Combining Chain-Ladder and Additive Loss Reserving Methods for Dependent Lines of Business

Transcription

1 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness by Mchae Merz an Maro V Wüthrch BSTRCT Often n non-fe nsurance, cam reserves are the argest poston on the abty se of the baance sheet Therefore, the estmaton of aequate cam reserves for a portfoo consstng of severa run-off subportfoos s reevant for every non-fe nsurance company In the present paper we prove a framework n whch we unfy the mutvarate chan-aer () moe an the mutvarate atve oss reservng (LR) moe nto one moe Ths moe aows for the smutaneous stuy of nvua run-off subportfoos n whch we use both the metho an the LR metho for fferent subportfoos Moreover, we erve an estmator for the contona mean square error of precton (MSEP) for the prector of the utmate cams of the tota portfoo KEWORDS Cams reservng, mutvarate chan-aer metho, mutvarate atve oss reservng metho, mean square error of precton 27 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

2 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness Introucton an motvaton Cams reservng for severa correate run-off subportfoos Often, cam reserves are the argest poston on the abty se of the baance sheet of a non-fe nsurance company Therefore, gven the avaabe nformaton about the past eveopment, the precton of aequate cam reserves as we as the quantfcaton of the uncertantes n these reserves s a maor task n actuara practce an scence (eg, Wüthrch an Merz (28), Casuaty ctuara Socety (2), or Teuges an Sunt (24)) In ths paper we conser the cam reservng probem n a mutvarate context More precsey, we conser a portfoo consstng of severa correate run-off subportfoos On some subportfoos we use the chan-aer () metho an on the other subportfoos we use the atve oss reservng (LR) metho to estmate the cam reserves Snce n actuara practce the contona mean square error of precton (MSEP) s the most popuar measure to quantfy the uncertantes, we prove an MSEP estmator for the overa reserves Ths means that we prove a frst step towars an estmate of the overa MSEP for the prector of the utmate cams for aggregate subportfoos usng fferent cams reservng methos for fferent subportfoos These stues of uncertantes are cruca n the eveopment of new sovency guenes where one exacty quantfes the rsk profes of the fferent nsurance companes 2 Mutvarate cams reservng methos The smutaneous stuy of severa correate run-off subportfoos s motvate by the fact that: In practce t s qute natura to subve a non-fe run-off portfoo nto severa correate subportfoos, such that each subportfoo satsfes certan homogenety propertes (eg, the assumptons or the assumptons of the LR metho) 2 It aresses the probem of epenence between run-off portfoos of fferent nes of busness (eg, boy nury cams n auto abty an n genera abty busness) 3 The mutvarate approach has the avantage that by observng one run-off subportfoo we earn about the behavor of the other run-off subportfoos (eg, subportfoos of sma an arge cams) 4 It resoves the probem of atvty (e, the estmators of the utmate cams for the whoe portfoo are obtane by summaton over the estmators of the utmate cams for the nvua run-off subportfoos) Homberg (994) was probaby one of the frst to nvestgate the probem of epenence between run-off portfoos of fferent nes of busness Braun (24) an Merz an Wüthrch (27; 28) generaze the unvarate moe of Mack (993) to the mutvarate case by ncorporatng correatons between severa run-off subportfoos nother feasbe mutvarate cams reservng metho s gven by the mutvarate LR metho propose by Hess, Schmt, an Zocher (26) an Schmt (26a) whch s base on a mutvarate near moe Uner the assumptons of ther mutvarate LR moe Hess, Schmt, an Zocher (26) an Schmt (26a) erve a formua for the Gauss-Markov prector for the nonobservabe ncrementa cams whch s optma n terms of the cassca optmaty crteron of mnma expecte square oss Merz an Wüthrch (29) erve an estmator for the contona MSEP n the mutvarate LR metho usng the Gauss-Markov prector propose by Hess, Schmt, an Zocher (26) an Schmt (26a) 3 Combnaton of the mutvarate an LR methos In the seque we prove a framework n whch we combne the mutvarate moe an the VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 27

3 Varance vancng the Scence of Rsk mutvarate LR moe nto one mutvarate moe The use of fferent reservng methos for fferent subportfoos s motvate by the fact that n genera not a subportfoos satsfy the same homogenety assumptons; an/or 2 sometmes we have a pror nformaton (eg, premum, number of contracts, externa knowege from experts, ata from smar portfoos, market statstcs) for some seecte subportfoos whch we want to ncorporate nto our cams reservng anayss That s, we use the metho for a subset of subportfoos on the one han an we use the LR metho for the compementary subset of subportfoos on the other han From ths pont of vew t s nterestng to note that the metho an the LR metho are very fferent n some aspects an therefore expot fferng features of the ata beongng to the nvua subportfoos: The metho s base on cumuatve cams whereas the LR metho s appe to ncrementa cams 2 Unke the metho, the LR metho combnes past observatons n the upper trange wth externa knowege from experts or wth a pror nformaton 3 The LR metho s more robust to outers n the observatons than the metho Organzaton of ths paper In Secton 2 we prove the notaton an ata structure for our mutvarate framework In Secton 3 we efne the combne moe an erve the propertes of the estmators for the utmate cams wthn the framework of the combne metho In Secton 4 we gve an estmaton proceure for the contona MSEP n the combne metho an our man resuts are presente n Estmator 47 an Estmator 48 Secton 5 s ecate to the estmaton of the moe parameters, an, fnay, n Secton 6 we gve an exampe n ntereste reaer w fn proofs of the resuts n Secton 7 2 otaton an mutvarate framework We assume that the subportfoos consst of run-off tranges of observatons of the same sze However, the mutvarate metho an the mutvarate LR metho can aso be appe to other shapes of ata (eg, run-off trapezos) In these tranges the nces n, n, refer to subportfoos (tranges),, I, refer to accent years (rows),, J I, refer to eveopment years (coumns) The ncrementa cams (e, ncrementa payments, change of reporte cam amounts or number of newy reporte cams) of run-off trange n for accent year an eveopment year are enote by (n), an cumuatve cams (e, cumuatve payments, cams ncurre or tota number of reporte cams) are gven by, k (n),k : () Fgure shows the cams ata structure for nvua cams eveopment tranges escrbe above Usuay, at tme I, we have observatons D (n) I f, ; Ig, (2) for a run-off subportfoos n 2f,:::,g Ths meansthatattmei (caenar year I) wehavea tota of observatons over a subportfoos gven by D I [ D (n) I, (3) 272 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

