Analysis of the provisions for claims outstanding for non-life insurance based on the run-off triangles

Transcription

1 OFFE OF THE NSURANE AND PENSON FUNDS SUPERVSORY OMMSSON Analyss of the provsons for clams outstandng for non-lfe nsurance based on the run-off trangles Ths Report has been prepared n the nformaton Systems and Supervson Standards Department of Offce of the nsurance and Penson Funds Supervsory ommsson Warsaw may

2 The report on Analyss of the provson for clams outstandng for non-lfe nsurance based on the run-off trangles has been prepared n the nformaton Systems and Supervson Standards Department (DSS) drected by Ms. wona WOŹNAK Department Drector. Subect-matter preparaton: Wocech BJAK Proect Manager Marusz SMĘTEK Grzegorz SZYMAŃSK Any comments proposals and queres related to ths Report should be drected at DSS department secretarat: Tel. (48 ) Fax (48 ) E-mal. sekretarat.dss@knufe.gov.pl - -

3 Table of contents: ntroducton... 4 Rsk related to the establshng techncal provsons... 6 lassfcaton of models of provson for clams outstandng... 7 Methods of calculaton of the BNR provsons n Polsh nsurance undertakngs... 9 V Methodology of the analyss... 0 V. Scope of the analyss... 0 V. Reportng duty n relaton to nformaton concernng clams accordng to the year of loss occurrence and the year of loss reportng... 0 V.3 Run-off trangle... V.4 Methodology of data supplementaton and correcton... 3 V.5 Dvson of run-off trangles... 5 V.6 Models used n the analyss... 6 V.7 Verfcaton of resduals... 7 V Lnk ratos and ther applcaton... 8 V Provsons adequacy analyss percentle approach... V. Emprcal rsk margn... V. Theoretcal rsk margn... 5 V.3 onfdence level of provsons... 6 V Provsons adequacy analyss cost of captal approach... 8 V. Methodology... 8 V. Algorthm of determnaton of the market value margn... 9 V.3 Requred market value margn V.4 Equvalence of rsk margns... 3 V Analyss of future clam payments volatlty V. Regresson functon of future payments volatlty V. oeffcent of provson underestmaton rsk X Other use of statstcal nformaton by the year of loss occurrence X. Ultmate loss rato X. Average clam amount X.3 Rato of provson development onclusons... 4 References

4 Annexes Annex. Lnk ratos accordng to occurrence year determned on the bass of cumulatve pad clams accordng to nsurance classes Annex. han-ladder Methods... 5 Annex 3. ost of captal approach Annex 4. harts of the regresson functons fttng to emprcal data accordng to nsurance classes Annex 5. lassfcaton of rsks accordng to classes n non-lfe nsurance

5 ntroducton Snce 999 the European Unon Member States have been undertakng actvtes amed at establshng an unform effectve and globally compettve market of fnancal servces. The Fnancal Servces Acton Plan (FSAP) whch provdes for mplementaton of sgnfcant changes n the nsurances sector n the near future s beng realsed. Accordng to FSAP new solutons concernng nsurance accountancy and nsurance undertakngs solvency montorng system are to be adopted. Wth regard to the above many ssues requre prompt acton. From the pont of vew of nsurance practce t s very mportant to defne a new approach to valuaton of nsurance labltes. n the EU Member States a general rule for valuaton of techncal labltes accordng to ther market value s acceptable for ts adoptng for the purposes of the new nsurance undertakngs solvency montorng system (Solvency proect). nternatonal accountng standards wthn the scope of valuaton of nsurance labltes have not been developed yet and therefore some solutons whch are consstent wth general rules of defnng techncal labltes accordng to ther market value should be adopted (perhaps temporarly) for the purposes of Solvency proect. One of proposed approaches defnes that value on the bass of probablty dstrbuton of the future clam payments. Such dstrbuton may be obtaned wth the use of.a. statstcal methods and stochastc models. These methods and models may consttute a bass for a new solvency system. The need of further harmonzaton of European law provsons wthn the scope of techncal provsons s emphaszed n numerous EU papers for example n Müller report Manghett report KPMG report 3 and report of the Workng Group on non-lfe techncal provsons to the Solvency Subcommttee. 4 The papers ssued by the ommttee of European nsurance and Occupatonal Pensons Supervsors (EOPS) contanng the answers to questons 7 and 8 from the so-called second wave of European ommsson s Specfc alls for Advce drected to EOPS wth regard to the need for more detaled solutons as regards Solvency 5 proect also concern techncal provsons. Solutons proposed as answers to questons 7 and 8 were subect to emprcal verfcaton by the EOPS. n 005 the Quanttatve mpact Study (QS) devoted to the methodology of estmaton of the market value of nsurance labltes was conducted n the EU Member States. For the purposes of the study t was assumed that the market value of nsurance labltes may be defned as a value of the best estmate of provsons (wth the use of a gven measure of central tendency mean medan etc.) and rsk margn (addtonal component establshed on the bass of specfed confdence level of provsons provsons at a specfed percentle set above the mean or rsk margn dependent on standard devaton of Müller H. (charmanshp) Report Solvency of nsurance undertakngs onference of the nsurance Supervsory Authortes of the Member States of the European Unon Aprl 997. Manghett G. (charmanshp) Report Techncal provsons n non-lfe nsurance onference of the nsurance Supervsory Authortes of the Member States of the European Unon KPMG European ommsson Study nto the methodologes to assess the overall fnancal poston of an nsurance undertakng from the perspectve of prudental supervson May European ommsson Report of the workng group on non-lfe techncal provsons to the Solvency Subcommttee MARKT/59/0-EN September The fnal verson of EOPS reply to the European ommsson can be found at the EOPS webste:

