both for the Poisson d istrib u tio n and for the negative binom ial d istrib u tio n there is no ground for rejecting the null hypothesis o f conform ity o f the num ber o f losses in p o rtfo lio in com parison with exam ined distribution. Some cases were also noticed when the chi-square goodncss-of-fit test rejects the null hypothesis both for the Poisson distribution and for the negative binom ial distribution. In those cases, searching for the form o f d istrib u tio n o f num ber o f losses seems to be reasonable. T hen, the foregoing m eth o d s can n o t be applied.
O ne o f the m eth o d s o f p ortfolio hom ogeneity exam in atio n is variance analysis (D om ań ski 2001). H ow ever, ap plication o f this m ethod needs the assu m p tio n th a t distrib u tio n s o f the num ber o f losses fo r p artic u la r policies are no rm al, w ith equal variances. In such case th e test statistics has the F -S nedecor distribution.
T h e issue o f hom ogeneity assessm ent o f po rtfo lio groups requires further researches, because hom ogeneity is the fundam ental assum ption in estim ation o f fu tu re losses, and - in consequence - in calculations o f insurance rates.
S earching for o th er m eth o d s o f portfo lio division into ta riff groups to achieve hom ogeneity o f ta riff classes m ay be a solution. In practice, however insurance com panies d o n o t look for such m eth ods, despite the fact th at losses in auto m o b ile insurances confirm incorrect co n stru ctio n o f portfolios.
R E F E R E N C E S
D o m a ń sk i Cz. (red.) (2000), N ieklasyczne m elody sta tystyczn e, PW E , W arszaw a. D o m a ń sk i Cz. (red.) (2001), M eto d y sta tysty c zn e, W yd. U Ł , Ł ódź.
H o ssack I.B ., P o llard J .H ., Z eh n w irth B. (1999), Introductory S ta tistics with A pplications in
General Insurance, C am bridge.
A p p licatio n o f Selected S tatistical M eth o d s in A ssessing H o m o g en eity ... 223
A n n a S z y m a ń s k a
ZASTOSOWANIE TESTÓW STATYSTYCZNYCH DO BADANIA JEDNORODNOŚCI PORTFELA UBEZPIECZEŃ
Streszczenie
P o d sta w ą p raw idłow ego fu n k cjo n o w an ia to w a rzy stw a ubezpieczeniow ego je s t odpow iednie d o p a so w a n ie w ysokości składek d o po zio m u ry zy k a, jak ie re p rezen tu ją ubezpieczani. U b ez pieczyciel najczęściej grupuje ko n trak ty ubezpieczeniowe w portfele charakteryzujące się zbliżonym poziom em ryzyka.
Istn ieją jed n a k czynniki bezp o śred n io nieobserw ow alne, w pływ ające n a wielkość i czę stość szkód. D la te g o isto tn y m zagadnieniem jest ocena jed n o ro d n o ś ci p o rtfe la ubezpiecze niow ego.
Celem referatu je s t ocena w ybranych m etod, służących d o sp raw d zan ia jed n o ro d n o ści portfeli ubezpieczeniow ych n a przykładzie d an y ch ubezpieczeń k o m u n ik acy jn y ch .
