Generalized Linear Models for Traffic Annuity Claims, with Application to Claims Reserving

Transcription

1 Mathematcal Statstcs Stockholm Unversty Generalzed Lnear Models for Traffc Annuty Clams, wth Applcaton to Clams Reservng Patrca Mera Benner Examensarbete 2010:2

2 Postal address: Mathematcal Statstcs Dept. of Mathematcs Stockholm Unversty SE Stockholm Sweden Internet:

3 Mathematcal Statstcs Stockholm Unversty Examensarbete 2010:2, Generalzed Lnear Models for Traffc Annuty Clams, wth Applcaton to Clams Reservng Patrca Mera Benner Aprl 2010 Abstract Currently the standard/common reservng technques used by the general nsurance actuares are based on the assumptons that future clams are gong to behave wth the same pattern/trend as they dd n the past, no allowance s made for any ndvdual clam nformaton, any change n speed wth whch the clams are settled or for any factors that may change the pattern. The man objectve of ths thess s to provde the reservng actuary wth an alternatve method than can be appled n cases when the common reservng methods fal to delver a sutable result, but also to nvestgate the outcome of usng an alternatve model that allows for ndvdual clams nformaton. In ths paper we wll try to predct the loss reserve for personal njury clams that compensates for the loss of ncome a clamant wll have as a consequence of traffc njury. Also known as traffc annuty clams, these types of clams are rather dffcult to estmate; gven that the vctms personal crcumstances wll have a sgnfcant effect on how the compensaton wll turn out to be. We wll apply a Generalzed Lnear Model to ndvdual data, and test f ths type of ndvdual clams method can work as a support for the actuares to mprove the reserve estmaton and at the same facltate the understandng of whch factors that are mportant for predctng the loss reserve for traffc annuty clams Postal address: Mathematcal Statstcs, Stockholm Unversty, SE , Sweden. E-mal: patrca.benner@trygghansa.se. Supervsor: Ola Hössjer.

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5 Generalzed Lnear Models for Traffc Annuty Clams, wth Applcaton to Clams Reservng Patrca Mera Benner Aprl 2010 Abstract Currently the standard/common reservng technques used by the general nsurance actuares are based on the assumptons that future clams are gong to behave wth the same pattern/trend as they dd n the past, no allowance s made for any ndvdual clam nformaton, any change n speed wth whch the clams are settled or for any factors that may change the pattern. The man objectve of ths thess s to provde the reservng actuary wth an alternatve method than can be appled n cases when the common reservng methods fal to delver a sutable result, but also to nvestgate the outcome of usng an alternatve model that allows for ndvdual clams nformaton. In ths paper we wll try to predct the loss reserve for personal njury clams that compensates for the loss of ncome a clamant wll have as a consequence of traffc njury. Also known as traffc annuty clams, these types of clams are rather dffcult to estmate; gven that the vctm s personal crcumstances wll have a sgnfcant effect on how the compensaton wll turn out to be. We wll apply a Generalzed Lnear Model to ndvdual data, and test f ths type of ndvdual clams method can work as a support for the actuares to mprove the reserve estmaton and at the same facltate the understandng of whch factors that are mportant for predctng the loss reserve for traffc annuty clams. 1

6 Foreword Acknowledgement I started to wrte ths report n the late sprng of 2006 at the Non-Lfe Insurance Company Trygg-Hansa, Nordc Actuaral dvson, Stockholm. In March 2008, I re-establsh my thess after workng at Trygg-Hansa for almost three years. I decded to make some changes n the report gven that I felt more mature n my role as an actuary. Ths report consttutes my master thess and conssts of 30 credts for a master degree of a total of 240 credts n mathematcal statstcs, from the Mathematcs Economy program at the Stockholm Unversty. Frst of all I would lke to thank my current manager and supervsor Lars Klngberg, Reservng Chef Actuary at Trygg-Hansa, for all hs support, effort and understandng. Lars has constantly provded good deas and dscussons for what ths thess should nclude. I wll also lke to thank my second supervsor Olov Dahlberg, Senor Actuary at Trygg Hansa, for hs contnuous support and gudance; Olov has played and plays an essental role n my contnuous development as an actuary and my understandng of the nsurance busness. I would as well want to express my grattude to my supervsor at the Unversty Ola Hössjer, for hs valuable comments, gudance and supervson durng the completon of ths thess. Lots of thanks also to Trygg Hansa s reservng and captal modelng team for beng such good workng mates. At last, but not least I would lke to thank my mother, rest n peace, for all the encouragement she gave me through my lfe. 2

7 Tables of contents Abstract... 1 Foreword... 2 Acknowledgement Introducton Purpose Background Personal Injury Clams Settlement Swedsh Road Traffc Injury Commsson Traffc Annuty Clams Swedsh Socal Insurance Method Clams Reservng The Chan Ladder Method Generalzed Lnear Model (GLM) The Exponental dsperson famly of dstrbuton Maxmum Lkelhood Estmaton of GLM Goodness of Ft Actuaral use of GLM Data Analyss Selectng a GLM Varables Number Observatons The response varable - Monthly annuty payment Gender Age at the tme of the accdent Salary Income Medcal Dsablty Percentage Varable Occupaton/Professon Addtonal Varables Not Included In The Analyss Income Dsablty Percentage Workmen s njury Compensaton from the Swedsh Socal Insurance Agency (Försäkrngskassan) Lnk Functon Model Selecton and Parameter Estmaton for Selected GLM

