Introduction to Reserving

Transcription

1 Itroducto to Reservg Patr Dahl orrected edto 003

2 Itroducto to reservg 0. Preace The ollog text s ust a troducto to reservg etods. I practce reservg ll cota portat o-atheatcal eleets. The terology ll ote use the ord reserve. I urope the accoutg terology s oadays provsos aybe eve techcal provsos. or reasos o tradto ad ease the orer ord s ost ote used the sequel. The reader that sh to becoe a cogoscete should ollo the ogog dscusso th asualty Actuaral Socety here both practtoers ad scholars tae actve part. Those ho beleve that all these ssues ust have bee settled log ago ll be or a surprse. There s a lst o sybols used secto 5 ad a vocabulary secto 4.. Itroducto What s a o-le surace ro a acal perspectve? Brely: or a preu a surace copay cots tsel to pay a su a evet has occurred. I e troduce a te axs e d that rst the polcyholder sgs up or a surace the pays a preu ad he receved by the surace copay the copay starts to ear the preu. Durg the durato o the polcy as preus are eared there ght or ght ot occur a cla. I a cla has occured t ll evetually be o by the surer. Whe the cla s o by the surer the surer reserves the cla ad later possbly pays out a aout. Scheatcally Preus rtte --> Preus pad -- > Preus reserved -- > Preus eared -- > las curred -- > las reported -- > las pad There are several probles to solve: Ho uch preu s eared? Ho uch preu s ueared? Ho do e easure the uber ad sze o uo clas? Ho do e o the reserves o o clas are sucet? The devce that solves the to rst probles are tradtoally called preu reserve. The soluto to the to last probles are called curred but ot reported reserve or IBNR reserve. soetes there s a splt ad e tal about totally uo clas curred but ot yet reported IBNYR ad curred but ot eough reported IBNR he reported reserves are beleved to be sucet. A surace copay has to a reasos or dg out ho large the clas o rtte busess are. rst ad ost portat to eed bac ths to the prcg. The secod reaso s to produce acal statstcs or aalyss ad to produce coe stateets ad balace sheets or the copay. The value o a correct balace sheet could be oud accoutg theory. We ll ot deal th ho to eed bac the orato to prcg but cocetrate o the estates produced by a e establshed ethods.. Preu reserve The preu reserve s splt to to parts that accoutg terology are called: - Provso or ueared preus - Provso or uexpred rss To start th the rst o these t s assued that rtte preus are eared evely/uorly over the cover perod. I e are sde ths perod the the share o the preu that has bee eared s the past te s proporto o the total perod. Ths ay o apportog s called pro rata tepors lat.. p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

