Modelling The Claims Development Result For Solvency Purposes

Transcription

1 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes Mchael Merz Maro V. Wüthrch Abstract We assume that the clams lablty process satses the dstrbuton-ree chan-ladder model assumptons. For clams reservng at tme we predct the total ultmate clam wth the normaton avalable at tme and smlarly at tme we predct the same total ultmate clam wth the (updated) normaton avalable at tme. The clams development result at tme or accountng year ( ] s then dened to be the derence between these two successve predctons or the total ultmate clam. n [6 0] we have analyzed ths clams development result and we have quanted ts predcton uncertanty. Here we smply mody and llustrate the results obtaned n [6 0]. We emphasze that these results have drect consequences or solvency consderatons and were (under the new rs-adusted solvency regulaton) already mplemented n ndustry. Keywords. tochastc lams Reservng han-ladder Method lams evelopment Result Loss Experence ncurred Losses Pror Accdent Years olvency Mean quare Error o Predcton.. NTROUTON We consder the problem o quantyng the uncertanty assocated wth the development o clams reserves or pror accdent years n general nsurance. We assume that we are at tme and we predct the total ultmate clam at tme (wth the avalable normaton up to tme ) and one perod later at tme we predct the same total ultmate clam wth the updated normaton avalable at tme. The derence between these two successve predctons s the so-called clams development result or accountng year ( ]. The realzaton o ths clams development result has a drect mpact on the prot & loss (P&L) statement and on the nancal strength o the nsurance company. Thereore t also needs to be studed or solvency purposes. Here we analyze the predcton o the clams development result and the possble luctuatons around ths predcton (predcton uncertanty). Bascally we answer two questons that are o practcal relevance: asualty Actuaral ocety E-Forum Fall

2 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes (a) n general one predcts the clams development result or accountng year ( ] n the budget statement at tme by 0. We analyze the uncertanty n ths predcton. Ths s a prospectve vew: how ar can the realzaton o the clams development result devate rom 0? Remar: we dscuss below why the clams development result s predcted by 0. (b) n the P&L statement at tme one then obtans an observaton or the clams development result. We analyze whether ths observaton s wthn a reasonable range around 0 or whether t s an outler. Ths s a retrospectve vew. Moreover we dscuss the possble categorzaton o ths uncertanty. o let us start wth the descrpton o the budget statement and o the P&L statement or an example we reer to Table. The budget values at an. year are predcted values or the next accountng year ( ] o ths accountng year ( ].. The P&L statement are then the observed values at the end Postons a) and b) correspond to the premum ncome and ts assocated clams (generated by the premum lablty). Poston d) corresponds to expenses such as acquston expenses head oce expenses etc. Poston e) corresponds to the nancal returns generated on the balance sheet/assets. All these postons are typcally well-understood. They are predcted at an. year (budget values) and one has ther observatons at ec. 3 year n the P&L statement whch descrbes the nancal closng o the nsurance company or accountng year ( ]. asualty Actuaral ocety E-Forum Fall

3 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes budget values at an. year P&L statement at ec. 3 year a) premums earned b) clams ncurred current accdent year c) loss experence pror accdent years d) underwrtng and other expenses e) nvestment ncome ncome beore taxes Table : ncome statement n $ 000 However poston c) loss experence pror accdent years s oten much less understood. t corresponds to the derence between the clams reserves at tme t and at tme t adusted or the clam payments durng accountng year ( ] or clams wth accdent years pror to accountng year. n the sequel we wll denote ths poston by the clams development result (R). We analyze ths poston wthn the ramewor o the dstrbuton-ree chan-ladder (L) method. Ths s descrbed below. hort-term vs. long-term vew n the classcal clams reservng lterature one usually studes the total uncertanty n the clams development untl the total ultmate clam s nally settled. For the dstrbuton-ree L method ths has rst been done by Mac [7]. The study o the total uncertanty o the ull run-o s a long-term vew. Ths classcal vew n clams reservng s very mportant or solvng solvency questons and almost all stochastc clams reservng methods whch have been proposed up to now concentrate on ths long term vew (see Wüthrch-Merz [9]). However n the present wor we concentrate on a second mportant vew the short-term vew. The short-term vew s mportant or a varety o reasons: asualty Actuaral ocety E-Forum Fall

