Stochastic Claims Reserving under Consideration of Various Different Sources of Information

Transcription

1 Stochastc Clams Reservng under Consderaton of Varous Dfferent Sources of Informaton Dssertaton Zur Erlangung der Würde des Dotors der Wrtschaftswssenschaften der Unverstät Hamburg vorgelegt von Sebastan Happ geb. am n Tübngen Hamburg, Jul 2014

2 Vorstzender: Prof. Dr. Bernhard Arnold (Unverstät Hamburg) Erstgutachter (Supervsor): Prof. Dr. Mchael Merz (Unverstät Hamburg) Zwetgutachter (Co-Supervsor): Prof. Dr. Maro V. Wüthrch (ETH Zürch) Datum der Dsputaton:

3 Acnowledgements Durng my dploma studes n mathematcs and busness admnstraton at the Eberhard Karls Unverstät Tübngen Prof. Dr. Mchael Merz started teachng at unversty as an assstant professor at the department of busness admnstraton. Ths gave me the chance to attend hs lecture seres on selected topcs n actuaral scence. Vstng these lectures I got a fundamental nsght n practcal problems n quanttatve rs management and nsurance and how they can be approached by mathematcal statstcal concepts. Ths awaened my deep nterest n actuaral scence. At ths pont I would le to express my deepest grattude to my supervsor Mchael Merz for gvng me the chance to pursue a PhD at the faculty of busness admnstraton n Hamburg. Not only dd he support me n scentfc ssues but also n personal matters. I am deeply grateful to my co-supervsor Prof. Dr. Maro V. Wüthrch from ETH Zürch for hs great support and nvaluable advce n our jont contrbutons. He permanently supported me wth hs vast nowledge and experence n actuaral scence. Moreover, I would le to than my co-author René Dahms for hs valuable collaboraton. My thans also go to the whole team, namely T. Gummersbach, J. Heberle, Nha-Ngh Huynh, A. Johannssen, A. Ruz-Merno and A. Thomas at the char of mathematcs and statstcs n busness admnstraton at Unverstät Hamburg under the admnstraton and supervson of Mchael Merz for ts support. I would le to than Marco Bretg for hs valuable dscussons. Fnally, I than my wfe Svetlana for her confdence and support n all those years. Sebastan Happ

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5 Contents 1 Introducton 1 2 Reservng Problem Insurance Contracts and Process of Clams Settlement Data Bass n a Non-lfe Insurance Company Classcal Vew Extended Vew Predcton Problem Inflaton Predcton Precson Mean Squared Error of Predcton Clams Development Result Classcal Dstrbuton-Free Clams Reservng Methods General Notaton Chan Ladder Method Bayes Chan Ladder Method Complementary Loss Rato Method Bornhuetter Ferguson Method Munch Chan Ladder Method (Bayesan) Lnear Stochastc Reservng Methods Lnear Stochastc Reservng Methods Classcal Clams Reservng Methods as LSRMs Parameter Estmaton for LSRMs Predcton of Future Clam Informaton Bayesan Lnear Stochastc Reservng Methods Classcal Bayesan Clams Reservng Methods as Bayesan LSRMs Predcton of Future Clam Informaton Credblty for Lnear Stochastc Reservng Methods I

6 II Contents Mean Squared Error of Predcton Specal Case: Mean Squared Error of Predcton for the Bayes CL Method Clams Development Result Specal Case: Clams Development Result for the Bayes CL Method Example Bayesan LSRM Conclusons Pad-Incurred Chan Reservng Method Notaton and Model Assumptons One-year Clams Development Result Expected Ultmate Clam at Tme J Mean Squared Error of Predcton of the Clams Development Result Sngle Accdent Years Aggregated Accdent Years Example PIC Reservng Method Conclusons Pad-Incurred Chan Reservng Method wth Dependence Modelng Notaton and Model Assumptons Ultmate Clam Predcton for Known Parameters Θ Estmaton of Parameter Θ Predcton Uncertanty Example PIC Reservng Method wth Dependence Modelng Conclusons Solvency Regulatory Requrements on Reserves Maret-Value Margn Solvency Captal Requrements Fnal Regulatory Reserves Smplfcatons for Regulatory Solvency Requrements Example for Regulatory Reserves Conclusons and Outloo 125 Data Sets 133

7 Lst of Fgures 2.1 Generc tme lne of the clams settlement process Classcal vew (extended vew): Generc run-off trapezod of the m-th LoB (clam nformaton) for m {1,..., M} and ncremental clams payments (clam nformaton) of accdent year and development year wth + = I Data set D I observable at tme I and data set D I+1 observable at tme I Reserves R I based on D I at tme I, updated reserves R I+1 based on D I+1 at tme I + 1 and the resultng clams development result CDR M,I σ-felds (sets of observatons): B, - all clam nformaton n accdent year up to development year, D - all clam nformaton up to development year, D n - all clam nformaton up to accountng year n and D n - the unon of all nformaton n D and D n σ-felds (sets of observatons): D - all clam nformaton up to development year, D n - all clam nformaton up to accountng year n and D n - the unon of all nformaton n D and D n Development factors for BUs 1 3 n the classcal LSRM and credblty development factor F 0 I,Cred {0,..., 10} for BU Cumulatve clams payments P,j and ncurred losses I,j observed at tme t = J both leadng to the ultmate loss P,J = I,J Updated cumulatve clams payments P,j and ncurred losses I,j observed at tme t = J Emprcal densty for the one-year CDR (blue lne) from smulatons and ftted Gaussan densty wth mean 0 and standard devaton (dotted red lne) QQ-plot for lower quantles q (0, 0.1) to compare the left tal of the emprcal densty for the one-year CDR wth the left tal of the ftted Gaussan densty wth mean 0 and standard devaton III

