Best Estimates for Reserves

Transcription

1 Best Estmates for Reserves Glen Barnett and Ben Zehnwrth 7 Ma 998 Abstract In recent ears a number of authors (Brosus 99; Mac ; and Murph 994 have shown that ln rato technques for loss reservng can be regarded as weghted regressons of a certan nd. We extend these regresson models to handle dfferent exposure bases and modellng of trends n the ncremental data and develop a varet of dagnostc tools for testng the assumptons these technques carr wth them. The new 'extended ln rato faml' (ELRF of regresson models s used to test the assumptons made b the standard ln rato technques and compare ther predctve power wth modellng (trends n the ncremental data. Not onl does the ELRF of regresson models ndcate that for most f not all cumulatve arras the assumptons made b the standard ln rato technques are not satsfed b the data but that modellng the trends n the (log ncremental data has more predctve power. The ELRF modellng structure creates a brdge to a statstcal (probablstc modellng framewor where the assumptons are more n eepng wth what we see n actual data. There s a paradgm shft from the standard ln rato technques to the statstcal modellng framewor; and the ELRF can be regarded as the brdge from the 'old' paradgm to the 'new'. There are three (crtcal stages nvolved n arrvng at a reserve fgure namel extracton of nformaton from the data n terms of trends and stablt thereof and dstrbutons about trends; formulaton of assumptons about the future leadng to forecastng of dstrbutons of pad losses; and correlaton between lnes and securt level sought. Fnall other benefts of the new statstcal paradgm are dscussed ncludng segmentaton credblt and reserves or dstrbutons for dfferent laers.

2 Introducton and Summar A model that s used to forecast reserves cannot nclude ever varable that contrbutes to the varaton of the fnal reserve amount. The exact future pament (beng a random varable s unnown and unnowable. Consequentl a probablstc model for future reserves s requred. If the resultng predctve dstrbuton of reserves s to be of an use or have an meanng the assumptons contaned n that probablstc model must be satsfed b the data. An approprate probablstc model wll enable the calculaton of the dstrbuton of the reserve that reflects both the process varablt producng the future paments and the parameter estmaton error (parameter uncertant. The regresson models based on ln ratos developed b Brosus (99 Murph (994 and Mac ( are descrbed n Secton and extended to nclude trends n the ncremental data and dfferent exposure bases. We refer to that faml of models as the extended ln rato faml (ELRF. The ELRF provdes both dagnostc and formal tests of the standard ln rato technques. It also facltates the comparson of the relatve predctve power of ln ratos vs-a-vs modellng the trends n the (log ncremental data. Ver often for real data even the best model wthn the ELRF s not approprate because the data doesn t satsf the assumptons of that model. The common causes of ths falure to satsf assumptons motvate the development of the statstcal modellng framewor dscussed n Secton 3. The rch faml of statstcal models n the framewor contans assumptons more n eepng wth realt. In Secton 3 a statstcal modellng framewor based on the analss of the log ncremental data s descrbed where each model n the framewor has four components of nterest. The frst three components are trends n each of the drectons: development perod accdent perod and pament/calendar perod. The fourth component s the dstrbuton of the data about the trends. Each model fts a dstrbuton to each cell n the loss development arra and relates cell dstrbutons b trend parameters. Ths rch faml of models we call the Probablstc Trend Faml (PTF. We descrbe how to dentf the optmal model n the statstcal modellng framewor va a step b step model dentfcaton procedure and llustrate that n the presence of an unstable pament/calendar ear trend formulatng assumptons about the future ma not be straghtforward. The statstcal modellng framewor allows separaton of parameter uncertant and process varablt. It also allows us to:. chec that all the assumptons contaned n the model are satsfed b the data. calculate dstrbutons of reserve forecasts ncludng the total reserve 3. calculate dstrbutons of and correlatons between future pament streams 4. prce future underwrtng ears ncludng aggregate deductbles and excess laers 5. easl update models and trac forecasts as new data arrve.

3 The fnal part of the paper dscusses how the combnaton of nformaton extracted from the data and busness nowledge allow the actuar to formulate approprate assumptons for the future n terms of predctng dstrbutons of loss reserves. Correlatons between dfferent lnes and a prescrbed securt level are mportant nputs nto a fnal reserve fgure. Fnall other benefts of the statstcal paradgm are alluded to ncludng segmentaton credblt and prcng dfferent laers. Extended Ln Rato Faml. Introducton Brosus (99 ponts out that the use of regresson n loss reservng s not new datng bac to at least the 950 s and sas that usng ln rato technques corresponds to fttng a regresson lne wthout an ntercept term. Mac (993 derves standard errors of development factors and forecasts (ncludng the total for the chan ladder regresson ratos. He mentons the connecton to weghted least squares regresson through the orgn and he presents dagnostcs that ndcate that an ntercept term ma be warranted on the data he analses. Worng drectl n a regresson framewor Murph (994 derves results for models wthout an ntercept such as the chan ladder ratos and also models wth an ntercept. Under the assumpton of (heteroscedastc normalt we derve results for a more general faml of models that also nclude accdent ear trends for each development ear. Ths extended faml we call the Extended Ln Rato Faml (ELRF. We dscuss calculatons and dagnostcs for fttng and choosng between models and checng assumptons. Standard errors of forecasts for both cumulatves and ncrementals are also derved. In the current secton we analse a number of real loss development arras. Dagnostcs ncludng graphs of the data and formal statstcal testng both ndcate that models based on ln ratos suffer several common defcences and frequentl even the optmal model n the ELRF s napproprate. Moreover models based on the log ncremental data have more predctve power than the optmal model n the ELRF. The standard ln rato models carr assumptons not usuall satsfed b the data. Ths can lead to false ndcatons and low predctve power so that the standard errors of forecasts are meanngless. Hence we relegate the calculaton of standard errors to the Appendx.. Calculatng Ratos usng Regressons Suppose x ;... n represent the cumulatve at development perod for accdent perods... n and are the correspondng cumulatve values at development perod.