4 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness Fgure Cams eveopment of trange n 2 f,:::,g an we nee to prect the ranom varabes n ts compement D,c I f, ; I, >I, n g: (4) In the seque we assume wthout oss of generaty that we use the mutvarate metho for the frst K (e, K ) run-off tranges n,:::,k an the mutvarate LR metho for the remanng n K,:::, tranges Therefore, we ntrouce the foowng vector notaton C, C, C (), C (K), C, C (K), C (), C, (), (K), an, C, (K), (), (5) C for a 2f,:::,Ig an 2f,:::,Jg In partcuar, ths means that the cumuatve/ncrementa cams of the whoe portfoo are gven by the vectors C, C, C, an,,, : (6) We efne the frst k coumns of observatons by Bk K fc, ; I an kg (7) for k 2f,:::,Jg Fnay, we efne L-mensona coumn vectors for L, K, K consstng of ones by L (,:::,) 2 R L, an enote by a D(a) C an a L (8) c b D(c) b C cl b the L L-agona matrces of the L-mensona vectors a(a,:::,a L ) 2 R L an (c b,:::,cb L ) 2 R L, where b 2 R an c (c,:::,c L ) 2 R L 3 Combne mutvarate an LR metho The foowng moe s a combnaton of the mutvarate moe an the mutvarate LR moe presente n Merz an Wüthrch (28) an Merz an Wüthrch (29), respectvey SSUMPTIOS 3 (CombneanLRmoe) ² Incrementa cams, of fferent accent years are nepenent ² There exst K-mensona constants f (f (),:::,f (K) ) an (9) ¾ (¾ (),:::,¾ (K) ) > ank-mensona ran- wth f (k) >, ¾ (k) om varabes ", ("(),,:::,"(K), ), () such that for a 2f,:::,Ig an 2f,:::, J g we have C, D(f ) C, D(C, )2 D(", ) ¾ : () VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 273

5 Varance vancng the Scence of Rsk ² There exst ( K)-mensona constants m (m (),:::,m (K) ) an (2) ¾ (¾(K),:::,¾ () ), wth ¾ (n) > an ( K)-mensona ranom varabes ", (" (K),,:::," (), ), (3) such that for a 2f,:::,Ig an 2f,:::,Jg we have, V m V 2 D(", ) ¾, (4) We ntrouce the notaton ¾ (¾,¾ ), E[D(", ) ¾ ¾ D(", )], (6) E[D(", ) ¾ (¾ ) D(", )], (C) () (C,) (,C) E[D(", ) ¾ E[D(", ) ¾ E[D(", ) ¾ ( (C,) ) : Thus, we have (¾ ) D(", )], (¾ ) D(", )], (¾ ) D(", )] (7) (¾ () ) 2 ¾ () ¾ (2) ½ (,2) ¾ (2) ¾ () ½ (2,) (¾ (2) ¾ () ¾ () ½ (,) ¾ () ¾ (2) ¾ () ¾ () ) 2 ¾ (2) ¾ () ½ (,2) (¾ () ) 2 ½ (,) ½ (2,) C (C) (,C) (C,) () : (8) where V 2 R (K) (K) are etermnstc postve efnte symmetrc matrces ² The -mensona ranom varabes ", ", ", an " k, " k, " k, are nepenent for 6 k or 6,wthE[", ] an Cov(",,", ) E[", ", ] ½ (,2) ½ (,) ½ (2,), C ½ (,2) ½ (2,) ½ (,) for fxe ½ (n,m) 2 (,) for n 6 m (5) The Mutvarate Moe 3 s sutabe for portfoos of correate subportfoos n whch the frst K subportfoos satsfy the homogenety assumptons of the metho, an the other K subportfoos satsfy the homogenety assumptons of the LR metho Uner Moe ssumptons 3, the propertes of the cumuatve cams C, an the ncrementa cams, are consstent wth the assumptons of the mutvarate tme seres moe (see Merz an Wüthrch (28)) an the mutvarate LR moe (see Merz an Wüthrch (29)) In partcuar for K an K Moe ssumptons 3 reuce to the moe assumptons of the mutvarate tme seres moe an the mutvarate LR moe, respectvey REMRK 32 ² The factors f are cae K-mensona eveopment factors, factors, age-to-age factors or nk-ratos The K-mensona con- 274 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

6 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness stants m are cae ncrementa oss ratos an can be nterprete as a mutvarate scae expecte reportng/cashfow pattern over the fferent eveopment years ² In most practca appcatons, V s chosen to be agona so as to represent a voume measure of accent year, a pror known (eg, premum, number of contracts, expecte number of cams, etc) or externa knowege from experts, smar portfoos or market statstcs Snce we assume that V s a postve efnte symmetrc matrx, there s a we-efne postve efnte symmetrc matrx V 2 (cae square root of V )satsfyngv V 2 V 2 ² Wthn the an LR framework, Braun (24) an Merz an Wüthrch (27; 28; 29) propose the eveopment year-base correatons gven by (5) Often correatons between fferent run-off tranges are attrbute to cams nfaton Uner ths pont of vew t may seem more reasonabe to aow for correaton between the cumuatve or ncrementa cams of the same caener year (agonas of the cams eveopment tranges) Ths wou ntrouce epenences between accent years However, at the moment t s not mathematcay tractabe to treat such yearbase correatons wthn the an LR framework That s, a caener year-base epenences shou be remove from the ata before cacuatng the reserves wth the or LR metho However, after correctng the ata for the caener year-base correatons, further rect an nrect sources for correatons between fferent run-off tranges of a portfoo exst an shou be taken nto account (cf Houtram (23)) Ths s exacty what our moe oes ² Matrx (C) refects the correaton structure between the cumuatve cams of eveopment year wthn the frst K subportfoos an matrx () the correaton structure between the ncrementa cams of eveopment year wthn the ast K subportfoos The matrces (C,) an (,C) refect the correaton structure between the cumuatve cams of eveopment year n the frst K subportfoos an the ncrementa cams of eveopment year n the ast K subportfoos ² There may occur ffcutes about postvty n the tme-seres efnton (), whch can be sove n a mathematcay correct way We omt these ervatons snce they o not ea to a eeper unerstanng of the moe Refer to Wüthrch, Merz, an Bühmann (28) for more etas ² The nces for ¾ an " ffer by, snce t smpfes the comparabty wth the ervatons an resuts n Merz an Wüthrch (28; 29) We obtan for the contonay expecte utmate cam E[C D I ]: LEMM 33 Uner Moe ssumptons 3: we have for a I: a) E[C D I ]E[C C,I ]E[C C,I ] b) E[C J I D I ]E[C D(f ) C,I, C,I V C,I ]E[C J I m : C,I ] PROOF Ths mmeatey foows from Moe ssumptons 3 Ths resut motvates an agorthm for estmatng the outstanng cams abtes, gven the observatons DI IftheK-mensona factors f an the ( K)-mensona ncrementa oss ratos m are known, the outstanng cams abtes of accent year for the frst K an the ast K correate run-off tranges are pre- VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 275

