POSSIBILITIES OF INDIVIDUAL CLAIM RESERVE RISK MODELING

Transcription

1 POSSIBILITIES OF INDIVIDUAL CLAIM RESERVE RISK MODELING Pavel Zimmermann * 1. Introduction A ignificant increae in demand for inurance and financial rik quantification ha occurred recently due to the fact that the implementation of a complex of rule of international reporting tandard (IFRS) and olvency reporting (Solvency II) ha tarted. It appear that the key quetion for olvency meauring i the etimation of the probability ditribution of future cah flow of an inurance company. That mean it i required to model how much and when the company i liable to the inured. Solvency i then reported through an appropriate rik meaure baed, e.g., on a percentile of thi ditribution. While the preent popular model (ee, e.g., Mack, 1993; England and Verrall, 2002; England and Verrall, 2006; or (Merz and Wuthrich, 2008) are baed olely on aggregated data (uch a total lo development for a certain time period), it i certainly challenging from both the theoretical and the practical point of view to crutinize poibilitie of modeling the reerve rik (i.e., roughly aid, the ditribution of the ultimate incurred value of claim that have already happened in the pat) baed directly on individual claim. Although it i obviou that thi kind of modeling i neceary to properly include ome individual propertie uch a limit or the re-inurer hare, thi kind of model ha not yet become popular. Several uch approache have already been publihed. One can ditinguih article focued on theoretical apect of the underlying procee and one focued on the practical apect. An intereting inconitency appear between thee two group of model. While the theoretically oriented model (uch a Norberg, 1993 or Arja, 1989) tre the continuou time apect of the individual claim random procee approach, model that are oriented more on the practical ide of the problem (motly thoe publihed on the ground of conultancy companie, uch a Taylor et al., 2008 or Murphy and McLennan, 2006) contruct their model baed on dicrete time point (uually conecutive development year). In thi paper, an attempt to overcome thi paradox i made. The development of an individual claim (the ettlement proce) i not modeled in conecutive development year. Intead, the o-called tage of the ettlement proce are introduced and defined a the time period between change * Univerity of Economic, Prague, Faculty of Informatic and Statitic (pavel.zimmermann@ve.cz). Thi paper wa upported by internal grant of Univerity of Economic, Prague, number F4/25/

2 AOP 19(6), 2011, ISSN in the incurred value. It i hown that under practically acceptable aumption, the handling time model (model of the time from occurrence to ettlement) i eparated from the model of the ultimate incurred value (final lo) of the claim. It i apparent that the key element of uch model i the ditribution of the future liability that tem from an individual claim. In thi paper, we will compile the theoretical background of the modeled random procee and then we will focu on poibilitie of practical modeling of an individual claim. Thu, we will focu on: 1. Aumption that would ignificantly implify the model on the one hand and could till be conidered acceptable from the practical point of view on the other hand; 2. Decompoition of the problem into ub-problem that can be olved under real condition. The main focu will be on a model of the ettlement proce of an individual claim. Aggregation of the rik will not be within the cope of thi paper. It i, however, anticipated that Monte Carlo imulation will be ued for thi purpoe. In Section 2, ome baic term of the ettlement proce are introduced and the proce i decribed formally. Main variable and their notation are introduced here. In Section 3, the baic tructure of the model i outlined and implification that allow eparating the component of the incurred value change (i.e., rik of the reerve reviion) and the timing component are et. In Section 4, the model of the incurred value change i derived. Some further implifying aumption are made and commented on from the practical point of view. One of the main reult of thi article, the probability denity function of the ultimate incurred value (i.e., the ultimate lo) i derived at the end of thi ection. The other component of the model, i.e., the ditribution of the time to ettlement, i conidered in Section 5. The decompoition of the problem into the above mentioned component allow uing tandard tatitical procedure for etimating them. One of the component, however, eem to be particularly challenging: the model of the ize of the incurred value change. In Section 6, a joint modeling procedure i applied to thi particular component and ome intereting practical propertie are demontrated on real data. 2. The Ultimate Incurred Value 2.1 Practical Decription Firt, we formulate the problem from the practical point of view and we briefly introduce ome baic term. The inurance company ha to et up reerve (technical proviion) until all claim are paid out to preerve it olvency. We can ditinguih three main type of reerve: 1. The reerve on claim that occurred, were reported but are not yet ettled. 2. The reerve on claim that occurred but were not enough reported. 3. The reerve on claim that occurred but have not yet been reported. The firt type of reerve i uually the reponibility of the claim handling department. Thi type of reerve i alway aigned to a pecific claim. Thee reerve 47

