On the Estimation of Outstanding Claims. Abstract. Keywords. Walther Neuhaus

Transcription

1 On he Esimaion of Ousaning Claims Walher euhaus Gabler & Parners AS PO Box 88 Vika -3 Oslo orway Tel (47) Fax (47) 437 Absrac This paper presens a iscree ime moel for he esimaion of ousaning claims ha comprises elay in wo imensions: reporing elay an valuaion elay This moel allows a sric isincion beween he cos of repore claims an he cos of unrepore claims Keywors Ousaning claims Loss reserving Creibiliy IBR RBS

2 Inroucion Esimaion of ousaning claims is an essenial par of acuarial work in general insurance Due o he naure of general insurance conracs an he claim selemen process almos any acuarial ask mus aress he quesion: have ousaning claims been aken ino accoun? Mos of he common acuarial mehos for esimaing he cos of ousaning claims involve exrapolaion of a wo-imensional evelopmen riangle The row or verical imension of he evelopmen riangle is normally he accien year or unerwriing year an he column or horizonal imension is he elay beween he accien year (unerwriing year) an successive valuaion aes The eveloping quaniy which is subec o moelling an preicion is usually one of he following: he number of repore claims he accumulae claim paymens or he amoun of repore incurre claims For a survey of raiional acuarial mehos for loss reserving see Taylor () While he wo-imensional moels may be effecive ools o preic he ousaning cos of claims per accien year hey o no allow he acuary o make a sric isincion beween he ousaning cos of claims ha are repore bu no sele (RBS) an claims ha are incurre bu no repore (IBR) The reason for his failing is ha claim evelopmen beween wo valuaion aes comprises wo separae ypes of evelopmen: changes in he assessmen of repore incurre claims an repors of new claims ha are receive by he insurer An explici isincion beween repore an unrepore claims is mae by Aras (989) who provies a srucural framework for claim reserving bu no operaional moels Aras framework forms he basis of papers by Haasrup & Aras (996) an orberg (999a 999b) who also provie skeches of operaional moels Implemening hose moels may sill be a formiable ask as hey are formulae in coninuous ime This paper presens a iscree ime moel ha comprises elay in wo imensions: elay beween he accien year an he reporing year (hereafer calle he reporing elay) an elay beween he reporing year an he valuaion year (hereafer calle he valuaion elay) This moel allows a sric isincion beween he cos of repore claims an he cos of unrepore claims

3 The main feaures of moel follow orberg ( a 999b) The occurrence of acciens is moelle by a mixe Poisson process wih a possibiliy for moelling serial correlaion beween accien frequencies in consecuive years The reporing elay is assume o be governe by a fixe paern of elay probabiliies The severiy of iniviual claims is assume o be inepenen of he claim number process an he moel allows for a sochasic epenence beween he reporing elay an he claim severiy The process of parial paymens an reassessmens is mae coniional on he severiy of claims repore The propose preicors of ousaning claim cos are of he creibiliy-weighe form which inclues as limiing cases he Chain laer meho an he Bornhueer- Ferguson meho I is no he purpose of his paper o inrouce new creibiliy moels bu o show how he exising ones can be exploie in an inegrae moel To make he moel operaional one nees o quanify subecively or by esimaion several ses of fixe parameers This paper aresses he problem of esimaing hose parameers only cursorily My main concern is o argue ha esimaion of ousaning claims shoul be conuce using hree raher han only wo ime imensions A hree-imensional moel was firs propose an analyse by Ørse (999) who evelope Kalman-filer echniques o upae is esimaes The moel playe a role in he recogniion of an subsequen recovery from he orwegian Workers Compensaion ebacle of he mi-99s A ha ime he ulimae cos of Workers Compensaion insurance claims sill was very uncerain Being able o separae claims IBR from claims RBS an o show convincingly ha he cos of claims RBS was likely o escalae far beyon wha mos people expece an wih i he cos of claims IBR was crucial o gaining accepance for he ire acuarial preicions A moel of claim evelopmen he observables Conforming wih sanar acuarial erminology he iscree ime perios will be calle "years" hroughou his paper In pracice i is enirely possible an usually avisable o buil he moel wih shorer ime perios (quarers or monhs) The iniial invesmen in oing so is more han compensae by he faciliy wih which one can calculae upae esimaes a shorer ime inervals an using a consisen se of assumpions ow le us ge on wih "years" 3