4 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes the short-term behavour s not adequate the company may smply not get to the long-term because t wll be declared nsolvent beore t gets to the long term. A short-term vew s relevant or management decsons as actons need to be taen on a regular bass. Note that most actons n an nsurance company are usually done on a yearly bass. These are or example nancal closngs prcng o nsurance products premum adustments etc. Relected through the annual nancal statements and reports the short-term perormance o the company s o nterest and mportance to regulators clents nvestors ratng agences stoc-marets etc. ts consstency wll ultmately have an mpact on the nancal strength and the reputaton o the company n the nsurance maret. Hence our goal s to study the development o the clams reserves on a yearly bass where we assume that the clams development process satses the assumptons o the dstrbutonree chan-ladder model. Our man results Results and 3.5 below gve an mproved verson o the results obtaned n [6 0]. e Felce-Morcon [4] have mplemented smlar deas reerrng to the random varable representng the Year-End Oblgatons o the nsurer nstead o the R. They obtaned smlar ormulas or the predcton error and vered the numercal results wth the help o the bootstrap method. They have notced that ther results or aggregated accdent years always le below the analytcal ones obtaned n [6]. The reason or ths s that there s one redundant term n (4.5) o [6]. Ths s now corrected see ormula (A.4) below. Let us menton that the deas presented n [6 0] were already successully mplemented n practce. Predcton error estmates o Year-End Oblgatons n the overdspersed Posson model have been derved by VAP [5] n a eld study on a large sample o talan MTPL companes. A eld study n lne wth [6 0] has been publshed by AAM-AME []. Moreover we would also le to menton that durng the wrtng o ths paper we have learned that smultaneously smlar deas have been developed by Böhm- Glaab []. asualty Actuaral ocety E-Forum Fall

5 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes. METHOOLOGY. Notaton We denote cumulatve payments or accdent year { 0 K } { 0 K } untl development year by. Ths means that the ultmate clam or accdent year s gven by. For smplcty we assume that (note that all our results can be generalzed to the case > ). Then the outstandng loss labltes or accdent year { 0 K } at tme t are gven by and at tme t they are gven by R (.) R. (.) Let denote the clams data avalable at tme { ; and } (.3) t and { and } { } ; ; (.4) denote the clams data avalable one perod later at tme t. That s we go one step ahead n tme rom to we obtan new observatons { } ; on the new dagonal o the clams development trangle (c. Fgure ). More ormally ths means that we get an enlargement o the -eld generated by the observatons generated by the observatons.e. ( ) ( ) to the -eld. (.5). strbuton-ree chan-ladder method asualty Actuaral ocety E-Forum Fall

6 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes We study the clams development process and the R wthn the ramewor o the wellnown dstrbuton-ree L method. That s we assume that the cumulatve payments satsy the assumptons o the dstrbuton-ree L model. The dstrbuton-ree L model has been ntroduced by Mac [7] and has been used by many other actuares. t s probably the most popular clams reservng method because t s smple and t delvers n general very accurate results. accdent development year accdent development year year 0 K K year 0 K K 0 0 M M M M Fgure : Loss development trangle at tme t and t Model Assumptons. umulatve payments n derent accdent years { 0 K } are ndependent. ( ) 0 are Marov processes and there exst constants > 0 > 0 such that or all and 0 we have [ ] E (.6) asualty Actuaral ocety E-Forum Fall