3 3 I a larger preu has bee receved the derece s the the ueared preu. Ths belogs to the provso or ueared preu. xaple: Suppose e have a surace polcy startg o Septeber gog through 3 August ext year. The preu or the total perod s 400. At 3 Deceber e have receved to quarterly preus or 00. We have the eared the What s ueared o the receved preu s To uderstad the secod part the uexpred rs reserve e loo upo the hole perod covered by the surace. ro a pot te e loo orard to all the clas ad expeses that could occur ater ths pot. The surer should reserve uds to cover the expected value o the. I there are uture preus ot yet due these could be deducted. I ths aout s larger tha the aout gve by the pro rata tepors calculato the derece should be accouted as uexpred rs reserve. I North Aerca t s aybe ore approprately called preu dececy reserve. To retur to our prevous exaple assue e at 3 Deceber beleve the uture costs are 800. We have to quarterly preus ot yet due or receved each o 600. Thus e eed to have provso or the reag exposure. As ths s hgher tha the prevously calculated 400 e eed a uexpred rs reserve o the derece.e. 00. The preus a surer receves ay cota varable acqusto expeses. Accordg to the la th U they should be allocated te the sae ay as the preu. I these acqusto expeses are cotget o a ucerta uture expected values should be used. 3. las reserves I the sequel soe ethods to calculate the ultate clas or provso or IBNR ll be preseted. Beore usg these ethods ot oly statstcal cosderatos should be tae. It s portat to d out the purpose o the gure. I t should be used prcg t should be realstc oly a sall bas should be alloed beore loadgs. I used traserrg a portolo o polces t should also be realstc but the sg o the bas ll deped t s evaluated or the seller or the buyer. Whe used to obta provsos or acal stateets soe coservatve bas s alloed. urtherore t should agree th accoutg prcples ad applcable las. Oe ll have to detere t s gross.e. beore resurace or et.e. ater resurace provsos that should be calculated. It s ot geerally obvous hch ay to go. We have the equato NetGross-eded. But you ll probably get deret results depedg o hch to o these three you evaluate. The ethods presuppose or at least avour hoogety. Ths lac o addtvty also aes the subdvso o the portolo o a surace copay a delcate atter. Ths ll be sho exercse 6.. Beore usg ay ethod oe should ae approprate adustet or lato that s ot otherse cosdered the odel. It s portat that the relevat rate o lato s used. I re surace t could be buldg cost dces. I there are aards or persoal ury they ore ote ollo the geeral developet o ealth the socety. urtherore there s also a tred ro the dea that the uortuate should ot be let poverty toards copesatg people or hat they ght have becoe the ury had ot happeed. That could result a socal or superposed lato eve hgher tha both age cost dces or GNP developet rug prces. Sel-evdetly t s hard to orecast such lato ad uzg t th the rght vestet ll hardly be possble. I the cost o hat s replaced depeds o aother currecy the exchage rate chage should also be cosdered. It should also be reebered that lato ll have a pact o the uber o clas as deductbles do ot usually ove pace th lato. The ethods assue that all polces questo have the sae perod o exposure. Ths s soetes ot true. Hoever t s usually ot a good dea to ae the sple adustet o lettg all polces start at the sae te as perls ll ote vary by seaso ad reportg ght also deped o seaso. The best s ote possble to or o accdet year. p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

4 4 Not oly actors outsde the surace copay aects the tg ad sze o clas. There s usually a clas departet too. It s o great portace to uderstad the procedures o the clas departet ad ts sta. Ho do they set the provsos? Whe do they set the provso? Ho log does t tae to set the reserve ro the reportg o a cla to the recordg o a provso? Ho ll the lead te be aected by the sze o the cla? Are there ay baclogs? Are there ay vacaces? Whe do they chage the reserves? Whe ad hy do they reve the reserves? Is there ay ocal or ocal pressure to eep do ot oly the payets but also the case reserves? Have there bee ay chages procedures? Have there bee ay chages sta? las people usually tal about ope ad closed clas. The latter category should actuarally be thought o as ot yet reopeed clas. The provsos set by the clas people are approxate. Thus t could be teptg to dsregard the ad ust go o the hard acts the payets. Hoever ost experece speas or also ad aybe oreost usg curred clas.e. both the pad ad the reserved aout. Ths does ot ea to say that a aalyss o pad should ot be perored. Hoever there s o obvous ay to recocle aalyss by pad ad curred respectvely the geeral case. Ote the ords log-taled clas are etoed. By ths people do ot alays ea the sae thg. They could ea that the cla s reported very late or that t taes a log te to ally settle or pay t ater t becoes o. There could also be chages the legal evroet ad setets o courts ad socety here evets that as ot thought to be covered at the te o polcy ssuace later s cosdered to be covered. It could be reasoable to tae out soe large clas ad estate the separately as they ght develop deretly. I ths s doe there should be good crteras or dog t such as that t ll be recovered by a excess-o-loss resurace. To ust tae the out ad gore the by labelg the as outlers s use. xpese reserve A surace copay should also have the uds to hadle the clas the uture order or the polcyholder to receve ther rghtul aouts eve the copay stops to rte busess. The calculato o uture hadlg expeses s ore o a exercse or accoutats. But he t s calculated also the uo clas should be tae to cosderato. I North Aerca they use the cocept uallocated loss adustet expeses UA or the IBNR clas as opposed to allocated loss adustet expeses AA or the o clas. 4. About the ethods The cha ladder ethod could be sad to buld purely o past experece. The Borhuetter-erguso ethod bulds o exposure. The ape od ethod s bascally the sae as the Borhuetter-erguso ethod but uses clas experece to replace the a pror loss rato. The Betader/Hove Method tres to ae a credblty coprose betee the cha ladder ethod ad the Borhuetter-erguso ethod by eghg the together th the assued proporto o ad uo clas. The cha ladder ethod or versos thereo has bee use or decades. The presetato gve here s based o the artcle [Mac 994] by Thoas Mac. The Borhuetter-erguso ethod s aed ater to US actuares ad as orgally preseted 975 [Borhuetter&erguso 975]. The presetato gve here leas o a presetato by [Gluc 997]. The ape od ethod as veted depedetly by J Staard ad Has Bühla ad s North Aerca ote called the Staard- Bühla ethod. Ths presetato s based upo [Patr 996] ad [Gluc 997]. The Betader ethod or Hove ethod s aed ater Guar Betader [Betader 976] ad sa Hove [Hove 98] ho depedetly veted t. The separato ethod as orulated by Greg Taylor [Taylor 977] o hch ths presetato bulds. p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