8 5.5.1 Model Selecton Includng All Varables Model Selecton Excludng Gender Model Selecton Excludng Chldren Model Excludng Chldren and Outlers Dscusson Summary Data Qualty Chldren Concluson and Outlook Gong Forward The Post Retrement Annuty Chldren Remanng Issues References A. Appendx A.1 THE ML-ESTIMATION OF Β AND Ø IN GLM A2. SALARY GROUP CLASSIFICATION A3. MODEL SELECTION INCLUDING FULL MODEL WITHOUT GENDER AND CHILD GROUP A4. CHILDREN GLM OUTPUT

9 1. Introducton One of the most mportant thngs that an nsurance company needs to be famlar wth s how much money t needs to set asde to face future labltes, clams ncurred but not reported or not enough reported. The assessment of the reserve plays a vtal role n the nsurance company s management, prcng and fnancal state and t s therefore crucal that the company can recognze at any gven tme how much ts ultmate responsblty s gong to be. It s also essental to understand the mportance of settng the rght clam cost. The reserve s classfed as a lablty n the company balance sheet and a reserve that s wrongly estmated ether excessvely or thnly could represent a false pcture of the company s fnancal state and lead to consderable problems for the company. For example a reserve that s nadequate can lead to under-prcng of rsk and as a result the premum rate wll be based on an optmstc estmaton of the company s current labltes and wll damage ts compettve poston n the market. The man objectve of reservng s to predct a best estmate of the ultmate cost of clams and the responsblty of the actuary s to choose a model/technque that produces an ultmate cost estmate that s as close to realty as possble. Unfortunately the best estmate of the clam cost s not always related to precson, as the standard devaton n the more common technques s rather wde and the outcome of the methods s used as a dstrbuton of possble outcomes rather than precse cost estmaton. 2. Purpose There exst a number of statstcal methods avalable for actuares for use to estmate clams reserves, each method requres dfferent assumptons and acheves dverse level of prudence. The selecton of model wll depend on the actuary and on what type of clams we wsh to estmate. One of the most common use reservng technques s the Chan Ladder. It s a relatvely smple reservng method based on the assumpton that clams on average are gong to developed at the same rate as they developed n the past. The man dlemma the actuares face by usng Chan Ladder s that t works rather poorly when the hstorcal pattern s not stable, t doesn t allow for any ndvdual clams nformaton or for any change n the speed wth whch ams are settled, or for any other factors whch may change. In many cases legal, envronmental or socety changes can dsturb the development of future clams, ndcatng that the use of the past to predct the future may be a rather dffcult task and wll probably not work n practce as well as other methods. One of the reason that Trygg-Hansa wants to nvestgate f a new alternatve model for traffc annuty clams may gve a better estmate of the future loss reserve s that the estmated average total cost usng payment trend dffers consderably from the one obtaned when we trend the ncurred cost (payments + expected payments). As a consequence of ths the company feels an uncertanty wth the models that they are currently usng and wants to nvestgate the factors that are creatng ths uncertanty. To ths end we ntroduce a generalzed lnear model wth11 covarates and ft ths model to a data set of 778 observatons. We leave for future work to apply ths model to clams reservng,.e. summng predcted loss reserves for a number of ndvduals wthn the reservng class. 5

10 3. Background To reserve accurately the actuary can not only rely on her/hs mathematcal/statstcal sklls, he/she also needs to spend great tme understandng the busness, prcng, clams handlers and the data. An actuary who gnores these factors would probably not be able to explan movements or devatons when they occur. How precse the reserve estmaton wll be may depend on what types of clams we are estmatng and on the effort the actuary has made to understand the data and the busness. It s wdely known that there are some busness classes whch are easer to predct than others, and the clams are usually classfed nto two groups, short and long taled classes. The short taled classes are the clams reported and pad to the nsurance company shortly after the accdent occurs. Mnor car accdents not resultng n any bodly njury damage are good examples of these types of clams. Generally all property damage type of polces are short taled. These types of clams are normally well behaved and have a stable pattern over tme makng t deal for the actuary to estmate. On the contrary long taled clams are those that take many years to pay, the tme between when the clams are reported and settled can be as long as 40 years, creatng a natural varaton n the reserve estmaton. Car accdents that result n bodly njury clams are usually long taled, a clam where a chld s severely njured wll probably take many years to settle. The complexty of the long taled clams and the lack of precson n the standard models used make reserve estmaton a sgnfcant challenge for the actuary. In concluson there are some busness classes/groups of clams where the standard reservng technques work better and can be appled wth great confdence by the actuary and there are some classes where the standard technques can be challenged and maybe should be challenged by the actuares n the struggle to predct the best estmate of the future cost of the company. The traffc annuty clams presented n ths thess are a perfect example of clams that are long taled, very dffcult to predct, wth a great varaton from clam to clam and where the standard reservng methods work rather poorly. To be able to understand the complexty of these types of clams we need to truly understand what t s that makes these types of clams so dffcult to predct. I wll therefore spend the rest of ths secton explanng the characterstcs of the market before we get nto mathematcal detals n the next secton. 3.1 Personal Injury Clams Settlement To understand the dffculty of estmatng the total cost of traffc annuty clams, t s mportant to frst of all understand how the Swedsh Motor Insurance busness works and how the legal enttes work wth t. The socal reforms play a very mportant role on how much compensaton the njured wll receve and should therefore not be underestmated. Accordng to Swedsh law, anyone that suffers from a personal njury clam as a result of a traffc accdent s enttled bascally to full compensaton for ther damages. The nsurance company s accountable to cover for damages cause by the nsured vehcle but also for labltes to thrd partes. Traffc accdents often occur n a fracton of a second but ther consequences can last for days, months, years or n the worst case for the rest of lfe. A large number of road users nvolved n traffc accdents recover from ther njures, but some of them never recover fully and suffer from some knd of permanent dsablty. 6

