Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness by Mchae Merz and Maro V Wüthrch ABSTRACT Often n non-fe nsurance, cams reserves are the argest poston on the abty sde of the baance sheet Therefore, the predcton of adequate cams reserves for a portfoo consstng of severa run-off subportfoos from dependent nes of busness s of great mportance for every non-fe nsurance company In the present paper, we consder the cams reservng probem n a mutvarate context that s, we study a speca case of the mutvarate addtve oss reservng mode proposed by Hess, Schmdt, and Zocher (26) and Schmdt (26a) Ths mode aows for a smutaneous study of the ndvdua run-off subportfoos and enabes the dervaton of an estmator for the condtona mean square error of predcton (MSEP) for the predctor of the utmate cams of the tota portfoo We ustrate the resuts usng the data gven n Braun (24) and compare them to the resuts derved by the mutvarate chan-adder methods of Braun (24) and Merz and Wüthrch (28) KEYWORDS Cams reservng, sovency, uncertanty, dependent nes of busness, mutvarate addtve oss reservng method, mutvarate chan-adder method, process varance, estmaton error, mean square error of predcton VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 3
Varance Advancng the Scence of Rsk Introducton and motvaton Cams reservng Often n non-fe nsurance, cams reserves are the argest poston on the abty sde of the baance sheet Therefore, gven the avaabe nformaton about the past, the predcton of an adequate amount of cam abty assumed by the non-fe nsurance company, as we as the quantfcaton of the uncertantes n these reserves, s a maor task n actuara practce and scence [eg, Tayor (2); Wüthrch and Merz (28); Casuaty Actuara Socety (2); Teuges and Sundt (24); Engand and Verra (22)] 2 Mutvarate cams reservng methods and ther condtona MSEP In the present paper, we consder the cams reservng probem for a portfoo consstng of severa correated run-off subportfoos Ths smutaneous study of severa ndvdua run-off subportfoos s motvated by the foowng consderatons: ² In practce t s qute natura to subdvde a non-fe run-off portfoo nto severa correated subportfoos, such that each subportfoo satsfes certan homogenety propertes (eg, the chan-adder assumptons or the assumptons of the addtve method) ² It addresses the probem of dependence between the run-off portfoos of dfferent nes of busness (eg, between auto abty and genera abty busness) ² The mutvarate approach has the advantage that by observng one run-off subportfoo we can earn about the behavor of the other runoff subportfoos (eg, subportfoos of sma and arge cams) ² It resoves the probem of addtvty (e, the estmators of the utmate cams for the whoe portfoo are obtaned by summaton over the estmators of the utmate cams for the ndvdua run-off subportfoos) However, n the case of correated run-off subportfoos, the cacuaton of the condtona mean square error of predcton (MSEP) for the predctor of the utmate cam sze of the tota portfoo s more sophstcated than the cacuaton of the condtona MSEP for the predctor of the utmate cam sze of a snge run-off subportfoo An aternatve dea to the smutaneous study of severa ndvdua run-off subportfoos s to cacuate the reserves and ther uncertantes ony for the tota aggregated run-off portfoo However, one shoud pay attenton to the fact that f the subportfoos satsfy, for exampe, the assumptons of the chan-adder or the assumptons of the addtve method, the aggregated run-off portfoo does not n genera satsfy these assumptons (Ane 994; Kemmt 24) Therefore, n most cases t s not a promsng souton to study the aggregated portfoo for the cams reservng probem of severa run-off subportfoos Homberg (994) was probaby the frst one to nvestgate the probem of dependence between run-off portfoos of dfferent nes of busness Later Hawe (997) and Quarg and Mack (24) [see aso Merz and Wüthrch (26)] proposed the frst bvarate modes whch express the dependence between the pad and ncurred osses of a snge run-off subportfoo Braun (24) generazed the we-known unvarate chan-adder mode of Mack (993) to the bvarate case by ncorporatng correatons between two run-off subportfoos In ths setup he derved an estmate for the condtona MSEP for the predctor of the utmate cam sze of two correated run-off subportfoos Usng a mutvarate tme-seres mode for the chan-adder method Merz and Wüthrch (27) gave an estmator for the condtona MSEP n the case of N correated run-off subportfoos However, both the Braun (24) approach and the Merz and Wüthrch (27) approach have the dsadvantage that the chan-adder factors are estmated 32 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness n a unvarate way Ths means the estmaton of the chan-adder factors s restrcted to the data of the respectve ndvdua run-off subportfoo and therefore does not take nto account the correaton structure between the dfferent runoff subportfoos Pröh and Schmdt (25) and Schmdt (26a) showed that these unvarate estmates of the chan-adder factors are not optma n terms of a cassca optmaty crteron n the case of correated run-off subportfoos and therefore one shoud repace the unvarate estmators wth mutvarate estmators of the chanadder factors refectng the correaton structure However, ther study dd not go beyond best estmators; that s, they dd not derve an estmator for the condtona MSEP for the predctor of the utmate cam sze of the tota portfoo Fnay, usng a mutvarate chan-adder tmeseres mode, Merz and Wüthrch (28) derved an estmate for the condtona MSEP, n whch the chan-adder factors are estmated n a mutvarate way That s, Merz and Wüthrch (28) studed the condtona MSEP for the mutvarate chan-adder estmates proposed by Pröh and Schmdt (25) and Schmdt (26a) 3 Mutvarate addtve oss reservng method The mutvarate addtve oss reservng method proposed by Hess, Schmdt, and Zocher (26) and Schmdt (26a) s based on a mutvarate near mode whch s sutabe for certan portfoos consstng of severa correated run-off subportfoos The addtve oss reservng method has the foowng features: It s a very smpe cams reservng method whch can easy be mpemented n a spreadsheet 2 Unke the chan-adder method, the addtve oss reservng method combnes past observatons n the upper cams deveopment trange wth externa knowedge from experts or wth a pror nformaton (eg, premum, number of contracts, data from smar run-off