8 IV Lst of Fgures 6.1 Correlaton estmators ˆρ l for ρ l for l {0, 1, 2, 3} as a functon of the number of observatons used for the estmaton Reserves consst of BEL, MVM (together satsfyng accountng condton) and SCR (satsfyng the nsurance contract condton) The calbrated log-normal dstrbuton wth µ = and σ = used as an approxmaton for the dstrbuton of the quantty S21 M + BEL21 and correspondng expected value, VaR and ES for the securty level α = Best-estmate valuaton of labltes BEL 20, maret-value margn MVM 20 (together satsfyng accountng condton) and solvency captal requrements SCR 20 (satsfyng the nsurance contract condton) leadng to the overall reserves

9 Lst of Tables 2.1 Classcal rs characterstcs: Reserves and CDR and the correspondng frst two moments Reserves and predcton uncertanty Indvdual LoB and overall CDR uncertanty Ultmate clam predcton and predcton uncertanty for the one-year CDR calculated by the ECLR method for clams payments and ncurred losses (cf. Dahms [16] and Dahms et al. [18]) and by the PIC method, respectvely Ratos msep 1/2 CDR /msep1/2 Ultmate calculated by the ECLR method for clams payments and ncurred losses (cf. Dahms et al. [18]) and calculated by the PIC method, respectvely Left-hand sde: development trangle wth cumulatve clams payments P,j ; rghthand sde: development trangle wth ncurred losses I,j ; both leadng to the same ultmate clam P,J = I,J Uncorrelated case and three explct choces for correlatons Clams reserves n the classcal PIC model and PIC model wth dependence Predcton uncertanty msep 1/2 for the classcal PIC model and the PIC model wth dependence Predcted ncremental clam nformaton for LoB 1, 2 and Expected pattern of BEL for calendar years n = 20,..., Cumulatve clams payments Incurred losses Busness unt Busness unt Busness unt Cumulatve clams payments P,j, + j 21, from a motor thrd party lablty Incurred losses I,j, + j 21, from a motor thrd party lablty V

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11 1 Introducton Recent developments n (fnancal) marets have shown that unexpected negatve events may have a tremendous mpact on a wde range of fnancal nsttutons such as bans, funds, nvestment and nsurance companes. Often such events are followed by serous problems rangng from economc depresson wth hgh unemployment rates, a decrease n common wealth and bad medcal mantenance to socal rots. Governmental authortes and regulatory nsttutons have been establshed to adopt and develop regulatory framewors for the fnancal ndustry, n order to reduce negatve mpact of such events and to avod collateral damage on other parts of the economy n the future. In Germany the Federal Fnancal Supervsory Authorty (BaFn) supervses bans, fnancal servces provder, nsurance companes as well as securtes tradng. Moreover, n response to the fnancal crss the European Unon (EU) created the European System of Fnancal Supervson, whch conssts of three European Supervsory Authortes: 1. European Banng Authorty (EBA) for the European banng sector 2. European Insurance and Occupatonal Pensons Authorty (EIOPA) for the nsurance sector 3. European Securtes and Marets Authorty (ESMA) for securtes tradng For the banng sector the correspondng regulatory framewor called Basel II was developed by the Basel Commttee on Banng Supervson and s currently replaced by ts successor Basel III. Insurance companes n Europe are controlled by the regulatory framewor called Solvency II. In Swtzerland the regulaton of all fnancal nsttutons ncludng nsurance companes s provded by the Swss Fnancal Marets Authorty (FINMA) wth the correspondng regulaton framewors Basel II and Basel III for the banng sector and the Swss Solvency Test (SST) for the nsurance ndustry. For an nsurance company there are two ways a regulatory framewor can be looed at. a) From the perspectve of nvestors and the management: The functon and the exstence of the company must be mantaned n the md/long term run to generate earnngs for the nvestors and the management. Moreover, these earnngs should be maxmzed 1

12 2 1 Introducton (proft maxmzaton). b) From the perspectve of regulatory authortes: Fnancal lqudty of the nsurance company must be provded even n tmes of extreme fnancal dstress and phases of an extraordnary accumulaton of clam compensaton payments. The ablty of the nsurer to pay losses has to be mantaned n almost all realstc scenaros to prevent losses for the polcyholders and to elmnate wde-rangng negatve effects on the whole economy. Smlar to Basel II, the Solvency II regulatory framewor s subdvded nto three man pllars to ncorporate the man deas of the regulatory authortes pont of vew: Pllar I: Mnmum Standard and Implementaton Maret consstent valuaton of assets and labltes Internal models, best-estmate reserves, techncal provsons, solvency captal requrements, target captal and own funds Pllar II: Supervsor Revew and Control Group supervson Supervsory revew process Governance Pllar III: Dsclosure Supervsory transparency Accountablty Reportng and dsclosure For detals on the techncal standards, further gudelnes and nformaton, see the Webste of EIOPA 1. For the basc structure of the SST we refer to the Webste of FINMA 2. In ths thess we focus on the the frst pllar. Moreover, one has to dstngush between lfe and non-lfe nsurance busness, snce the contract specfcatons, rs drvers and payoff patterns and hence the methodologc means of approachng and modelng lfe and non-lfe contract labltes dffer substantally. For an llustraton of ths fact we refer to the examples gven n Chapter 7 n Wüthrch Merz [62]. An ntroducton on stochastc models n lfe nsurance can be found n Gerber [26] and Koller [35]. It s crucal to eep n mnd that from now on throughout the thess we wll strctly deal wth non-lfe nsurance busness. The frst pllar n non-lfe nsurance has been subject to many quanttatve scentfc studes, see Wüthrch Merz [63] and [62] for an overvew, snce t s drectly assocated wth the