4 x A graph of versus x ma appear as follows. x trend x A ln rato x s the slope of a lne passng through the orgn and the pont x. So each rato s a trend. Accordngl a ln rato (trend average method s based on the regresson ( = bx( + ( Var[(] = x( (. The parameter b represents the slope of the best lne through the orgn and the data ponts x ;... n. The varance of about the lne depends on x va the functon weghtng parameter. x where s a

5 bx Fgure. Chan Ladder Ratos Regresson In the above fgure Var[(] = x( where =. Interestngl the assumpton that condtonal on x( the average value of ( s bx( s rarel true for real loss development arras. Consder the followng cases: Case (: = The weghted least squares estmator of b s x x n x b x x x. (. Ths s the weghted average b volume.e. the chan ladder average method or chan ladder rato. Case (: = The weghted least squares estmator of b s. (.3 b x n Ths s the smple arthmetc average of the ratos. Case (: =0 Ths elds a weghted average weghted b volume squared.

6 So b varng the parameter we obtan dfferent ln-rato methods (averages. One of the advantages of estmatng ln-ratos usng regressons s that both standard errors of the average method selecton and standard errors of the forecasts can be obtaned. Another more mportant advantage s that the assumptons made b the method can be tested. One mportant assumpton s that (/x( / =... n are normall dstrbuted. Otherwse the weghted least squares estmator of b s not necessarl effcent and the reserve forecasts consequentl ma be based for the mean and wll have a large varance. The normalt assumpton can be tested b examnng the three dagnostc dsplas: normal probablt plot Box-plot and hstogram of the weghted standardsed resduals. The Shapro-Ws test based on the normalt plot s a formal test. The ln rato method also maes other assumptons that should alwas be tested. Another basc assumpton s that bx Ex. (.4 That s n order to obtan the mean cumulatve at development perod tae the cumulatve at the prevous development perod -and multpl t b the rato. A quc dagnostc chec of ths assumpton s gven b the graph of ( versus x(. Ver often a (non-zero ntercept s also requred. See Fgure.4. Equaton (.4 can be re-cast E x x b x. (.5 That s the mean ncremental at development perod equals the cumulatve at development perod - multpled b the ln rato b mnus. What are the dagnostc tests for ths assumpton? If the assumpton (.4 s vald then the weghted standardsed resduals versus ftted values should appear random. Instead what ou wll usuall see s a downward trend depcted n the Fgure. below representng the chan ladder ratos resduals for the Mac (994 data. (See Example below.

7 Fgure. Ths ndcates that large values are over ftted and small values are under ftted so that E x bx s not true. Comparson of graphs of weghted standardsed resduals wth graphs of the data wll ndcate that accdent perods that have 'hgh' cumulatves are over ftted and those wth 'low' cumulatves are under ftted. Here are the two dsplas for the Mac (994 data. Note that as a result of the equvalence of equatons (.4 and (.5 the resduals of the cumulatve data are also the resduals of the ncremental data. Fgure.3a

8 Fgure.3b If ou thn of the wa the ncrementals are generated and the fact that there are usuall pament perod effects the cumulatve at development perod rarel s a good predctor of the next ncremental (after adustng for trends. Y Cumulatve vs. X Cumulatve Fgure.4. Cumulatve Development Perod versus Cumulatve Development Perod 0 Murph (995 suggested to extend the regresson model (. to nclude the possblt of an ntercept. ( = a + bx( + ( (.6 such that Var[(] = x(. If the ntercept a s sgnfcant and we do not nclude t n the regresson model then the estmate of the ln rato b (slope s based. Note that n the above graph (Fgure.4 of cumulatve at development perod versus cumulatve at development perod 0 the ntercept appears sgnfcant. Indeed t s sgnfcant between ever par of contguous development perods. (See the data of Example below.

9 We can rewrte (.6 thus: ( x( = a + (b x( + ( (.7 So here ( x( s the ncremental at development perod. Consder the followng two stuatons:. b and a 0 Here to forecast the mean ncremental at development perod we tae the cumulatve x at development perod and multpl t b b.. b = and a 0 Ths means that x( has no predctve power n forecastng ( x(. The estmate of a s a weghted average of the ncrementals n development perod. So we would forecast the next accdent perods ncremental b averagng the ncrementals down a development perod. Accordngl the standard ln rato technque s abandoned n favour of averagng ncrementals for each development perod down the accdent perods. If b= then the graph of ( x( aganst x( should be flat as depcted below n Fgure.5 whch represents the ncrementals versus prevous cumulatves (development perod 0 for the Mac (994 data. It s clear that the correlaton s zero. Ths s also true for ever par of contguous development perods. Fgure.5 Incrementals Development Perod versus Cumulatve Development Perod 0 In concluson f the ncrementals ( x( n development perod sa appear random t s ver lel that the graph of ( x( versus x( s also random. That s there s zero correlaton between the ncrementals and the prevous cumulatves.

