Simpson s Paradox, Confounding Variables and Insurance Ratemaking. John Stenmark, FCAS, MAAA. Cheng-sheng Peter Wu, FCAS, ASA, MAAA ABSTRACT

Transcription

1 Spo Paradox Cooudg Varale ad Iurae Rateakg Joh Steark FCAS MAAA Cheg-heg Peter Wu FCAS ASA MAAA ASTRACT The urae proe oplex th uerou ator og to produe oth preu ad loe. Whle oplg rate atuare ote aggregate data ro ore tha oe oure hle at the ae te tratg the data to aheve hoogeet. Hoever uh exere a lead to aed ad oete eve urprg reult alled Spo paradox eaue the varale volved the aggregato proe or the tratato proe are oouded the preee o other varale. I th paper e ll dere Spo paradox ad ooudg ad the tattal uderpg aoated th thoe pheoea. We ll urther du ho uh a a ext P&C atuaral ratg applato ad oluto that a reolve the a.

2 . INTRDUCTIN A atuar aked the CE or a all urae opa to exae the good tudet dout that the opa oerg. The dout urretl tee peret ut everal opettor oer a tet peret dout or qualg outhul operator. A uual the CE a hurr o the atuar ople the experee ad develop a relatvt aed o the pure preu or all outhul operator (Age 5 to 5). Iage the atuar hok he the experee date ot the tet peret dout or hh the CE had ee hopg ut a tet peret urharge. The lo experee appear TALE. TALE Wthout Good Studet Dout Expoure % Loe Pure Preu % $006 $9 Wth Good Studet Dout Expoure % Loe Pure Preu Relatvt 00.7% $8759 $806 0% The atuar ko o the prole uet th pure preu ut ertal the a t aue th agtude o a dpart. The atuar dede to reve the experee drver age that avalale ro the opa la pla. TALE dpla that experee. TALE Wthout Good Studet Dout Expoure % Loe Pure Preu Age % $66 $98 Age % $88608 $8 Age % $00600 $7

3 Total 8980 $006 $9 Wth Good Studet Dout Expoure % Loe Pure Preu Relatvt Age % $ $06 -% Age % $70500 $679-5% Age % $660 $66-8% Total 00 $8759 $806 0% The relatvte la appear ore reaoale ut the atuar tll ha a oer. Ho a the average o thee three dout produe a urharge? The atuar alo oered aout the varato the dated relatvte. The atuar requet data drver age ro the opa IS departet ad reve the experee hh dplaed TALE. TALE Wthout Good Studet Dout Age Expoure % Loe Pure Preu % $ $ % $ $ % $ $ % $75969 $ % $85658 $ % $659 $ % $50790 $ % $07750 $ % $ $ % $669 $ % $9687 $98 Total 8980 $006 $9 Wth Good Studet Dout Age Expoure % Loe Pure Preu Relatvt % $65000 $750-5% % $50 $87-5% % $7609 $709-5% % $066 $0-5% % $705 $958-5% % $95 $66-5% 0 7.0% $9800 $ -5% 80.0% $967 $0-5%

4 0.0% $07 $0-5% - 0.0% - - 0% % Total 00 $8759 $806 0% urther tratg the data eve ore preo appear to e aheved ad t appear that a eve hgher dout juted. I addto the ae dout ee to e upported or all drver age. Neverthele the queto rea: Ho doe the auulato o all thee dout produe a urharge? The aer Spo paradox.. SIMPSN S PARADX E. H. Spo rt dered the paradox 95 a paper ttled "The Iterpretato o Iterato Cotge Tale" []. It a teretg tattal pheoeo that aue a potetal a erta data aale. The paradox our he a relatohp or aoato etee to varale revere he a thrd ator alled a ooudg varale trodued. The paradox alo our he a relatohp/aoato revere he the data aggregated over a ooudg varale... The College Ado Exaple The la llutrato o the paradox volve ollege ado geder hh a e llutrated the exaple TALE []. TALE Male Feale Shool Applg Aepted % Applg Aepted % Egeerg % %

5 Art % % Total % % I TALE the overall aeptae rato or eale applat 9% loer tha the rato or the ale applat 5%. Hoever th relatohp revere he the ator o the hool to hh the appl trodued. Whe th varale odered the aeptae rato or eale applat 5% hgher tha ale applat or oth the egeerg hool (50% to 0%) ad the art hool (% to 0%). The reao h Spo paradox our that ore eale applat appl to the art hool hh ha a overall loer aeptae rate tha the egeerg hool. The egeerg hool ha a 0% to 50% aeptae rate hle the art hool ha a 0% to % aeptae rate. I the aove exaple aout 8% o eale applat appl to the art hool hle 8% o ale applat appl to the egeerg hool. Let var the peretage o the eale applat applg to the art hool ad aue all the other paraeter the exaple rea the ae. The alulate the rato o the overall eale applat to the ale applat. 5

6 FIGURE Spo' Paradox % 0% 0% 0% 0% 50% 60% 70% 80% 90% 00% Rato o verall Feale Aeptae Rato to Male Aeptae Rato Rato o verall Feale Aeptae Rate to Male Aeptae Rate Atual Rato Revere Le % o Feale Applg to Egeerg Shool I FIGURE the old le repreet the rato o the overall eale aeptae rate to the overall ale aeptae rate varg the peretage o eale applg to the egeerg hool. We ko that the uderlg rato.5 he e aalze the aeptae hool ad the dahed le repreet the atual rato o.5. We a ee that ol he the peretage o eale tudet applg to the egeerg hool 8% the overall rato the ae a the true rato. Th 8% the ae peretage a the ale tudet applg to the egeerg hool. For all the other peretage the overall rato deret ro the true rato. Aother teretg pot dated FIGURE that he the peretage o eale tudet applg to the egeerg hool le tha 60% the rato o the overall aeptae rate o eale to ale le tha.00 repreeted the dotted le uggetg that the overall eale aeptae rate loer. Th a reveral o the at 6

7 that the eale aeptae rate hgher tha the ale aeptae rate ad Spo paradox []. Fro the aove exaple e a ee that Spo paradox our he the dtruto o the aple populato are ot uor aro the to predtve varale. Whe th take plae the varale o hool ooudg the aeptae rate ad oug the relatohp etee the aeptae rate ad applat geder. We ll du the oept o ooudg varale detal later... The Sple Math o Spo Paradox Spo paradox are ro oe ple atheatal truth. Gve eght real uer: a d A C D th the ollog properte: a d > ad > the t A C D ot eearl true that a A C > d. I at t a e true that: D a A C < d. D Th Spo paradox. Th a ovou ath realt et t ha gat raato aea aal edal reearh ee ad egeerg tude oetal tattal aal ad e urae rateakg. It o oer or a tattal atvt volvg the alulato ad aal o rato o to eaureet. Th atvt prevalet urae; lo rato pure preu reque evert ad lo developet ator are jut oe o the tatt volvg the rato o to eaure.. CNFUNDING VARIALES A varale a ooud the reult o a tattal aal ol t related (o-depedet) to oth the depedet varale ad at leat oe o the other 7