7 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes ( ) Var. (.7) Remars. n the orgnal wor o Mac [7] there were weaer assumptons or the denton o the dstrbuton-ree L model namely the Marov process assumpton was replaced by an assumpton only on the rst two moments (see also Wüthrch-Merz [9]). The dervaton o an estmate or the estmaton error n [0] was done n a tmeseres ramewor. Ths mposes stronger model assumptons. Note also that n (.7) we requre that the cumulatve clams are postve n order to get a meanngul varance assumpton. Model Assumptons. mply (usng the tower property o condtonal expectatons) [ ] E and [ ] E. (.8) Ths means that or nown L actors we are able to calculate the condtonally expected ultmate clam gven the normaton and respectvely. O course n general the L actors ramewor o the L method ths s done as ollows: are not nown and need to be estmated. Wthn the. At tme t gven normaton the L actors are estmated by 0 where. (.9) 0. At tme t gven normaton the L actors are estmated by asualty Actuaral ocety E-Forum Fall

8 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes where 0 0. (.0) Ths means the L estmates at tme use the ncrease n normaton about the clams development process n the new observed accountng year ( ] based on the addtonal observaton. and are thereore Mac [7] proved that these are unbased estmators or and moreover that m and m l ( m or ) are uncorrelated random varables or l (see Theorem n Mac [7] and Lemma.5 n [9]). Ths mples that gven s an unbased estmator or [ ] L (.) E wth and gven L (.) s an unbased estmator or [ ] E wth. Remars.3 The realzatons o the estmators realzatons o 0 K are nown at tme t but the 0 K are unnown snce the observatons K durng the accountng year ( ] are unnown at tme. asualty Actuaral ocety E-Forum Fall

9 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes When ndces o accdent and development years are such that there are no actor products n (.) or (.) an empty product s replaced by. For example and. The estmators are based on the L estmators at tme and thereore use the ncrease n normaton gven by the new observatons n the accountng year rom to..3 ondtonal mean square error o predcton Assume that we are at tme that s we have normaton and our goal s to predct the random varable. Then gven n (.) s a -measurable predctor or. At tme we measure the predcton uncertanty wth the so-called condtonal mean square error o predcton (MEP) whch s dened by E (.3) That s we measure the predcton uncertanty n the L ( P[ ]) -dstance. Because s -measurable ths can easly be decoupled nto process varance and estmaton error: ( ) ( [ ] Var E ). (.4) Ths means that s used as predctor or the random varable and as estmator or the expected value [ ] [ ] E at tme. O course the condtonal expectaton E s nown at tme (.e. the L actors are nown) t s used as predctor asualty Actuaral ocety E-Forum Fall

10 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes or and the estmaton error term vanshes. For more normaton on condtonal and uncondtonal MEP s we reer to hapter 3 n [9]:.4 lams development result (R) We gnore any prudental margn and assume that clams reserves are set equal to the expected outstandng loss labltes condtonal on the avalable normaton at tme and respectvely. That s n our understandng best estmate clams reserves correspond to condtonal expectatons whch mples a sel-nancng property (see orollary.6 n [8]). For nown L actors thereore used as predctor or expectaton [ ] the condtonal expectaton [ ] E s used as predctor or result (true R) or accountng year ( ] E s nown and s at tme. mlarly at tme the condtonal s dened as ollows.. Then the true clams development enton.4 (True clams development result) The true R or accdent year { K } n accountng year ( ] s gven by ( ) E [ R ] ( X E [ R ] ) E [ ] E [ ] R (.5) where by X denotes the ncremental payments. Furthermore the true aggregate s gven R ( ). (.6) Usng the martngale property we see that asualty Actuaral ocety E-Forum Fall

11 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes [ R ( ) ] 0 E. (.7) Ths means that or nown L actors equal to zero. Thereore or nown L actors the expected true R (vewed rom tme ) s we reer to ( ) R as the true R. Ths also ustes the act that n the budget values o the ncome statement poston c) loss experence pror accdent years s predcted by $0 (see poston c) n Table ). The predcton uncertanty o ths predcton 0 can then easly be calculated namely R ( 0) Var R ( ) ( ) E [ ]. (.8) For a proo we reer to ormula (5.5) n [0] (apply recursvely the model assumptons) and the aggregaton o accdent years can easly be done because accdent years are ndependent accordng to Model Assumptons.. Unortunately the L actors are n general not nown and thereore the true R s not observable. Replacng the unnown actors by ther estmators.e. replacng the expected ultmate clams E [ ] and E [ ] wth ther estmates and respectvely the true R or accdent year ( ) n accountng year ( ] predcted/estmated n the L method by: s enton.5 (Observable clams development result) The observable R or accdent year { K } n accountng year ( ] ( ) R X R R s gven by (.9) asualty Actuaral ocety E-Forum Fall