7 Varance vancng the Scence of Rsk cte by E[C D I ] C,I an E[C J I D I ] C,I V D(f ) C,I C,I J I m, (9) (2) respectvey However, n practca appcatons we have to estmate the parameters f an m from the ata n the upper tranges Pröh an Schmt (25) an Schmt (26a) propose the mutvarate factor estmates for f (,:::,J ) ˆf (ˆf () (K),:::, ˆf ) I I D(C D(C, )2 ( (C), )2 ( (C) ) D(C, )2 ) D(C, )2 C, : (2) In the framework of the mutvarate LR metho Hess, Schmt, an Zocher (26) an Schmt (26a) propose the mutvarate estmates for the ncrementa oss ratos m (,:::,J) ˆm (ˆm (),:::, ˆm (K) ) I V 2 ( () V 2 ( () ) V 2 ) V 2, : (22) REMRK 34 ² In the case K (e, ony one run-off subportfoo) the estmator (2) conces wth the cassca unvarate estmator of Mack (993) naogousy, n the case K (e, ony one atve run-off subportfoo) the estmator (22) conces wth the unvarate ncrementa oss rato estmates I, ˆm (23) I V k k wth etermnstc one-mensona weghts V (see, eg, Schmt (26a; 26b)) ² Wth respect to the crteron of mnma expecte square oss the mutvarate factor estmates (2) are optma unbase near estmators for f (cf Pröh an Schmt (25) an Schmt (26a)) an the mutvarate ncrementa oss rato estmates (22) are optma unbase near estmators for m (cf Hess, Schmt, an Zocher (26) an Schmt (26a)) ² For uncorreate cumuatve an ncrementa cams n the fferent run-off subportfoos (e, we set I, wherei enotes the entty matrx) we obtan the (unbase) estmators for f an m an ˆf() D(C ˆm () ) V I C, I ˆf () (24), : (25) For a gven, both ˆf an as we as ˆm an ˆm () are unbase estmators for the mutvarate factor f an mutvarate ncrementa oss rato m, respectvey (see Lemma 36 beow) However, ony ˆf an ˆm are optma n the sense that they have mnma expecte square oss; see the secon buet of these remarks In the seque we prect the cumuatve cams C, of the frst K run-off tranges an the cumuatve cams C, of the ast K run-off tranges for >Iby the mutvarate pre- 276 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

8 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness ctors C, ( C (), I,:::, C (K), ) Ê[C, D I ] D(ˆf ) C,I (26) an the mutvarate LR prectors C, ( C (K), C,I V,:::, C (), ) Ê[C, DI ] I ˆm : (27) Ths means that we prect the -mensona utmate cams C by C : (28) C ESTIMTOR 35 (Combne an LR estmator) The combne an LR estmator for E[C, DI ] s for >Igven by C C, Ê[C, DI, ]@ : C, The foowng emma coects resuts from Lemma 3:5 n Merz an Wüthrch (28) as we as from Property 3:4 an Property 3:7 nmerz an Wüthrch (29) LEMM 36 Uner Moe ssumptons 3 we have: a) ˆf s, gven B K, an unbase estmator for f, e, E[ˆf B K]f ; b) ˆf an ˆf k are uncorreate for 6 k, e, E[ˆf ˆf k ]f f k E[ˆf ] E[ˆf k ] ; c) ˆm s an unbase estmator for m, e, E[ ˆm ]m ; ) ˆm an ˆm k are nepenent for 6 k; e) Var( ˆm ) ³ PI V2 ( () ) V 2 ; f) C s, gven C,I,anunbaseestmator for E[C D I ], e, E[ C C,I ]E[C D I ] E[C C,I ] REMRK 37 ² ote that Lemma 36 f) shows that we have unbase estmators of the contonay expecte utmate cam E[C DI ] Moreover, t mpes that the estmator of the aggregate utmate cams for accent year K C K C K C s, gven C,I, an unbase estmator for P E[ C,I] ² ote that the parameters for the metho are estmate nepenenty from the observatons beongng to the LR metho an vce versa That s, here we cou even go one step beyon an earn from LR metho observatons when estmatng parameters an vce versa We omt these ervatons snce formuas get more nvove an negect the fact that one may even mprove estmators Our goa here s to gve an estmate for the overa MSEP for the parameter estmators (2) an (22) 4 Contona MSEP In ths secton we conser the precton uncertanty of the prectors K I K an, gven the observatons DI, for the utmate cams Ths means our goa s to erve an estmate of the contona MSEP for snge accent years 2f,:::,Ig whch s efne as K msep C (n) C(n) D C (n) I 2 K 2 3 E 4 C (n) C (n) D 5 I, (29) VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 277

9 Varance vancng the Scence of Rsk as we as an estmate of the contona MSEP for aggregate accent years gven by I K msep,n C I (n) D I E " I K I I 2 D I 4 Contona MSEP for snge accent years 3 5: (3) We choose 2f,:::,Ig The contona MSEP (29) for a snge accent year ecomposes as K msep D I K msep K D I msep D I " K 2 E K D I # : (3) The frst two terms on the rght-han se of (3) are the contona MSEP for snge accent years f we use the mutvarate metho for the frst K run-off tranges (numbere by n,:::,k) an the mutvarate LR metho for the ast K run-off tranges (numbere by n K,:::, ), respectvey Estmators for these two contona MSEPs are erve n Merz an Wüthrch (28; 29) an are gven by Estmator 4 an Estmator 42, beow ESTIMTOR 4 (MSEP for snge accent years, metho, cf Merz an Wüthrch (28)) Uner Moe ssumptons 3: we have the estmator for the contona MSEP of the utmate cams n the frst K run-off tranges for a snge accent year 2f,:::,Ig K msep K D I J J K D(ˆf k ) ˆ J, C D(ˆf k ) K I k k K D(C (n,m),i ) ( ˆ ) n,m K D(C,I ) K, wth ˆ, C D( C, ) 2 J ˆ (n,m) I J I ˆ (C) D( I ˆf (n) (m) ˆf k (32) C, ) 2, (33) â k (C) n ˆ (â k m ) ˆf (n) (m) ˆf, (34) where â k n an âk m are the nth an mth row of I Â D(C, )2 (C) ( ˆ ) D(C, )2 D( C k, ) 2 (C) ( ˆ ) (35) (C) an the parameter estmates ˆ are gven n Secton 5 ESTIMTOR 42 (MSEP for snge accent years, LR metho, cf Merz an Wüthrch (29)) Uner Moe ssumptons 3: we have the estmator for the contona MSEP of the utmate cams n the ast K run-off tranges for a snge accent year 2f,:::,Ig msep D I K V2 K V J I I J ˆ () V2 I K V 2 () ( ˆ ) V 2 V K, (36) 278 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