4 expose by We enoe accien years by For an accien year we enoe he amoun of risk p The number of claims repore wih elay is enoe by The ( k ) iniviual severiies of hose claims we enoe by { Y : k = L } an heir sum as Y For a given claim is ulimae severiy ha occur a elay afer he reporing ae: (k ) Y is mae up of a series of parial paymens (k ) U ( k ) () ( k ) Y = U = In aiion o parial paymens we may observe ousaning case esimaes Denoe by (k ) V he change in he ousaning case esimae a elay afer he reporing ae Finally le W = U + V enoe he change in he repore incurre claim cos oe ha ( k ) ( k ) ( k ) ( k ) ( k ) ( k ) () Y = U = W = = which saes he obvious fac ha he oal change in he ousaning case esimae from he ime when he claim is repore o he ime when i is sele is zero ow assume ha he las calenar year an he curren valuaion ae is A ha ime we will have recore he repore number of claims { : = L = L } while { : = L + } will sill be unrepore The only parial paymens ha we have ha he chance o observe are hose for which paymens o he en of year are + + The accumulae ( + ) = ( (3) + ) ( k ) ( k ) U U = wih corresponing formulas for he curren ousaning case esimae an curren repore incurre claim cos The ousaning paymens in respec of claims RBS are (4) = RBS U = = = ( + ) + an he fuure cos of claims IBR is (5) = IBR U = = + = 4

5 The evelopmen eraheron (below) illusraes he hree imensions of claim evelopmen Claims ha are RBS a ime have been repore insie he horizonal riangle given by + as inicae by a iamon The observe evelopmen of a repore claim is inicae by a soli verical line lying insie he eraheron which is elimie by + + an is fuure evelopmen is inicae by he oe exension of ha line The evelopmen of a claim ens a selemen inicae by a bulle A claim ha is IBR sars is observe evelopmen ousie he horizonal riangle an is evelopmen lifeline is oe all he way o selemen or course The curren saus of repore claims can be "rea off" on he simplex given by + + = Figure The evelopmen eraheron Valuaion elay Reporing elay RBS IBR Accien perio CBI The following abbreviaion will be use in he res of his paper: a variable wih a subscrip omie enoes he sum of he unerlying variables across all values of he subscrip ha has 5

6 been omie A variable wih a subscrip replace by an inequaliy (eg ( ) ) enoes U + he sum of he unerlying variables ha saisfy he inequaliy A variable wih a subscrip replace by is usually he sum of he unerlying variables ha lie insie he eraheron while a variable wih a subscrip replace by is he sum of he unerlying variables ha lie ousie he eraheron 3 A moel of claim evelopmen sochasic assumpions Having efine he necessary noaion for he observe quaniies le us now skech ou a sochasic moel of heir behaviour an ineracions More specific assumpions will be propose in laer secions We assume ha coniional on unknown claim frequencies { Θ : = L } he claim numbers are inepenen ranom variables each wih a Poisson isribuion (3) Θ = θ ~ Poisson( p θ ) wih fixe non-negaive elay probabiliies { : = L} ha a o one The evoluion of claim frequencies will be governe by some or oher sochasic process Denoe he mean of he vecor Θ = ( Θ L Θ )' by τ is covariance marix by Λ The severiies of iniviual claims repore in year + in respec of acciens ( k ) incurre in year we enoe by { Y : k = L } We assume ha ranom variables wih a isribuion (k ) Y are inepenen G ha may epen on he reporing elay We also assume ha he severiies are inepenen of he claim couns Denoe he mean an variance of (k ) Y by ξ an σ an le ρ = σ + ξ enoe he non-cenral secon orer momen Unil such ime as all claims are finally an irrevocably sele he aggregae severiy ( k ) Y of he claims { Y : k = L } will be an unknown an mus be esimae Denoe he unknown average severiy by = Y / (zero if = ) Coniionally on he number of claims he average severiy has mean ξ an variance σ / To moel he evelopmen of parial paymens { : = L} coniionally on he number of claims an he unknown average severiy an obvious caniae is he Dirichle isribuion Thus we will assume ha U 6