13 3 problem of the management and quantfcaton of (random) future cash flows. These cash flows typcally arse from assets and clams payments, see Wüthrch Merz [62]. The correspondng feld of study to analyze (random) rs outcomes and assocated loss lablty cash flows n nsurance wth mathematcal and statstcal methods s called actuaral scence. Actuaral scence comprses the followng aspects: 1. Evaluaton of (random) outstandng loss lablty cash flows and settng-up of suffcent reserves to meet these labltes 2. Evaluaton of assets and ts assocated rs 3. Level of premums n polces 4. Rensurance 5. Asset lablty management (ALM) comprsng all prevous aspects All stated aspects have an mpact on the process of future cash flows and are therefore crucal for management purposes n nsurance companes. That means that actuaral scence s drectly assocated wth the central problem n nsurance companes of predctng future cash flows. Therefore, the crucal tas and man goal of actuares s the predcton of (random) future cash flows. Among the fve aspects stated above we focus n ths thess on the frst aspect,.e. the feld of predctng future outstandng loss labltes. In actuaral scence ths feld s called clams reservng. Clams reservng belongs to the man tass of a non-lfe actuary, snce clams reserves are the bggest poston on a balance sheet of a non-lfe nsurance company and must therefore be predcted very precsely. Therefore, n ths doctoral thess we wll focus on the tas of predctng future loss labltes and calculatng the correspondng reserves needed to cover these outstandng loss labltes n non-lfe nsurance companes. For ths predcton problem there are often varous sources of nformaton avalable. Most classcal clams reservng methods are very lmted w.r.t. the sources of nformaton they can ncorporate. We present n ths thess two powerful models whch can cope wth several sources of nformaton n a mathematcally consstent way. The frst model generalzes most wdely used dstrbuton-free clams reservng methods. Ths provdes a new perspectve and new possbltes for dstrbuton-free clams modelng and s subject to Part II of ths thess. The second method s an mportant representatve of the class of dstrbutonal clams reservng methods whch can cope wth two dfferent data sources often avalable n nsurance practce. Ths s subject to Part III. The thess s closed up by Part IV dscussng some central aspects of clams reservng under new solvency requrements le Solvency II or SST.

14 4 1 Introducton Outlne Ths thess s dvded nto four parts: Part I: In the frst part (Chapter 2) the classcal clams reservng problem s ntroduced. We consder the assocated general predcton problem and pont out whch data sources have been used n classcal as well as n state-of-the-art clams reservng methods for the predcton of future loss labltes. Moreover we show how the ncorporated predcton uncertanty s classcally quantfed n long term as well as n short term rs consderatons. Part II: In the second part (Chapters 3 and 4) we brefly present wdely used classcal clams reservng methods. Followng Dahms [17] and Dahms Happ [15] all these methods are then merged n a general state-of-the-art dstrbuton-free clams reservng framewor n Chapter 4. Ths model framewor comprses almost all dstrbuton-free clams reservng methods. Moreover, t allows for the ncorporaton of varous sources of nformaton for the predcton process and hence provdes a new perspectve and possbltes of dstrbuton-free clams reservng. Part III: In contrary to Part II ths part s subject to dstrbutonal clams reservng. In the model class of dstrbutonal clams reservng methods we consder n Chapters 5 and 6 an mportant representatve, the pad-ncurred chan (PIC) reservng method presented n Merz Wüthrch [46]. Followng Happ et al. [30] and Happ Wüthrch [31] we consder for ths method the quantfcaton of the one-year reservng rs and generalze the classcal PIC method so that dependence structures n the data can be approprately captured. Moreover, the whole predctve dstrbuton of the clams development result s derved va Monte-Carlo (MC) methods. Part IV: In ths part (Chapter 7) we pont out central regulatory requrements ncluded n recent solvency framewors le SST or Solvency II. These solvency requrements are not coherent wth most classcal clams reservng methods. We pont out smplfcaton methods proposed n the SST and show how they mae most clams reservng methods accessble for these solvency requrements. We close up ths part by presentng an example where reserves are calculated regardng the SST reservng requrements.

15 2 Reservng Problem 2.1 Insurance Contracts and Process of Clams Settlement An nsurance contract s an agreement of two partes: For a fxed payment (nsurance premum) the nsurer (nsurance company) oblges to pay a fnancal compensaton to the nsured (polcyholder) n the case of an occurrence of some well defned (random) future event n a well specfed tme perod. In the case of such an event at a certan date (occurrence date) durng the nsured perod, the nsured person reports the clam to the nsurance company at the so-called reportng (notfcaton) date. The tme between the occurrence and the reportng date s called reportng delay. After the reportng of the clam the nsurance company verfes whether all nsurance contract specfcatons are fulflled so that the nsurer has to provde coverage of the clam. If ths s the case, the nsurance company starts payments for the fnancal compensaton of the clam n accordance to the contract specfcatons. Ths clams settlement process typcally conssts of one or more payments to the polcyholder. It ends wth the closure date where no further clams payments are expected and the clam s (presumably) completely settled and closed. The tme lne of typcal non-lfe nsurance clams from occurrence to the fnal settlement s llustrated n Fgure 2.1. Tme delays from occurrence to notfcaton and from settlement process to the premum nsured perod occurrence notfcaton closure clams payments Fgure 2.1: Generc tme lne of the clams settlement process tme closure date are typcal for non-lfe nsurance clams and can be caused by dfferent reasons: Delays when ncurred clam events are not mmedately reported to the nsurance company Fnal clam amounts are determned over a long perod of tme (up to several decades) Jurdcal nspecton of a clam. The lablty of the nsurance company to pay for the clam 5