6 provsons) 6. Wthn the framework of QS study performed by EOPS the possblty of valuaton of lablty on the bass of cost of captal approach was tested addtonally. The ssue of provsons s also a focus of actuares nterest whch s proved by the number of publcatons on ths topc. The paper Schmdt [005] Bblography on Loss Reservng contans a lst of about 600 tems (books artcles and studes) concernng loss provsons n nsurance. Ths report consttutes a contnuaton and extenson of the report developed by Offce of the nsurance and Penson Funds Supervsory ommsson (UKNUFE) n 00 7 contanng an analyss of the provsons for clams outstandng n non-lfe nsurance. Snce the report was ssued UKNUFE ganed new data provded by nsurance undertakngs the data qualty mproved sgnfcantly and nsurance market made a progress towards ts maturty. Therefore the am of ths analyss s not only to present the usage of run-off trangles for estmatng ultmate loss amounts for estmatng the provsons and for determnng characterstcs of partcular non-lfe nsurance classes (as t was n the prevous report) but also:. examnaton of the possblty of usng fnancal and statstcal nformaton concernng pad clams accordng to year of loss occurrence and year loss development provded to the supervsory authorty by the non-lfe nsurance undertakngs runnng ther actvty n Poland. examnaton of the possblty and effectveness of applcaton on the bass of avalable fnancal and statstcal data and takng nto account the present stage of market development of dfferent statstcal methods based on run-off trangles for provsons estmaton and analyss. The Report conssts of nne hapters. Factors nfluencng the process of estmatng provsons for clams outstandng are presented n the frst hapter. The revew of models of provsons for clams outstandng s presented n the second hapter. The subsequent hapter contans the lst of methods appled n Poland by nsurance undertakngs for estmatng provsons for clams outstandng n partcular BNR provsons. The methodology of analyss s presented n the fourth harter. The subsequent hapter contans the characterstcs of partcular nsurance classes on the bass of lnk ratos. hapters sx and seven concern the estmaton of adequacy of techncal provsons for clams outstandng n the non-lfe nsurance undertakngs performed wth the use of dfferent approaches (percentle and cost of captal approaches). The eghth hapter contans an analyss of future clam payments volatlty. n the last hapter the results of the analyss of provsons adequacy have been supplemented wth results of the statstc and ndcator analyss. The report ends wth conclusons and a lst of references. The analyss of the provsons for clams outstandng has been prepared wth the use of statstcal software (Statstca) actuaral software (ROS) as well as standard offce software (Mcrosoft Excel wth VsualBasc for Applcatons module bult - n). 6 The fnal examnaton report can be found at the EOPS webste: 7 Analyss of the provsons on the bass of run-off trangles prelmnary analyss for Polsh nsurance market W. Bak M. Pawlak M. Smętek n Fnancal nvestments and nsurances world tendences and Polsh market edted by: K. Jauga W. Ronka-hmelowec Prace Naukowe AE we Wrocławu Nr 990 Wroclaw 003 publshng house of Wroclaw Unversty of Ecnomcs pages Ths work s avalable also at the KNUFE webste:

7 Rsk related to the establshng techncal provsons An nsurance undertakng ust as any other busness undertakng s exposed to dfferent rsks whch may endanger ts exstence and lead to ts bankruptcy. However apart from the rsk categores whch are typcal for all nsttutons (.a. nvestment rsk assets and labltes matchng rsk or other rsks of general nature for example bad management) the actvty of nsurance undertakngs s lnked wth such categores of rsks whch are specfc for nsurers only. Partcularly mportant s the rsk related to estmaton of current and future labltes whch may arse from concluded nsurance contracts.e. techncal provsons and manly provsons for clams outstandng. The rsk for nsurance undertakng s generated n partcular by underestmaton of provsons. Underestmaton of the value of techncal provsons may lead to the stuaton n whch the nsurance undertakng may not be able to meet all ts labltes arsng from nsurance contracts or wll have to realze them wth the use of addtonal funds. Ths s one of basc techncal rsks for the solvency of nsurance undertakngs. Provsons overestmaton whch s postve from the pont of vew of solvency may on the other hand cause an nterventon of revenue offces amed at dsclosng and taxng the actual costs of nsurance undertakng (as the overestmated provsons cause the decrease of nsurance undertakng s techncal results). Provsons cannot be treated as an nstrument of nsurance undertakng s balance polcy whch s used for shftng the profts to the future perods; they should be adequate to nsurance labltes of an nsurance undertakng. The rsk of napproprate provsons may be caused by many factors. Whle estmatng the value of provsons for clams outstandng t s necessary to take nto account numerous factors nfluencng them n partcular: amounts of loss or clams any expenses and costs of court proceedngs whch can be estmated n the case when dspute s brought to court the costs whch wll have to be returned to the nsured person and whch were ncurred by the latter n order to lmt the loss amount f such a clause exsts n the nsurance contract (for example costs of fre extngushng n the case of fre nsurance contracts or costs of rescue acton n the case of marne and avaton nsurance) other costs lnked drectly to takng clam nto consderaton. Among the factors whch are dependent on nsurance undertakng (nternal factors) nadequate premum tarffs (underestmaton of premums) nadequate methods of provsons calculaton lack of approprate relable and complete hstorcal data or ther wrong nterpretaton are of the greatest mportance. More techncal factors nclude for example: dffcultes related to establshng the rensurers share n losses and loss provsons and dffcultes related to estmatng the clam settlement costs. External factors are lnked mostly wth real loss hstory e.g.: unpredcted fluctuaton of clams frequency (sudden changes of rsk factors taken nto account) losses dsproportonal to the sze of nsurance portfolo (e.g. as a result of natural dsasters) or changes of legal regulatons poltcal or economc changes. The lst of such factors can be supplemented among others by the followng tems: general nflaton and nflaton concernng partcular classes of rsk - 6 -