5 5 5. Notato Usually e preset the data the or o a clas tragle. The ros ould represet accdet years ad the colus developet perods. I the developet perods are years the dagoals ro the upper rght corer to the loer let corer ould represet caledar years. A exaple o a clas tragle o a cuulatve bass th three accdet years ad three developet perods ould thus loo le The ollog otato ll be used oe or ore places the sequel: uulatve clas ro accdet year reported through the ed o perod. D Icreetal clas ro accdet year reported perod. Ultate clas here the last developet perod that s o R Reserve [ ] Oe perod loss developet actor. Also called age-to-age actor or l rato. Developet actor ro accdet year perod to ultate. las relatve to a exposure P A easure o exposure A xperece up to developet perod The ollog otato relates oly to secto ad s ore precsely deed there: c a sgle cla aout λ A caledar year actors q cuulatve clas through perod to total clas or oe accdet year. r creetal clas perod to total clas or oe accdet year. They su to uty. d dagoal su N Nuber o clas or accdet year A estate o the expected uber o clas ro accdet year N B deed secto. We ll use crculex ^ to deote a estate. 6 The ha adder Method The cha ladder ethod bulds o that cuulatve clas a perod are proportoal to the clas the precedg perod. The proportoalty actor depeds o the uber o perods sce outset but s expected to be the sae or all accdet years. More orally e assue p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

6 p:\dos\y docuets\urser\sa\dahl\reservtexte.doc : Observe that does ot deped o accdet year. The vectors {... } ad {... } are depedet ust brgs us oe step ahead hereas e at to get to the ed. To get there e are gog to utlze the ollog ello result: MMA 6. I Z s te the X Z Z 6. Usg ths lea ad e d * The orula suggests a procedure ad e shall deed sho that t could be used. We could rerte o the ollog or... / Thus e could use observed ratos / as ubased estators o. Beore cobg estates o the sae e ae a urther assupto 3... Var σ Observe that the last actor the varace s ot depedg o accdet year. We ll also use the ollog lea MMA 6. Suppose X are ucorrelated rado varables th the sae ea but th varaces σ. The the best lear ubased estator o the ea s gve by X 6.3 here σ ad Proo: setch or the agraga... λ σ λ 6.4 Solve the syste

7 p:\dos\y docuets\urser\sa\dahl\reservtexte.doc : λ λ λ hch gves σ σ 6.6 Rertg 3 gves hy? 3 Var... σ ad the eghts are thus by 6.6 σ σ 6.7 ad 6.8 To be able to use the algorth suggested by orula 6. th the estators ro 6.8 e eed to prove that the estates are ucorrelated. Dee the set o experece up to developet perod by { } A The e have A 6.9a A 6.9b A 6.9c A 6.9d 6.9d 6.9e g