7 (3) ( U U L) ~ Dirichle( α L) wih non-negaive fixe parameers α L α α The Dirichle isribuion no suie o moel he coniional evelopmen of repore incurre claims { : = L} because is incremens are sricly non-negaive while he W incremens of repore incurre claims may be negaive Therefore we will propose a moel for he evelopmen of repore incurre claims where he { W : = L} are coniionally inepenen given he number of claims an he unknown average severiy an where W is a compoun Poisson ranom variable wih a frequency parameer ha is proporional o an a ump size isribuion H ha allows negaive umps: (33) W ~ Compoun Poisson ( H ) The assumpions ha have been skeche above will be uilise in he secions ha follow One more assumpion mus be menione being ha for every repore claim is evelopmen (consising of is parial paymens ousaning case esimaes an ulimae severiy) is sochasically inepenen of everyhing else ie claim numbers unerlying claim frequencies an he evelopmen of all oher claims This assumpion is a consequence of he marke Poisson process assumpion of orberg (999a 999b) I allows us o preic he amoun of claims IBR by preicing heir number an o preic separaely he evelopmen of each cohor of claims RBS ha have been repore a ime + in respec of accien year If you nee a sringen formulaion see orberg's papers 4 Esimaion of he number of claims IBR 4 General formulaion We sar wih a general formulaion using he noaion efine in he wo previous secions Define he iagonal marix (4) V p = M p L L O p M 7

8 A any ime he vecor of repore claim couns L )' is linearly = ( regresse on he vecor Θ = Θ L Θ )' of claim frequencies hrough he equaion ( (4) E ( Θ ) = V Θ an has a covariance marix given by (43) Cov( Θ ) = V iag( Θ ) Using he apparaus of linear greaes accuracy creibiliy heory we know ha he bes linear esimaor of (44) Θ = Z Θ ˆ + ( I Z ) τ Θ base on he vecor of observaions is ie i is a creibiliy-weighe average of he "chain-laer esimaes" (45) ˆ Θ = = V L p p an he prior mean τ where he creibiliy marix is (46) Λ ( Λ + iag( τ ) V ) Z = I is relaively easy o verify ha he mean square error marix of he esimaor ' (47) Q( Z ) = E( Θ Θ )( Θ Θ )' = Z V iag( τ ) Z + ( I Z ) Λ ( I Z )' Θ is The creibiliy preicor of he number of claims IBR in respec of acciens incurre in year is (48) = p Θ an is mean square error is (49) E( ) = ( p ) [ ( )] + p τ Z Q The creibiliy preicor of he oal number of claims IBR is p Θ = (4) = wih mean square error (4) E( ) = ( p )[ ( Z )] ( p + ' ' ' ) = ' = Q p τ = given We now urn o esimaing he cos of claims IBR In he coniional isribuion Θ = ( θ L θ )' he amouns { Y : = L } of claims IBR are inepenen 8

9 9 ranom variables an Y has a compoun Poisson isribuion wih frequency parameer p θ an a mixe severiy isribuion (he ail severiy isribuion) (4) + = = G G Slighly abusing noaion we le he inequaliy subscrip in conuncion wih a bar enoe a weighe average The non-cenral firs an secon orer momens of he ail severiy isribuion are hen (43) + = = ξ ξ an (44) + = = ρ ρ The creibiliy preicor of he amoun of claims IBR in respec of acciens incurre in year is (45) p Y Θ = ξ an is mean square error is (46) ( ) ( ) [ ] p p Y Y + = ρ τ ξ ) ( E Z Q The creibiliy preicor of he oal amoun of claims IBR in respec of all accien years is (47) p Y = Θ = ξ wih mean square error (48) [ ] p p p Y Y = = = = + ρ τ ξ ξ ' ' ' ' ' ) ( ) ( ) ( ) ( E Z Q Having wrien up general formulas for he creibiliy preicors an heir mean square error we will now propose a hanful of moels for he process = } { Θ ha can be use o eermine he mean τ an he covariance marix Λ The purpose in his paper is no o suy hese moels in any eail only o show how hey fi ino he general framework 4 Bühlmann-Sraub moel The Bühlmann-Sraub moel makes he assumpion ha he single-year accien frequencies } {Θ = are inepenen an ienically isribue ranom variables wih a known mean τ

