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

30 20 2 Reservng Problem R I for aggregated outstandng loss labltes R I s calculated. In classcal clams reservng the predcton uncertanty of the predctor R I for the outstandng loss labltes R I, see (2.5c), as well as the predctor 0 for the CDR M,I+1, see (2.13), s usually measured by the (condtonal) MSEP, see Defnton 2.3. The CDR became the central quantty n recent solvency framewors, see FOPI [24], snce t reflects the adjustments n the predctons of outstandng loss labltes that wll (possbly) be necessary n the tme step from I to I + 1. As motvated n the last secton the clams development result CDR M,I+1 s predcted by 0 at tme I,.e. no adjustment at tme I + 1 are to be expected. As a concluson, we note that n classcal clams reservng the aggregated outstandng loss labltes R I and the clams development result CDR M,I+1 are predcted by the predctors R I and 0, respectvely. The correspondng predcton uncertanty s measured by the MSEP, see Table 2.1 for an overvew. These quanttes can be calculated (or quantty predctor predcton uncertanty [ ] R I RI msep R I D RI I CDR M,I+1 0 msep CDR M,I+1 [0] D I Table 2.1: Classcal rs characterstcs: Reserves and CDR and the correspondng frst two moments estmated) for many clams reservng methods. Especally the clams reservng methods whch belong to the class of LSRMs or Bayesan LSRMs allow for the dervaton of these quanttes, see Chapter 4. In recent regulatory solvency framewors more sophstcated rs measures le hgher moments or quantle based rs measures such as VaR or ES are requred. The calculaton of such rs measures overcharges the possbltes of dstrbuton-free clams reservng methods. They are only accessble under smplfcatons or approxmatons, whch are subject to Chapter 7. These dfferent accessblty levels of clams reservng methods w.r.t. the MSEP (basc rs measure) and VaR or ES (more sophstcated rs measures) are mportant n new solvency framewors, see European Commsson [23] and Fop [24]. Therefore, we dvde the set of stochastc clams reservng methods nto two groups: 1. (Bayesan) dstrbuton-free clams reservng methods: (Bayesan) dstrbuton-free clams reservng methods comprse many classcal clams reservng methods used n actuaral practce. We present some mportant representatves of ths class n Chapter 3. These methods can be essentally summarzed n the wde class of (Bayesan) LSRMs ntroduced n Dahms [17] and Dahms Happ [15]. Ths unfcaton and generalzaton of dstrbuton-free clams reservng methods s subject to Chapter 4.

31 2.6 Clams Development Result 21 In ths class only the rs characterstcs gven n Table 2.1 can be derved. More sophstcated rs measures are accessble n ths model class only under smplfcatons and approxmatons, see Smplfcatons I II n Chapter Dstrbutonal methods: In actuaral scence varous dstrbutonal clams reservng methods have been ntroduced. Among others generalzed lnear model (GLM) technques were used n England Verrall [21] and [22], Haberman Renshaw [29], Taylor McGure [58] and Ala et al. [3], generalzed lnear mxed models (GLMM) were appled n Antono Berlant [5], Frees Sh [56] and De Jong [19] used copula based models and dstrbutonal Bayesan models were appled n Salzmann Wüthrch [54], Wüthrch [60], Merz Wüthrch [46], Happ Wüthrch [31] and others. These methods often allow for the dervaton of the whole predctve dstrbuton of outstandng loss labltes and not only of the frst two (condtonal) moments gven n Table 2.1. In Chapter 5 we hghlght one mportant representatve of the class of dstrbutonal methods, namely the pad-ncurred chan (PIC) reservng method by Merz Wüthrch [46]. Ths dstrbutonal clams reservng method combnes n an elegant way clams payments and ncurred losses data. Followng Happ et al. [15] we recaptulate the PIC reservng method and show how the MSEP of the CDR can be derved analytcally. Moreover, we derve the whole predctve dstrbuton of the CDR va Monte-Carlo (MC) smulatons. A generalzaton of the PIC model whch allows to model dependence structures n the data presented n Happ Wüthrch [30] s subject to Chapter 6.

32

33 3 Classcal Dstrbuton-Free Clams Reservng Methods In ths chapter we present classcal dstrbuton-free clams reservng methods commonly used n actuaral practce. We state model assumptons underlyng each method and present predctors R I for outstandng aggregated loss labltes (clams payments) R I. Predcton uncertanty s not consdered n ths chapter. Ths has the followng reason: As wll be shown n Chapter 4 almost all classcal dstrbuton-free clams reservng methods can be embedded n the general (Bayesan) LSRM framewor. For ths model class the (condtonal) MSEP of aggregated outstandng loss labltes for several accdent years R I and the CDR, see Table 2.1, are derved n Dahms [17] for LSRMs and n Chapter 4 for Bayesan LSRMs. In the sequel of each method presented we state some remars on advantages and dsadvantages and pont out to what extend the dsadvantages could be tacled n state-of-the-art clams reservng methods. 3.1 General Notaton In ths chapter we wor under the extended vew gven n Chapter 2. For the general formulaton of the stochastc dynamcs of classcal dstrbuton-free clams reservng methods we defne the lnear subspaces L n and L denotng the lnear spaces generated by all ncrements S,j m up to accountng year n and development year, respectvely. Furthermore, the lnear subspace generated by L n and L s denoted by L n,.e. M L n := m=1 M L := m=1 M L n := I =0 I m=1 =0 (n ) J =0 j=0 I x m,js,j m : x m,j R, j=0 x m,js,j m : x m,j R, ((n ) J) j=0 x m,js,j m : x m,j R, 23

34 24 3 Classcal Dstrbuton-Free Clams Reservng Methods 1 development year 0 J 0 D n B, accdent year n I D I accountng year Fgure 3.1: 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 where a b and a b denote the mnmum and maxmum of the real numbers a and b, respectvely. Moreover, the correspondng σ-felds (sets of observatons) are defned by B, := σ ( S,j m : 1 m M, 0 j ) ( I ), D := σ (L ) = σ B,, ( I ) ( I ) D n := σ (L n ) = σ B,(n ) J, D n := σ (Ln ) = σ B,((n ) J) =0 =0 and llustrated n Fgure 3.1. =0 3.2 Chan Ladder Method The CL method s the most popular and wdest used clams reservng method. The orgnal verson of the CL method was a purely determnstc procedure for calculatng reserves wthout consderng the clams reservng problem n a stochastc framewor. In 1993, Mac [38] was the frst to formulate a dstrbuton-free stochastc framewor where the reserves resultng from the orgnal determnstc method are gven a meanngful stochastc foundaton. Wthn the stochastc framewor, Mac [38] was able to present estmates for the predcton uncertanty n terms of the (condtonal) MSEP. The classcal model assumptons of Mac [38] for the CL

35 3.2 Chan Ladder Method 25 method are formulated n the cumulatve clams payments representaton,.e. we consder C, m nstead of S, m, and the method uses ts own past clams settlement process as the only source of nformaton. That means n the CL method we consder one cumulatve clam nformaton C,,.e. M = 1, M = {1} (the superscrpt 1 n the exponent wll be omtted, because there s only one clam nformaton). Model Assumptons 3.1 (Dstrbuton-free CL model of Mac) There exst constant factors g 0,..., g and varance parameters σ0 2,..., σ2 > 0 such that a) Cumulatve clams payments {C, 0 J} from dfferent accdent years {0,..., I} are ndependent. b) For all 0 I and 0 J 1 holds E[C,+1 B, ] = E[C,+1 C, ] = g C,, Var[C,+1 B, ] = Var[C,+1 C, ] = σ 2 C,. The factors g are called development or age-to-age factors, because they descrbe the expected one step transton of (C, ) 0. The condtonal expectaton E [ C,+1 D I] s the best D I - measurable predctor for C,+1 ( {0,..., J 1}), see Wllams [59, p. 86], page 86. In accountng year I (.e. gven D I ) the condtonal expected cumulatve clams payments for accdent year I J + 1 I up to development year I < + 1 J are gven under Model Assumptons 3.1 by E [ C,+1 D I] = C,I j=i see Wüthrch Merz [63] for a proof. For the ultmate clam,.e. = J 1, we obtan E [ C,J D I] = C,I j=i In practce the development factors {g 0 J 1} are unnown and have to be estmated from the data D I avalable n accountng year I. A (condtonal) unbased estmator wth mnmum (condtonal) varance n the class of lnear unbased estmators s gven by ĝ I,CL := I 1 C,+1 =0 I 1 =0 g j, g j. C,, {0,..., J 1}, (3.1)

36 26 3 Classcal Dstrbuton-Free Clams Reservng Methods see Wüthrch Merz [63] for a proof. The clam predctors are then gven by Ĉ I,CL,+1 := C,I j=i ĝ I,CL j. (3.2) For these predctors n (3.2) we obtan by the (condtonal) unbasedness of the estmators ĝ I,CL n (3.1) [ĈI,CL ] E,+1 C,I = E C,I ĝ I,CL j j=i C,I = E E C,I ĝ I,CL j D C,I = E C,I g. = C,I j=i j=i 1 ĝ I,CL j j=i g j = E [ C,+1 D I]. C,I Ths shows that the predctor (3.2) s condtonally gven C,I unbased for E [ C,+1 D I]. Ths motvates the explct clam predctor (3.2). Fnally, we obtan the CL predctor of aggregated outstandng loss labltes of accdent year I J + 1 at tme I (note that M = 1) R I,CL = ĈI,CL,J C,I = C,I j=i ĝ I,CL j C,I = C,I j=i j 1. (3.3) ĝ I,CL The CL predctor of aggregated loss labltes for several accdent years follows by summaton of (3.3) over accdent years {I J + 1,..., J} R I,CL := I =I J+1 R I,CL. (3.4) The predcton uncertanty n terms of the condtonal MSEP of the CL predctors R I,CL s gven under slghtly varyng approaches n Mac [38], Murphy [48] and Buchwalder et al. [10]. For the quantfcaton of the CDR uncertanty by means of the (condtonal) mean squared error of predcton msep CDR 1,I+1 [0] we refer to Merz Wüthrch [47]. D I Remars 3.2 (Dstrbuton-free CL model of Mac) ) The CL method s by far the most popular clams reservng method, snce t s easy to use, very ntutve and smple to mplement.

37 3.3 Bayes Chan Ladder Method 27 ) The CL method s based on only one source of nformaton (the settlement process tself) and does not respect other sources of nformaton avalable. In nsurance practce addtonal sources of nformaton, for example ncurred losses data, are often avalable. The problem of consderng cumulatve payments and ncurred losses smultaneously s addressed n Chapters 5 and 6. ) Accordng to (3.4) and (3.3) the reserves R I n the CL method completely rely on the last observable entres {C I J+1,,..., C I,0 }. Ths maes the method very senstve wth respect to outlers or zeros on the dagonal leadng to nonsense reserve estmates. Such scenaros are not unusual n excess-of-loss rensurance. v) The classcal CL method does not allow for the ncorporaton of expert nowledge or nformaton from ndustry-wde data n the development factors g. Ths problem s addressed n the Bayes CL method, see Secton 3.3. In Chapter 4 the general case s consdered of ncorporatng such nformaton n all models whch belong to the wde class of LSRMs. Ths ncludes the CL method and the class of Bayesan LSRMs s therefore a generalzaton of the Bayes CL method. 3.3 Bayes Chan Ladder Method Model Assumptons 3.1 requre unnown development (age-to-age) factors {g 0 J 1} whch have to be estmated approprately from the data. If there s addtonal portfolo data avalable, for nstance ndustry-wde data or expert nowledge concernng the development factors of the clams reservng method, new models are needed to cope wth those new nformaton sources. The Bayes chan ladder (Bayes CL) model n Gsler Wüthrch [27] allows for the ncorporaton of addtonal nformaton on the development factors n the CL method wthn a credblty based approach. In the Bayes CL model t s assumed that the development factors g wth 0 J 1 n the classcal CL model are random varables denoted by G wth nown (condtonal) mean and varance. The model assumptons are formulated for the development ratos Y, := C,+1 C, as follows: Model Assumptons 3.3 (Bayes CL model) a) Condtonally, gven G := (G 0,..., G ), the cumulatve clams payments {C, 0 J} from dfferent accdent years {0,..., I} are ndependent. b) Condtonally, gven G and B,, the dstrbuton of Y, depends only on C, and t holds E[Y, G, B, ] = G Var[Y, G, B, ] = σ2 (G ) C,,

38 28 3 Classcal Dstrbuton-Free Clams Reservng Methods for {0,..., I} and {0,..., J 1}. c) {G 0, G 1,..., G } are ndependent. In the Bayes CL method a credblty theory based approach s used for the predcton of the development factors G, see Bühlmann Gsler [14]. The resultng credblty predctor Ĝ I,Cred for the development factor G s then gven by a credblty weghted average of the classcal CL estmate ĝ I,CL n (3.1) and the pror mean E[G D ] where α [0, 1] s gven by Ĝ I,Cred wth the pror structural parameters := α ĝ I,CL + (1 α )g, (3.5) α := I 1 =0 I 1 =0 C,+1 C, + σ2 τ 2 g := E[G D ] σ 2 := E[ σ 2 (G ) D ], (3.6) τ 2 := Var[G D ]. (3.7) Ths leads to credblty based clam predctors for accdent year {I J + 1,..., I} and development year ( + 1) {I + 1,..., J} gven by Ĉ I,Cred,+1 := C,I j=i Ĝ I,Cred j. (3.8) Ths s exactly the predctor n (3.2) n the classcal CL model, but wth the estmate ĝ I,CL j replaced by the credblty predctor ĜI,Cred j. By Equaton (3.8) we obtan the predctor for aggregated outstandng loss labltes for accdent year {I J + 1,..., J} R I,Cred = Ĉ I,Cred,J C,I = C,I j=i j 1. (3.9) Ĝ I,Cred Fnally, by summaton of (3.9) over all accdent years {I J +1,..., I} we get the credblty based CL predctor for aggregated outstandng loss labltes gven by I I R I,Cred = R I,Cred = C,I Ĝ I,Cred j 1. =I J+1 =I J+1 j=i [ ] For the dervaton of the predcton uncertanty msep RI,Cred R I D I n terms of the (condtonal) MSEP we refer to Gsler Wüthrch [27]. For the predcton uncertanty of the CDR, see Chapter 4, where the Bayes CL method s looed at as a Bayesan LSRM. In ths general model framewor the (condtonal) MSEP of the CDR as well as the MSEP of the predctor for outstandng loss labltes are derved. R I,Cred

39 3.4 Complementary Loss Rato Method 29 Remars 3.4 (Bayes CL model) ) The credblty predctor (3.5) s a credblty weghted average of the classcal CL estmate ĝ I,CL purely based on the data and the structural parameter g. The structural parameters (3.7), whch are requred for the calculaton of the credblty weghts (3.6), can be estmated from data, see Bühlmann Gsler [14], or deduced from external sources of nformaton. ) Snce the CL method belongs to the class of LSRMs, see Chapter 4, one can as whether ths ncorporaton of pror nowledge of the development pattern by means of credblty theory does not only wor n the CL method but also for all other methods whch belong to the class of LSRMs. Ths queston s answered n Chapter 4. Moreover, estmates of [ ] (condtonal) mean squared error of predcton msep R I D RI I and msep CDR M,I+1 [0] D I are derved. 3.4 Complementary Loss Rato Method In the complementary loss rato (CLR) method, see Bühlmann [12], ncremental clams payments S, 1 are consdered as the frst clam nformaton. An external gven exposure P (e.g. earned premum, volume measure, ultmate clam predcton, number and sze of contracts, etc.) defned by S, 2 := P for = 0 0 otherwse and ndependent of tme s used as a second source of nformaton. That means n ths model there are M = 2 clam nformaton S, 1 and S2, and we are nterested n the predcton of the frst clam nformaton S, 1. The second clam nformaton S2, s used for the predcton of S1, but s not predcted tself,.e. M = {1} and M c = {2}. Model Assumptons 3.5 (CLR model) There exst constant weghts g 0,..., g and varance parameters σ0 2,..., σ2 > 0 such that a) Incremental clams payments {S, 1 0 I, 0 J} are ndependent. b) For 0 I and 0 J 1 holds E [ S,+1 1 ] B, = g P, Var [ S,+1 1 ] B, = σ 2 P.

40 30 3 Classcal Dstrbuton-Free Clams Reservng Methods In the CLR method estmates of the unnown model parameters g are gven by Ths leads to the CLR predctor ĝ I,CLR := 1 I 1 P =0 I 1 =0 P S 1,+1 P. for + I. Ŝ 1,+1 := ĝi,clr P (3.10) We then obtan for the predctor of aggregated outstandng loss labltes of accdent year {I J + 1,..., J} R I,CLR := =I and the correspondng predctor for several accdent years s gven by Ŝ 1,+1 (3.11) R I,CLR := I =I J+1 R I,CLR = I =I J+1 =I Ŝ 1,+1. (3.12) Note that f we choose n Model Assumptons 3.5 P := V where V s a volume measure we obtan the ALR method n Merz-Wüthrch [44]. Ths shows that the ALR method s a specal case of the CLR method. Moreover, n Chapter 4 we wll see that the CLR method belongs to the class of LSRMs. Therefore, estmates of the (condtonal) mean squared error of predcton [ ] msep R I D RI I and msep CDR 1,I+1 [0] for the CLR method are gven n Dahms [17]. D I 3.5 Bornhuetter Ferguson Method Besde the CL method the Bornhuetter Ferguson (BF) method presented n Bornhuetter Ferguson [8] s one of the most popular clams reservng methods. In 1972, Bornhuetter and Ferguson ntroduced the BF method n order to solve the man problem of the CL method that the reserve of accdent year {I J + 1,..., J} n (3.3) s proportonal to the last dagonal entry C,I. The central dea of the BF method s that for each accdent year the ncremental clams payments S 1, correspond to a fxed percentage y of a pror ultmate clam estmate x. We consder the BF method n ts ncremental representaton. A cumulatve verson s gven n Wüthrch Merz [63]. The ncremental clams payments S, 1 consttute the frst clam nformaton and the second clam nformaton s gven by S 2, := { x for = 0 0 otherwse,

41 3.5 Bornhuetter Ferguson Method 31 where x s a pror ultmate clam estmate. That means that n the BF method there are M = 2 clam nformaton whereas the set of clam nformaton generatng cash flows s gven by M = {1}. There are many slghtly varyng model assumptons underlyng the BF method. The assumptons n Mac [39] are gven by: Model Assumptons 3.6 (BF model of Mac) There exst constant weghts y 0,..., y J and varance parameters s 2 0,..., s2 J > 0 such that a) Incremental clams payments {S, 1 0 I, 0 J} are ndependent. b) For 0 I and 0 J holds E [ S 1,] = y x and y y J = 1 Var [ S 1,] = s 2 x. By settng g := y +1, P := x and σ 2 := s2 +1 for 0 J 1 we obtan for the BF model E [ S,+1 1 ] [ ] B, = E S 1,+1 = y+1 x = g P, Var [ S,+1 1 ] [ ] B, = Var S 1,+1 = s 2 +1 x = σ 2 P. Ths shows that the BF method can be looed at as a specal case of the CLR method. Therefore, the predctor of outstandng loss labltes of accdent year {I J+1,..., I} n the BF method s gven by (3.11), wth ĝ and P replaced by ŷ +1 and x, respectvely. For the dscusson of the assocated predcton uncertanty n the BF method we refer to the dscusson of the predcton uncertanty n the CLR method gven above. As already mentoned above there are varous versons of the BF model wth slghtly varyng model assumptons. For the correspondng reserves and predcton uncertanty n these cases we refer to Wüthrch Merz [63], Ala et al. [3], Ala et al. [4] and Saluz et al. [52]. Remars 3.7 (CLR and BF method) ) In the CLR method and therefore n the BF method too the second clam nformaton S, 2 s used for the predcton of the clams payments process S, 1, see Model Assumptons 3.5 and 3.6. Ths s the classcal case, where S, 2 s used for predctng S1, but not predcted tself,.e. M = 2 and M = {1}. ) The BF method requres and allows for a pror estmate of the ultmate clam x for each accdent year. These pror estmates are often deduced from prcng arguments. In the CLR method a pror exposure P s requred.

42 32 3 Classcal Dstrbuton-Free Clams Reservng Methods ) The CLR and BF methods solve the basc problem of the CL method that the reserves strongly depend on the observatons on the last observed dagonal. Ths becomes obvous by comparng the CL reserves n (3.3) and the CLR and BF reserves n (3.11). v) The classcal BF method does not allow for ncludng other sources of nformaton, for example ncurred losses data, and s therefore to some extend nflexble n nsurance practce. Ths problem s addressed n Chapters 5 and 6. v) Pror expert nowledge or ndustry wde data can not be ncluded n the estmaton of the parameters y n the BF method and g n the CLR method. Ths tas s consdered for the wde class of LSRMs n Chapter 4. All so far presented clams reservng methods (and much more) belong to the class of (Bayesan) LSRMs. Therefore, best-estmate predctors for outstandng loss labltes, ther predcton uncertanty and the one-year reservng rs can be calculated n the general framewor of (Bayesan) LSRMs n Dahms [17] and Dahms Happ [15] presented n Chapter 4. A dstrbuton-free clams reservng method whch does not ft nto the LSRM model framewor s brefly presented n the followng secton. 3.6 Munch Chan Ladder Method Besde the clams payments data, ncurred losses as a second data source are often avalable n nsurance companes. Applyng the CL method, see Secton 3.2, to the clams payments data leads to CL reserves based on clams payments data. On the other sde, f one apples the CL method to the ncurred losses data we obtan CL reserves based on ncurred losses data. However, ths strategy leads to dfferng ultmate clam predctons whch generally do not concde and there remans a gap between the predcton based on the cumulatve payments and the predcton based on ncurred losses. Ths gap s reduced by applyng the Munch chan ladder (MCL) method ntroduced by Quarg Mac [50] n However, ths method does not completely close the gap between the predctons. Let C, 1 be the cumulatve clams payments and C, 2 the cumulatve ncurred losses of accdent year {0,..., I} and development year {0,..., J},.e. M = 2 and M = {1}. As shown n Merz Wüthrch [43] the model assumptons for the MCL method can be formulated as follows: Model Assumptons 3.8 (MCL model) a) The sets {C,j 1, C2,j 0 j J} are ndependent for dfferent accdent years {0,..., I}. b) There exst g m 0,..., gm > 0 and σm 0,..., σm > 0 for m {1, 2} such that for all

43 3.6 Munch Chan Ladder Method 33 0 I and 0 J 1 holds E [ C,+1 1 ] C = g 1 C, 1 E [ C,+1 2 ] I = g 2 C, 2 Var [ C,+1 1 ] ( ) C = σ 1 2C 1, Var [ C,+1 2 ] ( ) I = σ 2 2C 2, wth C := σ(c 1,j : 0 I, 0 j ) I := σ(c 2,j : 0 I, 0 j ). c) There are constants λ 1, λ 2 R such that [ ] [ ] [ ] C C 1,+1 C 1 1/2, 2 C E 2, E C, 1 C, I = g 1 +,+1 λ1 Var C, 1 C C, 1 C, 1 C [ ] C, Var 2 1/2 C C 1, and [ ] [ ] [ ] C C 2,+1 C 2 1/2, 1 C E 1, E C, 2 C, I = g 2 +,+1 λ2 Var C, 2 I C, 2 C, 2 I [ ] C, Var 1 1/2 I C 2, for 0 I and 0 J 1. Remars 3.9 (MCL method) ) To the best of our nowledge, estmates for the predcton uncertanty as well as for the one-year reservng rs CDR n terms of the (condtonal) MSEP could not be derved, yet. ) On the contrary to the MCL method, the extended complementary loss rato (ECLR) method by Dahms [16] provdes one unfed predctor for outstandng clams payments based on the clams payments and ncurred losses data smultaneously. The correspondng predcton uncertanty and the one-year reservng rs s derved n Dahms et al. [18]. ) Another clams reservng method beng able to ncorporate cumulatve clams payments and ncurred losses data leadng to one unfed ultmate clam predcton s the PIC reservng method by Merz Wüthrch [46]. In the PIC reservng method unfed ultmate clam predctors as well as the predcton uncertanty n terms of the (condtonal) MSEP are derved. Moreover, the MSEP of the one-year reservng rs CDR and the whole predctve dstrbuton of the CDR can be derved, see Chapter 5.

44 34 3 Classcal Dstrbuton-Free Clams Reservng Methods Summary In ths chapter we gave a bref ntroducton n classcal dstrbuton-free clams reservng methods often used n actuaral practce. The CL method can cope wth only one source of nformaton (.e. the clams payments process tself). The Bayes CL method addtonally ncludes pror nowledge of the development factors usng credblty theory. However, the man problem of the CL method remans that predctors for the outstandng loss labltes are very senstve w.r.t. outlers on the last observed dagonal, see (3.3) and (3.8). Thus, we proceeded wth the CLR method. Ths method allows to ncorporate pror external nowledge, for example ultmate clam estmates, as a second source of nformaton and s more robust than the CL method w.r.t. outlers on the last observed dagonal. We saw that the well-nown BF method s a specal case of the CLR method. As shown n Dahms [17] all these methods can be embedded nto the class of LSRMs, see Chapter 4 for detals. Ths new general class of reservng methods provdes a very flexble and powerful framewor for dstrbuton-free clams reservng modelng. Many dfferent sources of nformaton can be ncluded for the predcton of outstandng loss labltes. However, clams reservng methods whch belong to the class of LSRMs do not allow for the ncorporaton of pror expert nowledge or ndustry wde data n the clams reservng process. In Gsler Wüthrch [27] ths problem s solved for the classcal CL method resultng n the Bayes CL method. We consder ths problem for the whole class of LSRMs and generalze the approach from the Bayes CL method to all LSRMs. Ths leads to the new class of Bayesan LSRMs presented n Dahms Happ [15]. Ths very general and powerful class of reservng methods s subject to Chapter 4.