10 Now f the ncrementals possess a trend down the accdent perods the estmate of the parameter b n equaton (.7 wll be sgnfcant and so the ln rato (b plus the ntercept (a wll have some predctve power. We should however ncorporate an accdent perod trend parameter for the ncremental data namel ( x( = a 0 + a + (b x( + ( Var[(] = x(. (.8 For most real cumulatve loss development arras that possess a constant trend down the development perod the trend parameter (a wll be more sgnfcant than the rato mnus (b. Indeed ver often b wll be nsgnfcant. That s the trend wll have more predctve power than the rato and the resdual predctve power of the rato after ncludng the trend wll be nsgnfcant. We use the followng namng conventon for the three parameters: a 0 - Intercept a - Trend b - Rato (Slope Here are some models ncluded n the ELRF descrbed b equaton (.8. Chan Ladder Ln Ratos Here Intercept = Trend = 0 and =. Cape Cod Intercept Onl Here t s assumed that the Rato = and the Trend = 0. The Cape Cod estmates a weghted (dependng on average of the ncrementals n each development perod. It also forecasts a weghted average down the accdent perods for each development perod. The model can be wrtten Trend wth Rato = ( x( = a 0 + ( Var[(] = x(. The model estmates a weghted (dependng on trend (parameters a 0 and a down the accdent perods for each development perod. It also forecasts a weghted trend down the accdent perods for each development perod.

11 Example : The Mac data The data for the frst example s from Mac (994. The data are ncurred losses for automatc facultatve busness n general lablt taen from the Hstorcal Loss Development Stud 99 publshed b the Rensurance Assocaton of Amerca Table.. Incurred loss arra for the Mac data. Rows are accdent ears and columns are development ears. Note that 98 accdent ear values are low. We frst ft the chan ladder ratos regresson model. That s we ft equaton (. wth = for ever par of contguous development perods. The standardsed resduals are dsplaed n Fgure.6. Note that the equvalence of equatons (.5 and (.6 means that the resduals of the cumulatve data are dentcal to the resduals of the ncremental data. Wtd. Std. Resduals vs. Dev. Year Wtd. Std. Resduals vs. Acc. Year Wtd. Std. Resduals vs. Pa. Year Wtd. Std. Resduals vs. Ftted Values Fgure.6 Resdual plot for the chan ladder ratos model.

12 We have alread observed the downward trend n the ftted values (Fgure. and that the hgh cumulatves are overftted whereas the low cumulatves are underftted. Ths s manl due to the fact that ntercepts are requred. So we now ft models (.6 wth ntercepts except for the last two pars of contguous development perods as there s nsuffcent data here. See Table. for the regresson output. Note that none of the slope (rato parameters are sgnfcantl dfferent from and f both parameters are nsgnfcant the slope (rato s less sgnfcant. Ths means that the prevous cumulatve s not reall of much help n predctng the next ncremental ncurred loss. Intercept and Rato Regresson Table = Develop. Intercept Slope Perod Estmate Std.Error p value Estmate Slope - Std. Error p value (AIC=760.8 Table.. Ft of the model wth ntercept and rato wth at. (There s no ntercept ftted for the last two ears. The model s overparameterzed so we elmnate the least sgnfcant parameter n each regresson. We fnd that n each case the ntercept s the parameter retaned: that s for ever par of contguous development perods the model reduces to Cape Cod that s ( x( a0. The resdual plots for the reduced model (Cape Cod are gven n Fgure.7.

13 Fgure.7. Resdual plot for = model wth ntercepts and wth slopes set to. The lne ons mean resduals. Note that resduals versus ftted values are 'straght' now and that we do not have the hgh low effect n resduals versus accdent perods. Snce resduals versus accdent ears do not exhbt a trend f we were to nclude a trend that s estmate ( x( a a 0 we would fnd that the estmate of a s nsgnfcant. We now present forecasts and coeffcents of varaton based on the Cape Cod (ntercept onl wth = model and compare ths wth the forecasts and coeffcents of varaton for the chan ladder ratos.

14 Cape Cod Chan Ladder Accdent Year Mean Forecast Standard Error Coeff. of Varaton Mean Forecast Standard Error Coeff. of Varaton Total Table.4. Comparson of Cape Cod coeffcents of varaton wth those for the Chan Ladder Ratos. Note that for the Cape Cod model the standard errors are generall decreasng as a percentage of the accdent ear forecast totals as we proceed down to the later ears. Ths s because the model relates the numbers n the trangle to a certan degree- t assumes that the ncremental values n the same development perod are random from the same dstrbuton. Ths does not happen wth the chan ladder ratos because the model does not relate the ncrementals n the trangle n an meanngful wa. For example how are the values n the development perod 0 related? Consequentl the coeffcents of varaton are substantall hgher for the chan ladder ratos model and moreover volate the fundamental statstcal prncple of nsurance - rs reducton b poolng. It does not mae sense that the coeffcent of varaton for 990 s 50% but for the prevous ear 989 t s 59% when 990 has onl one more ncremental value to forecast than 989. For the Mac data the model wth ntercepts s reasonable as there s no accdent ear trend n the ncrementals. For data where a constant trend (on a dollar scale does exst then the trend wll be sgnfcant but ver often the rato - wll be nsgnfcant..3 Summar We have so far consdered two modellng cases: ncrementals for a partcular development perod have a zero trend and ncrementals have a constant trend (after possbl adustng the data b accdent ear exposures. In both these real data cases ln ratos are often nsgnfcant and therefore also lac predctve power. The case encountered most often n practce however nvolves a trend change along the pament/calendar perods (dagonals. Ths means that as ou loo down each

15 development perod the change n trend wll occur n dfferent accdent perods. Consequentl none of the above models n the ELRF can capture these trends. The weghted standardsed resduals depcted n Fgure.8 and Fgure.9 are those of the chan ladder ratos and model (.8 respectvel appled to proect ABC (Worer's Compensaton Portfolo dscussed n Secton 3. Note that the chan ladder ratos ndcate a pament ear trend change and model (.8 that fts a constant trend down the accdent ears for each development ear ndcates that the trend before pament ear 984 s lower than the trend after 984. Ths proect (ABC s analsed n more detal n Secton 3. Wtd. Std. Resduals vs. Dev. Year Wtd. Std. Resduals vs. Acc. Year Wtd. Std. Resduals vs. Pa. Year Wtd. Std. Resduals vs. Ftted Values Fgure.8. Resdual plot for chan ladder ratos. The lne ons the means of the resduals.