12 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes where R and aggregate R s gven by R are dened below by (.) and (.) respectvely. Furthermore the observable ( ) R. (.0) Note that under the Model Assumptons. gven R s an unbased estmator or [ R ] s an unbased estmator or [ R ] ( ) (.) E and gven R ( ) (.) E. Remars.6 We pont out the (non-observable) true clams development result ( ) approxmated by an observable clams development result R ( ) R s. n the next secton we quanty the qualty o ths approxmaton (retrospectve vew). Moreover the observable clams development result R ( ) s the poston that occurs n the P&L statement at ec. 3 year. Ths poston s n the budget statement predcted by 0. n the next secton we also measure the qualty o ths predcton whch determnes the solvency requrements (prospectve vew). We emphasze that such a solvency consderaton s only a one-year vew. The remanng run-o can or example be treated wth a cost-o-captal loadng that s asualty Actuaral ocety E-Forum Fall

13 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes based on the one-year observable clams development result (ths has or example been done n the wss olvency Test). 3. MEP OF THE LAM EVELOPMENT REULT Our goal s to quanty the ollowng two quanttes: R ( ) E R ( ) 0 (3.) R 0 R ( ) ( ( E R )) R. (3.) The rst condtonal MEP gves the prospectve solvency pont o vew. t quantes the predcton uncertanty n the budget value 0 or the observable clams development result at the end o the accountng perod. n the solvency margn we need to hold rs captal or possble negatve devatons o ( ) R rom 0. The second condtonal MEP gves a retrospectve pont o vew. t analyzes the dstance between the true R and the observable R. t may or example answer the queston whether the true R could also be postve ( we would now the true L actors) when we have an observable R gven by $ (see Table ). Hence the retrospectve vew separates pure randomness (process varance) rom parameter estmaton uncertantes. n order to quanty the condtonal MEP s we need an estmator or the varance parameters. An unbased estmate or s gven by (see Lemma 3.5 n [9]) 0. (3.3) asualty Actuaral ocety E-Forum Fall

14 Modellng The lams evelopment Result For olvency Purposes 3. ngle accdent year n ths secton we gve estmators or the two condtonal MEP s dened n (3.)-(3.). For ther dervaton we reer to the appendx. We dene Δ (3.4) Φ (3.5) Ψ (3.6) and Φ Ψ Φ Γ. (3.7) We are now ready to gve estmators or all the error terms. Frst o all the varance o the true R gven n (.8) s estmated by R ar V Ψ. (3.8) The estmator or the condtonal MEP s are then gven by: Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety E-Forum Fall

15 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes Result 3. (ondtonal ME estmator or a sngle accdent year) We estmate the condtonal MEP s gven n (3.)-(3.) by R ( 0) ( ) ( ) Γ Δ (3.9) R ( ) ( Φ Δ ) R ( ). (3.0) Ths mmedately mples that we have ( 0) R ( ) R Var R ( ) ( R ) ( ) R R. (3.) Note that ths s ntutvely clear snce the true and the observable R should move nto the same drecton accordng to the observatons n accountng year ( ]. However the rst lne n (3.) s slghtly msleadng. Note that we have derved estmators whch gve an equalty on the rst lne o (3.). However ths equalty holds true only or our estmators where we neglect uncertantes n hgher order terms. Note as already mentoned or typcal real data examples hgher order terms are o neglgble order whch means that we get an approxmate equalty on the rst lne o (3.) (see also dervaton n (A.)). Ths s smlar to the ndngs presented n hapter 3 o [9]. 3. Aggregaton over pror accdent years When aggregatng over pror accdent years one has to tae nto account the correlatons between derent accdent years snce the same observatons are used to estmate the L actors and are then appled to derent accdent years (see also ecton 3..4 n [9]). Based on the denton o the condtonal MEP or the true aggregate R around the aggregated observable R the ollowng estmator s obtaned: asualty Actuaral ocety E-Forum Fall