16 6 2 Reservng Problem s to be determned Court decsons leadng to payment adjustments, reverse transactons of already pad compensatons or addtonal clams payments These tme delays often lead to a very slow clams settlement process wth clams payments far n the future (up to several decades). Ths shows that the very nature of nsurance busness (.e. underwrtng rss through nsurance contracts) often causes a very slow settlement process and the predcton of ths process becomes a central pont of nterest. For a more detaled dscusson on that topc, see Wüthrch Merz [63]. General Remar: In non-lfe nsurance busness many clam characterstcs (occurrence date, frequency of clams, severty of a clam, clam settlement pattern, clams payments, etc.) are subject to randomness and can not be predcted wthout uncertanty. Hence, probablty theory and statstcs provde sutable mathematcal tools for dealng wth those clam characterstcs. Thereby, t s assumed that the very nature and the behavor of these clam characterstcs do not change too fast over tme. Ths assumpton s requred to utlze past observatons for predctng purposes and to reveal systematc propertes (behavor) of the quanttes under consderaton. For ths reason we model all quanttes of nterest n a stochastc framewor as random varables, whch are defned on a common probablty space (Ω, D, P). 2.2 Data Bass n a Non-lfe Insurance Company In general, nsurance companes group polces (nsurance contracts) wth smlar rs characterstcs or comparable contract specfcatons nto suffcently homogeneous nsurance portfolos. Ths s often done by Lnes of Busness (LoB), but can be subdvded further nto smaller unts. Typcal LoBs are: Motor thrd party, product lablty, prvate and commercal property, commercal lablty, health nsurance, etc. An nsurance company has to put provsons asde, n order to cover future loss labltes arsng from these grouped nsurance portfolos. For ths reason an accurate predcton of future loss labltes and the assocated cash flows n the clam settlement process s of central nterest. Ths predcton can be based on varous sources of nformaton Classcal Vew In the classcal vew the predcton of future loss labltes s often based on the nformaton of the past observed development of the settlement process tself. Classcal clams reservng

17 2.2 Data Bass n a Non-lfe Insurance Company 7 lterature often assumes that an nsurance company has, after groupng of ndvdual contracts, M 1 nearly homogeneous portfolos. All clams, whch occur n year, are called clams n accdent year {0,..., I}, where I s the current year. The number of years between accdent year and the year of the actual clams payment s called development year {0,..., J}, wth J beng the total number of development years. It s usually assumed that I J and that all clams are completely settled n development year J,.e. there are no clams payments beyond development year J. For models consderng clams payments beyond development year J by means of so-called tal factors, see Mac [40] and Merz Wüthrch [42]. We denote all payments for accdent year and development year n the m-th portfolo (m {1,..., M}) by S, m and say that all clams payments Sm, wth + = n and n {0,..., I + J} belong to accountng year n. Ths notaton s called ncremental clams representaton n the actuaral lterature, because we consder clams payments S, m n accdent year and development year of the m-th portfolo. In the actuaral lterature (cf. Wüthrch Merz [63]) the cumulatve clams payments representaton of the clam settlement process s also used. In ths representaton one consders cumulated amounts n accdent year up to development year defned by C m, := S,j, m (2.1) j=0 where all clams payments whch belong to accdent year up to development year n the m-th portfolo are aggregated. At tme n {0,..., I + J} all clams payments S, m wth + n and 1 m M are observed and generate the σ-feld D n := σ { S, m + n, 0 I, 0 J, 1 m M } = σ { C, m + n, 0 I, 0 J, 1 m M }. Moreover, we denote the resultng fltraton by D := (D n ) 0 n I+J leadng to the probablty space wth fltraton (Ω, D, D, P). The two representatons (ncremental or cumulatve representaton) are commonly used n the clams reservng lterature, and t manly depends on the model choce whether the ncremental or the cumulatve representaton s used. The settlement process of the m-th portfolo n the ncremental as well as n the cumulatve clams payments representaton s llustrated n clams development (run-off) trapezods where accdent years {0,..., I} and development years {0,..., J} are gven by the rows and the columns, respectvely. Ths means the ncremental clams payments n accdent year and development year of the m-th portfolo are postoned n the -th row and the -th column n the m-th development trapezod, see Fgure 2.2. (2.2) We wll see n Chapter 4 that the ncremental clams payments representaton s an approprate choce for almost all dstrbuton-free clams reservng methods. Moreover, the ncremental representaton s advantageous f one s nterested n the valuaton of outstandng loss labl-

18 8 2 Reservng Problem development years 1 m J 0 1 accdent years... D I S m, I... to be predcted Fgure 2.2: Classcal vew (extended vew): Generc run-off trapezod of the m-th LoB (clam nformaton) for m {1,..., M} and ncremental clams payments (clam nformaton) of accdent year and development year wth + = I tes va valuaton portfolos, see Wüthrch Merz [62]. However, we swtch to the cumulated clams payments representaton, f helpful (Chapters 5 and 6) Extended Vew Besde the clam settlement process data there are often other sources of nformaton avalable for the predcton of loss labltes: Settlement processes of other correlated portfolos Data of collectves whch may nfluence the settlement process under consderaton Incurred losses: Clams payments plus ndvdual case dependent loss reserves Pror ultmate clam estmates: Ths nformaton may nclude prcng arguments Insured volume Number and sze of contracts etc. Recent publcatons n actuaral scence consder new models whch allow for ncludng some of these sources of nformaton n a mathematcally consstent way, see for example Dahms [17] and Merz Wüthrch [46]. In these models S, m and Cm, do not necessarly only correspond

