Application of Discriminant Analysis on Romanian Insurance Market

Transcription

1 Applcaon of Dscrmnan Analyss on Romanan Insurance Marke n Consann Anghelache Dan Armeanu Academy of Economc Sudes, Buchares Absrac. Dscrmnan analyss s a supervsed learnng echnque ha can be used n order o deermne whch varables are he bes predcors of he classfcaon of obecs belongng o a populaon no predeermned classes. A he same me, dscrmnan analyss provdes a powerful ool ha enables researchers o make predcons regardng he classfcaon of new obecs no predefned classes. he man goal of dscrmnan analyss s o deermne whch of he N descrpve varables have he mos dscrmnaory power, ha s, whch of hem are he mos relevan for he classfcaon of obecs no classes. In order o classfy obecs, we need a mahemacal model ha provdes he rules for opmal allocaon. hs s he classfer. In hs paper we wll dscuss hree of he mos mporan models of classfcaon: he Bayesan creron, he Mahalanobs creron and he Fsher creron. In hs paper, we wll use dscrmnan analyss o classfy he nsurance companes ha operaed on he Romanan marke n We have seleced a number of egh (8) relevan varables: gross wren premum (GR_WRI_PRE), ne mahemacal reserves (NE_M_PES), gross clams pad (GR_CL_PAID), ne premum reserves (NE_PRE_RES), ne clam reserves (NE_CL_RES), ne ncome (NE _IN- COME), share capal (SHARE_CAP) and gross wren premum ceded n Rensurance (GR_WRI_PRE_CED). Before proceedng o dscrmnan analyss, we performed cluser analyss on he nal daa n order o denfy classes (clusers) ha emerge from he daa. Key words: dscrmnan analyss; classfer; classfcaon cos; predcon Fsher classfer; Bayesan classfer; Mahalanobs classfer; nsurance. JEL Codes: G22. REL Codes: B, C. n Applcaon of Dscrmnan Analyss on Romanan Insurance Marke 5

2 heorecal and Appled Economcs In order o undersand how dscrmnan analyss operaes, frs we mus sae he classfcaon problem. Consder a fne populaon P whose obecs are descrbed by N varables. he fne populaon s called ranng se (or learnng se) and he varables are called descrpve varables. hese varables are called predcve varables. Le v, v 2,..., v N be he descrpve varables and v ', v',..., ' 2 v be he predcve n varables. herefore we have ' ' ' {,v,...,v } { v,v,...,v } v 2 n Í 2 N () Generally speakng, he classfcaon problem requres an algorhm o denfy he crera accordng o whch obecs are assgned o classes. Recall he populaon P we defned earler. Le us now consder ha P s paroned no classes p, p 2,..., p called nal classes. he nal classes sasfy he properes: p Ì P =, 2,..., (2) U = p = P (3) I s mporan o noe ha we do no requre he nal classes o be dson subses of P: p I p ¹ F ¹ (4) As menoned prevously, dscrmnan analyss s a predcon ool. he core of hs echnque consss of deermnng an effcen way of paronng he ranng se P no dson classes (subses) called predcve classes. Le p, p 2,..., p be he predcve classes. We have p Í P =,2,..., (5) U = p = P (6) p I p = F (7) ¹ I follows mmedaely from (7) ha, n general, he predcve class p and he nal class p are dfferen: p ¹ p (8) hs happens because he predcve classes resul from he runcaon of he nal classes. As such, he predcve class p represens a subse of he correspondng nal class p : p Í p =,2,..., (9) Of course, perfec classfcaon requres ha p = p =,2,..., (0) In order o classfy obecs, we need a mahemacal model ha provdes he rules for opmal allocaon. hs s he classfer. In hs paper we wll dscuss hree of he mos mporan models of classfcaon: n he Bayesan creron; n he Mahalanobs creron; n he Fsher creron. he Bayesan creron s based on he mnmzaon of classfcaon coss. A correc classfcaon has cos zero and an ncorrec classfcaon has cos c: C ( p, p ) ì0, p = í î c, p where ( p, ) = p ¹ p () C p s he cos generaed by he classfcaon no class p of an obec acually belongng o class p and c s a posve consan. Obvously, he perfec classfcaon requres ha he predcve and he nal classes be dencal. For ease of compuaon we wll make he followng noaons: n P ( p, p ) = he probably of classfcaon no class of an obec p acually belongng o class p ; 52

