Claims Reserving Estimation for BPJS Using Archimedean Copulas

Transcription

1 Claims Reservig Esimaio for BPJS Usig Archimedea Copulas Yuciaa Wiladari,, a) Sri Haryami Kariko 3, b) 3, c) ad Adhiya Roie Effedie Ph.D Sude Deparme of Mahemaics Uiversias Gadjah Mada, Yogyakara. Deparme of Saisics Uiversias Dipoegoro, Semarag. 3 Deparme of Mahemaics Uiversias Gadjah Mada, Yogyakara. a) Correspodig auhor: yuciaa.wiladari@gmail.com b) s_kariko@yahoo.com c) adhiyaroie@yahoo.com Absrac. I some ypes of o-life isurace, claims paymes are made more ha oce ad require cosiderable ime from he ime of claim. Due o he ime-lag bewee a claim is repored uil he claim payme is seled resuls i a ousadig claims liabiliy. The compay ca solve he problem by preparig he fuds used o pay he ousadig claim called reserves. Calculaio of claim reserve is usually based o wo radom variables ha is claim ad developme ime. This radom variable is aalyzed which will resul i esimaio of claim reserve. The claim reserve esimaio mehod used i his aricle is he Copula Archimedea mehod, as i is very powerful i modelig he joi disribuio ad does o assume he ormaliy of daa so ha i is flexible for muliple daa disribuios. The purpose of his aricle is o esimae he claim reserve by usig he bes Archimedea class copula ad apply i o he daa. The daa used i his aricle is BPJS Kesehaa (BPJS Healhcare Securiy) daa. The resuls obaied have a correlaio bewee claim ad developme ime, ad he bes mehod is family o. of Copula Archimedea. INTRODUCTION No-life isurace policy is a corac bewee he policyholder ad he isurace compay. The isurace compay will assig a amou of moey o be paid by he policyholder called premium, while he policyholder will receive a cerai amou of isurace coverage from he desigaed isurace compay called he claim []. I his ype of log-ailed busiess isurace, claims paymes are made more ha oce ad ake cosiderable ime from he mome of claim. Because of he ime-lag bewee a claim is repored uil he claim payme is seled resuls i he ousadig claims liabiliy i he isurace compay. The compay ca resolve he ousadig claim by preparig a special fud used o pay claims payable. The fuds ha are prepared are called claims reserves. Esimaed claims reserves have a very impora role i he isurace compay, because i is he compay's obligaios agais policyholders i he fuure. Various esimaio mehods for calculaig claims reserves have bee widely proposed for boh idividual ad aggregae daa. These mehods iclude he chai ladder mehod iroduced by Mack []. Nzoufras ad Dellaporas ad de Alba uderook a sochasic claims reserve sudy usig he Bayesia approach for aggregae daa [3,4]. While [5] wroe a aricle abou Reservig by Deailed Codiioig mehod for idividual daa. I his aricle, we will discuss mehods for calculaig claims reserves by he copula mehod. The copula mehod is a very powerful mehod i modelig mulivariae disribuios or joi disribuios because i does o assume he ormaliy of he daa so ha i is flexible o use for various daa disribuios. Copula is paricularly useful for modelig depedecies i fiace, acuarial sciece ad survival aalysis. Copula was firs i 959 by Sklar, bu was firs applied i fiace by Embrechs i 999 [6]. While he copula applicaio for geeral isurace daa has bee proposed by Frees ad Valdez [7] ad Klugma ad Parsa [8] for he loss ad allocaed loss adjusme expeses model. Frees ad Wag [9] used M - 3