19 2.2 Data Bass n a Non-lfe Insurance Company 9 anymore to ncremental clams payments and cumulatve clams payments (.e. nformaton from the clam settlement process). They may also represent some other sources of nformaton stated above, for example ncurred losses data, see the PIC reservng method n Merz Wüthrch [46], or pror ultmate clam estmates, see the Bornhuetter Ferguson (BF) method n Mac [39]. Therefore, t s necessary to extend the denotaton of S, m of the classcal vew, snce we focus n the actuaral contrbutons of ths thess on such new model classes, see Chapters 4 6. Throughout the thess S, m denotes the m-th (m {1,..., M}) clam nformaton of accdent year {0,..., I} and development year {0,..., J} and not necessarly only the clams payments as t s convenent n classcal clams reservng methods. These clam nformaton may besde the clams payments process contan ncurred losses, see Merz Wüthrch [46] and Dahms [16], receved premum and the average loss rato, see Bühlmann [11], pror ultmate clam estmates, see Mac [39] and Arbenz Salzmann [6], clam volume nformaton, see Dahms [17], or other addtonal sources of nformaton. By a slght abuse of notaton we wll call also m {1,..., M} the m-th clam nformaton by dentfyng the ndex m wth ts assocated clam nformaton S, m. In the extended vew some clam nformaton S, m do not generate any loss lablty cash flows n the future and thus do not have to be predcted. Therefore, we defne M := { m M S m, generates loss lablty cash flows }. (2.3) By defnton M s the set of clam nformaton whch generate cash flows, see (2.3), and s therefore of central nterest for clams reservng and rs management. Remars 2.1 (Set M) In most classcal clams reservng methods, each clam nformaton m M s gven by the clams payments of an nsurance portfolo of a specfc LoB, see Chapter 3 for examples. However, ths s not always the case. In Example 1 n Dahms [17] there s a clam nformaton m M of subrogaton payments. Ths shows that M may besde the clams payments of dfferent LoBs also contan other clam nformaton whch also generate cash flows. That means that the clam nformaton n M are not explctly restrcted to clams payments of dfferent nsurance portfolos. However, for a smpler nterpretaton of the set M one may thn of each clam nformaton m M as clams payments arsng from an nsurance portfolo of a certan LoB. As a consequence of the defnton of M, the set of all clam nformaton {1,..., M} s dvded nto dsjont subsets M {1,..., M} and M c = {1,..., M}\M. The clam nformaton m M have already been dscussed above. The set M c of clam nformaton s not of central nterest for rs management and clams reservng, because t does not generate any loss lablty cash flows. However, clam nformaton out of M c are utlzed n many models for the predcton of clam nformaton m M under consderaton,.e. they contan nformaton, whch are requred for

20 10 2 Reservng Problem the predcton of clam nformaton m M. To name only a few of them, an ultmate clam estmate (as a clam nformaton m M c ) s ncorporated for the predcton of clams payments (as a clam nformaton m M) n the BF method, see Secton 3.5, ncurred losses are used for the predcton of clams payments n the extended complementary loss rato (ECLR) method, see Dahms [16], and n the PIC reservng method, see Chapters 5 and 6, or volume measures are ncluded for clams payments predctons n the addtve loss reservng (ALR) method n Merz Wüthrch [44]. In analogy to the classcal vew, clam nformaton m {1,..., M} n the extended vew are also llustrated n development (run-off) trapezods, see Fgure 2.2. Notatonal Conventon: Unless otherwse ndcated we wor n ths thess wthn the extended vew,.e. we assume that a set of M 1 clam nformaton (sources of nformaton) s avalable today,.e at tme I. In ths extended vew all clam nformaton m M generate loss lablty cash flows and hence have to be predcted, whereas clam nformaton m M c are used only for the predcton of clam nformaton m M. 2.3 Predcton Problem As mentoned n the prevous secton nsurance companes often have varous sources of nformaton (clam nformaton) for the predcton of future loss labltes cash flows S, m wth m M. We wor n the extended vew,.e. we assume that M 1 clam nformaton m {1,..., M} (as mentoned above we dentfy m by ts correspondng clam nformaton S, m ) are avalable today (at tme I). The set of clam nformaton generatng cash flows s denoted by M {1,..., M}. A reservng actuary has to predct today (at tme I) and at all future tmes up to the fnal run-off,.e. at tmes n {I,..., I + J 1}, the outstandng loss lablty cash flows. These are gven for clam nformaton m M and accdent year {I J + 1,..., I} at tme n {I,..., I + J 1} by (an empty sum s defned by zero) R m n := J j=n +1 S m,j. By summaton of (2.4a) over all clam nformaton of nterest,.e. aggregated outstandng loss labltes of accdent year gven by R n := m M R m n = m M J j=n +1 S m,j (2.4a) m M, we obtan the (2.4b)

21 2.3 Predcton Problem 11 and the aggregated outstandng loss labltes for several accdent years are gven by I I J R n := R n = =n J+1 =n J+1 m M j=n +1 S m,j. (2.4c) It manly depends on the stuaton whch of the quanttes n (2.4a) (2.4c) s of nterest. In an accountng vew the actuary often consders at tme n {I,..., I + J 1} the aggregated outstandng loss lablty cash flows R n gven by (2.4c). However, n some stuatons a more detaled analyss at tme n of the outstandng loss labltes of a specfc clam nformaton m M and a certan accdent year n (2.4a) s requred. One such stuaton s that most classcal clams reservng model framewors consder clams payments (n these models only clams payments are consdered and hence we spea about clams payments nstead of loss lablty cash flows) on the level of each ndvdual clam nformaton of a specfc accdent year (cf. Wüthrch Merz [63]). The aggregated clams payments R n are then derved by aggregaton over dfferent ndvdual clam nformaton as n (2.4c). In order to buld up suffcent reserves for outstandng loss labltes an nsurance company s oblged to predct precsely all outstandng loss labltes, based on nformaton D n n (2.2) avalable at tme n of predcton. In most well-nown clams reservng methods predctors for the ncremental clam nformaton S, m for + > n and m M are derved. Ths s descrbed for some well-nown clams reservng methods n Chapter 3 and n a more general model framewor n Chapter 4. Throughout ths thess we wll denote those predctors for S, m based on the data D n at tme n by Ŝm n. At tme n {I,..., I + J 1} the outstandng loss labltes n (2.4a), (2.4c) consst of (sums of) ncremental clam nformaton S, m wth + > n and m M. Therefore, the predcton of these loss labltes s equvalent to the predcton of ncremental clam nformaton S, m for + > n and m M. For the resultng predctors for Rm n, R n and R n based on the data D n we then obtan R m n := R n := m M J j=n +1 R m n Ŝ m n,j, (2.5a) = m M J j=n +1 Ŝ m n,j and the predctor for aggregated outstandng loss labltes for all accdent years s gven by (2.5b) R n := I =n J+1 R n = I J =n J+1 m M j=n +1 Ŝ m n,j. (2.5c) So far, we do not state any requrements w.r.t. propertes of the predctors of loss labltes Ŝm n, and hence for (2.5a) (2.5c), except that the predctor Ŝm n, at tme n must be D n -measurable. In Chapter 7 we state a regulatory requrement for these predctors to be so-called best-estmate valuaton of labltes (BEL).