45 4 (Bayesan) Lnear Stochastc Reservng Methods 4.1 Lnear Stochastc Reservng Methods As outlned n the prevous chapter, many classcal clams reservng methods are formulated n dstrbuton-free model framewors usng varous sources of nformaton for the descrpton of the stochastc dynamcs of the underlyng model. The most popular representatves among them are the well-nown CL, BF and CLR methods. In actuaral scence t was not completely understood what all these dstrbuton-free clams reservng methods have n common. In 2012, Dahms [17] ntroduced the class of lnear stochastc reservng methods (LSRMs) and ponted out that many of the well-nown dstrbuton-free clams reservng methods can be looed at as LSRMs. Among them are the CL, BF, and CLR methods presented n Chapter 3, but also the hybrd chan ladder (HCL) method by Arbenz Salzmann [6] and the ECLR method by Dahms [16]. Ths means that from a mathematcal pont of vew LSRMs are a state-of-the-art generalzaton of the models mentoned above. Ths provdes a complete new vewng angle on the large class of dstrbuton-free clams reservng methods. Benefts for practtoners from ths generalzaton are lsted n Conclusons 4.3. Notatonal conventon: For reasons of notatonal consstency wth Dahms [17] the countng for m starts wth 0, M 0 and the number of clam nformaton s M + 1 n ths chapter. In ths secton we follow Dahms [17]. In the LSRM framewor ncremental clam nformaton S m, are consdered for m {0,..., M}, {0,..., I} and {0,..., J},.e. we wor under the extended vew. Model Assumptons 4.1 (LSRM) a) There exst f m R such that for all, m and the expectaton of the ncremental clam nformaton S,+1 m under the condton of all nformaton of ts past D+ s proportonal to an exposure R, m contaned n L+ L,.e. [ E S,+1 m D + ] = f m Rm, L+ L. 35

46 36 4 (Bayesan) Lnear Stochastc Reservng Methods b) There exst σ m 1,m 2 > 0 such that for all, m 1, m 2 and the covarance of the ncremental clam nformaton S m 1,+1 and Sm 2,+1 under the condton of all nformaton of ther past D + s proportonal to an exposure R m 1,m 2, contaned n L + L,.e. [ Cov S m 1,+1, Sm 2,+1 ] D + = σ m 1,m 2 R m 1,m 2, L + L. For an llustraton of the lnear spaces (or the generated σ-felds, respectvely) used n Model Assumptons 4.1, see Fgure development year 0 J 0 D n accdent year D n n I I D accountng year Fgure 4.1: 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 Remars 4.2 (LSRM) ) Besde the clams payments process LSRMs can nclude varous sources of nformaton, for example external gven exposures le pror ultmate clam estmates (BF method) or ncurred losses as a second source of nformaton (ECLR method), see Dahms [16]. ) In Model Assumptons 4.1 a) and b) R, m and Rm 1,m 2, are assumed to be elements out of L + L. That means t s mplctly assumed that there exst (unque) exposure parameters γ m,l,,h,j R and γm 1,m 2,l,,h,j R such that R m, = M I l=1 h=0 (+ h) j=0 γ m,l,,h,j Sl h,j and R m 1,m 2, = M I l=1 h=0 (+ h) j=0 γ m 1,m 2,l,,h,j S l h,j, (4.1)

47 4.1 Lnear Stochastc Reservng Methods 37 respectvely. These exposure parameters are called LSRM defnng parameters, snce they defne the stochastc dynamcs of the LSRM. ) The LSRM defnng parameters γ m,l,,h,j and γm 1,m 2,l,,h,j n (4.1) have to be determned. There are often heurstc and/or LoB based nformaton that motvate a specfc parameter choce. An excellent example for such a scenaro s gven n Example 1 n Dahms [17]. In other cases one can use bactestng technques for verfyng whether the LSRM would have wored well n the past Classcal Clams Reservng Methods as LSRMs To get an dea of the flexblty of the class of LSRMs we frst analyze whch of the classcal clams reservng methods belong to the class of LSRMs. CL Method Model Assumptons 3.1 for the dstrbuton-free CL method are: There exst constant factors g 0,..., g and varance parameters σ0 2,..., σ2 > 0 such that a) Cumulatve clams payments {C, 0 J} from dfferent accdent years {0,..., I} are ndependent. b) For all 0 I and 0 J 1 holds E[C,+1 B, ] = E[C,+1 C, ] = g C,, Var[C,+1 B, ] = Var[C,+1 C, ] = σ 2 C,. Snce n the CL method only one source of nformaton, namely the clams payments process tself, s ncorporated we have that M = 0. The ncremental clams payments n the CL method are gven by Wth R 0, := C, = S 0,+1 := C,+1 C,. and wth Model Assumptons 3.1 a) and b) we obtan [ E S,+1 0 ] D + = E [ S,+1 0 ] B, = (g 1) } {{ } =: f 0 [ Var S,+1 0 D, ] j=0 = Var [ S,+1 0 ] B, = S 0,j =: R 0,0, L+ L (4.2) σ 2 }{{} =: σ 0,0 C, }{{} = R 0, C, }{{} = R 0,0, L + L, L + L.

48 38 4 (Bayesan) Lnear Stochastc Reservng Methods Ths shows that the CL method belongs to the class of LSRMs. CLR Method Model Assumptons 3.5 for the CLR method are gven by: a) Incremental clams payments {S, 0 0 I, 0 J} are ndependent. b) For 0 I and 0 J 1 holds E [ S,+1 0 ] B, = g P, Var [ S,+1 0 ] B, = σ 2 P. If we tae S 1, := { P for = 0 0 otherwse we see that the CLR method belongs to the class of LSRMs. Snce the BF method s a specal case of the CLR method, ths mples that the BF method also belongs to the class of LSRMs. As already mentoned above the ECLR method can also be looed at as a LSRM, see Dahms [17] Parameter Estmaton for LSRMs In the LSRMs defned above the model parameters f m and σ m 1,m 2 are unnown and have to be estmated from the data. Note that the LSRM defnng exposure parameters γ m,l,,h,j R and R n (4.1) are requred by the model to provde a well-defned LSRM, see Remars γ m 1,m 2,l,,h,j 4.2. An (D condtonal) unbased estmator for f m s gven by (we set 0 0 := 0) f m I 1 := =0 w, m S,+1 m R, m, (4.3) where w m, 0 are Dn D -measurable weghts wth ) R m, = 0 mples wm, = 0 and ) I 1 =0 w m, = 1 f at least one Rm, 0. Wth the choce of explct weghts w m, := ( ) 2 R, m R m,m, I 1 h=0 ( R m h, R m,m h, ) 2 1 (4.4)

49 4.1 Lnear Stochastc Reservng Methods 39 the estmators (4.3) have mnmum varance n the class of all lnear estmators of the form (4.3),.e. they are (homogeneous) credblty estmators, see Bühlmann Gsler [14] for a defnton. In the case of the CL method, the CLR method or the ECLR method these mnmum varance estmators are the well-nown standard estmators, see for example Mac [38] and Dahms [16]. Estmators for the second model parameter σ m 1,m 2 are not requred for the predcton of clam nformaton S, m n LSRMs, but for the quantfcaton of the predcton uncertanty. An unbased estmator for σ m 1,m 2 where wth σ m 1,m 2 := Z m 1,m 2 := s gven by I 1 1 Z m 1,m 2 =0 I 1 =0 ( R m ( 1, Rm 2, S m 1 R m 1,m 2,,+1 R m 1, 1 w m 1, wm 2, + wm 1 ( I 1 j w,j m := (Rm,j )2 R m,m,j h=0, wm 2, f m 1 R m 1,m 2, R m 1, Rm 2, ) 1 (Rh,j m )2 R m,m. h,j ) ( ) S m 2,+1 R m f m 2 2, (4.5), I 1 h=0 R m ) 1 h, Rm 2 h, R m 1,m 2 h, For a proof of the stated unbasedness of the estmators (4.3) and (4.5), see Dahms [17] Predcton of Future Clam Informaton By (4.1) t becomes evdent that R m, s a lnear combnaton of Sl h,j L+ L. Ths mples that the lnear projecton P n : L n L n+1, x (P n x) m, := where wth for + n,, 1 x for + = n + 1 x m, P m P m, : L+ R, x P m, x := f m Γm, x Γ m, : L+ R, x Γ m, x := M I l=1 h=0 (n h) () j=0 γ m,l,,h,j xl h,j generates, based on all clam nformaton n D n, the next dagonal (accountng year) n + 1 and projects all clam nformaton out of D n on ths next dagonal. The concatenaton of these lnear projectons flls up several dagonals at once and s gven by P n 2 n 1 : L n 1 L n2+1, x P n Π n 2 n 1 x := L 2 +1 x for n 2 < n 1 P n 2 P n 2 1 P n 1 x for n 2 n 1

50 40 4 (Bayesan) Lnear Stochastc Reservng Methods wth Π L n denotng the projecton on the frst n dagonals and we defne Wth ) P m n, : Ln R, x P m n m, (P x := + n x. (4.6),+1 S n := ( S, m ) 0 m M + n we obtan for 1 I and + n + 1 J the best predctor for S,+n+1 m gven by [ E S,+n+1 m ] D + = P m +,+n S+. (4.7) Replacng the unnown development factors f m n (4.6) by ther estmates f m (4.7) for I < J the LSRM predctor we obtan wth Ŝ m I m I,+1 := P, SI. (4.8) Besde the LSRM predctor (4.8) the predcton uncertanty of Ŝm I,+1 and CDRM,I+1 quantfed by means of the MSEP are of nterest. However, ths requres cumbersome notaton and long calculatons and we refer therefore to Dahms [17]. Concluson 4.3 ) In most dstrbuton-free clams reservng methods accdent year ndependence s a central model assumpton. Ths s not requred n LSRMs. ) LSRMs possess (condtonal) uncorrelated dagonals, see Lemma 2.3 c) n Dahms [17]. Ths mples that calendar year effects le nflaton whch have an mpact on the whole dagonal can not be captured by LSRMs. Ths ssue should be subject to further research. ) A regme change can be modeled by an exchange of external gven exposures leadng to a much faster calbraton of the model to the new regme than n the CL method. One example for such an exposure change s llustrated n Example 1 n Dahms [17]. In LSRMs there s no mathematcally consstent way to ncorporate pror expert nowledge or ndustry-wde data nowledge nto the development factors f m. Ths s exactly the same ssue that s consdered n the Bayes CL method n Gsler Wüthrch [27] for the CL method. We generalze ths aspect to the whole class of LSRMs what s subject to the next secton. 4.2 Bayesan Lnear Stochastc Reservng Methods In nsurance practce some LoBs show large devatons and rregulartes n the clams development. Therefore, t s dffcult for a reservng actuary to fnd relable estmates for the development factors for hs model. In classcal LSRMs the development factors f m are estmated n

51 4.2 Bayesan Lnear Stochastc Reservng Methods 41 (4.3) based on data of the trapezods only. However nformaton about typcal development factors n smlar LoBs or from ndustry-wde data could be helpful and ncluded n the estmaton of the development factors. To solve ths problem one has to as the followng queston: What can we learn from the collectve (ndustry-wde data) for the development pattern of the LoB under consderaton? Ths queston has ts orgn n credblty theory where nowledge of the collectve and ndvdual loss records are combned to mprove estmates, see Bühlmann Gsler [14]. In our reservng context we combne experence of the ndustry-wde data and the ndvdual LoB s clams record to calculate credblty development factors. Ths leads to the class of Bayesan LSRMs. In ths secton we follow Dahms-Happ [15]. In the Bayesan LSRM setup we assume that the development factor f m of clam nformaton m {0,... M} n development year {0,... J 1} s a realzaton of a random varable F m. We denote the random matrx contanng all development factors F m F := (F m )0 m M 0. (4.9) The vector collectng all development factors of development year {0,..., J 1} s denoted by by and we defne n an analog way F := (F 0,..., F M ) and Moreover, for an arbtrary mappng f := (f m )0 m M 0 R(J) (M+1) f := (f 0,..., f M ) R M+1. h a : A B, x h a (x), (for arbtrary sets A and B) dependng on a fxed parameter (vector) a we denote by the functon h, but wth a replaced by b. h a=b (x) := h b (x) For the constructon of Bayesan LSRMs we assume that condtonally, gven F, Model Assumptons 4.1 for classcal LSRMs are fulflled.

52 42 4 (Bayesan) Lnear Stochastc Reservng Methods Model Assumptons 4.4 (Bayesan LSRM) a) Condtonally, gven F = (F m)0 m M 0, for all, m and the expectaton of the ncremental clam nformaton S,+1 m under the condton of all nformaton of ts past D+ s proportonal to an exposure R, m L+ L,.e. [ E S,+1 m D + ], F = F m Rm,. b) Condtonally, gven F, the covarance of the ncremental clam nformaton S m 1 S m 2,+1 under the condton of all nformaton of ther past D+ L + L,.e. exposure R m 1,m 2, [ Cov S m 1,+1, Sm 2,+1,+1 and s proportonal to an ] D +, F = σ m 1,m 2 (F)R m 1,m 2, wth R m 1,m 2, L + L. c) For all n {I,..., I + J}, j J 1 and 0 0 < 1 <... < j J 1 t holds [ j ] j E Ω D n = E[Ω D n ], where, for {0,..., J 1}, Ω { F m, σm 1,m 2 (F), F m 1 F m 2 =0 =0 0 m, m1, m 2 M }. Remars 4.5 (Bayesan LSRM) ) The covarance coeffcents σ m 1,m 2 (F) n Model Assumptons 4.4 b) have to be chosen, so ( ) that σ m 1,m 2 (F)R m 1,m 2, s postve semdefnte almost sure for all and. m 1,m 2 ) Model Assumpton 4.4 c) s a nd of uncorrelatedness assumpton, whch s actually less restrctve than a pror ndependence of {F : 0 J 1}. Indeed, one can show that uncondtonal ndependence of the development factors {F : 0 J 1} together wth Model Assumpton 4.4 a) satsfy c), f n b) σ m 1,m 2 (F) depends on F only and n a) the slghtly stronger assumpton holds that not only the expected value but the dstrbuton of S,+1 m depends on F and D+ va F m and R, m only (compare ths wth the Model Assumptons 3.3 for the Bayes CL method). ) The Bayes CL method presented n Gsler-Wüthrch [27] belongs to the class of Bayesan LSRMs, see the next subsecton Classcal Bayesan Clams Reservng Methods as Bayesan LSRMs We frst analyze whch of the classcal Bayesan clams reservng methods belong to the class of Bayesan LSRMs.

53 4.2 Bayesan Lnear Stochastc Reservng Methods 43 Bayes CL Method Model Assumptons 3.3 for the Bayes CL method are: a) Condtonally, gven G := (G 0,..., G ), the cumulatve clams payments {C, 0 J} from dfferent accdent years {0,..., I} are ndependent. b) Condtonally, gven G and B, the dstrbuton of Y, depends only on C, and t holds by for {0,..., I} and {0,..., J 1}. c) {G 0, G 1,..., G } are ndependent. E[Y, G, B, ] = G Var[Y, G, B, ] = σ2 (G ) C,, In the Bayes CL method we have that M = 0. The ncremental clams payments are gven We defne componentwse F := G 1 and R 0, := C, = S 0,+1 := C,+1 C,. j=0 S 0,j =: R 0,0, L+ L. (4.10) Wth Model Assumptons 3.3 a) and b) we obtan [ E S,+1 0 ] [ D +, F = E S,+1 0 ] D +, G = E [ S,+1 0 B,, G ] = (G 1)C, = F 0 R0, [ Var S,+1 0 ] D,, F = Var [ S,+1 0 B,, G ] = σ 2 (G )C, = σ 2 (F 0 } {{ + 1) C, = σ 0,0 } (F 0 )R0,0,. =: σ 0,0 (F 0) Ths shows that Model Assumpton 4.4 a) and b) are fulflled and c) follows by Theorem 3.2 n Gsler Wüthrch [27]. It follows that the Bayes CL method belongs to the class of Bayesan LSRMs. Credblty-Based Addtve Loss Reservng Method The credblty-based ALR method n Merz Wüthrch [44] s based on the classcal ALR method, whch s a specal case of the CLR method, see Secton 3.4. Snce the CLR method belongs to the class of LSRMs we expect that the credblty-based ALR method also belongs to class of Bayesan LSRMs. The model assumptons of the credblty-based ALR method are gven by:

54 44 4 (Bayesan) Lnear Stochastc Reservng Methods Model Assumptons 4.6 (Credblty-based ALR model) There exst constant postve volume measures V 0,..., V I such that a) Condtonally, gven G = (G 0,..., G J ) the ncremental clam payments {S 0, 0 I, 0 J} are ndependent. b) Condtonally, gven G, the dstrbuton of S 0,+1 only depends on G and the constant V, and for all = 0,..., I and = 0,..., J holds E [ S,+1 0 G ] = G +1 V Var [ S,+1 0 ] G = σ 2 +1 (G +1 )V. c) {G 0,..., G J } are ndependent wth pror dstrbutons U(g). In the credblty-based ALR method we have M = 1 and M = {0}. We defne S, 1 := V for = 0. 0 otherwse Settng F 0 := G +1 for = 0..., J 1 and σ 0,0 (F ) := σ+1 2 (G +1) we obtan [ E [ Var S,+1 0 S 0,+1 D + D + ], F ], F [ = E = Var S,+1 0 [ S 0,+1 D + D + ], G = G +1 V = F 0 V ], G = σ 2 +1 (G +1)V = σ 0,0 (F 0 )V. Model Assumptons 4.4 a) and b) are satsfed, snce V L + L. Model Assumpton 4.4 c) follows by Theorem 3.3 n Merz Wüthrch [44]. Ths shows that the credblty-based ALR method also belongs to the class of Bayesan LSRMs Predcton of Future Clam Informaton The Bayesan LSRM s constructed n such a way that condtonally, gven F, Model Assumptons 4.1 of the classcal LSRM are fulflled. Ths mples that, n analogy to (4.7) for classcal LSRMs, we obtan for Bayesan LSRMs for {0,..., I}, + I, +n+1 J and m {0,..., M} where P m n,f,,+n+1 [S := E,+n+1 m ] D +, F Ŝ m +,F = P m +,F,+n S +, (4.11) := P m n. An mmedate consequence s the followng result: f = F,

55 4.2 Bayesan Lnear Stochastc Reservng Methods 45 Predctor 4.7 (Bayesan predctor of S,+1 m at tme I) Under Model Assumptons 4.4 we obtan for {0,..., I}, + {I,..., I + J 1}, + n + 1 J and m {0,..., M} where P m n,bayes, := P m n. f = E[ F D n ], [ E S,+n+1 m ] D + = P m +,Bayes,+n S +, Proof: Condtonally, gven D + and F, we obtan wth (4.11) Because each mappng P, m s lnear n f j m Assumpton 4.4 c) [ E S,+n+1 m ] D +, F = P m +,F,+n S +. [ E S,+n+1 m D +] = E, we get wth the second equaton n (4.11) and Model [ [ = E [ E S,+n+1 m ] ] D +, F D + P m +,F,+n S + D + ] = P m +,Bayes,+n S Credblty for Lnear Stochastc Reservng Methods Our goal n ths secton s to derve a predctor for the unnown ncremental clam nformaton S m,+1 for + I. It s a well nown result n probablty theory that the best square-ntegrable predctor for the ncremental clam nformaton S m,+1, gven the data DI at tme I, s gven by Ŝ m I,Bayes,+1 := E [ S,+1 m D I] = P m I,Bayes, S I. (4.12) In order to calculate P m I,Bayes, one needs to now the jont pror dstrbuton of F and the explct form of the condtonal jont dstrbuton of S, m for + I, gven F. These dstrbutons are often unnown n practce and t s not obvous, how reasonable estmates of these dstrbutons can be derved. Thus, we choose a so-called credblty based approach, where only the frst two moments (or approprate estmates) of the condtonal dstrbuton of F, gven D, are requred. That means nstead of calculatng E [ F D I] contaned n P m I,Bayes, (cf. (4.12)) we use a so-called credblty predctor for F for {0,..., J 1}. That means that for all development years {0,..., J 1} we replace ( E [ F D I]) = E[ F D I] by best predctors, whch are affne-lnear n the observatons Y, := ( Y 0,, Y 1,,..., Y, M ) wth Y m, := Sm,+1 R m, for = 0,..., I 1.

56 46 4 (Bayesan) Lnear Stochastc Reservng Methods Ths mples that at tme I we use best predctors from the lnear subspace { } I 1 L nd (Y 0,,..., Y I 1, ) := F F = a 0 + A Y,, a 0 R (M+1), A R (M+1) (M+1), =0 whch contans all affne-lnear combnatons of the observatons Y, wth {0,..., I 1} at tme I. The best predctor from ths subspace s gven by Defnton 4.8 (Credblty predctor of F at tme I) The credblty predctor at tme I for the development factor F for {0,..., J 1} s defned by F I,Cred := Pro(F L nd (Y 0,,..., Y I 1, )), where Pro(F L nd (Y 0,,..., Y I 1, )) denotes the orthogonal projecton operator on the lnear subspace L nd (Y 0,,..., Y I 1, ). For many dervatons t s easer to wor wth the followng so-called normal equatons whch are necessary and suffcent condtons for orthogonal projectons. Lemma 4.9 (Normal equatons) A predctor F L nd (Y 0,,..., Y I 1, ) s the credblty predctor for F,.e. F I,Cred = F, f and only f for {0,..., I 1} the followng equatons hold true: ) ( ) 1. E [( F F Y, ]= 0 R (M+1) (M+1) I,Cred denoted by: F F Y,, [ ) ] 2. E ( F F 1 = 0 R (M+1) (M+1). Proof: For a proof of Lemma 4.9 we refer to Brocwell-Davs [9]. Lemma 4.9 s often called Hlbert projecton theorem. We wll use the normal equatons for the F I,Cred dervaton of the credblty predctor n Theorems 4.13 and 4.15 below. m I,Cred In the same way as n (4.9) we collect all credblty predctors F ( FI,Cred ) := for m clam nformaton m {0,..., M} and development year {0,..., J 1} n F I,Cred defned by ( ) F I,Cred m I,Cred 0 m M := F. 0 Defnton 4.10 (Credblty based predctor of S,+1 m at tme I) The credblty based predctor of the ncremental clam nformaton S,+1 m at tme I s gven by where P m I,Cred, := P m I,. f = FI,Cred Ŝ m I,(Cred),+1 := P m I,Cred, S I,

57 4.2 Bayesan Lnear Stochastc Reservng Methods 47 Remars 4.11 (Credblty based predctor) ) Credblty predctors are the best predctors, whch are affne-lnear n the observatons. We base the predcton of F m at tme I on the normalzed observatons {Y, = 0, 1,..., I 1}, snce they are the only observatons contanng nformaton on F. ) For the credblty predctor t holds.e. F I,Cred [ ) ) F I,Cred := arg mn E ( F F ( F ] F D, F L nd (Y 0,,...,Y I 1, ) mnmzes the condtonal, gven D, MSEP n the lnear subspace L nd (Y 0,,..., Y I 1, ). ) The superscrpt Cred n bracets n Defnton 4.10 means that S m I,(Cred),+1 s a predctor based on the credblty predctors F I,Cred and not a credblty predctor tself. Ths follows from the defnton, because credblty predctors are affne-lnear functons of the observatons. In our multplcatve model structure, t would not mae sense to restrct to affne-lnear predctors of S m,. In order to calculate the credblty based predctor Ŝm I,(Cred),+1 stated n Defnton 4.10 we have I,Cred to derve the credblty predctor F for F gven n Defnton 4.8. We start our calculatons wth some basc results on Y, : Lemma 4.12 Under Model Assumptons 4.4 t holds for {0,..., I}, {0,..., J 1} and m 1, m 2 {0,..., M}: ) Y, are condtonally, gven D and F, unbased predctors for F,.e. E[Y, D, F] = F. ) Y, are condtonally, gven D and F, uncorrelated for dfferent accdent years,.e. Cov[Y 1,, Y 2, D, F] = 0 R (M+1) (M+1) for 1 2. ) We have [ ] Cov Y m 1,, Y m 2, D, F = Rm 1,m 2, σ m 1,m 2 R m 1, Rm 2, (F) =: (Σ, (F)) m1,m 2. (4.13) Proof: The frst and thrd clam s a drect consequence of Model Assumptons 4.4. For the second clam we assume wthout loss of generalty that 1 < 2. Then Y 1, s D 2+ -measurable

58 48 4 (Bayesan) Lnear Stochastc Reservng Methods and [ [ ] Cov[Y 1,, Y 2, D, F] = E Cov Y 1,, Y 2, D 2+, F] D, F [ [ ] [ ] + Cov E Y 1, D 2+, F, E Y 2, D 2+, F] D, F = 0 + Cov[Y 1,, F D, F] = 0. By Defnton 4.8 the credblty predctor F I,Cred for the development factor F s an affnelnear predctor n the normalzed observatons Y,. We compress the data Y 0,,..., Y I 1, lnearly to one (M + 1)-dmensonal vector C (see Theorem 4.13) and show that the credblty predctor F I,Cred depends on the data Y 0,,..., Y I 1, only va C,.e. C s a suffcent statstcs (see Theorem 4.15). For that reason, we defne where W 1, := ( I 1 =0 W, ) 1 I 1 C := W 1, W, Y,, (4.14) and =0 W, := dag ( R 0, R 0, R 0,0,,..., RM, RM, R M,M, For ths compressed data vector C we obtan wth Lemma 4.12 and (4.13) that ). (4.15) E[C D, F] = F (4.16) ( I 1 ) W, Cov[Y,, Y, D, F] W, Cov[C, C D, F] = W 1, = W 1, =0 ( I 1 =0 W, Σ, (F)W, ) W 1, W 1,. (4.17) Theorem 4.13 (Credblty predctor of F at tme I based on compressed data) Under Model Assumptons 4.4 the credblty predctor for F vector C s gven by F I,Cred = A I C + (I A I )µ, wth the dentty matrx I, the structural parameter vector (pror mean) and the credblty weght µ := E[F D ] based on the compressed data A I := T (T + U I ) 1, (4.18)

59 4.2 Bayesan Lnear Stochastc Reservng Methods 49 where U I := W 1, and the structural parameter matrx ( I 1 =0 W, E[Σ, (F) D ] W, ) T := Cov[F, F D ]. W 1, Proof: Condtonally on D the random varable C fulflls Model Assumptons 7.1 n Bühlmann-Gsler [14] and s therefore the credblty predctor for F at tme I based on the compressed data vector C. Remars 4.14 (Credblty predctor F I,Cred ) ) The vector C s the estmator for f n the classcal (non-bayesan) LSRM framewor n Secton 4.1. Ths becomes clear by comparng C n (4.14) and assocated weghts (4.15) to f n (4.3) wth correspondng weghts (4.4) componentwse. ) The credblty predctor s a credblty weghted average of the pror mean µ and the compressed data vector C consstng of credblty weghted observatons. ) Note, that we only need estmators for the frst two moments of F, gven D, and not the full jont dstrbuton. Ths s the great advantage of credblty theory. v) The credblty predctor n Theorem 4.13 s based on the three structural parameters: µ, T and U I. These parameters can ether be estmated by ncludng pror expert nowledge or by portfolo data (see Secton n Bühlmann-Gsler [14]). Note that the estmaton of U I can be reduced to an estmator σm 1,m 2 of E [ σ m 1,m 2 (F) ] D, because all remanng terms n U I are DI -measurable and can be observed. An unbased estmator for E [ σ m 1,m 2 (F) ] D s gven by (4.5). The (condtonal) unbasedness of the estmator follows by E [ σ m 1,m 2 ] [ [ D = E E σ m 1,m 2 ] ] [ F, D D = E σ m 1,m 2 (F) ] D. Now we wll prove that the data compresson C s an admssble compresson,.e. that the credblty predctor F Cred for F based on all data Y 0,,..., Y I 1,. for F based on C n Theorem 4.13 s also the credblty predctor Theorem 4.15 (Credblty predctor of F at tme I based on all data) Under Model Assumptons 4.4 the credblty predctor for F based on Y 0,,..., Y I 1, s gven by F I,Cred = A I C + (I A I )µ, where A I, C and µ are defned as n Theorem 4.13.