8 change of legal regulatons n partcular concernng changes n the manner of loss management and changes n the clam settlement process changes n awareness and behavour of nsured persons changes n the courts mode of conduct and n ther case-law severty socal and other changes whch nfluence the ultmate loss amount. Due to the mportance of the ssue of establshng provsons for clams outstandng the rsk of underestmatng or overestmatng of the provsons may be lmted both by the nsurance undertakngs through the use of adequate procedures and methods for settng the level of provsons (tasks related to these processes should be carred out by approprate persons n most cases they shall be actuares) as well as through other enttes: audtors and nsurance supervson authorty. Most of rsk factors related to the above may be taken nto account merely n the analyses conducted at the level of nsurance undertakng. Only some of them may be taken nto account n the examnaton conducted by the nsurance supervson. n the further analyss only the mpact of nflaton was consdered. lassfcaton of models of provson for clams outstandng Estmaton of provsons for clams outstandng s a process of forecastng future values of pad clams and other cash flows relatng to the settlement of clams e.g. clams settlement costs shares of rensurers n clams or returns and recourses. Varous forecast methods and models may therefore be used. lassfcaton of models of clams settlement n tme may be performed accordng to many crtera. A study by G. Taylor et al. [003] 8 ntroduces the dvson of models accordng to: occurrence of random varables (determnstc and stochastc models) occurrence of lagged varables (statc and dynamc models) structure of the model nature of relatons between varables (the whole range of models from phenomenologcal models to mcro-structural ones basng on mcrodata) methods of assessment of model parameters (models wth parameters assessed by way of optmsaton and heurstc methods). The dvson nto determnstc and stochastc models depends on the ncluson of the varable determnng samplng errors (random component) n the clams settlement model n case of stochastc or non-ncluson n determnstc models. The dvson nto statc and dynamc models depends on the fact whether the lagged varables are ncluded n the gven year clams settlement model n case of dynamc model or relate to the year of clam occurrence only statc models. The dvson on account of the model structure ncludes the whole range of models from phenomenologcal models descrbng only certan noted relatons to mcro-structural models explanng the mechansm of occurrence of loss to a specfed amount dependng on the rsk 8 Taylor G. G. McGure A. Greenfeld Loss Reservng: Past Present and Future Research Paper Number 09 The Unversty of Melbourne September

9 factors relatng to respectve nsurance contracts (e.g. tarff-related a pror or a posteror factors) basng on mcrodata (mcro-structural models). Dependng on the methods of estmaton of unknown model parameters there are two types of models: models estmated by optmsaton of some statstcal crteron e.g. usng the maxmum lkelhood method and models estmated wth the use of heurstc methods estmated wth set algorthms. Models dscussed later n ths report (n Secton V.6) are the followng:. Mack s model stochastc statc phenomenologcal estmated heurstcally. Hertg s model stochastc statc phenomenologcal estmated optmally. Another dvson of the models of clams settlement n tme relates to the way of takng the rsk margn of provsons nto account. Ths dvson s mportant as t s connected wth dfferent ways of understandng the market way of valuaton of nsurance labltes. The followng models can be dstngushed: takng nto account the rsk margn of provsons by makng conservatve assumptons guaranteeng suffcent provsons wth the pre-establshed probablty (confdence level) guaranteeng the transfer of the nsurance portfolo based on market prncples. n-depth and synthetc descrpton of models classes s provded n the letter of 3 February 006 of the European Actuaral onsultatve Group (EAG) Solvency : Rsk Margn omparson to EOPS 9. The class of models whch facltate takng nto account the surcharge of provsons safety by makng conservatve assumptons ncludes above all classcal determnstc models commonly used n Poland. Models guaranteeng suffcent provsons wth the pre-establshed probablty must belong to the class of stochastc models as the provsons are determned at the level of pre-establshed percentle n the forecast values of nsurance labltes dstrbuton. Models guaranteeng transfer of the nsurance portfolo based on market prncples are economc models valuatng nsurance labltes usng cost of captal approach. The cost of captal approach s descrbed n more detal n hapter V. 9 Avalable under the address:

10 Methods of calculaton of the BNR provsons n Polsh nsurance undertakngs Estmaton of the BNR provsons (provsons for the ncurred but not reported clams) s regulated to certan yet lttle extent by the legslaton. Pursuant to Artcle 34() of the Ordnance of the Mnster of Fnance of 8 December 003 on specal accountng prncples for nsurance undertakngs (Dz. U. of 003 No. 8 tem 44 as amended) when determnng the value of BNR provson the nsurance undertakng should take nto account the current course of clams settlement n a gven nsurance class ncludng the number and amount of clams made n the next reportng perods after the perod for whch t was establshed. n addton Artcle 44() ndcates that the prncples of creatng and methods of establshng techncal provsons appled by the nsurance undertakng as well as assumptons as to data and statstcal ndcators used to determne provsons should be appled n a contnuous manner. t s unacceptable to modfy prncples methods or assumptons wthout ustfcaton. Takng nto account the experence learned relatng to the establshng BNR provsons the nsurance undertakng may choose any method of valuaton of those provsons. Ths secton presents methods of calculaton of BNR provsons n Polsh nsurance undertakngs and has been prepared on the bass of a report of the state of the portfolo of non-lfe nsurance undertakngs for 004. The analyss was performed for the ten selected nsurance undertakngs wth dversfed structure and scope of actvty. Nearly all the analysed undertakngs use one of the two methods presented below for estmaton of the BNR provson.. Bornhuetter-Ferguson s method. han-ladder s method (classcal or any modfed verson) or a method based on the weghted average of the results obtaned wth the use of han- Ladder s and Bornhuetter-Ferguson s method.e. Gunnar Benktander s method. n solated cases ape od methods loss ndcator method or the lump-sump method are appled. Pursuant to Artcle 3 of the Ordnance on specal accountng prncples for nsurance undertakngs respectve methods have been dvded by the nsurance undertakngs nto one of the two types of methods: actuaral method consstng n the establshng the provson wth the use of nsurance and fnancal mathematcs and statstcs lump-sump method consstng n the establshng the overall provson for the whole nsurance portfolo or ts part as a gven percent (lump-sump ndcator) on the premum or clams amount. 0 0 The lump-sump method may only be used when results obtaned wth the use of ths method are close to results obtaned wth the use of the ndvdual method. The lump-sump ndcator should be determned mantanng the contnuty prncple