16 Wtd. Std. Resduals vs. Dev. Year Wtd. Std. Resduals vs. Acc. Year Wtd. Std. Resduals vs. Pa. Year Wtd. Std. Resduals vs. Ftted Values Fgure.9 Resdual plots for trends plus rato model The tpes of models descrbed b equaton (.8 can be used to dagnostcall dentf pament perod trend changes but cannot estmate these trend changes or forecast wth them. These models n the ELRF form a brdge to models that also nclude pament perod trend parameters that s statstcal models n the PTF. It s mportant to note that ELRF models also mae the mplct assumpton that the weghted standardsed errors come from a normal dstrbuton. If the assumpton s true the estmates of the regresson parameters are optmal. If the assumpton s not true the estmates ma be ver poor. Ths normalt assumpton s rarel true for loss reservng data. In fact the weghted standardsed resduals are generall sewed to the rght suggestng that the analss should be conducted on the logarthmc scale. The graph below llustrates the sewness of a set of weghted standardsed resduals based on chan ladder ratos for Proect Pan6 analsed n detal n Secton 3. The postve weghted standardsed resduals are further from zero than the negatve ones. If the normalt assumpton were correct the plot would loo roughl smmetrc about the zero lne.

17 Wtd. Std. Resduals vs. Ftted Values In summar usng the regresson methodolog of ELRF ou wll dscover that for most real loss development arras of an data tpe standard development factor (ln-rato technques are napproprate. Analsng the ncrementals on the logarthmc scale wth the ncluson of pament perod trend parameters has more predctve power. Fnall but mportantl the estmate of a mean forecast of outstandng (reserve and correspondng standard devaton based on a model are meanngless unless the assumptons made b the model are supported b the data.

18 3 Statstcal Modellng Framewor 3. Introducton Clearl we requre a model that s able to deal wth changng trends; trends n the data on the orgnal (dollar scale are hard to deal wth snce trends on that scale are not generall lnear but move n percentage terms for example 5% supermposed (socal nflaton n earl ears and 3% n later ears. It s the logarthms of the ncremental data that show lnear trends. Consequentl we ntroduce a modellng framewor for the logarthms of the ncremental data that allows for changes n trends. The models of ths tpe provde a hgh degree of nsght nto the loss development processes. Moreover the facltate the extracton of maxmum nformaton from the loss development arra. The detals of the modellng framewor and ts nherent benefts are descrbed n Zehnwrth (994. However gven that there s a paradgm shft from the standard ln rato methodolog to the statstcal modellng framewor we revew the salent features of the statstcal modellng framewor. 3. Trend Propertes of Loss Development Arras Snce a model s suppose to capture the trends n the data t behoves us to dscuss the geometr of trends n the three drectons vz. development ear accdent ear and pament/calendar ear. Development ears are denoted b ; = 0... s-; accdent ears b ; =... s; and pament ears b t; t =... s. Fgure 3. The pament ear varable t can be expressed as t = +. Ths relatonshp between the three drectons mples that there are onl two ndependent drectons. The two drectons development ear and accdent ear are orthogonal equvalentl the have zero correlaton. That s trends n ether drecton are not proected onto the other. The pament ear drecton t however s not orthogonal to ether the development or accdent ear drectons. That s a trend n the pament ear drecton s also proected onto the development ear and accdent ear drectons. Smlarl accdent ear trends are proected onto pament ear trends.

19 The man dea s to have the possblt of parameters n each of the three drectons development ears accdent ears and pament ears. The parameters n the accdent ear drecton determne the level from ear to ear; often the level (after adustng for exposures shows lttle change over man ears requrng onl a few parameters. The parameters n the development ear drecton represent the trend from one development ear to the next. Ths trend s often lnear (on the log scale across man of the later development ears often requrng onl one parameter to descrbe the tal of the data. The parameters n the pament ear drecton descrbe the trend from pament ear to pament ear. If the orgnal data are nflaton adusted before beng transformed to the log scale the pament ear parameters represent supermposed (socal nflaton whch ma be stable for man ears or ma not be stable. Ths s determned n the analss. Consequentl the (optmal dentfed model for a partcular loss development arra s lel to be parsmonous. Ths allows us to have a clearer pcture of what s happenng n the ncremental loss process. The mathematcal formulaton of the models n the statstcal modellng framewor s gven b equaton (3.6 below. We now llustrate the geometr of trends wth a smulaton example. Example - Smulated Data: To llustrate the trend propertes of a loss development arra let us examne a stuaton where we now the trends because we have selected them. Consder a set of data where the underlng pad loss (at ths pont wthout an pament ear trends or even randomness ust the underlng development s of the form lnp On a log scale ths s a lne wth a slope of -0.. The accdent ears are completel homogeneous. Let's add some pament/calendar ear trends. A trend of 0. from 978 to from 98 to 983 and 0.5 from 983 to Fgure 3.. Dagram of the trends on the log scale n the data arra.