3 ACTA OECONOMICA PRAGENSIA 6/2011 are uually called the cae reerve or the RBNS reerve (reported but not ettled). When a claim occur and i reported to the inurance company, the claim handler collect all available information and et up thi reerve. Thi reerve baically reflect the claim handler opinion on the unknown random ultimate lo paid to the inured. A more and more information about the claim become available the claim handler update hi etimate and change the reerve. When all the neceary information i collected (or the trial i ended, etc.), the claim i ettled and paid out. Thi proce will be referred to a the ettlement proce. Sometime the claim may be partially paid out before the ettlement. Then, the correponding amount of the reerve i releaed. For thi reaon, another term i commonly ued: the o-called incurred value. The incurred value of a claim i the amount already paid out plu the appropriate reerve aigned to thi claim. That mean that the incurred value i a general term that repreent the preently etimated value of the claim (no matter whether the claim i paid out or not). The potential unfavorable evolution of the incurred value need to be covered from the reerve on claim that are incurred but not enough reported (IBNER). The third type of reerve, i.e., the reerve on claim that incurred but have not yet been reported (alo called the IBNR reerve ) i the reponibility of the actuarie. Baed on the hitorical obervation and tatitical method, the total amount of claim that are yet to be reported i etimated. Thi type of reerve i not aigned to individual claim but rather to ome aggregate. From a rik management point of view, it i needed to determine the rik that all thee reerve will not be ufficient to cover future liabilitie. That mean that an additional econd level etimate i commonly produced. Thi etimate concentrate on the etimate of volatility and perhap alo on the potential bia of the firt level etimate (i.e., meaure whether the previouly mentioned claim handler and actuarie etimate are not ytematically under- or overtated and what i their accuracy). 2.2 Formal Decription of the Settlement Proce Paper on thi topic (uch a Arja, 1989 or Norberg, 1993) generally ue marked procee for the decription of the ettlement proce. We will adopt a imilar decription of the problem in thi ection. We aume that for a claim that occurred at a certain time point T, there are two marked procee: 1. {( XT) 1}, where X are change in the incurred value (the adjutment) of the claim and T are the time point of the change in the incurred value for which T T1 T2 T 3 (1) 2. The delay DT1 T i then the reporting delay. The period between two conecutive event will be referred to a the tage of the proce in thi paper. (The term tage i not an official term.) 3. {( P T ) 1}, where P i the payment correponding to that claim, and i the correponding time point for which T 48

4 AOP 19(6), 2011, ISSN T T T1T2T 3 (2) 1 4. Notice that T1 T 1 due to the fact that ome amount of reerve ha to be et up prior to the firt payment. Thee procee are inter-related in many apect. Some baic relation hold between the cumulative procee defined a: and Yt () X (3) T t Qt () One practical property i that the inequality P (4) T t Qt () Yt () (5) hold for each t. Another ueful relation i that after a ufficient amount of time, the whole claim will be paid out and the two cumulative procee will be equal, i.e., Q( ) Y( ) (6) The variable Y ( ) i called the ultimate incurred value; it repreent the final lo generated by the claim and therefore i the mot important variable to be modeled. For the ake of implicity, thi variable will be denoted imply a Y. In thi article, we will conider the o-called tage of the ettlement proce. The -th tage will be defined a the time period between change in the incurred value, i.e., the interval T t T 1. During the -th tage, the incurred value remain contant and will be denoted a Y, i.e., Y Y() t for T t T 1 (7) 2.3 Cloed Claim Practically, there i only a finite number of nonzero adjutment { X } or payment { P } repectively. The number of the nonzero adjutment or payment will be denoted a S or S repectively. The claim i uually referred to a cloed or ettled after TS. Formally, we can conider (a in Arja, 1989) that after the ettlement, all change in the incurred value or payment equal to zero and their time point are equal to infinity. In thi article, we add extra mark to the procee. Namely, we add a dummy variable ( indicator ) A (or A repectively) repreenting the open/cloed tatu of the claim, i.e., defined a 49