60 50 4 (Bayesan) Lnear Stochastc Reservng Methods Proof: Choose {0,..., I 1}. All followng calculatons are done condtonally gven D. Snce F I,Cred = A I C + (I A I )µ, t follows straghtforward that L nd (Y 0,,..., Y I 1, ). Moreover, usng (4.16) we get E[C ]= E[E[C F]]= E[F ]= µ. F I,Cred Ths mples E [ FI,Cred ] = E [ A I C + (I A I )µ ] = µ = E[F ],.e. Condton 2. n Lemma 4.9 s fulflled. It remans to show that It holds [( E F F F I,Cred Y,. ) ] I,Cred F Y, = E [( F A I C ( I A I ) ) ] µ Y, = A I E[ (F C ) Y,] + ( I A I ) E [ (F µ ) Y,]. (4.19) We have (see Theorem A.3 of Appendx A n Bühlmann-Gsler [14] for a proof) that C s the orthogonal projecton of F on the affne-lnear subspace L nd e (Y 0,,..., Y I 1, ) { := F F = I 1 =0 } [ ] A, Y,, A, R (M+1) (M+1), E F F = F, of all condtonally, gven F, unbased estmators for F, whch are lnear n Y,, = 0,..., I 1. Ths mples and together wth (4.17) we obtan E [ (F C )(Y, C ) ] = 0 R (M+1) (M+1) E [ (F C )Y,] [ = E (F C )C ] = E [ E [ (F C )C ]] F = E [ E [ (F C )(C F ) F ]] [ ( I 1 ) = E W, Σ, (F )W, Usng the same arguments, we get = U I. W 1, =0 E [ (F µ )Y,] = T. W 1, ]

61 4.2 Bayesan Lnear Stochastc Reservng Methods 51 By puttng ths nto (4.19) and usng the defnton of A I we get A I E[ (F C )Y,] + (I A I )E [ (F µ )Y,] = A I U I + (I AI )T [(.e. E F = A I (T + U I ) + T = T (T + U I ) 1 (T + U I ) + T = T T = 0, ) ] I,Cred F Y, = 0. Ths completes the proof. Theorem 4.15 provdes an explct formula for the credblty predctor F I,Cred and allows for a drect calculaton of the credblty based predctor Ŝm I,(Cred),+1 stated n Defnton In the followng subsecton our goal s to quantfy the MSEP of ths credblty based predctor. We wll base several defntons n the followng subsectons on Ŝn n 1 L n wth n 1 {I, I + 1} defned by (Ŝn n ) m 1 :=, Ŝ m I, for n 1 < + n, (4.20) S, m for 0 + n 1 for an arbtrary σ(d n 1, F)-measurable predctor Ŝm I, for S m I,. That means that the matrx S I contanng all observatons up to tme I s extended by addtonal predcted dagonals up to accountng year n Mean Squared Error of Predcton One s often nterested n weghted sums of ncremental clam nformaton of the form =I αm Sm,+1 for fxed m M, {I J + 1,..., I} and αm R. The credblty based predctor for these sums at tme I s gven by =I αm uncertanty, we consder the (condtonal) MSEP gven by Ŝ m I,(Cred),+1. For the predcton msep α m Sm,+1 m M =I DI [ m M =I α m Ŝ m I,(Cred),+1 ( = E ] m M =I α m ( S m,+1 Ŝm,I,(Cred),+1 ) ) 2 DI.

62 52 4 (Bayesan) Lnear Stochastc Reservng Methods We decompose the (condtonal) MSEP nto [ msep α m Sm,+1 m M =I DI m M =I α m Ŝ m I,(Cred),+1 [ [ ] ] = E Var α m S,+1 m F, DI DI m M =I } {{ } + E average process varance ( [ ] E α m S,+1 m F, DI ) 2 α m Ŝ m,i,(cred),+1 m M =I m M =I DI } {{ } average estmaton error At frst we analyze the average estmaton error (4.22). ] (4.21). (4.22) Estmaton Error for Sngle Accdent Years For the average estmaton error (4.22) of a sngle accdent year {I J + 1,..., I} we fnd (usng to ndcate that the equaton s approxmately fulflled) ( [ ] E E α m S,+1 m F, DI α m m M =I m M =I ( E α m Ŝ m I,F,+1 m M =I m M [ ] = Var m M =I α m Ŝ m I,F,+1 DI The condtonal varance (4.23) can further be decomposed nto [ ] Var m M =I α m Ŝ m I,F,+1 DI ( = E = m M =I α m Ŝ m I,F,+1 α m 1 α m 2 m 1,m 2 M 1, 2 =I ) 2 DI Ŝ m I,(Cred),+1 E E 1 I,F [Ŝm α m 1 α m 2 m 1,m 2 M 1, 2 =I =I α m [, 1 +1 Ŝm 2 I,F, 2 +1 E ) 2 DI Ŝ m I,Bayes,+1 m M =I 1 I,F [Ŝm, 1 +1 α m ) 2 DI Ŝ m I,F,+1 DI (4.23) (4.24) ] 2 D I] (4.25) D I] E 2 I,F [Ŝm, 2 +1 D I]. In order to fnd an estmator for (4.25), we have to mae some computatons for expectatons of products of Ŝm I,F,+1. Let I and 2 > 1. Then we get wth Model Assumpton 4.4 c) E 1 I,F [Ŝm D I] = E 1 I,F [Ŝm Ŝ 2+ 2 I,F D I], 2 I,F 1, 1 +1Ŝm 2, , 1 +1 Pm ,Bayes 2, 2

63 4.2 Bayesan Lnear Stochastc Reservng Methods 53 where (cf. (4.20)) Ŝ n I,F := Ŝn I Ŝm I,. =Ŝm I,F, For 1 = 2 =: and 1 +, 2 + I we calculate agan wth Model Assumpton 4.4 c) E 1 I,F [Ŝm wth 1,+1 Ŝm 2 I,F 2,+1 D I] = E [ F m 1 = ( E [ F m 1 F m 2 D I] E [ F m 2 D I ] [ E D I] + ϱ m 1,m 2 Γ m 1 1,Ŝ 1+ I,F Γ m 2 ϱ m 1,m 2 := ( Cov [ F, F D I]) m 1,m 2. 2,Ŝ 2+ I,F D I] ) E [ Γ m 1 1,Ŝ 1+ I,F Γ m 2 2,Ŝ 2+ I,F D I], To smplfy notaton of the terms above, we defne for n {I,..., I+J 1} and {0,..., J 1} the lnear operators (the symbol denotes the tensor product, see Lang [37]) wth (H n (τ ) xy ) m1,m 2 H n (τ ): Ln Ln Ln +1 Ln +1, xy Hn (τ ) xy := ( 1, 1, 2, 2 P m 1 n ( 1 +) 1, 1 1 f m 1 f m 2 + τ m 1,m 2 1, 2, x P m 2 n ( 2 +) 2, 2 1 y for 1 2 n 1 or 1 2 ) [ ], E Γ m 1 1, xγm 2 2, y D I otherwse where τ s a (M +1) (M +1) I I J tensor and τ m 1,m 2 1, 2, are the entres of τ. Concatenatons of these operators wll be denoted by H n H n 2 1 (τ ) := 2 (τ )H n 2 1 (τ ) Hn 1 (τ ) for 2 1, otherwse H m 1,m 2 n 1, 1, 2, 2 (τ )xy := Π L n 2 +1 Ln 2 +1 (H n (1 2 ) 0 (τ )xy ) m1,m 2 where Π L n 2 +1 Ln 2 +1 denotes the projecton onto Ln 2 +1 Ln By replacng n (4.26) f by E [ F D I] and τ m 1,m 2 1, 2, := ϱ m 1,m 2 H m 1,m 2 n,bayes 1, 1, 2, 2 (ϱ ) := H m 1,m 2 n 1, 1, 2, 2 (τ ) 1, 1 +1, 2, 2 +1, (4.26) we obtan f =E[ F D I ] τ m 1,m 2 1, 2, =ϱ m 1,m 2. (4.27) Wth n = I n (4.27) we get for each summand n (4.25) E 1 I,F 2 I,F [Ŝm 1, 1 +1Ŝm 2, 2 +1 D I] = H m 1,m 2 I,Bayes 1, 1, 2, 2 (ϱ )S I S I. (4.28) However, H m 1,m 2 I,Bayes 1, 1, 2, 2 (ϱ ) n (4.28) stll depends on Cov [ F, F D I] and E [ F D I], see (4.27). In a frst step we estmate E [ F D I] by F I,Cred,.e. Ê [ F D I] := F I,Cred. (4.29)

64 54 4 (Bayesan) Lnear Stochastc Reservng Methods Usng the approxmaton Cov [ F, F D I] [ (F = E E [ F D I]) ( F E [ F D I]) ] D I [ ( ) ( ) ] I,Cred I,Cred E F F F F D I (4.30) A I UI, where n the last approxmaton we used the loss matrx of the credblty predctor Theorem 7.5 n Bühlmann-Gsler [14]), we fnd the followng estmator for ϱ m 1,m 2 1, 2, ϱ m 1,m 2 1, 2, F I,Cred (see := ( A I ) UI m 1,m 2. (4.31) Puttng the estmates (4.29) and (4.31) nto (4.27) we obtan for n {I,..., I + J 1} Ĥ m 1,m 2 n,cred 1, 1, 2, 2 ( ϱ ) := H m 1,m 2 n 1, 1, 2, 2 (τ ). f = FI,Cred τ = ϱ For n = I ths leads to the followng estmator of the estmaton error (4.22). Estmator 4.16 (Estmaton error for sngle accdent years) Under Model Assumptons 4.4 at tme I an estmator for the estmaton error (4.22) of accdent year {I J + 1,..., I} s gven by ( Ê E [ m M =I α m S,+1 m := where 0 denotes the 0-tensor. F, DI ] m 1,m 2 M 1, 2 =I m M α m 1 α m 2 =I α m Ŝ m I,(Cred),+1 (Ĥm 1,m 2 I,Cred, 1,, 2 ) 2 DI ( ϱ ) Ĥm 1,m 2 I,Cred, 1,, 2 (0) ) S I S I, Process Varance for Sngle Accdent Years For the process varance (4.21) we see that, condtonally gven F, Model Assumptons 4.1 for the classcal LSRM are fulflled and wth Lemma 4.2 n Dahms [17] follows for I 1 + 1, = [ Cov S m 1 1, 1 +1, Sm 2 ( ) ( )+1 n=i+1 2, 2 +1 M F, D I] J l 1,l 2 =0 j=n I wth the couplng exposure Γ m 1,m 2, S + := [ ] ( ) σ l 1,l 2 j 1 (F) E Γ l 1,l 2 n j,j 1 Sn 1 F, D I P m l1 ( ) 1 n,f 1, 1 P m l2 2 n,f n j,j 2,. 2 n j,j M I l=1 h=0 (+ h) j=0 γ m 1,m 2,l,,h,j S l h,j = Rm 1,m 2,,

65 4.2 Bayesan Lnear Stochastc Reservng Methods 55 see (4.1). Thus, wth Model Assumptons 4.4 c) we obtan for a fxed accdent year {I J + 1,..., I} for the process varance (4.21) [ [ M ] ] E Var α m S,+1 m F, DI DI m=0 =I ( = where m 1,m 2 M α m 1 α m 2 1, 2 =I +( 1 2 )+1 n=i+1 M J l 1,l 2 =0 j=n I [ ] E σ l 1,l 2 j 1 (F) D I Γ l 1,l 2 n j,j 1Ŝn 1 I,Bayes E Ŝ n I,Bayes := Ŝn I Ŝm I, [ ( P m 1 n,f, 1. =Ŝm I,Bayes, ) l1 n j,j ( P m 2 n,f, 2 ) l2 n j,j (4.32) (4.33) ] ), DI Snce n the dervaton of (4.28) no specal property of S I s used, except that t s contaned n L I, we get wth the same arguments for 1 + 1, I [ ( ( ] E P m 1 n,f 1, 1 P m [ 2 n,f ] 2, 2 DI = E P m 1 n,f 1, 1 e l 1 n j,j P m 2 n,f 2, 2 e l 2 n j,j D I ) l1 n j,j ) l2 n j,j = H m 1,m 2 n,bayes 1, 1, 2, 2 (ϱ )e l 1 n j,j e l 2 n j,j, (4.34) where e l 1 n j,j L n wth e l 1 n j,j = 1 n the entry (n j, j, l 1 ), 0 otherwse. Usng the approxmatons (estmates) Ŝ + I,(Cred) σ l 1,l 2 j 1 := Ê Ŝ+ I,Bayes [ σ l 1,l 2 j 1 (F) D I ] Ĥ m 1,m 2 n,cred, 1,, 2 ( ϱ ) H m 1,m 2 n,bayes, 1,, 2 (ϱ ), (4.35) see Remar 4.14 for the estmate σ l 1,l 2 j 1, and puttng the estmates (4.35) nto (4.33) and (4.34), respectvely, we get the followng estmator: Estmator 4.17 (Process varance for sngle accdent years) Under Model Assumptons 4.4 at tme I an estmator for the process varance (4.21) of accdent year {I J + 1,..., I} s gven by Ê [ Var [ m M =I α m S,+1 m := m 1,m 2 M F, DI ] DI ] ( α m 1 α m 2 1, 2 =I +( 1 2 )+1 n=i+1 M J l 1,l 2 =0 j=n I σ l 1,l 2 j 1 Γl 1,l 2 n j,j 1Ŝn 1 I,(Cred) Ĥ m 1,m 2 n,cred, 1,, 2 ( ϱ )e l 1 n j,j e l 2 n j,j ).

66 56 4 (Bayesan) Lnear Stochastc Reservng Methods Mean Squared Error of Predcton for Sngle Accdent Years Combnng the Estmators 4.16 and 4.17 we get an estmator for the (condtonal) MSEP for sngle accdent years. Estmator 4.18 (MSEP for sngle accdent years) Under Model Assumptons 4.4 at tme I an estmator for the (condtonal) MSEP of accdent year {I J + 1,..., I} s gven by msep α m Sm,+1 m M =I DI := m 1,m 2 M +( 1 2 )+1 + n=i+1 α m 1 α m 2 M [ m M =I ( l 1,l 2 =0 j=n I 1, 2 =I J α m Ŝ m I,(Cred),+1, 1,, 2 ] [ (Ĥm ) 1,m 2 I,Cred ( ϱ ) Ĥm 1,m 2 I,Cred, 1,, 2 (0) S I S I σ l 1,l 2 j 1 Γl 1,l 2 n j,j 1Ŝn 1 I,(Cred) Ĥ m 1,m 2 n,cred, 1,, 2 ( ϱ )e l 1 n j,j e l 2 n j,j ]). Mean Squared Error of Predcton for Aggregated Accdent Years Now we tae a closer loo at the predcton uncertanty of sums of credblty based predctors Ŝ m I,(Cred) +1 for dfferent accdent years. Snce these predctors depend on data of all accdent years they are not ndependent. Agan, we decompose the (condtonal) MSEP nto msep = E I α m Sm,+1 =I J+1 m M =I DI [ Var [ ( + E I =I J+1 m M =I E [ I =I J+1 m M =I [ I =I J+1 m M =I α m S,+1 m α m S,+1 m F, DI ] F, DI DI ] ] α m Ŝ m I,(Cred),+1 I ] =I J+1 m M =I α m Ŝ m I,(Cred),+1 ) 2 (4.36) (4.37) DI, (4.38) where (4.37) corresponds to the process varance and (4.38) to the estmaton error, respectvely. Usng the same technques as n the prevous secton leads to the followng estmator:

67 4.2 Bayesan Lnear Stochastc Reservng Methods 57 Estmator 4.19 (MSEP for aggregated accdent years) Under Model Assumptons 4.4 at tme I an estmator for the (condtonal) MSEP for aggregated accdent years (4.36) s gven by := msep I α m Sm,+1 =I J+1 m M =I DI I 1, 2 =I J+1 m 1,m 2 M ( ) ( )+1 + n=i+1 α m 1 1 α m 2 2 M J l 1,l 2 =0 j=n I [ ( I =I J+1 m M =I 1 =I 1 2 =I 2 α m 1, 1, 2, 2 Ŝ m I,(Cred),+1 ] [ (Ĥm ) 1,m 2 I,Cred ( ϱ ) Ĥm 1,m 2 I,Cred 1, 1, 2, 2 (0) S I S I σ l 1,l 2 j 1 Γl 1,l 2 n j,j 1Ŝn 1 I,(Cred) Ĥ m 1,m 2 n,cred 1, 1, 2, 2 ( ϱ )e l 1 n j,j e l 2 n j,j ]) Specal Case: Mean Squared Error of Predcton for the Bayes CL Method We saw n Subsecton that the Bayes CL method n Gsler Wüthrch [27] belongs to the class of Bayesan LSRMs. In the followng we show that f the Bayesan LSRM s the Bayes CL method the LSRM estmate for the (condtonal) MSEP gven by Estmator 4.18 concdes wth the estmate n the Bayes CL method gven n Theorem 4.4 n Gsler Wüthrch [27]. Agan we use the dentty G := (G 0,..., G ) = ( F ,..., F ) = F + 1. We frst study the average process varance (4.21) for sngle accdent years gven by formula (4.33). Process Varance For the Bayes CL method looed at as a LSRM, see Subsecton 4.2.1, we frst analyze the nner part of the average process varance (4.32) [ M ] Var α m S,+1 m F, DI = m=0 =I 1, 2 =I +( 1 2 )+1 n=i+1 J j=n I (4.39) ) σ 0,0 j 1 (F)Γ0,0 n j,j 1Ŝn 1 I,F( P 0 n,f 0 ( ), 1 P 0 n,f 0,. (4.40) n j,j 2 n j,j A short calculaton yelds (the empty product s set to 0) 0 for n j ( ) P 0 n,f 0, = 1 for = n j 1 = j 1 1 n j,j 1 G l 1 1 G l for = n j 1 j l=n l=n (4.41)

68 58 4 (Bayesan) Lnear Stochastc Reservng Methods and Γ 0,0 n j,j 1Ŝn 1 I,F = C n j,i n+j j 2 l=i n+j G l. (4.42) Wth (4.41) and (4.42) we obtan for (4.40) 1, 2 =I J+ = C,I +( 1 2 )+1 n=i+1 n=i+1 J+ = C,I n=i+1 J+ = C,I n=i+1 J+ = C,I n=i+1 = C,I n=i ) σ 0,0 n 1 (F)Γ0,0,n 1Ŝn 1 I,F( P 0 n,f 0 (, 1,n n 2 σn 1(G 2 n 1 ) n 2 σn 1(G 2 n 1 ) n 2 σn 1(G 2 n 1 ) σ 2 n 1(G n 1 ) σ 2 n(g n ) n 1 l 1 =I G l1 P 0 n,f, 2 ) 0,n ( ) G l P 0 n,f 0, 1,n l=i 1 =n 1 2 =n 1 ( ) G l P 0 n,f 0, 1,n l=i 1 =n 1 2 =n 1 2 ( ) G l P 0 n,f 0, 1,n l=i 1 =n 1 n 2 l 1 =I l 2 =n+1 G l1 l 2 =n (G l2 ) 2 ( ( P 0 n,f, 2 P 0 n,f, 2 ) 0,n ) 0,n (4.43) (G l2 ) 2. (4.44) Tang the condtonal (on D I ) expectatons of (4.43) and (4.44) and usng Model Assumptons 4.4 c) mples for the process varance E [ Var [ M m=0 =I = C,I n=i = C,I n=i α m S,+1 m F, DI ] DI E [ σn(g 2 n ) n 1 D I] ] l 1 =I E [ σn(f 2 n 0 + 1) n 1 D I] E [ G l1 D I] l 1 =I l 2 =n+1 E [ F 0 l D I] (4.45) [ E (G l2 ) 2 ] D I (4.46) l 2 =n+1 E [ (F 0 l 2 + 1) 2 D I]. (4.47) Formula (4.46) for the process varance n the LSRM s dentcal wth Formula (4.13) n Gsler Wüthrch [27] for the process varance n the Bayes CL method. Hence, we have to chec that n the Bayesan LSRM and n the Bayes CL method each component of (4.46) s estmated n

69 4.2 Bayesan Lnear Stochastc Reservng Methods 59 the same way. In the Bayesan LSRM the followng estmates are used, see (4.29) (4.31), wth F 0 I,Cred Ê [ F D I] := + 1 (4.48) (F 0 E[ + 1 ) 2 ] D I (F 0 = E[ E [ F 0 D I]) 2 ] D I + E [ F D I] 2 [ ( ) ] E F 0 F 0 I,Cred 2 ( ) D + F 0 I,Cred ( ) = A I UI + F 0 I,Cred [ ] E σ 0,0 (F 0 ) D ( 2, = α + F 0 I,Cred I 1 + 1) (4.49) C, =0 α := I 1 =0 I 1 C, =0 E[ σ0,0 (F 0) D ] Var[ F 0 D ] C, +. (4.50) The approxmatons (4.48) and (4.49) for the LSRM concde wth the approxmatons (4.15) and (4.16) n Gsler Wüthrch [27] for these quanttes for the Bayes CL method. By puttng the approxmatons (4.48) and (4.49) nto (4.47) and usng the fact that F 0 I,Cred + 1 = G I,Cred and σ 0,0 (F 0 ) = σ2 (G ) we obtan the Bayesan LSRM estmate for the process varance (cf. Estmator 4.17) n the case of the Bayes CL method [ [ M Ê Var α m S,+1 m m=0 =I = C,I n 1 n=i l 1 =I F, DI ] DI G I,Cred l 1 Ê [ σn(g 2 n ) D I ] ] l 2 =n+1 ( ) 2+ Ê [ σl 2 2 (G l2 ) ] D l2 l αl2 2, (4.51) G I,Cred I l 2 1 where Ê[ σ 2 (G ) D I ] and Var[G D ] are approprate estmates for E [ σ 2 (G ) D I ] and E [ σ 2 (G ) D ] as well as Var[G D ] and α := α E[ σ 2 (G ) D I ]=Ê[ σ2 (G ) D I ] Var[ G D ]= Var[ G D ]. If we use n the Bayesan LSRM as well as n the Bayes CL method the same estmators Ê [ σ 2 (G ) D I] and Var[G D ] for the structural parameters E [ σ 2 (G ) D I] and Var[G D ] (ths s the case, snce n both methods the Bühlmann-Straub estmators, see Bühlmann-Gsler [14], are proposed) the estmator (4.51) s the well-nown estmator for the process varance n =0 C,l2

70 60 4 (Bayesan) Lnear Stochastc Reservng Methods the Bayes CL method gven n Theorem 4.4 n Gsler Wüthrch [27]. For non-nformatve (vague) prors for the development factors F 0,.e. Var[G D ] = Var [ F 0 ] D, we have that α 1 and for the credblty predctor F 0 I,Cred holds n ths case G I,Cred = F 0 I,Cred + 1 α 1 f = ĝi,cl,.e. for non-nformatve prors the credblty predctor G I,Cred concdes wth the classcal CL estmator ĝ I,CL, see (3.1). Ths then results n the estmate for the process varance wth nonnformatve prors gven by Ê [ Var [ M m=0 =I = C,I n=i l 1 =I α m S,+1 m n 1 F, DI ] DI ] ĝ I,CL l 1 Ê [ σn(g 2 n ) D I] l 2 =n+1 (4.52) ( ) 2 ĝ I,CL Ê [ σ 2 l 2 + l 2 (G l2 ) ] Dl2 I l 2 1. (4.53) =0 C,l2 The estmator (4.53) s exactly the estmator for the process varance for non-nformatve prors n the Bayes CL method, see Theorem 4.4 n Gsler Wüthrch [27]. The estmator (4.53) s slghtly hgher then the estmator gven n Mac [38] and Buchwalder et al. [10]. For a detaled comparson of the dfferent estmators for the process varance n the (Bayes) CL method we refer to Gsler Wüthrch [27] and Wüthrch Merz [63]. Estmaton Error Wth the approxmaton F m I,Cred F m I,Bayes (4.54)

71 4.2 Bayesan Lnear Stochastc Reservng Methods 61 the estmaton error (4.22) can be rewrtten by ( E E = [ m M =I 1, 2 =I 1, 2 =I ( = E =I = C,I 2 = C,I 2 α m S,+1 m H 0,0 I,Bayes, 1,, 2 F, DI [Ŝ0 I,F E, 1 +1Ŝ0 I,F, 2 +1 j=i Ŝ 0 I,F,+1 C,I 2 = C,I 2 = C,I 2 C,I 2 j=i j=i j=i j=i ) 2 ] (ϱ )S I S I DI m M D I] ( [ E (G j ) 2 ] D I 2 j=i [ E (G j ) 2 ] D I ( ( ( F 0 I,Cred =I =I 1, 2 =I 1, 2 =I E j=i α m [Ŝ0 I,F,+1 E [ G j D I] 2 2 j + 1 G I,Cred j G I,Cred j j=i Ŝ m I,(Cred),+1 ) 2 H 0,0 I,Bayes, 1,, 2 (0)S I S I [Ŝ0 I,F E, D I]) E [ G j D I] + 1 j=i E [ G j D I] 2 [ ) 2 E + αj [ ) 2 E + αj [ ) 2 Ê + αj I j 1 =0 σ 0,0 D I] E E [ G j D I] + 1 DI [Ŝ0 I,F, 2 +1 ] j (Fj 0 ) Dj I j 1 =0 C,j C,j ] σj 2(G j) D j ( I j 1 =0 C,j j=i ] σj 2(G j) D j ( j=i j=i G I,Cred j G I,Cred j D I] ( ) F 0 I,Cred 2 j + 1 ) 2 (4.55) ) 2. (4.56) Formula (4.55) s exactly the formula for the estmaton error of the Bayes CL method n Theorem 4.4 n Gsler Wüthrch [27]. In the same way as for the process varance, f we use n the Bayesan LSRM and n the Bayes CL method the same estmators Ê[ σ 2 (G ) D ] and Var[G D ] for the structural parameters E [ σ 2 (G ) D ] and Var[G D ] (ths s the case, snce n both methods the Bühlmann-Straub estmators, see Bühlmann-Gsler [14], are proposed) the estmator (4.56) s the well-nown estmator for the process varance n the Bayes CL method gven n Theorem 4.4 Gsler Wüthrch [27].