8 (depedet) varale the aal. More peall a varale a ooud the reult o a urae rate truture aal ol t related (o-depedet) to oth the experee eaure (lo rato pure preu et.) ad at leat oe o the other ratg varale the aal... Experetal Deg Cooudg ad Spo paradox o great oer the deg o reearh tude. For exaple a tpal deg o edal reearh reearher ould lke to ko the pat o a terveto eaure. Ug the otato trodued Seto. aue that A ad C are the uer o oervato here the terveto ha take plae. ad D are the uer o oervato the group here the terveto ha ot ee exeuted (the otrol group). The dtto etee the A ad C (ad alo ad D) oervato the potetal ooudg varale. For exaple Cate [5] A ad C ould repreet oker atteptg to qut th ure terveto (the terveto) ro to deret tude (the potetal ooudg varale). Alo our prevou ollege ado exaple A ad C ght repreet the uer o eale (the terveto) applg to the art ad egeerg hool (the potetal ooudg varale) repetvel. Cate [5] dered the eta-aal o oker atteptg to qut th ad thout hgh tet ure terveto. Cate llutrated everal ethod o og tude ro depedet oure. Method luded Maetel-Heel xed eet ethod ad a rado eet ethodolog. oth o thee ethodologe produed eght that ere ued to oe the rk deree rather tha the uderlg data. Cate hoed that a reveral (Spo paradox) ourred he the ra data ere oed. 8

9 Further the uer o evet repreeted a ad d ad the rato a/a the proporto o evet per uer o oervato e.g. the peretage o eale eg adtted to art hool or the proporto o oker Stud # (o the Cate paper) ho qut th the ad o a ure. Whle oth the ollege ado exaple ad the okg terveto exaple volve tude here extg data are oerved ad aalzed aue or a oet that th ot the ae; that e a deg a experet uh aer that e a ze the a o a potetal ooudg varale. Ultatel e d the a elated the ooudg varale ad the varale uder tud are depedet. The a alo elated ether the group are alaed (poe a equal uer o oervato) or are proportoall dtruted (there the ae rato o oervato o the varale uder tud or eah value o the ooudg varale). It pole to llutrate th ug the ollog arguet. TALE 5 Cooudg Varale Value Cooudg Varale Nuer o Evet Nuer o ervato Nuer o Evet Varale Uder Stud Exaple Feale Male Art Shool Egeerg Shool a A d oure the alae odto a peal ae o the proportoal odto. The alae odto epeall portat experet deg. 9

10 0 Value Nuer o ervato C D Coder a experet th group A C D a dered aove. Alo aue that the rato deree are ko ad are equal to oe K: D d C K A a. Ho a the experet e deged o that K D d C A a? Frt aue that the potetal ooudg varale depedet o the varale uder tud.e. that C A a ad D d. Thereore ac A ad d D ad D d D d C ac a ( ) ( ) K D d C d d D d a C a. Thereore the potetal ooudg varale ad the varale uder tud are depedet the there o ooudg. No tead o aug depedee aue that the experet ha a alaed dtruto.e. there the ae uer o oervato eah group relatve to the varale uder tud e.g. the ae uer o eale applg to the art hool ad the egeerg hool ad the ae uer o ale applg to oth hool. The C A ad D. Ad D d C A a D d D C A C A a D D d C C A A a D d C A a D d C A a [ ] K K K. So there o ooudg the oervato poe a alaed dtruto.

11 No aue that the experet proportoall dtruted.e. there the ae rato o oervato o the varale uder tud or eah value o the ooudg varale A C.e. the uer o eale applg to the art hool dvded the D uer o ale applg to the art hool the ae a the uer o eale applg to the egeerg hool dvded the uer o ale applg to the egeerg hool. I A C A the dee K. The A CK DK Thereore D C D a d A C D a d a d CK C DK D K C D a K C C D d D a K K C D a K A K K K a K K K A K ( K K K ) K K K. Thereore the oervato are proportoall dtruted the K there o ooudg. I the exaple detaled the troduto o the paper the good tudet pure preu ad ultatel the dated good tudet dout ere oouded drver age. It ot urprg that there the oerved relatohp etee the dtruto o drver age ad thoe th the good tudet dout. A drver age approahe 5 eer are tudet uh le good tudet. The reveral our e there a hgher dtruto o oug drver th good tudet dout ad oug drver have hgher pure preu.

12 Iportat Prple: I there depedee etee the potetal ooudg varale ad the varale uder tud or the tud alaed or proportoall dtruted the there o ooudg. Iurae rateakg der ro ot tattal tude a uer o a:. It geerall ot pole to deg the akeup o group o ured o that laato are alaed.. Geerall there are ar ore value or eah varale ad proal ore varale urae tha reearh aal.. I ot tattal tude the ojetve to aept or rejet a hpothe. The prar oer urae rateakg to properl alulate a rate hh requre a otuou rather tha ar output. I the ext our eto e ll urther exae ad exted the aove Iportat Prple o ooudg to ore tha to varale ug geeral tattal odel ad experetal deg theore. The to tattal odel that e ll ue are the ple addtve odel ad ultplatve odel oth thout a terato ter. Suh addtve ad ultplatve ultvarale odel are the deal odel ad are lar to a urae ratg ad la pla truture []. For llutratve purpoe e ll ue a ratg exaple th age o drver (outhul drver v. adult drver) ad terrtor (ura terrtore v. uura terrtore) throughout the eto. For ore detal o the addtve ad ultplatve tattal odel ad experetal deg theore pleae ee Motgoer [8] ad Neter et al [9]... The Cooudg Eet o a Addtve Model th o Iterato Ter

13 Let tart th a addtve odel. Aue that the oervato or expoure dtruto o eah ell. Dee: ( ) ; e.g. () () () (). Note: Whle th otato a e ualar pleae aept th veral terpretato. I repreet the expoure ell the ( ) repreet the argal expoure dtruto o ell or ell th For a learl depedet addtve odel the ea value (uderlg rate) or eah o the ell a e repreeted a ollo: :. learl depedet e ea that there o terato ter. I the odel ere ot learl depedet the ea value (uderlg rate) or eah o the. ell ould e repreeted a: ε : here ε the terato ter. More peall e dee the ollog or the age o drver ad terrtor exaple: (Age o Drver) (Vehle Terrtor) here dot dex date the ea aro that dex. No e at to opare the deree the aggregate rate etee adult ad outhul drver: The the aggregate rate or eah ( ) ( ) Ad () () () ().