8 8 Where e repeatedly used the lea 6. the assupto the estator 6.8 ad that e could tae out hat s o. Ths ust proves that the to deret estates are ucorrelated. But the reader ll observe that e ever dd aythg th. I act could be replaced by a product o several s ad e could repeat the procedure that e used or or the oe by oe utats utadu. I e cobe ths th 6. t shos that the ollog ultate estator s ubased [ ] xaple Ultate year 999: Reserve year 999: Ultate year 000: Reserve year 000: Reserve both years: 5 Observadu Whe e use the product o the actors t s ade up ro the last actors. I e vert ths cuulatve actor e ll get the percetage that s reported. To see ths you could th o the ultate as 00% ho do e get there? By ultplyg th the last actor hece by dvdg 00% by the last actor e get the percetage reported up to ths perod. Ths s a coveet oto he coucatg th o-actuares. I the exaple you ll d that e thus have % reported ater oe year ad % ater to years. xercses 6. Geeralze the assupto 3 so that the codtoal varace s equal to σ or a c > 0. c Detere the best lear estato xae especally the cases c Detere the varace o the cha ladder estato. xpress t as a expected value hch e caot calculate. Ht: odto o the sae A as o page The Nave oss Rato Method Ths ethod assues that e a pror o the ultate losses share o the preu P. Ths s usually reerred to as the ultate loss rato. Oe could o course use soe other preerrably better easure o exposure but ths s the stadard oe. Ho ths share s o s outsde the p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

9 9 ethod t could coe ro the prcg calculatos or ro guesstates by e.g. accout executves or re egeers accordg to ther experece the aous uderrtg udget [ ] P 7. Thus the ecessary IBNR reserve ll be the derece betee the ultate losses ad the reported clas R P 7. It s obvous that ths ethod does ot presupppose aythg about the clas locato te or does t deretate betee actual clas or expected clas t sply sees the as coucatg vessels. It s true that ths ethod s splstc ad have ts ost propoets aog the practcal e. It has lted value outsde the case the early le o a accdet year he ust a e ad sall clas are o. 8. The Borhuetter-erguso Method The Borhuetter-erguso ethod s ore sophstcated tha the Naïve oss Rato ethod. It loos o here te clas ll be reported or pad. It s very slar to a ordary budgetg odel used by busesses. You could say that you budget or uture clas by perod. The su o these uture budgeted clas s the IBNR reserve. As te goes o estated clas or past perods are replaced by outcoes thout aectg the estate o uture clas. Ths could also be cast a statstcal raeor. More orally the ollog prcples apply: B xpected clas are cosdered o.e. e have a predctor o al clas detcal to the N. ea [ ] N I practce Ĉ could be coputed by the Naïve oss Rato ethod. B Ueerged clas are depedet o eerged clas or s depedet o et. be the actor that ould develop losses ro developet perod to the ed or accdet year B3 The Ho the are o the eag that e o [ ] are detered s outsde the ethod but practce the detered by the ha adder techque. No t s readly see that the ollog s a ubased predctor o al clas. B 8. could have bee Ths predctor o as the Borhuetter-erguso ethod has the ert over the Naïve oss Rato ethod that t taes eerged clas to accout as t saps past expected th real eergece. We could also rerte 8. o the ollog or p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

10 p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0 0 N N B W W 8. Wth W. We d that ths s a eghtg o o a cha ladder-type estate ad the o expected clas. We ae a urther assupto B4 Var Var Theore The eghts plctly deed 8. produces the best cobato o the to predctors ad N Ĉ the eag o zg the quadratc loss Proo We shall d the eghts W that solves the proble N W W W 8.3 Due to the ubasedess ths s equvalet to zg the ollog varace N W W Var 8.4 To sho the ucorrelatedess o the copoets e eed the auxllary result : 0 v Var Var Var ov ov ov ov ov 8.5 Usg the calculato rules or covaraces ad reeberg that covaraces beetee a r.v. ad a costat vashes e d 0 v v ov ov ov N 8.6 Whch shos the ucorrelatedess. alculatg the varace o the copoets gves v Var N 8.7 ad