72 62 4 (Bayesan) Lnear Stochastc Reservng Methods For non-nformatve prors we have that α 1 and for the estmaton error we obtan ( [ ] Ê E α m S,+1 m F, DI ) 2 α m Ŝ m I,(Cred),+1 m M =I m M =I DI := C,I 2 ( ) 2 ĝ I,CL Ê [ σ 2 j + l 2 (G j ) ] D j ( ) 2 ĝ I,CL j. j=i I j 1 =0 C,j j=i Ths estmator for the estmaton error s the same as n Buchwalder et al. [10] for the classcal (non-bayesan) CL method, but dfferent from the one n Mac [38]. For detals see Chapter 3 n Wüthrch Merz [63]. Wth the same technques as for sngle accdent years, we also obtan for several accdent years the estmates for the MSEP n the Bayes CL method n Gsler Wüthrch [27] as a specal case of CL method looed at as a Bayesan LSRM Clams Development Result Now we turn bac to general Bayesan LSRMs and consder the one-year predcton uncertanty n terms of the CDR. In the current busness perod I, we observe the data D I and use the credblty based predctors Ŝm I,(Cred),+1 for the predcton of outstandng ncremental clam nformaton. In the next busness perod (.e. at tme I +1 wth observed data D I+1 ) we calculate the credblty based predctors Ŝm I+1,(Cred),+1 for outstandng ncremental clam nformaton. The at tme I + 1 observed CDR for accdent year {I J + 1,..., I} and M {0,..., M} measures the dfference between these two predctons: CDR M,I+1 = m M α m =I (Ŝm I,(Cred),+1 ) Ŝm I+1,(Cred),+1. For the estmaton of the at tme I expected clams development result CDR M,I+1 and ts (condtonal) MSEP we state a theorem, whch provdes an updatng-formula for the credblty predctors F I,Cred that wll be crucal n further calculatons. I,Cred F ) Under Model Assumptons 4.4 for the cred- Theorem 4.20 (Updatng-formula for ) blty predctors the followng updatng-formula holds: F I+1,Cred ( FI,Cred = I F I,Cred + Z I ( Y I, ) I,Cred F = Z I Y I, + ( I Z I ) FI,Cred, wth Z I := AI UI [ A I U I + E[ Σ I,(F) D ] ] 1, where A I and UI are defned n Theorem 4.13 and Σ,(F) s gven n Lemma [B] 1 denotes a generalzed nverse of the matrx B.

73 4.2 Bayesan Lnear Stochastc Reservng Methods 63 Proof: For the loss matrx of the credblty predctor Bühlmann-Gsler [14]: [ ( E F ) ( I,Cred F F F I,Cred ) ] I,Cred F D = A I UI. we obtan by Theorem 7.5 n Now the clam follows by the Kalman-Flter Algorthm n Theorem 10.1 n Bühlmann-Gsler [14] or by Proposton n Brocwell-Davs [9]. Remars 4.21 (Updatng-formula for F I,Cred ) ) The credblty predctor F I+1,Cred based on the data D I+1 at tme I + 1 s a credblty weghted average of the new observaton Y I, at tme I + 1 and the prevous credblty I,Cred predctor F at tme I. ) Theorem 4.20 s central for the dervaton of the CDR uncertanty, because t allows to F I,Cred separate the credblty predctor at tme I and the new observaton Y I, at tme I+1,Cred I + 1, together leadng to the new credblty predctor F at tme I + 1. F I+1,Cred Now we consder what we can say about condtonally gven D I at tme I. Wth I,Cred Theorem 4.20, Model Assumpton 4.4 a) and the fact that F s D I -measurable we get ] F := E[ FI+1,Cred D I = Z I E[ Y I, D I ] ( ) + I Z I FI,Cred (4.57) = Z I E[ E [ Y I, D I, F ] D I ] ( ) + I Z I FI,Cred = Z I E[ F D I ] ( ) + I Z I FI,Cred. In the same way as n (4.9) we defne wth F m := ( F )m For further calculatons we use F := ( F m ) 0 m M 0. F : = E [ F D I ] = Z I E[ F D I] + ( I Z I ) FI,Cred F : = ZI E[ F D I] + ( I Z I ) E[F D ], F : = E[F D ]. F I,Cred, (4.58) Note that by (4.58) follows that F I,Cred can be estmated at tme I by F. The approxmaton F = E[ FI+1,(Cred) D I] I,Cred F, at tme I the est- see (4.58), motvates for the expected clams development result CDR M,I+1 mate [ Ê CDR M,I+1 D I] := 0. (4.59)

74 64 4 (Bayesan) Lnear Stochastc Reservng Methods For the dervaton of an estmator for the (condtonal) MSEP of the clams development result CDR M,I+1 msep CDR M,I+1 for sngle accdent years, we use the predctor n (4.59). Ths mples [ ( [0] := E CDR M,I+1 D I 0 [ = Var + m M ( E α m [ m M =I α m ) 2 D I ] Ŝ m I+1,(Cred),+1 =I DI ] (Ŝm I+1,(Cred),+1 (4.60) ) ]) 2 Ŝm I,(Cred),+1 D I, (4.61) where we used that Ŝm I,(Cred),+1 s D I -measurable. The frst term (4.60) corresponds to the process varance, whereas the second term (4.61) s a nd of estmaton error. Process Varance for Sngle Accdent Years We decompose the process varance (4.60) as follows Var [ m M α m ( = E m M Ŝ m I+1,(Cred),+1 DI =I α m =I Ŝ m I+1,(Cred),+1 ] ) 2 DI ( E [ m M α m Ŝ m I+1,(Cred),+1 DI =I ]) 2. (4.62) We begn wth the calculaton of (4.62) by computng products of condtonal expectatons of Ŝ m I+1,(Cred),+1. For 1 < 2 and I we get wth (4.57) E 1 I+1,(Cred) [Ŝm 1, 1 +1 Ŝ m 2 I+1,(Cred) 2, 2 +1 D I] [ = E E = E 1 I+1,(Cred) [Ŝm 1, I+1,(Cred) [Ŝm 1, 1 +1 P m 2, F 2, 2 Ŝ m 2 I+1,(Cred) 2, 2 +1 Ŝ 2+ 2 I+1,(Cred) ] ] D I 2 D I (4.63) D I] wth P m, F, := P, m f = F P, m f =E[ F D] I + > I. + I For R m I+1,(Cred), := Γ m,ŝ+ I+1,(Cred) := M I (+ h) γ m,l,,h,jŝl I+1,(Cred) l=0 h=0 j=0

75 4.2 Bayesan Lnear Stochastc Reservng Methods 65 we compute n the case of 1 = 2 =: for 1, 2 I E 1 I+1,(Cred) [Ŝm 1,+1 Ŝ m 2 I+1,(Cred) 2,+1 D I] [ ] = E E 1 I+1,(Cred) [Ŝm 1,+1 Ŝ m 2 I+1,(Cred) ] 2,+1 D I D I ) ] E[( F m 1 F m 2 + ϱ m 1,m 2 1 I+1,(Cred) 1, 2, Rm 1, R m 2 I+1,(Cred) 2, D I [( E E [ F m ) 1 D] I F m 2 + ϱ m 1,m 2 1 I+1,(Cred) = 1, 2, Rm 1, 2, E[( F m 1 E [ F m ) 2 D] I + ϱ m 1,m 2 1 I+1,(Cred) 1, 2, Rm 1, 2, [( E E [ F m ] [ 1 D I E F m 2 ] ) D I + ϱ m 1,m 2 1 I+1,(Cred) 1, 2, Rm 1, R m 2 I+1,(Cred) R m 2 I+1,(Cred) ] D I ] D I R m 2 I+1,(Cred) 2, ] D I for 1, 2 > I for 2 > 1 = I for 1 > 2 = I for 1 = 2 = I (4.64) wth ϱ m 1,m 2 1, 2, := [ m Cov F 1 I+1,Cred, F ] m 2 I+1,Cred D I [ Cov S m 1 1,+1, F ] m 2 I+1,Cred D I [ m Cov F 1 I+1,Cred, S m 2 R m 1 1, R m 2 2, [ Cov S m 1 1,+1, Sm 2 A short calculaton yelds ( ϱ m 1,m 2 1, 2, = 2,+1 R m 1 1, Rm 2 2, 2,+1 ] D I ] D I for 1, 2 > I for 2 > 1 = I for 1 > 2 = I for 1 = 2 = I 0 otherwse or denomnator equals zero Z I ( [ E ΣI, (F) D I ] [ ) + Cov F, F D]) I Z I m 1,m 2 ] [ ) + Cov F, F D]) I Z I m 1,m 2 ] [ + Cov F, F D])) I ( (E [ ΣI, (F) D I ( Z I ( E [ ΣI, (F) D I m 1,m 2 for 1, 2 > I for 2 > 1 = I. (4.65) for 1 > 2 = I ( [ E ΣI, (F) D I ] [ + Cov F, F D]) I m 1,m 2 for 1 = 2 = I 0, otherwse or denomnator equals zero Replacng all unnown components n (4.65) (note that by Remar 4.14 v) we already have an estmate for the structural matrx E[Σ, (F) D ]) by the estmates. Ê [ Σ I, (F) D I ] := E[ΣI, (F) D ] Ĉov [ F, F D I ] := A I U I (4.66)

76 66 4 (Bayesan) Lnear Stochastc Reservng Methods leads to an estmator ϱ of ϱ. Replacng ϱ by ts estmate ϱ and usng the estmate Ê [ F D I ] := F I,Cred (4.67) n (4.64) and (4.63) we obtan the followng estmator for each summand n the frst addend of (4.62) Ê 1 I+1,(Cred) [Ŝm 1, 1 +1 Ŝ m 2 I+1,(Cred) 2, 2 +1 D I] = Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ)s I S I. Estmator 4.22 (Process varance for sngle accdent years) Under Model Assumptons 4.4 at tme I an estmator for the process varance (4.60) for sngle accdent years {I J + 1,..., I} s gven by Var [ α m Ŝ m I+1,(Cred),+1 DI m M =I := m 1,m 2 M α m 1 α m 2 ] ( 1, 2 =I (Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ) Ĥm 1,m 2 I,Cred, 1,, 2 (0) ) S I S ). I Estmaton Error for Sngle Accdent Years We get wth (4.58), Model Assumptons 4.4 c) and Lemma 4.12 ) S,+1 m := E [Ŝm I+1,(Cred),+1 D I] m I, F = P, S I, where P m I, F, := P m I m, F, m P 1 and P 1, = P m 1, F 1 1, 1, := P, m f = F P, m f =E[ F D I ] + > I + I and we defne ( ) 0 m M S n := Sm,. + n Havng ths notaton the estmaton error (4.61) can be decomposed nto M : = ( E = [ α m (Ŝm I+1,(Cred),+1 m M =I m 1,m 2 M 1, 2 =I ) ]) 2 Ŝm I,(Cred),+1 D I [ Sm,1+1 S m,2+1 S m,1+1ŝm 2 I,(Cred), 2 +1 (4.68) Ŝm 1 I,(Cred), 1 +1 S m, Ŝm 1 I,(Cred), 1 +1 ] Ŝ m 2 I,(Cred), 2 +1.

77 4.2 Bayesan Lnear Stochastc Reservng Methods 67 For the estmaton of (4.68) we apply the resamplng approach, see Wüthrch Merz [63],.e. we defne the resamplng probablty measure as the product measure of P ( F I,Cred ) ( A := P F I,Cred ) A D (4.69) and estmate the estmaton error n (4.68) by ts expectaton under ths resamplng probablty measure. We denote the correspondng expected value under P by E and the covarance by Cov, respectvely. For 1 > 2 and I we obtan for the terms n (4.68) E [ Sm 1 1, 1 +1 S m 2 E [ Ŝ m 1 I,Cred 1, 1 +1 S m 2 2, , 2 +1 E [ Sm 1 1, 1 +1Ŝm 2 I,Cred E [ Ŝ m 1 I,(Cred) 1, , 2 +1 Ŝ m 2 I,(Cred) 2, 2 +1 ]= E [ ] m P 1, F 1, S Sm 2 1 2, 2 +1 ]= E [ P m 1,F 1, 1 Ŝ 1+ 1 I,Cred Sm 2 ]= E [ ] m P 1, F 1, S Ŝ m 2 I,Cred 2, 2 +1 ]= E [ P m 1,F 1, 1 2, 2 +1 Ŝ 1+ 1 I,Cred Ŝ m 2 I,(Cred) 2, 2 +1 ] ] (4.70) wth P m,f, := P m, f =E[ F D ] and m, F P, := P, m f = F P, m f =E[ F D I ] + > I. + I Moreover, n the case of 1 = 2 =: the frst two denttes n (4.70) stll hold f at least one clam property les on or above the dagonal I + 1. Otherwse, for 1 = 2 =:, we get after short calculatons E [ Sm 1 S ] m 2 1,+1 2,+1 = E [ ( F m 1 E [ Ŝ m 1 I,(Cred) 1,+1 S m 2 2,+1 E [ Sm 1 2 I,(Cred) 1,+1Ŝm 2,+1 E [ Ŝ m 1 I,(Cred) 1,+1 Ŝ m 2 I,(Cred) 2,+1 F m 2 + ϱ 11 m 1,m 2 1, 2, ) R m 1 R ] m 1 1, 2, ] = E [ (E [ F m ] 1 D F m 2 + ϱ 12 m 1,m 2 1, 2, ] = E [ ( F m 1 E [ F m 2 ] = E [ (E [ F m ] [ 1 D E F m 2 ) R m 1 I,(Cred) 1, ] R m 2 2, ] ] D + ϱ 21 m 1,m 2 1, 2, ) R m 1 R m 2 I,(Cred) 1, 2, D ] + ϱ 22 m 1,m 2 1, 2, ) R m 1 I,(Cred) 1, ] R m 2 I,(Cred) 2, (4.71) where R m I,(Cred), := Γ m,ŝ+ I,(Cred) and Rm, := Γ m, S +

78 68 4 (Bayesan) Lnear Stochastc Reservng Methods and ϱ 11 m 1,m 2 1, 2, := ϱ 12 m 1,m 2 1, 2, := ϱ 21 m 1,m 2 1, 2, := ϱ 22 m 1,m 2 1, 2, := ( (I ) ( Z I A I T + U I ( ) ) A I ( ) ) I Z I m 1,m 2 for 1 + > I and 2 + > I 0 otherwse ( ( T + U I ( ) ) A I ( ) ) I Z I for 1 + I, 2 + > I A I m 1,m 2 0 otherwse ( (I ) ( Z I A I T + U I ( ) ) ) A I for 1 + > I, 2 + I m 1,m 2 0 otherwse ( ( T + U I ( ) ) ) A I for 1 + I and 2 + I A I m 1,m 2 0 otherwse. (4.72) Summarzng all parts and replacng all unnown parameters n (4.70), (4.71) and (4.72) by ther estmates leads to the followng estmator: Estmator 4.23 (Estmaton error for sngle accdent years) Under Model Assumptons 4.4 at tme I an estmator for the estmaton error (4.61) for sngle accdent years {I J + 1,..., I} s gven by M := m 1,m 2 M ( α m 1 α m 2 1, 2 =I (Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 11 ) Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 12 ) ) Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 21 ) + Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 22 ) S I S ). I Mean Squared Error of Predcton for Sngle Accdent Years Combnng the Estmators 4.22 and 4.23 mples Estmator 4.24 (MSEP for sngle accdent years) Under Model Assumptons 4.4 at tme I an estmator for the (condtonal) mean squared error of predcton for sngle accdent years {I J + 1,..., I} s gven by msep CDR M,I+1 D I [0] := α m 1 α m 2 m 1,m 2 M ( 1, 2 =I (Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ) Ĥm 1,m 2 I,Cred, 1,, 2 (0) + Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 11 ) Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 12 ) Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 21 ) + Ĥm 1,m 2 I,Cred, 1,, 2 ( ϱ 22 ) ) S I S ). I

79 4.2 Bayesan Lnear Stochastc Reservng Methods 69 Mean Squared Error of Predcton for Aggregated Accdent Years In the same way as for sngle accdent years we decompose the MSEP nto msep I CDR M,I+1 =I J+1 ( I := E = Var + [ ( =I J+1 m M I DI [0] (4.73) α m =I (Ŝm I+1,(Cred),+1 α m Ŝ m I+1,(Cred),+1 DI =I J+1 m M =I E [ I α m (Ŝm I+1,(Cred),+1 =I J+1 m M =I We decompose the process varance (4.74) by Var [ I =I J+1 m M α m Ŝ m I+1,(Cred),+1 DI =I ( I = E and the estmaton error (4.75) nto ( E [ I =I J+1 m M α m (Ŝm I+1,(Cred),+1 =I J+1 m M =I = I 1, 2 =I J+1 m 1,m 2 M α m ( E ( α m 1 1 α m 2 2 ] =I [ ) Ŝm I,(Cred),+1 ] Ŝ m I+1,(Cred),+1 I =I J+1 m M 0) 2 DI (4.74) ) ]) 2 Ŝm I,(Cred),+1 D I. (4.75) α m ) ]) 2 Ŝm I,(Cred),+1 D I 1, 2 =I ) 2 DI =I Ŝ m I+1,(Cred),+1 DI [ Sm 1,1+1 S m 2,2+1 S m 2,1+1Ŝm 2 I,(Cred), 2 +1 Ŝm 1 I,(Cred) 1, 1 +1 S m 2, Ŝm 1 I,(Cred) 1, 1 +1 We obtan n the same way as for sngle accdent years the followng result: ]) 2, ] ) Ŝ m 2 I,(Cred) 2, 2 +1.

80 70 4 (Bayesan) Lnear Stochastc Reservng Methods Estmator 4.25 (MSEP for aggregated accdent years) Under Model Assumptons 4.4 at tme I the mean squared error of predcton for aggregated accdent years can be estmated by msep := I CDR M,I+1 =I J+1 I 1, 2 =I J+1 m 1,m 2 M DI [0] ( α m 1 1 α m =I 1 2 =I 2 (Ĥm 1,m 2 I,Cred 1, 1, 2, 2 ( ϱ) Ĥm 1,m 2 I,Cred 1, 1, 2, 2 (0) + Ĥm 1,m 2 I,Cred 1, 1, 2, 2 ( ϱ 11 ) Ĥm 1,m 2 I,Cred 1, 1, 2, 2 ( ϱ 12 ) Ĥm 1,m 2 I,Cred 1, 1, 2, 2 ( ϱ 21 ) + Ĥm 1,m 2 I,Cred 1, 1, 2, 2 ( ϱ 22 ) ) S I S ). I Specal Case: Clams Development Result for the Bayes CL Method We saw n Subsecton that the Bayes CL method n Gsler Wüthrch [27] belongs to the class of Bayesan LSRMs. In Subsecton we showed that n the case of the Bayes CL method the Bayesan LSRM estmate for the MSEP concdes wth the estmate derved n Gsler Wüthrch [27] for the Bayes CL method. Now we consder whether ths also holds true for the MSEP of the CDR. The MSEP for the CDR n the Bayesan CL method was derved n Bühlmann et al. [13]. In ths dervaton only the process varance s taen nto account, whereas the estmaton error s set to 0. Ths becomes clear by Formula (4.5) n Bühlmann et al. [13], where n the frst step the approxmaton G n,cred E[G D n ] for n {I, I + 1} (exact credblty case) s used leadng to [ ( msep CDR 0,I+1 [0] = E CDR 0,I+1 D I 0 [ = Var [ Var CDR 0,I+1 CDR 0,I+1 ) 2 D I ] D I] [ + E CDR 0,I+1 D I] 2 D I], (4.76).e. the calculaton of the MSEP of the CDR s reduced to the calculaton of the condtonal varance (4.76). In the Bayesan LSRM we addtonally quantfy the estmaton error, see Estmator Thus, Estmator 4.24 for the MSEP of the CDR n the Bayesan LSRM and the estmator n Bayes CL method n Result 4.1 n Bühlmann et al. [13] do not concde. However, we show that the Estmator 4.22 for the process varance n the Bayesan LSRM s dentcal wth the estmator of the MSEP n the Bayes CL method n Result 4.1 n Bühlmann et al. [13]. In order to prove ths equalty we have to verfy that n the dervaton of the process varance (4.76) n both methods the same approxmatons are used. In the Bayes CL method n Bühlmann et

81 4.3 Example Bayesan LSRM 71 al. [13] Lemmata 4.5 and 4.6 use the followng approxmatons (wth slghtly other notaton) E [ σ 2 (G ) D I] Ê[ σ 2 (G ) D I] Var [ G D I] [( ) ] E G I,Cred G D I D E [ G D I] G I,Cred [ ( ) ] [ 2 ( Var G I+1,Cred D I E G I+1,Cred Recallng the denttes ) ] 2 ( G I,Cred D I D + F = G 1 and σ 2 (G ) = σ 0,0 (F 0 ) G I,Cred (cf. Subsecton 4.2.1) we see that exactly the same approxmatons are used n the Bayesan LSRM, see (4.66) and (4.67), for the estmaton of the process varance of the CDR. Consequently, the Estmator 4.22 and the estmator n Result 4.1 n Bühlmann et al. [13] concde. By the same arguments as above also follows that the MSEP of the CDR for several accdent years n the Bayes CL method, see Result 4.7 n Bühlmann et al. [13], concdes wth the Bayesan LSRM estmator for the process varance for several accdent years, gven by the frst lne of Estmator For a dscusson of the case that non-nformatve prors are used and the ln to the MSEP of the CDR n the classcal CL method we refer to Bühlmann et al. [13], Merz Wüthrch [45] and Wüthrch et al. [64]. ) Example Bayesan LSRM For a detaled nowledge of proftablty and a better understandng of prcng for dfferent busness unts (BU) we have to calculate best-estmate reserves and ts correspondng predcton uncertanty n terms of the MSEP for each BU. Furthermore, we consder the varablty (MSEP) of the CDR as a measure for the one-year reservng rs, what s requred under Solvency 2 and SST. For our example we revst the buldng engneerng data set of Wnterthur Insurance Company presented by Gsler-Wüthrch [27]. It contans trapezods of ncremental clams payments of sx BUs multpled by a constant due to confdentalty reasons. For smplcty and llustraton purposes we pc out BU 1 3 (of totally 6 BU) and apply the Bayesan LSRM. The data used s provded n Tables For the detaled specfcaton of the Bayesan LSRM we choose the exposure R, m to be the sum over all payments from all BUs n accdent year up to development year. In a smlar way we use for the couplng R m 1,m 2, of these three BUs all payments from all BUs n accdent year up to development year. For the structural parameter σ m 1,m 2 (F) we use the unbased estmator gven n (4.5).

82 72 4 (Bayesan) Lnear Stochastc Reservng Methods Method Reserves MSEP 1/ CL (135%) 288 (122%) 411 (58%) Cred CL (99%) 402 (164%) 520 (100%) LSRM (316%) 273 (80%) 346 (58%) Bayesan LSRM (195%) 222 (64%) 248 (42%) Table 4.1: Reserves and predcton uncertanty The pror covarance matrx T of the development factors F s, see (4.18), a component of the weghts A I gven to the pror mean µ and the observaton C. That means that T can be nterpreted as an nput parameter reflectng the actuares confdence n the data n comparson to the beleve n pror expert nowledge (see Theorem 4.13). In our example the confdence n the data of development year s the hgher, the more observatons are avalable n development year and thus we choose T = (10 ) I. In order to get a smoothng effect for the development pattern, we choose the pror mean µ to be the mean over all BUs of development factor estmates resultng from the classcal LSRM,.e. µ m := 1 3 ( f 0 + f 1 + f 2 ) for m {0, 1, 2}. Applyng the Bayesan LSRM for ths parameter constellaton leads to reserves and correspondng MSEP gven n the last lne of Table 4.1. Compared to the classcal LSRM results, we observe a smoothng effect (lower fluctuatons n the reserves of dfferent BUs) for the reserves n the credblty case. Ths s a drect consequence of the smoothng effect of credblty for each ndvdual development factor. The ncorporaton of pror nowledge for the mean of the development factors has a common nfluence on the development factors of all BUs,.e. the credblty development factors are nbetween the classcal LSRM development factors and the pror means. Ths smoothng effect of pror nformaton on the credblty development factors s llustrated for BU 1 n Fgure 4.2. For BU 2 and 3 we obtan smlar results (not shown here). Smlar smoothng effects were observed n the Bayes CL method n Gsler-Wüthrch [27]. For the predcton uncertanty (MSEP) we obtan slghtly lower values as n the classcal LSRM n all BUs. Ths s not always the case, because the predcton uncertanty depends drectly on the pror (co)varance for the development factors (see Estmator 4.16 for that). Now we tae a loo on the one-year reservng rs uncertanty n the CDR presented n Table 4.2. The one-year predcton uncertanty n the Bayesan LSRM only slghtly dffers from the uncertanty n classcal LSRM and the estmates are qute robust wth respect to dfferent pror choces for the development factor covarance matrx.