10 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness where the parameter estmates Secton 5 ˆ () are gven n REMRK 43 ² The frst terms on the rght-han se of (32) an (36) are the estmators of the contona process varances an the secon terms are the estmators of the contona estmaton errors, respectvey ² For K Estmator 4 reuces to the estmator of the contona MSEP for a snge run-off trange n the unvarate tme seres moe of Buchwaer et a (26) ² For K Estmator 42 reuces to the estmator of the contona MSEP for a snge run-off trange n the unvarate LR moe (see Mack (22)) In aton to Estmators 4 an 42 we have to estmate the cross prouct terms between the estmators an the LR metho estmators, namey (see (3)) " K E K # D I K Cov(C,C D I ) K K ( C E[C D I ]) ( C E[C D I ]) K : (37) That s, ths cross prouct term, agan, ecoupes nto a process error part an an estmaton error part (frst an secon term on the rght-han se of (37)) 4 Contona cross process varance In ths subsecton we prove an estmate of the contona cross process varance The foowng resut hos: LEMM 44 (Cross process varance for snge accent years) Uner Moe ssumptons 3: the contona cross process varance for the utmate cams C of accent year 2f,:::,Ig, gven the observatons DI, s gven by K Cov(C,C DI ) K where K J J I D(f ) C, K, (38), C E[D(C, )2 (C,) C,I ] V2 : PROOF See appenx, Secton 7 (39) If we repace the parameters f an C, n (38) by ther estmates (cf Secton 5), we obtan an estmator of the contona cross process varance for a snge accent year 42 Contona cross estmaton error In ths subsecton we ea wth the secon term on the rght-han se of (37) Usng Lemma 33 as we as efntons (26) an (27), we obtan forthecrossestmatonerrorofaccentyear 2 f,:::, Ig the representaton K ( C E[C D I ]) ( C E[C D I ]) K I J I D(ˆf ) J I (, E[ D(f ) C, ]),I K K D(C,I ) (ĝ J g J ) ( ˆm m ) V K, I where ĝ J an g J are efne by ĝ J D(ˆf I ) ::: D(ˆf J ) K, g J D(f I ) ::: D(f J ) K : (4) (4) VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 279

11 Varance vancng the Scence of Rsk In orer to erve an estmator of the contona cross estmaton error we wou ke to cacuate the rght-han se of (4) Observe that the reazatons of the estmators ˆf I,:::,ˆf J an ˆm I,:::, ˆm J are known at tme I, but the true factors f I,:::,f J an the ncrementa oss ratos m I,:::,m J are unknown Hence (4) cannot be cacuate expcty In orer to etermne the contona cross estmaton error we anayze how much the possbe factor estmators an the ncrementa oss rato estmators fuctuate aroun ther true mean vaues f an m In the foowng, anaogousy to Merz an Wüthrch (28), we measure these voattes of the estmators ˆf an ˆm by means of resampe observatons for ˆf an ˆm For ths purpose we use the contona resampng approach presente n Buchwaer et a (26), Secton 42, to get an estmate for the term (4) By contonay resampng the observatons for ˆf I,:::,ˆf J an ˆm I,:::, ˆm J, gven the upper tranges DI, we take nto account the possbty that the observatons for ˆf an ˆm cou have been fferent from the observe vaues Ths means that, gven, we generate new observatons C DI,, an for 2f,:::,Ig an 2f,:::,J g usng the formuas (contona resampng) C, D(f ) C, D(C, )2 D( ", ) ¾ an, V m V2 wth " ",, ",, ", D( ", ) ¾, ", ", (42) (43) (44) are nepenent an entcay strbute copes We efne I D(C ) D(C k k, )2 ( (C) k, )2 an I k V 2 k ( () ) V 2 k The resampe representatons for the estmates of the mutvarate factors an the ncrementa oss ratos are then gven by (see (2) an (22)) I ˆf f W D(C an I ˆm m U, )2 ( (C) V 2 ) D( ", )¾, : (45) ( () ) D( ", )¾ : (46) ote, n (45) an (46) as we as n the foowng exposton, we use the prevous notatons ˆf an ˆm for the resampe estmates of the mutvarate factors f an the ncrementa oss ratos m, respectvey, to avo an overoae notaton Furthermore, gven the observatons DI, we enote the contona probabty measure of these resampe mutvarate estmates by P For a more etae scusson DI of ths contona resampng approach we refer to Merz an Wüthrch (28) We obtan the foowng emma: LEMM 45 Uner Moe ssumptons 3: an resampng assumptons (42) (44) we have: a) ˆf,:::,ˆf J are nepenent uner P, ˆm DI, :::, ˆm J are nepenent uner P,anˆf DI an ˆm k are nepenent uner P f k 6, DI b) E [ˆf DI ]f an E [ ˆm DI ]m for J an c) E (m) [ ˆf DI ˆm (n) ]f(m) m (n) T (m,n), where T (m,n) s the entry (m,n) of the K ( K)-matrx I T W D(C, )2 ( (C) ) (C,) ( () ) V 2 U : (47) 28 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

12 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness PROOF See appenx, Secton 72 Usng Lemma 45 we choose for the contona cross estmaton error (4) the estmator 2 3 J K D(C,I ) E D 4(ĝ I J g J ( ˆm m ) 5 V K K D(C,I ) Cov D J, I J I ˆm V K : (48) We efne the matrx ª k, (ª (m,n) k, ) m,n Cov DI J kj, J I ˆm Cov DI (ĝ kj, ˆm ) (49) for a k, 2f,:::,Ig The foowng resut hos for ts components ª (m,n) k, : LEMM 46 Uner Moe ssumptons 3: an resampng assumptons (42) (44) we have for m,:::,k an n,:::, K ª (m,n) k, J J (I)_(Ik) rik f (m) r PROOF See appenx, Secton 73 f (m) T (m,n): Puttng (3), (37), (38) an (48) together an repacng the parameters by ther estmates we motvate the foowng estmator for the contona MSEP of a snge accent year n the mutvarate combne metho: ESTIMTOR 47 (MSEP for snge accent years, combne metho) Uner Moe ssumptons 3: we have the estmator for the contona MSEP of the utmate cams for a snge accent year 2f,:::,Ig msep D I wth K msep K D I msep D I 2 K J I K J D(ˆf ) ˆ C, K 2 K D(C,I) (â (m,n), ) m,n V K, (5) ˆ, C D( C, ) 2 (C,) ˆ V2, (5) â (m,n) k, ĝ (m) kj J (I)_(Ik) ˆT ˆf (m) (m,n): (52) Thereby, the frst two terms on the rght-han se of (5) are gven by (32) an (36), ĝ (m) kj enotes the mth coornate of ĝ kj (cf (4)) an the parameter (C,) estmates ˆ as we as ˆT (m,n) (entry (m,n) of the estmate ˆT for the K ( K)-matrx T )aregvennsecton5 42 Contona MSEP for aggregate accent years ow, we erve an estmator of the contona MSEP (3) for aggregate accent years To ths en we conser two fferent accent years < I We know that the utmate cams C an C are nepenent but we aso know that we have to take nto account the epenence of the estmators C an C The contona MSEP for two aggregate accent years an VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 28