10 an a known variance λ In ha case one fins ha τ τ an = Λ = λ I an easily erives he opimal creibiliy marix λp Z = iag an is mean square error λp + τ marix Q( Z ) = τ Z V Z ' + λ ( I Z )( I Z )' oe ha since he opimal creibiliy marix is iagonal each accien year's claim frequency is esimae on he basis of ha accien year's claim numbers alone 43 Hierarchical moel One can replace he known mean τ of he Bühlmann-Sraub moel wih an unknown ranom variable T ha has mean τ an variance λ an assume ha coniionally on T = τ he Θ are ii ranom variables wih mean τ an a known variance λ In ha case one fins ha τ τ an Λ = λ ' + λ I The opimal creibiliy esimaor may be wrien up = explicily bu in my opinion one may us as well sick o he marix formulas (44)-(46) 44 Ranom walk moel The Bühlmann-Sraub moel sipulaes ha he claim frequencies are saisically consan in he sense ha each accien year s claim frequency a priori has he same expece value I also sipulaes ha he claim frequencies Θ L Θ are inepenen; herefore in esimaing he claim frequency of a specific accien year nohing can be gaine by incluing aa from oher accien years The hierarchical moel allows for ransfer of informaion beween accien years bu i sill reains he unerlying assumpion ha claim frequencies are saisically consan In real-life siuaions claim frequencies are neiher consan nor inepenen bu raher behave like a correlae ime series A simple assumpion ha reflecs ha observaion woul be ha he claim frequencies follow a ranom walk ε ε Θ = Θ + ε where L are inepenen an ienically isribue error erms wih mean zero an variance λ Assume also pro forma ha here exiss an iniial ranom variable Θ ha has mean τ = E( Θ ) an variance λ = Var( Θ) Then i is easy o verify ha he ranom vecor

11 Θ = ( Θ L Θ )' has a mean vecor τ τ an a covariance marix Λ = λ ) wih = elemens λ = λ min( ' ) λ These can be insere ino (44)-(47) ' + One coul argue ha sricly posiive claim frequencies canno be moelle as a ranom walk ie a maringale ha will converge almos surely when boune In my opinion he error ha one commis in making he ranom walk assumpion is of he same naure as he error one commis by moelling recruis' heigh by a normal isribuion - ie negligible for pracical purposes One can evelop more sophisicae moels for he ime series of claim frequencies For example if he basic ime perio is shorer han a year i may be necessary o moel seasonal variaion This can be one a he expense of having o specify a larger number of moel parameers ( ' 45 Kalman filer Several auhors have propose he Kalman filer as a ool in he esimaion of ousaning claims In my opinion he Kalman filer is an elegan ool bu no paricularly well suie in he esimaion of ousaning claims I will pu forwar some argumens for my view I goes beyon he scope of his paper o inrouce he Kalman filer for reaers who are no familiar wih i Le i suffice o say ha he Kalman filer upaing formula is of he form (using he same moel an noaion as before) / p (49) Θ = K M + ( I K ) Θ / p Here Θ enoes he creibiliy esimaor of Θ = Θ L Θ )' a valuaion ae The ( ' vecor ( ) ' Θ consiss of he creibiliy esimaor of Θ Θ L Θ )' = Θ Θ a valuaion ae - an a creibiliy preicor of = ( Θ base on wha was known a ime - The creibiliy preicor of Θ epens of course on he ynamics of he unerlying process moel; in he ranom walk moel i is Θ = Θ The Kalman gain marix K can be calculae recursively by formulas ha are similar o he formula for he creibiliy esimaor (46) which involves he inversion of a marix The vecor of observaions in big brackes consiss of he incremenal claim couns - ie new claims repore in perio -