22 12 2 Reservng Problem Remars 2.2 (Dscountng) In the defnton of outstandng loss labltes at tme n {I,..., I + J 1} n (2.4a) (2.4c) loss labltes S, m occurrng at dfferent tmes + > n are aggregated. In these aggregatons S, m are not weghted by a dscount factor and hence tme value of money (dscountng) s not ncorporated. Ths shows that we wor on a nomnal scale as almost all classcal clams reservng methods. The problem of the ncorporaton of stochastc dscountng n clams reservng methods s an mportant topc n recent actuaral research and leads to the concept of maret-consstent valuaton va valuaton portfolos. Snce a detaled dscusson s beyond the scope of ths thess we wll not dscuss ths further here and refer to Wüthrch Merz [62]. Concludng at tme n = I, an reservng actuary calculates wthn a certan model framewor predctors Ŝm I, for loss labltes S, m wth m M and + > I. Ths leads to the predctors of outstandng loss labltes gven by (2.5a) (2.5c). 2.4 Inflaton For the dscusson on nflaton we partly follow Taylor [57] and Wüthrch Merz [63]. Inflaton n clams reservng has not been often dscussed n classcal clams reservng lterature. For the tme beng we assume that each clam nformaton m M corresponds to clams payments of a specfc LoB. Most clams reservng methods are based on the assumpton that the observed outcome of the clam settlement process of clam nformaton out of M n the past plus other addtonal sources of nformaton of clam nformaton out of M c can be used to predct future outcomes of a clam nformaton m M. Therefore, we have to dfferentate between the development of the clam settlement process tself and the nflaton nose whch overlays the clam settlement process. The crucal pont s that dependng on the clam nformaton m M under consderaton the development of clam costs may vary over tme. The clams payments S m, n accountng year + = n and m M and ts development over tme may therefore be affected not only by the pure severty and other characterstcs of the clam but also by clams nflaton. In general, ths clams nflaton does not concde wth (but may be effected by) the classcal nflaton. Moreover, the mpact and the severty of clams nflaton may dffer n each specfc LoB under consderaton. Therefore, for each LoB we try to exclude the nflaton from the clams payments S m, at tme + = n. Let λm (n) be an nflaton ndex that measures clams nflaton of LoB (clam nformaton) m M at tme n relatve to tme 0. Then the ndexed clams payments S m,nd, are gven by S m,nd, := 1 ϕ m I ϕ m n S, m, for + = n, (2.6)

23 2.4 Inflaton 13 where ϕ m n := (λ m (n)) 1. Note that ϕ m n play the role of stochastc deflators as dscussed n Wüthrch Merz [62]. The ndexed payments S m,nd, n (2.6) should be the bass for clams reservng modelng, snce they contan the pure nformaton of the clams payments development wthout the nflaton nose. However, t s dffcult n practce to model such an nflaton process, because λ m (n) s not drectly observable. Moreover, sgnfcant changes n λ m (n) are manly caused by new developments, nnovatons n certan ndustres, for example n health care, and also by common nflaton. Thus, t s dffcult to calbrate a tme seres model for the clams nflaton rate by data of the past. We propose two strateges: 1. In the clams reservng model non-ndexed clams payments S, m are consdered. Ths s an acceptable assumpton as long as there s no perod of hgh clams nflaton followed by a perod of low clams nflaton or vce versa,.e. as long as there s no regme swtch n the clams nflaton process. 2. All observatons n D I are adjusted at tme I by the observed (clams) nflaton rate and nflaton-adjusted clams payments S m,nd, are modeled, leadng to the predctor of nflaton-adjusted clams payments Ŝm,nd I,. In ths case, the nflaton ndex λ m (n) s modeled ndependently from the clams payments leadng to a predctor of the nflaton ndex λ m (n) for n > I. The predcted values of the nflaton-adjusted clams payments Ŝ m,nd I, and the nflaton ndex λ m (n) are then combned to the predctor of clams payments Ŝ m I, := 1 ϕ m n ϕ m I Ŝ m,nd I, for + = n > I, (2.7) where λ m (ϕ m n ) 1 (n) for n > I :=. (2.8) λ m (n) for n I As mentoned above, t s dffcult n general to predct the clams nflaton process λ m (n), snce changes n ths process are manly caused by exogenous shocs. Hence, t s dffcult to calculate (2.8) and (2.7). Therefore, Strategy 1. s preferred and non-ndexed clams payments are modeled. The restrcton n the begnnng of ths secton that all clam nformaton m M correspond to clams payments of a specfc LoB can be removed, snce the arguments above hold true not only for clams payments, but also for nformaton m M. Thus, we model throughout ths thess non (clams) nflaton-adjusted quanttes. Ths s n lne wth almost all classcal clams reservng methods.