20 Patterns of change le ths are qute common n real data. Trends n the pament/calendar ear drecton proect onto the other two drectons. The resultant trends for the frst sx accdent ears are shown below. Fgure 3.3. Plot of the log(pad data aganst dela for the frst sx accdent ears. Note that each lne n the graph s the resultant development ear trend for a sngle accdent ear. As ou go down the accdent ears (978 to 983 the 30% trend alwas cs n one development perod earler. The pament ear trends also proect onto the accdent ears whch s wh the earl ears are at the bottom and the later ears are at the top. Note how the n moves bac as we go up to the more recent accdent ears. The resultant development ear trends are dfferent for each accdent ear now. We can t model even ths smple stuaton wth ln ratos or an model n ELRF. Of course real data s never so smooth. On the same log scale we add some nose random numbers wth mean zero and standard devaton 0.. Fgure 3.4. Trend plus randomness for the frst sx accdent ears.

21 Now the underlng changes n trends are not at all clear for two reasons. The pament ear trends proect onto development ears and the data alwas exhbts randomness that tends to obscure the underlng trend changes. It has man of the propertes we observe n real data and et t s plan that even wth the extensons the regresson models n ELRF from Secton are nadequate for ths data. We nstead model the trends (n the three drectons and the varablt. We measure these thngs on the log scale. In ths Secton let ( be the natural log of the ncremental pament data n accdent ear and development ear. Ths s dfferent from our use of ( n Secton but we do t for consstenc wth the lterature approprate to the models n each Secton. We wll analze ths data shortl. Consder a sngle accdent ear. We represent the expected level n the frst development ear b a parameter (. We can model the trends across the development ears b allowng for a (possble parameter to represent the expected change (trend between each par of development ears. We model the varaton of the data about ths process wth a zero-mean normall dstrbuted random error. That s: (3. Ths probablstc model s depcted below (for the frst sx development ears. ( Fgure 3.5. Probablstc model for trends along a development ear on the log scale. For ths probablstc model s not the value of observed at dela 0. It s the mean of 0. Indeed 0 has a normal dstrbuton wth mean and varance. Smlarl s not the observed trend between development ear - and but rather t s the mean trend between those development ears E[((] =. The parameters of the probablstc model represent means of random varables. Indeed the model (on a log scale comprses a normal dstrbuton for each development ear where the means of the normal dstrbutons are related b the parameter and the trend parameters....

22 Based on the model n equaton 3. the random varable p( has a lognormal dstrbuton wth d Medan exp (3. Mean = medan exp[0.5 ] (3.3 and Standard Devaton = mean exp (3.4 The probablstc model for p( comprses a lognormal dstrbuton for each development ear where the medans of the lognormal dstrbutons are related b equaton 3. and the means are related b equaton (3.3. So n fttng or estmatng the model we are essentall fttng a lognormal dstrbuton to each development ear. The trend (on a log scale comprsng the straght lne segments s onl one component of the model. A prncpal component comprses the dstrbutons about the trends. P( Fgure 3.6. Model for trends along a development ear (dollar scale. Means and medans of the dstrbutons are mared. Note equaton (3. that exponentatng the mean on the log scale gves the medan on the dollar scale (whch s wh the lne above ons the medans. We wll normall use the mean as our forecast rather than the medan but the uncertant (measured b the standard devaton of the lognormal dstrbuton s ust as mportant a component of the forecast. If we compute expected values of the logs of the development factors on the ncremental data wth ths model we obtan E[ln(p(/p(] = E[( + - ] =. That s trend parameters also underpn ths new model but n a wa that wll allow t to appropratel model the trends n the ncremental data n the three drectons.

23 The model descrbed so far onl covers a sngle accdent ear. We have not et accounted for the pament ear and accdent ear trends. Let the mean of the (random nflaton between pament ear t and t+ be represented b t (ota-t. Hence the faml of models can be wrtten: t t (3.6 We call ths faml of models the probablstc trend faml (PTF. Note that the mean trend between cells (- and ( s + + and the mean trend between cells (- and ( s A member of ths faml of models relates the lognormal dstrbutons of the cells n the trangle. On a log scale the dstrbuton for each cell s normal where the means of the normal dstrbutons are related b the trends descrbed b the member. If the error terms comng from a normal dstrbuton wth mean zero do not have a constant varance then the changng varance also has to be modelled. Note that there are numerous models n PTF even f we do not nclude the varng (stochastc parameter models dscussed n Secton 3.3. The actuar has to dentf the most approprate model for the loss development arra beng analsed. The assumptons made b the 'optmal' model must be satsfed b the data. In dong so one extracts nformaton n terms of trends stablt thereof and the dstrbutons of the data about the trends. Example contnued - Estmaton: Let s now tr to dentf the model that created the data. We begn b fttng a model wth all the development ear trends equal (one and all pament ear trends equal (one and wth no accdent ear trends (one. That s wth = t = and = for all parameters. The parameter estmates are gven n Table 3.. Parameter Estmate Std. Error t-rato s = 0.9 R = 97.0% Table 3.. Parameter estmates for the model wth constant trends. The estmate of (ota s the weghted average of the three trends and 0.5. Removng constant trends maes an changes n trend more obvous. The resduals are shown n Fgure 3.7.