portfoos, and market statstcs) 3 It s apped to ncrementa data and thus aows for modeng negatve ncrementa cams n contrast to some other modes such as the (overdspersed) Posson mode [cf Wüthrch and Merz (28)] Ths makes the addtve oss reservng method sutabe for the use of ncurred data, whch often exhbts negatve ncrementa vaues n ater deveopment years due to earer overestmaton of case reserves 4 Unke the chan-adder method, the predcton for the utmate cam does not depend competey on the ast observaton on the dagona Ths means an outer on the dagona w not be proected drecty to the utmate cam Therefore, the addtve oss reservng method s more robust to outers n the ast observatons than the chan-adder method Under the assumptons of ther mutvarate addtve oss reservng mode, Hess, Schmdt, and Zocher (26) and Schmdt (26a) derved a formua for the Gauss-Markov predctor for the nonobservabe ncrementa cam szes whch s optma n terms of a cassca optmaty crteron The components of these predctors are dfferent from the predctors of the unvarate addtve oss reservng method f the subportfoos are correated (eg, see Schmdt (26a; 26b) for the unvarate addtve oss reservng method) Ths means that the predctors of the unvarate method are not optma n the case of correated subportfoos However, Hess, Schmdt, and Zocher (26) and Schmdt (26a) dd not derve an estmator of the condtona MSEP for the mutvarate addtve oss reservng method Snce n actuara practce and scence the condtona MSEP s a very popuar measure to quantfy the uncertantes n cams reserves, ths paper ams to f that gap These studes of uncer- VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 33
Varance Advancng the Scence of Rsk Fgure Cams deveopment trange number n tanty are especay cruca n the deveopment of new sovency gudenes where one exacty quantfes the rsk profe of the dfferent nsurance companes More precsey, we formuate a stochastc mode for the mutvarate addtve oss reservng method to derve an estmator for the condtona MSEP usng the Gauss-Markov predctor proposed by Hess, Schmdt, and Zocher (26) and Schmdt (26a) Furthermore, by means of a detaed exampe, ths estmator s then compared to the estmator for the condtona MSEP of the unvarate predctor (e, f we gnore the correaton structure between ndvdua subportfoos) as we as to the estmator for the condtona MSEP of the mutvarate chan-adder methods consdered by Braun (24) and Merz and Wüthrch (28) 2 Notaton and mutvarate framework In the seque we assume that the data for the N run-off subportfoos consst of run-off tranges of observatons of the same sze However, the mutvarate addtve oss reservng method can aso be apped to other shapes of data (eg, run-off trapezods) In these N tranges the ndces n, n N, refer to subportfoos (tranges),, I, refer to accdent years (rows), and, J, refer to deveopment years (coumns) Fgure shows the cams data structure for the N cams deveopment tranges descrbed above The ncrementa cams (e, ncrementa payments, change of reported cam amount, or number of reported cams wth reportng deay ) of run-off trange n for accdent year and deveopment year are denoted by (n), and cumuatve cams (e, cumuatve payments, cams ncurred, or tota number of reported cams) of accdent year up to deveopment year are gven by C (n), = k= (n),k : () We assume that the ast deveopment year s gven by J, thats (n), = for a >J,and the ast accdent year s gven by I Moreover, our assumpton that we consder run-off tranges mpes I = J 34 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness Usuay, at tme I, we have observatons D (n) I = f (n), ; + Ig, (2) for a run-off subportfoos n 2f,:::,Ng Ths meansthatattmei (caendar year I) wehavea tota of observatons over a subportfoos D N I = N[ D (n) I, (3) and we need to predct the random varabes n ts compement D N,c I = f (n), ; I, + >I, n Ng: (4) For the dervaton of the condtona MSEP for severa run-off subportfoos, t s convenent to wrte the data of the N subportfoos n vector form Thus, we defne the N-dmensona random vectors of ncrementa and cumuatve payments by, =( (),,:::,(N), ) and (5) C, =(C (),,:::,C(N), ) for 2f,:::,Ig and 2f,:::,Jg Moreover,we defne the N-dmensona coumn vector consstng of ones by =(,:::,) 2 R N (6) and denote by a D(a)= B @ C A (7) a N the N N-dagona matrx of the vector a = (a,:::,a N ) 2 R N 3 Mutvarate addtve oss reservng method The addtve oss reservng method s easy to appy It s based on the study of ndvdua ncrementa oss ratos We defne for 2f,:::,Ig and 2f,:::,Jg the N-dmensona vector of ndvdua ncrementa oss ratos for accdent year and deveopment year by M, =(M (),,:::,M(N), ) =V,, (8) wthavoumemeasure V = B @ V (,) V (,2) V (,N) V (2,) V (2,2) V (2,N) V (N,) V (N,2) V (N,N), C A (9) whch s a determnstc postve defnte symmetrc N N-matrx The component M (n), of M, denotes the ndvdua ncrementa oss rato (reatve to V ) for accdent year and deveopment year of subportfoo n In the unvarate case N =wehave M, =, =V, () where V s an approprate (determnstc) voume measure If, denotes ncrementa payments and V s the tota premum receved for accdent year, thenm, tes how the tota oss rato s pad over tme 3 Mutvarate addtve oss reservng mode The foowng mutvarate addtve oss reservng mode s a speca case of the mutvarate cams reservng mode studed by Hess, Schmdt, and Zocher (26) and Schmdt (26a) MODEL ASSUMPTIONS 3 (MULTIVARIATE DITIVE MODEL) ² Incrementa payments of dfferent accdent years are ndependent ² There exst N N-dmensona determnstc postve defnte symmetrc matrces V,:::,V I and N-dmensona constants ( =,:::,J) m =(m (),:::,m (N) ) and () ¾ =(¾ (),:::,¾(N) ) VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 35