16 Modellng The lams evelopment Result For olvency Purposes Result 3. (ondtonal MEP or aggregated accdent years part ) For aggregated accdent years we obtan the ollowng estmator R R sep m (3.) > > Λ Φ R R 0 wth Λ. (3.3) For the condtonal MEP o the aggregated observable R around 0 we need an addtonal denton. Φ Φ Ξ. (3.4) Result 3.3 (ondtonal MEP or aggregated accdent years part ) For aggregated accdent years we obtan the ollowng estmator 0 R sep m (3.5) > > Λ Ξ R 0 0. Note that (3.5) can be rewrtten as ollows: Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety E-Forum Fall

17 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes ( 0) m sep (3.6) R ( ) R ( ) R ( ) Var ( R ( ) ) > > 0 ( ) R R ( ). ( Ξ Φ ) Hence we obtan the same decouplng or aggregated accdent years as or sngle accdent years. Remars 3.4 (omparson to the classcal Mac [7] ormula) n Results we have obtaned a natural splt nto process varance and estmaton error. However ths splt has no longer ths clear dstncton as t appears. The reason s that the process varance also nluences the volatlty o and hence s part o the estmaton error. n other approaches one may obtan other splts e.g. n the credblty chan ladder method (see Bühlmann et al. [3]) one obtans a derent splt. Thereore we mody Results whch leads to a ormula that gves nterpretatons n terms o the classcal Mac [7] ormula see also (4.)-(4.3) below. Result 3.5 asualty Actuaral ocety E-Forum Fall

18 Modellng The lams evelopment Result For olvency Purposes For sngle accdent years we obtan rom Result 3. R 0 Δ Γ (3.7). / / / For aggregated accdent years we obtan rom Result 3.3 R R ) ( (0) 0 (3.8). / / 0 > > We compare ths now to the classcal Mac [7] ormula. For sngle accdent years the condtonal MEP o the predctor or the ultmate clam s gven n Theorem 3 n Mac [7] (see also Estmator 3. n [9]). We see rom (3.7) that the condtonal MEP o the R consders only the rst term o the process varance o the classcal Mac [7] ormula and or the estmaton error the next dagonal s ully consdered ) ( but all remanng runo cells ) ( are scaled by /. For aggregated accdent years the condtonal MEP o the predctor or the ultmate clam s gven on page 0 n Mac [7] (see also Estmator 3.6 n [9]). We see rom (3.8) that the condtonal MEP o the R or aggregated accdent years consders the estmaton error or the next accountng year ) ( and all other accountng years ) ( are scaled by /. Hence we have obtaned a derent splt that allows or easy nterpretatons n terms o the Mac [7] ormula. However note that these nterpretatons only hold true or lnear approxmatons (A.) otherwse the pcture s more nvolved. Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety E-Forum Fall

19 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes 4. NUMERAL EXAMPLE AN ONLUON For our numercal example we use the dataset gven n Table. The table contans cumulatve payments or accdent years { 0 K8} at tme 8 and at tme 9. Hence ths allows or an explctly calculaton o the observable clams development result Table : Run-o trangle (cumulatve payments n $ 000) or tme 8 and 9 asualty Actuaral ocety E-Forum Fall

20 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes Table summarzes the L estmates and o the age-to-age actors as well as the varance estmates or 0 K 7. nce we do not have enough data to estmate Usng the estmates clams labltes 7 (recall and R at tme ) we use the extrapolaton gven n Mac [7]: 4 { }. (4.) 7 mn we calculate the clams reserves R or the outstandng t and X R or X R at tme t respectvely. Ths then gves realzatons o the observable R or sngle accdent years and or aggregated accdent years (see Table 3). Observe that we have a negatve observable aggregate R at tme o about $ (whch corresponds to poston c) n the P&L statement n Table ). R X R R ( ) Total Table 3: Realzaton o the observable R at tme t n $ 000 The queston whch we now have s whether the true aggregate R could also be postve we had nown the true L actors at tme t (retrospectve vew). We thereore asualty Actuaral ocety E-Forum Fall