83 4.3 Example Bayesan LSRM LSRM BU 1 LSRM BU 2 LSRM BU 3 Cred LSRM BU development year Fgure 4.2: Development factors for BUs 1 3 n the classcal LSRM and credblty development 0 I,Cred factor F {0,..., 10} for BU 1 Method MSEP 1/ CDR MSEP 1/2 CDR /MSEP1/ LSRM % 76% 69% Bayesan LSRM % 93% 96% Table 4.2: Indvdual LoB and overall CDR uncertanty

84 74 4 (Bayesan) Lnear Stochastc Reservng Methods 4.4 Conclusons The classcal LSRMs presented n Dahms [17] consttute a wde class of dstrbuton-free stochastc clams reservng methods coverng many popular dstrbuton-free stochastc clams reservng methods such as the CL, BF and (E)CLR method. As already mentoned n Dahms [17] for settng up an adequate clams reservng method t s crucal to dentfy approprate exposures for the stochastc dynamcs n order to specfy a LSRM. If one s nterested n usng pror expert nowledge or nformaton from ndustry-wde data n the LSRM framewor the Bayesan LSRMs presented n ths chapter provde an approprate mathematcally consstent bass for the ncorporaton of such nformaton. Conservatve pror means for the development factors can generate rs-margns n the resultng credblty predctors and hence n the correspondng reserves. Moreover, Bayesan LSRMs provde the welcome effect of smoother development factors as shown n Fgure 4.2. Man results of (Bayesan) LSRMs: For solvency consderatons n Chapter 7 we summarze all quanttes of nterest derved n the LSRM framewor: 1. The predctor R I for outstandng loss labltes R I, see (2.5c) and (2.4c), gven by R I = I =I J+1 m M =I α m Ŝ m I,(Cred),+1 (4.77) 2. The estmator for the predcton uncertanty n terms of the (condtonal) MSEP msep I α m Sm,+1 =I J+1 m M =I DI [ I =0 m M =I α m Ŝ m I,(Cred),+1 ] (4.78) gven by Estmator The estmator for the CDR uncertanty n terms of the (condtonal) MSEP gven by Estmator msep I CDR M,I+1 =I J+1 DI [0] (4.79)

85 5 Pad-Incurred Chan Reservng Method In the prevous chapter we consdered the class of LSRMs whch covers many popular dstrbutonfree clams reservng methods and gves a new perspectve on these methods. In a second step we extended the LSRMs to the class of Bayesan LSRMs that allows for the ncorporaton of pror nowledge of the development pattern. In ths Bayesan LSRM framewor we stated explct predctors for the outstandng loss labltes R I and estmates for ts assocated predcton and CDR uncertanty. Ths shows that all classcal rs characterstcs n Table 2.1 can be calculated n the Bayesan LSRM framewor. However, wth respect to recent solvency regulaton n Solvency II and SST nsurance labltes have often to be evaluated by rs measures le VaR and ES, see AISAM ACME [2] and FOPI [24], and not only by the rs characterstcs gven n Table 2.1. The nowledge of the predctve dstrbuton of outstandng loss labltes an the CDR allows for the calculaton of such rs measures, see Robert [51], Merz Wüthrch [46] and Happ et al. [30]. That means that we are nterested n clams reservng modelng where the predctve dstrbuton of outstandng clams payments and the dstrbuton of the CDR can be derved (analytcally or smulatvely). A second mportant aspect whch s assgned to the choce of a clams reservng method s the data whch can be ncorporated n the method. In nsurance practce cumulatve clams payments and ncurred losses data are often avalable and should therefore be utlzed for the predcton of outstandng clams payments. Thus, we are loong for a flexble model ) whch s able to cope wth these two data sources and ) allows for the dervaton of the predctve dstrbuton of outstandng clams payments and the CDR. At frst, we consder the tas of modelng cumulatve clams payments and ncurred losses smultaneously. The MCL method n Quarg Mac [50] addresses ths problem, see also Secton 3.6. Ths method reduces the gap between the CL ultmate clam predctor based purely on cumulatve clams payments data and the CL ultmate clam predctor based on ncurred losses, respectvely, but does not close ths gap. However, snce the MCL method s dstrbuton-free no predctve dstrbuton can be derved. Moreover, to the best of our nowledge, even estmates for the (condtonal) MSEP of the ultmate clam and the CDR have not been found up to now. Another dstrbuton-free approach to the problem of the ncorporaton of cumulatve clams payments and ncurred losses n clams reservng s presented n Dahms [16] wth the ECLR 75

86 76 5 Pad-Incurred Chan Reservng Method method. The ECLR method was the frst clams reservng method whch can cope wth cumulatve clams payments and ncurred losses smultaneously leadng to one unfed ultmate clam predcton. It allows for the dervaton of predctors for the ultmate clam and the CDR and estmates for ther correspondng (condtonal) MSEP. Unfortunately, the ECLR method also does not allow for the dervaton of a predctve dstrbuton. An alternatve s presented n Merz Wüthrch [46] by a dstrbutonal approach. Based on Hertg s log-normal clams reservng method (cf. Hertg [32]) and Gogol s Bayesan clams reservng method for ncurred losses (cf. Gogol [28]), Merz and Wüthrch ntroduced the PIC reservng method and provded an ultmate clam predctor as well as the correspondng predcton uncertanty. In ths chapter we derve the uncertanty of the CDR for the PIC reservng method and calculate the predctve dstrbuton of the CDR. Ths s crucal for new solvency consderatons, see Chapter 7 as well as AISAM ACME [2] and FOPI [24], [25]. In ths chapter we follow Happ et al. [30]. Notatonal conventon: In ths and n the followng chapter we use the notaton employed n the PIC reservng method n Merz Wüthrch [46] for consstency reasons. Therefore, we recaptulate the notaton used n ths paper. 5.1 Notaton and Model Assumptons The PIC reservng method combnes two channels of nformaton: ) clams payments, whch correspond to the payments for reported clams; ) ncurred losses, whch refer to the reported clam amounts. In the followng, we assume I = J for notatonal smplcty, but all results hold true also n the case I > J. Cumulatve clams payments n accdent year after j development years are denoted by P,j and the correspondng ncurred losses by I,j. The crucal observaton s that the clams payments and ncurred losses tme seres must reach the same ultmate value, because these two tme seres both converge to the total ultmate clam. Therefore, we assume that all clams are settled and closed after development year J,.e. P,J = I,J holds wth probablty 1 for all {0,..., J}, see Model Assumptons 5.1. After accountng year t = J we have observatons n the pad and ncurred trangles gven by (see Fgure 5.1) D J := {P,j, I,j ; 0 J, 0 j J, 0 + j J}. After accountng year t = J + 1 we have observatons n the pad and ncurred trapezods gven by (see Fgure 5.2) D J+1 = {P,j, I,j ; 0 J, 0 j J, 0 + j J + 1}. Ths means the updatng of nformaton D J D J+1 adds a new dagonal to the observatons.

87 5.1 Notaton and Model Assumptons 1 77 development years J clams payments ncurred losses 1 accdent years J... P,j P,J =I,J I,j J... Fgure 5.1: 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 1 development years J clams payments ncurred losses 1 accdent years J... P,j P,J =I,J I,j J... Fgure 5.2: Updated cumulatve clams payments P,j and ncurred losses I,j observed at tme t = J + 1 Our goal s to predct the ultmate losses P,J = I,J, = 1,..., J, based on the nformaton D J and D J+1, respectvely. We state the PIC model, whch combnes both cumulatve payments and ncurred losses nformaton: Model Assumptons 5.1 (PIC model) a) Condtonally, gven the parameter vector Θ := (Φ 0 ; Φ 1, Ψ 1, Φ 2, Ψ 2,..., Φ J, Ψ J ), we assume: - the random vectors Ξ := (ξ,0 ; ξ,1, ζ,1, ξ,2, ζ,2,..., ξ,j, ζ,j ) are..d. wth multvarate Gaussan dstrbuton Ξ N (Θ, V) for {0, 1,..., J} and postve defnte covarance matrx V as well as ndvdual development factors ξ,j := log P,j P,j 1 and ζ,l := log I,l I,l 1,

88 78 5 Pad-Incurred Chan Reservng Method for j {0, 1,..., J} and l {1, 2,..., J}, where we have set P, 1 := 1; - P,J = I,J, P-a.s., for all {0, 1,..., J}. b) The components of Θ are ndependent wth pror dstrbutons Φ j N (φ j, s 2 j) for j {0,..., J} and Ψ l N (ψ l, t 2 l ) for l {1,..., J} wth pror parameters φ j, ψ l R and s 2 j > 0, t2 l > 0. Remars 5.2 (PIC reservng method) ) In Model Assumptons 5.1 we can choose any arbtrary postve defnte covarance matrx V. Ths allows for modelng dependence structures between clams payments ratos and ncurred losses ratos I,l I,l 1. ) Expert opnon should be ncluded to structure the covarance matrx V. P,j P,j 1 For a more detaled dscusson on ths topc and sutable choces for V we refer to Happ Wüthrch [31]. However, the problem of fndng statstcally optmal estmators should be subject to further statstcal research. ) We defne pror dstrbutons for the components of the mean vector Θ and assume V to be a gven covarance matrx. Ths Bayesan approach guarantees closed form results. If we also put a pror on V we have to use Marov-Chan-Monte-Carlo (MCMC) methods for the calculaton of the posteror dstrbuton (see Merz Wüthrch [46]). 5.2 One-year Clams Development Result We consder the short term (one-year) run-off rs ntroduced n Secton 2.6. Ths means, we study the uncertanty n the one-year CDR for accountng year J + 1 gven by CDR J+1 = E[P,J D J ] E[P,J D J+1 ], = 1,..., J, between the best estmates for the ultmate clam P,J at tmes J and J + 1. The one-year CDR n accountng year J + 1 measures the change n the predcton by updatng the nformaton from D J to D J+1. Wth the tower property of the condtonal expectaton we obtan for the expected one-year CDR for accdent year, vewed from tme J, [ ] E D J = 0, CDR J+1 whch s the martngale property of successve predctons. Ths justfes the fact that, n the budget statement, the one-year CDR s usually predcted by 0 at tme J. In the followng we

89 5.3 Expected Ultmate Clam at Tme J study the uncertanty n ths predcton by means of the condtonal MSEP, gven the observatons D J. In other words we calculate, see Wüthrch Merz [63], Secton 3.1, [ ( ) ] 2 [ ] msep CDR J+1 D J [0] = E CDR J+1 0 D J = Var CDR J+1 D J = Var[E[P,J D J+1 ] D J ]. The condtonal MSEP s probably the most popular uncertanty measure n clams reservng practce and has the advantage that t can be derved analytcally n the PIC model. Moreover, we also present the full predctve dstrbuton below, whch also allows to evaluate other rs measures. (5.1) 5.3 Expected Ultmate Clam at Tme J + 1 In ths secton we derve the condtonal expected ultmate clam E[P,J D ] for {J, J + 1} n two steps. E[P,J D ], see Corollary 5.7. In the frst step we derve E[P,J Θ, D ] and n the second step we calculate In the followng we can ether wor wth the random vector Ξ R 2J+1 (see Model Assumptons 5.1) or wth the logarthmzed observatons of accdent year, namely, X := (log P,0, log I,0, log P,1,..., log P,, log I,, log P,J ) R 2J+1. Ths s possble, snce there exst an nvertble matrx B R (2J+1) (2J+1) such that X = B Ξ,.e. there s a one-to-one correspondence between X and Ξ. Ths mples X Θ = B Ξ Θ N (µ := BΘ, Σ := BVB ). (5.2) Let {J, J + 1} and defne n := 2J + 1 and q := q () := 2( + 1). To smplfy notaton we defne: X (1), := (log P,0, log I,0, log P,1, log I,1..., log P,, log I, ) R q for < J, X otherwse; X (2), := (log P, +1, log I, +1,..., log P,, log I,, log P,J ) R n q for < J, (log P,J ) otherwse. X (1), descrbes the observatons at tme {J, J +1},.e. t corresponds to the σ-feld generated by D, see Fgures 5.1 and 5.2. X (2), at tme for > J. s the part of clams development that needs to be predcted For < J we decompose the transformaton matrx B n a smlar way nto ( ) (1) B, B := B (2), (5.3),

90 80 5 Pad-Incurred Chan Reservng Method where B (1), Rq n. For J we set B (1), := B and B(2), n ths case. We obtan, for < J, a decomposton of the mean vector, where [ µ (1), := E X (1), ] Θ µ = BΘ = ( µ (1),, µ(2), ) R n [ = B (1), Θ and µ(2), := E := B(2) 1,J, but (5.3) does not hold X (2), ] Θ = B (2), Θ. For < J the covarance matrx s decomposed n a smlar way such that ( (11) Σ Σ = BVB, Σ (12) ), = Σ (21), Σ (22) R n n, (5.4), wth Σ (11), R q q. For J we set Σ (11), = Σ, Σ (12), = Σ (12) 1,J and Σ(22), = Σ (22) 1,J, but (5.4) does not hold n ths case. Now havng ths notaton we provde the followng lemma: Lemma 5.3 (Condtonal dstrbuton) Choose {J, J + 1} and > J. Under Model Assumptons 5.1 we obtan for the condtonal dstrbuton of X (2),, gven {Θ, D }, where For = J we obtan X (2), {Θ,D } = X (2), {Θ,X (1), } N ( µ (2), µ (2),, Σ (22), ), ( ) := µ(2), + Σ(21), (Σ(11), ) 1 X (1), µ(1), R n q, Σ (22), := Σ (22), Σ(21), (Σ(11), ) 1 Σ (12),. (log P,J +1, log I,J +1 ) {Θ,DJ } N (µ, Σ ) for {2,..., J}, wth µ := ( ) e 1 e 2 µ (2),J and Σ := ( ) e 1 e 2 ) Σ (22),J (e 1 e 2, (5.5) where e := e () R n q s the -th canoncal bass vector of dmenson n q. Moreover, for = 1 we have log P 1,J {Θ,DJ } N ( µ 1 := µ (2) 1,J, Σ 1 := ) (22) Σ (1,J). Proof: Condtonally gven the parameter vector Θ, the random vectors X are ndependent for dfferent accdent years. Therefore, the condtonal dstrbuton of X (2), depends on D only through X (1),. Ths shows the frst equalty n the frst clam. The dstrbutonal clam s a well-nown result for multvarate normal dstrbutons usng the Schur complement for the calculaton of the condtonal covarance matrx. The second clam s a drect consequence of the frst clam. Ths proves the lemma.

91 5.3 Expected Ultmate Clam at Tme J Remars 5.4 (Condtonal dstrbuton) ) The second clam n Lemma 5.3 s used to derve the dstrbuton of the elements n the next dagonal D J+1 \D J. Ths s needed for the calculaton of the full predctve dstrbuton of the CDR va Monte-Carlo methods. For detals see Secton 5.5. As a drect consequence of the frst clam n Lemma 5.3 we get for the ultmate clam, > J, ( ) log I,J {Θ,D } = log P,J {Θ,D } N e n q µ(2),, e (22) n q Σ, e n q. (5.6) Ths mmedately mples the followng corollary: Corollary 5.5 (Condtonal dstrbuton) For the predctor of the ultmate clam P,J, gven {Θ, D J }, we obtan for > J E[P,J Θ, D ] = exp { } e n q µ(2), + e (22) n q Σ, e n q /2. Proof: The clam s a drect consequence of Lemma 5.3 and (5.6). We see that the ultmate clam predctor n Corollary 5.5 stll depends on Θ, namely through ( ( ) ) e n q µ(2), = e n q µ (2), + Σ(21), (Σ(11), ) 1 X (1), µ(1), ( ( ) ) = e n q B (2), Θ + Σ(21), (Σ(11), ) 1 X (1), B(1), Θ = Γ, Θ + e n qσ (21), (Σ(11), ) 1 X (1),, (5.7) where Γ, s gven by Γ, := e n q ( ) B (2), Σ(21), (Σ(11), ) 1 B (1),. Our am now s to calculate the posteror dstrbuton of Θ, condtonally gven observatons D for {J, J + 1}. The lelhood of the logarthmzed observatons at tme, gven Θ, s gven by l D (Θ) J =0 { exp 1 ( ) ( X (1) ) } 2, B(1), Θ (11) (Σ, ) 1 X (1), B(1), Θ. (5.8) Wth Model Assumptons 5.1 and Bayes theorem follows that the posteror dstrbuton u(θ D ) has the form wth pror mean { u(θ D ) l D (Θ) exp 1 } 2 (Θ ϑ) T 1 (Θ ϑ), (5.9) ϑ := (φ 0 ; φ 1, ψ 1, φ 2, ψ 2,..., φ J, ψ J ) R n

92 82 5 Pad-Incurred Chan Reservng Method and pror covarance matrx T := dag(s 2 0; s 2 1, t 2 1, s 2 2, t 2 2,..., s 2 J, t 2 J) R n n. (5.10) Theorem 5.6 (Posteror dstrbuton of Θ) Under Model Assumptons 5.1 the posteror dstrbuton u(θ D ) s a multvarate Gaussan dstrbuton wth posteror mean ϑ(d ) := T(D ) and posteror covarance matrx T(D ) := ( [ T 1 + T 1 ϑ + J =0 J =0 (B (1), ) (Σ (11), ) 1 X (1), (B (1), ) (Σ (11), ) 1 B (1), ) 1, Proof: From (5.8) mmedately follows that the posteror dstrbuton u(θ D ) s a multvarate Gaussan dstrbuton. Therefore, t remans to calculate the frst two moments of u(θ D ). Ths s done by squarng out all terms and analyzng quadratc and lnear terms. From (5.7) we see that the exponent of the predctor gven n Corollary 5.5 s a affne-lnear functon of Θ. Usng Theorem 5.6 ths mples the followng corollary: ]. Corollary 5.7 (Ultmate clam predctor) The predctor for the ultmate clam for accdent year > J and {J, J + 1}, gven D, s gven by { } E[P,J D ] = exp Γ, ϑ(d ) + Γ, T(D )(Γ, ) /2 + e n qσ (21), (Σ(11), ) 1 X (1), + e (22) n q Σ, e n q /2. Proof: The proof s a drect consequence of Corollary 5.5 and Theorem 5.6. Remars 5.8 (Ultmate clam predctor) ) For = J and dagonal covarance matrx V we obtan the same ultmate clam predctor as n Merz Wüthrch [46]. ) For = J + 1 we get a closed formula for the ultmate clam predctor n the case that nformaton D J+1 s avalable at tme J + 1. Ths allows for the smulaton of the full predctve dstrbuton of the CDR. Ths s done n detal n Secton 5.5. ) For other choces of pror dstrbutons MCMC methods can be appled to calculate the posteror dstrbuton n Theorem 5.6. For detals see Merz Wüthrch [46].

93 5.4 Mean Squared Error of Predcton of the Clams Development Result Mean Squared Error of Predcton of the Clams Development Result Sngle Accdent Years In the last secton we have calculated the expected ultmate clam n the PIC reservng model, gven the observatons D for {J, J + 1}. Our am now s to calculate the predcton uncertanty of the CDR n terms of the condtonal MSEP. From (5.1) we see that the problem to derve the condtonal MSEP for the one-year CDR s solved by calculatng Var[E[P,J D J+1 ] D J ]. Snce (E[P,J D J ]) 2 s gven by Corollary 5.7 for = J, ths condtonal varance can be derved by calculatng E [(E[P,J D J+1 ]) 2 ] DJ. We see that for = J + 1 the exponental term from Corollary 5.7, namely, Γ,J+1 ϑ(d J+1 ) + Γ,J+1 T(D J+1 )(Γ,J+1 ) /2 + e n qσ (21),J+1 (Σ(11),J+1 ) 1 X (1),J+1 + e (22) n q Σ,J+1e n q /2, s affne-lnear n the observatons D J+1 \D J gven by Y := (log P 1,J, log P 2,, log I 2,,..., log P J,1, log I J,1 ). That means that for all > 1 there exst a matrx L and a D J -measurable random varable g (D J ) such that L Y + g (D J ) = Γ,J+1 ϑ(d J+1 ) + Γ,J+1 T(D J+1 )(Γ,J+1 ) /2 + e n qσ (21),J+1 (Σ(11),J+1 ) 1 X (1),J+1 + e (22) n q Σ,J+1e n q /2. For = 1 we set L 1 to be the projecton on the frst component,.e. L 1 Y := log P 1,J and g 1 (D J ) = 0. Ths mples for the ultmate clam predctor n Corollary 5.7 E[P,J D J+1 ] = exp{l Y + g (D J )} for = 1,..., J. (5.11) Dfferent accdent years are ndependent, gven Θ. Thus, Lemma 5.3 leads to the jont dstrbuton of Y, gven {D J, Θ}: Lemma 5.9 (Condtonal dstrbuton of Y) Under Model Assumptons 5.1 we have Y {DJ,Θ} = (log P 1,J, log P 2,, log I 2,,..., log P J,1, log I J,1 ) {DJ,Θ} N (µ, Σ), where µ 1 µ 2 µ := R 2 and Σ :=. µ J wth µ and Σ defned n Lemma 5.3. Σ Σ Σ J R (2) (2),

94 84 5 Pad-Incurred Chan Reservng Method In Lemma 5.9 the dstrbuton of Y {DJ,Θ} stll depends on Θ va µ = ( µ 1, µ 2,..., µ ) J R 2 and recallng the defnton of µ (see Lemma 5.3) we obtan, for = J, e 1 Σ (21) γ := e,j (Σ(11),J ) 1 X (1),J for 2 2 e n qσ 21 1,J (Σ(11) 1,J ) 1 X (1) 1,J for = 1 Γ,J Θ + γ for 2 µ = Γ 1,J Θ + γ 1 for = 1 and Γ,J := e 1 e 2 e n q ( ) B (2),J Σ(21),J (Σ(11),J ) 1 B (1),J ( ) B (2) 1,J Σ(21) 1,J (Σ(11) 1,J ) 1 B (1) 1,J for 2. for = 1 Next, we defne the matrx Γ wth rows Γ,J,.e. Γ := ( Γ 1,J Γ 2,J... Γ J,J ) R (2) n. and γ := (γ 1,..., γ J) R (2). Ths shows that µ = ΓΘ + γ s a affne-lnear functon of Θ. Ths mples together wth (5.11) the followng theorem. Theorem 5.10 (Condtonal expectaton) Under Model Assumptons 5.1 we obtan for, l {1,..., J} E[E[P,J D J+1 ] E[P l,j D J+1 ] Θ,D J ] = exp { (L +L l )µ+(l +L l )Σ(L +L l ) /2+g (D J )+g l (D J ) }, and E[E[P,J D J+1 ] E[P l,j D J+1 ] D J ] = E[P,J D J ] E[P l,j D J ] exp{l ΓT(D J )Γ L l + L ΣL l }. Proof: Usng standard propertes of log-normal dstrbuton, the frst clam mmedately follows by Lemma 5.9 and (5.11). The second clam follows wth the dentty µ = ΓΘ +γ and Theorem 5.6. By means of ths relatonshp between E [(E[P,J D J+1 ]) 2 ] DJ and E[P,J D J ] 2 t s straghtforward to derve the (condtonal) MSEP of the one-year CDR for sngle accdent years, whch s gven n the next theorem:

95 5.4 Mean Squared Error of Predcton of the Clams Development Result 85 Theorem 5.11 (Condtonal MSEP for sngle accdent years) Under Model Assumptons 5.1 the condtonal MSEP, gven D J, of the one-year CDR for sngle accdent years {1,..., J} s gven by msep CDR J+1 D J [0] = (E[P,J D J ]) 2 ( exp{l ΓT(D J )Γ L + L ΣL } 1 ). In the followng secton we consder the condtonal MSEP for aggregated accdent years Aggregated Accdent Years We study the condtonal MSEP of the one-year CDR for aggregated accdent years: msep J CDR J+1 =1 D J ( J [0] = E = Var =1 ( J =1 CDR J+1 0 CDR J+1 ) 2 D J ) ( J ) D J = Var E [P,J D J+1 ] D J. =1 (5.12) Usng the tower property of condtonal expectatons and Theorem 5.10 we obtan for (5.12): Theorem 5.12 (Condtonal MSEP for aggregated accdent years) Under Model Assumptons 5.1 the condtonal MSEP, gven D J, of the one-year CDR for aggregated accdent years s gven by msep J CDR J+1 =1 D J [0] = J =1 msep CDR J+1 D J [0] + 2 l> E[P,J D J ] E[P l,j D J ] ( exp { L ΓT(D J )Γ L l + L ΣL l} 1 ). Proof: Wth (5.12) and Theorem 5.10 we obtan msep J = CDR J+1 =1 D J J =1 [0] = J Var [E[P,J D J+1 ] D J ] + 2 =1 msep CDR J+1 D J [0] + 2 J E [E[P,J D J+1 ] E[P l,j D J+1 ] D J ] l> 2 J E[P,J D J ] E[P l,j D J ] J E[P,J D J ] E[P l,j D J ] ( exp { L ΓT(D J )Γ L l + L ΣL ) l} 1. l> l>