3 n () x f p = he probably densy of obecs belongng o class p (). x denoes a vecor of m values of varables ha defne he obecs n he populaon; n R = he subse of R m n whch he vecors ha defne obecs belongng o class p ake values; n f (x) = he uncondonal probably densy of obecs. Accordng o probably heory, can be compued usng he formula: ( p, p ) = ò f p () x P dx (2) R For each class p we defne he cos of classfcaon (C(p )) as follows: C ( p ) = å C( p, p ) P( p, p ) = (3) he expeced oal cos of classfcaon (C) s: C = åå = = C ( p, p ) P( p, p ) P( p ) (4) where P(p ) s he a pror probably of occurrence for class p. Unless furher nformaon s provded, we can consder P = 2, ( p ) P( p ) =... = P( p ) =... = P( p ) = whch means ha all classes have equal probables of occurrence. We can rewre (4) as é å ( ) å ( ) ( ) = = ú ú ù C = P p ê C p, p P p, p ê ë û (5) akng no accoun (2), (5) becomes: C = é åp( p ) å ( ) ò ( ) ê ê C p, p f p x ù dx = = ú úú ê ë R û whch can be wren as é å ò å ( ) ( ) () = = ú ú ù C = ê P p C p, p f p x dx êë û R (6) (7) where ò êå P( p ) C( p, p ) f p () x dx R é êë = ù ú ú û denoes he cos of classfcaon no class p of all he obecs acually belongng o class p. Now defne S () x = å P( p ) C( p, p ) f p () x = herefore, we have: C = å ò = R S () x dx. (8) As saed before, he classfcaon rule s gven by he mnmum cos prncple. Gven he defnon of S () x, R can be expressed as R { x Î RP S () x - S () x <,() " ¹ } where S () x S () x = 0 (9) - represens he ecuaon of he separaon surface beween classes p and p. I can be shown ha he mnmzaon of classfcaon coss s acheved when each obec from he nal populaon s allocaed o he class whch has he greaes a poseror probably of occurrence. he a poseror probables are compued usng Bayes heorem: P ( p x) where ( p x) () x P( p ) f () x f p = (20) P s he a poseror probably of occurrence for class p gven x. Le us consder agan a fne populaon P whch s paroned no classes. he man dea underlyng he Mahalanobs classfer s he dsance beween he cenrods of he classes and he obecs subec o classfcaon. he Mahalanobs creron requres ha each class comprse he obecs Applcaon of Dscrmnan Analyss on Romanan Insurance Marke 53

4 heorecal and Appled Economcs ha are closes o he cenrod of he class n erms of Mahalanobs dsance. More formally, he algorhm of he Mahalanobs classfer consss of fve seps: SEP. Esmae he cenrods of he classes: ˆ m m 2 m.,ˆ,...,ˆ SEP 2. Esmae he covarance marx Ŝ. SEP 3. Evaluae he Mahalanobs dsance ( ( x,ˆ ) d m ) beween every obec and he cenrods. hs s done usng he followng formula: d ( ) ( ) ˆ - x,ˆ m = x - mˆ S ( x - mˆ ) =,2,...,, (2) SEP 4. Classfy he obecs accordng o he mnmum dsance prncple. SEP 5. Re-compue he cenrods and repea he algorhm unl all obecs wll have been classfed. I s mporan o noe ha sep 2 s no necessary anymore. he Fsher classfer s a smple, ye robus dscrmnaon mehod based on he analyss of varance. I s well known ha he man purpose of paern recognon s he classfcaon of obecs no classes so ha he beween-class varance s maxmzed and he whn-class varance s mnmzed. Fsher (933) addresses hs ssue usng lnear classfcaon funcons: d ( ) ( ) ( ) ( ) = a 0 + a x + a 2 x a m xm (22) where d s dscrmnan funcon and ( ) = a,,2,...,m represen he coeffcens of lnear combnaon. Consderng ( ) ( ) ( ) ( ) ) ( ( x... ) a = a a... a and 0 m x = x m, relaon (22) can be rewren as ( ( ) d = a ) x (23) he coeffcens of he lnear combnaons wll be deermned bearng n mnd ha we have o maxmze he beweenclass varance whle mnmzng he whnclass varance. he covarance marx S can be wren as sum beween he beween-class covarance marx ( S b ) and he whn-class covarance marx ( S w ): S =S b + S w (24) Now consder a dscrmnan funcon (or dscrmnan varable) d and a vecor of coeffcens. Assumng all dscrmnan varables are cenered, follows mmedaely from (23) ha he varance of varable d s: VAR ( d ) = a E = a S é = E ( a x) ( a x) êë ( x x ) a = a b a + a S w S a = a ù = úû (25) he coeffcens of he lnear combnaon wll be deermned so ha he followng condon s sasfed: a Sb a max y = a (26) a S a w Dfferenang wh respec o a, we have: y a 2 = ( S b a) ( a S w a) - 2 ( S w a) ( a S b a) ( a S a) 2 y he condon = 0 leads o a w ( S b a) ( a S w a) -( S w a) ( a Sb a) 2 ( a S a) w = 0 (27) (28) 54