11 When an ndvdual gets njured as a consequence of a traffc accdent, the person s enttled to compensaton for medcal emergency and for future loss of ncome the vctm may have as result of the accdent. In Sweden the clams settlement for personal njury clams s often dvded nto two stages, medcal emergency and permanent dsablty. Fgure 1. Tme At the Accdent Medcal Dsablty (Temporary Dsablty) Permanent Dsablty Annuty Medcal emergency represents the perod from when the accdent occurred untl the tme when the njurer s medcal stuaton has become stable. The length of the perod wll generally depend on the complexty of njury. To be able to categorze the degree of an njury, n Sweden the dsablty s often measured as a percentage (1-99%). In the medcal emergency phase the dsablty percentage s more or less consdered to be temporary, mostly due to the uncertanty of the njury. The percentage wll only be consdered defntve when the vctm s resdual adverse effect from the accdent s more or less permanent. The compensaton durng ths perod wll embrace non-fnancal expenses such as pan and sufferng, but also compensatons for loss of ncome wthn the perod. Permanent dsablty represents the perod when the dsablty percentage of the vctm s consdered permanent. From ths stage the vctm wll clam compensaton for the annual loss of ncome that s attrbutable to the njury. The process to calculate the annual loss of ncome the vctms have s done based on factors such as llness, unemployment and resdual earnng capacty as well as wder economc factors such as nflaton and taxaton. The nsurance company s man ratonale s to try to estmate what the work stuaton of the vctm would have been f the accdent never occurred, basng ths on how the njured work and salary stuaton was pror to the accdent but also on what the vctm s carrer pattern would have been f the accdent never occurred. The assumptons made to calculate the amount pad for loss of ncome s calculated after wthdrawng any socal compensaton or beneft (ncludng workers compensaton agreements). The amount pad wll n most cases be pad as a monthly annuty durng the vctm s lfetme and then reduced by a certan amount. 7

12 3.2 Swedsh Road Traffc Injury Commsson In Sweden, the socal securty system plays a key role n regulatng the clam. For ths reason, t s mportant that we understand how ths entty works. The Traffc Injury Commsson TSN (Traffc Injury Commsson, Swedsh abbrevaton) s an entty created by the Swedsh government wth the purpose of actng as a moderator between the Insurance Company and the clamant. The man objectve s that the clamant gets hs compensaton as equally and unformly as possble. The nsurance company s oblgated to consult TSN for a revewed opnon before reachng a fnal settlement n cases where the clamant and the nsurance company dsagree or when the medcal dsablty of the vctm s 10 percent or more. In theory, the vctm can appeal the decson taken by TSN; however n the vast majorty of the cases the clamant accepts the compensaton and the case s consdered settled. Is also worth mentonng that n Sweden the compensaton agreed by the vctm and the nsurance company can be re-examned for up to ten years after the clam was settled. The reopenng of the case s prmarly due to deteroraton n the vctm s medcal condton, as result, the njured work capacty has worsened and the compensaton for loss of ncome needs to be recalculated. 3.3 Traffc Annuty Clams A traffc annuty clam s a personal njury clam that emerges when a polcy-holder s njured (or a thrd party njured) as a result of a traffc accdent and fnd hmself ncapable to work as before. In Sweden ths annuty amount covers the future loss of ncome that the clamant may have, the annuty s set, so that t can be pad out over the entre lfetme of the njured. A traffc annuty usually deals wth heavly dsabled vctms of accdents of varous knds, makng t very dffcult to even attempt to predct the future loss of ncome for the vctm. The complexty of estmatng the cost of the annuty makes these types of clams a great challenge for the clams handler and actuares to predct. An annuty n whch the payment s connected to the lfe loss of ncome of one person s usually adjusted to an ndexaton mechansm. In Sweden the annuty ndexaton depends on the status of the annuty. If the annuty s open the monthly value of the annuty s ndexed by a wage ndex and f closed t s ndexed by the offcal CPI (Consumer prce ndex, KPI, Swedsh abbrevaton) When an annuty s calculated t s usually dvded nto two stages, pre retrement and post retrement. The frst one s the stage where calculaton s made for the loss of ncome the njured wll get untl he or she retres and the second represents the amount pad when the njured retres. It s mportant to separate these two stages, gven that the calculaton for each perod wll dffer. For the pre retrement stage, the calculaton s done based on parameters such as age, salary, gender, medcal dsablty, worker compensaton, occupaton and CPI. The post retrement wll be related to the amount calculated n stage 1, but wll depend on how many years of compensaton the njured wll get pror to retrement. The post retrement compensaton also known as dsablty penson s payable n full, three quarters, two thrd, half of one quarter of the full rate basc penson and supplementary penson. For example, full dsablty penson wll be pad for ndvduals that correspond to the correct full old age penson. For smplcty n ths paper we wll only revew the annuty amount calculated for pre retrement. 8

13 3.4 Swedsh Socal Insurance Accordng to Swedsh law f you lve and work n Sweden you are covered by Swedsh socal nsurance (Försäkrngskassan). The Swedsh socal nsurance agency compensates ndvduals for a number of dfferent benefts, such as sckness, actvty, parental, chld allowance and penson. For ths thess t s mportant to understand how the Swedsh socal nsurance works together wth nsurance companes, when they compensate vctms njured as a consequence of a traffc accdent. The compensaton the nsurance company pays out wll to a large extent depend on how much compensaton the vctm receves from the socal nsurance. In other terms, what s not covered by the socal nsurance wll be covered by the nsurance company, There are two beneft that are essental for vctms njured as a result of a traffc accdent; Sckness and actvty compensaton. The njured wll receve one or the other compensaton dependng on the crcumstances. The sckness compensaton s a beneft for people age that wll almost certanly not be able to work full-tme due to a dsablty, njury or llness. The compensaton s pad out as full, three-quarter, half or a quarter dependng on how your work capacty has been reduced. The amount of compensaton receved wll be related to the vctm s yearly salary, the socal nsurance entty wll use ths nformaton to calculate an assumed ncome estmatng how the vctm s earnng would have been f he/she had contnued to work. If the vctm has low or no ncome the compensaton wll be based on the njurer s age (guarantee beneft). The Actvty compensaton s a beneft granted to young people age that are not able to work due to llness, njury or dsablty. The beneft s smlar to the sckness compensaton, wth the only dfference that the njured can partcpate n recovery actvtes. We wll come back later n ths paper and dscuss the mpact and mportance the socal nsurance has n the GLM analyss done n ths thess, 9