14 The I ()( ) ()( ) ( ) ()( ). () ( ) () ad ( ) () the ()( ) () ( ) ( ) ()( ) () ( ) () ( ). ( ) Se or the X ae: ( ) () e a derve the ae reult or the other ator the vehle terrtor. I () ( ) ad () () the. Thereore terrtor doe ot ooud the age o drver relatvte or th learl depedet addtve odel terrtoral dtruto o expoure depedet o the age o drver dtruto o expoure. That ( ) () ( ) () () () ad () (). Th a proportoal dtruto. oure a peal ae or uh a dtruto our he eah ell ha the ae uer o data pot. Th a alaed dtruto. The ollog a ueral exaple that llutrate uh a addtve odel. The tatt or the exaple are a ollo: (Age o Drver) Let $00 (Vehle Terrtor)

15 $ 00 or outhul drver ad $ 00 or adult drver 00 or ura drver ad $ 00 $ or uura drver. Thereore the pure preu or eah o the our oato are: 600 $00 $00 $00. Alo aue that $.5% 7.5%.5% 7.5%. TALE 6 Ura Suura Total Youthul Total Lo $000 $6000 $9000 Expoure Dtruto.5% 7.5% 50.0% Pure Preu $600 $00 $50 Adult Total Lo $000 $000 $5000 Expoure Dtruto.5% 7.5% 50.0% Pure Preu $00 $00 $50 Total Total Lo $5000 $9000 $000 Expoure Dtruto 5.0% 75.0% 00.0% Pure Preu $500 $00 $50 I e tud TALE 6 e a ee that the deree etee outhul drver uderlg rate ad the adult drver uderlg rate : $ 00 hh the ae a the deree etee the aggregate rate $50-$50$00. Thereore th ae ooudg doe ot our. 5

16 The data or the other ator the vehle terrtor eld the ae reult. The deree etee uderlg rate or the ura terrtor ad the uura terrtor : $00 hh the ae a e ue the aggregate rate $500-$00$00. Thereore th ae a ell ooudg doe ot our. No oder a deret dtruto:.5% 7.5% 7.5%.5%. Th dtruto ether alaed or proportoal. The ooudg eet o terrtor o la (ad ve vera) eoe apparet. TALE 7 dpla that th ae or the age o the drver ator e a ee that the deree etee the uderlg rate or outhul drver ad adult drver : $ a eore. Hoever the aggregate rate deree $50-$50$00. TALE 7 Ura Suura Total Youthul Total Lo $000 $6000 $9000 Expoure Dtruto.5% 7.5% 50.0% Pure Preu $600 $00 $50 Adult Total Lo $6000 $000 $7000 Expoure Dtruto 7.5%.5% 50.0% Pure Preu $00 $00 $50 Total Total Lo $9000 $7000 $6000 Expoure Dtruto 50.0% 50.0% 00.0% Pure Preu $50 $50 $00 6

17 .. The Cooudg Eet or a -Deoal Addtve Model th o Iterato Ter No e at to exted the learl addtve odel ro to deo to - deo. Alo e ll exted the uer o value or eah varale to ore tha to that value. Th eaue a tpal urae ratg truture ha a varale th ultple value. Aga aue that the aple dtruto o eah ell.... Dee: ( ) For a learl addtve odel the ea value or eah o the x ell a e repreeted a ollo: : here a dot ( ) dex date the ea aro that dex. Aga e at to opare the deree the aggregate rate ad the uderlg rate etee a to value or the rt ator.... ( )... The the expeted rate or eah ( )... ad The ()... 7

18 ( ) ( ) ( ) ( ) I ( )... ()... the ( ) () ( ) ( ) [( ) ( )] ()... ()... ( )... (......)... ().... ut... ( )... So The dtruto o the aple populato deed a proportoal he: ( )... ()... or all ( )... ()... or all ( )... ()... or all...( )...() or all. 8

19 Cooudg ll ot our or the -deo learl addtve odel the aple dtruto proportoal... The Cooudg Eet o a Multplatve Model th o Iterato Ter Let tart th a ultplatve odel thout a terato ter. Aue that the aple dtruto o eah ell a eore. Aga dee: ( ) ; e.g. () () () (). For a ultplatve odel th o terato ter the ea value or eah o the ell a e repreeted a ollo: :. More peall e dee the ollog or the age o drver ad terrtor exaple: (Age o Drver) (Vehle Terrtor) here a dot ( ) dex date the ea aro that dex. No e at to opare the deree the aggregate rate ad the uderlg rate etee adult ad outhul drver. The the expeted rate or eah ( ) ( ) ad () () () (). The 9

20 () () () () () () () () () () ad ( ) () ( ) ( ) ( ) ( ). I ( ) () ad ( ) () the () () ( ( ( ) ) () () () () ( () () ) ) ( ( ) ad ). Thereore terrtor doe ot ooud the age o drver relatvte or th ultplatve odel the terrtoral dtruto o expoure depedet o the age o drver dtruto o expoure. That () ( ) () ( ) () () ad () (). Th our he the dtruto aog the predtve varale are depedet ad proportoal to eah other. oure a peal ae or uh depedet dtruto he eah ell ha the ae uer o data pot.e.. Aga th a alaed dtruto. The ollog a ueral exaple that llutrate uh a ultplatve odel. The tatt or the exaple are a ollo: (Age o Drver) (Vehle Terrtor) Let $00 0

21 .5 or outhul drver ad or adult drver.50 or ura drver ad or uura drver. Thereore the pure preu or eah o the our oato are: 750 $50 $50 $ 50. $ Alo aue that.5% 7.5%.5% 7.5%. TALE 8 Ura Suura Total Youthul Total Lo $750 $750 $7500 Expoure Dtruto.5% 7.5% 50.0% Pure Preu $750 $50 $75 Adult Total Lo $50 $50 $500 Expoure Dtruto.5% 7.5% 50.0% Pure Preu $50 $50 $5 Total Total Lo $6000 $6000 $000 Expoure Dtruto 5.0% 75.0% 00.0% Pure Preu $600 $00 $00 I e tud TALE 8 or the age o the drver ator e a ee that the uderlg rate or the deree etee outhul drver ad adult drver : $75.67 hh the ae a e ue the aggregate rate. 67. $5 Thereore th ae ooudg doe ot our.