13 Varance vancng the Scence of Rsk s gven by msep ( C(n) )D I msep D I " K 2 E K K K msep C(n) D I K K D I # : (53) The frst two terms on the rght-han se of (53) are the contona precton errors for the two snge accent years < I, respectvey, whch we estmate by Estmator 47 For the thr term on the rght-han se of (53) we obtan the ecomposton " K E K " K E E K " K " K E E K " K K K D I D I D I D I # # # D I # # : (54) Usng the nepenence of fferent accent years we obtan for the frst two terms on the rght-han se of (54) " K E K D I K ( C E[C D I ]) ( C E[C DI ]) K J K D(ˆf ) D(f ) I J I I (, E[, ]) #,I K K D(C,I ) (ĝ J g J ) ( ˆm m ) V K, (55) I an anaogousy " E K K D I K D(C,I ) (ĝ J g J ) ( ˆm m ) V K : I # (56) 282 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

14 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness Uner the contona resampng measure P DI these two terms are estmate by (see aso Lemma 46), s, an t,, " # K D(C s,is ) E D I V t K (ĝ sj g sj ) J It K D(C (m,n) s,is ) (ª s,t ) m,n V t K : ( ˆm m ) ow we conser the thr term on the rghthan se of (54) gan, usng the nepenence of fferent accent years we obtan " K E K K K # D I K ( C E[C D I ]) ( C E[C D I ]) K K D(C,I ) (ĝ J g J ) (ĝ J g J ) D(C,I ) K : (57) Ths term s estmate by K D(C,I ) E [(ĝ DI J g J ) (ĝ J g J ) ] D(C,I ) K where (n,m) ˆ (n,m) K D(C,I) ( (n,m) J I ) n,m K I D(C,I ) D(f k ) K, (58) ki s estmate by I ˆf (n) (m) ˆf J I k â k (C) n ˆ (â k m ) ˆf (n) (m) ˆf : (59) The parameter estmates â k n an âk m are the nth an mth row of (35) an the parameter estmate ˆ (C) sgvennsecton5(seeasomerzan Wüthrch (28)) Fnay, we obtan for the ast term on the rghthan se of (54) E " # D I K ( C E[C D I ]) ( C E[C D I ]) K, (6) whch s estmate by (see aso Merz an Wüthrch (29)) K E[( C E[C D I ]) ( C E[C I D I ]) ] K K V J I V 2 k ( () ) V 2 7 k 5 k V K : (6) Puttng a the terms together an repacng the parameters by ther estmates we obtan the foowng estmator for the contona MSEP of aggregate accent years n the mutvarate combne metho: ESTIMTOR 48 (MSEP for aggregate accent years, combne metho) Uner Moe ssumptons 3: we have the estmator for the contona MSEP of the utmate cams for aggregate accent years I K msep n C I (n) D I I 2 msep n D I < I K K D(C (m,n),i ) (â, ) m,n V K VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 283

15 Varance vancng the Scence of Rsk 2 2 < I < I K D(C (m,n),i ) (â, ) m,n V K K D(C (m,n),i ) ( ˆ ) m,n I D(C,I ) D(ˆf ) K 2 < I I J I I k K V V 2 () k ( ˆ ) V 2 k V K : (62) 43 Contona MSEP wth () ˆf an ˆm () In some cases, t may be more convenent to use estmators (24) an (25) to estmate f an m, respectvey, nstea of (2) an (22) Estmators (24) an (25) o not refect the correaton among subportfoos an are thus smper to cacuate, but beng ess than optma, w have greater MSEP than estmators (2) an (22) The changes that occur when estmators (24) an (25) are use are note here In Estmator 4, (35) becomes I Â D(C D( C k, ) 2 :, ) In Estmator 42, the ast term of (36) becomes 2 J I K V 4 V I I V 2 () ˆ V2 V K : W becomes D(C k k, ), I 3 V 5 U becomes an T becomes I I V k k D(C, )2 (C,) V 2 U, wth anaogous changes to ther estmators The rght-han se of (6) an the expresson to the rght of the frst summaton sgn n the ast term of (62) become 2 J I K V 4 V k I I k k V 2 () k ˆ V2 k V K : 5 Parameter estmaton, I k V k 3 5 For the estmaton of the cam reserves an the contona MSEP we nee estmates of the K- mensona parameters f,the( K)-mensona parameters m as we as the covarance matrces (C), () an (C,) Observe the fact that the mutvarate factor estmates an ncrementa oss rato estmates ˆf an ˆm,respectvey, can ony be cacuate f the covarance matrces (C) an () are known (cf (2) an (22)) On the other han, the covarance matrces (C), () an (C,) are estmate by means of ˆf an ˆm Therefore, as n the mutvarate metho (cf Merz an Wüthrch (28)) an the mutvarate LR metho (cf Merz an Wüthrch (29)), n the foowng we propose an teratve estmaton of these parameters In ths sprt, the true estmaton error s sghty arger because t shou aso nvove the uncertantes n the estmates of the varance parameters How- 284 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