14 4. Method The man purpose of ths thess s to frst of all, fnd a model that fts the data on an ndvdual bass and then apply t to expected future annutes. 4.1 Clams Reservng The nsurance company receves premums from ts costumers when they purchase an nsurance cover. As compensaton the nsurer wll cover for all damages occurrng durng the nsured perod. Generally the company wll receve the premums long before the clam occurs, the company wll then need to set asde a reserve that covers future payments. Most clams are reported to the company wthn a few days after the accdent occurs but there are also clams that take months and sometmes years before they are reported to the nsurance company. Clams reservng can be descrbed by the followng key elements: Case Reserve, whch s the amount set asde for clams that have been reported to the company but are not yet fully settled. Ths amount s usually set by clams handlers. Incurred But Not Yet Reported (IBNYR), whch s the amount set asde by the actuary for clams that are not yet reported to the company. Incurred But Not Enough Reported (IBNER) s the allowance for changes n the clams handlers estmate. Usually companes do not dstngush between IBNYR and IBNER, nstead they report the reserve IBNYR+IBNER as IBNR (ncurred but not reported). 4.2 The Chan Ladder Method The Chan Ladder method s one of the oldest and most well-known reservng methods used to estmate the ultmate clam cost. It s based on a smple algorthm whch bascally bulds on past experence. The Chan ladder technque s determnstc n nature, gven that the outcome s a pont estmate of the ultmate clams rather than a range of estmates. For several years the Chan Ladder methodology has been explaned from a determnstc algorthm whch was not derved from a stochastc model. But n more recent years a varety of artcles have been wrtten tryng to fnd the stochastc model that motvates the basc Chan Ladder algorthm, all n the search of quantfyng the varablty of the estmated ultmate clams amount, see for nstance Mack (1994), Mack and Venter (2000) and Taylor (2000). Generally the data s presented n form of clams trangles, where the rows represent accdent perods and the columns represent development perods. The data n the trangle can be of dfferent types such: Numbers of clams, ncurred clams, pad clams, case reserve etc. 10

15 4.3 Generalzed Lnear Model (GLM) Generalzed lnear models play a very mportant role n statstcal nference. They represent a mathematcal way of quantfyng the relatonshp between a response varable and a set of ndependent varables, ncludng a general class of statstcal models. The use of GLM provdes an mportant advantage over the multple lnear regresson model. One of the benefts s that we now go beyond the assumpton of a normal dstrbuton for response varables and can use any member of the exponental dsperson famly of dstrbutons. Ths ncludes Posson, gamma, bnomal and log- normal dstrbutons. Another mportant advantage s that the relatonshp between the expected response varable (Y) and the dependent varables does not longer need to be lnear. Instead of (Y) βx (4.1) we can allow for the coordnate wse transformaton g ( (Y)) βx (4.2) of (Y). Here X, β, Y and µ (Y) are defned as X X X X n 1... X... X X 1k 2k nk., β, k Y Y. Y, µ Y n n, where X s the n (k 1) matrx of covarates, usually called the desgn matrx. It contans all the values of ndependent varables and one column of 1: s correspondng to the ntercept. β s a (k 1) 1 matrx of parameters, contanng k+1 regresson parameters that need to be estmated. Generally the functon g( ) s referred as the lnk functon, snce t ratonalty s to lnk the expected value µ (Y) to the lnear predctor Xβ n a general, flexble way. 11

16 4.3.1 The Exponental dsperson famly of dstrbuton The dstrbuton of a random varabley, for observaton number ( 1,2,..., n) belongs to the exponental famly of dstrbuton f ts probablty densty functon can be wrtten n the form y b( ) f y ;, ) exp c( y,, ). (4.3) Y ( We assume here that the dsperson parameter s a postve constant, 0, and that the weghs are also postve, 0, for all observaton numbers 1,2,..., n. Further, s the canoncal parameter that depends on the observaton number, the functons b( ) and c ( ) depend on the dstrbuton and y s the multplcatve term between the response varable and the parameters. The mean and varance of Y are specfed by (cf. McCullagh and Nelder 1989) [Y ] b '( ) (4.4) where Var [Y ] b''( ) / 2 (4.5) V ) b''( ) (4.6) ( s known as the varance functon. Combnng (4.2) and (4.4) we fnd that parameters through s related to the regresson k g( b'( )) (4.7) where g ( ) (( b') ( )), assumng ( ) for all and some functon ( ). 4.4 Maxmum Lkelhood Estmaton of GLM j 0 The maxmum lkelhood method s used to determne the parameters that maxmze the probablty of sampled data. It also provdes an effcent method for quantfyng uncertanty by means of confdent ntervals. The method s consdered to be flexble and t can be appled to most models and dfferent types of data. More about The Maxmum Lkelhood method n Appendx A.2 j, j 12