12 scale by he appropriae exposures To sum up he Kalman filer is a evice o upae he esimae of Θ in he ligh of new informaion as i emerges Why on' I like i hen? In normal ime-series applicaions wih a long ime series an observaion vecors of fixe imension he Kalman filer is an algorihm ha allows one o calculae he laes sae esimaes wihou having o inver large marices In esimaion of ousaning claims however he imension of he marix o be invere is always be equal o he lengh of he ime series Therefore he Kalman filer oes no reuce compuaional effor compare wih (44)-(47) Seconly he Kalman filer is inene for auomaic upaing of esimaes as new aa becomes available I have ye o see a line of insurance where he esimaion of ousaning claims can be lef o he auomaic pilo for any lengh of ime Any ausmen in he parameers necessiaes a whole new run of he filer hrough all ime poins = which can be more easily accomplishe by a sraigh applicaion of (44)-(47) Afer hese criical commens abou he Kalman filer I mus a ha ynamic linear moelling of which he ranom walk moel is he very simples example fis perfecly ino he framework of esimaing he number of claims IBR 5 Esimaion of he amoun of claims RBS Le us now urn o he problem of esimaing he ulimae cos of a cohor of claims ha has been repore in calenar year + an was incurre in accien year where of course + We know wih cerainy he number of claims ha have been repore ( ) an any aciviy ha has alreay been recore on he claims We preen o know he ulimae cos of claims ha are close bu le s face i hey coul be reopene In fac he ulimae claim cos of hose claims will never be known wih full cerainy In his secion wo moels will be propose o esimae he ulimae cos One moel is base on paymens an he oher moel is base on repore incurre claims ie paymens plus case esimaes I woul be nice o have formulae a moel ha uilises paymen informaion an case esimae informaion simulaneously bu I have no foun any elegan an racable moel ye Anyone who has care o rea so far is hereby invie o oin he search pary

13 5 Esimaion of claims RBS by paymen aa For he cohor of claims ha has been repore in calenar year + an was incurre in accien year we enoe he paymens a elay afer he reporing year by U The unknown ulimae claim cos we enoe by Y an he unknown average severiy by To moel he evelopmen of parial paymens { : = L} coniionally on he number of claims an he unknown average severiy an obvious caniae is he Dirichle isribuion Thus le us make he assumpion ha U (5) ( U U L) ~ Dirichle( α L) wih non-negaive fixe parameers coniional momens of he parial paymens are hen (5) ( U ) = υ E an α ' ' (53) ( ) ( ) severiy Cov U U ' Coniional on only α α L summing o α Le υ = α / α The δ υ υ υ = α + an before any paymens have been recore he average has a "prior mean" of ξ an a variance of σ / We now use he apparaus of linear greaes accuracy creibiliy heory o fin he bes linear preicor of in he coniional moel I is (54) = z ˆ + ( z ) ξ wih chain laer esimae (55) ˆ = U υ ( + ) an a creibiliy facor of (56) z σ ( α + ) υ = σ ( α + ) υ + ( + ) ( + ) ( σ + ξ ) υ ( + ) The coniional mean square error of he preicor (54) is (57) q z ) = E ( ) ) ( σ + ξ ) υ ( + ) ( = + z ( ) z σ ( α + ) υ ( + ) The bes linear preicor of he ousaning paymens is (58) ( ) = U ( ) U + + 3