Varance Advancng the Scence of Rsk wth ¾ (n) > foran =,:::,N as we as N- dmensona random varabes ", =(" (),,:::,"(N), ), (2) such that for a 2f,:::,Ig and 2f,:::,Jg we have, =V m +V =2 D(", ) ¾ : (3) Moreover, the random varabes ", are ndependent wth E[", ]= and Cov(",,", ) = E[", ", ]= B @ ½ (,2) ½ (,N) ½ (2,N) C ½ (2,) ½ (N,) ½ (N,2), C A (4) knowedge for subportfoo m nfuences the ncrementa payments for another subportfoo n n accdent year by choosng V (n,m) 6=Inths case we obtan a nondagona matrx V In the unvarate case N =, the addtve mode satsfes, =V = m + V =2 ¾ ",, (5) wth E[, ]=V m and Var(, )=V ¾ 2 : (6) Hence ths mode can aso be nterpreted as a GLM mode wth Gaussan varance functon (e, V(x) = ), voume measure V and dsperson parameter ¾ 2 [cf McCuagh and Neder (989)] Under Mode Assumptons 3 we have Cov(,,, )=V =2 V =2, (7) where =E[D(", ) ¾ ¾ D(",)] =D(¾ ) Cov(",,", ) D(¾ ) (¾ () )2 ¾ () ¾(2) ½(,2) ¾ () ¾(N) ¾ (2) ¾() ½(2,) (¾ (2) )2 ¾ (2) ¾(N) = B @ ½(,N) ½(2,N) ¾ (N) ¾() ½(N,) ¾ (N) ¾(2) ½(N,2) (¾ (N) )2 : (8) C A where ½ (n,m) 2 (,) for n,m 2f,:::,Ng and n 6= m Ceary, n most practca appcatons V s chosen to be dagona so as to represent a voume measure of accdent year, known a pror (eg, premum, number of contracts, expected number of cams, etc), or an estmate from externa knowedge such as experts, smar portfoos, or market statstcs (see Exampe n Secton 6) However, we can aso take nto account that the voume measure or estmate from externa By Mode Assumptons 3 we restrct any assumpton regardng the correaton between the N run-off subportfoos to each of the correspondng deveopment years ( =,:::,J) nthe N run-off tranges Matrx refects the correaton structure between the ncrementa cams of deveopment year n the N dfferent subportfoos Often correatons between dfferent run-off subportfoos are attrbuted to cams nfaton Under ths pont of vew, t may seem more reasonabe to aow for correaton between 36 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness the ncrementa cams of the same caender year (dagonas of the cams deveopment tranges) However, ths woud contradct the assumpton of ndependent accdent years whch s common to most cams reservng methods, and n fact aso necessary to deveop reasonabe estmators from a mathematca pont of vew The Mutvarate Addtve Mode 3 s a speca case of the mutvarate cams reservng mode proposed by Hess, Schmdt, and Zocher (26) and Schmdt (26a), n contrast to whch we assume that ncrementa payments, are ndependent (nstead of ony uncorreated) and generated by the tme seres (3) REMARK 32 ² The ncrementa cams, and k, are ndependent for 6= k or 6= ² The N-dmensona expected ncrementa oss ratos (m ) J can be nterpreted as a mutvarate scaed expected reportng/cashfow pattern over the dfferent deveopment years ² In (7) we use the notaton nstead of snce t smpfes the comparabty wth the dervatons and resuts n Merz and Wüthrch (28) ² Snce we assume that V s a postve defnte symmetrc matrx, there s a we-defned postve defnte symmetrc matrx V =2 (caed square root of V )satsfyngv =V =2 V =2 We obtan for the condtona expectaton (best estmate) E[C,J DI N ] of the utmate cam C,J : PROPERTY 33 Under Mode Assumptons 3 we have for a I J + I E[C,J D N I ]=E[C,J C,I ] = C,I +V =I + m : (9) PROOF Usng the ndependence of the ncrementa cams we obtan 2 3 E[C,J DI N ]=C,I + E 4, D I N 5 = C,I + =I + =I + = C,I +V Ths fnshes the proof E[, ] =I + m = E[C,J C,I ]: (2) QED Ths resut motvates an agorthm for estmatng the expected utmate cams gven the observaton DI N IftheN-dmensona expected ncrementa oss ratos (m ) J are known, the expected outstandng cams abtes of accdent year for the N correated run-off tranges based on the nformaton DI N are estmated by E[C,J D N I ] C,I =V =I + m : (2) However, n most practca appcatons we have to estmate the ratos m from the data n the upper eft trange Hess, Schmdt, and Zocher (26) and Schmdt (26a) propose the foowng mutvarate estmates, for =,:::, J ˆm =(ˆm () I = @ = I =,:::, ˆm (N) ) V =2 V=2 (V =2 V=2 A ) M, : (22) The varabe ˆm (n) denotes the estmated ncrementa oss rato for deveopment year and runoff trange n 2f,:::,Ng basedonthenformaton DI N Note that the covarance structure between the ncrementa cams n the dfferent runoff subportfoos s ncorporated nto the estmaton of m through the matrx VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 37
Varance Advancng the Scence of Rsk Hess, Schmdt, and Zocher (26) and Schmdt (26a) showed the foowng property, whch states that the mutvarate ncrementa oss rato estmates (22) are optma estmators of m wth respect to the crteron of mnma expected squared oss PROPERTY 34 Under Mode Assumptons 3, the estmator ˆm s an unbased estmator for m, whch mnmzes the expected squared oss among a N-dmensona near combnatons of the unbased estmators (M, ) I for m, e, E[(m ˆm ) (m ˆm )] 2 I = mn 4@m W, M A, W, 2R N N E = 3 I @m W, M A, 5: = (23) PROOF See proof of Theorem 4 n Schmdt (26a) QED Note, n Property 34 we assume that the covarance matrx sknownhowever,fwe do not have a reabe estmate for ths covarance matrx t s often more approprate n practce to use the unvarate estmators Property 34 motvates the foowng estmator for the condtonay expected utmate cam: ESTIMATOR 35 (Mutvarate addtve estmator) The mutvarate addtve estmator for E[C, D N I ] s for + I gven by d dc, =( C (),,:::, C d(n), ) = Ê[C, D N I ]=C,I +V =I + ˆm : (24) Ths means that n the mutvarate addtve method we predct the normazed cumuatve cams V C, by the sum of the ast observed normazed cumuatve cams V C,I and the weghted estmated ratos ˆm I +,:::, ˆm,gven the nformaton DI N From (24) we obtan for the ncrementa payments, wth + >Ithe predctors d d, =( (),,:::, d(n), ) REMARK 36 =V ˆm : (25) ² In the case = J (note that we assume I = J) we have ˆm J = M,J ² Estmator (22) s a weghted average of the observed ndvdua normazed ncrementa cams M, InthecaseN = (e, ony one run-off subportfoo), the estmators (22) concde wth the unvarate estmated ncrementa oss ratos I ˆm = = V P I k= V k M, (26) wth determnstc weghts V, whch are used n the unvarate addtve oss reservng method, and from Estmator 35 we obtan the unvarate addtve estmator P I dc,j = k= C,I + k, P I =I + k= V V k (27) [see, for exampe, Schmdt (26a; 26b)] ² If we negect the covarance structure between the ncrementa cams n the dfferent run-off subportfoos [e, n (22) we set =I, where I denotes the dentty matrx], we obtan the foowng (unbased) estmator I ˆm () I @ A V M, : (28) = = V = Moreover, f the voumes V are dagona matrces, then the components of (28) are gven by ˆm (n)() I = = V (n,n) P I k= V(n,n) k M (n), : (29) Ths means that n ths case the components of ˆm () are gven by the estmators of the unvarate addtve oss reservng method Itcaneasybeseenthat ˆm does not depend on the matrx f = J or f and V,:::, V I are dagona In ths case the N components 38 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness ˆm (),:::, ˆm (N) of (22) concde wth the unvarate estmators (29) for the N run-off subportfoos Ths means that f,:::, J 2 