16 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness ever, n orer to obtan a feasbe MSEP formua we negect ths term of uncertanty Estmaton of f an m s startng vaues for the teraton we use the unbase estmators ˆf() an ˆm () efne by (24) an (25) for,:::,j Fromˆf () an ˆm () we erve the estmates ˆ (C)() ()() an ˆ of the covarance matrces (C) an () for,:::,j (see estmators (64) an (67) beow) Then these estmates ˆ (C)() ()() an ˆ areusetoetermneˆf () an ˆm () va (s ) ˆf(s) an I D(C, )2 (C)(s) ( ˆ ) D(C, )2 I D(C, )2 (C)(s) ( ˆ ) D(C, )2 C, ˆm (s) I V 2 ()(s) ( ˆ V 2 ()(s) ( ˆ ) V 2 ) V 2, : (63) Ths agorthm s then terate unt t has suffcenty converge Estmaton of (C), () an (C,) The covarance matrces (C) an () are estmate teratvey from the ata for,:::,j Forthe covarance matrces (C) we use the estmator propose by Merz an Wüthrch (28) (s ) Q D(C, )2 (F, ˆ (C)(s) I ˆf (s) ) (F (s), ˆf ) D(C, )2, (64) where enotes the Haamar prouct (entrywse prouct of two matrces), F, D(C, ) C, I w (n,m) an n,m K (65) wth w (n,m) r, r C (m) I, I, C (m), 2 : (66) For more etas on ths estmator see Merz an Wüthrch (28), Secton 5 For the covarance matrces () we use the teratve estmaton proceure suggeste by Merz an Wüthrch (29) (s ) ˆ ()(s) I I V 2 (, V ˆm (s) ) (, V ˆm (s) ) V 2 : (67) For more etas on ths estmator see Merz an Wüthrch (29), Secton 5 Motvate by estmators (64) an (67) for matrces (C) an (), we propose for the covarance matrx (C,) ( (,C) ) estmator ˆ (C,) I I D(C, )2 (F, ˆf ) (, V ˆm ) V 2 : (68) Estmaton of, C an T Wth these estmates we obtan as estmates of the matrces, C an T C, D( C, ) 2 ˆ (C,) V 2, ˆT Ŵ ( where an I k I (C) ˆ ) D(C, )2 (C,) ˆ D(C k, )2 ( I k () ( ˆ ) V 2 Û, (C) ˆ ) D(C V 2 () k ( ˆ ) V 2 k k, )2 : VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 285

17 Varance vancng the Scence of Rsk ˆ (C) ˆ () The matrces an are the resutng estmates n the teratve estmaton proceure for the parameters (C) an () (cf (64) an (67)) REMRK 5 ² For a more etae motvaton of the estmates for the fferent covarance matrces see Merz an Wüthrch (28; 29) an Sectons 825 an 835 n Wüthrch an Merz (28) ² If we have enough ata (e, I>J), we are abe to estmate the parameters (C) J, () J an (C,) J ( (,C) J ) by (64), (67) an (68) respectvey Otherwse, f I J, we o not have enough ata to estmate the ast covarance matrces In such cases we can use the estmates ˆ' (m,n) for J (e, ˆ' (m,n) ¾ (m) ¾(n) ½(m,n) of the eements '(m,n) of (C) s an estmate of '(m,n), cf (6)) to erve estmates ˆ' (m,n) J of the eements ' (m,n) J of (C) J for a m, n K For exampe, ths can be one by extrapoatng the usuay ecreasng seres ˆ' (m,n),:::, ˆ' (m,n) J2 (69) by one atona member ˆ' (m,n) J for m, n K naogousy, we can erve estmates for () J, (C,) J an (,C) J ( (C,) J ) (see Merz an Wüthrch (28; 29) an the exampe beow) However, n a cases t s mportant to verfy that the estmate covarance matrces are postve efnte ² Observe that the K K-mensona estmate ˆ (C)(s) s snguar f I K 2 snce n ths case the menson of the near space generate by any reazatons of the (I ) K- mensona ranom vectors D(C, )2 (F, (s) ˆf ) wth 2f,:::,I g (7) s at most I I (I K 2)K naogousy, the ( K) ( K)- ()(s) mensona estmate ˆ s snguar when I ( K) 2 Furthermore, the ranom (C)(s) matrx ˆ an/or ˆ ()(s) may be -contone for some <I K 2 an <I ( K) 2, respectvey Therefore, n practca appcaton t s mportant to verfy whether (C)(s) ()(s) the estmates ˆ an ˆ are we-contone or not an to mofy those estmates (eg, by extrapoaton as n the exampe beow) whch are -contone (see aso Merz an Wüthrch (28; 29)) 6 Exampe To ustrate the methooogy, we conser two correate run-off portfoos an B (e, 2) whch contan ata of genera an auto abty busness, respectvey The ata s gven n Tabes an 2 n ncrementa an cumuatve form, respectvey Ths s the ata use n Braun (24) an Merz an Wüthrch (27; 28; 29) The assumpton that there s a postve correaton between these two nes of busness s ustfe by the fact that both run-off portfoos contan abty busness; that s, certan events (eg, boy nury cams) may nfuence both run-off portfoos, an we are abe to earn from the observatons from one portfoo about the behavor of the other portfoo In contrast to Merz an Wüthrch (28) (mutvarate metho for both portfoos) an Merz an Wüthrch (29) (mutvarate LR metho for both portfoos) we use fferent cams reservng methos for the two portfoos an B We now assume that we ony have estmates V of the utmate cams for portfoo an use the LR metho for portfoo The metho s appe for portfoo B Ths means we have K K, an the parameters f, m, (C), (), (C,) as we as the a pror estmates V of the utmate cams n the fferent accent years n portfoo are now scaars Moreover, t hos that (C) (¾ ) 2 (¾ () ) 2, () (¾ ) 2 (¾ (2) ) 2 an (C,) ½ (,2) ¾ () ¾ (2) ½ (,2) (,C) ¾ ¾ 286 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

18 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness Tabe Portfoo (ncrementa cams (2), ), source Braun (24) Genera Labty Run-Off Trange /D ,966 3,86 9,36 95,2 83,74 42,53 37,882 6,649 7,669,6,738 3,572 6,823,893 49,685 3,659 9,592,43 75,442 44,567 29,257 8,822 4, ,73 2, ,94 8,34 49,56 5,825 78,97 4,77 4,76 7,95,97 2,643,3, ,937 88,246 34,35 39,97 74,45 65,4 49,65 2, ,48 2, ,92 79,48 7,2 3,6 79,64 8,364 2,44,324 6, ,78 73,879 7,295 44,76 93,694 72,6 4,545 25,245 7, ,35 93,57 8,77 53,86 2,96 86,753 45,547 23, ,73 27,43 24,558 22,276,88 4,966 59, ,58 245,7 232,223 93,576 65,86 85,2 9 73, ,73 262,92 232,999 86,45 39,82 297,37 372, ,27 54, ,5 54, ,24 576, ,325 Tabe 2 Portfoo B (cumuatve cams C (), ), source Braun (24) uto Labty Run-Off Trange /D , ,96 32, ,34 37,479 37,2 38,99 385, ,52 392,26 39,225 39,328 39,537 39,428 52,296 35,75 376,63 48,299 44,38 465, , , ,34 479,97 48, ,38 483, ,325 37,244 43,69 464,4 59, ,26 535,45 536, ,92 539, ,765 54, ,94 37, ,94 433, ,2 478,93 482, ,56 485, ,34 485,6 4 7,333 34,5 434,2 47, ,2 5,96 54,4 57,679 58,627 57, ,643 36,23 446,857 58,83 526,562 54,8 547,64 549,65 549, ,22 396, ,34 553,487 58,849 6,64 622, , ,98 46,47 52, ,72 6,69 63,82 648, ,95 426, ,47 587,893 64, , ,426 59, ,433 73,692 79,9 249,58 58, 722,36 844,59 258, ,2 95, ,762 99, ,997 Tabe 3 shows the estmates of the utmate cams for the two subportfoos an B as we as the estmates for the whoe portfoo consstng of both subportfoos Snce I J 3 we o not have enough ata to erve estmates of the parameters (C) 2, () 2 an (C,) 2 (,C) 2 by means of the propose estmators Therefore, we use the extrapoatons ˆ (C) 2 ˆ () 2 (C) (C) mnfˆ,(ˆ )2 (C) ˆ g, () () mnfˆ,(ˆ )2 () ˆ g, an (7) ˆ (C,) (,C) (C,) 2 ˆ 2 mnf ˆ,(ˆ (C,) ) 2 (C,) ˆ g to erve estmates of (C) 2, () 2 an (C,) 2 (,C) 2 Moreover, so that ˆ an ˆ 2 are postve efnte, we estmate () ˆ () an (C,) (,C) by () () mnfˆ 9,(ˆ )2 () ˆ 9 g, an (72) ˆ (C,) (,C) (C,) ˆ mnf ˆ 9,(ˆ (C,) ) 2 (C,) ˆ 9 g: Tabe 4 shows the estmates for the parameters () The one-mensona estmates ˆm an ( ˆ ) 2 are the parameter estmates use n the unvarate LR metho appe to the nvua subportfoo naogousy, the one-mensona estmates ˆf (C) an ( ˆ ) 2 are the parameter estmates use n the unvarate metho appe VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 287