11 V Methodology of the analyss V. Scope of the analyss Run-off trangles ncludng data on the value and number of premums and pad clams n respectve non-lfe nsurance classes obtaned on the bass of annual fnancal and statstcal statements of nsurance undertakngs for are subect to analyss. Run-off trangles were prepared on the bass of data on gross pad clams (ncludng clams n form of annutes) takng nto account clams settlement costs returns recourses and recoveres accordng to the year of loss clams occurrence and the perod of loss development. n addton the analyss covers the value of provsons for clams outstandng accordng to the year of loss occurrence. The scope of nformaton submtted by nsurance undertakngs to Offce of the nsurance and Penson Funds Supervsory ommsson relatng to clams accordng to the year of loss occurrence and year of loss reportng has been presented n detal n hapter V.. The analyss was performed for the aggregated value of clam payments and the provsons for each of 8 non-lfe nsurance classes. The aggregated values took nto account the results of 43 nsurance undertakngs whch carry out operatonal actvty n non-lfe nsurance n the studed reportng perods. The term of a unt wll be herenafter used to determne the subect of analyss. The unt shall mean the nsurance undertakng carryng out actvty n a sngle nsurance class at least n one year covered by the analyss. Every unt s matched wth a run-off trangle prepared on the bass of data relatng to pad clams by a sngle nsurance undertakng n one nsurance class. Eventually run-off trangles for 485 unts have been analysed. V. Reportng duty n relaton to nformaton concernng clams accordng to the year of loss occurrence and the year of loss reportng Whle acknowledgng the mportance of the provsons for clams outstandng arsng from non-lfe nsurance contracts for the safety of nsurance undertakng operatons the nsurance supervsory body advsed the nsurance undertakngs collectng and submttng to the supervsory body the reports on relevant clams statstcal detaled data on provsons for clams outstandng.e. concernng clams clams settlement costs returns recourses and recoveres and data on the level of provsons accordng to the year of loss occurrence and the year of loss reportng wthn the scope of the addtonal annual report. The reportng oblgaton concernng the submsson to the supervsory body of the nformaton on clams accordng to the year of loss occurrence and year of loss reportng was frst mposed on nsurance companes n Poland by the Ordnance of the Mnster of Fnance of 3 December 998 on methods of preparaton of quarterly and annual fnancal reports submtted to supervsory body by the nsurance undertakngs ( ) (Journal of Laws of 998 No 66 tem 5). At present the Ordnance of the Mnster of Fnance of 30 March 005 (Journal of Laws of 005 No 5 tem 465 as amended) on quarterly and addtonal annual fnancal and statstcal reports of nsurance undertakngs consttutes the legal bass for the Dvson nto nsurance classes results from the Annex to the Act of May 003 on nsurance actvty (Journal of Laws of 003 No 4 tem 5 as amended) and has been presented n Annex 5. Ths order was defned n the letter of harman of PUNU (The State Offce for nsurance Supervson) of September 998 (NS/4/40//JB/98 and NS/40/90/JB/98). Full text of ths letter s avalable n the PUNU bulletn nsurance sector results n

12 reportng oblgaton. The reportng forms (forms..3.4) to be used were defned n the Annex 3 to the Ordnance of the Mnster of Fnance of 30 March 005. Templates for the forms are presented below. Table. Reportng forms template for the clam nformaton accordng to the year of loss occurrence and the year of loss reportng Total Year Detals Number Number Value of losses of losses Value. Gross pad clams excludng returns recourses and recoveres (a) pad clams (b) clams settlement costs. ncludng: gross reopened clams (a) pad clams (b) clams settlement costs (c) ncludng: clams settlement costs wthout payment of clams. The rensurer s share n pad clams (a) pad clams (b) clams settlement costs 3. Pad clams - net of rensurance (a) pad clams (b) clams settlement costs. Returns recourses and recoveres ncluded n the gross pad clams from the techncal account. Returns recourses and recoveres receved (a) on the bass of rensurer s share (b) net of rensurance. Returns recourses and recoveres due (a) on the bass of rensurer s share (b) net of rensurance. Provsons for gross clams outstandng. Provsons for clams reported and estmated excludng provsons for clam settlement costs as of the end of the year (a) rensurer s share (b) net of rensurance. Provsons for clams reported and not estmated excludng provsons for clam settlement costs as of the end of the year (a) rensurer s share (b) net of rensurance 3. Provsons for clams ncurred but not reported (BNR) excludng provsons for clam settlement costs as of the end of the year (a) rensurer s share (b) net of rensurance 4. Provsons for clams settlement costs estmated and not estmated as of the end of the year (a) rensurer s share (b) net of rensurance 5. Provsons for clams settlement costs ncurred but not reported (BNR) as of the end of the year (a) rensurer s share (b) net of rensurance V. Amount of gross clams and gross provsons as of the end of the year V. Amount of gross provsons as of the end of prevous reportng year V. Overestmaton/underestmaton of provsons - -