21 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes perorm the varance and MEP analyss usng the results o ecton 3. Table 4 provdes the estmates or sngle and aggregated accdent years. On the other hand we would le to now how ths observaton o $ corresponds to the predcton uncertanty n the budget values where we have predcted that the R s $ 0 (see poston c) n Table ). Ths s the prospectve (solvency) vew. We observe that the estmated standard devaton o the true aggregate R s equal to $ 65 4 whch means that t s not unlely to have the true aggregate R n the range o about $ ± Moreover we see that the square root o the estmate or the MEP between true and observable R s o sze $ (see Table 4) ths means that t s lely that the true R has the same sgn as the observable R whch s $ Thereore also the nowledge o the true L actors would probably have led to a negatve clams development experence. Moreover note that the predcton 0 n the budget values has a predcton uncertanty relatve to the observable R o $ whch means that t s not unlely to have an observable R o $ n other words the solvency captal/rs margn or the R should drectly be related to ths value o $ R V ar ( R) m sep R R cov Total Table 4: Volatltes o the estmates n $ 000 wth: Mac asualty Actuaral ocety E-Forum Fall

22 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes R estmated reserves at tme t c. (.) V ar estmated std. dev. o the true R c. (3.8) ( R) estmated R (3.0) and (3.) between true and observable R c. m predcton std. dev. o 0 compared to R ( ) sep R 0 Mac c. (3.9) and (3.5) o the ultmate clam c. Mac [7] and (4.3) Note that we only consder the one-year uncertanty o the clams reserves run-o. Ths s exactly the short term vew/pcture that should loo ne to get to the long term. n order to treat the ull run-o one can then add or example a cost-o-captal margn to the remanng run-o whch ensures that the uture solvency captal can be nanced. We emphasze that t s mportant to add a margn whch ensures the smooth run-o o the whole labltes ater the next accountng year. Fnally these results are compared to the classcal Mac ormula [7] or the estmate o the condtonal MEP o the ultmate clam by n the dstrbuton-ree L model. The Mac ormula [7] gves the total uncertanty o the ull run-o (long term vew) whch estmates Mac ( ) E (4.) and Mac E (4.3) see also Estmator 3.6 n [9]. Notce that the normaton n the next accountng year (dagonal ) contrbutes substantally to the total uncertanty o the total ultmate clam over pror accdent years. That s the uncertanty n the next accountng year s $ and asualty Actuaral ocety E-Forum Fall

23 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes the total uncertanty s $ Note that we have chosen a short-taled lne o busness so t s clear that a lot o uncertanty s already contaned n the next accountng year. Generally speang the porton o uncertanty whch s already contaned n the next accountng year s larger or short-taled busness than or long-taled busness snce n long-taled busness the adverse movements n the clams reserves emerge slowly over many years. one chooses long-taled lnes o busness then the one-year rs s about /3 o the ull run-o rs. Ths observaton s nlne wth a European eld study n derent companes see AAM-AME []. APPENX A. PROOF AN ERVATON Assume that a are postve constants wth >> a then we have ( ) a a (A.) where the rght-hand sde s a lower bound or the let-hand sde. Usng the above ormula we wll approxmate all product terms rom our prevous wor [0] by summatons. ervaton o Result 3.. We rst gve the dervaton o Result 3. or a sngle accdent year. Note that the term Δ s gven n ormula (3.0) o [0]. Henceorth there remans to derve the terms Φ and Γ. For the term Φ we obtan rom ormula (3.9) n [0] ( ) ( ) asualty Actuaral ocety E-Forum Fall