5 ACTA OECONOMICA PRAGENSIA 6/2011 A 1 for T T (8) 1 S A 0 for T T 1 S that i, A 1 for tage in which the claim i open and A 0 for tage in which the claim i cloed. can be defined analogouly. A 3. The Model Structure 3.1 The Tak The time point up to which the information about the hitory i available will be referred to a the preent time and will be denoted a. At thi time, the main tak of the modeler i to etimate the amount yet to be paid out and the timing of the future payment. The latter problem need to be anwered due to the fact that the time value of money hould be conidered. (Determination of the invetment trategy i not within the cope of thi paper.) Thi mean we need to model the conditional ditribution of a random variable which i ome tranformation (e.g., preent value) of the random vector S PP P conditioning on the hitory of the claim known T 1 T 2 T 1 2 S up to the modeling time. Thi hitory will be denoted a H. 3.2 Payment or Incurred Value In the above paragraph, the incurred value i not mentioned explicitly a the required output of the model. However, in portfolio with longer development, it i beneficial to bae the etimate of the ultimate incurred value on the development of the incurred value Yt (), which typically converge to the ultimate incurred value ignificantly fater than the development of the payment Qt. () Thu, it i aumed that there i more information about the ultimate incurred (or, equivalently, ultimate paid) value contained in the incurred value hitory than in the paid amount hitory. Thi approach, however, till require the etimate of the payment a well a their timing in order to be able to include the time value of money (dicounting). That mean that the vector TT T SPP P Y would need to be etimated (conditioning on H ) Potential Simplification S It i quite obviou that a model with o many mutually dependent variable would be very difficult to contruct. Therefore, uitable implification hould be earched for to implify the model a much a poible while keeping the aumption more or le realitic. Since under normal economic condition it can be expected that dicounting i connected with lower rik than the reviion of the claim value, it eem reaonable to implify mainly the time variable. The tak can be greatly implified if we can aume that the ultimate incurred value of an individual claim i paid all at once at ome time T *. Thi reduce the tak to modeling only the random vector T Y (conditioning on H ). Thi implification further allow decompoing the problem into a reerve reviion rik problem (i.e., the change in the claim handler etimate of the incurred value) and the timing problem. It i poible to take advantage of the relation 50

6 AOP 19(6), 2011, ISSN T Y Y T f t yh f yh f t y H (9) where f denote the correponding probability ditribution function. Thi immediately ugget modeling the two component equentially. 4. Reerve Reviion Rik In thi ection, we will conider the firt multiple of the right-hand ide of equation (9), i.e., fy y H. The hitory H conit of the knowledge of value of everal random variable, namely the knowledge of: 1. the (change in the) incurred value { XT } of all claim in previou tage; 2. the time of the change in the incurred value { TT } of all claim in the previou tage; 3. the previou payment { P T } of all claim; 4. the time of the previou payment { T T } ; 5. potential external information. To determine the ditribution of the ultimate incurred value, we need to chooe the information that i relevant on the one hand and till manageable in term of modeling on the other hand. At the time of modeling (the time ), every claim i in a different tage and ha a different hitory. Therefore, every claim will have a different conditional ditribution of the ultimate incurred value. The elementary property i that once the claim i cloed, there i no randomne or rik in the claim and the ultimate incurred value equal the preent (i.e., at ) incurred value (or, equivalently, paid value). Augmenting the model for re-opening of claim i technically poible but thi feature would require ignificantly more operation and the materiality of re-opening i often negligible, therefore re-opening are not conidered in thi paper. We denote the preent tage of the claim a. We can write for the conditional ditribution of the ultimate incurred value of the preently open claim: (Notice that the value of i degenerated for.) Y Y S (10) fy S f yh f yh Y i known at the time. Therefore, the ditribution Thi probability denity can be further rewritten conidering the following relationhip f yh f yh f Y yh (11) Y S Y S Thi approach eem to be beneficial for modeling becaue (analogouly to traditional aggregate model) it allow modeling of the claim development in each future tage. (Notice, however, that the aggregate model conider dicrete time interval 51