5 or Mulplyng by a S ( S a) y = 0 w a, we ge: S b a - w (29) I follows mmedaely ha mus sasfy: ( S - S y) a = 0 b w (30) ( S - S - y) a = 0 w b I m (3) where I m s he deny marx. Equaon (3) shows ha a s an egenvecor of marx S - w Sb. In order for a o be a non-zero vecor, y mus sasfy he characersc equaon: S - w Sb - I m y = 0 (32) he maxmum number of egenvalues of marx S - w Sb s m (provded he marx s non-sngular (2) ). Le l, l 2,..., l m be he m egenvalues and assume furher ha l ³ l 2 ³... ³ l m (33) I can be easly proven ha, for each egenvalue l, we have: y = l (34) herefore, he maxmum value of y (he beween-class varance o whn-class varance rao) corresponds o he greaes Class PREMIUM A B C D E egenvalue of marx S - w Sb, whch s l. As such, egenvecor ( a ) (correspondng o egenvalue l ) defnes dscrmnan funcon d, whch has he greaes dscrmnaory power. Egenvecor ( 2 a ) (correspondng o egenvalue l 3 ) defnes dscrmnan funcon d 2, whch has less dscrmnaory power han d, and so on. In hs paper, we wll use dscrmnan analyss o classfy he nsurance companes ha operaed on he Romanan marke n We have seleced a number of egh (8) relevan varables: gross wren premum (GR_WRI_PRE), ne mahemacal reserves (NE_M_PES), gross clams pad (GR_CL_PAID), ne premum reserves (NE_PRE_RES), ne clam reserves (NE_CL_RES), ne ncome (NE _INCOME), share capal (SHARE_CAP) and gross wren premum ceded n Rensurance (GR_WRI_PRE_CED). Before proceedng o dscrmnan analyss, we performed cluser analyss on he nal daa n order o denfy classes (clusers) ha emerge from he daa. he resuls are dsplayed n he able below: Class members ASIROM, ALLIANZ-IRIAC, ING ASIGURARI DE VIAA BCR ASIGURARI, OMNIASIG, ASIBAN GENERALI, UNIA, B ASIGURARI RANSILVANIA, ASRA, ARDAF GARANA, CARPAICA ASIG, ASIRANS, AIG ROMANIA INERAMERICAN, OMNIASIG VIAA, GRAWE, BCR ASIGURARI DE VIAA, AVIVA, AIG LIFE HE RES OF HE COMPANIES As we can see, we have sx classes of nsurance companes. PREMIUM class ncludes he larges and mos profable companes on he marke (ALLIANZ- IRIAC, ASIROM, ING ASIGURARI DE VIAA). Class A comprses companes wh mporan marke share and good levels of profably (BCR ASIGURARI, OMNIASIG, ASIBAN). Class B ncludes companes wh sgnfcan marke share bu Applcaon of Dscrmnan Analyss on Romanan Insurance Marke 55