and V,:::,V I are dagona matrces, the foowng estmates concde: ) the estmaton for the whoe portfoo based on the unvarate estmators (26) for every ndvdua run-off subportfoo, 2) the mutvarate predcton based on the estmators (28), and 3) the mutvarate predcton based on the mutvarate estmators (22) However, Property 34 shows n other cases t s more reasonabe to use the mutvarate estmators (22) Moreover, under Mode Assumptons 3 t hods: PROPERTY 37 Under Mode Assumptons 3 we have a) ˆm and ˆm k are ndependent for 6= k; ³ PI ; b) Var( ˆm )= = V=2 V=2 c) gven C,I,theestmatorC d,j s an unbased estmator for E[C,J DI N]=E[C,J C,I ], e, E[ C d,j C,I ]=E[C,J DI N]; d) C d,j s an unbased estmator for E[C,J ], e, E[ C d,j ]=E[C,J ] PROOF a) Foows from the ndependence of the normazed ncrementa cams M, =V, and M k, =V k k, for 6= b) Usng (7) we obtan Var(M, )=V Var(, ) V =V =2 V =2 : (3) Wth the ndependence of the M, ths eads to where c) We have I A = @ V =2 = E[ d C,J C,I ] = C,I +V = C,I +V =I + =I + V=2 E[ ˆm ] A : (32) m = E[C,J D N I ]: (33) d) Foows mmedatey from c) Ths fnshes the proof QED Observe that Property 37 c) shows that the Estmator 35 s an unbased estmator for E[C,J DI N ] Furthermore, ths mmedatey mpes that the estmator for the aggregated utmate cam of one snge accdent year N d C (n) = C d,j (34),J s, gven C,I, an unbased estmator for P N E[C (n),j DN I ] 4 Condtona MSEP In ths secton we consder the uncertanty n the cams reserves predcted by the estmators P N d C (n),j and P I P N d = C (n),j, gven the ob- I Var( ˆm )=A Var @ (V =2 = V=2 ) M A, A 2 3 I =A 4 (V =2 V=2 ) Var(M, ) (V =2 V=2 ) 5 A = 2 I =A 4 V =2 = V=2 3 5 A =A, (3) VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 39
Varance Advancng the Scence of Rsk servatons DI N Ths means our goa s to derve an estmate of the condtona MSEP for ndvdua accdent years 2f,:::,Ig whch s defned as à msep P N d C (n) n C(n),J DN,J I 2à N = E 4 d C (n),j N C (n),j 2 DN I 3 5 = E[( d C,J C,J ) ( d C,J C,J ) D N I ] (35) as we as an estmate of the condtona MSEP for aggregated accdent years msep P @ d C (n) A,n C(n),J DN,J I,n 2 3 6 = E 4@ d C (n),j C (n),j A2 D N 7 I 5:,n,n (36) 4 Condtona MSEP for snge accdent years We choose 2f,:::,Ig Snce the estmator P N d C (n),j s known at tme t = I (e, t s based on observatons from DI N ), the condtona MSEP (35) can be decouped nto condtona process varance and condtona estmaton error, that s we derve estmates for both the condtona process varance and the condtona estmaton error for N correated run-off tranges 4 Condtona process varance In ths subsecton we derve an estmate for the condtona process varance of a snge accdent year Var(C,J DI N ) We obtan the foowng resut: PROPERTY 4 (Process varance for a snge accdent year) Under Mode Assumptons 3 the condtona process varance for the utmate cam C,J of accdent year 2f,:::,Ig s gven by Var(C,J DI N ) = V =2 @ =I + A V =2 : (38) PROOF Usng the ndependence of the ncrementa cam payments, we have Var(C,J DI N ) = Var @ A = @ = V =2 =I + @ Var(, ) A =I +, =I + for >I J Ths competes the proof A V =2 (39) QED msep P n C(n),J DN I à N d C (n),j = Var(C,J D N I ) {z } condtona process varance + ( C d,j E[C,J DI N ]) ( C d,j E[C,J DI N ]) : (37) {z } condtona estmaton error The condtona process varance orgnates from the stochastc movement of C,J, whereas the condtona estmaton error refects the uncertanty n the estmaton of the condtona expectaton (best estmate) E[C,J DI N ] In the seque If we repace the parameter n (38) by ts estmate (cf Secton 5), we obtan an estmator of the condtona process varance for accdent year Moreover, from (39) we obtan the recursve formua for the condtona process varance of 4 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness PROOF Usng Propertes 37 a) b) we obtan E[( C d,j E[C,J DI N ]) ( C d,j E[C,J DI N ]) ] 2 = E 4@ V ( ˆm m ) A @ V ( ˆm m ) A 3 5 = V @ 2 = 6 V 4 =I + =I + =I + =I + Var( ˆm ) A V (43) I @ V =2 = V=2 A 3 7 5 V : (44) accdent year Var(C, D N I ) = (Var(C, D N I ) +V =2 V =2 ), (4) for = I +,:::,J wth Var(C,I DI N)= 42 Condtona estmaton error Now we estmate the uncertanty n the estmaton of E[C,J DI N] by the estmator C d,j Ths means we derve an estmator for the second term on the rght-hand sde of (37) We estmate the condtona estmaton error by ts expected vaue E[( C d,j E[C,J DI N ]) ( d C,J E[C,J D N I ]) ] : (4) We obtan the foowng resut: PROPERTY 42 (Estmator of the estmaton error for a snge accdent year) Under Mode Assumptons 3 the estmator (4) of the condtona estmaton error for P N d C (n),j wth 2 f,:::,ig s gven by E[Var( C d,j C,I )] 2 Ã I 3 = V 4 V =2 V=2 5 =I + = V : (42) On the other hand, usng Property 37 c), we have E[( d C,J E[C,J D N I ]) ( d C,J E[C,J D N I ]) ] = E[Var( d C,J C,I )] : (45) Ths fnshes the proof QED Note, we can rewrte (42) n the recursve form E[Var( d C, C,I )] = E[Var( d C, C,I )] + V Ã I = V =2 V=2 V (46) for = I +,:::,J wth Var( d C,I C,I ) = Fnay, repacng the parameters n (38) and (42) by ther estmates (see Secton 5), we obtan the foowng estmator of the condtona MSEP for a snge accdent year: RESULT 43 (Condtona MSEP for a snge accdent year) Under Mode Assumptons 3 we VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 4
Varance Advancng the Scence of Rsk have the estmator for the condtona MSEP of the utmate cam for a snge accdent year 2 fi J +,:::,Ig à dmsep P N d C (n) n C(n),J DN,J I = V =2 + V 4 =I + 2 =I + ˆ V =2 à I 3 V =2 ˆ V=2 5 = V, (47) where the estmated covarance matrx ˆ s gven n (59), beow For N = formua (47) reduces to the estmator of the condtona MSEP for a snge portfoo n the unvarate addtve oss reservng method msep d C,J D I ( C d,j ) = V =I + ˆ¾ 2 + V2 =I + ˆ¾ 2 P I = V, (48) where V s a known one-dmensona voume measure for accdent year [cf Mack (22)] 42 Condtona MSEP for aggregated accdent years In the foowng we consder the condtona MSEP for aggregated accdent years Our goa s to derve an estmate for (36) From Mode Assumptons 3 we know that the utmate cams C,J and C k,j of two accdent years and k wth <k I are ndependent However, snce the estmators C d,j and C d k,j use the same observatons DI N for estmatng the parameters m,dfferent accdent years are no onger ndependent We start wth the consderaton of two accdent years <k à N msep P P C d (n) N n C(n),J + n C(n) k,j DN,J + C d (n) k,j