13 The relevant regulatons concern the mode of how nsurance undertakngs keepng the loss regster are ncluded n the ordnance of the Mnster of Fnance of 8 December 003 on detaled rules of accountng applcable n relaton to nsurance undertakngs (Journal of Laws of 003 No 8 tem 44 as amended). Accordng to Artcle nsurance undertakngs shall regster the clams reported n a way enablng the establshng of the nformaton n relaton to every clam separately on the date of the loss occurrence the date of the loss reportng class and type of nsurance relevant to the loss value of clam or estmaton of that value the value (n part or total) of the pad clams ncludng the payment date the value of the clams outstandng ncluded n the provsons. Moreover n accordance wth the Artcle 3 nsurance undertakngs shall keep regsters of recourses and recoveres for drect actvty n a way that enables obtanng of the followng nformaton: the value of clams the value of recourses and recoveres receved. These regsters shall be kept n a way that enables the establshng the value of recourses and recoveres n relaton to a specfc clam as well as establsh the value of recourses and recoveres accordng to the year of loss reportng and year of loss occurrence relevant for recourses and recoveres. V.3 Run-off trangle For the purpose of the estmaton of the value and number of the ultmate losses and estmaton of the provsons for clams outstandng the analyss uses hstorcal nsurance data. Hstorcal data can be presented n form of a run-off trangle 3 : Fgure. nput data n form of a run-off trangle Detals Years of loss development () ^R Y Y Y Y... Y - Y ^R Y Y Y Y Y - ^Y ^R 3 Y 3 Y 3 Y 3 Y 3... ^Y 3- ^Y 3 ^R 3... Y Y Y ^Y ^Y - ^Y ^R - Y - Y - ^Y - Y - ^Y -- ^Y - ^R - Years of loss occurrence () Where: Y ^Y ^Y ^Y ^Y - ^Y ^R... (the earlest reportng year s desgnated as whereas the latest reportng year desgnated as ) Y the clam amount 4 arsng from losses ncurred n the year and pad n year ^Y the forecasted value of Y for > ^R estmate of clam amounts for losses ncurred n year and not pad untl calendar year (the value of provson for clams outstandng for losses ncurred n year ). for 3 n the run-off trangle used n the analyss the years of loss development correspond to yearly reportng perods. t would be necessary to have data on every sngle loss n order to defne precsely the perod of loss development. 4 Y may also ndcate the rensurers share n the clams number of ncurred losses reported or fully pad clams etc. dependng on the data as placed n the run-off trangle. - -

14 Total amount of loss calculated along the sde Y reflects the value of clams pad n calendar year k for losses occurrng n years...k. k k Y k V.4 Methodology of data supplementaton and correcton The verfcaton and modfcaton of data taken from the regsters and placed n the run-off trangle consttutng a bass to calculate provsons for clams outstandng s a part of natural actuaral and statstcal practce. There are many reasons why ths s done for example: to remove the results that concern atypcal observatons (sngle bg clams) to remove dscrepances between dfferent regsters and to supplement the mssng data. The value of pad clams wthn a specfc nsurance class n a relevant year as shown n the clams forms accordng to the year of loss occurrence shall equal the value of gross pad clams as shown n the techncal nsurance account of the same class for the same year. Nevertheless ths equaton was not reached for every unt analysed. n case the dfference of value among the reports was sgnfcant (more than 0%) the value of the pad clams as stated n the run-off trangle was supplemented or corrected usng the value as stated n the techncal nsurance account. n order to supplement or correct the poston Y the rato of clams settled n perod n relaton to total clams pad n calendar year K was created. ϑ Y Y. Smlarly the rato ϑ was created usng the total value for ZU K k k K k TRU TRU each class. The supplemented value s calculated as follows Y ϑ YK where Y K s the value of gross pad clams as shown n techncal nsurance account for year K- ϑ ϖϑ (8 ϖ ) ϑ ϖ s a number of the run-off trangle poston ( ) 8 OG ZU supplemented or corrected n relaton to the analysed unts. The supplementaton or correcton was conducted n approxmately /3 of the unts analysed. The data relatng to the number of losses 5 was also smlarly supplemented. n order to supplement the mssng data wth the number of losses the nformaton on number of payments accordng to specfc classes collected by the supervsory body (statstcal report on nsurance actvtes of nsurance undertakngs KNUFE 0) was used. The carred out comparson of number of losses and number of payments showed that for the maorty of unts analysed the numbers dffered nsgnfcantly and therefore t was possble to use them as a bass for further analyss. The supplementaton of data on the number of losses was done n accordance wth the ratos estmated smlarly as n the case of clam amounts. Although only the supplementaton of data was conducted wthout the correcton the supplementaton concerned approxmately ½ of the unts undergong analyss. n order to defne the qualty of conducted correctons and supplementaton a data qualty rato was constructed OG 5 n accordance wth the explanatory notes to reportng forms the number of losses s the number of events for whch the clam amount s hgher than

15 λ k N TRU ( Y Y ) S TRU ( Yk Yk ) k k k defnng the share of errors remanng to be corrected where Y s the value of gross pad clams as presented n techncal nsurance account for year k whereas Y k and Y k are the values of gross pad clams n year k on the bass of run-off trangle respectvely before and after correcton. The total value of ratos for aggregates n ndvdual classes s presented n the table. TRU k Table. Rato of data qualty accordng to nsurance classes Specfcaton nsurance class Value Numbers Value Numbers By supplementaton and correctons of the run-off trangles ncludng clam amounts almost 3/4 of errors were corrected whle the supplementaton of the run-off trangles ncludng number of losses resulted n correcton of /3 of errors 6. Many repeatng errors were observed whle analysng the data n the fnancal statement and statstcal statement n the part concernng the number of losses accordng to the year of loss occurrence and the year of loss reportng. These concerned mostly cases when the specfc postons were not flled n and errors whle conductng the summaton. Startng from 00 the number of errors made decreased sgnfcantly. Ths was caused by addton of explanatory notes to the report that explaned the rules of ndcatng losses accordng to the year of loss occurrence and the year of loss reportng. onsderng the number of errors and partal removal of errors the run-off trangles ncludng number of losses were only used to calculate the average value of clams for specfc nsurance class. The report that ncludes the lst of clams of non-lfe nsurance of the drect actvtes accordng to the year of loss occurrence the loss ncurred untl 998 were presented together as one class. For the needs of the analyss the losses were dvded nto two groups: losses ncurred n 998 and losses ncurred before 998. The dvson was made on the bass of the estmated ratos 998 ϑ ϑ ϑa. Only estmated losses of 998 were used n the further analyss. a S N 6 n few cases the value of rato s hgher than. Ths does not mply that the value of errors for any analysed unt ncreased. Example: n one class for one nsurance undertakng an error was removed by means of ncreasng the value of clams as placed n run-off trangle by almost 0.6 mllon. Before the correcton was made for another nsurance undertakng the value as placed n the trangle was almost.6 mllon hgher than n the techncal account but there was no need to correct t snce the error was nsgnfcant. After the supplementaton was carred out n relaton to the former nsurance undertakng and not carred out n relaton to the latter undertakng the value of error for aggregate ncreased from app. mllon to app..6 mllon. The values of rato hgher than can be smlarly explaned n relaton to other cases