2 copula for he credibiliy of aggregae loss, Kaishev ad Dimirova [0] for reisurace, Bregma ad Kluppelberg [] for rui probabiliy, amog ohers. Peere [] as well as Peere ad Kollo [3] discussed he IBNR model by usig he copula of he size of claims ad delays. Zhao ad Zhou [4] used copula o esimae he reserves of idividual claims. Shi ad Frees [5] used copula for depede loss reservig. The calculaio of he claim reserve is usually based o wo radom variables, he size of he claim ad he ime of developme where he developme ime is he ime from he mome of claim o claim payme. Two radom variables will be aalyzed so ha i will resul i he esimaed claim reserve. Some ypes of copula ha is o- Codiioal Copula, Codiioal Copula, Archimedea Copula, ad Ellipical Copula. Copula Archimedea is he mos widely used copula i various applicaios. The purpose of his aricle is o deermie he bes Archimedea copula ha will be used o esimae he claim reserve, o esimae he claim resevig usig he bes Archimedea copula ad apply i o BPJS Kesehaa (BPJS Healhcare Securiy) daa. LITERATURE I liabiliy isurace, seleme of claims is usually o doe immediaely. Accordig o [] his is due o several reasos:. There is a claims reporig delay or a ime-lag bewee claims occurece ad claims reporig.. Afer a claim is repored, here is a delay uil he claim is fially seled. 3. There is a case ha a closed claim mus be reopeed wih he addiio of a claim payme. The seleme of a o-immediae claim will resul i a deb o he isurer. The compay ca resolve he ousadig claim by preparig a special fud used o pay claims payable. The fuds ha are prepared are called claims reserves. Copula Copula has bee popular i fiace ad acuary, very useful for modelig depedecies i fiace, acuarial sciece ad survival aalysis. The firs copula was i 959 by Sklar [6], bu i was firs applied i fiace by Embrechs i 999. While he copula applicaio for geeral isurace daa has bee proposed by Frees ad Valdez [7] ad Klugma ad Parsa [8] for he loss ad allocaed loss adjusme expeses model. Peere ad Peere ad Kollo discussed he IBNR model usig he copula of he size of claims ad delays [, 3]. While Zhao ad Zhou [4] use copula o esimae he reserves of idividual claims. Copula is a fucio ha ca combie several margial disribuios io a joi disribuio. Accordig o [6] Copula -dimesioal is a mulivariae disribuio fucio F of radom variables U,..., U wih he margial disribuios F,..., F is sadard uiform disribued Ui : F i wih Fi : U(0,); i,..., [4]. The copula fucio deoed by C is a fucio ha has he domai [0,] [0,]x[0,]x...x[0,] ad rage [0,] wih oher words he fucio C :[0,] [0,]. Copula -dimesioal has he followig properies [6]:. Domai C I [0,] [0,]x[0,]x...x[0,]. C( u,..., u ) is a icreasig fucio for all compoes u i 3. C(0,...,0, ui,0,...,0) 0 for every i {,,..., }, u i [0,] 4. C(,...,, ui,,...,) ui for every i {,,..., }, u i [0,] 5. For all ( a,..., a ),(,..., ) [0,] b b wih ai bi applies: i... i... ( ) C u,..., u 0 i i m i i where u a ad u, {,,..., } j j j b j j. Suppose ha a umber of uivariae disribuio fucios F ( x ), F ( x )..., F( x ) he he copula fucio [6]: C F ( x ), F ( x )..., F ( x ) F( x, x,..., x ) M - 4