11 Var Var v By usg ea 6. e d that the optal eghts are W [ ] [ ] Var[ ] ov[ ] v v v v v 8.9 v 8.8 hch s hat as asserted. It should be oted ro 8.6 that e should have. As the Naïve oss Rato Method secto 7 t s ote coveet to express loss rato tes a preu c orula 7.. Ê as a a pror xaple We have the sae clas tragle as the cha ladder exaple but also suppleeted th rs preu. Preus Assue that e th that expected clas to rs preu should be 00% or both 999 ad 000. et us also accept that 385% s expected to be reported through perod oe ad 769% through perod to c the observadu at the ed o secto 6 or both years. The e have Reserve 999: Ultate 999: Reserve 000: Ultate 000: Reserve both years: 7 xercses 8. Prove that X ad Y are rado varables th exstg varaces ad X ad Y-X are ucorrelated the ov [ X Y ] Var[ X ] 8. By troducg the otato that Var[ ] Var[ D ] D prove that gve [ D ] [ D ]. Var ad 8.3 Suppose e have estates o or all ad e th e could estate the e.g. by the aïve loss rato ethod. Detere the orulas or the aual costs D. p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

12 p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0 9. The ape od Method Ths ethod s slar to the B- ethod. Istead o requrg a a pror loss rato t estates oe th the help o a easure o exposure ad clas to date. et us retur to orula 8.. By usg 7. e could rerte t as P 9. We o assue that the do ot deped o accdet year.e. Thus a IBNR or all years s gve by P IBNR 9. No assue that or all.e. P IBNR 9.3 At the sae te e have P IBNR 9.4 ro 9.3 ad 9.4 e could coclude P 9.5 Ths straghtorard exercse s let to the reader A closer loo ll reveal that e stead o blog up the clas as cha ladder e stead apporto the preus. We could rerte orula 9.5 as { } P P P / * / * 9.5 We could terpret ths as a dcato o a varace structure c.. ea 6.. Thus orula 9.5 says the varace s proportoal to ho ar e are ro the ultate easured o a developet scale ad versely proportoal to the exposure. Ths ll urther be coeted upo chapter. xaple The data s stll the sae:

13 3 Preus We ll also use 385% as the gure expected to be reported through perod oe ad 769% through perod to c the observadu at the ed o chapter * Reserve 999: Ultate 999: Reserve 000: Ultate 000: Reserve both years: 5 xercse 9. Very orula The Betader Hove Method A advatage th the Borhuetter-erguso ethod copared to the cha ladder ethod s that t does ot let early clas drve the reserve. I act the reserve does ot tae the to accout at all. I a year has developed qute deretly ro the a pror expected t does ot see se to gore that ths ght have a bearg o the reserves. The geeral experece also tells us that thgs get ore stable over te ad accdet years ted to develop ore ale hch s the prary assuptos the cha ladder ethod. Thus a eghtg credblty theory style o the estates ould be teptg: R R R 0. B Moreover e sh the eght to crease by te. Istead o selectg te o a caledar scale e select t as the expected proporto o o clas. Ths gves the orula: R p q R q R 0. B Ths eghtg could ot be usted by ea 6. as there s o coo varace structure. The value o ths pragatc ethod ll urther be dscussed secto. xaple We use the gures ro the cha ladder ad Borhuetter-erguso exaple. 0 Reserve 999: Ultate 999: Reserve 000: Ultate 000: p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

14 4 Reserve both years: 37. The Separato Method The separato ethod assues that creetal clas are products o actors that deped o the accdet year the developet year ad the caledar year. et c the ea sgle cla aout the accdet year the developet year r the expected proporto o clas developet year o caledar year eect exsts. Thus the r s sus to uty. λ the caledar year eect actor e.g. lato s the uber o clas or the accdet year. The estato o s outsde ths ethod but oe ethod could be oud exercse. the [ D ] c r [ D ] λ. c r λ. Ths last expresso could be see as etry a tragle. or ths tragle e dee the dagoal sus d 0 c r0 λ0 c r0 λ c r λ c r0 r λ c r0 λ c r λ c r λ c r0 r r d d... d d λ c r0 r... r λ c r λ c r0 r... r λ c λ We have a observed tragle th etres D e dvde each ro th the predcted B D.. Dee We ca o or observed dagoal sus d. Startg ro the last equato e could recursvely calculate c λ c d λ ad r / c λ r B c λ / d r r / B B c λ c λ c λ / d r r r B B B / c λ c λ c λ 3 etc. p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