16 For all analysed unts the data of the pad clams was corrected usng consumer prce ndex of goods and servces. The value of pad clams was expressed wth prces as n 004. Lnk ratos accordng to the year of loss occurrence and year of loss development determned on the bass of cumulatve pad clams and total provsons accordng to nsurance classes as of the end of 004 are presented n Annex. V.5 Dvson of run-off trangles Run-off trangles have been dvded nto four categores: full trangles ncludng the trangles of the unts that have pad out clams for the losses ncurred n all 7 occurrence years whle the unt shall start to make the payments n the same year as the occurrence year and the amount of the payments for losses ncurred n the gven year may not be negatve (.e. return and recourses may not exceed the amount of the payments made) ncomplete trangles ncludng trangles of these analysed unts that have made payments at least n three successve ntal years or at least durng the two successve fnal years lack of payments n a trangle ncludng trangles of these analysed unts that have made no payments sporadc payments n a trangle ncludng the remanng trangles (only a few payments ntervals n the occurrence perods recourses and returns exceedng the amount of the pad clams). The analyss of the run-off trangles contanng the values of payments shows that slghtly more than 3% of the analysed unts made no payment whle the next 3% made only sporadc clam payments n the years Therefore the run-off trangles regardng these unts have not been subect to analyss. Nearly 35% of the unts analysed n respect of whch t was possble to create full run-off trangles were subect to full analyss of provsons adequacy. The remanng 0% of the unts made ther payments ether n the ntal or fnal perod of the analyss. These unts were only subect to a smplfed analyss of provsons adequacy. Detaled breakdown of run-off trangles by nsurance classes s shown n the table below. Table 3. Analysed unts by the type of trangles and nsurance classes nsurance class Type of trangle Full trangles ncomplete trangles Sporadc payments No payments Dependng on the range of nformaton regardng the partcular unts the followng assumptons have been made: Analyss of adequacy of provson for clams outstandng for unts wth ncomplete runoff trangles was carred out based on the comparson of the created provsons wth provsons ndcated by means of the han-ladder method. Lnk ratos regardng the ndvdual unts were determned as weghted average values of lnk ratos of cumulatve payments of the analysed unt and of the total aggregate n a gven class; - 5 -

17 The analyss of adequacy of provsons for clams outstandng as well as analyss of volatlty of provsons for unts wth full run-off trangles was carred out based on lnk ratos ndcated by means of the han-ladder method. The volatlty of the pad clams n the successve years of loss development the value of the establshed provsons as well as the value of the rsk margn were subect to analyss. V.6 Models used n the analyss n the analyss of the full run-off trangles two approaches were used. The approaches were based on the followng models: parametrc model proposed by J. Hertg 7 non-parametrc model proposed by T. Mack 8. J. Hertg and T. Mack have proposed and developed estmators for the future clam payments standard devaton of such clams as well as estmators for provsons and ther standard devaton. The knowledge of such characterstcs allows for a statstcal descrptons of the value of provsons establshed by the analysed unts. Both Mack s model and Hertg s model are based on run-off trangle of cumulatve payments Y a a where stands for the accdent year... stands for the development year of clam payments. The ultmate loss amounts ncurred n year equals R N Y. n T. Mack s non-parametrc model the followng assumptons have been made: there are lnk ratos ξ for whch ndependent E [... ] * ξ - losses of dfferent occurrence years {... } { } where are J... J varance equals V[... ] σ - where s an unknown parameter. Hence the lnk rato ξ equals the quotent.e. the quotent of values of payments made wthn years to the value of payments made wthn years. n J. Hertg's parametrc model the followng assumptons have been made: values of cumulatve losses are postve >0 logarthm of the quotent of the cumulatve losses wthn two successve clams development perods s constant rrespectve of the occurrence year accurate to the error term σ 7 J. Hertg A statstcal approach to the BNR-reserves n marne nsurance. ASTN Bulletn p and G. Taylor Loss Reservng An Actuaral Perspectve. Kluwer Academc Publshers Boston/Dordrecht/London 000 p T. Mack Dstrbuton-free calculaton of the standard error of han Ladder reserve estmates. ASTN Bulletn 993; 3; p. 3-5 and G. Taylor Loss Reservng An Actuaral Perspectve. Kluwer Academc Publshers Boston/Dordrecht/London 000 p