96 86 5 Pad-Incurred Chan Reservng Method 5.5 Example PIC Reservng Method We revst the data gven n Dahms [16]. In Model Assumptons 5.1 we can choose any covarance matrces V as long as t s postve defnte. Ths allows for modelng dependence between pad and ncurred data. The tas of structurng a sutable covarance matrx V based on expert opnon and data s dscussed n detal n Happ-Wüthrch [31], see Chapter also 6. In ths example we choose V as a dagonal matrx and estmate the varances on the dagonal wth standard sample estmates. Ths s n-lne wth the choce of V n Merz Wüthrch [46]. More detaled, for the estmaton of V we use for j {0,..., J 1} and {1,..., J 1} J j 1 J 1 Φ j := ξ,j, Ψ := ζ,, J j + 1 J + 1 =0 =0 σ ξ 2 j := 1 J j (ξ,j J j Φ j ) 2 and σ ζ 2 := 1 J (ζ, J Ψ ) 2. Snce we have for the estmaton of the two parameters σ 2 ξ J and σ 2 ζ J only one observaton we use the extrapolaton formula, see Wüthrch Merz [63], and set σ 2 ξ J := mn{ˆσ 2 ξ J 2, ˆσ 2 ξ, ˆσ 4 ξ J 2 /ˆσ 2 ξ } and σ 2 ζ j := mn{ˆσ 2 ζ J 2, ˆσ 2 ζ, ˆσ 4 ζ J 2 /ˆσ 2 ζ } V := dag ( σ 2 ξ 0, σ 2 ξ 1, σ 2 ζ 1,..., σ 2 ξ J, σ 2 ζ J ). Because we do not have any pror nowledge of the pror dstrbuton parameters φ l and ψ j we choose non-nformatve prors,.e. we let s 2 j and t2 l. Ths mples that n Theorem 5.6 the matrx T 1, see (5.10), s the matrx consstng of zeros and no pror nformaton s used n our calculatons. In Table 5.1 we compare the predcton uncertanty measured by the square root of the condtonal MSEP for the one-year CDR calculated by the PIC method and the ECLR method (cf. Dahms [16]). Under Model Assumptons 5.1, these values for the PIC method are calculated analytcally wth Theorem 5.11 for sngle accdent years and wth Theorem 5.12 for aggregated accdent years. Note that for the ECLR method we obtan two dfferent values for the (condtonal) MSEP because we can estmate the varance n two ways, namely based on pad data or based on ncurred data, respectvely. We observe n the PIC method for most sngle accdent years and aggregated accdent years a lower predcton uncertanty for the CDR than n the ECLR method based on pad or ncurred data (see Table 5.1). Ths can partly be explaned by the fact that n the ECLR method we have to estmate 44 parameters (cf. Dahms [16]) whereas n the Bayesan PIC model only 19 varance parameters have to be estmated leadng to a lower standard error. Moreover, we observe that n the PIC method t s not unlely that the total clams reserves ncrease about 3% n the one-year horzon. Ths s smlar to the fndngs for the CDR uncertanty for the ECLR method n Dahms et al. [18]. =0 =0

97 5.5 Example PIC Reservng Method 87 clams msep 1/2 CDR clams msep 1/2 CDR n % accdent reserves ECLR method ECLR method reserves PIC method reserves year ECLR Pad Incurred PIC PIC ,78% ,51% ,31% ,65% ,86% ,34% ,13% ,93% ,23% Total ,76% Table 5.1: 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 Table 5.2 provdes the ratos of the square root of the condtonal MSEP for the one-year CDR and the square root of the condtonal MSEP for the ultmate clam. We observe that for later accdent years (.e. 7) and aggregated accdent years the values for the ECLR method and for the PIC method only slghtly dffer. Moreover, we see that for aggregated accdent years the one-year uncertanty s about 75% of the uncertanty of the ultmate clam predcton. Ths result s n lne wth the feld study conducted by AISAM ACME [2]. msep 1/2 CDR /msep1/2 Ultmate msep 1/2 CDR /msep1/2 Ultmate accdent ECLR method ECLR method PIC method year Incurred Pad Pad & Incurred % 100.0% 100.0% % 84.5% 87.6% % 52.7% 83.7% % 89.6% 62.4% % 69.1% 94.3% % 65.4% 80.8% % 93.9% 93.1% % 70.1% 70.3% % 65.3% 66.4% Total 74.1% 74.3% 75.2% Table 5.2: 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 As already mentoned n the PIC method we can not only calculate the condtonal MSEP for the one-year CDR but also the full predctve dstrbuton of the one-year CDR by means of MC smulatons. Frstly, we apply Theorem 5.6 to u(θ D J ) to generate Gaussan samples Θ (n) wth

98 88 5 Pad-Incurred Chan Reservng Method Normal Densty Emprcal Densty Fgure 5.3: Emprcal densty for the one-year CDR (blue lne) from smulatons and ftted Gaussan densty wth mean 0 and standard devaton (dotted red lne) mean ϑ(d J ) and covarance matrx T(D J ). Secondly, we generate ndependent two-dmensonal Gaussan samples (log P,J +1, log I,J +1 ) {DJ,Θ} and fll up the off-dagonal entres n the pad and ncurred trapezods (see Lemma 5.3). Ths way we obtan the data avalable at tme J + 1,.e. D J+1, and can calculate E[P,J D J+1 ] by means of Corollary 5.7. Ths provdes Fgure 5.3, where we compare the emprcal densty from smulatons (blue lne) to the Gaussan densty wth mean µ = 0 and standard devaton σ = (dotted red lne), see Table 5.1. We observe that these two denstes loo qute smlar. To get a closer loo on the left tal of the emprcal densty for the one-year CDR we show a QQ-plot for quantles q (0, 0.1). We observe that the tal behavour of the emprcal densty of the one-year CDR and the ftted Gaussan densty wth mean 0 and standard devaton only slghtly dffer (see Fgure 5.4). Ths s smlar to the fndngs for the dstrbuton of the ultmate clam n Merz Wüthrch [46]. Ths means that usng a Gaussan approxmaton for the densty of the one-year CDR provdes wthn the PIC method and for the gven data a good approxmaton for the shortfall rs of the one-year CDR. 5.6 Conclusons The PIC reservng method provdes a framewor, where unfed ultmate clam predctons can be calculated based on cumulatve payments and ncurred losses data smultaneously. It allows for the dervaton of the (condtonal) MSEP for the ultmate clam n the long run as well as for

99 5.6 Conclusons 89 Emprcal Quantles Theoretcal Quantles Fgure 5.4: 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 the CDR n the one-year tme horzon. Merz Wüthrch [46] derved the MSEP formula for the ultmate clam uncertanty. In ths chapter we dd the same for the one-year CDR uncertanty. In contrast to the ECLR method by Dahms [16], where also MSEP formulas for the ultmate clam and the CDR uncertanty exst, the PIC method allows for the calculaton of the full predctve dstrbuton of the ultmate clam and the CDR va Monte-Carlo smulatons. Ths mples that any other rs measure for example VaR or ES can be calculated for the ultmate clam uncertanty (long term rs) as well as for the CDR uncertanty (one-year rs). Man results of the PIC reservng method: For solvency consderatons n Chapter 7 we summarze all quanttes of nterest derved n the PIC reservng method: 1. The predctor R I for outstandng loss labltes R I, see (2.5c) and (2.4c) gven by R I = J (E[P,J D J ] P,J ), (5.13) =1

100 90 5 Pad-Incurred Chan Reservng Method see Corollary 5.7 and Ŝ 0 I,Bayes, := E[P, D J ] E[P, 1 D J ] 2. The estmator for the predcton uncertanty n terms of the (condtonal) MSEP [ ] msep R I D J RI, (5.14) gven by Theorem 4.1 n Merz Wüthrch [46]. 3. The estmator for the CDR uncertanty n terms of the (condtonal) MSEP gven by Theorem msep J CDR J+1 =1 D J [0], (5.15)

101 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng As mentoned n the prevous secton, the classcal PIC reservng method ntroduced n Merz Wüthrch [46] s one of the frst clams reservng methods whch can cope wth three sources of nformaton: () clams payments for reported clams; () ncurred losses whch correspond to the reported clam amounts; () pror expert opnon whch can be used to desgn the pror covarance matrx V and pror means. The ntal verson of the PIC reservng method assumes V to be dagonal and hence does not allow for dependence modelng between clams payments and ncurred losses data. We revst the problem of the classcal PIC reservng method and generalze t to allow for approprate dependence modelng. In ths secton we follow Happ Wüthrch [31]. 6.1 Notaton and Model Assumptons For the PIC model we consder three channels of nformaton: () clams payments, whch refer to the payments done for reported clams; () ncurred losses, whch correspond to the reported clam amounts; () pror expert opnon. As already mentoned n Chapter 5 the crucal observaton s that the clams payments and ncurred losses tme seres must reach the same ultmate value, because these two tme seres both converge to the total ultmate clam. By choosng approprate model assumptons we force ths property to hold true n our model. In the same way as n Chapter 5 we denote accdent years by {0,..., J} and development years by j {0,..., J}. We assume that all clams are settled after the J-th development year. Cumulatve clams payments n accdent year after j development perods are denoted by P,j and the correspondng ncurred losses by I,j. Moreover, for the ultmate clam we assume (force) P,J = I,J wth probablty 1, whch means that ultmately (at tme J) they reach the same ultmate value. For an llustraton we refer to Table 6.1. The PIC model wth dependence s defned as follows: 91

102 Pad-Incurred Chan Reservng Method wth Dependence Modelng development years J clams payments ncurred losses 1 accdent years J... P,j P,J =I,J I,j J... Table 6.1: 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 Model Assumptons 6.1 (PIC model wth dependence) a) Condtonally, gven the parameter vector Θ := (Ψ 0 ; Ψ 1, Φ 1, Ψ 2, Φ 2,..., Ψ J, Φ J ), we assume: - the random vectors Ξ := (ζ,0 ; ζ,1, ξ,1, ζ,2, ξ,2,..., ζ,j, ξ,j ) R 2J+1 are..d. wth multvarate Gaussan dstrbuton Ξ N (Θ, V) for {0,..., J}; and postve defnte covarance matrx V R (2J+1) (2J+1) as well as ndvdual development factors ζ,j := log I,j I,j 1 and ξ,l := log P,l P,l 1, (6.1) for j {0,..., J} and l {1,..., J}, where we have set I, 1 := 1; - P,J = I,J, P-a.s., for all {0,..., J}. b) The components of Θ are ndependent wth pror dstrbutons Ψ j N ( ψ j, t 2 j) for j {0,..., J} and Φl N ( φ l, s 2 l ) for l {1,..., J} wth pror parameters ψ j, φ l R and t 2 j > 0, s2 l > 0. The only dfference between Model Assumptons 5.1 of the PIC model and Model Assumptons 6.1 of the PIC model wth dependence s that I,j and P,j have changed roles and I,j are used as prors for P,j, see Remar 6.2 ) for detals.

103 6.1 Notaton and Model Assumptons 93 Remars 6.2 (PIC model wth dependence) ) For V = dag(τ0 2; τ 1 2, σ2 1,..., τ J 2, σ2 J ) we obtan the PIC reservng model from Merz Wüthrch [46]. In the followng we allow for general covarance matrces V (as long as they are postve defnte). In (6.3) below, we gve an explct choce that wll be appled to a motor thrd party lablty portfolo. ) The PIC model combnes both cumulatve payments and ncurred losses data to get a unfed predctor for the total ultmate clam that s based on both sources of nformaton. Thereby, the model assumpton P,J = I,J guarantees that the total ultmate clam concdes for clams payments and ncurred losses data. In partcular, we obtan by (6.1) the denttes I,j = I,j 1 exp {ζ,j }, wth ntal value I,0 = exp {ζ,0 }, and by bacwards recurson P,j 1 = P,j exp { ξ,j }, wth ntal value P,J = I,J. (6.2) Note that n comparson to Merz Wüthrch [46] we have exchanged the role of I,j and P,j. In the orgnal model of Merz Wüthrch [46] the resultng clams reserves are completely symmetrc n the exchange of I,j and P,j. If we consder the model wth dependence, as n Model Assumptons 6.1 above, t s more natural to use ncurred losses I,J as pror for clams payments P,j. Ths means that Hertg s log-normal model [32] for I,j plays the role of the pror for Gogol s clams reservng model [28] for P,j, see also Merz Wüthrch [46]. ) If we have pror (expert) nowledge (as a thrd nformaton channel) ths can be used to desgn the pror dstrbuton of Θ. If there s no pror nowledge we choose non-nformatve prors for Θ, that s we let t 2 j and s2 l for j {0,..., J} and l {1,..., J}. v) The assumpton P,J = I,J means that all clams are assumed to be settled after J development years and there s no so-called tal development factor. If there s a clams development beyond development year J, then one can extend the PIC model for the estmaton of a tal development factor, see Merz Wüthrch [42] for more detals. v) Under Model Assumptons 6.1 the dstrbuton of the ultmate clams I,J are a pror equal across accdent years. However, gven the observed data, we observe dfferent posteror dstrbutons for clams of dfferent accdent years. Therefore, the PIC reservng method allows for accdent year varaton (see Corollary 6.6). However, f nowledge of pror dfferences s avalable t should be ncorporated n the pror means. Ths relaxaton of the model assumpton wll stll lead to closed form solutons. A smlar effect can be acheved by consderng (volume-) adjusted observatons.

104 94 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng v) Condtonal..d. guarantees that we obtan a model of CL type, see (6.2), where CL factors do not depend on accdent year. Of course, ths model assumpton requres that the data consdered need to be suffcently regular. If ths s not the case, one can ntroduce pror dfferences between accdent years (see also last bullet pont). These more general assumptons stll lead to a closed form soluton. The drawbac s that the model mght become over-parametrzed and/or t requres extended expert nowledge. v) The covarance matrx V allows for modelng dependence wthn Ξ. In partcular, we wll choose ths covarance matrx such that the correlaton between ζ,j and ξ,j s postve because P,j s contaned n I,j (and hence they are dependent). Ths last bullet pont s motvated by the followng argument: a postve change (an ncrease) from I,j 1 to I,j means that the clams adjusters ncrease ther expectaton n future clams payments. One part of ths ncreased expectaton s mmedately pad n development perod j (and hence contaned n both I,j and P,j ) and the remanng ncreased expectaton s pad wth some settlement delay, whch means that we also have hgher expectatons for P,l, l > j. Ths argument leads to the followng possble explct choce for the correlaton matrx Ṽ (note that we have to dfferentate between the covarance matrx V and ts assocated correlaton matrx Ṽ of the random vector Ξ ) ζ,0 ζ,1 ξ,1 ζ,2 ξ,2 ζ,3 ξ,3 ζ,4 ξ,4 ζ,j ξ,j Ṽ := ζ,0 1 0 ρ 1 0 ρ ζ,1 0 1 ρ 0 0 ρ 1 0 ρ ξ,1 ρ 1 ρ ζ, ρ 0 0 ρ 1 0 ρ ξ,2 ρ 2 ρ 1 0 ρ ζ, ρ 0 0 ρ ξ,3 0 ρ 2 0 ρ 1 0 ρ ζ, ρ ξ, ρ 2 0 ρ 1 0 ρ (6.3) ζ,j ρ 0 ξ,j ρ 0 1 The ratonal behnd ths correlaton matrx s that the ncurred losses ncrements ζ,j are (postvely) correlated to the clams payments ncrements ξ,j, ξ,j+1 and ξ,j+2 wth postve correlatons ρ 0, ρ 1 and ρ 2, respectvely. ζ,0 plays the specal role of the ntal value for ncurred losses I,0 (on the log scale), whereas the ntal value for clams payments P,0 (on the log scale) can be defned by ξ,0 = J j=0 ζ,j J l=1 ξ,l. Notatonal remar: In comparson to Chapter 5 there wll be many smlartes n the notaton and dervatons n

105 6.2 Ultmate Clam Predcton for Known Parameters Θ 95 the followng sectons of ths chapter. Most of the proofs n ths chapter follow n a smlar way as the correspondng proofs n Chapter 5 and we often refer to these proofs. However, there are some dfferences whch are not obvous at frst glance. Note that on the contrary to the Model Assumptons 5.1 n Chapter 5 the role of the pad and ncurred ratos has changed n Model Assumptons 6.1. Ths mples that the matrx B n ths chapter does not concde wth the matrx B n Chapter 5, although we choose the same symbols. Besde ths, we use the bass vectors e n ths chapter n a slghtly dfferent meanng n order to further smplfy notaton. Moreover, we focus n ths chapter on the dependence structure of the pad and ncurred ratos and hence do not derve the MSEP for the CDR (ths can be done n the same way as n Chapter 5). Ths allows to leave out the tme ndex {J, J +1} n ths chapter smplfyng the notaton and calculatons n comparson to Chapter 5 and mang the dervatons easer to understand. 6.2 Ultmate Clam Predcton for Known Parameters Θ We can ether wor wth the random vector Ξ R 2J+1 (see Model Assumptons 6.1) or wth the logarthmzed observatons gven by the random vector X := (log I,0, log P,0, log I,1, log P,1,..., log I,, log P, ; log I,J ) R 2J+1. The consderaton of Ξ was easer for the model defnton and for the nterpretaton of the dependence structure; but often t s more straghtforward f we drectly wor wth X (under the explct logarthmzed cumulatve observatons). Smlar to Secton 5.3 there s a lnear oneto-one correspondence B between Ξ and X, such that X = B Ξ. By ths correspondence we obtan the followng condtonal multvarate Gaussan dstrbuton for X : X Θ = B Ξ Θ N ( µ := µ(θ) := BΘ, Σ := BVB ). (6.4) Condtonally, gven the parameter vector Θ, the random vector X s multvarate Gaussan dstrbuted. Our frst am s to study the condtonal dstrbuton of the ultmate clam P,J = I,J, condtonally gven the parameter vector Θ and the observatons D J = {I,j, P,j : + j J, 0 J, 0 j J} n the upper pad and ncurred trangles, see Table 6.1. For accdent years {1,..., J}, defne n := 2J + 1 and q := q() := 2(J + 1) {2,..., 2J}. At tme J we have for accdent year observatons (gven n the upper trangles D J ) X (1) := (log I,0, log P,0, log I,1, log P,1,..., log I,J, log P,J ) R q, and we would le to predct the lower trangles gven by X (2) := (log I,J +1, log P,J +1,..., log I,, log P, ; log I,J ) R n q. (6.5)

106 96 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng Ths provdes, for {1,..., J}, the followng decomposton µ = (µ (1), µ (2) ) = BΘ R n of the condtonal mean: [ ] [ ] E X (1) Θ = µ (1) = B (1) Θ R q and E X (2) Θ = µ (2) = B (2) Θ R n q, wth partton of B R n n gven by, for {1,..., J}, ( ) (1) B B = B (2) wth B (1) R q n and B (2) R (n q) n. (6.6) In the same way we also decompose the covarance matrx whch provdes ( (11) Σ Σ (12) ) Σ = Σ (21) Σ (22) wth Σ (11) R q q. (6.7) For = 0 we set q(0) := n, X (1) 0 := X 0 R n, Σ (11) 0 := Σ and B (1) 0 := B, but (6.6) and (6.7) do not hold n ths case. Havng ths notaton, we provde the predcton of X (2), condtonally gven {Θ, D J }: Lemma 6.3 (Condtonal dstrbuton) Choose an accdent year {1,..., J}. Model Assumptons 6.1 we have X (2) = {Θ,DJ } X(2) N (1) {Θ,X } wth the condtonal mean and covarance ( matrx ) µ (2) := µ (2) + Σ (21) (Σ (11) ) 1 X (1) µ (1) and ( ) µ (2), Σ(22), Proof: The proof follows n the same way as the proof of Lemma 5.3. Under Σ(22) := Σ (22) Σ (21) (Σ (11) ) 1 Σ (12). An mmedate consequence of Lemma 6.3 s the followng corollary, whch consttutes an analog of Corollary 5.5 for the PIC model wth dependence. Corollary 6.4 (Condtonal dstrbuton) Under the assumptons and notaton of Lemma 6.3 we obtan for the ultmate clam I,J = P,J, for {1,..., J}, ( log I,J {Θ,DJ } N e µ(2), e Σ (22) ) e. Proof: By Lemma 6.3 the random vector X (2) has (condtonal) dstrbuton ( ) N µ (2), Σ(22). {Θ,DJ } X (2) Snce log I,J s the last entry of X (2), see (6.5), we have that log I,J = e X(2). Ths corollary mples that, condtonally gven the parameter vector Θ and the observatons D J, we get the ultmate clam predctor, for {1,..., J}, { } E[I,J Θ, D J ] = exp e µ(2) + e (22) Σ e /2. (6.8) In the specal case of a dagonal correlaton matrx (6.3),.e. ρ 0 = ρ 1 = ρ 2 = 0, ths s exactly the predctor derved n Corollary 2.5 of Merz Wüthrch [46].

107 6.3 Estmaton of Parameter Θ Estmaton of Parameter Θ The ultmate clam predctor (6.8) s stll based on the unnown parameter vector Θ, namely ( ( )) e µ(2) = e µ (2) + Σ (21) (Σ (11) ) 1 X (1) µ (1) ( ( )) = e B (2) Θ + Σ (21) (Σ (11) ) 1 X (1) B (1) Θ (6.9) = Γ Θ + e Σ (21) (Σ (11) ) 1 X (1), where we have defned Γ := e ( ) B (2) Σ (21) (Σ (11) ) 1 B (1). In partcular, we see that e µ (2) s an affne-lnear functon n Θ. We am to calculate the posteror dstrbuton of Θ, condtonally gven the observatons D J. The σ-feld generated by D J s the same as the one generated by D J = {X (1) 0,..., X(1) J }. Therefore, by a slght abuse of notaton, we dentfy the observatons D J wth D J. The lelhood of the logarthmzed observatons, condtonally gven Θ, s then wrtten as, see also (6.4), l DJ (Θ) J =0 { exp 1 ( ) ( X (1) B (1) (11) Θ (Σ ) 1 X (1) B (1) Θ) }. 2 Under Model Assumptons 6.1 the posteror densty of Θ, gven D J, s gven by { u (Θ D J ) l DJ (Θ) exp 1 } 2 (Θ ϑ) T 1 (Θ ϑ), (6.10) where the last term s the pror densty of Θ wth pror mean gven by and (dagonal) covarance matrx defned by ϑ := (ψ 0 ; ψ 1, φ 1, ψ 2, φ 2,..., ψ J, φ J ) R n, T := dag(t 2 0; t 2 1, s 2 1, t 2 2, s 2 2,..., t 2 J, s 2 J) R n n. Ths mmedately mples the followng theorem: Theorem 6.5 (Posteror dstrbuton of Θ) Under Model Assumptons 6.1, the posteror dstrbuton of Θ, gven D J, s a multvarate Gaussan dstrbuton wth posteror mean [ ] J ϑ(d J ) := T(D J ) T 1 ϑ + (B (1) ) (Σ (11) ) 1 X (1), and posteror covarance matrx T(D J ) := ( T 1 + J =0 =0 ) 1 (B (1) ) (Σ (11) ) 1 B (1).

108 98 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng Proof: The proof follows n the same way as the proof of Theorem 5.6. Remars on credblty theory: We defne the matrx S := ( J =0 ) 1 (B (1) ) (Σ (11) ) 1 B (1). The exstence of S follows by the fact that J =0 (B(1) ) (Σ (11) ) 1 B (1) s as sum of symmetrc postve defnte (s.p.d) matrces also s.p.d. and hence nvertble. Moreover, we defne the credblty weghts (I s the dentty matrx) A := ( T 1 + S 1) 1 S 1 and I A = ( T 1 + S 1) 1 T 1, (6.11) see also formula (7.11) n Bühlmann Gsler [14]. Ths mples for the posteror covarance matrx T(D J ) = AS = (I A)T and we obtan the credblty formula for the posteror mean ϑ(d J ) = (I A) ϑ + A Y, (6.12) wth compressed data Y := S [ J =0 (B (1) ) (Σ (11) ) 1 X (1) ], (6.13) see Chapters 7 and 8 n Bühlmann Gsler [14]. That s, the posteror mean ϑ(d J ) s a credblty weghted average between the pror mean ϑ and the observatons Y wth credblty matrx A. The crucal pont why we obtan dentcal results usng Theorem 6.5 and credblty theory s that the normal pror dstrbuton and the normal condtonal dstrbuton n Model Assumptons 6.1 lead to the exact credblty case, see Bühlmann Gsler [14] for detals. Therefore, the compressed data vector (6.13) n the PIC method has a smlar structure as the compressed data vector (4.14) used for the credblty predctors F I,Cred n the (Bayesan) LSRM and (6.12) corresponds to the credblty predctor F I,Cred for Bayesan LSRMs, see Subsecton Now we state an analog to Corollary 5.7 for the PIC model wth dependence. Corollary 6.6 (Ultmate clam predctor) Under Model Assumptons 6.1 we obtan for I,J = P,J the ultmate clam predctor, gven observatons D J, { E[I,J D J ] = exp Γ ϑ(d J ) + Γ T(D J ) Γ /2 + e Σ (21) (Σ (11) ) 1 X (1) + e Proof: Σ (22) } e /2. The proof follows by Theorem 6.5 and (6.8) n the same way as the proof of Corollary 5.7. Corollary 6.6 gves the ultmate clam predctor that s now based on clams payments, ncurred losses and pror expert nformaton. In contrast to Merz Wüthrch [46] we can now easly choose any meanngful covarance matrx V for Ξ.