15 5 We have o obtaed alost all paraeters or predctg the loer hal o the tragle. The reag actor ould be the λ s that represet uture lato or ay other slar caledar year eect. Ths could buld upo the ratos c λ / c λ ad possbly o acroecooc cosderatos. xaple We have the sae data as beore but ths te creetal aouts. We also have the estated ultate cla ubers: 0 est. o Dvdg by the estated uber gves: ol. sus Observed dagoal sus: d d d recursvely e calculate c λ 85 r 88 / c λ 53/ r 59/ c λ 375 / r / To ae predctos e eed uture cλ s. We choose to tae the ro the last developet c λ 3 cλ cλ / cλ / c λ cλ cλ / cλ / No e are able to calculate the creetal clas D r cλ * D r cλ * D r cλ * ro hch e d a reserve o p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

16 6 xercses. Assue that the creetal uber o clas oted perod o accdet year s Po a b ad that each cell s depedet o each other. Derve the axu lelhood equatos or the paraeters. or that you thout loss o geeralty could assue b. d a recursve procedure or the paraeters. Ht: reuse the etoed detty ad th about ho the tragle goes to the M equatos. Apply ths to the ollog tragle o creetal cla ubers: Model selecto ad tests All predcto rests o the assupto that the uo has soethg coo th the o. I the uture outcoe s ot the sae as predcted by the provso t could eaate ro several sources. No estato procedure the orld could orce a rado varable to stop beg rado. Thus oe should alays try to see the devato s ro the process or ro a estato error. Moreover there ght also be a odel error. It should also be boure d that gures produced by a actuary ll ot alays be tae or grated. It s hard to argue or the result o a odel the assuptos are ot reasoably et or t gores relevat actors. A orula ght press soe people but ot all ad a arguet that starts ro data descrpto usually s ore covcg. There are a uber o odels or IBNR publshed actuaral papers. They ght cota advaced atheatcs ad ay paraeters. Soe authors ay have very good acadec credetals. Hoever ay have ot bee tested practce. It s portat to evaluate both ther explct ad plct assuptos as ell as ther robustess both th regard to data ad to paraeters. I there s a bas both sg ad sze should be checed. O course all ths also goes or ay odel you buld yourselves. There s very ote a dead or early arg odels that at the sae te gore sudde rado luctuatos. It s o ro cotrol theory that such deads caot both be et a ucoprosg ay. The portat thg s to ae a good predcto by hch oe usually ea soethg that s MVU. Icludg ore paraeters gve a better t th the past but orses the precso o the predcto. To decde o the sze o odels the pragatc Aae Iorato rtera AI cae up soe decades ago. Sce the several authors have tred to prove ad usty slar odels. The terested reader s reerred to the oder statstcal theory lterature. At a rst loo t ould be easy to dscard the ave loss rato ethod but t has ts advatages he the experece s lted te ad ubers ad too e clas have occured or ay cocluso. Hoever aagers a surace copay ght also be tepted to reduce the IBNR reserve th a crease curred as that s hat t s or. But the t s portat or a actuary to deed the derece betee a average ad outcoe. The developet actors could be see as a regresso l betee to developet perods. It could be possble that clas develop ater the ollog odels here x s the past value ad y the uture value: y b x ε. y a b x ε. p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