22 No aue a deret dtruto:.5% 7.5% 7.5%.5%. TALE 9 Ura Suura Total Youthul Total Lo $750 $750 $7500 Expoure Dtruto.5% 7.5% 50.0% Pure Preu $750 $50 $75 Adult Total Lo $6750 $750 $5000 Expoure Dtruto 7.5%.5% 50.0% Pure Preu $50 $50 $50 Total Total Lo $0500 $500 $5000 Expoure Dtruto 50.0% 50.0% 00.0% Pure Preu $55 $5 $75 Th dtruto ether alaed or proportoal ad the ooudg eet o terrtor o la (ad ve vera) aga ovou. TALE 9 dpla that th ae or the age o the drver ator e a ee that the relatohp etee the uderlg.5 rate or outhul drver ad the adult drver :. 67 a eore $75 Hoever the aggregate rate aed:. 50. $50.5. The Cooudg Eet o a -Deoal Multplatve Model th o Iterato Ter

23 No e at to exted the ultplatve odel ro to deo to - deo. I addto or eah varale e ll exted the uer o value to ore tha to that value. Aga aue that the aple dtruto o eah ell.... Dee: ) (... For a ultplatve odel the ea value or eah o the x ell a e repreeted a ollo: : here a dot dex date the ea aro that dex. Aga e at to opare the deree the aggregate rate ad the uderlg rate etee a to value or the rt ator. The the expeted rate or eah... ) ( ad... () ) ( The )... (... )... ( () ) ( I... ()... ) ( or all the )... (... )... ( () ()......

24 ( ( ) () )... () Cooudg ll ot our or the -deo ultplatve odel the aple dtruto eet the aove depedet or proportoal odto.. TYPES F CNFUNDING VARIALES A varale that ooud the reult o a tud doe o eetall the ae a regardle o the ature o the varale uder tud or the ooudg varale tel. Hoever the ature o the varale a aet t detato ad treatet. For the purpoe o th paper ooudg varale ll e ategorzed a oe o three tpe: tratato ooudg varale aggregato ooudg varale or lurkg ooudg varale... Stratato Cooudg Varale I order to properl pre a pool o rk t a e eear to trat thoe rk to aller ore hoogeeou group. te a truture trated ug ore tha oe rtero. A exaple that ha alread ee dued peroal autoole hh uuall rated terrtor ad laato. Eah o thee ratg varale utoarl aalzed eparatel ad ratg ator developed reletg pat lo experee. I terrtor depedet o laato the the rate developed ll e approprate. I the dtruto laato vare etee terrtore that laato ot depedet o terrtor the uh a ple aal ll eld aed

25 rate. For exaple there a dproportoatel hgh uer o outhul operator a partular terrtor ad outhul operator are uder pred a uvarate aal o eah ratg varale ll eld rate that are too hgh or the outhul drver that terrtor. I the ratg varale uder aal terrtor the laato a potetal tratato ooudg varale... Aggregato Cooudg Varale It a ell aepted rule o thu that the larger the data et the ore relale the oluo dra. Spo paradox hoever la a haer do o the rule ad the reult a good deal ore tha a ore thu. Uortuatel Spo paradox deotrate that a great deal o are ha to e take he og all data et to a large oe. Soete oluo ro the large data et are exatl the oppote o oluo ro the aller et. Uortuatel the oluo ro the large et are alo uuall rog. [6] I order to trat data to aller ad ore hoogeeou lae atuare gather data ro a a oure a pole. Addg tate opae ad ear o experee are three a that a atuar a ata la hoogeet hle reag la ze (ad thu redlt). I the varale alog hh data aggregated orrelated th oe or ore ratg varale the that varale a ooud the reult o a aal o thoe ratg varale. For exaple aue that tate lo experee to e oed th tate A lo experee to reae the volue o data avalale or a la relatvt aal. Alo aue that tate ha a hgher proporto o outhul operator a ell a ore lo experee overall. Whle a 5

26 aal o eah tate outhul operator experee aloe ght eld the ae approprate relatvt he oed the aal ll produe a dated outhul operator relatvt that appropratel hgh. Exht [] llutrate the eet o to aggregato ooudg varale. I th earo oth lo rato ad expoure dtruto la are related to oth ear ad tate. The lo experee dplaed Exht (eod page) are ro the requred ator o.00 or la 0 ad.0 or la 0. The dervato o the dated la 0 relatvt dplaed o the rt page o Exht. The dated relatvte are.7 ug the lo rato ethod.6 ug the pure preu ethod ad.7 ug the oded lo rato ethod. Although eah o the dated relatvte are aed the pure preu ethod ore ueptle to a tha ether o the other to ethod. Aggregato ooudg varale (though ot deted a uh) ere dued at legth Steark []. The exaple o aggregato ooudg varale gve Exht ll e dued urther Seto Lurkg Cooudg Varale A dplaed the Itroduto to th paper t pole that a ooudg varale a ot e uder exaato. Whle a reeree ue the ter lurkg varale ad ooudg varale terhageal a ore arro deto o lurkg ooudg varale eg ued here. A lurkg ooudg varale the a varale that ha ot et ee uovered a a tratato ooudg varale or a aggregato ooudg varale. A lurkg ooudg varale a ext outde o a atuar rateakg data pol to e deteted ug oe o the a data g tehque avalale. A lurkg ooudg varale a e a data eleet that 6

27 avalale ol through deograph data ot aptured through a opa proeg te. Mot douragg o all the pee o orato a ot ext ahere a data. Iurae opae have ee olletg ore ad ore orato ad uderrter ad atuare have eoe etve to rtera that ght aet the lo proe. Hopeull the there are ot too a udovered ooudg varale lurkg our data that ll gatl dtort our rate. Regardle oe ol eed to pot at the ue o redt ore to reogze a portat lurkg ooudg varale that ha ol reetl ee utlzed to t ull potetal. There are to ue relatve to the duo o ooudg prevoul uued ratg varale. Frt pror to t ue a a ratg varale the alure to eget ured aordg to a redt eaure a have aued ooudg o thoe ratg varale atuall ue. For exaple aue that a erta la o ured ote dpla a poor redt ratg ad a a reult that la aet poor lo experee. The rate or ured that la th a etter redt ore ould reeve a appropratel hgh rate. Seod oe redt ore ha ee etalhed a a ratg varale proper ethod ut e udertake to prevet the otued ooudg o the la rate through the ue o oe o the treatet dered the ext eto. For exaple a opa that provde a dout autoole or ured th a hoeoer pol ght d that ater trodug a dout or good redt the rate or autoole rk th a aopag hoeoer pol are too lo. Th hallege dued at legth Wu [5]. 7