18 the error term s ndependent and has a normal dstrbuton η ~ N(0 σ ). Thus where: log[ / ] log[ u / u ] η log[ u ξ - development factor / u ] u - loss payment factor n development perod. Thus log[ / ] ~ N ( ξ σ ). For detaled descrpton of the appled models see Annex. V.7 Verfcaton of resduals The maorty of models that are used n the analyss of run-off trangles s based on the assumpton regardng dstrbuton of losses or dstrbuton of lnk ratos. n order to examne the types of these dstrbutons standardzed resduals derved upon the applcaton of Mack s model were subected to analyss. n the model t s not necessary to make any assumptons n respect of the form of loss dstrbuton and n respect of lnk ratos. The analyss of derved resduals was a startng pont for verfcatons of hypotheses regardng the functon formula of loss dstrbuton and loss rato dstrbuton. Endeavours amed at adustment of the theoretcal dstrbutons to emprcal dstrbuton of resduals provded unclear results 9. Accordng to Kolomogorov-Smrnov statstcs among 8 nsurance classes t s the dstrbuton other than the normal one that s the most frequent correct dstrbuton. However on the other hand t was the normal dstrbuton that was the best ftted theoretcal dstrbuton. n accordance wth statstc χ the normal dstrbuton s more sutable than the lognormal dstrbuton used n Hertg's model. Basng on the attempt amed at the verfcaton of hypotheses regardng the form of resduals dstrbuton several conclusons may be drawn. Frstly market data does not always correspond to the assumptons made n ndvdual models and methods whch may lead to nadequacy of the appled methods and results that are not credble. Secondly methods of resduals verfcaton do not provde unversal and clear solutons and they may not be used as a crteron for selecton of the most adequate method of analyss. When choosng the method of analyss market specfcty and type of nsurance n ndvdual nsurance classes should be taken nto account n the frst place. However n the case of some ndvdual nsurance classes verfcaton of resduals as well as dstrbutons fttng proved to be successful. Despte the fact that the normal dstrbuton seems to be a more sutable dstrbuton for characterzng the loss value than lognormal one the provsons analyss was also carred out based on Hertg s model as the lognormal dstrbuton s used more wdely n lterature on provsons for clams outstandng. 9 Analyss of adustment of dstrbutons was carred out based on dstrbuton contaned n the statstcal package Statstca.a.: normal lognormal gamma beta logstc exponental and Webull dstrbutons dstrbuton of extreme values and dstrbuton other than the normal

19 V Lnk ratos and ther applcaton The am of ths secton s to estmate the average relatonshp between the payments made n precedng development years.e. the so called lnk ratos for ndvdual drect nsurance classes wthn non lfe nsurance. Basng on these lnk ratos and on the value of the pad clams values of future payments of nsurance undertakngs have been estmated. These values were used.a. for establshng classfcaton of ndvdual nsurance classes by classes wth long and short settlement perod. Assumng that the process of lnk ratos devatons n respect of payments made n ndvdual development perods was relatvely stable lnk ratos were estmated usng Mack s model n accordance wth the followng model: 7 for ξ Ult for 7 7 where stands for cumulatve payments made n the years: to - for losses ncurred n year Ult stands for the ultmate value of losses ncurred n year ndcated as a sum of cumulatve value of payments made n years from to 004 and establshed provsons for clams outstandng for losses ncurred n the year. Lnk ratos were estmated on the bass of the aggregaton of payments made by the nsurance undertakngs n ndvdual nsurance classes usng: trangles contanng data on payments for losses ncurred snce 998 for all nsurance classes trangles contanng data on payments for losses ncurred snce 999 for 4 nsurance classes: for class 7 (for negatve values of payments for two development years) classes and (for consderable devatons of values payments made over ndvdual years) and class 7 (no payments made n 998 occurrence perod). Table 4. Estmated values of lnk ratos nsurance Next years / year /3 year 3/4 year 4/5 year 5/6 year 6/7 year class n total Based on the trangles wth data on payments for losses ncurred snce 998 lass 47.76% 0.48% 00.48% 00.0% 00.7% 00.03% 00.05% lass 9.80% 0.38% 00.55% 00.3% 00.03% 00.03% 00.60% lass % 00.0% 00.4% 00.% 00.% 00.0% 00.0% lass % 5.46% 00.00% 00.49% 00.00% 00.00% 00.00% lass % 03.3% 04.4% 0.4% 0.6% 00.05% 00.00% lass % 3.06% 03.5% 00.78% 00.% 00.6% 00.53% lass % 08.54% 04.93% 00.0% 99.78% 00.3% 5.0% lass % 03.88% 0.44% 00.6% 00.7% 00.5% 0.36% lass % 03.4% 0.4% 0.% 00.9% 00.9% 00.94% lass % 05.68% 0.80% 0.76% 0.54% 0.58% 6.3% lass 05.08% 0.08% 00.3% 00.8% 00.6% 00.4% 0.43% - 8 -

20 nsurance Next years / year /3 year 3/4 year 4/5 year 5/6 year 6/7 year class n total lass 39.47% 7.56%.40% 03.04% 00.58% 00.40% 3.7% lass % 0.0% 06.40% 04.79% 08.3% 0.4% 46.79% lass % 5.08% 03.0% 94.54% 98.38% 95.79% 0.7% lass % 0.6% 99.53% 05.% 98.3% 99.50% 0.67% lass % 7.49% 00.50% 0.59% 00.63% 00.6% 0.6% lass % 0.88% 04.36% 00.00% 00.00% - - lass % 0.58% 00.05% 00.05% 00.00% 00.00% 00.4% Based on the trangles wth data on payments for losses ncurred snce 999 lass % 08.54% 04.93% 00.0% 99.78% 00.3% 5.0% lass 05.08% 0.08% 00.3% 00.8% 00.6% 00.4% 0.43% lass 39.47% 7.56%.40% 03.04% 00.58% 00.40% 3.7% lass % 0.88% 04.36% 00.00% 00.00% 00.00% 00.00% The estmated values of lnk ratos allowng for the results for the classes 7... and 7. derved on the bass of the trangles contanng data on payments for losses ncurred snce 999 were used to forecast the value of the future payments for ndvdual development years and the ultmate value of payment. The results derved n respect of payments made n the frst development year are shown n the table below. Table 5. The ultmate value of payments n relaton to payments made n the frst development year Value of rato for the Average value of rato for nsurance aggregated amount of loss amounts after class losses ncurred untl ndvdual occurrence years 004 lass 5.68% 5.73% lass 34.80% 34.76% lass 3.6%.6% lass % 307.6% lass % 69.05% lass % 59.45% lass % 85.0% lass % 48.59% lass % 47.99% lass % 97.80% lass 5.35%.7% lass 490.7% % lass % 35.5% lass % 83.99% lass % 87.4% lass % 3.9% lass 7 5.0% 4.5% lass 8.37%.04% - 9 -