17 4.5 Goodness of Ft Two statstcs that are useful n assessng the goodness of ft are the scale devance D defned as D ( ( ˆ,, ˆ, ˆ)) (4.19) 2 sat 0 k where ˆ, ˆ, o sat s the log lkelhood obtaned from the saturated model wth y, ˆ 1 k are maxmum lkelhood estmated of o,, k defned n Appendx A.1. ˆ for all., 1, and Pearson s ch-square 2 statstc 4.6 Actuaral use of GLM Generalzed lnear models have proved to be a very useful tool n general nsurance, see for nstance Brockman and Wrght (1992) and Ohlsson and Johansson (2004). It has been wdely used by actuares as a prcng nstrument for many years, and n more recent years t has also become a powerful tool for other areas n General Insurance. The benefts of generalzed lnear models are beng recognzed n many areas of actuaral practce, several actuaral papers have been wrtten n the last 30 years descrbng the use of GLM wthn loss reservng, clam cost estmaton, average clam cost and much more. A number of statstcal software s have been developed for actuaral use and as actuares contnue to become more famlar wth GLM we expect the number to ncrease. 13

18 5. Data Analyss The data used n ths paper was extracted from Trygg-Hansa s clams and annuty database. From ths regster we have only selected the clams that have an annuty monthly payment as consequences of labltes n a traffc accdent. The amount set as an annuty s only based on the clamant s nablty to work n the future. Due to the long tal nature of traffc personal njury clams t was qute a challenge to obtan a set of data that could be used n ths thess. The clams stagng envronment has changed on several occasons, makng t very complcated to get hstorcal data wth the qualty needed. We were able to create a dataset wth 902 ndependent observatons, where we only ncluded data wth accdent years between 1990 and We ddn t nclude any data for more recent accdent years due to the smple fact that t takes at least three years before an annuty can be settled. Clams older than 1990 were also excluded snce development changed consderably from the 70 s to the 90 s. The frst step was to calculate the monthly payment of each observaton as f the annuty was settled n We needed to do so due to the annuty ndexaton mechansm descrbed n secton 3.3. We acheved ths by calculatng the monthly value backwards n tme to the value t had when the accdent occurred. By dong ths we could then easly apply a wage ndex to calculate the monthly value as f the annuty was settled today. The wage ndex used depended on what type of occupaton the clamant had when the accdent occurred, and was gathered from Statstcs Sweden (SCB). Unfortunately we couldn t fnd any statstcs pror to For those years we decded to use the government s offcal CPI. 5.1 Selectng a GLM An mportant aspect of ths thess s model buldng. It s essental that we ensure that the selected model fts data well. For that reason, t s crucal that we begn wth an exploratory data analyss n order to get a better pcture of how the varables nteract wth each other. The purpose of ths exploratory analyss s to select a dstrbuton for the response varables, a lnk functon for the GLM and a set of predctor varables (covarates). We plot the response varable aganst each of the canddate predctor varables (properly dvded nto classes) and nspect vsually whether or not there s any correlaton. 14

19 5.2 Varables Number Observatons The number of annuty clams n each accdent year s a crucal statstc gven that several of the clams that are open today are as old as It s therefore very mportant to understand the dstrbuton of number of clams by accdent year for our dataset. Our frst thought was to nclude all statstcs avalable from the system. We were able to get data from accdent years 1973 to 2006, but we could at an early stage determne that clams older than 1990 were n many cases not representatve of the types of clams we beleve wll occur n the future. Dfferent crcumstances, such as law, governmental and envronment changes make t very complcated to rely on the past to predct the future. The number of observatons per accdent year durng s shown n Fgure 2. We can also observe from Fgure 2 that a great number of the annutes are dstrbuted between accdent years Ths s not surprsng gven that we know that annutes take a long tme to settle. Annutes are often settled wthn years, and as a result, annutes from 2001 may not be fully represented n the annuty system untl We can also observe that t takes at least three years before an annuty s settled see Fgure 2 and the low values for years , annutes that are settled quckly are usually cases where the njured medcal stuaton s stable. Fgure 2: Numbers of Annutes per Accdent Year Accdent Year 15

20 Fgure 3 shows the average number of years t takes to settle an annuty clam, the average year s expect to be around years dependng on what type of njury the clamant may have. The average number of years decreases for more recent accdent years. Ths s as expected as we expect the number of clams to ncrease consderably n future years when they remanng open clams are settled. Fgure 3: Average Tme to Settled Annutes Accdent Year The response varable - Monthly annuty payment One of the mportant characterstcs of the generalzed lnear model s that we don t need to lmt our response varable to be quanttatve and normally dstrbuted. We can use a varety of dstrbutons from the exponental dsperson famly to ft the model (as mentoned n Secton 4.3). The response varable can be of dfferent types such as bnary, ordnal, contnuous etc. By examnng data we can determne what type of response varable y we are dealng wth. It s clear from the graphs below that the monthly annuty payments have a dstrbuton where most of the compensaton s less than SEK. We can also observe that the dstrbuton s skewed to the rght and that there are some outstandng values that we mght consder excludng later on. 16

21 Probablty Fgure 4: Observed Monthly Annuty Payments as at ,300 0,250 0,200 0,150 0,100 0,050 0,000 Monthly annuty payment (000 SEKs) Is well-known that the Gamma dstrbuton s a flexble model for a skewed dstrbuton on the postve axs. We thus assume that the densty of Y s gamma dstrbuted,.e. where f 1 ) y ( y;, ) y e, 0,1,2,3 ( 1 for 0 s the Gamma functon. The mean and the varance are defned as ] [y Var 2 ( y ) If s fxed and vares, we thus get 2, a quadratc varance functon V ( ) and the dsperson parameter to ts mean. 1. Ths mples that the standard devaton of an observaton s proportonal 17