24 Wtd. Std. Resduals vs. Development Year Wtd. Std. Resduals vs. Accdent Year Wtd. Std. Resduals vs. Pament Year Wtd. Std. Resduals vs. Ftted Values Fgure 3.7. Plots of standardzed resduals aganst the three drectons and aganst ftted values for the sngle pament ear trend model. The lnes on mean resduals. The resduals need to be nterpreted as the data adusted for what has been ftted. Accordngl the resduals versus pament ears represent the data mnus the ftted Immedatel the changes n trends n the pament ear drecton become obvous. We can see that the trend n the earl ears s substantall less than the estmated average of that the trend from 98 to 983 s much larger than t and after that the trend s prett close to the ftted trend as s approxmatel zero. Ths suggests that we should ntroduce another (ota between and another between (that wll contnue to 99. Wtd. Std. Resduals vs. Development Year Wtd. Std. Resduals vs. Accdent Year Wtd. Std. Resduals vs. Pament Year Wtd. Std. Resduals vs. Ftted Values Fgure 3.8. Plots of standardzed resduals aganst the three drectons and aganst ftted values for the model wth three pament ear trends.

25 The resduals of the model wth three pament ear trends are gven n Fgure 3.8. Ths model seems to have captured the trends. Parameter Estmate Std. Error t-rato s = R = 97.7% Table 3.3. Parameter estmates for the model wth three pament ear trends. Note that the estmates of the trend parameters are not equal to the true values ndeed (±0.044 s a bt off the mar (but not sgnfcantl. That s because n the pament ears 98 and 983 there aren't man data ponts. Gven the trend of 0.5 s n the data snce 983 we would expect stablt of forecasts and trend parameter estmates as we remove ears. The forecasts are stable f we remove the most recent data the forecasts of ths model don t change much relatve to the standard error n the forecast as we can see n Table 3.4. Yrs n Estm. N (83-9 std. err. (83-9 std. err. Mean Fcst std. error. Fcst Table 3.4. Forecasts and standard errors and trend estmates (and ther standard errors for the selected model as the later pament ears are removed. Note that the estmate of (= -0. s prett stable as we remove the latest ears.

26 The dspla below Fgure 3.9 gves the predcton errors (on a log scale for the four pament ears based on the model estmated at ear end 987. Predcton Errors vs. Dev. Year Predcton Errors vs. Acc. Year Predcton Errors vs. Pa. Year Predcton Errors vs. Ftted Values Fgure 3.9 Predcton errors for based on model estmated at ear end 987. So the estmated model at the end of pament ear 987 slghtl over-predcts the pament perods That s because the trend estmate (snce 983 s now 5.63%.03% n place of 4.46% 0.46% when we use all the ears n the estmaton. Hence the forecast of $5.89M ($.87M s 'hgher' than $3.4M ($0.93M. When ou test for a trend change between 987 and 988 t s not sgnfcant (as we would expect. Note that removal of pament ears (valdaton analss s part of the model dentfcaton procedure and extracton of nformaton process. Pament Years n Estmaton Estmate of gamma Estmate of ota ( Forecast SE $M

27 Example 3 - Real data wth maor pament ear trend nstablt We now analze a real data set Acc. Yr Exposure Table 3.5. Incremental pad losses and exposures for ABC. Ths loss development arra has a maor trend change between pament ears 984 and 985 even though the data and ln ratos are relatvel smooth. Indeed t needs to be understood that n general trend nstablt has nothng to do wth volatlt or smoothness of the data and ln ratos. Formulaton of the assumptons about the future trend wll depend on the explanaton for the trend change (when there s one. The ndvdual ln ratos for the cumulated data are ver stable as can be seen n Fgure 3.0 below. It s ver dangerous to tr to mae udgements about the sutablt of development factor technques from the ndvdual ln ratos on the cumulated data. Indvdual Ln Ratos b Dela Fgure 3.0. Plot of ndvdual ln ratos b dela. The lne ons Chan Ladder ratos.

28 We frst conduct some dagnostc PTF analss and then show how the ELRF modellng structure also ndcates pament ear trend change and moreover that an method based on ln ratos s qute meanngless. Fgure 3. below shows the standardsed resduals of the statstcal chan ladder n PTF.e. the statstcal chan ladder fts all the gamma parameters and all the alpha parameters (no otas. So the resduals are the data adusted for the (average trend between ever par of contguous development perods and ever par of contguous accdent perods. Ths s wh the resduals versus development ears and resduals versus accdent ears are centred on zero! We use ths model onl as a dagnostc tool to determne (speedl whether there are pament ear trend changes whch can be attrbuted solel to the pament ears. Contrast the smoothness of the above ratos wth the plot of the resduals from ths model. Wtd. Std. Resduals vs. Dev. Year Wtd. Std. Resduals vs. Acc. Year Wtd. Std. Resduals vs. Pa. Year Wtd. Std. Resduals vs. Ftted Values Fgure 3. Standardsed resduals of the statstcal chan ladder model. We can now see dramatc changes n the pament ear drecton. It mght be ver dangerous to use forecasts from an model assumng no changes n pament ear trend such as a model from the ELRF t would correspond to forecastng along the zero lne n Fgure.8. (the resduals of the standard chan ladder ratos. There s a dfference between Fgures 3.0 and.8. The statstcal chan ladder shows the pament ear trends after adustng for the trends n the other two drectons. The chan ladder ratos (Fgure.8 do not do that. But the change n trend s clear n ether graph. In the current statstcal modellng framewor we are able to model ths change; we have a lot more control over how we ncorporate the trend changes nto our model and hence nto the forecasts. Even the best ELRF model here hardl uses ratos and s defcent because t gves us no control n the pament ear drecton. It turns out that the trend before 984 s approxmatel 0% whereas the trend past 984 s approxmatel 0%. So whch trend should we assume for the future? Ths depends on the explanaton for the change. If the trend nstablt s due to new legslaton that apples retrospectvel (to all accdent