17 7 y b x ε.3 y b x x ε.4 or each o these odels oe assues that the error ter has expected value zero that they are ucorrelated th the sae varace across accdet years but aybe ot betee perods. The reader edately? recogzes that.4 gves the ordary cha ladder actor. Soe urther thoughts o these odels could be oud [Murphy 994]. [Gogol 995] have soe obectos to [Murphy 994] that ould ht aybody that gore Jese s equalty perorg estatos ad calculatg expectatos. [Veter 998] has tred to suarze hch tests should be passed order to use a odel. I oe beleve that oe o these odels are vald u varace estato s the ost useul. As ot all are stadard odels the burde o proo rests th the user. To chec regresso dagostcs could be used. It s also portat to avod a dless use o tests to test hether actors der sgcatly ro oe. Oly usg those oes ould lead to a bas as dstrbutos ost lely are se. That the Gauss-Marov theore s applcable should ot lead to the cocluso that dstrbutos are syetrcal. I several loss developet actors are o ther o cosdered ot to be deret ro oe that does ot ecessarly go or ther product. Usg odels based o several ubased estators do ot eccesarly lead to a ubased outcoe. Oe should alays bear d that the ucto o a expected value s ot equal to the expected value o the ucto uless the ucto s lear. The Betader-Hove ethod as as sad secto 0 ot ustable by eghg together th verse varaces as the copoets dd ot have a coo varace structure. Ths does ot ea that the results o the odel are bad. I act t could be sho that t uder a varace structure le the Borhuetter-erguso s qute close to a credblty estate. The artcle [Mac 000] shos ths detal ad t eve clas that the ethod beats the copoets ost cases. It s portat to chec the eergece o clas to hat as predcted. I these result ro process error estato error or odel error should be cosdered. I errors o the later ds are ot deterable t could be orthhle to ece o overterpretato by laye by sple tests such as usg the boal dstrbuto or the ubers o ups ad dos. What s preseted here are ethods or deterg reserves or ultate clas. Iplctly they restrct the uber o odels that ould t th these ethods. To llustrate ths the separato ethod ad ethods that buld o that the logarth o a creetal cla reles o a dstrbuto th a ro actor ad a colu actor have soetes bee called overdspersed Posso odels. I soe cases they ould have the sae estators as the cha ladder but other ot. There has also bee qute a cotroversy about hch odel that uderles the cha ladder ethod. Whether t s better to odel the eergece o clas ro botto up or to use a odel ll have to be decded based o evdece. We caot o hch ethod/odel s the true oe. We should hoever see to that the ethods/odels used are testable ad hece alsable. I practce t s ot ucoo to gore observatos that are ay caledar years bac. The reaso or dog ths s that oe ght o loger be able to assue te hoogety. Soetes oe also ds that extree actors are dscarded e.g. d three o last ve. Apart ro the alays questoable practce o dog aay th true data oe ght also troduce a bas hch should be corrected. The terested reader could chec hat correcto ould apply ths case the l ratos are assued to coe ro a expoetal dstrbuto. May practtoers also polsh actors to obta soothess ad robustess. O course ths volates soe depedece assuptos. I ths s doe oe should do t cosderg that t s a geoetrc average that should apply ad that ther eect ll be eghted. The ore toards the ed the share p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