7 ACTA OECONOMICA PRAGENSIA 6/2011 intead of tage.) It i poible to conider the ditribution of the incurred value a an integral over all trajectorie, i.e., f yh f Y yh Y S 1 1 Y Y Y 1 1 f y y yh f y y yh dy dy dy (12) S Since the event that the claim will be cloed in the -th tage, i.e., the event S i equivalent to the event the claim will remain open until and will be cloed in, that i A1 1A 2 1 A 1 1A 0. (Remember that no reopening wa conidered here, therefore thi event already implie that A 1 0A 2 0.) Thi mean that we can write further f yh f Y y H Y S 1 1 Y Y Y 1 1 f y y yh f 11 0 y y y H dy dy dy (13) A A 1 A Thi mean that we can further eparate the problem into a model of the incurred value development given that the claim i open and the probability that the claim remain open until, when it i cloed. At thi tage, ome further implification will have to be employed. In imilar cae, ome independence aumption are ued for the vector Y1Y 2 Y H. The mot common i probably the Markovian property. Of coure the validity of thi aumption i not guaranteed for all portfolio. In general it eem to be a reaonable approximation of reality which implifie the model ignificantly. If we accept thi aumption, we can write f y y y H f y H f y y H Y Y Y 1 1 Y Y 1 1 f y y H (14) Y Y 1 f y y H 14) Thi form finally allow for fitting the ditribution on the data uing a more or le tandard tatitical procedure. Of coure, the implet choice in thi cae i to aume that only the previou tage incurred value i relevant for the modeling. More complex aumption are alo poible if felt neceary. Some variable contained in H can be conidered. Notice the analogy of the right-hand ide multiple with the tochatic development factor common in aggregate model. 52

8 AOP 19(6), 2011, ISSN Now, we will conider the ditribution Again, it eem reaonable to model the probability of the claim remaining open or getting cloed tage by tage ; therefore, we can write f 11 0y y y H A A 1 A 1 f 11 0y y y H. A A 1 A 1 f 1 y y y H A 1 f 1 A 1 y y y H A 1 1 f 1 A 1 A 1 y y y H A f 0 A 1 A 1 A 1 y y y H (15) A The implification to aume here are traightforward. In the implet cae, the independence of the probability of a claim cloure on it incurred value can be aumed. Thi would implify the calculation to the greatet degree. Thi independence can, however, only be aumed in ome pecial cae. In general, one hould expect that larger claim are more complicated to handle and tend to pa through more tage before cloure. In uch a cae, it i probably reaonable to aume that the probability of a claim cloure (given that the claim i open) depend at leat on the preent incurred value. (Of coure, ome further variable contained in H might be added if felt neceary.) Thu, we can aume that f 11 0y y y H A A 1 A 1 f 1 y f 1 A 1y f 1 A 1A 1y A A 1 1 A f 0 A 1 A 1 A 1 y A 1 1 f 1 y f 1 A 1y f 1 A 1y A A 1 1 A A 1 f 0 A 1y (16) We will implify thi further uing the notation f 1 A 1y z y (17) A j j1 j j j 53

9 ACTA OECONOMICA PRAGENSIA 6/2011 Notice alo that f 0 A 1y 1z y (18) A j j1 j j j If we now plug equation (13) - (17) back into (10), we get fy yh 2 f 1 y f y y z y A Y f y y z y Y f y y z y Y f y y z y dy dy dy Y (19) Although an analytical olution to equation (19) will not be achievable in the general cae, thi equation reveal a way for a potential imulation model. The ultimate incurred value of a claim can be imulated by generating the future trajectorie of the claim. That i, for each claim and each future tage until cloure, we need to generate: 1. Whether the claim wa cloed or remained open in the given tage (given the incurred value in that tage) from the ditribution z j y j ; 2. Given that the claim wa open, the jump in the incurred value given the incurred value in the previou tage from the ditribution fy y y j j1. Once the claim i cloed (and therefore the ultimate incurred value i known), the ettlement time t can be generated given thi value. 5. The Settlement Time The previou ection were devoted to the firt multiple of the right-hand ide of equation (9), i.e., the ditribution fy y H. Now we will make ome note on the econd multiple, i.e., f t y H T. Thi component repreent the ditribution of the time to ettlement t given the hitory and the ultimate incurred value. Of coure, the choice of the covariate to condition on will be portfolio dependent. In general, one may expect dependence on the ultimate incurred value y ince (a tated above) larger claim are more complicated to ettle and tend to be ettled later than attritional claim. There i alo ome information contained in the hitory H. An obviou candidate to include i at leat the time already pent in the ettlement proce. An obviou relation for unettled claim i then T. 54