28 5. TREATMENT F PTENTIAL CNFUNDING VARIALES We have preeted the epral ad theoretal evdee or the extee o Spo paradox ad ooudg varale. I th eto e preet everal alteratve or the treatet o th pheoeo. 5.. No Treatet Pearl [] olude that there o tet or ooudg. Muh o Pearl rtg oer the prple o aualt [0]; preual eaue ooudg o uh great oer edal reearh ad that reearh aualt o pre portae. Se urae e are ore oered th tattal orrelato tha aualt e allo a ore leral tet or ooudg. Thereore e a that a varale urelated to the varale uder tud or to the lo eaure the ooudg ll ot reult ad o treatet eear. Hoever t ll adved or a atuar to aue that there o ooudg thout exteve exaato o the relatohp o all the varale aetg the lo proe. 5.. Cotrollg Cooudg through Experet Deg A dued Seto. e a areull deg a aal ad the ollet the data aordgl the e a otrol ooudg. Whether e have pror koledge o the relato etee the potetal ooudg varale ad the target orato or ot e a otrol t eet the ooudg varale urelated to the varale() uder tud. Th a e aheved through opletel alaed deg or proportoal deg o the experet. That or eah oato o the ooudg varale ad the varale() o teret the ae or proportoal aout o data 8

29 olleted. Th oept ool ued a reearh area uh a edal egeerg ad et reearh projet. Hoever he a atuar aalze urae data the atuar tpall aot deg the aal. The atuar ha to aalze hatever he or he gve. The data are otl detered the opa ook o ue hh largel detered the arket eget that the opa erve. Moreover e there are ultple ratg varale ad or eah ratg varale there ext a deret value t pole that a oato o the varale ll have g or ver lttle data. I other ord urae data hghl o-deal or the ooudg ue ad t dult ot pole or u to otrol the a through the experetal deg approah. 5.. Cotrollg Cooudg through Multvarate Aal I the urae data hghl o-deal ad e aot otrol ooudg through tadard experetal deg e a otrol t ug ultvarate aal. That e a peror the ale [] u a aal or GLM aal [] [7] ludg the ooudg varale alog th the varale() o teret. dog o the ooudg varale relato th the target varale ad the varale() teret ll e properl detered ad e alloated through the ultvarate approah. Thereore the true relatohp o the varale() uder tud a e revealed. Whle ultvarate aal a e a deal oluto to deal th the ooudg ue there a ext pratal ue or ug the approah th urae applato. e ue that urae applato otatl aggregate or trat data th regard to tate ear ad opae. I theor e a lude thee potetal 9

30 ooudg varale the aal ut the luo o thee o-ratg varale the ultvarate aal a lead to other ue uh a redlt o the data or aal ad reaoalt ad terpretato o the aal reult or the varale. Thereore later e propoe a alteratve approah ug alg ator or atuare to oder or addreg ooudg. The alteratve approah ll e dued detal Seto Cotrollg Cooudg through the ue o Meta-aal Reearher are ote aed th tuato that opel the ue o data ro everal tude. I urae e trve to reae the volue o our data to reae redlt ad edal reearher attept to do the ae through oplato o ore tha oe tud alled eta-aale [5]. A reearh tud tpall lude oervato ro to group: a terveto group ( N ) ad a otrol group ( N ). Fro thee oervato our pee o data are derved: A terveto th a evet ( ) terveto thout a evet ( N ) otrol th a evet ( ) ad otrol thout a evet ( N ). Fro thee a tatt geerall alled a ze eet alulated. The to ze eet geeral ue are tered the rk deree ad the odd rato. The rk deree the deree etee the rato o the uer o terveto th a evet to the total oervato o all terveto ad the rato o the uer o otrol ujet th a evet to the total oervato o the otrol group. Rk Deree. The reproal o the rk N N deree tered the uer eeded to treat (or har ) ad repreet the uer o terveto requred to aheve oe evet. The odd rato the rato o the rato o the uer o terveto th a evet to the uer o terveto thout a evet 0

31 to the rato o the uer o otrol ujet th a evet to the uer o otrol ujet thout a evet. dd Rato. N N I a aalt avel oe all o the oervato ooudg a reult ad lead to aed dg eaue there a deret dtruto o oervato etee tude. For exaple Cate [5] eve out o te tude reulted a potve uer eeded to treat ad the three that dd ot repreeted ol 89 o the 6 oervato the eta-aal. Regardle og the ra data produed a uer eeded to har otrat to the uer eeded to treat the ajort o tude. A dple ha re etered aroud the optu ethod to e ued to oe uh tude. I geeral ethodologe ou o alulatg a varae or eah tud. The reproal o th varae ued to eght the ze eet theelve rather tha the ra oervato. Th treatet aalogou to alulatg la relatvte or eah ear ad tate ad eghtg thoe relatvte to arrve at a opote relatvt or eah la. A uh t ha oe larte to redlt eghtg. Hoever oe ajor deree etee tpal edal reearh ad urae rateakg that edal reearh reult are ar output ad urae rateakg reult are relatvte or rate o a otuou ale. Thereore although eta-aal provde a teretg exaple o the eet ad treatet o ooudg edal reearh t doe ot appear to have a dret applato to urae prg Cotrollg Cooudg through the ue o Salg Fator I th eto e trodue a pratal approah alled alg ator to treat the ooudg eet that a ool ext urae ratg applato. Th

32 approah a rt propoed Steark h 990 paper [] ad e are revtg the approah ro the perpetve o ooudg varale ad Spo paradox. Th approah portat eaue there are oe ooudg varale that are ot optall addreed ug a o the treatet etoed aove. It ot uuall derale or pratal to lude a ultvarate aal o ot aggregato ooudg varale a dered Seto. e data ro everal tate are luded a ultpler tate proal ot a eear ratg odel output. Th eaue eah tate overall rate hage requreet ll e alulated through a tatede dato pol at oe deterate te the uture. I addto a ultpler or eah experee ear ha o dret applato or terpretato. Regardle reogto o uh varale ultvarate aal through the ue o du varale a aepted ad eetve prate a ll e dued Seto 5.6. A alteratve to that approah ll e dued th eto. I there a a that data ro everal experee ear ad everal tate a e aggregated to reae data volue thout pol ooudg the reult o the tud ad thout the eet o luo o the ooudg varale the aal? A tated prevoul there are to odto eear or a varale to ooud the reult o a aal:. There ut e a relatohp etee that varale ad the experee varale.. There ut e a relatohp etee that varale ad at leat oe o the ratg varale uder aal. I ether o thoe to odto ot et the there o ooudg o the reult.