3 is a mulivariae disribuio fucio wih margial F ( x ), F ( x )..., F( x ) The cocep of copula was firs popularized i 959 by a mahemaicia amed Abe Sklar hrough his heorem of he Sklar s heorem. I he heorem, copula is described as a fucio ha ecompasses various forms of margial disribuio o a form of joi disribuio. Theorem (Sklar s Theorem) [6] Le H be a joi disribuio fucio wih margis F ad G. The here exiss a copula C such ha for all x,y: H ( x, y) C F( x), G( y) () If F ad G are coiuous he copula C is uique. The H fucio is a joi disribuio fucio wih he margial disribuio of F ad G. Based o he Sklar s heorem ca be defied copula i he form of joi disribuio H so ha () ca be wrie as: H ( x, y) C F( x), G( y) P X x, Y y Archimedea Copula Copula class Archimedea is oe of he may copulas ha are widely used i various applicaios. This is caused by several facors, amog ohers [6]:. Ease i cosrucig is copula.. The copula families who belog o his class vary ad each model a differe depedecy srucure. I geeral, he form of Copula Archimedea is C( u, v) ( u) ( v),0 u, v which is a geeraor of C wih (0) ad () 0. The geeraor fucio defies he specificaios of he Archimedea copula. If copula is geeraed by usig pseudo-iverse of : 0, if (0) Copula Archimedea has he followig characerisics [6]:. C is symmerical, C( u, v) C( v, u), u, v [0,]. C is associaive, 3. If c 0,if 0 (0) C C( u, v), w C u, C( v, w), u, v, w [0,] is a cosa he c is also a geeraor of C. 4. If he margial disribuio U C(, v) v for all 0v. ad V uiform a ierval [0,] he C( u,) (0) is fiie, he Archimedea u for all 0u. By aalog Accordig o [6], if UV, is a uiform radom variable (0,) wih a joi disribuio fucio is he Archimedea copula, he he K () fucio is defied as: C is a uivariae disribuio fucio of he radom variable T C( U, V ). () KC ( ),0 () '( ) Kedall Correlaio wih Copula Archimedea To kow depedecies bewee variables used Kedall correlaio. Here is he calculaio of Kedall correlaio wih Copula. Schweizer ad Wolff [6] provide soluios i he calculaio of ρ τ by usig copula fucios ha is: 0 0 ( X, Y) 4 C( u, v) dc( u, v) 4 E C( u, v) (3) M - 5

4 I equaio (3), Kedall correlaio calculaios use he double iegral cocep. Wih Archimedea copula, he calculaios ca be simplified usig he geeraor, by usig Gees ad Rives proposiio. Proposiio (Proposiio Gees ad Rives) [7] Le X ad Y be coiuous radom variables wih Archimedea copula C geeraed by geeraor. The he Kedall correlaio coefficies for X ad Y are: () 4 d (4) '( ) Deermiig he Bes Archimedea Copula To deermie he bes copula, he firs sep is o deermie some families of he Archimedea copula. The copula of he Archimedea class cosiss of 0 families. Of he 0 families ha oly 8 families who have aalyic soluio. So i his aricle oly used 8 families ha family umber,, 4, 8,, 4, 5 ad 8. The basic assumpio is a sample of radom bivariae sized, ( X, Y ),( X, Y ),...( X, Y ) geeraed from he bivariae disribuio wih a margial disribuio fucio F( X) ad GY ( ) 0 H( x, y) each of which is coiuous ad he Archimedea Copula C. Esimaed Archimedea Copula Parameers The eigh family of Archimedea Copula class is formed of fucio geeraor which is a fucio of he ukow parameers, so ha afer deermiig a Copula models, he ex sep is o esimae. Parameer esimaio ca be performed usig parameric ad oparameric approaches. I he parameric approach, is esimaed usig he likelihood mehod, whereas i he oparameric approach, is esimaed usig he procedure proposed by Gees ad Rives. I his aricle, which will be used o esimae he parameer is a oparameric approach. The copula of he Archimedea class ca geerally be expressed i he geeraor fucio, C( u, v) ( u) ( v). From he previous discussio, he uivariae disribuio fucio (margial) radom () variable T C( U, V ) is KC (). Similarly kow for he Archimedea copula, he Kedall correlaio ca '( ) be calculaed from (3) which depeds o he geeraor fucio. Usig his iformaio, Gees ad Rives iroduces a procedure o esimae he parameer, he parameer esimae is doe by solvig (4). The geeral form of Copula fucio C ( u, v), geeraor Kedall correlaio, parameer esimaio ad disribuio fucio KC (, ) of he eigh Copula Archimedea families are preseed i Table. Afer 8 fucios of Archimedea copula cosruced usig, he relaively bes copula seleced o model he daa. For his purpose ca be used graph mehod. I is kow ha he uivariae disribuio fucio of he Archimedea copula radom variable is: () KC ( ) PC( U, V ) P C F( X ), G( Y) '( ) Because K C (.) is a disribuio fucio, he K ( ), ( ) (.) sadard uiform disribued. Thus, usig he C F X G Y cosruc QQplo K ( ), ( ) (.) o he quarile of sadard uiform disribuio fucio, QQplo will show a sraigh C F X G Y lie of 45 0 if he copula is cosruced from is ideed explorig empirical daa well. M - 6