10 AOP 19(6), 2011, ISSN A tated in Section 3, it i aumed that the whole claim i paid at once at T intead of modeling the whole payment pattern 1 2 S 1 2 S TT T PP P (20) Thi mean that T hould be ome ort of average of the payment time T1T2 T S. Since the purpoe of the incluion of the ettlement time in the model i to include the time value of money, an obviou requirement for thi implification i that the dicounted value of the claim (to the time point repreenting the time of modeling) remain unchanged when uing thi implification. Therefore, we require S P S l1 l Pl ( T ) (21) ( T l ) (1 r) l1 (1 r) where r denote the interet rate aumed for dicounting. Thi mean we can et log( Y) log( Y D ) T (22) log(1 r) where Y D i the dicounted ultimate incurred value and P S l (23) ( T l ) l1 (1 r) S Y P (24) l1 l i the (undicounted) incurred value. Thi relation can be ued to fit the ditribution of T to the cloed claim data. In practical tak, everal extenion are needed ince one alway work with cenored data. Some further idea onthi topic can be found in (Zimmermann, 2010). 6. Illutration and Propertie of the Incurred Value Change In order to be able to ue the tochatic model outlined above, one mut elect appropriate model and data fitting technique for the component of the model. Wherea calibrating the probabilitie of a claim remaining open zj( y j) and the (conditional) ditribution of the time to ettlement f T t y H concern motly common tatitical method (e.g., logitic regreion in the former cae and urvival analyi in the latter cae), etting up an appropriate model for the component fy y y j j1 eem to be more challenging. Therefore, we will focu only on thi component here. Further information on the other component can be found in (Zimmermann, 2010). 55

11 ACTA OECONOMICA PRAGENSIA 6/2011 Although imilaritie can be expected for imilar product to ome extent, pecific behavior may of coure occur for pecific data. Therefore, the following analyi ha no ambition to be valid in general. The purpoe of thi chapter i to: 1. outline an appropriate method (which, however, may not be the only option for the given tak); and to 2. illutrate the method on a pecific portfolio with real propertie and interpret the reult to uncover propertie that might be earched for in other practical portfolio. The data ued have real propertie (real data with ome noie). In order to preerve confidentiality, actual value of the parameter are motly not publihed. Intead, a graphical repreentation of the reult i preented. We believe that the pecific value are not really important. Intead, we focu on propertie or technique that could be tranferable to other portfolio. On purpoe, we elected a group of claim that appear to be one of the mot complex to model the liability bodily claim. The maximum poible number of nonzero adjutment for each claim wa empirically et to m 11. There were a few claim with more than eleven oberved tage but eleven i believed to be ufficiently large to cover the abolute majority of claim. All adjutment in tage higher than eleven were therefore treated a the adjutment in the eleventh tage. Annuitie were excluded from thi analyi ince annuitie require an entirely different model baed on life inurance technique. Since the generalized linear model (GLM) are now commonly ued for analogou purpoe in the cae of aggregate model, one wa ued a the modeling technique in the cae of our model a well. A uggeted in Anderon et al., 2007, and ued, e.g., in Laren, 2007, only factor (no covariate) were ued. That mean, intead of uing the value y 1 directly, a dicretized variable Y i defined with Q dicrete value denoted a q 12 Q for each oberved claim (claim index i i dropped) and each tage 1. The dicretization i diplayed in Table 1. The reaoning for thi approach i decribed in Anderon et al., 2007: Although variate do not require any artificially impoed categorization, the main diadvantage i that the ue of polynomial may mooth over intereting effect in the underlying experience. Often it i better to begin modeling all variable a narrowly defined categorical factor (enuring ufficient data in each category) and if the categorical factor preent GLM parameter etimate which appear appropriate for modeling with a polynomial, then the polynomial in the variate may be ued in place of the categorical factor. Baed on a pre-tudy, two highly ignificant interaction were identified. Namely the expected incurred value for tage 3 with a previou incurred value higher than 13.3 thouand (i.e., 3 and q 3 ) and for tage 11 and a previou incurred value higher than 22 thouand (i.e., 11 and q 4 ). In an ideal cae, thee anomalie hould be confirmed from a practical point of view (e.g. by claim handler). However, thi wa not our cae, o no interpretation can be provided for thee interaction. 56