33 Th lead to the queto: oth odto are et a the data e oded o that oe o the odto o loger et elatg the ooudg? Th ut e doe uh a aer that the portat uderlg relatohp the data are ot dtured. I the ollog eto e ll ho the alg ator approah ug a la pla aal exaple th to potetal ooudg varale tate ad ear The Lo Experee Model To elate the ooudg eet t rt eear to quat that eet o a laato lo odel. The odel eed ot e oplex ad opoed at the ato level o expoure ae rate urret ratg ator requred ratg ator ad ae la lo rato. Appedx A outle th odel ad the quatato o the pat o ooudg. For exaple the total eared preu or la o preet rate P e ad the total urred loe or la L e r. The otato trodued Appedx A ll e ued throughout the reader o Seto 5. The pat o dated la relatvte due to the ooudg eet o aggregato o experee ear ad tate dplaed or three laato rateakg ethod: the lo rato ethod the pure preu ethod ad the oded lo rato ethod. The oded lo rato ethod ear oe derpto. The preu are alulated at ae la rate o that the output o the ethod the la relatvt ot the dated hage to that relatvt. I addto to the three ethod preeted there aother utle varato ethodolog. It pole to develop eah la relatvt a a rato o the eleted tatt (e.g. lo rato) to that o a ae la (peal ae) or to the tatt o the all la experee (geeral ae). The ord peal or geeral are ued to det

34 eah ethod. For exaple Peroal Autoole Iurae t oo to dvde the la lo rato the lo rato or adult drver (pleaure ue). Th the peal ae. It ot ala the ae that the ae la ha a large porto o the ue o the all la lo rato a provde a ore tale ae. Th the geeral ae. The la relatvte a e oralzed ak to the ae la ater the dated relatvte have ee redlt eghted ad eleto have ee ade ro thoe redlt eghted relatvte. The odel trodued Steark [] a or the peal ae ol. Iludg the geeral ae add urther lexlt. elo: The a produed ooudg derved Appedx A ad reprodued a arg ro ooudg ug pure preu ethod (peal ae) Equato 5- g r r E r E L a arg ro ooudg ug lo rato ethod (peal ae) Equato 5- g r r P r P L a arg ro ooudg ug oded lo rato ethod (peal ae) Equato 5- g r r P r L

35 Eah o the aove utlze the ae la experee a the ae. I the relatvt alulated utlzg tead the all la experee (geeral ae) the the a or the oded lo rato ethod ho Equato 5-. Equato 5- g r r 5.5. Dervato o the Salg Fator I t pole to ale the preu or loe (or oth) uh a aer that the a reoved he the data ro oe tate ad/or ear are oed th that o aother tate or ear? What haratert hould uh a alg ator poe? To rtera ut e et a alg ator addate: Crtero : The alg ator hould ata the relatohp etee la lo rato ear ad tate (The alg ator hould ot hage the uderlg relatvte). Crtero : The alg ator hould redue the a due to ooudg to zero. A alg ator that appled uorl to eah la th a pe tate or a partular ear or appled to oth preu ad loe or a pe la ll ulll the requreet o Crtero. Whe ether the expoure dtruto or the ae la lo rato rea otat the dtorto ot preet ad a alg ator that talze ether the ae or total la lo rato ( the peal ae or geeral ae repetvel) or the expoure dtruto hould ulll the requreet o Crtero. A lue a to ho to approah the dervato o a alg ator a dued the eto o experet deg. I the experee alaed or there o relatohp 5

36 etee the experee ad the ooudg varale the ooudg doe ot our. I a alg ator addate a proote ether haratert the ooudg hould e tgated. Appedx dpla the evaluato o our tpe o alg ator that eet the eed o oth rtera. Thee alg ator a e roke to to ategore. e ategor apple to the peal ae ad the other apple to the geeral ae. Eah ategor ha oe alg ator that ued to addre the o-depedee o the ooudg varale ad the lo tatt (lo rato pure preu et.). The other alg ator addree the o-depedee o the ooudg varale ad the ratg eleet() uder tud (alae). l oe tpe o alg ator eed e ued a rate aal. The hoe o hh tpe o ator to ue the hoe o the atuar. Pleae ote that thee ator ere arrved at peto. Th a ot a trval proe ut t eleved the author that a atheatal dervato o the ator ot pole. The ator are teted th Appedx to dpla that the a ro ooudg elated. The rt alg ator that odered the reproal o the ae la lo rato or eah tate ad ear. applg th ator uorl to the loe or eah la the relatohp etee eah o the la lo rato ataed (Crtero ) hle the ethod error redued to zero (Crtero ). Th ho Appedx. The exaple gve Exht -5 ued to exae the alg ator. Exht dpla the eet o alg loe th the reproal o Salg Fator the ae la lo rato. oth Exht ad utlze put paraeter that ere et orth Exht. The oded lo rato ethod utlzed th exht. The preu 6

37 oded to the ae la rate level dvdg the la ator pror to alulatg the lo rato. For eah la the loe are aled the ae la adjuted lo rato or that ear ad tate. For exaple the urred loe or tate 0 ear (500000) are ultpled the reproal o la 0 lo rato (.00/ ) to eld the aled loe o $ The la 0 urred loe (55000) are alo ultpled th ator to eld the aled loe or that la o $ Thee aled loe ata the relatohp etee the la lo rato ut loe a orato regardg the atual ae la lo rato. It pole to appl a alg ator (the ae la lo rato th ae rather tha t reproal) to the preu rather tha the loe. Th ethod hould e ued ol or larger ore tale le o ue. I ae here eve the ae la lo rato a lutuate ldl t ore approprate to ale the loe. The reao that aled loe are equal total to preu. I the alg ator ere appled to preu the reult ould e equal to (the ore volatle) loe. The eod alg ator derved Appedx addree the deret expoure dtruto ear ad tate. The rato o the total expoure or eah la to the total expoure or the ae la ultpled the rato o the ae la expoure eah tate ad ear to the la expoure eah tate ad ear to provde the alg ator (algeraall e e ). A oppoed to the rt alg ator the eod alg ator uque or eah la ear ad tate. Hoever e the ator appled to oth preu ad loe th alg ator alo ate the requreet o Crtero. Whe e' replae e the equato or the error developed Appedx 7

38 A that error redued to zero thu atg the requreet o Crtero. Exht dpla the eet o utlzg the eod alg ator. The thrd alg ator or the geeral ae ad t addree the odepedee o the ooudg varale ad the lo experee a dd Salg Fator. Appedx dpla the dervato o th ator a ell. The reproal o the lo rato or the tate ad ear ho to elate the a the lo experee. The ourth alg ator larl teted Appedx. Th ator appled the geeral ae ad addree alae. A dplaed the appedx th alg ator e. e The advatage o Salg Fator ad are:. Eae o ue: The ae la ad tatede lo rato dretl otaale ro the data alread eear or the oded lo rato ethod.. Se the alg ator appled uorl or eah la the preu dtruto la or eah ear ad tate let ualtered.. Ma o the tradtoal adjutet to preu ad lo data are o loger eear. A adjutet that apple uorl to the preu or loe o all lae ulled the applato o that alg ator. Thee adjutet ould lude preet level adjutet or overall rate hage developet ator ad tred ator. I hoever a adjutet ot appled uorl la t ll tll e eear. For exaple tred ator are appled aue o lo thee ator ll eed to e appled pror to the alg proe. 8