109 6.4 Predcton Uncertanty Predcton Uncertanty In order to analyze the predcton uncertanty, we can now study the posteror predctve dstrbuton of I J = (I 1,J,..., I J,J ), whch exactly corresponds to the column of unnown ultmate clams, gven the observatons D J. If g( ) s a suffcently nce functon we obtan E[g(I J ) D J ] = g(x) f (x D J ) dx = g(x) f (x Θ, D J ) u (Θ D J ) dx dθ, x R J x R J,Θ where u (Θ D J ) denotes the posteror densty of Θ, gven D J (cf. (6.10)). Because the denstes f (x Θ, D J ) and u (Θ D J ) are explctly gven by Corollary 6.4 and (6.10) and the condtonal ndependence of accdent years, gven Θ, we can calculate the predctve values E[g(I J ) D J ] numercally, for example usng MC smulatons. Ths allows for the analyss of any uncertanty and rs measure. If we consder the total ultmate clam J =1 I,J and the correspondng predctor J =1 E[I,J D J ] the condtonal MSEP s gven by [ J ] ( J 2 [ J J ] msep J E[I,J D J ] = E I,J E[I,J D J ]) D J = Var I,J D J. I,J D J =1 =1 =1 Henceforth, we need to calculate ths last condtonal varance n order to obtan the condtonal MSEP. Theorem 6.7 Under Model Assumptons 6.1 the condtonal MSEP s gven by msep J I,J D J =1 = [ J ] E[I,J D J ] =1 J E[I,J D J ] E[I,J D J ],=1 wth E[I,J D J ] gven by Corollary 6.6. =1 Proof: Wth the varance decouplng formula we obtan [ J ] J Var I,J D J = Cov[I,J, I,J D J ] =1 =,=1 ( { exp Γ T(D J ) Γ + e J Cov[E[I,J Θ, D J ], E[I,J Θ, D J ] D J ] +,=1 =1 Σ (22) } ) e 1 {=} 1, J E[Var[I,J Θ, D J ] D J ], (6.14) where for the second term on the rght hand sde of (6.14) we have used the condtonal ndependence of dfferent accdent years, gven Θ. Thus, we need to calculate these last two terms. Usng Corollary 6.4 and (6.9), we obtan { E[I,J Θ, D J ] = exp Γ Θ + e Σ (21) (Σ (11) ) 1 X (1) + e =1 } (22) Σ e /2, (6.15)

110 100 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng and ( { Var[I,J Θ, D J ] = E[I,J Θ, D J ] 2 exp e } ) (22) Σ e 1. (6.16) We frst treat the second term n (6.14). Usng (6.16) leads to E[Var[I,J Θ, D J ] D J ] [ = E exp { 2Γ Θ + 2e Σ (21) (Σ (11) ) 1 X (1) = E[I,J D J ] 2 exp { Γ T(D J ) Γ } ( exp + e { e For the frst term n (6.14) we need to consder Cov[E[I,J Θ, D J ], E[I,J Θ, D J ] D J ] { = exp e Σ (21) (Σ (11) ) 1 X (1) + e Σ (22) Cov[exp {Γ Θ}, exp {Γ Θ} D J ], where the last covarance term s gven by Cov[exp {Γ Θ}, exp {Γ Θ} D J ] Σ (22) } ] e DJ (exp ). Σ (22) e } 1 { e e /2 + e Σ(21) (Σ (11) ) 1 X (1) + e = E[exp {Γ Θ} D J ] E[exp {Γ Θ} D J ] ( exp { Γ T(D J ) Γ } 1 ). Henceforth, usng (6.16) we obtan for the frst term n (6.14) } ) (22) Σ e 1 } (22) Σ e /2 Cov[E[I,J Θ, D J ], E[I,J Θ, D J ] D J ] = E[I,J D J ] E[I,J D J ] ( exp { Γ T(D J ) Γ } 1 ). Collectng all the terms completes the proof. Thus, we obtan a closed form soluton for both, the ultmate clam predctors E[I,J D J ] and the correspondng predcton errors, measured by the condtonal MSEP. 6.5 Example PIC Reservng Method wth Dependence Modelng We apply the PIC model wth dependence to the motor thrd party lablty data gven n Table 7.8 and 7.9 below. In Model Assumptons 6.1 we wor wth logarthmzed pad clams ratos ζ, and logarthmzed ncurred losses ratos ξ,, respectvely (cf. (6.1)). That means that we have to transform the data n Table 7.8 and 7.9 nto ζ,j and ξ,l. Due to the fact that there s no expert nowledge for the specfc choce of the means n the pror dstrbutons for Ψ l and Φ j we choose n Model Assumptons 6.1 non-nformatve prors,.e. we let t 2 j and s2 l. Ths mples asymptotcally for the credblty matrx A = I n (6.11) and no pror nowledge s used n our calculatons.

111 6.5 Example PIC Reservng Method wth Dependence Modelng 101 For Model Assumptons 6.1, t remans to choose a sutable covarance matrx V. Here we present three dfferent choces of correlaton matrces Ṽ of the type (6.3), whch are motvated by an ad-hoc estmate. The correspondng covarance matrx V s then gven by V := Var 1/2 Ṽ Var 1/2, (6.17) where Var := dag ( σζ 2 0 ; σζ 2 1, σξ 2 1,..., σζ 2 J, σξ 2 ) J. The estmator for the correlaton matrx of the type (6.3) should be seen as an ntutve proposal for a correlaton structure and not as an estmator beng optmal n some mathematcal sense. Correlaton Matrx Choce The choce of a correlaton matrx of type (6.3) reduces the number of parameters to be estmated n comparson to the estmaton of a general correlaton matrx. Note that we have decded for structure (6.3) by pure expert choce. For a correlaton matrx of type (6.3), we have to choose ρ l for l {0, 1, 2} as ρ l = Cor[ζ,, ξ,+l ] for = 1,..., J and = 1,..., J l, and ρ l = Cor[ζ,0, ξ,l ] for l {1, 2}. We propose the followng ad-hoc estmators for ρ l. 1: For the unnown means Ψ = E[ζ, ] and Φ = E[ξ, ] as well as varances σζ 2 σξ 2 = Var[ξ, ] we use sample estmates = Var[ζ, ] and J 1 ˆΨ := ζ, for = 0,..., J ˆσ ζ 2 J + 1 := 1 J (ζ, J ˆΨ ) 2 for = 0,..., J 1 and =0 J 1 ˆΦ := ξ, for = 1,..., J ˆσ ξ 2 J + 1 := 1 J (ξ, J ˆΦ ) 2 for = 1,..., J 1. =0 Snce for the estmaton of the last varance parameters ˆσ 2 ζ J and ˆσ 2 ξ J there s only one observaton n the observed trangle we use the well-nown extrapolaton formula ˆσ 2 ζ J := mn{ˆσ 2 ζ J 2, ˆσ 2 ζ, ˆσ 4 ζ J 2 /ˆσ 2 ζ } and ˆσ 2 ξ J := mn{ˆσ 2 ξ J 2, ˆσ 2 ξ, ˆσ 4 ξ J 2 /ˆσ 2 ξ }, =0 =0

112 102 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng estmated correlaton rho_0 (lag 0) rho_1 (lag 1) rho_2 (lag 2) rho_3 (lag 3) number of observatons Fgure 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 see Wüthrch Merz [63]. 2: We consder for accdent year {0,..., J} the standardzed logarthmzed ratos 3: ζ, := ζ, ˆΨ ˆσ ζ for = 0,... J and ξ, := ξ, ˆΦ ˆσ ξ for = 1,... J. We use the correlaton estmator for ρ l gven by J l ˆρ l := =1 J l =1 ζ, ξ,+l for l {0, 1, 2, 3}. (6.18) Accordng to the correlaton estmators (6.18) we obtan for ˆρ l wth l = 0, 1, 2, 3 as a functon of the number of observatons the values gven n Fgure 6.1. We see n Fgure 6.1 that the assumpton of postve correlatedness between ζ, and ξ,+l for l {0, 1, 2} s evdent. For l = 3 or hgher tme lags the correlaton estmaton s comparably small (about 5%) and wll therefore be neglected n our followng consderatons. For the sample estmators (6.18) we obtan: ˆρ 0 ˆρ 1 ˆρ 2 ˆρ 3 (6.19) 23% 27% 28% 5% In order to study correlaton senstvtes, we mae three explct correlaton choces, see Table 6.2, based on the correlaton estmates n (6.19) and compare t to the uncorrelated case (case 0) treated n Merz Wüthrch [46]. For these cases we have to chec whether the correspondng covarance matrx V s postve defnte (see Model Assumptons 6.1). To calculate V, we use the dentty (6.17) and obtan V := Var 1/2 Ṽ l Var 1/2,

113 6.5 Example PIC Reservng Method wth Dependence Modelng 103 case 0 case 1 case 2 case 3 ρ 0 = ρ 1 = ρ 2 = 0% ρ 0 = 30%, ρ 1 = 25%, ρ 2 = 40% ρ 0 = 30%, ρ 1 = 25%, ρ 2 = 30% ρ 0 = 25%, ρ 1 = 25%, ρ 2 = 30% Table 6.2: Uncorrelated case and three explct choces for correlatons where Var denotes the varance estmates Var := dag (ˆσ 2 ζ 0 ; ˆσ 2 ζ 1, ˆσ 2 ξ 1,..., ˆσ 2 ζ J, ˆσ 2 ξ J ). Snce Var 1/2 s dagonal we only have to chec, whether the matrx Ṽl s postve defnte. The egenvalues of the estmated correlaton matrx Ṽl are for the four cases strctly postve, the smallest beng and the largest beng and hence the correspondng covarance matrx V fulflls Model Assumptons 6.1. Based on these choces for the covarance matrx V, we calculate the ultmate clam reserves and the condtonal MSEP. Remars 6.8 (Example PIC reservng method wth dependence modelng) ) We choose by means of exploratve data analyss explct covarance matrces V. Ths was partly done by ntutve expert nowledge. ) The dervaton of an optmal estmator Ṽ for the covarance matrx V wth good statstcal propertes s not trval and should be subject to more statstcal research. Therefore, we present an ad-hoc estmator for the correlaton matrx and use the resultng estmates as an orentaton for dfferent explct choces for the correlaton structure (case 1-3). ) Model Assumptons 6.1 allow for arbtrary covarance matrces as long as they are postve defnte. If suffcent data for a robust estmaton of ther n(n + 1)/2 entres s avalable, there s no need to reduce to correlatons up to lag 2. However, we beleve (due to overparametrzaton) that an arbtrary correlaton structure s not a feasble alternatve and expert opnon always needs to specfy addtonal structure. v) Postve defnteness of V should always be checed because most ntutve choces do not provde a postve defnte covarance matrx. Clams Reserves and Predcton Uncertanty 1) Clams reserves at tme J: We consder the expected outstandng loss labltes (clams reserves) ˆR(D J ) := E[I,J D J ] P,J

114 104 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng acc. ˆR(DJ) ˆR(DJ) D( ˆR(D J)) ˆR(DJ) D( ˆR(D J)) ˆR(DJ) D( ˆR(D J)) year case 0 case 1 case 1 case 2 case 2 case 3 case ,0% ,0% ,0% ,0% ,0% ,0% ,2% ,5% ,3% ,4% ,6% ,1% ,7% ,0% ,7% ,8% ,8% ,6% ,2% ,3% ,1% ,6% ,4% ,9% ,6% ,2% ,9% ,5% ,2% ,9% ,6% ,7% ,5% ,8% ,6% ,0% ,9% ,3% ,3% ,9% ,1% ,6% ,2% ,7% ,9% ,6% ,7% ,8% ,1% ,8% ,8% ,0% ,0% ,3% ,4% ,9% ,6% ,4% ,3% ,6% ,9% ,2% ,3% total ,8% ,0% ,8% Table 6.3: Clams reserves n the classcal PIC model and PIC model wth dependence at tme J. The percental dfference between clams reserves wth and wthout dependence s denoted by D( ˆR(D J )). We observe n Table 6.3 that n the frst case the clams reserves are about 6% lower than the clams reserves wthout dependence. In the other two cases the dfference s stll about 3%. Ths shows that the specfc choce of correlaton structure has a crucal mpact on the sze of clams reserves. 2) Predcton uncertanty at tme J: In Table 6.4 we provde the MSEP for our four explct correlaton structures, see Table 6.2. We observe that the predcton uncertanty measured n terms of the condtonal MSEP for the PIC msep 1/2 msep 1/2 msep 1/2 msep 1/2 case 0 case 1 case 2 case 3 total n % of clams reserves 2,44% 3,06% 3,04% 3,02% Table 6.4: Predcton uncertanty msep 1/2 for the classcal PIC model and the PIC model wth dependence

115 6.6 Conclusons 105 model wth dependence s hgher than n the classcal PIC model. The reason s that postve correlatons of the type (6.3) between pad and ncurred ratos n Model Assumptons 6.1 ncrease the correlatons n the ultmate outcomes, and hence the uncertantes. Ths means that not consderng dependence n the PIC model clearly underestmates the total uncertanty. 6.6 Conclusons In the orgnal PIC model of Merz Wüthrch [46] a unfed predctor of the ultmate clam based on ncurred losses and clams payments as well as the correspondng predcton uncertanty n terms of the condtonal MSEP can be derved analytcally. The man crtcsm s that the orgnal PIC model does not allow for modelng dependences between clams payments and ncurred losses as t s observed n the data. In ths paper, we generalze the orgnal PIC model so that t allows for modelng dependence between clams payments and ncurred losses data. Ths s motvated by the fact that on the one hand, clams payments are contaned n ncurred losses data and on the other hand, ncurred losses contan addtonal nformaton, whch nfluences future clams payments data. The data n our example (see Tables 7.8 and 7.9) confrms ths hypothess of dependence between clams payments and ncurred losses data (see Table 6.1). We have seen n the senstvty analyss that dependence modelng n the PIC method has a crucal mpact on the clams reserves and the correspondng (condtonal) MSEP (see Tables 6.3 and 6.4). Therefore, the classcal PIC model of Merz Wüthrch [46] underestmates the predcton uncertanty, see Table 6.4, due to the mssng dependence structure wthn accdent years. For a better understandng of the nfluence of pror choces on the reserves and ts uncertanty t mght be useful to provde a senstvty analyss of the method to the choce of prors, whch should be subject to an extended case study n future wor. Summarzng, the benefts of the PIC method wth dependence modelng are that two dfferent channels of nformaton are combned to get a unfed ultmate loss predctor; dependence structures between pad and ncurred data can be modeled approprately; pror expert nowledge can be used to desgn the pror dstrbutons of the parameter vector Θ, otherwse we can choose non-nformatve prors for Θ. Pror expert opnon should also be used for the desgn of approprate correlaton structures; we can calculate the ultmate clam and the condtonal MSEP analytcally; the CDR predcton uncertanty can be calculated, see Happ Wüthrch [30] or Chapter 5; the full predctve dstrbuton can be derved va MC smulatons. Ths allows for the

116 106 6 Pad-Incurred Chan Reservng Method wth Dependence Modelng calculaton of any rs measure le VaR or ES. Man results of the PIC reservng method wth dependence: For solvency consderatons n Chapter 7 we summarze all quanttes of nterest derved n the PIC reservng method: 1. The predctor R I for outstandng loss labltes R I, see (2.5c) and (2.4c) gven by R I = J (E[I,J D J ] P,J ), (6.20) =1 see Corollary 6.6 and the predctor for ncremental clam payments Ŝ 0 J,Bayes,+1 := E[P,+1 D J ] E[P, D J ]. (6.21) 2. The estmator for the predcton uncertanty n terms of the (condtonal) MSEP [ ] msep R I D J RI (6.22) gven by Theorem The estmator for the CDR uncertanty n terms of the (condtonal) MSEP msep J CDR J+1 =1 D J can be derved n the same way as n Theorem [0] (6.23)

117 7 Solvency Recently, regulatory authortes have establshed new solvency requrements n order to mantan the long-term functon of nsurance companes and to protect polcyholders from losses. As already mentoned n the ntroducton the European supervson authorty EIOPA wll regulate nsurance companes n Europe through the Solvency II framewor. Solvency II wll presumably be oblgatory for European nsurance companes from January In Swtzerland the regulatory solvency framewor Swss Solvency Test (SST) s already mplemented snce 2006 and oblgatory for all nsurance companes n Swtzerland. In ths chapter we consder the ssue whch condtons are requred by the regulator w.r.t. approprate reserves n solvency framewors le Solvency II and SST. Moreover, we pont out how these requrements can be fulflled n clams reservng framewors. A central tas n nsurance companes s to buld up approprate reserves to meet future loss labltes and to provde solvency. In the regulatory pont of vew reserves should be such that they provde protecton aganst almost all possble adverse events. The Internatonal Assocaton of Insurance Supervsors [34] states: Solvency ablty of an nsurer to meet ts oblgatons (labltes) under all contracts at any tme. Due to the very nature of nsurance busness, t s mpossble to guarantee solvency wth certanty. In order to come to a practcable defnton, t s necessary to mae clear under whch crcumstances the approprateness of the assets to cover clams s to be consdered,... Followng ths regulatory statement we consder the ssue whch regulatory condtons are requred for approprate reserves. In the prevous chapters we ntroduced dstrbuton-free as well as dstrbuton-based clams reservng methods, namely LSRMs n Chapter 4 and the PIC reservng method (wth dependence) n Chapters 5 and 6. All these methods provde predctors R I for the outstandng loss labltes R I at tme I, see (4.77) for LSRMs, (5.13) and (6.20) for the PIC reservng method (wth dependence). Of course, dependng on the method used for clams reservng (.e. for the predcton of R I ), there may result rather dfferent reserves. However, the regulatory authortes do not explctly requre the usage of certan clams reservng methods, see European 107

118 108 7 Solvency Commsson [23], FOPI [24] and FOPI [25]. As already mentoned, for example, the regulatory authorty EIOPA and FINMA only requre the reserves to be best-estmate valuaton of labltes (BEL). Mathematsch ausgedrüct snd de verscherungstechnschen Bedarfsrücstellungen ene bedngt erwartungstreue Schätzung des bedngten Erwartungswertes der zuünftgen Zahlungsflüsse aufgrund der zum Zetpunt der Schätzung vorlegenden Informaton. Se gelten damt als Best-Estmate, snd also weder auf der vorschtgen noch auf der unvorschtgen Sete und enthalten nsbesondere ene bewussten Verstärungen. (FINMA Rundschreben 2008/42) Remars 7.1 (BEL) ) Note that nether n Solvency II nor n the SST exact mathematcal defntons are gven for BEL. Of course, the condtonal expectaton E [ R I D I] s the best D I -measurable predctor for R I when usng the condtonal MSEP as rs measure. However, many wdely used clams reservng methods do not allow for the exact dervaton of E [ R I D I], for example the CL method, the CLR method and the class of (Bayesan) LSRMs, see Chapters 3 and 4. ) The analytcal dervaton of the condtonal expectaton E [ R I D I] of the outstandng clams payments at tme I s possble for some models such as the PIC model (wth dependence) n Merz Wüthrch [46] and Happ Wüthrch [31], Hertg s [32] log-normal model and the gamma-gamma model n Wüthrch [60]. For more models whch allow for the analytcal dervaton of the condtonal expectaton E [ R I D I] of the outstandng clams payments at tme I, see Wüthrch Merz [63]. Followng clams reservng tradton we have used the term reserves for the value of the predctor R I of outstandng loss labltes at tme I. From now on we only consder predctors R I whch are BEL. In the followng we wll use the term reserves for the amount (current value of all assets hold by the nsurance company) avalable to cover all nsurance labltes. Ths amount at tme I wll be denoted by RES I. The acceptablty of reserves of an nsurance company w.r.t regulatory solvency requrements s specfed n Defntons 7.2 and Regulatory Requrements on Reserves In order to buld up reserves for outstandng loss labltes an nsurance company s oblged to predct very carefully all future loss lablty cash flows based on the data avalable. So far we have dscussed the concept of BEL. From the regulatory pont of vew, t s not acceptable to

119 7.1 Regulatory Requrements on Reserves 109 tae the amount of BEL I as reserves at tme I. BEL I only reflects the average outcome of loss labltes and does not provde any protecton aganst shortfalls n the CDR. At frst a consensus has to be found and formulated whch functons and requrements reserves for outstandng loss labltes should fulfll. There s a general agreement that reserves should support the short- and longterm functon and solvency of an nsurance company. In the short run reserves are requred to provde fnancal strength and solvency n the current accountng year I,.e. reserves should be hgher than the current value of labltes. The Solvency II gudelne [23] states: Labltes should be valued at the amount for whch they could be transferred, or settled, between nowledgeable wllng partes n an arm s length transacton. Ths requrement wll be later referred to as accountng condton, see Wüthrch Merz [62], and ensures that there are suffcent reserves to transfer the run-off to a nowledgeable thrd party. The BEL does not fulfll ths requrement, because no rs-averse maret partcpant would bear the run-off rs for the prce of BEL Maret-Value Margn A rs-averse maret partcpant s only wllng to tae over the run-off of the nsurance portfolo and to transfer the assocated labltes on hs balance sheet, f he s pad the amount of BEL plus a maret-value margn (MVM) for bearng the run-off rs. Therefore, the MVM accounts for the rs averson of maret partcpants and s a compensaton payment for bearng the runoff rs. The MVM s also often called rs-margn (RM), safety margn (SF) or cost-of-captal (CoC) margn. The regulatory authorty FOPI states for the MVM: The Maret Value Margn (MVM) s the addtonal amount on top of the best estmate whch s requred by a wllng buyer n an arms-length transacton to assume the labltes the loss reserves are held to meet... In accordance to the last quotaton the far value of labltes (FVL) n accountng year I s defned by FVL I := BEL I + MVM I. In ths context the FVL I s the (pseudo-maret) prce to transfer the run-off of the nsurance portfolo to a thrd party. The ablty of an nsurance company to transfer the nsurance run-off portfolo to a thrd party at tme I s requred by regulatory authortes, see FOPI [24] and FOPI [25]. Defnton 7.2 (Accountng Condton) In accountng year I reserves RES I fulfll the accountng condton, f the nsurance company s able to transfer the nsurance run-off portfolo

120 110 7 Solvency to a wllng nowledgeable party,.e. RES I FVL I = BEL I + MVM I. By defnton of the MVM I, the FVL I s the (maret) prce for whch the run-off portfolo can be transferred to a thrd party n accountng year I. At ths pont arses the problem that there s no maret for tradng nsurance contract labltes of nsurance run-off portfolos and we can not tae the MVM as the dfference between maret prce FVL I of the run-off portfolo and ts BEL I. Thus, mared-to-model approaches have to be used to determne MVMs. The Internatonal Actuaral Assocaton (IAA) [33] dvdes all approaches to the problem of determnng (maredto-model) MVMs nto the followng four classes: Quantle based methods usng rs measures le the VaR, the Condtonal Tal Expectaton (CTE) and the ES also called Condtonal VaR (TVaR). Cost-of-Captal (CoC) approach. CoC s defned as the cost of fnancng the Solvency Captal Requrements (SCR) (protecton aganst adverse events, see the defnton below) n all future accountng years up to the complete settlement of the nsurance portfolo. Dscount related methods: The MVM s defned as the dfference between ) the dscounted cash flows usng rs-free nterest rates and ) the dscounted cash flows usng the rs-free nterest rate plus a rs-adjustment. Often probablty dstortons, see Wüthrch Merz [62], are used to ncorporate rs averson leadng to MVMs. Smple methods usng a fxed percentage of the BEL as MVM. The IAA prefers among these four categores the CoC approach, snce ths approach s most rs senstve and used n lfe as well as n non-lfe nsurance. However, ths approach s the most sophstcated one and s feasble only under restrctve model assumptons or smplfcatons, see Robert [51] and Salzmann Wüthrch [53]. In recent publcatons probablty dstortons and deflators (to model rs-averson) have been appled to calculate MVM, see Wüthrch et al. [61] or Wüthrch-Merz [62] n a maret-consstent full balance sheet approach. In the followng we present the CoC approach for the calculatons of MVMs. Maret-Value Margn n a Cost-of-Captal Approach In Secton 2.6 we saw that the clams development result CDR M,I+1 consttutes the amount by whch BEL I s to be adjusted at tme I + 1 to have best-estmate valuaton of labltes BEL I+1 also n accountng year I + 1. In order to protect aganst shortfalls n the clams development result CDR M,I+1 the nsurance company needs to hold suffcent reserves. These requred reserves are determned by a condtonal (gven D I ) rs measure ρ I whch evaluates

121 7.1 Regulatory Requrements on Reserves 111 the shortfall rs of the clams development result CDR M,I+1 and provdes protecton aganst shortfalls. For the defnton of condtonal rs measures, let (Ω, D, D, P) be a fltered probablty space wth σ-feld D and fltraton D := (D n ) 0 n I+J wth D n D. Let L 0 (Ω, D n, P) := { X X s D n -measurable and X < P-a.s.}. be the space of almost sure fnte D n -measurable random varables. For all tmes n {I,..., I + J 1} from now we defne the followng condtonal rs measures whch determne the amount requred at tme n for protecton aganst shortfalls n the clams development result CDR M,n+1. Let A n L 0 (Ω, D, P) be such that ) c n A n for all D n -measurable c n L 0 (Ω, D n, P), ) X + Y A n for X, Y A n and ) λ n X A n for X A n and D n -measurable λ n. Defnton 7.3 (Condtonal rs measure) A mappng ρ n : A n L 0 (Ω, D n, P) : X ρ n (X) s called condtonal rs measure, f t s fnte a.s. on D n -measurable random varables. In a general CoC approach at each tme n {I,..., I + J 1} from today (I) to the last year before the fnal run-off (I + J 1) a condtonal rs measure determnes the amount ρ n whch the nsurance company needs to hold at tme n for protecton aganst shortfalls n the clams development result CDR M,n+1 at tme n + 1. However, the nsurance company does not hold the amount ρ n n ts boos, but the prce for provdng ths captal. The dea behnd that s to buy at tme n a rensurance contract for the prce (premum) of c ρ n where c > 0 s the cost-of-captal rate. Accordng to ths rensurance contract the seller of the contract (another nsurance company or an nvestor) provdes the nsurance company wth the requred amount ρ n at tme n. Ths money wll be used to compensate losses n the case of a shortfall n the CDR M,n+1. Alternatvely, one can thn of the ssue of an nsurance bond wth the nomnal value ρ n wth rate of return c at tme n. Vewed from tme I, ths strategy generates CoC cash flows up to the fnal settlement of the run-off portfolo c ρ I,..., c ρ I+. (7.1) Note that n the general CoC settng the CoC loadng ρ n n (7.1) at tme n {I,..., I + J 1} s D n -measurable (.e. observable only at tme n) but must be evaluated at tme I n. Ths means that the nsurance company has to buld up reserves ĈoCI at tme I for the CoC cash flows n (7.1). By settng MVM I,CoC := ĈoCI the far value of labltes (rs-adjusted reserves) FVL I at tme I s then gven by FVL I = BEL I + MVM I,CoC. (7.2)