12 AOP 19(6), 2011, ISSN Table 1 The dicretization of the variable Y 1 : The value q of the variable Y 1, the mean obervation in each category q q Y -1 Y y. The (unconditional) ditribution of the incurred value in the firt tage Y1 X1 involve only well-known tatitical method uch a hitogram or parametric ditribution fitting. Therefore, we focu in the following text only on modeling the conditional ( tranition ) ditribution tarting from 2. The etimate performed in thi chapter were calculated baed on 7744 obervation. The logarithmic link function and the gamma error ditribution were conidered. Thu, for each claim i we aume a model of the form: where and EY i i (25) i exp( ) (26) i m Q 2 i 0 j j q y qi 1 ur ui (27) j2 q1 u1 where i i the linear predictor, 0 i the intercept, j i the dummy variable indicating in which tage the obervation wa oberved, i.e., j 1 for j, j 0 otherwie, y qi i the dummy variable indicating whether the i -th claim 57

13 ACTA OECONOMICA PRAGENSIA 6/2011 wa in the tage ( 1) in the q -th category of the incurred value, and the lat term incorporate the impact of the two interaction mentioned above (i.e., r ui indicate occurrence of the u -th interaction). j q and u are the parameter. Notice that ome parameter (of the reference level) are redundant and will be et to 0. From a practical point of view, there i one property which can not be captured by a imple GLM model. The common GLM pecification of the variance of the repone variable i: var Y V V exp (28) i i i where i the o-called cale (or diperion) parameter and V i i the variance function, which i for example in the cae of logarithmic link function and the gamma 2 ditributed error V i i. Thi mean that the etimated repone variance depend on the factor (and/or covariate) only through the etimated mean of the repone variable. The cale parameter i a contant. Therefore, thi model can not capture cae where the impact of the factor on the mean i different than the impact on the variance. From a practical point of view, we can expect a trong impact of the factor on the variance, which might be rather different from the impact of the factor on the mean repone. Namely, it i probably natural that, in later tage, when more and more information about the claim arrive to the claim handler, their etimate (preent incurred value) of the ultimate incurred value hould be more and more precie the adjutment hould only repreent ome fine tuning of the lo. Thi can be tranlated a: We hould expect the variance in the repone to be decreaing with the tage. Since there i no reaon to aume that the mean repone EY i hould exhibit uch behavior, we can deduce from equation (28) that to capture thi property, the cale parameter hould be dependent on the tage. (Namely the cale parameter can be expected to be decreaing with the tage ). Similar behavior might be oberved for Y 1, i.e., it might be oberved that the proportionality of the variance to the expected repone i, e.g., lower for larger claim than for attritional claim. Thi mean that we alo need a model of the cale parameter. However, the model of the mean and the model of the cale parameter mut be interlinked. Thee two problem can not be eparated. Thi tak i called joint modeling of mean and diperion. The literature on thi topic i relatively limited. Some note can be found in the article by England and Verrall (2002. A few page on thi topic are alo mentioned in the monograph by McCullagh and Nelder (1989), which we ummarize in the following text. The main principle of the joint modeling i that for mean repone, we ue the ame pecification a in (25) - (27). Thi time, however, the pecification of the repone variance allow a factor-dependent (namely tage and previou incurred value dependent) cale parameter: It i aumed that var Y E d V (29) i i i i i, (30) 58