39 The advatage o Salg Fator ad that the expoure dtruto ore tale tha the lo rato ro ear to ear the Salg Fator ad ll reult le arupt adjutet or ot lae tha ll Salg Fator ad Coparo o Multvarate Aal v. Salg It oo prate to lude du varale or potetal aggregato ooudg varale a ultvarate aal. Iluo o a du varale or oth ear ad tate or exaple ould allo the o-depedee o thoe varale th the depedet varale (e.g. lo rato) to e releted the du varale. Doe th ethodolog opeate or the ooudg oerved prevoul? I t doe th ethod ore or le eetve tha the ue o oe o the alg ator dued the prevou eto? Tale 0 dpla the reult o uh a oputato. The reultg ator or State ad Year ad ad Cla 0 ad 0 are ho. I eleve terato the u a equato overged to the ra output dplaed TALE 0. 9

40 TALE 0 The ra output a the oralzed to ae la (ear ad la) ad the tate ator ere adjuted to orret or the oralzato. The oralzed la 0 ator equal to the orret value.0. It appear that oth the alg ator dued eto 5.5 ad the ultvarate aal dued aove eld the orret ator th detert earo. Reult o Mu a Ug Du Varale or State ad Year Ra utput Nuer o Iterato State.5.85 Year Cla Noralzed State Year Cla Hoever the orld hh e lve hardl detert. It eear to tet eah ethod a tohat odel. The detert odel a ued to paraeterze uh a odel. Separate reque ad evert average ere derved aug a reque o.0 adjuted the la ad ear lo rato. The tate lo rato a releted the evert. The reque dtruto a aued to e Poo ad the evert dtruto a aued to e logoral. Exht dpla the odel output. e thouad terato ere ulated. Wth eah terato or eah expoure or eah ear tate ad la a uer o la a derved ro the Poo. For eah o thee la a la ze a detered ro the logoral dtruto. The lo rato or eah ear tate ad la a detered ad ro thee the Cla 0 relatvt a derved ug the uvarate (tradtoal) ethod eah o the our alg ator 0

41 ethod a ell a ale u a. The author akoledge that the ue o a lear odel aed o the logoral ght have ee ore approprate. The value that eerged ro the detert odel are dplaed a the expeted value. elo thee are the average value ro all oe thouad terato. Fall the ext ro dpla the ea quare error (MSE) or eah olu. The value ued to alulate th error or the uvarate ethod a the orret la relatvt rather tha the relatvt eergg ro the detert odel (.e..0 rather tha.958). The preee or aee o a lo lt ght aet the etvt o eah ethod to varalt loe. Thereore the odel a repeated ut th te loe ere lted to $5000. oure the logoral paraeter had to e adjuted upard to opeate or the exluded loe at the top ed o the dtruto. The ea quare error or the uvarate ethod a oehat hgher tha that or the other ethod th or thout the lo ltato. Th a expeted e the ethod poeed a relatvel large a the rt plae. the other had there a o gat deree etee the error or ale u a ad the our alg ethod. It appear that hle ue o a teratve a redug ethodolog doe at redue a o do eah o the alg ator dered earler th paper. 6. CNFUNDING VARIALES AND CURRENT ACTUARIAL PRACTICE 6.. Area here ooudg varale have ee reogzed. ale ad So [] rt reogzed the potetal or a ro ooudg 960 though the dd ot det t a uh. Are there other area here atuare have reogzed th a ad opeated or t?

42 e aer the tredg proe that atuare requetl eplo ther ratg ad reervg applato to adjut preu ad lo data. It utoar he preparg a rate dato to tred loe to reogze the reae evert ad hagg reque. It alo eear to tred preu to reogze that oe lo tred ro ator that ll reae the preu over te. Thee lato ad overage etve ratg ator ooud the lo tred eetatg a adjutet. Se dedutle or exaple related to oth the tred eaure pure preu or reque ad evert a ell a related to te (dedutle ted to reae over te) dedutle a ooudg varale or tred data. ther ooudg varale or tred ght e ol odel ear lt o lalt aout o urae to ae jut a e. 6.. Area here ooudg a e a ureogzed prole Cooudg a requet ad erou prole rateakg. voul alot all the ratg varale a ooud eah other eaue ther dtruto are hardl depedet. A dued aove the preu ad lo o-levelg ad tredg a proe that atuare eplo to eouter uh ooudg to the et e a. Hoever the proe a ot e ale to reove all the potetal ooudg relatohp etee the varale. Moreover there are other potetal ooudg varale that ext outde the ratg varale that a ot e ull reogzed ad explored.e. lurkg ooudg varale. The ollog are a e exaple oe o hh have ee dued prevoul:

43 Geograph Iorato: Whle a ratg pla a lude geograph ratg varale uh a tate ad terrtor uh varale a ot e eough to ull expla the ooudg relatohp the ratg data. The real uderlg drver or uh geograph ator lude the uderlg deograph ouer eoo tra ad eather orato. Suh orato lude ut ot lted to orato uh a eduato eploet redt le tle ouer pedg tra volue re old heat hal tor et. Epeall or oeral le o ue uh geograph orato uuall uder-repreeted the ratg proe. Market Seget: The ratg varale dtruto gatl lueed arket eget. For exaple a o-tadard ook o ue ght e expeted to have a uh hgher dtruto o ouger drver ore rk th pror la ad volato ad urae th loer overage. Thereore t ght e prudet to aggregate or trat data alog deret arket eget. I a tae opae or terg ll e ued to eparate deret arket eget. It hghl lkel that laato experee ll e oouded ratg ter or opa. Varale ued opa plaeet or ter tpall lude oth ratg varale ad o-ratg varale. Copa or ratg ter a e ued a a varale th laato a lear odel or the experee hould e treated th oe o alg ator trodued Seto 5.5.

44 Dtruto Chael: ur experee date that dtruto hael ll alo aet the opoto ad the orato gathered or a ook o ue. Th ue ha eoe eve ore gat a a opae are gog o-le addto to the to tradtoal hael o dret rter ad depedet aget. We have oud that ue log through deret hael a e o ver deret qualt ad ota derg aout o orato. Exteral Evroet: The urae dutr ot operatg th a olated orld ad t perorae a part o the reagl ore tegrated atoal or eve orldde eoo. Thereore the athagg orld ue uh a tehologal developet eoo le ad reet terror atvt ll pat the urae dutr. The urret hard arket odto a lear pee o evdee ho the urae uderrtg le lueed the exteral orld. Thereore og ultple ear th pole ear to ear hage ad urae le requre peal are. Addtoal are ut e redered he projetg the htoral orato to the uture. 7. CNCLUSINS E. H. Spo trodued the oept o ko a Spo paradox. It the extree ae o a pheoeo ko a ooudg. Whle uh extree ae a ot our requetl atuaral alulato the hage relatohp due to ooudg doe.