14 wih coniional mean square error ( ( + ) ( + ) ) = q ( z ) (59) E ( U U ) Due o he inepenence beween he ifferen cohors he mean square error of he overall amoun of ousaning paymens for repore claims is aiive The assumpion of he paymen paern being he same for claims a all noificaion elays is no necessarily realisic To see why his nee no be he case conras claims noifie in he accien year (=) wih claims noifie in he subsequen year (=) If acciens are sprea evenly over he accien year claim noificaions in he accien year will be skewe owars he en of he year because of he noificaion elay On he oher han unless he reporing paern is very fla-aile claim noificaions in he subsequen year will occur mosly a he sar of he year before hey sar ailing off Thus on average claims ha are repore in he accien year will have less ime for he firs bach of paymens (=) o be processe han claims repore in he subsequen year Therefore one shoul expec ha υ is smaller for = han for = The formulas above exen reaily o a moel wih paymen paerns ha epen on ie { υ : = L} However his comes a he expense of having o se more parameers 5 Esimaion of claims RBS by repore incurre claims For he cohor of claims ha has been repore in calenar year + an was incurre in accien year we enoe he change in he repore incurre claim amoun a elay afer he reporing year by W As in he previous secion we enoe he unknown ulimae claim cos by Y an he unknown average severiy by To moel he evelopmen of { : = L} coniionally on he number of claims W an he unknown average severiy one nees a isribuion ha allows negaive as well as posiive incremens Tha requiremen exclues he Dirichle moel Consier he following moel: given he number of repore claims an he average claim amoun we assume ha he W a ifferen elays are coniionally inepenen an ha (5) W Compoun Poisson ( H ) ~ 4

15 The assumpion (5) implies ha he expece number of claim reassessmens a elay (a claim reassessmen being a parial paymens an/or a change o ousaning case esimae) is proporional o he unknown overall claim amoun iniviual reassessmens have a size isribuion Y = an ha he H Le us briefly iscuss his assumpion To assume ha he expece number of claim reassessmens is proporional o he number of claims repore is quie reasonable To assume ha i is acually proporional no o he number of claims bu o he amoun of claims sreches he imaginaion a bi more Tha coul be wrong bu i coul also be approximaely righ I will posulae here ha i is approximaely righ because his assumpion makes for nice mahemaics I am of course no saying ha he expece number of claim reassessmens is equal o he aggregae claim amoun (expresse in some currency or oher); i is only he proporionaliy ha couns The isribuion funcion H will have a high poin mass a zero so ha he number of acual claim reassessmens will be much smaller One coul generae he same compoun Poisson isribuion using a ifferen moel formulaion wih an explici proporionaliy facor in he claim frequency parameer an a isribuion funcion H ha is sricly non-zero Also ake noe ha we are no consraining he aggregae claim evelopmen o equal he aggregae severiy ie we are no emaning ha W = = as we i in he paymen moel Thus he aggregae severiy akes on he role of he expece level of ulimae paymens given he (absrac) severiies of claims repore raher han he efiniive level of ulimae paymens Thinking abou i i srikes me as quie a realisic assumpion ha wih given severiies here is resiual ranomness in he compensaion pai o he claimans So le us ge on wih he moel Denoe he firs an secon orer momens of he isribuion (5) ω = u H (u) an (5) η = u H ( u) Then we can easily esablish he following coniional momens: H by (53) E ( W ) = ω an 5

16 (54) Var( W ) = η = = We are assuming ha ω = an η < bu no all ω nee o be non-negaive Coniional on only an before any paymens have been recore he average severiy has a "prior mean" of ξ an a variance of σ / Using he apparaus of linear greaes accuracy creibiliy heory one can show ha he bes linear esimaor of in he coniional moel given is (55) = z ˆ + ( z ) ξ wih ( + ) ( + (56) ˆ ω = = η = ) ω W η an (57) z ( + ) ( + ) ω ω = σ + ξ σ = η = η I is ineresing o noe ha he number of claims oes no ener ino he creibiliy facor z The reason for his lies in he assumpion ha he "prior" variance of he unknown is inversely proporional o of is in he coniional moel The coniional mean square error (58) ) r = = + ( z ) E ( ) z ξ ( z ) σ = η ( + ) ω Having esimae he average severiy by he creibiliy formula (55) he esimaor of ousaning claim evelopmen becomes (59) W ( + ) = ω ( + ) wih mean square error W ( + ) ( + ) ( + ) + ( + ) (5) E ( W ) ) = η ( ω ) r ( z ) ξ 6