14 AOP 19(6), 2011, ISSN where where di i ome diperion tatitic and g ( ), (31) i D i g D i the diperion link function and m Q i j j q y qi 1 j2 q1 (32) i the diperion linear predictor and j and q are the parameter. There are everal option to chooe from for the diperion tatitic paper, the quared Pearon reidual wa choen, i.e., d 2 2 i i i i i 2 V i i di. In thi ( Y ) ( Y ) (33) In thi paper, we again work with the logarithmic link function, i.e., we aume the multiplicative effect of the tage and the previou incurred value on the cale parameter and we again aume the gamma ditributed error term. The pecification of diperion variance i var di VD( i ), (34) where VD( ) i the diperion variance function, which i again in the cae of logarithmic link function and the gamma ditributed error VD i 2 i. In McCullagh and Nelder (1989), the author tate: The two model are interlinked; that for the mean require an etimate of 1 i to be ued a the prior weight, while the diperion model require an etimate of i in order to form the diperion repone variable d i. The form of the interlinking ugget an obviou algorithm for fitting thee model, whereby we alternate between fitting the model for the mean for given weight 1 ˆ i, and fitting the model for the diperion uing the repone variable di diyi ˆ i. It i tated in England and Verrall (2006) that: If there i evidence that the cale parameter i not contant, imultaneou or joint modeling of the mean and variance can be performed. Uing maximum likelihood method, thi i an iterative proce, whereby initial parameter etimate are obtained uing arbitrary initial value for the cale parameter. The cale parameter are then updated, and revied parameter etimate obtained. The proce iterate until convergence, although after the firt iteration, the change are uually mall. The reulting parameter etimate provide an intereting inight into the propertie of the oberved proce. The etimate of the parameter of the model of the mean repone, i.e. model (27), are diplayed in Figure 1 and 2 and 3 repectively. Baed on 59

15 ACTA OECONOMICA PRAGENSIA 6/2011 the confidence interval, one can tate that in uch a pecified model, all the parameter etimate are tatitically ignificant. Figure 1 Etimate of the parameter (vertical axi) which are aigned to the dummy variable j coding the tage (horizontal axi). Dotted line repreent the upper and lower 95% confidence interval. Figure 2 Etimate of the parameter (vertical axi) which are aigned to the dummy variable y q coding the value of the dicretized previou tage incurred value Y 1. Dotted line repreent the upper and lower 95% confidence interval. On the horizontal axi, the category q i repreented by the mean of all the obervation oberved in the given category y q. 60

16 AOP 19(6), 2011, ISSN The etimate of the parameter aigned to the dummy variable repreenting the value of the tage (ee Figure 1) ugget that in tage 3, one can generally expect a higher mean incurred value than in the reference tage (tage 2). Thi i alo true for tage 4 to 10, but the difference from the reference category i not o dramatic. In the lat tage (tage 11), one can expect a lower mean incurred value than in the reference category. The etimate of the parameter aigned to the dummy variable repreenting the value of the previou incurred value Y 1 (Fig. 2) are again all tatitically ignificant. It eem that, if preferred, the original (undicretized) value could be ued a covariate with probably an additional effect of the mallet claim above the general trend. Both of the interaction parameter etimate were alo tatitically ignificant (ee Figure 3). Figure 3 Etimate of the parameter (vertical axi) which are aigned to the interaction dummy variable r u, u 12 (horizontal axi) with the upper and lower 95% confidence interval. The etimate of the diperion model parameter are diplayed in Figure 4 and 5 repectively. All the parameter etimate were tatitically ignificant. The parameter etimate aigned to the dummy variable repreenting the value of the tage ugget that the diperion ha a decreaing tendency with the tage, i.e., the reult confirm the aumption that in general the claim handler etimate become more and more precie during the ettlement proce. It i quite intereting that a imilar reult wa alo confirmed for the parameter etimate aigned to the dummy variable repreenting the dicretized previou tage incurred value. The reult can be interpreted a that the relative variability decreae with the ize of the claim if the claim i large, dramatic change are le probable than for mall claim. 61

17 ACTA OECONOMICA PRAGENSIA 6/2011 Figure 4 Model of the diperion Etimate of the parameter (vertical axi) which are aigned to the dummy variable j coding the tage (horizontal axi). Dotted line repreent the upper and lower 95% confidence interval. Figure 5 Model of the diperion Etimate of the parameter (vertical axi) which are aigned to the dummy variable y q coding the value of the dicretized previou tage incurred value Y. Dotted line repreent the upper and lower 95% confidence interval. On the horizontal axi, the category q i repreented by the mean of all obervation oberved in the given category y. q 62