45 A varale a ooud the reult o a urae rate truture aal ol t related (o-depedet) to oth the experee eaure (lo rato pure preu et.) ad at leat oe o the other ratg varale the aal. Cooudg varale a e ategorzed a ether a tratato ooudg varale a aggregato ooudg varale or a lurkg ooudg varale. Several ethod or the treatet o ooudg ere dued ludg o treatet experetal deg ultvarate aal eta-aal ad ue o alg ator. The oato o data ro ore tha oe ear a aue dtorto tradtoal laato rateakg tehque eah od o data repreet a deret ae rate adequa ad deret expoure dtruto la. The oato o data ro ore tha oe tate a aue dtorto the tradtoal pure preu ethod the ae rate ro eah tate deret ad poee a deret expoure dtruto la. The oato o data ro ore tha oe tate a aue dtorto oth o the tradtoal ethod the ae rate ro eah tate deret the ae la lo rato deret ad the tate/ear data exht a deret expoure dtruto la. It ore tha lkel that thee odto ll ext th ot ode o rateakg data. Thee dtorto a e reeded the applato o a alg ator to the data ro eah ear ad eah tate. Th alg ator a addre ether the expoure dtruto or the ae rate adequa. A vetgato o the eetvee o ultvarate aal oparo th the ue o alg ator reveal that oth ethodologe redue the eet o ooudg proal to the ae degree. 5

46 The author have eoutered the ooudg experee uerou te ther ork ad t th th otvato that e trodue Spo paradox ad the oept o ooudg to the atuaral out. We eleve that udertadg thee oept a ke or atuare udertadg the orrelato ue that ext requetl our atuaral ork ad the pat o uh orrelato o aal reult. 6

47 Exht Multple State - Multple Year Stuato Deret Lo Rato - Deret Dtruto Aupto Cla Fator Uderlg Experee Cla 0 Lo Rato Curret Requred Lo Rato Cla Fator Fator State Year Year % 75% % 90% Dtruto o Expoure State State Cla Year Year Year Year Total Total State ae Rate $00 State ae Rate $00 (The Derved Lo Experee ho o the ext page.) Idated Cla Relatvt Lo Rato Method: (8.56% / 7.67%) x.00.0 Pure Preu Method: / Moded Lo Rato Method: (69.% / 7.67%).0 7

48 Exht (ot.) Multple State - Multple Year Stuato Deret Lo Rato - Deret Dtruto Derved Lo Experee Moded Moded Eared Cla Eared Iurred Lo Pure Lo Cla Expoure Preu Fator Preu Loe Rato Preu Rato State Year $ $ $ % $ % $ $ $ % $ % Total 5000 $ $ $ % $ % Year $ $ $ % $ % $ $ $ % $ % Total 0000 $ $ $ % $ % All Year $ $ $ % $ % $ $ $ % $.8.8% Total 5000 $ $ $ % $ % State Year $ $ $ % $ % $ $ $ % $ % Total 5000 $ $ $ % $ % Year $ $ $ % $ % $ $ $ % $ % Total $ $ $ % $ % All Year $ $ $ % $ % $ $ $ % $ % Total $ $ $ % $ % All State Year $ $ $ % $ % $ $ $ % $5.5.00% Total 0000 $ $ $ % $ % Year $ $ $ % $ % $ $ $ % $ % Total $ $ $ % $ % All Year $ $ $ % $ % $ $ $ % $ % Total 0000 $ $ $ % $.6 5.8% 8

49 Exht Multple State - Multple Year Stuato Deret Lo Rato - Deret Dtruto Derved Lo Experee Eared Cla Adjuted Iurred Adjuted Saled Moded Cla Expoure Preu Fator Preu Loe Lo Rato Loe Lo Rato State Year $ $ $ % $ % $ $ $ % $ % Total 5000 $ $ $ % $ % Year $ $ $ % $ % $ $ $ % $ % Total 0000 $ $ $ % $ % All Year $ $ $ % $ % $ $ $ % $ % Total 5000 $ $ $ % $ % State Year $ $ $ % $ % $ $ $ % $ % Total 5000 $ $ $ % $ % Year $ $ $ % $ % $ $ $ % $ % Total $ $ $ % $ % All Year $ $ $ % $ % $ $ $ % $ % Total $ $ $ % $ % All State Year $ $ $ % $ % $ $ $ % $ % Total 0000 $ $ $ % $ % Year $ $ $ % $ % $ $ $ % $ % Total $ $ $ % $ % All Year $ $ $ % $ % $ $ $ % $ % Total 0000 $ $ $ % $ % Idated Cla Relatvt Moded Lo Rato Method: (0.00% / 00.00%) x.0 9

50 Exht Multple State - Multple Year Stuato Deret Lo Rato - Deret Dtruto Derved Lo Experee Eared Cla Iurred Salg Saled Saled Moded Cla Expoure Preu Fator Loe Fator Preu Loe Lo Rato State Year $ $ $ $ % $ $ $ $ % Total 5000 $ $05000 $ $ % Year $ $ $ $ % $ $ $00000 $ % Total 0000 $ $87500 $ $ % All Year $ $65000 $ $ % $ $ $ $ % Total 5000 $ $5500 $ $ % State Year $ $ $ $ % $ $ $00000 $ % Total 5000 $ $ $ $ % Year $ $ $ $ % $ $ $ $ % Total $ $ $ $ % All Year $ $ $ $ % $ $ $ $ % Total $ $ $ $ % All State Year $ $ $ $ % $ $05000 $ $ % Total 0000 $ $ $ $ % Year $ $85000 $ $ % $ $97500 $ $ % Total $ $97500 $ $ % All Year $ $ $ $ % $ $ $ $ % Total 0000 $ $90500 $ $ % Idated Cla Relatvt Moded Lo Rato Method: (5.70% / 7.67%) x.0 50

51 Stohat Model th Logorall Dtruted Loe (ulted) Exht Iterato: 000 Lo Rato Cla 0 Fator State State Uvarate Salg Salg Salg Salg Mu a Year Year Year Year Iterato Method Fator Fator Fator Fator State Year Cla Cla 0 Cla 0 Cla 0 Cla 0 Cla 0 Cla 0 Cla 0 Cla 0 Expeted erved MSE