5 No Family TABLE. Families of Archimedea Copula ad Associaed Properies Copula C ( u, v) Geeraor () / u v / ( u) ( v) / 4 exp ( l u) ( l v) 8 uv ( u)( v) ( ) ( u)( v) u v / / 4 u v / / 5 u v 8 u v l e e / / / ( ) ( l ) ( ) / / e Kedall Correlaio Parameer Esimaio Disribuio Fucio K (, ) C l 3 / Claim Saeme Esimaes Usig he Bes Archimedea Copula Before esimaig he claim reservig usig he bes Archimedea copula model, we firs esimaed he joi disribuio fucio for he claim size ad developme ime variable usig he copula fucio approach, usig () of he Sklar s heorem. Afer obaiig he value from copula, he claim reserve is calculaed by muliplyig he values of copula by he amou of he claim. The esimaed claim reserve o be calculaed is a reserve for claims icurred bu o ye repored for oe year. RESULTS AND DISCUSSION The daa used is daa obaied from documes or compay records for a cerai period. This daa is obaied from BPJS Kesehaa (BPJS Healhcare Securiy). The daa used i his case sudy amoued o 664 claims rasacios ha occurred i Jauary - December 04, wih measured variables cosisig of claims size daa, claims daa paid by isurace compaies ad developme ime daa, he ime from he mome of claim o payme. The daa is used o calculae he reserves for claims. Deermiaio of Variable Disribuio The variable used is claim size ad developme ime. Descripive saisics for hese variables are preseed i Table. From Table we ca see he skewess value of claim size ad developme ime variables respecively are 5.93 ad , he skewess values are greaer ha 0 he he graph is leaig o he lef raher ha he ormal M - 7

6 disribuio. So we eed o deermie he appropriae disribuio for he claim size ad developme ime variables. By usig kolmogorov-smirov es, he suiable disribuio for claim size variable is Gamma disribuio (0.584, ) ad for developme ime variable is Iverse gaussia disribuio (.7967,.388). TABLE. Descripive Saisics for Claim Size ad Developme Time Saisics Claim Size Developme Time Sampel Size Rage 37,54, Mea,879, Variace 6,687,755,87, Sd. Deviaio,586, Sadar Error 00, Skewess Kurosis Miimum 65, Maksimum 37,39, Esimaed Copula Archimedea To deermie he copula of he relaively bes Archimedea class for he claim size ad developme ime variables, experimes will be made o 8 models (families) of 0 models (families) i he Archimedea class of copula. Of he 0 families, oly 8 families were used because oly he eigh families had aalyical soluios. We will look for esimaio for copula parameer ( ) usig oparameric approach ha is Gees ad Rives approach. The resulig value is a esimae of ad expressed by ˆ. The values ˆ is obaied usig Table. Before lookig for esimaio from θ, firs sough correlaio of variable of claim size ad developme ime variables, obaied correlaio coefficie value is The correlaio coefficie is posiive, which meas ha he variable (claim size) has a posiive effec o he variable Y (developme ime). Esimaed resuls θ of 8 Copula Archimedea families are preseed i Table 3. TABLE 3. Esimaio Resul for The 8 families of Copula Archimedea No Family Esimaio Valid / No Valid Valid [, ) \{0} Valid Valid No Valid [, ) [, ) [, ) No Valid [, ) Valid [, ) Valid [, ) Valid [, ) From Table 3 i ca be see ha here are oly 6 valid families, so o deermie he bes copula will be doe usig oly he six families ha is family o.,, 4, 4, 5 ad 8. To deermie he bes relaively bes copula, we will use he graph mehod. The approach wih he graphical mehod is o compare he esimaio of he copula disribuio fucio wih he sadard uiform disribuio fucio hrough QQplo. If he copula maches he daa, he QQplo bewee KC C F( x), G( y) agais he uiform disribuio quarile (0.) will resemble a sraigh lie of The QQplo K C F( x), G( y) quarile (0,) is preseed i Figure. C agais he uiform disribuio X M - 8