18 AOP 19(6), 2011, ISSN Concluion An analyi of poibilitie of modeling the runoff of individual claim wa tudied in thi article. The problem wa decompoed into everal ub-problem. Thi decompoition followed two goal: 1. It i neceary that the conidered ditribution can be modeled from the practical point of view, i.e. baed only on data that can be expected to be available. 2. The model mut be reaonably complex in order to be applicable under real condition. One of the main focue of thi article wa to point out implifying aumption that are not obviouly wrong from a practical point of view on the one hand and that lead to a ignificant implification of the model on the other hand. For the reader convenience, we compile all the aumption uggeted in the above text here: 1. The ultimate incurred value of an individual claim i paid all at once at ome average time T. 2. A Markovian property of the erie of the incurred value Y 1 Y 2 Y given that the claim i open. 3. The probability of a claim cloure (given that the claim i open) only depend on the preent incurred value (Markovian property of the erie A1A 2 A 1). 4. The ettlement time T only depend on the ultimate incurred value Y. 5. Knowledge of the maximum number of tage a claim can pa through (i.e., knowledge of m ). Of coure, the aumption made here can not be conidered a alway valid. On the other hand, the aumption eem to be acceptable in a reaonable amount of practical cae and potential generalization of the aumption outlined can be conidered in many apect. Reference ANDERSON, D.; FELDBLUM S.; MODLIN, C.; SCHIRMACHER, D; SCHIRMACHER, E.; THANDIC, N A practitioner guide to generalized linear model. new/glmpaper/media/practitioner_guide.pdf. ARJAS, E The claim reerving problem in non-life inurance: Some tructural idea. ASTIN Bulletin, no. 19. ENGLAND, P.D.; VERRALL, R.J Stochatic claim reerving in general inurance. Britih Actuarial Journal vol. 8, no. 3, p ENGLAND, P.D.; VERRALL, R.J Predictive ditribution of outtanding liabilitie in general inurance. Annal of Actuarial Science vol. 1, no. 2, p LARSEN, C.R An individual claim reerving model. ASTIN Bulletin International Actuarial Aociation vol. 37, no. 1, p MERZ M. and WUTRICH M Modelling the claim development reult for olvency purpoe. Caualty Actuarial Society Forum Arlington, Virginia p

19 ACTA OECONOMICA PRAGENSIA 6/2011 MACK, T Ditribution-free calculation of the tandard error of chain-ladder reerve etimate. ASTIN Bulletin vol. 23, p MCCULLAGH, P.; NELDER, J.A Generalized Linear Model. Second Edition. Chapman and Hall, ISBN: MURPHY, K.; McLennan, A A method for projecting individual large claim. Caualty Actuarial Society Forum p NORBERG, R Prediction of outtanding liabilitie in non-life inurance. ASTIN Bulletin International Actuarial Aociation vol. 23, no. 1, p TAYLOR, G.; MCGUIRE G.; SULLIVAN, J Individual claim lo reerving conditioned by cae etimate. Annal of Actuarial Science vol. 3, p le/0016/24442/taylorreerving.pdf. ZIMMERMANN, P General Inurance Reerve Rik Modeling Baed on Unaggregated Data. Univerity of economic Prague, Czech Republic, PhD thei. POSSIBILITIES OF INDIVIDUAL CLAIM RESERVE RISK MODELING Abtract: Thi article outline poibilitie of modeling the ditribution of the future liabilitie of an inurance company that tem from a pat claim which ha not yet been ettled. Such a model might be ued a a key component of the internal model of the reerve rik of an inurance company. It focue on a probabilitic decription of the ettlement proce of an individual lo, i.e., on the development of the incurred value over the lifetime of the lo. Such a model allow etting up an internal model baed on an individual claim level intead of the aggregate claim level common nowaday. The propoed model repect two main retriction given by potential indutrial uage: Firtly, the model i et up in uch a way that neceary data can be aumed to be available from a practical point of view. Beide that, potential requirement on the complexity of the model are conidered and implifying aumption that allow etting up a model with reaonable complexity for practical ue are uggeted and commented on from a practical point of view. Calibration of the model of change in the anticipated lo (more preciely, the incurred value change) i illutrated on real data (adjuted for confi dentiality purpoe). The joint modeling procedure i applied, where a generalized linear model i aumed a a model of the incurred value (repone variable) a well a it repone variance. Intereting propertie which might be expected in imilar portfolio are revealed in the data. Keyword: Claim reerving, Settlement proce, Generalized Linear Model, Stochatic modeling JEL Claification: C40, G03 64