22 5.2.1 Gender The gender varable stands for the classfcaton between men and women. In Fgure 5 we plot the response varable aganst gender, to see f we could fnd any clear correlaton between them. We can observe that the monthly annuty value s hgher for men than for women. Ths s not a surprse gven the dfferences n the socal and economc status of men and women. One mport factor n the calculaton of the annuty s based on salary. Women frequently work wth publc related jobs that are usually less pad on average than jobs n the prvate sector, where a great number men work. Fgure 5: Box Plot of Loss of Income Compensaton Splt by Gender Class m o n t h l y _ v a l u e Woman 2 - Man Gender We let gender correspond to the frst covarate, 1 of the desgn matrx n secton 4.3 wth,1 0 for women and,1 1for men. Ths s descrbed n table 1. Table 1: Covarate for Gender Class Levels Class Values Covarates 2 Gender Woman, Man X, X 1,, 1 0, 1 18

23 5.2.2 Age at the tme of the accdent The process to calculate an annuty for future loss of ncome s very complex and a great challenge for the clams handlers. It can n many cases feel lke a guessng game as an attempt must be made to take all contngences nto account. These nclude factors such as llness, unemployment and resdual earnng capacty as well as wder economc factors such as nflaton and taxaton. When the njured s a chld, the uncertanty ncreases consderably gven that t does not have a relevant pre-njury salary to use as a startng pont to estmate future loss of ncome. In Sweden the chld s background, hs famly members, professon and socal status form the bass for calculatng the future loss of ncome. In other words, the assumpton made s that the chld wll follow a smlar career path as hs famly members. Thus the age of the vctm plays a sgnfcant role n how the annuty for future loss of ncome s calculated. We decded to classfy the varable nto four groups: Chldren, Young Adult, Adult and Mature Adult (see Table 2). Table 2: Age Group for Tme of Accdent Age Group Frequency Percentage Age Interval Chldren 31 3% 0-19 Young Adult % Adult % Mature Adult % >46 Total % In Fgure 6 we plot the response varable algned wth each age group. From the graph we can notce that vctms that were njured as a chld have a hgher compensaton compared to other age groups. There s strong evdence that the age of the vctm at the tme of the accdent plays an mport role n what compensaton the vctm wll get. It s reasonable to assume that chldren wll get hgher compensaton due to fact that t actually takes longer to settle these types of clams. Generally because we need to wat untl the chld s n a workng age before we even can make a patent statement on of how much the compensaton wll be. Table 3 gves an dea of how many years on average t takes to settle clams. We beleve that there s a clear relaton between the age of the vctm at the tme of the accdent and the tme t takes to settle the clam. Table 3: Average Age of Settlement of the Annuty for Dfferent Age Groups Chldren Young Adult Adult Mature Adult Chldren clams take on average at least eleven years to settle, eleven years s a rather optmstc number gven that we excluded all clams occurred pror to 1990 (see Secton 5.2.1). We beleve that the average age t takes to settle chldren clams s more lkely to be around 18 years. Unfortunately, by not ncludng clams occurred pror to 1990, the number of chldren n our data set s low (see Table 2). Takng all these factors n account t mght be reasonable to separate chldren annutes from the rest of the groups and do a separate analyss for ths type of annutes and nclude data from earler accdent years. We wll return to ths dscusson later n ths paper. 19

24 Fgure 6: Box Plot of Loss of ncome Compensaton Splt Age Group m o n t h l y _ v a l u e Mature Adult 4 - Adult 5 - Young Adult 6 - Chldren AgeGr oup In the desgn matrx n Secton 4.3 we use the covarates X, 2, X, 3 and X, 4 for the age at accdent of observaton, wth X, X, X ) (0,0,0) for mature adults, ( 1,0,0 ) for Adults, ( 0,1,0 ) for young adults and 0,0,1 ) (, 2,3, 4 ( for chldren. Ths s also descrbed n Table 4. Table 4: Covarates for Age Class Levels Class Values Covarates 4 Age Group Mature Adult, Adult Young Adult Chldren ( X, 2, X,3, X, 4) ( X, 2, X,3, X, 4) ( X, 2, X,3, X, 4) ( X, 2, X,3, X, 4) (0,0,0) (1,0,0) (0,1,0) (0,0,1) 20

25 Number of Annutes Salary Income Salary Income represents the yearly ncome the njured had at the tme when a settlement of the vctm s compensaton was reached between the nsurance company and the njured. Ths varable s extremely mportant gven that the compensaton the vctm wll get s to a great extent based on ths. Personal Injury clams, partcularly clams that become annutes are usually handled manually by clams handlers. On a case by case bass, the clams handler wll follow the case and close t when a settlement between the njured and nsurance company s reached. When the case s not settled the yearly ncome of the vctm wll be updated each year usng the wage ndex for that partcularly case. In many cases the settlement wll take years, the salary of the ndvdual would probably change, but gven that the cases are settled manually, clams handlers may fal to send ths nformaton nto the clams database. For smplcty we assume that salary n our sample dataset s the salary ncome the ndvdual had at the tme when a settlement was reached between the nsurance company and the vctm. We use an average wage ndex to ndex the salary to the current value. To group the varable nto a categorcal varable t was essental to analyze the observed salary values. In Fgure 7 we can study the dstrbuton of the yearly salary ncome for all annutes. Note that a great number of njures have salary zero and that there are some extreme values n the tal. It s rather crtcal to understand why some observatons have zero values. One of the reasons s that f a vctm at the tme of the accdent was a chld, a student or unemployed the yearly salary wll probably be zero, but the annuty wll stll be calculated on the bass of a yearly salary. We can also observe n Fgure 7 that there are some outstandng values n the tal that we mght consder excludng later. Fgure 7: Observed Yearly Salary Yearly Income(000' sek) 21