6 heorecal and Appled Economcs weak profably (ASRA) or companes ha ncur subsanal losses (GENERALI, UNIA and especally ARDAF). Class C groups companes wh average marke share and varable reurn on capal (CARPAICA ASIG, ASIRANS, GARANA, AIG ROMANIA). Class D consss of small companes whch are n he lfe nsurance busness (OMNIASIG VIAA, GRAWE, BCR ASIGURARI DE VIAA, AVIVA, AIG LIFE). Fnally, class E ncludes small companes wh weak fnancal performances (he res of he companes). he able below summarzes he role of he descrpve varables n he dscrmnan analyss performed: As we can see n he header, Wlks Lambda 3 s only , whch means ha he model has sgnfcan dscrmnaory power. he es-sasc F s and he p-value s below 0-4, hus assurng he goodness of he model. he second column of he able conans he Wlks Lambda sasc compued for each descrpve varable and shows ha all varables ncluded n he model have mporan dscrmnaory power. he Paral Lambda sasc (compued n he hrd column of he able) llusraes he conrbuon of he varables o he classfcaon and, n hs respec, s clear ha he descrpve varables aren very dfferen one from anoher. Dscrmnan analyss summary Wlks' Lambda: F-es: p-value < Wlks' Lambda Paral Lambda GR_WRI_PRE NE_M_RES GR_CL_PAID NE_PRE_RES NE_CL_RES NE_INCOME SHARE_CAP GR_WRI_PR_CED he Fsher dscrmnan funcons are: d d2 d3 d4 d5 GR_WRI_PRE NE_M_RES GR_CL_PAID NE_PRE_RES NE_CL_RES NE_INCOME SHARE_CAP GR_WRI_PR_CED Consan Egenval Cum.Prop

7 As he prevous able demonsraes, marx S - w S b has only fve posve egenvalues. he frs dscrmnan funcon (correspondng o he frs and greaes egenvalue) s, by far, he mos mporan, as accouns for over 95% of oal dscrmnaon. he nex able conans he means of he prevously defned dscrmnan varables: Means of dscrmnan varables Class d d2 d3 d4 d5 E he frs dscrmnan funcon dsngushes PREMIUM class from he oher classes of nsurance companes, he second dscrmnan funcons dsngushes companes peranng o class A and he hrd dscrmnan funcon dsngushes class B companes from he ohers. he fourh and ffh funcons have very lle dscrmnaory power. hs s normal consderng he egenvalues deermned earler. D C PREMIUM B A he dscrmnan scores are: Insurance company Class d d2 d3 d4 d5 ALLIANZ-IRIAC PREMIUM ASIROM PREMIUM ING ASIGURARI DE VIAA PREMIUM ASIBAN A BCR ASIGURARI A OMNIASIG A ARDAF B ASRA B B ASIGURARI RANSILVANIA B GENERALI B UNIA B AIG ROMANIA C ASIRANS C CARPAICA ASIG C GARANA C AIG LIFE D AVIVA D BCR ASIGURARI DE VIAA D GRAWE D INERAMERICAN D OMNIASIG VIAA D ABC ASIGURARI E AGRAS E ASIMED E Applcaon of Dscrmnan Analyss on Romanan Insurance Marke 57

8 heorecal and Appled Economcs Insurance company Class d d2 d3 d4 d5 ASIROM CONCORDIA E ASIO KAPIAL E AE INSURANCE E CERASIG E CIY INSURANCE E CLAL ROMANIA E DELA E DELA ADDENDUM E EUROASIG E FAA ASIGURARI AGRICOLE E GERROMA E IRASIG E KD LIFE ASIGURARI E NBG INSURANCE E OP GARANCIA ASIGURARI E R.A.I. E Even hough all fve dscrmnan funcons could be used n our analyss, we are gong o use only he frs wo of hem, because of he followng reasons: n ogeher hey accoun for over 98% of he oal dscrmnaon n hey enable us o plo he dscrmnan scores and hus emphasze he groupng of he obecs n he reduced dscrmnan space. he char s shown below: Fgure. Plo of dscrmnan scores 58