122 112 7 Solvency Snce there are many ways to defne the rs measures ρ n (from very smple to very sophstcated rs measures) n the CoC approach, there are varous dfferent ways to calculate rs-adjusted reserves n a CoC settng. For rs measures whch reflect the whole run-off rs t s generally far from straghtforward to calculate CoC maret-value margns, see Salzmann Wüthrch [53], Robert [51] and Wüthrch Merz [62]. Therefore, n practce the followng regulatory solvency approach s used for the calculaton of CoC maret-value margns. Ths approach meets the smplcty requrements n practce and s proposed n the SST [25] and dscussed n Salzmann Wüthrch [53]. Defnton 7.4 (Regulatory Cost-of-Captal approach) Let φ > 0 be a constant, c > 0 the cost-of-captal rate and R n I := m M I =0 =n Ŝ m I,+1 be the predcted at tme I outstandng loss labltes after tme n {I,..., I + J 1} and let S n+1 L n+1. The regulatory rs measure s then defned by ρ n := ρ n (S n+1 ) := φ R n I R I msep CDR M,I+1 D I [0] 1/2. (7.3) The regulatory CoC maret-value margn s then defned by MVM I,CoC := I+ n=i c ρ n = c φ I+ n=i R I R n I msep CDR M,I+1 D I [0] 1/2. (7.4) Remars 7.5 (Regulatory Cost-of-Captal approach) ) ρ n = ρ n (S n+1 ) wth n {I..., I+J 1} s the amount requred for protecton aganst CDR shortfalls at tme n + 1. In general ρ n s D n -measurable. However, n ths regulatory CoC approach ρ n are D I -measurable and hence observable at tme I for all n {I,..., I+J 1}. ) ρ n comprses only rs of the one-year clams development result CDR M,I+1. Run-off rss for accountng years I + 1,..., I + J 1 are not captured. ) The rs n the clams development result CDR M,I+1 vewed from tme I s measured by the condtonal MSEP, see (7.3), and not by the rs measures VaR or ES proposed n Solvency II and SST, see European Commsson [23] and FOPI [24]. The constant φ can be used to compensate for the dfference between MSEP and VaR or ES, respectvely. If, for example, the clams development result CDR M,I+1 s assumed to be condtonally normal dstrbuted CDR M,I+1 D I N (µ, σ 2 ) (7.5)

123 7.1 Regulatory Requrements on Reserves 113 wth mean µ := 0 and varance σ 2 := msep CDR M,I+1 D I [0] we choose the parameter φ := for any securty level q (0, 1), where Ψ µ,σ 2 Ψ 1 µ,σ 2 (q) msep CDR M,I+1 D I [0] 1/2 (7.6) denotes the dstrbuton functon of a normal dstrbuton wth mean µ and varance σ 2. The choce (7.6) replaces n (7.4) the MSEP by the rs measure VaR. In the same way, the MSEP n (7.4) can be replaced by any other rs measure le ES as requred n the SST. v) Equaton (7.3) shows that rs-bearng captal ρ n (S n+1 ) at tme n {I,..., I + J 1} s proportonal to the condtonal MSEP of the clams development result CDR M,I+1. The proportonalty factor depends only on the outstandng loss lablty predctors R I gven by (2.5c). Ths maes the regulatory CoC approach wdely applcable n nsurance practce. v) The smplfcatons n the regulatory CoC approach (only the clams development result CDR M,I+1 s consdered and the condtonal MSEP s used as rs measure) are necessary to mae the approach accessble to most clams reservng methods. In the case of a general CoC approach wth VaR or ES as rs measure the calculatons of MVMs become nfeasble for almost all clams reservng methods. For two exceptons under restrctve model assumptons, see Robert [51] and Secton n Wüthrch Merz [62]. For a detaled dscusson of more sophstcated CoC approaches we refer to Salzmann Wüthrch [53] and Wüthrch Merz [62] Solvency Captal Requrements The MVM n the accountng condton (cf. Defnton 7.2) guarantuees transferablty of the run-off portfolo to a thrd party n accountng year I. We presented n Defnton 7.4 the regulatory CoC approach whch s wdely used to provde such MVMs and s applcable to all methods (except for MCL method) n ths thess. Addtonally, the nsurance company has to fulfll the accountng condton also n the next accountng year I + 1 wth hgh probablty. Ths condton wll be referred to as nsurance contract condton, see Wüthrch Merz [62], and mantans that the nsurance company has the fnancal strength to fulfll the accountng condton wth hgh probablty also n accountng year I + 1. The requrement wth hgh probablty s determned by a condtonal rs measure ρ I based on the nformaton D I at tme I. Let ρ I be a condtonal rs measure as n Defnton 7.3 whch s addtonally translaton nvarant,.e. ρ I (X + c) = ρ I (X) + c for all X A I+1 and D I -measurable c, (7.7)

124 114 7 Solvency see McNel et al. [41]. Based on ths rs measure ρ I the solvency captal requrement (SCR), see European Commsson [23] and FOPI [24], for accountng year I s mplctly defned by FVL I + SCR I := ρ I ( S M I+1 + BEL I+1 + MVM I+1). (7.8) By defnton the solvency captal requrement SCR I s the mnmum amount needed at tme I n addton to the far value of labltes FVL I to pay out n accountng year I + 1 the loss labltes S M I+1 and to fulfll the accountng condton wth hgh probablty measured by ρ I. Defnton 7.6 (Insurance Contract Condton) Let ρ I be a translaton nvarant (condtonal) rs measure. In accountng year I the nsurance contract condton s fulflled, f RES I FVL I + SCR I = ρ I ( S M I+1 + BEL I+1 + MVM I+1), where RES I denotes the value at tme I of all assets hold by the nsurance company to meet outstandng loss labltes. Addng the solvency captal requrements SCR I to the far value of labltes FVL I creates an addtonal (besde the MVM) protecton aganst adverse events and mantans that the nsurance company holds wth hgh probablty (measured by ρ I ) at least the amount of FVL I+1 = BEL I+1 +MVM I+1 at tme I +1. That means that f the nsurance contract condton s fulflled at tme I the nsurance company fulflls the accountng condton, see Defnton 7.2, at tme I +1 (n the next accountng year) wth hgh probablty. In ths way the regulator can establsh a condton at tme I that provdes wth hgh probablty that an nsurance company has the fnancal strength to transfer ts run-off portfolo to a thrd nowledgeable party at tme I Fnal Regulatory Reserves The amount of the solvency captal requrements SCR I n (7.8) manly depends on the choce of the rs measure ρ I. Regulatory authortes propose the followng two rs measures: Value-at-Rs The Value-at-Rs (VaR) s a quantle-based rs measure and wdely used n the banng sector. It also became a central component of new regulatory requrements n Solvency II.

125 1 7.1 Regulatory Requrements on Reserves 115 SCR MVM RES BEL assets labltes Fgure 7.1: Reserves consst of BEL, MVM (together satsfyng accountng condton) and SCR (satsfyng the nsurance contract condton) Defnton 7.7 ((Condtonal) Value-at-Rs (VaR)) Let X be a random varable on a probablty space (Ω, D, P) and G D a σ-feld. For the dstrbuton of X G wth dstrbuton functon F G (n the case of ther exstence) the condtonal VaR for securty level α (0, 1) s defned by VaR α [X G] := nf{x R F G (x) α}. The VaR wth α = 99.5% wll be requred n Solvency II calculatons for determnng approprate reserves, see European Commsson [23]. Often the VaR 99.5% s assocated wth an one-n- 200-year event. For a detaled dscusson on VaR, see Artzner et al. [7]. Expected Shortfall (ES) The VaR wll be requred n Solvency II whereas the ES s the rs measure ncorporated n the SST framewor. Defnton 7.8 ((Condtonal) Expected Shortfall (ES)) Under the assumptons of Defnton 7.7 the ES s defned by ES α [X G] := 1 1 α 1 α VaR x [X G] dx.

126 116 7 Solvency The ES for securty level α (0, 1) s a normalzed average over all VaR x [X G] wth x (α, 1). If X descrbes losses the ES can be nterpreted as the average loss n the worst (1 α) 100 % of the cases. The ES s also called condtonal Value-at-Rs (CVaR). Snce the VaR s monotone ncreasng n α, we obtan VaR α [X G] ES α [X]. Ths mples that for the same securty level α the ES s a more conservatve rs measure than the VaR. Moreover, the ES has the advantage that t better reflects the tal behavour of a dstrbuton than the VaR. In contrary to the valueat-rs VaR α wth securty level α whch s by defnton the (1 α)-quantle of the dstrbuton F G the ES α also ncorporates the shape of the dstrbuton functon behnd the α-quantle,.e. F G (x) for x (α, 1). For detals and a further dscusson on VaR and ES, see Acerb-Tasche [1]. The nsurance contract condton n Defnton 7.6 s now formulated wth the explct rs measures requred under Solvency II and SST: Solvency II: VaR 99.5% s requred as rs measure. Ths means that RES I [( VaR 99.5% S M I+1 + BEL I+1 + MVM I+1) D I ] (7.9) At tme I only a one-n-200-year event leads to a fnancal stuaton where accountng condton n Defnton 7.2 can not be fulflled at tme I + 1. SST: The rs measure ES 99% s requred,.e. RES I [( ES 99% S M I+1 + BEL I+1 + MVM I+1) D I]. (7.10) Only an event worse than the average loss n the worst 1 % of the cases leads to a fnancal stuaton where accountng condton n Defnton 7.2 can not be fulflled at tme I + 1. However, the exact calculaton of the rght hand sde n (7.9) and (7.10) s nfeasble n most clams reservng methods, because many clams reservng methods allow only for the calculaton of the frst two moments (or approprate estmates) of R I and CDR M,I+1. The nowledge of the dstrbuton of SI+1 M + BELI+1 + MVM I+1 would allow for the calculaton of the rght hand sde of (7.9) and (7.10). Unfortunately, ths dstrbuton s unnown n most clams reservng methods. Ths shows that the regulatory requrements (accountng condton n Defnton 7.2 and nsurance contract condton n Defnton 7.6) for reserves exceed the possbltes of most current clams reservng methods. Therefore, an agreement has to be found whch smplfcatons are permtted to mae these requrements accessble to current clams reservng methods Smplfcatons for Regulatory Solvency Requrements We consder proposals for smplfcatons gven n the SST, see FOPI [24] and FOPI [25], to mae current clams reservng methods accessble for the accountng and nsurance contract

127 7.1 Regulatory Requrements on Reserves 117 condtons n Defntons 7.2 and 7.6. As already mentoned n the begnnng of ths chapter all methods presented n ths thess, namely the class of (Bayesan) LSRMs n Chapter 4 and the PIC reservng method (wth dependence) n Chapters 5 and 6 allow for the dervaton of the followng rs characterstcs and the MVM, accordng to the regulatory CoC approach n predctor predcton uncertanty [ ] R I RI msep R I D RI I CDR M,I+1 0 msep CDR M,I+1 [0] D I (7.11) Defnton 7.4. Smplfcaton I Vewed from tme I, t s dffcult to derve the dstrbuton of the maret-value margn MVM I+1 at tme I, f the method used for calculatng the MVM s sophstcated (for example a general CoC approach). Therefore, we approxmate MVM I+1 by omttng n (7.4) the frst summand,.e. MVM I+1 MVM I+1 := c φ I+ n=i+1 R I R n I msep CDR M,I+1 D I [0] 1/2. (7.12) Snce MVM I+1 s D I -measurable we obtan wth (7.12) for condtonal rs measures, whch are translaton nvarant, e.g. VaR and ES, for the rght hand sde of (7.8) ρ I ( S M I+1 + BEL I+1 + MVM I+1) ρ I ( S M I+1 + BEL I+1) + MVM I+1. (7.13) Smplfcaton II Applyng Smplfcaton I t remans to calculate ( ρ I S M I+1 + BEL I+1) (7.14) for ρ I = VaR 99.5% and ρ I = ES 99%. The smplfcaton strategy for the calculaton of (7.14) s as follows: Vewed from tme I most clams reservng methods allow for the dervaton of the frst two moments of the quantty SI+1 M + BELI+1. The unnown dstrbuton of the quantty SI+1 M + BEL I+1 s approxmated by a log-normal dstrbuton wth the same frst two moments usng the methods of moments for calbraton. Ths s llustrated n Fgure 7.2. The rs measures ρ I = VaR 99.5% and ρ I = ES 99% of the log-normal dstrbuton can then be calculated easly. Ths

128 118 7 Solvency approxmaton method smplfes the calculaton of the rght hand sde of (7.14) to the tas of calculatng (or estmatng) the frst two (condtonal) moments of the quantty SI+1 M + BELI+1 : Let us assume that the model used by the reservng actuary allows for the calculaton of the condtonal expectaton of outstandng loss labltes BEL n = R n := E[R n D n ] for n {I,..., I + J 1}, (7.15).e. the outstandng loss lablty predctors are best predctors. The tower property of condtonal expectatons (cf. Wllams [59]) then mples E [ S M I+1 + BEL I+1 D I ] = R I = BEL I Var [ S M I+1 + BEL I+1 D I] = msep CDR M,I+1 D I [0]. (7.16) For the PIC reservng method (wth dependence) the condton n (7.15) s fulflled, see (5.13) n Chapter 5 and (6.20) n Chapter 6 and hence we obtan the frst two (condtonal) moments of the quantty S M I+1 + BELI+1 by (7.16) (for the detaled structure, see (5.15) and (6.23)). For the class of Bayesan LSRMs we approxmate the rght hand sde of (7.16) by correspondng estmates gven n (4.77) and (4.79). Then we approxmate (7.14) by ( ρ I S M I+1 + BEL I+1) ( ) ρ I (X) wth X LN BEL I, msep CDR M,I+1 [0], D I where LN denotes a log-normal dstrbuton. 7.2 Example for Regulatory Reserves We revst the example gven for the Bayesan LSRMs n Chapter 4. In ths example we consder an nsurance company wth three nsurance portfolos 1, 2 and 3,.e. m {0, 1, 2} n the LSRM termnology, where each portfolo corresponds to an ndvdual LoB of the nsurance company, see Tables 7.5, 7.6 and 7.7. In ths secton we calculate reserves whch meet central regulatory solvency requrements n the SST. The best-estmate valuaton of labltes resultng from the Bayesan LSRM are gven n Table 7.1. The overall best-estmate valuaton of labltes are then gven by aggregaton over m {0, 1, 2}. For the calculaton of the maret-value margn MVM I we apply the regulatory CoC approach gven n Defnton 7.4 and as rs measure for the solvency captal requrements SCR I we use the expected shortfall ES 99% under Approxmatons I II as proposed n the SST. In the run-off trapezods n Tables there are 21 accdent years,.e. I = 20, and we consder 11 development years,.e. J = 10. Best-Estmate Valuaton of Labltes For the calculaton of the best-estmate valuaton of labltes BEL 20 at tme 20 we apply the specfc Bayesan LSRM used n Chapter 4. At tme 20 the best-estmate valuaton of labltes

129 7.2 Example for Regulatory Reserves 119 LoB BEL I Total 267 LoB BEL I Total 349 LoB BEL I Total 594 Table 7.1: Predcted ncremental clam nformaton for LoB 1, 2 and 3

130 120 7 Solvency BEL for each LoB s gven n Table 7.1. The overall BEL then result as BEL 20 = BEL BEL BEL 20 2 = = 1210, (7.17) where BEL 20 0, BEL 20 1 and BEL 20 2 denote the best-estmate valuaton of labltes for LoB 1, 2 and 3, respectvely. In order to fulfll the accountng condton n Defnton 7.2, whch guarantees transferablty of the run-off portfolo to a thrd party, we add to the BEL 20 n (7.17) a maretvalue margn MVM 20. Maret-Value Margn As proposed n the SST gudelnes we use the regulatory CoC approach gven n Defnton 7.4 wth a cost-of-captal rate c = 6% to calculate the MVM. We choose the parameter φ = 3, [ snce ths choce provdes VaR 99.5 CDR M,I+1 ] 3 msep CDR M,I+1 [0] 1/2 n the case that D I CDR M,I+1 s normally dstrbuted wth zero mean and varance σ 2 = msep CDR M,I+1 [0]. For D I the calculaton of the MVM accordng to Defnton 7.4 we apply the expected pattern of BEL gven by Table 7.2. The dentty msep CDR M,I+1 [0] 1/2 = 646 s gven n (7.18). We obtan for D I n R n I Table 7.2: Expected pattern of BEL for calendar years n = 20,..., 29 the MVM at tme I = 20 calculated by the regulatory CoC approach MVM 20,CoC = 29 n=20 = = ρ n ( ) 1210 Ths mples for the far value of labltes FVL 20 at tme I = 20 FVL 20 = BEL 20 + MVM 20,CoC = = Ths far value of the run-off labltes (n a mared-to-model vew) s the prce the run-off portfolo can be transferred to a thrd wllng and nowledgeable party. Hereby t s assumed that 646

131 7.2 Example for Regulatory Reserves 121 both the nsurance company and the thrd party (nvestor) agree to use the same clams reservng method for calculatng the best-estmate valuaton of labltes BEL 20 and the regulatory CoC approach for the calculaton of the maret-value margn MVM 20. Solvency Captal Requrements We follow the approxmaton outlne gven n the SST to derve the SCR. In the specfc Bayesan LSRM under consderaton we obtan BEL 20 = 1210, msep CDR M,21 D 20 [0] 1/2 = = 646, (7.18) where 480, 207 and 240 are the square roots of msep CDR 0,21 D 20 [0], msep CDR 1,21 D 20 [0] and msep CDR 2,21 D 20 [0], respectvely and 294 corresponds to the covarance terms between the dfferent LoBs (cf. Estmator 4.25). Now we ft by the method of moments a log-normally dstrbuted random varable X to these emprcal moments. Ths leads to E[X]= exp } {µ + σ2 = Var[X]= exp{2µ + σ 2 } ( exp{σ 2 } 1 ) = whch mply µ = and σ = The correspondng densty f(x) of X s then gven by { } f(x) = exp (log(x) ) 2 for x > 0 2πx for x 0, see Fgure 7.2 for an llustraton. Then the ES can be calculated by the well-nown formula ES α [X]= 1 1 α eµ+σ2 /2 Φ(σ Φ 1 (α)), where α (0, 1) denotes a securty level and Φ the dstrbuton functon of the standard normal dstrbuton. Usng the securty level α = 99% and the parameter estmates µ = and σ = resultng from the method of moments we obtan ES 99% [X]= By Defnton 7.6 we obtan for the solvency captal requrements SCR 20 = ρ 20 ( S M 21 + BEL 21 + MVM 21) FVL 20 ρ 20 ( S M 21 + BEL 21) + MVM 21 FVL 20 = ES 99% [ S M 21 + BEL 21 D 20] + MVM 21 FVL = (7.19)

132 122 7 Solvency f(x) = πx (log(x) ) 2 e f(x) 0e+00 2e-04 4e-04 6e-04 8e-04 E[X] = 1210 VaR 0.99[X] = 3421 ES 0.99[X] = x Fgure 7.2: 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 = 0.99

133 7.2 Example for Regulatory Reserves requred reserves RES 20 = 4164 SCR 20 = 2782 MVM 20 = 172 BEL 20 = 1210 assets labltes Fgure 7.3: 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 Reserves Summng up the results for best-estmate valuaton of labltes BEL 20, the maret-value margn MVM 20 and the solvency captal requrements SCR 20 we obtan RES 20 = BEL 20 + MVM 20 + SCR 20 = = Ths means that the overall reserves RES 20 must be at least to fulfll the regulatory solvency requrements (accountng condton, see Defnton 7.2, and nsurance contract condton, see Defnton 7.6). For an llustraton, see Fgures 7.2 and 7.3. At ths stage t becomes obvous that the Smplfcatons I II are crucal for the calculaton of MVM and SCR n all dstrbuton-free clams reservng methods. At a frst glance the requred amount of RES 20 = seems very hgh and we wll analyze ths fact n more detal. The bggest loadng of the reserves RES 20 s contrbuted by the solvency captal requrements SCR 20 [ = and the expected shortfall ES 99% S M 21 + BEL 21 D 20] of the log-normal dstrbuton s the man rs drver of ths quantty, see (7.19). The crucal pont s that the ES of the heavy-taled log-normal dstrbuton manly depends on the varance of the log-normal dstrbuton. In (7.16) the varance s replaced by the condtonal MSEP of the Bayesan LSRM. In our specfc Bayesan LSRM n Chapter 4 the square root of the condtonal MSEP s about 50% of the predcted outstandng loss labltes, see (7.18), what s much hgher than n many other clams reservng methods. Ths s due to the fact that the Bayesan LSRM s chosen n such a way that all three LoBs are hghly correlated, for detals see Secton 4.3. Hence the assocated MSEP of the CDR s much hgher then n most other clams reservng methods leadng to a hgh SCR. Ths explans the hgh value for the reserves RES 20.

134

135 Conclusons and Outloo In ths thess we consdered the problem of clams reservng as one of the man actuaral tass n non-lfe nsurance practce. In Chapter 2 we started wth the ntroducton of typcal loss lablty cash flows (clams payments) generated by classcal non-lfe nsurance run-off portfolos. For a complete rs assessment of these run-off portfolos all clams payments at any tme n the future have to be predcted based on all relevant nformaton avalable at tme of predcton (predcton step). In Chapter 3 we gave a bref ntroducton n classcal wdely-used dstrbuton-free clams reservng methods. We saw that most of these classcal methods are very lmted w.r.t. the nformaton sources they can ncorporate. To solve ths problem we presented three clams reservng methods whch can cope wth varous dfferent sources of nformaton. Model I: At frst we consdered the class of LSRMs n Chapter 4. Ths model class was recently presented by Dahms [17] and generalzes almost all dstrbuton-free clams reservng methods, gven n Chapter 3. However, expert nowledge w.r.t. the clams development pattern can not be ncluded n a mathematcally consstent way. We consdered LSRMs n a Bayesan model setup and approxmated the Bayes predctors by ther correspondng credblty based predctors. Ths led to the class of Bayesan LSRMs. Ths model class addtonally allows for ncludng expert nowledge/external data w.r.t. the development pattern. Such credblty based methods are often appled for prcng problems, f there s only a scarce data base avalable and extern pror nowledge s to be ncluded. Models II and III: Besde the class of dstrbuton-free clams reservng methods there are varous approaches to clams reservng based on dstrbutonal assumptons. An mportant representatve among them s the PIC reservng method ntroduced n Merz-Wüthrch [46] whch allows to combne pad and ncurred data smultaneously. These data sources are often avalable n nsurance practce and hence should be utlzed for predcton purposes. We recaptulated ths model and showed how the one-year CDR predcton uncertanty can be quantfed. The classcal PIC reservng method assumes the pad and ncurred ratos to be ndependent. Therefore, n a second step, we generalzed the classcal PIC reservng method n that way that t respects dependence structures often observed n practce n pad and ncurred data. Ths led to the PIC reservng method wth dependence modelng. Concludng, we consdered n ths thess three methods: The Bayesan LSRMs, PIC reservng 125

136 126 Conclusons and Outloo method and PIC reservng method wth dependence modelng. All three models allow for the ncorporaton of data sources often avalable n nsurance practce and hence tae account for the requrement that all predctons should be based on all data avalable. Moreover, all three methods allow for the dervaton of best-estmate valuatons of labltes BEL and estmates for the ultmate clam as well as the CDR predcton uncertanty. On top of that, the PIC reservng method (wth dependence modelng) provdes the whole predctve dstrbuton of the ultmate clam and CDR. As we ponted out n Chapter 7 BEL, MVM and SCR can be calculated n each model based on the BEL and the CDR predcton uncertanty. We state two questons to be answered n future research: 1. The (Bayesan) LSRMs s a large class of dstrbuton-free clams reservng methods. Therefore, t would be helpful to have a crteron for model selecton. Havng such a crteron the specfc LSRM s chosen whch provdes the best ft to the data w.r.t. ths model selecton crteron. A smlar problem remans n the PIC reservng method wth dependence. For the explct choce of the covarance matrx V an estmator has to be found whch s optmal w.r.t. to some optmalty crteron. 2. All calculatons n the three methods are provded on a nomnal scale,.e. tme value of money (stochastc dscountng) s not consdered. Of course, ths n-lne wth the status quo of classcal clams reservng lterature but does not accommodate recent developments of maret-consstent valuaton technques, see Wüthrch Merz [62]. Therefore, we put the queston: To what extend these methods can be generalzed to a full maret-consstent valuaton approach? All three methods allow for the calculaton of BEL. Based on these BEL the valuaton portfolo, see Wüthrch Merz [62], can be calculated for each method leadng to a maretconsstent BEL. The assessment of the predcton uncertanty n terms of the MSEP for the ultmate clams and for the CDR n the maret-consstent valuaton setup turns out to be more sophstcated. For the wde model class of (Bayesan) LSRMs t seems possble to extend ths model class w.r.t. maret-consstent valuaton. However, to the best of our nowledge no scentfc contrbuton consderng ths ssue exsts so far. Mang LSRMs accessble to maretconsstent valuaton would allow to consder most dstrbuton-free clams reservng methods (CL, BF, CLR methods), wdely used n practce, n a maret-consstent valuaton approach. Moreover, for the PIC reservng method (wth dependence modelng) and for most so far exstng dstrbutonal clams reservng methods the queston how these methods can be embedded n a valuaton approach should be subject to further research.

137 Accompanyng Publcatons Parts of ths thess have already been publshed: I: Credblty for the Lnear Stochastc Reservng Methods René Dahms Sebastan Happ Submtted II: Clams Developement Result n the Pad-Incurred Chan Reservng Method Sebastan Happ Mchael Merz Maro V. Wüthrch Insurance: Mathematcs and Economcs, Volume 51, Issue 1, July 2012, pp III: Pad Incurred Chan Reservng Method wth Dependence Modelng Sebastan Happ Maro V. Wüthrch Astn Bulletn, Volume 43, Issue 1, January 2013, pp Authors nformaton René Dahms: [email protected] Sebastan Happ: [email protected] Mchael Merz: [email protected] Maro V. Wüthrch: [email protected]

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143 Data Sets Data sets used n the contrbutons of ths thess: \ j Table 7.3: Cumulatve clams payments \j Table 7.4: Incurred losses 133

144 \ j Table 7.5: Busness unt 1 134

145 \ j Table 7.6: Busness unt 2 135

146 \ j Table 7.7: Busness unt 3 136

147 Table 7.8: Cumulatve clams payments P,j, + j 21, from a motor thrd party lablty 137

148 Table 7.9: Incurred losses I,j, + j 21, from a motor thrd party lablty

149 Edesstattlche Verscherung Hermt erläre ch, Sebastan Happ, an Edes statt, dass ch de vorlegende Dssertaton mt dem Ttel Stochastc Clams Reservng under Consderaton of Varous Dfferent Sources of Informaton selbständg und ohne fremde Hlfe verfasst habe. Andere als de von mr angegebenen Quellen und Hlfsmttel habe ch ncht benutzt. Ort und Datum: Hamburg, den Unterschrft: 139