26 After examnng the data we decded to group the varable nto three groups; Low, Medum and Hgh (also descrbed n Table 5) Table 5: Groups for Salary Income Salary Group Frequency Percentage Salary (000 SEKm) Low % Medum % Hgh 52 6% 351- Total % In Fgure 8 we can observe that the group wth hghest compensaton s the group wth hghest yearly earnngs. Ths s no surprse gven that the monthly annuty compensaton s calculated based on the vctm s ncome. We can also observe that the vctms wth lowest ncome have on average hgher compensatons than the group wth medum earnngs. Ths may seem a bt contradctory, but t can easly be explaned by two factors. The frst factor s that there are a number of observatons where the vctm s yearly salary may not be correct updated n the system and as a result the data can be msleadng. The other mportant factor s the mpact of socal nsurance (see Secton 3.4). Ths nformaton s mportant to keep n mnd when we choose our fnal model. Fgure 8: Box Plot of Loss of ncome Compensaton Splt by Salary Group m o n t h l y _ v a l u e low 8 - Medum 9 - Hgh Sal ar y_gr oup 22

27 In the desgn matrx n Secton 4.3 we use the covarates X, 5, X, 6 and X, 7 of observaton, wth, X ) (0,0) (, 5, 6 ncome. Ths s also descrbed n Table 6. to descrbe the salary ncome X for low ncome, ( 1,0 ) for medum ncome and 0,1 ) ( for hgh Table 6: Covarates for Salary Class Levels Class Values Covarates 3 Salary Group Low Medum ( X, 5, X, 6) X, X ) (, 5, 6 (0,0) (1,0) Hgh ( X, 5, X, 6) (0,1) Medcal Dsablty Percentage Medcal dsablty percentage represents the permanent medcal assessment that the vctm wll have as a consequence of a traffc njury, stated as a percentage 1-99%. The assessment s known to be an mportant factor n determnng the overall level of compensaton for non-fnancal damage, but no clear relatonshp has been establshed for the compensaton of loss of ncome that a person may get. Fgure 9 shows the medcal dsablty dstrbuton n our sample dataset. It s clear from the graph that most of the annutes have a medcal dsablty percentage that s less than 50 % and that there are sgnfcant number of annutes that have a dsablty less than 20%. Fgure 9: Number of Annutes Splt by Medcal Dsablty Percentage Medcal Dsablty Percentage 23

28 It s mportant to pont out that a medcal dsablty of 15 % does not necessary mean that the vctm s 15 % ncapable of workng, n many cases the vctm may be 100% compensated for loss of ncome, but the medcal dsablty percentage may be less than 50%. Consequently t s crucal to keep n mnd when we apply our model that usng medcal dsablty percentage to estmate future loss of ncome can be deceptve. It can gve over-compensaton as well as under-compensaton n comparson to the actual loss. We categorzed the varable nto 4 groups. Table 7 shows how the varable was classfed and dstrbuted wthn the chosen classes n our sample data. Table 7: Groups for Medcal Dsablty Medcal Dsablty Frequency Percentage Medcal Dsablty (%) Group Low % 0-25 Medum Low % Medum Hgh 28 3% Hgh 32 4% Total % The second step was to examne f ths varable had any statstcal sgnfcance n determnng the level of compensaton for the fnancal loss that the clamant wll get. We plot the varable aganst the response varable (see Fgure 10) and found that there was no clear lnear relatonshp between the two varables. However there seems to be a weak trend that hgher medcal dsablty leads to larger expected annuty payment. It s worth commentng, that the group of ndvduals wth the hghest annuty compensaton (wth a medcal dsablty nterval of 0-75%) have medcal dsablty between 26 50%. After dscussng ths wth clams handlers, they confrm that the clams that are more expensve are not the ones where the vctm s totally mpared, rather the cases where the vctm s % dsabled. Ths may come as a surprse, but the fact that ndvduals that are for nstance 70% dsable wll probably not be able to work for the rest of hs lfe, and the compensaton wll be decded at an earler stage. An ndvdual that s partally dsable s often more dffcult to estmate gven that the probablty of workng can be negotated wth the nsurance company. As a result the nsurance company would have to wat and see the development of the vctm s njury before a fnal settlement can be agreed. 24

29 Fgure 10: Box Plot of Loss of Income Compensaton for Medcal Dsablty Group m o n t h l y _ v a l u e Low 11 - Medum Low 12 - Medum Hg 13 - Hgh MedD sper _gr oup In the desgn matrx n Secton 4.3 we use the covarates X, 7, X, 8 and X, 9 of Observaton, wth, X, X ) (0,0,0) 3 and 0,0,1 ) (, 7,8, 9 to descrbe medcal dsablty X for Group 1, ( 1,0,0 ) for Group 2, 0,1,0 ) ( for Group 4. Ths s also descrbed n Table 8. Table 8: Covarates for Medcal Dsablty ( for Group Levels Class Values Covarates 4 Medcal Dsablty Percentage Low ( X, 7, X,8, X, 9 ) ( 0,0,0 ) Medum Low Medum Hgh Hgh ( X, 7, X,8, X, 9 ( X, 7, X,8, X, 9 ( X, 7, X,8, X, 9 ) ) ) (1,0,0 ) ( 0,1,0 ) ( 0,0,1) Varable Occupaton/Professon The varable represents the types of professon the njured had at the tme of the accdent. In Trygg-Hansa s database structure we could fnd thrteen dfferent classfcatons of the vctm s occupaton (for more nformaton see Appendx A2). The dstrbuton of the number of annutes splt by occupaton s descrbed n Fgure