I 2à N = E 4 ( C d(n),j + C d(n) k,j ) 2 N (C (n),j + C(n) k,j ) DN I 3 5 : (49) We obtan for the condtona MSEP of the sum of two accdent years the decomposton nto process varance and condtona estmaton error whch eads to msep P n C(n),J +P n C(n) k,j DN I =msep P n C(n),J DN I +msep P n C(n) k,j DN I à N d C (n),j à N d + C (n),j à N d C (n) k,j +2 ( d C,J E[C,J D N I ]) N d C (n) k,j ( d C k,j E[Ck,J D N I ]) : (5) Ths shows that we have to derve an estmator for the cross product [thrd term on the rght sde of (5)], whch comes from the dependence descrbed above Anaogousy to (4), we estmate ths cross product by ts expected vaue E[( d C,J E[C,J D N I ]) ( d C k,j E[Ck,J D N I ]) ] (5) and obtan the foowng resut: PROPERTY 44 (Estmator of the cross product) Under Mode Assumptons 3 the estmator (5) of the cross product of aggregated accdent years 42 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness and k wth <k I s gven by E[( C,J c E[C,J D N I ]) ( Ck,J d E[Ck,J D N I ]) ] 2 à I 3 = V 4 V =2 V=2 5 Vk : =I + = (52) PROOF Anaogousy to the proof of Property 42 we obtan for <k E[( C,J c E[C,J D N I ]) ( Ck,J d E[Ck,J D N I ]) ] " J # = V Var( ˆm ) V k 2 = V 4 =I + à I =I + = V =2 V=2 3 5 Vk : (53) QED Puttng (47) and (52) n (5) eads to the foowng estmator for the condtona MSEP of the utmate cam for aggregated accdent years: RESULT 45 (Condtona MSEP for aggregated accdent years) Under Mode Assumptons 3 we have the estmator for the condtona MSEP of the utmate cam for aggregated accdent years à I dmsep P N d C Pn (n) C(n),J DN,J I = à I = dmsep P N d C (n) n C(n),J DN,J I = +2 2 4 V <k I à I =I + = V =2 ˆ V=2 3 5 V k, (54) where the estmated covarance matrx ˆ s gven n (59), beow For N =, formua (54) reduces to the estmator of the condtona MSEP for aggregated accdent years n the unvarate addtve method à I msep d P C,J D I dc,j = = I msep d C,J D I ( C d,j ) = +2 <k I V V k =I + ˆ¾ 2 P I = V (55) wth known one-dmensona voume measure V for accdent year [cf Mack (22)] 5 Parameter estmaton For the estmaton of the cams reserves and the condtona MSEP we need estmates of the N-dmensona parameters m,:::,m J and of the N N-dmensona covarance parameters,:::, J Estmates of the mutvarate ncrementa oss ratos m are gven n (22) However, estmator (22) s ony an mpct estmator for m snce t depends on parameter, whch on the other hand s estmated by means of ˆm Therefore, as n the mutvarate chan-adder method [cf Merz and Wüthrch (28)], we propose an teratve estmaton of these parameters In ths sprt, the true estmaton error s sghty arger because t shoud aso nvove the uncertantes n the estmate of the varance parameters In order to obtan a feasbe MSEP formua we negect ths term of uncertanty Estmaton of m As startng vaues for the teraton we defne ˆm () by (28) for =,:::,J Estmator ˆm () s an unbased optma estmator for m f the N run-off subportfoos are uncorreated However, f the subportfoos are correated, t s st unbased but no onger optma (cf Property 34) From ˆm () we derve an estmate ˆ () of for =,:::,J [see estmator (59) beow] Then ths estmate s used to determne VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 43
Varance Advancng the Scence of Rsk ˆm () va ˆm (k) =(ˆm ()(k) I = @ = I =,:::, ˆm (N)(k) ) V =2 (k) ( ˆ (V =2 (k) ( ˆ ) V =2 A ) V =2 ) M, (56) for =,:::,J Ths agorthm s then terated unt t has suffcenty converged Estmaton of The N N-dmensona covarance parameters are estmated teratvey from the data for =,:::,J A postve semdefnte estmator of the postve defnte matrx s gven by ˆ = I I V =2 (, V ˆm () ) = (, V ˆm () ) V =2 (57) for =,:::,J If the matrces V are a dagona, the dagona eements of the random matrx (57) are unbased estmators of the correspondng dagona eements (¾ () )2,:::,(¾ (N) )2 (58) of Its nondagona eements sghty underestmate the absoute vaue of the correspondng nondagona eements of However,ths ack of unbasedness s not too mportant snce the random matrx (57) has to be nverted anyway and the nverse of an unbased estmator s n genera not unbased [cf Appendx of Merz and Wüthrch (28)] Ths eads to the foowng teraton for the estmator of : ˆ (k) = I I = V =2 (, V ˆm (k ) ) (, V ˆm (k ) ) V =2 (59) for =,:::,J and k If we have enough data (e, we have a runoff trapezod wth I>J),weareabetoestmate teratvey the parameter J by (59) Otherwse, we can use the estmates b' (n,m)(k) of the eements ' (n,m) of for J n teraton k [e,b' (n,m)(k) s an estmate of ' (n,m) = ¾ (n) ¾(m) ½(n,m) n teraton k, cf (8)] to derve estmates b' (n,m)(k) J of the eements of J for a n m N For exampe, ths can be done by extrapoatng the usuay exponentay decreasng seres b' (n,m)(k),:::,b' (n,m)(k) J 2 (6) by one addtona member b' (n,m)(k) J for n m N and k However, one needs to care- ˆ (k) J fuy check that the estmate s postve defnte In hgher dmensona cases ths s often nontrva, and n fact, many choces are not postve defnte, whch cas for addtona adustments Moreover, observe that the N N- ˆ (k) dmensona estmate s snguar when I N + 2, snce n ths case the dmenson of the near space generated by any reazatons of the (I +) N-dmensona random vectors V =2 (, V ˆm (k ) ) wth 2f,:::,I g (6) s at most I + I (I N +2)+= N Furthermore, the reazatons of (6) may be (neary) neary dependent for some <I N + 2 whch mpes that the correspondng ˆ (k) reazaton of the random matrx s -condtoned or even snguar Therefore, n practca appcaton t s mportant to verfy whether (k) the estmates ˆ are we-condtoned or not and to modfy those estmates (eg, by extrapoaton as n the exampe beow) whch are not we-condtoned Many methods have been suggested to mprove the estmaton of the covarance matrx so that the estmate s postve defnte and we-condtoned By producng a we-condtoned covarance es- 44 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness Tabe Genera abty run-off trange (ncrementa cams (), ), source Braun (24) Genera abty run-off trange AY/DY 2 3 4 5 6 7 8 9 2 3 59,966 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 4,355 879 4,73 2,727 776 2 5,94 8,34 49,56 5,825 78,97 4,77 4,76 7,95,97 2,643,3,44 3 84,937 88,246 34,35 39,97 74,45 65,4 49,65 2,36 596 24,48 2,548 4 98,92 79,48 7,2 3,6 79,64 8,364 2,44,324 6,24 265 5 7,78 73,879 7,295 44,76 93,694 72,6 4,545 25,245 7,497 6 92,35 93,57 8,77 53,86 2,96 86,753 45,547 23,22 7 95,73 27,43 24,558 22,276,88 4,966 59,46 8 97,58 245,7 