9 he plo shows ha a clear dsncon can be made only beween PREMIUM and A classes and he ohers. A he same me, s dffcul o dsngush beween classes C and E; once agan, we have clear proof ha he frs wo dscrmnan funcons are he mos sgnfcan. Dscrmnan varable d s negavely correlaed wh predcve varables gross wren premum, ne mahemacal reserves and gross clams Rows: observed classfcaons Columns: predced classfcaons % correc E D C PREMIUM B A E D C PREMIUM B A oal pad, ndcang ha companes wh low d values have a good chance of beng ncluded n he PREMIUM caegory. Dscrmnan varable d 2 s posvely correlaed wh gross wren premum and gross clams pad and negavely correlaed wh he oher sx predcve varables. Companes wh negave d 2 are very lkely o be allocaed o class A, bu hey can also be classfed n class B. As for classes C, D and E, hey are all characersed by smlar d and d 2 scores, makng very dffcul o dsngush beween hem. As menoned prevously, dscrmnan analyss s also an mporan predcon ool: based on he daa n he ranng se, produces a model ha can be used o classfy new obecs. he qualy of classfcaon can be assessed usng he classfcaon marx: As we can see, he oal percenage of correc classfcaon s 92.5%, whch can be consdered excellen. he applcaon of he model resuled n perfec classfcaon for obecs n classes PREMIUM, A and B. he percenage of correc classfcaon for obecs n classes C, D and E s no 00% because, as explaned n he prevous secon, s dffcul o dsngush beween companes belongng o hese classes. Sll, over 75% of obecs have been classfed correcly, whch proves ha we developed a robus predcon ool. he Bayesan creron produces exacly he same classfcaon as he Fsher lnear dscrmnan funcons. he a poseror probables of classfcaon are presened n he able below: A poseror probables Incorrec classfcaons are marked wh * Observ ed E D C PREMIUM B A E E D * 4 C PREMIUM B A E Applcaon of Dscrmnan Analyss on Romanan Insurance Marke 59

10 heorecal and Appled Economcs A poseror probables Incorrec classfcaons are marked wh * Observed E D C PREMIUM B A 9 PREMIUM E E C B E D A * 7 D B C E E E E E E E C B E D PREMIUM D E E E A D E * 39 E B I s mporan o menon ha he a pror probables of occurrence for each class have been chosen equal o 6. 60

11 Fnally, we ge he same resuls usng he Mahalanobs classfer: Squared Mahalanobs dsances from group cenrods Incorrec classfcaons are marked wh * Case Observed E D C PREMIUM B A E E D * 4 C PREMIUM B A E PREMIUM E E C B E D A * 7 D B C E E E E E E E C B E D PREMIUM D E E E A D E *39 E B Applcaon of Dscrmnan Analyss on Romanan Insurance Marke 6

12 heorecal and Appled Economcs o sum up, dscrmnan analyss provdes an mporan classfcaon ool. Regardless of he mehod used, he applcaon of he model produced 92.5% correc classfcaon, whch underlnes he Noes robusness of he analycal approach. I s also worh menonng ha dscrmnan analyss offers a powerful predcve ool whch can be used o descrbe fuure developmens n he Romanan nsurance marke. () We can also wre () x = f ( x x Î ). () x f p p f p s a condonal probably densy. (2) Recall ha s a posvely defned and symmercal marx. (3) Compued as sum he closer s value s o zero, he more dscrmnaory power he model has; he closer s value s o one, he less dscrmnaory power he model has. sum of squares of errors beweenclasses oal sum of squares of errors. References Armeanu, D. (2006). Managemenul rsculu în asgurãr, sera busness, Edura Unversãþ Romano-Brance, Bucureº Dscrmnan Funcon Analyss, hp:// www2.chass.ncsu.edu/garson/pa765/dscrm.hm Poulsen, J., French, A., Dscrmnan Funcon Analyss, hp://userwww.sfsu.edu/efc/classes/bol70/dscrm/ dscrm.pdf Ruxanda, Gh., Analza muldmensonalã a daelor, Maser Baze de Dae Supor penru Afacer Ruxanda, Gh. (200). Analza Daelor, Edura ASE, Bucureº Smar, L. (2004). Appled Mulvarae Sascal Analyss, Sprnger Sprcu, Llana (2006). Analza Daelor: Aplcaþ Economce, Edura ASE, Bucureº Wellng, Max, Fsher Lnear Dscrmnan Analyss, hp:/ / 62