19 Varance vancng the Scence of Rsk Tabe 3 Estmates of the utmate cams for subportfoo, subportfoo B, an the whoe portfoo Subportfoo Subportfoo B Portfoo V C c cc Tota 5,3 549,589 39,428 94,7 632, ,74 483,839,48, ,33 68,4 54,2,48, , , ,227,28, ,32 783,593 58,744,292, , ,88 552,825,389, , ,86 639,3,577,973 7,56,778,98,2 658,4,756,6 8,24,569,54,92 684,79,839,62 9,397,23,43,49 845,543 2,276,952,832,676,735, ,734 2,698,67 2,56,78 2,65,99,69,26 3,235,25 2 2,559,345 2,66,56,474,54 4,35,75 3 2,456,99 2,274,94,426,6 3,7, Tabe 5 Estmate reserves Subportfoo Subportfoo B Portfoo Reserves Reserves Reserves LR Metho Metho Tota 2, ,23 2 5, ,83 3 9,68,2,89 4 3, , ,386 3,32 29,58 6 4,96 3,66 44, ,946,45 9, ,95 2,567 65, ,823 54, , ,362 8,575 72,937,77,55 254,5,33,666 2,86, ,448 2,372,28 3 2,225,22,3,63 3,256,284 Tota 6,3,53 2,63,62 8,375,5 Tota 7,758,43 7,498,658,823,48 28,322,77 to the nvua subportfoo B From the estmates ˆ (C,) of the covarances (C,) (,C) we obtan estmates ˆ½ (,2) of the correaton coeffcents ½ (,2) (C,) q by ˆ ˆ () (C) ˆ ote: Snce both the metho an the LR metho are appe to one-mensona tranges, the parameter estmates ˆf an ˆm can be cacuate recty (usng the unvarate methos) an one can omt the teraton escrbe n Secton 5 The frst two coumns of Tabe 5 show for each accent year the reserves for subportfoos an B estmate by the (unvarate) LR metho an the (unvarate) metho, respectvey The ast coumn, enote by Portfoo Reserves Tota, shows the estmate reserves for the entre portfoo Tabe 6 shows for each accent year the estmates for the contona process stanar evatons an the corresponng estmates for the coeffcents of varaton The frst two coumns contan the vaues for the nvua subportfoos an B cacuate by the (unvarate) LR metho an the (unvarate) metho, respectvey The ast coumn, enote by Portfoo Tota, shows the vaues for the entre portfoo The same overvew s generate for the square roots of the estmate contona estmaton errors n Tabe 7 Tabe 4 Parameter estmates for the parameters m,f, ( () ) 2, ( (C) ) 2 an (C,) Portfoo /B ˆm ˆf ( ˆ () ) ( ˆ (C) ) ˆ (C,) 66: : : ˆ½ (,2) : : : CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

20 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness Tabe 6 Estmate contona process stanar evatons Subportfoo Subportfoo B Portfoo LR Metho Metho Tota 33 57% :8% % %,9 47:5%, % 3,64 7% 2,46 232% 2,85 26% 4 5,38 392% 2,78 273% 6, % 5 2,669 48% 4,75 57% 4,769 5% 6 4,763 36% 5,384 47% 7,45 39% 7 7,89 22% 6, % 2, % 8 23,84 66% 8,27 377% 27,258 65% 9 3,227 6% 4,69 267% 36,849 9% 43,67 72% 24,366 25% 55,63 77% 5,294 48% 33,227 3% 7,55 53% 2 64,43 36% 47,888 85% 96,2 4% 3 8,24 36% 7,293 4% 44,83 44% Tota 3,444 2% 34,676 65% 22,746 24% Tabe 7 Square roots of estmate contona estmaton errors Subportfoo Subportfoo B Portfoo LR Metho Metho Tota 49 63% :3% 5 226% % :3%,64 25% 3,74 2%, %,823 68% 4 2,96 23%,78 722% 3,67 245% 5 6,7 254% 2,66 832% 7, % 6 7,859 92% 3,5 85% 9,294 29% 7,49 3% 3,57 355%,92 3% 8 2,953 9% 4,44 92% 4,64 88% 9 6,473 58% 6,98 28% 9,467 58% 24,583 4%,22 93% 29,528 4% 3,469 28% 5,669 62% 38,363 29% 2 38,94 22% 23,625 42% 52,727 22% 3 42,287 9% 47,683 46% 66,27 2% Tota 72,74 27% 9,599 44% 24,339 26% n fnay the frst three coumns n Tabe 8 gve the same overvew for the estmate precton stanar errors Moreover, the ast two coumns n Tabe 8 contan the resuts for the estmate precton stanar errors assumng no correaton an perfect postve correaton between the corresponng cams reserves of the two subportfoos an B These vaues are cacuate by msep C DI msep C () D I 2cmsep 2 C () D I msep C (2) D I msep 2 C (2) D I (73) wth c anc, respectvey Except for accent year 3, for a snge accent years an aggregate accent years, we observe that the estmates n the thr coumn are between the ones assumng no correaton an perfect postve correaton ote that accountng for the correaton between subportfoos as about 9% to the estmate precton stanar error for the entre portfoo (295,38 vs 27,5) 7 ppenx: Proofs In ths secton we present the proofs for Lemmas 44, 45, an 46 Tabe 8 Estmate precton stanar errors Subportfoo Subportfoo B Portfoo Portfoo Portfoo LR Metho Metho Tota Correaton Correaton 2 85% :2% % % % %,436 94:2%,648 38%,557 3% 2,38 393% 3,96 24% 2,92 244% 3,353 3% 3,5 324% 4,872 45% 4 6,2 446% 3, % 7,432 55% 6,97 47% 9, % 5 4, % 5,48 73% 6,7 566% 5,326 59% 9, % 6 6,724 49% 6,22 699% 9,74 443% 7,844 4% 22,945 55% 7 2, % 7, % 23,735 26% 2,99 242% 28,6 39% 8 27,3 89% 9,23 423% 3,928 87% 28,624 73% 36,254 29% 9 34,424 2% 6,9 296% 4,675 23% 38,4 2% 5,65 5% 49,589 83% 26, % 62,569 88% 56,34 79% 76,33 7% 59,66 55% 36,737 45% 79,959 6% 7,64 53% 96,397 72% 2 75,25 42% 53,399 94% 9,72 46% 92,27 39% 28,649 54% 3 9,67 4% 26,65 23% 58,684 49% 55,73 48% 27,284 67% Tota 26,63 34% 62,874 79% 295,38 35% 27,5 32% 379,488 45% VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 289