29 perods then one would revert to the 0%. If there s no explanaton for the trend change except that the paments have ncreased then callng the future n terms of trends s more dffcult. Example 4 - Volatle data wth stable trends We now consder an arra where the pad losses are ver volatle but the trends are stable. Recall that trend stablt/nstablt s not dependent on the volatlt of the data nor of the ln ratos. Snce the random component s an ntegral part of the model ths model captures the behavour of ths volatle data ver well. We call ths arra PAN6. Development Year Acc. Yr **** **** **** 99 **** **** **** 996 **** Table 3.6. Pad loss arra for the PAN6 data for Example 4. A good model can be dentfed qucl for the logarthms of these data; t has no pament ear trends and onl two dfferent development ear trends; between development ears 0- and for all later ears. The resdual plot s gven n Fgure 3.. However note that the spread of the frst two development ears s wder than for the later ears and the spread for 'small' ftted values s larger than the spread for 'large' ftted values. If we estmate the standard devatons n the two sectons we fnd that the are and respectvel. Ths requres a weghted regresson; development ears 0 and are gven weght (0.805/3.077 and the other ears (+ have weght. The weghted standardzed resdual plots now loo fne; see Fgure 3.3. A chec of the plot of resduals aganst normal scores (not presented here ndcates that the assumpton of normalt of the logarthms of the data s ver reasonable; the squared correlaton s greater than The normal dstrbutons for ths model have relatvel large varances. The estmate of for development perods 0- s.93 and for development perods + s Note that f a normal dstrbuton has a varance then the correspondng lognormal dstrbuton has a coeffcent varaton of exp.

30 Ths model also has forecasts that are stable as we remove the most recent data as we see n Table 3.7. Ths s a ver mportant attrbute of ths dentfed model that captures the nformaton n the data f the trends n the data are stable then so are the forecasts based on the estmated model. In ths case we were able to remove almost half the (most recent data. The standard errors of the forecasts are large because the lognormal dstrbutons are sewed - nsurance s about measurng varance (not ust mean. Wtd. Std. Resduals vs. Development Year Wtd. Std. Resduals vs. Accdent Year Wtd. Std. Resduals vs. Pament Year Wtd. Std. Resduals vs. Ftted Values Fgure 3.. Plot of standardzed resduals for the model wth two gamma parameters and one alpha parameter. Wtd. Std. Resduals vs. Development Year Wtd. Std. Resduals vs. Accdent Year Wtd. Std. Resduals vs. Pament Year Wtd. Std. Resduals vs. Ftted Values Fgure 3.3. Plot of weghted standardzed resduals after the weghted regresson.

31 Whle the varablt of the data and hence the standard errors of the forecasts are large the message from the data has been consstent over man ears. We are predctng the dstrbuton of the data n each cell not merel ther mean and standard devaton so a large standard devaton does not mpl a bad model. Indeed the model s ver good. It captures the varances ndeed the dstrbutons n each cell; Years n Estmaton N Trend (dev perod + standard error Mean Fcst Standard Error Table 3.7. Forecasts and standard errors and the fnal trend estmates (and ther standard errors for the fnal model as the later pament ears are removed. The hgh standard errors of forecasts are due to large process varablt. As we remove recent ears (dagonals from the estmaton we note the stablt of forecasts (outstandng. Ths s further evdence of a stable trend n the data. Fgure Predcton errors for ears Note that at end of ear 99 the estmated model would have predcted the normal dstrbutons for the log(paments n ears and would have produced statstcall the same forecast outstandng.

32 Normalt Test Fgure 3.5. Normalt plot of predcton errors for based on model estmated at ear end 99. We now turn to ELRF analss. Snce the data are extremel sewed (lognormal wth large coeffcent of varaton the resduals of the chan ladder (regresson ratos n ELRF are extremel sewed to the rght. See Fgure 3.6 below. The plot of resduals aganst ftted values shows a downward trend ndcatng that we overpredct the large values and underpredct the small ones. The resduals also show strong ndcatons of non-normalt. Moreover all ratos have no predctve power (provded there s an ntercept. In an event resduals are sewed (not normal so even the best model n ELRF the Cape Cod ( x = a 0 + s not a good one. Recall that f model assumptons are not satsfed b the data then an forecast calculatons are qute meanngless. Fgure 3.6. Resduals of chan ladder ratos regresson model

33 Example 5 - Smulated arra based on a (smple model wth onl two parameters. Ths arra termed SDF contans a smulated data set where the ncremental pad losses have accdent ears that are completel homogeneous. The actual model drvng the data has one alpha ( = 0 one gamma ( = -0.3 and = 0.4. That s 0 0.3d where the ( are..d. from N( The smulated data s presented n Table 3.8. Acc. Development Year Year Acc. Development Year Year Table 3.8. Incremental pad loss data for smulated example SDF.