24 Modellng The lams evelopment Result For olvency Purposes (A.) Φ where the approxmatons are accurate because >> or typcal clams reservng data. For the term Γ we obtan rom (3.6) n [0] (A.3) Γ Φ Ψ. Henceorth Result 3. s obtaned rom (3.8) (3.4) and (3.5) n [0]. ervatons o Results 3. and 3.3. We now turn to Result 3.. All that remans to derve are the correlaton terms. We start wth the dervaton o Λ (ths ders rom the calculaton n [6]). From (4.4)- (4.5) n [6] we see that or < the cross covarance term o the estmaton error [ ] [ ] R E R E s estmated by resampled values gven whch mples Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety E-Forum Fall

25 Modellng The lams evelopment Result For olvency Purposes E (A.4) E E E E. Note that the last two lnes der rom (4.5) n [6]. Ths last expresson s now equal to (see also ecton 4.. n [6]). Next we use (A.) so we see that the last lne can be approxmated by Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety E-Forum Fall

26 Modellng The lams evelopment Result For olvency Purposes. Next we note that hence ths last term s equal to. Hence pluggng n the estmators or and at tme yelds the clam. Hence there remans to calculate the second term n Result 3.. From (3.3) n [0] we agan obtan the clam usng that >> or typcal clams reservng data. o there remans to derve Result 3.3. The proo s completely analogous the term contanng Λ was obtaned above. The term Ξ s obtaned rom (3.7) n [0] analogous to (A.3). Ths completes the dervatons. 5. REFERENE [] AAM-AME (007). AAM-AME study on non-le long tal labltes. Reserve rs and rs margn assessment under olvency. October [] Böhm H. Glaab H. (006). Modellerung des Kalenderahr-Rsos m addtven und multplatven chadenreserverungsmodell. Tal presented at the German ATN olloquum. [3] Bühlmann H. e Felce M. Gsler A. Morcon F. Wüthrch M.V. (008). Recursve credblty ormula or chan ladder actors and the clams development result. Preprnt ETH Zurch. [4] e Felce M. Morcon F. (006). Process error and estmaton error o year-end reserve estmaton n the dstrbuton ree chan-ladder model. Ale Worng Paper Rome November 006. Modelng the lams evelopment Result For olvency Purposes asualty Actuaral ocety E-Forum Fall

27 Modelng the lams evelopment Result For olvency Purposes Modellng The lams evelopment Result For olvency Purposes [5] VAP (006). Reserves requrements and captal requrements n non-le nsurance. An analyss o the talan MTPL nsurance maret by stochastc clams reservng models. Report prepared by e Felce M. Morcon F. Matarazzo L. avastracc. and Pasqualn. October 006. [6] Merz M. Wüthrch M.V. (007). Predcton error o the expected clams development result n the chan ladder method. Bulletn o wss Assocaton o Actuares No [7] Mac T. (993). strbuton-ree calculaton o the standard error o chan ladder reserve estmates. Astn Bulletn Vol. 3 No [8] Wüthrch M.V. Bühlmann H. Furrer H. (008). onsstent Actuaral Valuaton. prnger Berln. [9] Wüthrch M.V. Merz M. (008). tochastc lams Reservng Methods n nsurance. Wley Fnance. [0] Wüthrch M.V. Merz M. Lyseno N. (008). Uncertanty n the clams development result n the chan ladder method. Accepted or publcaton n cand. Act.. Bographes o the Authors Merz Mchael s Assstant Proessor or tatstcs Rs and nsurance at Unversty o Tübngen (Germany). He was awarded n 004 wth the OR Actuaral Prze or hs doctoral thess n rs theory. Wüthrch Maro V. s enor Researcher and Lecturer at ETH Zurch (wtzerland) n the eld actuaral and nancal mathematcs. He holds a Ph n Mathematcs rom ETH Zurch and serves on the board o the wss Assocaton o Actuares. Unversty Tübngen Faculty o Economcs Tübngen Germany. [email protected] ETH Zurch epartment o Mathematcs H-809 Zurch wtzerland. [email protected] asualty Actuaral ocety E-Forum Fall