21 hart. The ultmate value of payments n relaton to payments made n the frst development year 600% 500% 400% 300% 00% 00% 0% lass 8 lass 3 lass lass 8 Aggregated data after the years lass 9 lass lass 5 lass 5 lass 7 lass 0 lass 4 lass 7 lass lass 6 lass 4 Average value after the years lass 6 lass 3 lass nsurance of classes and. belongs to the types of nsurance under whch consderable number of payments s made n the occurrence year. nsurance of classes and 4. nclude nsurance under whch the lowest number of clams s fully settled. t should be also noted that n the case of almost all nsurance classes the value of the abovementoned rato for the ndvdual occurrence years s approxmate whch s also vsble when comparng the value of the rato after years wth the aggregated value. More dvergences may be observed only n the case of followng nsurance classes: and 6. n these classes the dstrbutons of payments made for gven development years may dffer slghtly as far as the ndvdual development years are concerned. The structure of the estmated cumulatve payments accordng to development perods s provded below. Table 6. Structure of the estmated payments accordng to development perods nsurance class year year 3 year 4 year 5 year 6 year 7 year Next years total lass 65.50% 96.78% 99.8% 99.65% 99.75% 99.9% 99.95% 00.00% lass 74.8% 96.9% 98.58% 99.% 99.35% 99.37% 99.40% 00.00% lass 3 8.3% 99.4% 99.44% 99.58% 99.68% 99.79% 99.80% 00.00% lass % 86.9% 99.5% 99.5% 00.00% 00.00% 00.00% 00.00% lass % 89.99% 9.89% 96.99% 98.37% 99.95% 00.00% 00.00% lass % 84.04% 95.0% 98.34% 99.% 99.% 99.48% 00.00% lass % 83.09% 90.7% 95.49% 95.65% 95.67% 95.67% 00.00% lass % 9.57% 96.7% 97.55% 98.4% 98.4% 98.66% 00.00% lass % 9.98% 96.5% 97.5% 98.59% 98.88% 99.07% 00.00% lass % 69.47% 73.4% 75.47% 76.80% 77.98% 79.% 00.00% lass 44.38% 67.57% 86.66% 89.05% 89.0% 89.0% 89.0% 00.00% lass 0.38% 6.65% 73.3% 80.7% 8.63% 83.95% 83.95% 00.00% lass 3 3.7% 49.98% 55.07% 58.60% 6.4% 66.5% 68.% 00.00% lass % 85.70% 98.63% 0.78% 96.3% 94.67% 90.68% 00.00% lass % 93.58% 95.09% 94.64% 99.58% 97.89% 97.40% 00.00% lass % 80.6% 94.8% 94.66% 97.% 97.7% 97.88% 00.00% lass % 86.4% 95.8% 00.00% 00.00% 00.00% 00.00% 00.00% lass % 97.94% 99.49% 99.53% 99.58% 99.58% 99.58% 00.00% - 0 -

22 hart. Structure of the estmated payments accordng to ndvdual development years 00% 90% 80% 70% 60% 50% 40% 30% 0% 0% 0% lass lass 3 lass 6 lass 4 lass 6 lass lass 7 lass 4 lass 0 lass 7 lass 5 lass 5 lass lass 9 lass 8 lass lass 3 lass 8 Followng t should be noted that n the case of nsurance classes 4. and 5. n whch the value of the obtaned recourses and recoveres exceeds the amount of payments n respect of some development years the total value of ratos provdng for the structure of payments by development years s lower than 00%. Havng obtaned the structure of the total payments made n ndvdual development years t was possble to dvde the nsurance nto nsurance wth long and short settlement perods as well as nto nsurance wth short and long ncubaton perods. For the purposes of ths document t has been assumed that: nsurance wth short loss settlement perod ncludes nsurance n respect of whch 95% of the clam amount s settled wthn the perod (average full settlement perod x 095 ) not longer that 4 development perods whereas nsurance wth long loss settlement perod ncludes nsurance n respect of whch the perod exceeds 4 development perods 0. nsurance wth short loss ncubaton perod ncludes nsurance n respect of whch 50% of the clam amount s settled wthn the perod not longer than development perod whereas nsurance wth long loss ncubaton perod ncludes nsurance n respect of whch the perod exceeds year. Table 7. Expected average full settlement perod and loss ncubaton perod Loss settlement perod Loss ncubaton perod nsurance class Development Development nsurance type year year nsurance type lass Short settlement perod Short ncubaton perod lass Short settlement perod Short ncubaton perod lass 3 Short settlement perod Short ncubaton perod 7 y ear 6 y ear 5 y ear 4 y ear 3 y ear y ear y ear 0 Lmted perod (4 years) corresponds to the rulng of Artcle 60()(g)() of the Drectve 9/674/EE (OJ No L374) accordng to whch t s possble to dscount provsons n non-lfe nsurance f an average perod of loss settlement conssts of at least four years from the day of ts accountng. Loss ncubaton perod stands for the tme when the loss has arsen up to the moment of ts start of settlement. The appled measure (50 th percentle) by means of whch the nsurance was classfed as nsurance wth long or short loss ncubaton perod s not precse. However t reflects the core dea of the phenomenon n queston. f the data on clam payments was provded more frequently (data provded monthly or quarterly) t would be possble to determne the loss ncubaton perod more precsely (lower percentle than the 50 th percentle). - -