17 6 Inflaion an iscouning I is easy o wrie own expressions for he inflae an possibly iscoune value of fuure paymens Denoe he rae of inflaion by ε an he iscoun rae by δ The inflae iscoune value of he esimae cos of claims IBR is (ID) (6) IBR = ( p Θ ξυ ) = = + = ( + + ) 5 + ε + δ an he inflae iscoune value of he esimae fuure paymens on claims RBS is (ID) (6) RBS = ( Y U ( + ) ) = = = ( + ) + υ υ ( + ) + ε + δ ( + + ) 5 By subracing 5 in he exponen we have mae allowance for he assumpion ha claim paymens will be sprea evenly over he paymen year These equaions can easily be exene o variable raes of inflaion or ineres 7 A numerical example The numerical example is aken from a small porfolio of liabiliy insurances The aa came on a file wih he following recors: Claim number Accien ae Reporing ae Valuaion ae Unique ienifier mmyyyy mmyyyy Every year en of every claim (saring 988) beween he reporing ae an 3 Accumulae paymens unil he valuaion ae Ousaning case esimae on he valuaion ae This file conains sufficien informaion o fill he evelopmen eraheron wih cumulaive paymen an case esimae aa an claim couns Tables -3 shows he raiional riangles In orer o proec informaion I have convere all amouns o a non-exisen currency ha will be enoe (euro) I shoul be obvious from he summary saisics ha preicing he claim evelopmen in he porfolio is no an easy ask In wha follows I will briefly ouline he esimaions ha have been mae 7

18 Table Claim couns by reporing+valuaion elay umber of claims Delay + Accien year Table Claim paymens by reporing+valuaion elay Pai claims ( ) Delay + Accien year Table 3 Repore incurre claims by reporing+valuaion elay Repore incurre claims ( ) Delay + Accien year The reporing paern { : = L} was esimae by he sanar chain laer proceure which involves calculaion of year-o-year evelopmen facors in Table smoohing he evelopmen facors an appening a ail beyon he observe aa an convering he cumulaive evelopmen facors o probabiliies Graph shows he esimae reporing paern The paymen paern { υ : = L} was esimae by he same ype of proceure using he riangle of accumulae paymens by reporing year an valuaion elay ha is shown in Table 4 Graph shows he esimae paymen paern 8

19 Table 4 Claim paymens by valuaion elay Pai claims ( ) Delay Reporing year The claim revaluaion paern { ω : = L} was esimae in he same way using he riangle of accumulae repore incurre claims by reporing year an valuaion elay ha is shown in Table 5 One can see a number of subsanial upwar revaluaions Graph 3 shows he esimae claim revaluaion paern A noional ail was appene o ha paern o allow for claims being re-opene Table 5 Repore incurre claims by valuaion elay Repore incurre claims ( ) Delay Reporing year For he claim frequencies he Bühlmann-Sraub moel was use an is parameers τ an λ were esimae wih he ieraive proceure of De Vyler (98) reaing he previously esimae reporing paern as a given The volume of risks expose ha been sable an was se o one hroughou he perio so ha Θ expresses he expece number of acciens in year De Vyler s proceure reurne esimaes of τ * = 5 an λ * = 6 The means { ξ : = L} an variances { σ : = L} in he severiy isribuion were calculae using iniviual claim aa ha ha been ause wih expece fuure revaluaions using he previously esimae claim revaluaion paern Graph 4 shows he esimae means as a funcion of he reporing elay The variances were esimae on he basis of all claims an linke o he means by assuming ha he coefficiens of variaion 9