7 FIGURE. QQplo K C F( x), G( y) C agais he uiform disribuio quarile (0,) From Figure i shows ha he QQplo idicaes he relaive o. family is beer ha he oher family. Thus, for furher discussio will be used family o. wih ˆ So obaied he relaively bes copula model ha is: / C( u, v) u v Esimaed Claim Reservig Usig Bes Archimedea Copula I will be esimaed he disribuio fucio of boh variables ha is claim size ad developme ime by usig approach of Archimedea copula fucio wih x = claim size ad y = developme ime. From he Sklar s heorem F( x ) disribued gamma ad Gy ( ) disribued iverse gauss. Afer obaiig value from copula, he claim reserve is calculaed by muliplyig he value of copula by he amou of he claim. The esimaed claim reserve is Rp 960,050,056. The esimaed reserve claim is a reserve for claims icurred bu o ye repored for oe year. CONCLUSION Based o he discussio ha has bee doe he obaied he coclusio by kowig he fucio of he margial disribuio of each radom variable, a disribuio fucio wih a se of radom variables ca be cosruced usig copula fucio where copula does o require he margial disribuio for radom variables mus be he same so i ca be used for ay radom variable. I he case sudy of BPJS Kesehaa (BPJS Healhcare Securiy) daa obaied M - 9

8 he bes claim esimaio resul is copula Archimedea family o. The resul of reserve claim esimaio represes ha reserves for claims icurred bu o ye repored for oe year. REFERENCES. M. Würich ad M. Merz, Sochasic Claims Reservig Mehods i Isurace (Joh Wiley ad Sos Ld.: Wes Sussex, 008).. T. Mack, ASTIN Bullei. 3, pp. 3-5 (993). 3. J. Nzoufras ad P. Dellaporas, Norh America Acuarial Joural, 6, pp. 3-8 (00). 4. E. de Alba, Norh America Acuarial Joural, 6, pp. -0 (00). 5. S. Roselud, ASTIN Bullei. 4, pp (0). 6. R. B. Nelse, a Iroducio o Copulas (Spriger: New York, 006). 7. E. W. Frees ad E. A. Valdez, Norh America Acuarial Joural, pp. -5 (998). 8. S. A. Klugma ad R. Parsa, Isurace: Mahemaics ad Ecoomics 4, pp (999). 9. E. W. Frees ad P. Wag, Isurace: Mahemaics ad Ecoomics 38, pp (006). 0. V. K. Kaishev ad D. S. Dimirova, Isurace: Mahemaics ad Ecoomics 39, pp (006).. Y. Bregma ad C. Kluppelberg, Scadiavia Acuarial Joural 6, pp (005).. G. Peere, Proceedigs of he 8h Ieraioal Cogress of Acuaries, ICA (006). 3. G. Peere ad T. Kollo, Proceedigs of he 8h Ieraioal Cogress of Acuaries, ICA (006). 4. X. B. Zhao ad X. Zhou, Isurace: Mahemaics ad Ecoomics 46, pp (00). 5. P. Shi ad E. W. Frees, ASTIN Bullei. 4, pp (0). 6. B. Schweizer ad E. F. Wolff, The Aals of Saisics 9, pp (98). 7. C. Gees ad L. Rives, Joural of he America Saisical Associaio 88, pp (993). M - 0