24 14 2 Reservng Problem 2.5 Predcton Precson As outlned n Secton 2.3, at tme n {I,..., I + J 1} a reservng actuary has to calculate R n to meet outstandng loss labltes n the run-off portfolos. Dependng on the data sources avalable, see Secton 2.2 for an overvew of possble sources of nformaton, and the structure of the data the actuary sets up a model framewor, see Chapters 4 6. The model s then calbrated to the data and the outstandng loss labltes are predcted n ths model framewor. Ths leads to (model based) reserves. Of course, there s a rs that the actual outcome of loss labltes R n sgnfcantly devates from the predcton R n. Ths may have the followng reasons: 1. Model msspecfcaton: The chosen model does not descrbe the stochastc dynamcs of the loss lablty process approprately 2. Parameter uncertanty: Wthn a gven model framewor unnown model parameters are replaced by estmates. These estmates may devate from the true values, due to randomness n the parameter estmaton 3. Process varance of the stochastc (random) process of loss labltes: Even f we assume that we have chosen the rght model and model parameters the realzaton of the stochastc process of loss labltes may be far from a typcal realzaton (the mean) by pure randomness The approprateness of the model under consderaton s to be verfed before usng the model. Ths can be done n some cases by statstcal methods smlar to Chapter 11 n Wüthrch Merz [63]. Of course, more sophstcated models requre other strateges for verfyng model assumptons. Statstcal tests have to be deduced n each ndvdual model under consderaton. Ths s not well developed so far n actuaral scence and should be subject to further research. Havng chosen a model framewor, one s nterested n the quantfcaton of the predcton uncertanty. However, for ths we must fnd an agreement n what sense the dstance between the predcton and the actual outcomes should be measured. For that reason we have to choose an approprate rs measure whch determnes a concepton of measurng the qualty of predcton. There s a large range of reasonable rs measures (cf. Artzner et al. [7]) whch could be used to quantfy predcton uncertanty. The choce of a sensble rs measure s not a pure mathematcal ssue and t manly depends on the applcaton at hand whch rs measure s most approprate. In actuaral tradton the most mportant rs measure s the (condtonal) mean squared error of predcton (MSEP). However, the (condtonal) MSEP has some conceptonal weanesses, see Remars 2.4. Therefore, the MSEP s supplemented by other rs measures le Value-at-Rs (VaR) or Expected Shortfall (ES) n recent regulatory solvency framewors le Solvency II and SST, see European Commsson [23], FOPI [24] and FOPI [25]. For a proper

25 2.5 Predcton Precson 15 defnton of VaR and ES, see Defntons 7.7 and 7.8. The ssue of the ncorporaton of VaR and ES n solvency consderatons s dscussed n Chapter Mean Squared Error of Predcton As already stated above the most popular rs measure n actuaral scence s the (condtonal) MSEP. Defnton 2.3 (MSEP) For a square ntegrable random varable X and a D I -measurable predctor X the condtonal MSEP s defned by msep X D I [ X] ( := E[ X X ) ] 2 D I. Remars 2.4 (MSEP) ) The (condtonal) MSEP s very popular n statstcs and actuaral scence, snce t corresponds to the squared norm of the Hlbert space of square ntegrable random varables L 2 wth respect to P( D I ). Ths allows for usng basc Hlbert space theory (cf. Kolmogorov Fomn [36]) what maes many calculatons easer to handle. ) In clams reservng practce one s bascally nterested n the shortfall rs,.e. n the rs of not havng adequate reserves to meet loss labltes. The MSEP uses a quadratc loss functon and therefore does not reflect ths rs potental, because upsde as well as downsde devatons are taen n the same way nto account. ) Replacng the MSEP by another more reasonable rs measure requres completely new models n clams reservng wth much stronger model assumptons. Moreover, analytcal closed form results would mostly be nfeasble, because other rs measures are often much harder to handle. Instead smulaton methods such as Marov-Chan-Monte-Carlo (MCMC) have to be used n those cases (cf. Scolln [55]). The (condtonal) MSEP has the useful property that t can be decomposed nto [ msep X D I X] = Var [ X D I] ( [ + X E X D I ]) 2. (2.9) }{{}}{{} process varance estmaton error Ths decomposton s a central technque to derve estmates for the MSEP of the outstandng loss labltes n varous clams reservng methods, see Mac [38], Wüthrch Merz [63], Dahms [17] and Dahms Happ [15]. Unless otherwse ndcated we wll use the (condtonal) MSEP as an optmalty crteron (rs measure) and the term best means wth the smallest (condtonal) MSEP.

26 16 2 Reservng Problem In the context of our clams reservng problem at tme I the (condtonal) MSEP of the aggregated outstandng loss labltes R I n (2.4c) s gven by [ ] I J ( ) 2 msep R I D RI I = E S,j m Ŝm I,j =I J+1 m M j=i +1 DI. (2.10) The condtonal MSEP (2.10) measures the mean quadratc devaton of the aggregated lablty predctors Ŝm I,j and the actual loss lablty outcomes S,j m up to the fnal settlement of the run-off n development year J. The condtonal MSEP n (2.10) s called long term or ultmate clam vew of the predcton uncertanty. In new solvency regulaton framewors such as Solvency II and SST the so-called one-year vew s of central nterest, see European Commsson [23] and FOPI [24], whch s qute dfferent from the ultmate clam vew. Ths short term vew focuses on the changes n the loss lablty predctons,.e. the change of the predcton n an one-year horzon (from tme I to tme I + 1). The stochastc quantty whch descrbes these changes n an one-year horzon s the so-called clams development result (CDR). 2.6 Clams Development Result We recaptulate the predcton problem of a reservng actuary at tmes I and I + 1. Accountng year I: The nformaton D I s avalable. Based on ths nformaton the actuary determnes the (model dependent) predctor of aggregated outstandng loss labltes Accountng year I + 1: All loss labltes R I = I J =I J+1 m M j=i +1 S M I+1 := m M I =I J+1 Ŝ m I,j. S m,i +1 (2.11) for accountng year I + 1 are pad out to the polcyholder (or pad to the nsurance company n the case of subrogaton payments). Snce new updated nformaton D I+1 s avalable at tme I + 1, updated predctors R I+1 are calculated based on D I+1, see Fgure 2.3. The resultng updated loss lablty predctor at tme I + 1 s then gven by R I+1 = I J =I J+2 m M j=i +2 Ŝ m I+1,j. The CDR descrbes the one-year change of predctons of aggregated outstandng loss labltes for several accdent years R I n the tme step from accountng year I to I + 1, adjusted by the loss lablty payments SI+1 M n (2.11) at tme I + 1:

27 2.6 Clams Development Result 17 1 development years J development years J accdent years... D I... D I+1 I... to be predcted I... to be predcted Fgure 2.3: Data set D I observable at tme I and data set D I+1 observable at tme I + 1 Defnton 2.5 (Clams Development Result) The clams development result (CDR) at tme I + 1 of the predctor R I of aggregated outstandng loss labltes for several accdent years s defned by CDR M,I+1 := R I ( RI+1 + S M I+1 ). (2.12) In many clams reservng methods the CDR s often consdered on the level of sngle accdent years {I J + 1,..., I} and clam nformaton m M. The CDR at tme I + 1 for the predctor of outstandng loss labltes gven by CDR m,i+1 := R m I R m I of accdent year and clam nformaton m M s ( ) Rm I+1 + S,I +1 m. Moreover, the CDR at tme I + 1 for the predctor of aggregated loss labltes R I accdent years {I J + 1,..., I} s defned by CDR M,I+1 := m M CDR m,i+1 = R ( ) I RI+1 + S,I +1 M, of sngle wth the aggregated loss labltes of accdent year and development year S M, := m M S m,. By Defnton 2.5 the clams development result CDR M,I+1 exactly corresponds to the change between ) the predctor R I at tme I and ) the predctor R I+1 at tme I + 1 plus the loss labltes SI+1 M pad out at tme I+1, see (2.12). A negatve clams development result CDRM,I+1

28 Reservng Problem reserves at tme I RI reserves+loss lablty at tme I + 1 RI+1 + S M I+1 CDR M,I+1 tme I I+1 Fgure 2.4: Reserves R I based on D I at tme I, updated reserves R I+1 based on D I+1 at tme I + 1 and the resultng clams development result CDR M,I+1 results n a loss n the poston loss experence pror accdent years on the balance sheet of an nsurance company, whereas a postve one leads to a proft n ths poston (cf. Merz Wüthrch [45]). Hence the CDR drectly effects the proft and loss statement n the balance sheet of an nsurance company. Ths reveals the drect ln of the clams development result CDR M,I+1 to re-adjustments of the predctor R I+1 n accountng year I + 1, see Fgure 2.4. We analyze the CDR n more detal. Propertes of the clams development result CDR M,I+1 In accountng year I the best D I -measurable estmator for S, [S m s gven by E, m D ]. I If for [ the predctor holds Ŝm I, = E S, m D ], I the lnearty and the tower property of condtonal expectatons (cf. Wllams [59]) mply for the expected clams development result CDR M,I+1 at tme I ) D I ] E [ CDR M,I+1 D I] ( = E[ RI RI+1 + SI+1 M I J = E = E = =I J+1 m M j=i +1 I J =I J+1 m M j=i +1 I =I J+1 m M j=i +1 = 0. J ( E Ŝ m I,j Ŝ m I,j [Ŝm I,j I J =I J+2 m M j=i +2 I J,j + SI+1 M Ŝ m I+1 Ŝ m I+1,j =I J+1 m M j=i +1 D I] E [Ŝm I+1,j D I]) The nterpretaton of ths result s as follows: Assume that the data generatng process of loss labltes s gven by the model used by the reservng actuary for clams reservng and the DI DI

29 2.6 Clams Development Result 19 [ model allows for the calculaton of E S m, D ]. I Then the predcton at tme I of aggregated outstandng loss labltes of several accdent years R I equals n the average the sum of R I+1 and S M I+1, vewed from tme I. That means that the amount of R I s such that, vewed from tme I, n the average the predcton R I+1 as well as the loss lablty payments S M I+1 can be fnanced by R I. Ths property s often called self-fnancng property. In most (classcal) dstrbuton-free clams [ ] reservng methods estmates of E D I are used as a predctor for outstandng loss labltes, S m, see the chan ladder (CL) method n Mac [38] or the lnear stochastc reservng methods (LSRMs) n Dahms [17] among others. Ths motvates the fact that the clams development result CDR M,I+1 s often predcted by 0 n the proft and loss statement of the balance sheet n a non-lfe nsurance company. Smlar to the quantfcaton of the predcton uncertanty of the aggregated outstandng loss labltes n terms of the (condtonal) MSEP, see (2.10), we measure the predcton uncertanty of the CDR by means of the (condtonal) MSEP gven by (( msep CDR M,I+1 [0] := E[ CDR M,I+1) ) ] 2 0 D I. (2.13) D I Sometmes the (condtonal) MSEP for the clams development result CDR M,I+1 accdent years {I J + 1,..., J} s consdered msep CDR M,I+1 [0] := E D I [ (( CDR M,I+1 ) ) ] 2 0 D I. for sngle For more nformaton on the CDR see Ohlsson Lauzenngs [49]. Remars 2.6 (CDR) ) The CDR s the rs drver n the one-year reservng rs (for the mult-year reservng rs, see Ders Lnde [20]). Therefore, the CDR s the central quantty n current regulatory solvency framewors, see Chapter 7 for detals. ) Regulatory solvency rules am to protect aganst shortfalls n the CDR, see European Unon [23] or FOPI [24]. In these rules the (condtonal) MSEP (2.13) s utlzed to calbrate a log-normal dstrbuton by the method of moments, see Chapter 7. Therefore, t s questonable, whether the choce of the MSEP as a rs measure s approprate, snce many dstrbutonal propertes can not be captured by the MSEP. Rs Characterstcs n Classcal Clams Reservng Methods In ths chapter we ntroduced predctors for aggregated outstandng loss labltes for several accdent years R I and the clams development result CDR M,I+1 as the central stochastc quanttes under consderaton for clams reservng at tme I (today). In a frst step a predctor