18 8 the product s pushed the ore accdet years ll be aected ad vce versa. Dog ths requres cratashp th ubers experece ad oledge. O course t should alays be ustable. xercse. d u varace estators or the paraeters o the odels Regulato Isurace copaes are requred to be able to eet all uture oblgatos arsg ro all surace polces rtte by the ths s actually the product sold. To sho that ths s the case the provsos or that should also be sho the balace sheet o the copay. The Sedsh Isurace opaes Act örsärgsrörelselage R starts the chapter o the actual busess chapter 7 by: tt örsärgsbolags örsärgstesa avsättgar sall otsvara belopp so erordras ör att bolaget vd vare tdput sall ua uppylla alla åtagade so sälge a örvätas uppoa ed aledg av gåga örsärgsavtal.... The ay ths s sho s govered by the accouts drectve ro U 9/674/ to hch the Sedsh la adheres: Artcle 60 Provsos or clas outstadg. No-le surace a A provso shall prcple be coputed separately or each case o the bass o the cost stll expected to arse. Statstcal ethods ay be used they result a adequate provso havg regard to the ature o the rss; Meber States ay hoever ae the applcato o such ethods subect to pror approval. b Ths provso shall also allo or clas curred but ot reported by the balace sheet date; ts aout shall be detered havg regard to past experece as to the uber ad agtude o clas reported ater the balace sheet day. c las settleet costs shall be cluded the provso rrespectve o ther org.... More or less the sae prcples apply the Iteratoal Accoutg Stadard IAS ad the US GAAP Geerally Agreed Accoutg Prcples ore speccally AS 60. Whe t coes to statutory accoutg the US ths s o state level ot ederal level although ost states have regulatos that coor closely th the North Aerca Isurace osoers odel la. I accoutg there are stadards ad prcples that should be olloed. These ould be oud acadec textboos o accoutg. They do ot alays ollo ro atheatcal reasog but ro cetures o coercal experece ad practce. These prcples are portat to uderstad. I you volate the you ll get to trouble th accoutats ad audtors. A exaple s that oe should ever set a loer reserve or balace sheet purposes tha you o beleve ll be ecessary soe te the uture or the o exstg busess. 4. Glossary Accdet year Sadeår AA Allocated loss adustet expeses Alloerade sadereglergsoostader Acqusto expeses asagsostader ATA Age-to-age Perod tll perod ATU Age-to-ultate Perod tll ulto Balace sheet Balasräg ase reserve Saderegleraras uppsattg av ratda sadeostader ör äda sadeall ar täa xcess-o-loss Återörsärg so ebär att a ersätter bara de del av e sada so överstger ett vsst belopp p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0

19 9 xpese reserve Oostadsreserv ör ratda adstratva ostader Icoe stateet Resultaträg Icurred Iträade Iaous Öäda IBNR Iträade e ullt rapporterade sador IBNR Iträade e rapporterade sador oss rato Sadevot Perls aror Polcyholder örsärgstagare Preu dececy reserve Se Uexpred rs provso Provso Avsättg Recocle örea Sastäa UA Uallocated loss adustet expeses Oalloerade sadereglergsoostader Uderrtg year Tecgsår Uexpred rs provso Avsättg ör vardröade rser Wrtte preu Tecad pree 5. Reereces Betader Guar 976: A Approach to redblty alculatg IBNR or asualty xcess Isurace. The Actuaral Reve Aprl 976 p. 7. Borhuetter R.. ad erguso R.. 97: The Actuary ad IBNR. Proceedgs o the asualty Actuaral Socety Vol. IX pp uropea Uo 99: Accouts Drectve 9/674/ örsärgsrörelselage SS 98:73. Gogol Dael.: Dscusso o Ubased oss Developet actors. PAS XXXII 995 p7-77. Gluc Specer M. 997: Balacg Developet ad Tred oss Reserve Aalyss. Proceedgs o the asualty Actuaral Socety Vol. XXXIV pp Hove sa 98: Addtve ad otuous IBNR. Proceedgs o the XV ASTIN oll. oe Noray 98. Mac Thoas 994: Measurg the Varablty o ha adder Reserve states. asualty Actuaral oru Sprg 994 Vol. Mac Thoas 000: redble las Reserve: The Betader Method ASTIN Bull. Vol. 30 No.. Mac Thoas ad Veter Gary 999: A oparso o Stochastc Models that Reproduce haadder Reserve states. Proceedgs o the XXXth ASTIN oll Toyo 999. Murphy Dael M.994: Ubased oss Developet actors. PAS XXXI 994 p Patr Gary S. 996: Resurace oudatos o asualty Actuaral Scece 3 rd edto. Taylor G.. 977: Separato o Ilato ad Other ects ro the Dstrbuto o No-e Isurace la Delays. ASTIN Bull. Vol. 9 Issue. Veter Gary 998: Testg the assuptos o age to age actors. Proceedgs o the asualty Actuaral Socety Vol. XXXV pp p:\dos\y docuets\urser\sa\dahl\reservtexte.doc :0