20 σ / ξ were inepenen of The esimae coefficien of variaion using all claims was ( / ) * = 3 58 σ ξ The parameer α in he Dirichle isribuion of parial paymens was esimae on he basis of he regression equaions υ ( υ ) + υ α + (7) ( ) ( ) E U = where he ulimae claim cos ha been approximae by repore incurre claims ause for expece revaluaions an he υ ha been replace by heir esimae values The esimae ha came ou of he proceure was α * = 3 37 To esimae he sequence { η : = L} in he compoun Poisson isribuions for repore incurre claims anoher simplifying assumpion was mae being ha η = ηω The parameer η was hen esimae on he basis of he regression equaions E W (7) ( ) ( ) ( ) = η + ω = ηω + ω again replacing unknown quaniies wih he esimaes a han an ignoring correlaions The proceure reurne an esimae of η * = 76 bu amiely he resul was highly uncerain a number of ouliers in (7) ha o be eliminae The level of censoring has significan influence on he resuling esimae Table 6 shows he esimaion of ousaning claims using he evelopmen paerns The moel esimae of ousaning claim paymens is 935 of which is ousaning case esimaes 879 is for expece revaluaion of claims RBS an 4 6 is for claims IBR The able also shows he square roo of he MSEP as compue by (5) (4) an (48) which of course oes no inclue he effec of parameer esimaion error Inflaion an iscouning have been ignore From a saisical poin of view one coul argue ha he moel is over-paramerise consiering he small volume of aa an he lengh of he evelopmen elays involve Tha is probably rue The aase was chosen for he example mainly because i was wellorganise an clearly illusraes he problems ha nee o be aresse long reporing elays slow paymens an unreliable case esimaes

21 Graph Esimae reporing paern * % 9 % Proporion of claims repore 8 % 7 % 6 % 5 % 4 % 3 % % % % Delay (Accien o reporing) Graph Esimae paymen paern * υ % 9 % 8 % Proporion of claims pai 7 % 6 % 5 % 4 % 3 % % % % Delay (Reporing o valuaion)

22 Graph 3 Esimae claim revaluaion paern * ω % Proporion of ulimae claim cos recognise 9 % 8 % 7 % 6 % 5 % 4 % 3 % % % % Delay (Reporing o valuaion) Graph 4 Esimae mean severiies * ξ 6 Average claim amoun ( ) Average una Average a Selece Delay (Accien o reporing)

23 Table 6 Esimaion of ousaning claims Moel esimaes Claim saisics RBS IBR Accouns Pricing esimaes Accien year Exposure Repore number of claims Pai claims Ous case esimaes Repore incurre claims Revaluaions umber of claims IBR Amoun of claims IBR Ous claim paymens Toal number of claims Ulimae claim cos Sum sqr(msep) The propose moel will no auomaically prouce more reliable esimaes han he raiional moels My poin is ha by separaing claims RBS from claims IBR one as a egree of ransparency o he ousaning claim esimaes which he raiional moels o no have This ransparency makes i much easier o convey he meaning of he esimaes an o es alernaive assumpions (eg in respec of fuure claim revaluaions) Separae moels for he evelopmen of repore an unrepore claims also faciliae he analysis of claim evelopmen as one can spli up he evelopmen ino is ifferen componens: umber of new claims repore (acual vs preice) severiy of new claims repore (acual vs preice) an revaluaion of ol claims (acual vs preice) I ll leave ha opic for anoher paper as i requires heavy noaion in heory in pracice i s very easy Acknowlegemen Discussions an cooperaion wih Moren Ørse have been invaluable in he evelopmen an implemenaion of he moel escribe in his paper Mos of he paper was wrien while he auhor was eaching a course on Loss Reserving a he Technical Universiy of Lisboa wih funing from ISEG an Cemapre 3

24 References Aras E (989) The Claims Reserving Problem in on-life Insurance: Some Srucural Ieas ASTI Bullein Volume 9 o De Vyler F (98) Pracical creibiliy heory wih emphasis on opimal parameer esimaion ASTI Bullein Volume 5-3 Haasrup S an Aras E (996) Claims Reserving in Coninuous Time; A onparameric Bayesian Approach ASTI Bullein Volume 6 o orberg R (986) A Conribuion o Moelling of IBR Claims Scaninavian Acuarial ournal 986 o 3-4 orberg R (999a) Preicion of Ousaning Liabiliies in on-life Insurance ASTI Bullein Volume 3 o orberg R (999b) Preicion of Ousaning Liabiliies II Moel Variaions an Exensions ASTI Bullein Volume 9 o Taylor GC () Loss Reserving An Acuarial Perspecive Kluwer Acaemic Publishers Boson / Dorrech / Lonon Ørse M (999) En flerimensional moel for ersaningsreservering basere på Kalmanfilere Thesis wrien for he Laboraory of Acuarial Mahemaics Copenhagen 4