34 The frst thng to note wth ths data s that once nose s added t loos le ncremental pad data for a real arra even though t was generated from a ver smple model. The relatvel large = 0.4 explans the hgh varablt n the observed pad losses. The ncremental data dsplaed n Table 3.8 appear volatle but the values n the same development perod are ndependent realzatons from the same lognormal dstrbuton. For example n development perod zero the smulated values 8045 and 907 come from a lognormal dstrbuton wth mean 6903 and standard devaton Snce a lognormal dstrbuton s sewed to the rght realzatons larger than the mean are tpcall 'far' awa whereas realzatons less than the mean are bounded b zero and the mean and so are 'closer' to the mean. The apparent volatlt n the data s not due to nstablt n trends - ndeed the realt s qute the opposte - though volatle the ncremental pad losses have stable trends. Snce we now the exact probablt dstrbutons drvng the data we can compute the exact mean and exact standard devaton for each cell n the rectangle and also the exact means and standard devatons of sums. The exact mean of the total outstandng s $845 wth an exact standard devaton of $ (So the process varance s When we analse the data n PTF we dentf onl two sgnfcant parameters = and = The estmate of s Resduals of ths estmated model are dsplaed n Fgure 3.7 below. The table below gves forecasts of total outstandng ncludng valdaton forecasts. Note stablt as expected. Wtd. Std. Resduals vs. Dev. Year Wtd. Std. Resduals vs. Acc. Year Wtd. Std. Resduals vs. Pa. Year Wtd. Std. Resduals vs. Ftted Values Fgure 3.7 Resduals based on the estmated parameters of the true model.

35 Pament Years n Estmate of Gamma Mean Forecast SE Estmaton We now stud the cumulatve arra. Unad usted D ata vs. D ev. Year Unad usted D ata vs. Dev. Year Fgure 3.8 Accdent ear 98 hgh development 979 low development. Even though the ncremental data was generated wth accdent ears homogeneous the cumulated data have each accdent ear at a completel dfferent level; the plot aganst accdent ears umps all over the place the values along an accdent ear tend to be hgh or low. Ths s a common feature wth cumulatve arras. The cumulatve values for 979 le entrel below those for 98 (Fgure 3.8 et most of the ncremental paments are 'close' together. One 'large' ncremental value from the tal of the lognormal has a maor mpact on the cumulatve data. The ln rato technques assume that the next ncremental pament wll be hgh f the current cumulatve s hgh and ths loos le what s gong on wth the cumulatve data. So the cumulatves delver a false ndcaton even for data where there are no pament ear trend changes. Note that for 979 cumulatve pad at development ear 5 s $45750 whereas for 98 t s $7635. So "current emergence s not a predctor of future emergence" a term used b Gar Venter. The chan ladder ratos model gves a mean outstandng forecast of $5430 and a standard error of the outstandng forecast of $5949. The plot of resduals aganst ftted values maes t clear where the problem les as we see n Fgure 3.9 below.

36 Wtd. Std. Resduals vs. Dev. Year Wtd. Std. Resduals vs. Acc. Year Wtd. Std. Resduals vs. Pa. Year Wtd. Std. Resduals vs. Ftted Values Fgure 3.9 Plot of weghted standardzed resduals for chan ladder ratos. Agan we have a ear wth hgh cumulatves over ftted and ear wth low cumulatves under ftted. So 979 accdent ear s under ftted and 98 accdent ear s over ftted. Note how there s a dstnct downward trend n the ftted values plot. It ndcates that the model overpredcts the hgh cumulatve values and underpredcts the low values whch t wll do f the cumulatves don t reall contan nformaton on the subsequent ncrementals. Normal scores plots show the non-normalt. If we loo at the plot of the ncremental pad losses aganst the prevous cumulatve we can see that models nvolvng ratos wll be napproprate snce there s no relatonshp. Incremental vs. Prevous Cumulatve Fgure 3.0. Plot of ncremental paments aganst prevous cumulatve.

37 The best model n ELRF sets ratos to and onl uses ntercepts. That s t taes averages of ncrementals n each development ear. But due to non-normalt ths s not good enough. At least ELRF analss nforms us that the ncrementals n a development perod are random from a dstrbuton and these ncrementals are not correlated to the prevous cumulatves - the wa the data were generated. It also tells us that the data are sewed and so we need to tae a transformaton. B wa of summar the ELRF analss nforms us that the data were created ncrementall accdent ears are homogeneous and we should be modellng the log ncremental data. It s tellng us the truth. If ou generate (smulate data usng ratos ELRF wll tell ou that ratos have predctve power and that the data were generated cumulatvel. But and ths s an extremel mportant qualfcaton for most real loss development arras ELRF analss wll ndcate that the data were generated ncrementall that ratos have no or much less predctve power than trends n the log ncrementals and that there ma be pament/calendar ear trend changes. 3.3 Varng (stochastc parameters In vew of the trend relatonshps between the three drectons development ear accdent ear and pament ear a model wth several parameters n the pament ear and accdent ear drectons wll suffer from mult-collneart problems. Zehnwrth (994 n Secton 7. dscusses the mportance of varng (stochastc parameter models especall the ntroducton of a varng alpha parameter (n place of addng parameters to overcome mult-collneart. Ths s an to exponental smoothng n the accdent ear drecton. Ths approach s necessar ver powerful and ncreases the stablt of the model especall f n the more recent accdent ears there are some slght changes n levels. The 'amount' of stochastc varatons n alpha s determned b the SSPE statstc whch s explaned n Zehnwrth ( Model Identfcaton The am s to dentf a parsmonous model n PTF that separates the (sstematc trends from the random fluctuatons and moreover determnes whether the trend n the pament/ calendar ear drecton s stable. The model dentfcaton procedure s dscussed n Secton 0 of Zehnwrth (994. We start off wth a model that onl has one parameter n each drecton model (sequentall the trends n the development ear drecton followed b pament ear or accdent ear drectons dependng on whch drecton exhbts more dramatc trend changes. Heteroscedastc adustments ma also be necessar. Valdaton analss s an ntegral component of model dentfcaton extracton of nformaton and testng for stablt of trends.