21 Varance vancng the Scence of Rsk 7 Proof of Lemma 44 By nucton we prove that Cov(C,k, k, C,I ) D(f ) C,, (74) where, C s efne by (39) for a k I an,:::,i a) ssume k Then, usng (7), we have Cov(C,,, C,I ) E[D(C, )2 D(", ) ¾ (V2 D(", ) ¾ ) C,I ] E[D(C, )2 E[D(", ) ¾ (V 2 D(", ) ¾ ) C, ] C,I ] E[D(C, )2 (C,) C,I ] V 2 C, : (75) Ths competes the proof for k b) Inucton step ssume that the cam s true for k We prove that t s aso true for k Usng the nucton step, we have contona on C,, k, Cov(C,k,, C,I ) D(f k ) Cov(C,k,, C,I ) k D(f ) C, : Ths fnshes the proof of cam (74) Usng resut (74) eas to the proof of Lemma Proof of Lemma 45 a) Foows from (45) an (46) an the fact that ",, ",k are nepenent for 6 k b) Foows from (45) an (46) an the fact that E [ " DI, ] c) Usng the nepenence of fferent accent years we obtan Cov DI (ˆf, ˆm ) Hence, E D I I W Cov DI (D( " ( () ) V 2 I W ( (C) (m) [ ˆf ˆm (n) D(C, )2 ( (C) ), )¾ U D(C, )2,D( ", )¾ ) ) (C,) ( () ) V 2 U T : ]f(m) m (n) Cov DI ( f (m) m (n) T (m,n), (m) ˆf, ˆm (n) ) where T (m,n) s the entry (m,n) of the K ( K)-matrx T Ths competes the proof of Lemma ProofofLemma46 The components ª (m,n) k, are efne by (49) Hence, we cacuate the terms Cov D (ĝ I kj, ˆm )E D [ĝ I kj ˆm ] E D [ĝ I kj ]E D [ ˆm ]: I Ths expresson s equa to (e, the K ( K)-matrx consstng of zeros) for < I k Hence J ª k, (ª (m,n) k, ) m,n Cov D (ĝ I kj, ˆm ): (I)_(Ik) For I k we have, usng Lemma 45, that the (m,n)-component of the covarance matrx on the rght-han se of the above equaty s equa to 2 rik f (m) r (f (m) J rik m(n) f (m) r J T (m,n)) f (m) T (m,n): r f (m) r J rik Ths competes the proof of Lemma 46 f (m) r m (n) 29 CSULT CTURIL SOCIET VOLUME 3/ISSUE 2

22 Combnng Chan-Laer an tve Loss Reservng Methos for Depenent Lnes of Busness References Braun, C, The Precton Error of the Chan Laer Metho ppe to Correate Run-off Tranges, STI Buetn 34, 24, pp Buchwaer, M, H Bühmann, M Merz, an M V Wüthrch, The Mean Square Error of Precton n the Chan Laer Reservng Metho (Mack an Murphy Revste), STI Buetn 36, 26, pp Casuaty ctuara Socety, Founatons of Casuaty ctuara Scence (4th e), rngton, V: Casuaty ctuara Socety, 2 Hess, K Th, K D Schmt, an M Zocher, Mutvarate Loss Precton n the Mutvarate tve Moe, Insurance: Mathematcs an Economcs 39, 26, pp 85 9 Homberg, R D, Correaton an the Measurement of Loss Reserve Varabty, Casuaty ctuara Socety Forum, Sprng 994, pp Houtram,, Reservng Jugement: Conseratons Reevant to Insurance Labty ssessment uner GPS2, Insttute of ctuares of ustraa IV Genera Insurance Semnar, 23 Mack, T, Dstrbuton-free Cacuaton of the Stanar Error of Chan Laer Reserve Estmates, STI Buetn 23, 993, pp Mack, T, Schaenverscherungsmathematk (2n e), Karsruhe, Germany: Verag Verscherungswrtschaft, 22 Merz, M, an M V Wüthrch, Precton Error of the Chan Laer Reservng Metho ppe to Correate Run-off Tranges, nnas of ctuara Scence 2, 27, pp 25 5 Merz, M, an M V Wüthrch, Precton Error of the Mutvarate Chan Laer Reservng Metho, orth mercan ctuara Journa 2, 28, pp Merz, M, an M V Wüthrch, Precton Error of the Mutvarate tve Loss Reservng Metho for Depenent Lnes of Busness, Varance 3, 29, pp 3 5 Pröh, C, an K D Schmt, Mutvarate Chan-Laer, paper presente at the STI Cooquum, 25, Zurch, Swtzeran Schmt, K D, Optma an tve Loss Reservng for Depenent Lnes of Busness, Casuaty ctuara Socety Forum, Fa 26a, pp Schmt, K D, Methos an Moes of Loss Reservng Base on Run-Off Tranges: Unfyng Survey, Casuaty ctuara Socety Forum, Fa 26b, pp Teuges, J L, an B Sunt, Encycopea of ctuara Scence, Vo, Chchester, UK: Wey, 24 Wüthrch,MV,anMMerz,Stochastc Cams Reservng Methos n Insurance, Chchester, UK: Wey, 28 Wüthrch, M V, M Merz, an H Bühmann, Bouns on the Estmaton Error n the Chan Laer Metho, Scannavan ctuara Journa, 28, pp VOLUME 3/ISSUE 2 CSULT CTURIL SOCIET 29