232,223 93,576 65,86 85,2 9 73,686 285,73 262,92 232,999 86,45 39,82 297,37 372,968 364,27 54,965 373,5 54,64 2 96,24 576,847 3 24,325 Tabe 2 Auto abty run-off trange (ncrementa cams (2), ), source Braun (24) Auto abty run-off trange AY/DY 2 3 4 5 6 7 8 9 2 3 4,423 33,538 65,2 3,358 27,39 377 9,889 4,477 36 7,8,35 3 29 9 52,296 52,879 7,438 4,686 22,9 25,35 7,96 4,843 3,593 848 4,383,64 2 44,325 62,99 6,365 5,432 55,224 7,95 8,234,49 2,6 669 76 977 3 45,94 6,732 79,458 46,642 29,384 5,8 3,598 5,527 2,484 462,8 4 7,333 7,68 92,6 36,227,872 8,76 3,8 3,538 948 875 5 89,643 7,48 85,734 6,226 8,479 3,556 7,523,964 88 6 79,22 27,22,8 56,83 28,362 29,79,244 2,568 7 25,98 2,39 4,397 45,277 34,888 3,93 7,563 8 2,95 25,478 98,68 62,846 52,435 22,824 9 23,426 295,796 4,2 82,259 59,29 249,58 33,52 42,26 22,23 258,425 427,587 229,97 2 368,762 54,34 3 394,997 tmate we automatcay get a we-condtoned estmate for the nverse of the covarance estmate Most of these approaches rey on the concept of shrnkage whch s qute smar to the we-known actuara concept of credbty For more detas and other advanced methods on covarance matrx estmaton we refer to Schäfer and Strmmer (25) 6 Exampe: two correated abty run-off subportfoos To ustrate the methodoogy, we consder two correated run-off portfoos A and B (e, N = 2), whch contan data of genera and auto abty busness, respectvey The data are gven n Tabes and 2 n ncrementa form These are the data used n Braun (24) and aso n Merz and Wüthrch (27; 28) The assumpton that there s a postve correaton between these two nes of busness s ustfed by the fact that both run-off portfoos contan abty busness; that s, certan events (eg, body nury cams) may nfuence both run-off portfoos, and we are abe to earn from the observatons from one portfoo about the behavor of the other portfoo We assume that the 2 2-matrces V are dagona and ther dagona eements V (,) and V (2,2) are pror estmates of the utmate cams n the dfferent accdent years n run-off portfoo A and B, respectvey Such pror estmates are usu- VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 45
Varance Advancng the Scence of Rsk Tabe 3 Pror estmates and chan-adder estmates of the utmate cams Run-off portfoo A V (,) CL C c(),j Run-off portfoo B V (2,2) CL C c(2),j 5,3 549,589 43,23 39,428 632,897 564,74 537,988 483,839 2 658,33 68,4 589,45 54,2 3 723,456 795,248 523,49 486,227 4 79,32 783,593 5,498 58,744 5 845,673 837,88 598,345 552,825 6 94,378 938,86 68,376 639,3 7,56,778,98,2 698,993 658,4 8,24,569,54,92 74,29 684,79 9,397,23,43,49 93,557 845,543,832,676,735,433 947,326 962,734 2,56,78 2,65,99,34,29,69,26 2 2,559,345 2,66,56,538,96,474,54 3 2,456,99 2,274,94,487,234,426,6 Tota 7,758,43 7,498,658,86,268,823,48 ay obtaned from budget fgures, pan vaues or from premum cacuaton parameters Tabe 3 shows these a pror estmates as we as the correspondng cassca unvarate chan-adder estmates C d() CL,J and C d(2) CL,J for comparson purposes We see that the pror estmates and the unvarate chan-adder estmates are cose together [for the unvarate chan-adder method see, eg, Mack (993) or Buchwader, Bühmann, Merz, and Wüthrch (26)] Snce I = J = 3 we do not have enough data to derve an estmate of the 2 2-matrx 2 usng estmator (59) Therefore, we use the extrapoaton b' (n,m) 2 =mnf(b' (n,m) ) 2 =b' (n,m),b' (n,m) g (62) to derve estmates of ts eements ' (n,m) 2 = ¾ (n) 2 ¾ (m) 2 ½(n,m) 2 for n,m =,2 (note ½ (,) 2 = ½ (2,2) 2 =) Moreover, snce estmator (59) woud ead to an -condtoned matrx ˆ,wehaveasoestmated the eements of the 2 2-matrx by b' (n,m) =mnf(b' (n,m) ) 2 =b' (n,m) 9,b' (n,m) 9 g: (63) Tabe 4 shows the estmates for the parameters m, ¾ and ½ (,2) after three teratons k =,2,3 We observe fast convergence of the two-dmensona estmates ˆm (k ), b¾ (k) and the one-dmensona estmates ˆ½ (,2)(k) (k =,2,3) n the sense that there are barey any changes n the estmates after three teratons The frst and second component of the estmates ˆm () and b¾ () are the parameter estmates used n the unvarate addtve method apped to the ndvdua run-off portfoos A and B, respectvey Except for deveopment years, 6, and, we observe postve estmates ˆ½ (,2)(k) for the correaton coeffcents The three negatve estmates shoud not be overstated snce they are cose to zero The frst two coumns of Tabe 5 show for each accdent year the cams reserves for run-off subportfoos A and B estmated by the (unvarate) addtve oss reservng method Coumn portfoo (k = ) shows the reserves for the whoe portfoo consstng of the two run-off subportfoos A and B estmated by the mutvarate addtve oss reservng method These vaues are based on the estmates ˆm () and therefore concde wth the sum of the cams reserves for the two ndvdua subportfoos Coumns portfoo (k = 2) and portfoo (k = 3) contan the cams reserves for the whoe portfoo based on the estmates ˆm () and ˆm (2), respectvey These estmates ead to a tota reserve whch s about 6,9 hgher than the one based on ˆm () The coumn denoted by overa cacuaton shows the estmated reserve when frst aggregatng both run-off tranges to one snge run-off trange and then estmatng the cams reserves wth the (unvarate) addtve oss reservng method Snce n ths approach two run-off tranges wth dfferent deveopment patterns are added together (cf components of estmates ˆm (k) n Tabe 4), ths approach s ony reasonabe f the proporton of exposures from each trange does not change sgnfcanty over the dfferent accdent years In our exampe ths approach eads to a 46 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness Tabe 4 Estmates ˆm (k ), b¾ (k) and ˆ½ (,2)(k) for the parameters m, ¾ and ½ (,2) n the frst three teratons k =,2,3 A/B 2 3 4 5 6 7 8 9 2 3 ˆm () 9969 2638 7528 27 8466 4852 2474 43 86 66 428 529 37 32897 629 954 5577 366 548 9 6 349 :5 355 : :26 b¾ () 358 23 442 892 364 39 579 75 22 69 84 56 7 2774 89 57 6 74 57 47 25 496 35 3 35 6 ˆ½ (:2)() :2644 84865 599 378 344 3249 :46 75342 3322 66573 :395 4397 4895 ˆm () 9974 264 7493 29 8452 4844 2476 44 95 64 428 529 37 32899 672 96 5572 37 55 9 7 354 :5 354 :97 :26 b¾ (2) 358 23 442 892 364 39 579 76 22 69 84 56 7 2774 82 57 6 74 57 47 25 496 35 3 35 6 ˆ½ (:2)(2) :2654 84893 5925 37 3434 3262 :467 75527 33235 6662 :392 4399 4894 ˆm (2) 9974 264 7493 29 8452 4844 2476 44 95 64 428 529 37 32899 672 96 5572 37 55 9 7 354 :5 354 :97 :26 b¾ (3) 358 23 