30 Fgure 11: Number of Annutes Splt by Professon Group Professon Group We classfed the varable nto fve groups dfferent groups, also descrbed n Table 9. We beleved that t was mportant to keep vctms that were unemployed or not work actve at the tme of the accdent as one group, the reason behnd ths s that we know that n cases where the vctm s a chld or a student the occupaton or carrer pattern wll change from what was ntally set as ther occupaton. In ths type of cases the professon group wll probably not be as sgnfcant as for the other groups. Table 9: Mergng of Professons nto Groups Professon Group Frequency Percentage Professon Unemployed/Student/Not work actve 90 10% 13 Publc employee % 3 and 4 Shop assstant/other manual workers % 7, 9, 10 and 12 Industral/Constructon/Polce % 1, 2, 5, 6 and 11 Whte-collar worker % 8 Total % In Fgure 12 we plot the monthly annuty value to the dfferent professon groups so that we could get a descrptve vew of how the varable nteracts wth the fnancal compensaton of the vctm. 26

31 Fgure 12: Box Plot of Loss of Income Compensaton for Professon Group m o n t h l y _ v a l u e unemployed/stu 15 - publc employe 16 - Shop assstant 17 - Industral/Con 18 - Whte-collar w occupat on_gr oup In the desgn matrx of Secton 4.3 we use the covarates X, 10, X, 11, X, 12 and X, 13 to descrbe professon group of observaton, wth X, X, X, X ) (0,0,0,0 ) for Group 1, ( 1,0,0,0 ) for Group 2, (, 10,11,12, 13 ( 0,1,0,0) for Group 3, ( 0,0,1,0 ) for Group 4 and 0,0,0,1 ) ( for Group 5. Ths s also descrbed n Table 8. Table 8: Covarate for Professon Group Levels Class Values Covarates 5 Professon Group 1 ( X, 10, X,11, X,12, X, 13) (0,0,0,0) ( X, 10, X,11, X,12, X, 13) ( X, 10, X,11, X,12, X, 13) ( X, 10, X,11, X,12, X, 13) ( X, 10, X,11, X,12, X, 13) (1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1) 27

32 5.3 Addtonal Varables Not Included In The Analyss Income Dsablty Percentage Income dsablty percentage s an mportant reference n determnng the sze of the monthly annuty payment. It represents the degree of reduced capacty to work after the vctm s exposed to a traffc accdent The varable s generally lnked to the medcal dsablty percentage (see Secton 5.2.4), but not n all cases. A 50% medcal dsablty does not necessary mply a 50% ncome dsablty. In many cases the medcal dsablty s only used as a gude n determnng the ncome dsablty of the vctm. The vctm s workng and lvng condton after the accdent plays a more sgnfcant role. Unfortunately ths varable can t be found n our system and can only be collected manually. Gven the lack of tme the varable could not be ncluded n the analyss. In future development of ths thess t would be mportant to nclude ths varable Workmen s njury Accordng to Swedsh law, workers compensaton s coordnated wth the Swedsh socal securty system. Injures experenced at work are often qualfed under the Work Injures Insurance Act. A traffc njury sustaned as a result of an accdent at work, or on the way to work s covered by the worker s compensaton nsurance. Ths means that the amount pad by the nsurance company wll only be the share that socal nsurance doesn t cover. As a result the sze of the annuty wll to a large extent depend on ths varable. It s therefore mportant to adjust for ths when creatng the mathematcal model The man data ssue s that we can t dentfy when an njury s covered by the Workers Injury Insurance Act or not. Unfortunately, ths nformaton can only be collected manually. To solve ths problem we dvded data nto two dfferent polcy groups, Commercal and Prvate nsured clams, based on what type of cover the clamant had at the tme of the accdent. Prvate clams are njures that are only covered by the nsurance company and wll therefore not face ths problem. On the contrary, a sgnfcant proporton of Commercal clams are covered by the workers compensaton. Unfortunately, we are not able to classfy when the compensaton s pad as an ntegrated part of the socal nsurance and when t s pad by the nsurance company. For smplcty, we wll therefore not nclude these clams. In the future t s mportant to nclude them to get the overall estmate for traffc annuty clams Compensaton from the Swedsh Socal Insurance Agency (Försäkrngskassan) The nsurance company covers for the loss of ncome that the socal nsurance doesn t cover. As a result, what the nsurance company pays out wll to a large extend depend on ths. For example n cases where the vctm s earnng s very hgh the socal nsurance wll only cover a top amount, the rest wll be covered by the nsurance company. These types of cases wll have a large annuty reserve and wll be very expensve for the nsurance company. Another mportant example s that f the vctm was young or unemployed when the accdent occurred, the socal nsurance wll pay out a guarantee beneft that may dffer from what not be what the vctm clams to be enttled to. As a consequence the nsurance company wll pay a large amount to cover ths. 28

33 To get a correct comparson between the amount the nsurance company pays out for each ndvdual we need to get full nformaton of how much compensaton the vctm receves from The Swedsh Socal nsurance agency. The problem s that we don t have ths nformaton n our database; unfortunately ths can only be gathered manually. We beleve t s essental that n future development of ths thess, ths varable s added to the dataset. 5.4 Lnk Functon The lnk functon provdes the relatonshp between the lnear predctor and the mean of the response varable. There are many commonly used lnk functons, and ther choce can be somewhat arbtrary. It can be convenent to match the doman of the lnk functon to the range of the dstrbuton functon's mean. The lnk functon s defned n secton 4.7 as g( ) x, j j. j For contnuous data wth no heteroscedastcty t s approprated to use an dentty lnk: =. If each covarate has a multplcatve effect one mean response, we express ths by means of a log lnk, log( ) wth ts nverse e. Ths s the standard choce of lnk functon n non-lfe nsurance. 29