442 892 364 39 579 76 22 69 84 56 7 2774 82 57 6 74 57 47 25 496 35 3 35 6 ˆ½ (:2)(3) :2654 84893 5926 37 3434 3262 :467 75529 33235 6662 :392 4399 4894 Tabe 5 Estmated reserves Addtve method Chan-adder method Mutvarate Unvarate Mutvarate Unvarate Unvarate portfoo portfoo portfoo portfoo portfoo portfoo subportfoo A subportfoo B (k =) (k =2) (k = 3) overa cac Braun (24) MW (28) 2,348 42 2,26 2,26 2,26 2,262,8,8 2 5,923 747 5,76 5,96 5,96 5,442 4,655 4,655 3 9,68,93,8,85,85,356,827,826 4 3,77 893 4,6 4,677 4,677 3,82 6,22 6,37 5 26,386 3,54 29,54 29,723 29,723 28,266 29,2 29,49 6 4,96 3,243 44,49 44,749 44,753 4,64 45,793 46,829 7 8,946,87 9,32 9,88 9,83 84,45 86,4 87,24 8 43,95 2,58 64,973 65,79 65,75 53,693 57,65 58,569 9 283,823 55,625 339,448 34,6 34,66 328,7 344,3 346,42 594,362,5 75,53 76,398 76,45 659,59 679,82 68,729,77,55 235,757,33,272,33,647,33,653,246,294,287,458,287,654 2,86,833 568,4 2,374,947 2,376,6 2,376,7 2,325,74 2,453,38 2,45,6 3 2,225,22,38,295 3,263,56 3,264,85 3,264,826 3,223,75 3,,679 3,92,98 Tota 6,3,53 2,47,68 8,359,83 8,366,62 8,366,9 8,23,852 8,28,874 8,25,35 tota reserve whch s about 235,3 242,3 ess than the one obtaned by separate cacuaton of the cams reserves n run-off subportfoos A and B The ast two coumns show the vaues cacuated by the mutvarate chan-adder reservng methods proposed by Braun (24) (e, chan-adder factors are estmated n a unvarate way) and Merz and Wüthrch (28) (e, chan-adder factors are estmated n a mutvarate way), respectvey We see that the mutvarate addtve oss reservng method eads to a tota reserve whch s about 47,2 5,8 hgher than the ones obtaned by the two mutvarate chan-adder methods Tabe 6 shows for each accdent year the estmates for the condtona process standard de- VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 47
Varance Advancng the Scence of Rsk Tabe 6 Estmated condtona process standard devatons Addtve method Chan-adder method Mutvarate Unvarate Mutvarate Unvarate Unvarate portfoo portfoo portfoo portfoo overa portfoo portfoo subportfoo A subportfoo B (k =) (k = 2) (k = 3) cacuaton Braun (24) MW (28) 33 57% 444 33:% 483 29% 483 29% 483 29% 52 226%,289 72%,289 72% 2 47 79%,34 5:8%,289 249%,289 248%,289 248%,275 234% 5,966 282% 5,966 282% 3,64 7% 2,48 227% 2,783 258% 2,783 257% 2,783 257% 2,85 275% 7,29 66% 7,29 66% 4 5,38 392% 2,552 2859% 6,42 439% 6,42 437% 6,42 437% 6,96 448% 9,8 65% 9,85 599% 5 2,669 48% 4,743 53% 4,78 5% 4,782 497% 4,782 497% 4,656 58% 6,43 554% 6,49 549% 6 4,763 36% 5,43 555% 7,227 39% 7,233 385% 7,234 385% 7,2 49% 9,2 48% 9,45 49% 7 7,89 22% 6,682 663% 2,537 226% 2,544 224% 2,544 224% 2,33 238% 2,9 255% 2,937 25% 8 23,84 66% 7,989 379% 27,2 64% 27,8 64% 27,8 64% 26,64 73% 28,933 84% 28,966 83% 9 3,227 6% 4,366 258% 36,978 9% 36,985 9% 36,985 9% 37,86 5% 39,28 4% 39,322 4% 43,67 72% 2,49 93% 53,848 76% 53,854 76% 53,854 76% 53,978 82% 63,663 94% 63,724 93% 5,294 48% 28,466 2% 67,39 5% 67,44 5% 67,44 5% 69,957 56% 99,98 78%,4 78% 2 64,43 36% 4,2 7% 9,552 39% 9,569 39% 9,569 39% 94,86 4% 99,543 8% 99,68 8% 3 8,24 36% 5,955 5% 7,567 33% 7,58 33% 7,58 33%,223 34% 36,2 2% 36,2 2% Tota 3,444 2% 77,62 38% 74,596 2% 74,624 2% 74,624 2% 79,43 22% 396,73 48% 396,85 48% Tabe 7 Square roots of estmated condtona estmaton errors Addtve method Chan-adder method Mutvarate Unvarate Mutvarate Unvarate Unvarate portfoo portfoo portfoo wthout portfoo overa portfoo portfoo subportfoo A subportfoo B (k =) (k =2) (k =3) corrn ˆm () cacuaton Braun (24) MW (28) 49 63% 57 357:2% 549 249% 549 249% 549 249% 549 249% 576 255%,32 729%,32 729% 2 375 63% 985 3:9%,3 23%,3 22%,3 22%,3 23%,86 99% 4,533 97,4% 4,533 974% 3,74 2%,538 289%,89 67%,89 67%,89 67%,89 67%,898 83% 6,87 55% 6,87 55% 4 2,96 23%,547 733% 3,55 24% 3,55 239% 3,55 239% 3,56 24% 3,383 245% 7,37 434% 7,34 43% 5 6,7 254% 2,65 829% 7,8 264% 7,8 263% 7,8 263% 7,8 264% 7,64 27% 9,796 336% 9,795 333% 6 7,859 92% 2,75 848% 9,87 26% 9,9 23% 9,9 23% 9,92 26% 8,87 22%,738 256%,742 25% 7,49 3% 3,584 355%,887 3%,89 3%,89 3%,892 3%,283 34% 3,99 63% 3,996 6% 8 2,953 9% 4, 9% 4,5 88% 4,53 88% 4,53 88% 4,56 88% 3,734 89% 6,637 6% 6,644 5% 9 6,473 58% 6,934 25% 9,523 58% 9,527 57% 9,527 57% 9,53 58% 9,446 59% 22,767 66% 22,776 66% 24,583 4% 9,52 86% 28,86 4% 28,865 4% 28,865 4% 28,87 4% 27,84 42% 34,3 5% 34,6 5% 3,469 28% 3,6 56% 36,975 28% 36,982 28% 36,982 28% 36,996 28% 36,798 3% 5,43 4% 5,386 4% 2 38,94 22% 2,38 36% 5,834 2% 5,843 2% 5,843 2% 5,956 2% 5,665 22% 99,933 4% 99,857 4% 3 42,287 9% 23,687 23% 54,274 7% 54,282 7% 54,282 7% 54,38 7% 54,98 7% 3,734 42% 3,59 43% Tota 72,74 27% 74,52 36% 27,9 25% 27,57 25% 27,57 25% 27,3 25% 23,99 25% 33,36 38% 33,74 38% vatons and the correspondng estmates for the coeffcents of varaton The frst two coumns of Tabe 6 contan the vaues for the ndvdua subportfoos A and B cacuated by the unvarate addtve oss reservng method Coumns portfoo (k = ) to portfoo (k = 3) show the estmated condtona process standard devatons for the portfoo consstng of the two subportfoos A and B f we use the mutvarate addtve oss reservng method (frst three teratons) In partcuar ths means that the vaues n coumn k = are based on the parameter estmates ˆm () The coumn denoted by overa cacuaton shows the resuts for the overa cacuaton The ast two coumns show the vaues cacuated by the mutvarate chan-adder reservng methods proposed by Braun (24) and Merz and Wüthrch (28), respectvey Tabe 7 shows the square roots of estmated condtona estmaton errors The frst two coumns contan the estmates for the ndvdua subportfoos A and B cacuated by the unvarate method Coumns portfoo (k = ), portfoo (k = 2) and portfoo (k = 3) show the est- 48 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE
Predcton Error of the Mutvarate Addtve Loss Reservng Method for Dependent Lnes of Busness Tabe 8 Estmated predcton standard errors Addtve method Chan-adder method Mutvarate Unvarate Mutvarate Unvarate Unvarate portfoo portfoo portfoo wthout portfoo overa portfoo portfoo subportfoo A subportfoo B (k =) (k =2) (k =3) corrn ˆm () cacuaton Braun (24) MW (28) 2 85% 674 475:% 73 33% 73 33% 73 33% 73 33% 77 34%,845 9%,845 9% 2 62 2%,52 2:%,696 328%,697 326%,697 326%,696 328%,675 38% 7,493 6% 7,493 6% 3,96 24% 2,866 243% 3,39 37% 3,39 37% 3,39 37% 3,39 37% 3,425 33% 9,497 83% 9,497 83% 4 6,2 446% 2,984 3343% 7,39 5% 7,32 499% 7,32 499% 7,32 5% 7,59 5% 2,66 744% 2,67 737% 5 4,337 543% 5,46 77% 6,77 566% 6,78 562% 6,78 562% 6,78 566% 6,528 585% 8,883 648% 8,887 642% 6 6,724 49% 5,744 77% 9,477 44% 9,484 435% 9,484 435% 9,479 44% 9,63 46% 22,435 49% 22,459 48% 7 2,677 255% 7,583 752% 23,729 26% 23,737 259% 23,737 259% 23,732 26% 23,79 273% 25,996 32% 26,22 298% 8 27,3 89% 8,935 424% 3,75 86% 3,757 86% 3,757 86% 3,753 86% 29,972 95% 33,376 22% 33,47 2% 9 34,424 2% 5,952 287% 4,85 23% 4,823 23% 4,823 23% 4,88 23% 42,562 29% 45,4 32% 45,442 3% 49,589 83% 23,44 2% 6,94 87% 6,2 86% 6,2 86% 6,99 87% 6,723 92% 72,222 6% 72,282 6% 59,66 55% 3,342 33% 76,868 59% 76,883 59% 76,883 59% 76,878 59% 79,45 63% 2,37 87% 2,434 87% 2 7525 42% 44,965 79% 4,78 44% 4,737 44% 4,738 44% 4,777 44% 8,7 46% 223,69 9% 223,92 9% 3 9,67 4% 57, 55% 2,484 37% 2,499 37% 2,499 37% 2,532 37% 23,74 38% 342,377 % 342,322 % Tota 26,63 34% 6,947 52% 27,89 32% 27,938 32% 27,939 32% 27,3 32% 27,358 33% 55,56 62% 55,44 62% mated condtona estmaton errors for the portfoo consstng of the two subportfoos A and B f we use the mutvarate addtve oss reservng method The new coumn wthout corr n ˆm () contans the estmated condtona estmaton errors f we do not take nto account correatons wthn the parameter estmates ˆm and use nstead the estmates ˆm () In contrast to the reserve and the condtona process standard devaton, these estmates do not concde wth the vaues n coumn portfoo (k = ) snce the estmator of the estmaton error for a snge accdent year and the cross product term [e, rght-hand sde of (42) and (52)] are now gven by 2 Ã I Ã I V 4 V V =2 V =2 Ã I and 2 V 4 =I + = Ã I = = V 3 5 V (64) =I + Ã I V = Ã I = V =2 V =2 3 V 5 V k, (65) = respectvey We see (as expected) that the estmaton error s arger (27,3 vs 27,57) f we estmate the parameters on the snge tranges However, the dfference n ths exampe s sma, whch woud ustfy workng wth ˆm () The coumn overa cacuaton shows the estmates for the overa cacuaton The ast two coumns show the vaues cacuated by the mutvarate chan-adder reservng methods proposed by Braun (24) and Merz and Wüthrch (28), respectvey Tabe 8 contans the estmated predcton standard errors and coeffcents of varaton for the same set of modes as above Tabe 9 contans the resuts for the estmated predcton standard errors assumng perfect postve correaton, no correaton, and perfect negatve correaton between the correspondng cams reserves of the two run-off subportfoos A and B These vaues are cacuated by msep d C,J DI N = msep d C (),J DN I +2c msep d =2 + msep d C (2),J DN I C (),J DN I msep d =2 C (2),J DN I (66) wth c =, c = and c =, respectvey Except for accdent year 3, we observe that the estmator n the mutvarate addtve oss reservng method eads to estmates of the predcton standard errors whch are between the ones assumng VOLUME 3/ISSUE CASUALTY ACTUARIAL SOCIETY 49
Varance Advancng the Scence of Rsk Tabe 9 Estmated predcton standard errors assumng correaton, and, respectvey Portfoo Portfoo Portfoo dmsep =2 C,J D N I dmsep =2 C,J D N I dmsep =2 C,J D N I correaton = correaton = correaton = 874 73 474 2 2,4,68 9 3 4,826 3,472 95 4 9,5 6,89 3,36 5 9,752 5,325 8,92 6 22,469 7,683,98 7 28,26 22,24 3,94 8 36,66 28,565 8,97 9 5,376 37,94 8,472 73,29 54,85 26,49 9,3 67,392 28,38 2 2,25 87,66 3,286 3 47,769 7,5 33,57 Tota 323,56 24,576 9,666 no correaton and a correaton equa to one for a accdent years and a accdent years together (cf coumns 3 5 n Tabe 8) Moreover, we see that an assumed correaton of or woud ead to an estmated predcton standard error that s about 29,5 ower and 52,5 hgher, respectvey, than the one takng the estmated correaton between the two subportfoos nto account 7 Concuson In ths paper we consder the cams reservng probem for a portfoo consstng of severa correated run-off subportfoos The smutaneous study of severa ndvdua run-off subportfoos s motvated by severa mportant facts and s especay cruca n the deveopment of new sovency gudenes However, the cacuaton of the condtona MSEP for the predctor of the utmate cam sze for a whoe portfoo of severa correated run-off subportfoos s more sophstcated snce now mutdmensona matrx cacuatons are nvoved and the mode parameters are nterdependent so that generay an teratve parameter estmaton procedure s requred In the present paper we study a speca case of the mutvarate addtve oss reservng mode proposed by Hess, Schmdt, and Zocher (26) and Schmdt (26a) Our derved formuas for the condtona MSEP n the addtve cams reservng method can be used to quantfy the uncertanty n the cams reserves for a snge runoff portfoo (e, N = ) or a whoe portfoo of severa correated run-off subportfoos (e, N> ) and can easy be mpemented n a spreadsheet By means of a detaed exampe, we compare our mutvarate estmator to the resutng estmator for the condtona MSEP f we gnore the correaton structure between ndvdua subportfoos as we as to the estmator for the condtona MSEP of the mutvarate chan-adder methods consdered by Braun (24) and Merz and Wüthrch (28) We obtan that n our exampe the predcton standard errors are substantay smaer n the mutvarate addtve method than n the mutvarate chan-adder cams reservng methods proposed by Braun (24) and Merz and Wüthrch (28) These fndngs may suggest that n the present case the mutvarate addtve method woud provde a better reserve estmate than the mutvarate chan-adder cams reservng method However, t s mportant to note that such a concuson woud be ony admssbe f we tested that the underyng mode assumptons of the addtve method are fufed Ths coud be done, for exampe, by the technques descrbed n Venter (998) Fnay, we want to emphasze that the condtona MSEP does not provde a compete pcture of the uncertanty assocated wth the predctor of the utmate cams of the tota portfoo Ths can ony be provded by the whoe predctve dstrbuton of the cams reserves [cf Engand and Verra (26) and Wüthrch and Merz (28)] Unfortunatey, n most cases one s not abe to cacuate the predctve dstrbuton anaytcay and one s forced to adopt numerca agorthms such as bootstrappng methods and Markov chan Monte Caro methods [cf Wüthrch and Merz (28)] Endowed wth the smuated predctve dstrbuton, one s not ony abe to cacuate estmates for the frst two moments of the cams reserves but one can aso derve predcton ntervas, quantes (eg, vaue at 5 CASUALTY ACTUARIAL SOCIETY VOLUME 3/ISSUE