DETERMINISTI AND STOHASTI MODELLING OF TEHNIAL RESERVES IN SHORT-TERM INSURANE ONTRATS Patrck G O Weke School of Mathematcs, Uversty of Narob, Keya Emal: pweke@uobacke ABSTART lams reservg for geeral surace busess has developed sgfcatly over the recet past Ths has bee occasoed by the growth of the surace market, wth the rsk uderwrtg process becomg more ad more complex New surace products have bee developed that cater for the more specfc eeds of the polcyholder Latet clams have also arse recet years, puttg maor stras o compay resources The case of asbestoss related clams testfes to ths, havg receved wdespread atteto Furthermore, recet dsasters, such as the floods Europe ad the September th terrorst attacks o the US have cotrbuted to the eed for more complex ways of aalyzg clams experece The sutablty of the models used clams reservg, have had to be revewed to esure that they do ot gve false mpressos The obect of ths paper, therefore, s to come up wth a comparso of dfferet determstc ad stochastc methods of clams reservg for a geeral surer wth a gve clams experece The sutablty of each of the estmates s oted to deped o the purpose of the reservg exercse The paper dscusses some of the methods (for stace, the basc cha ladder method, flato adusted cha ladder method, separato techque, Borhuetter-Fergusso techque ad copula) used clams reservg, ad for a partcular clams experece, t gves a aalyss of how well each of the methods models clams experece INTRODUTION The settlemet of clams s the prme obectve of surace Polcyholders effect surace so that retur for the paymet of a premum, a surace compay accepts the lablty to make a moetary paymet to the sured o the occurrece of a specfed evet wth a specfed perod of tme I theory, the surer's lablty to pay a clam crystallzes at the stat of occurrece of the sured cotgecy However, there are may factors whch ca lead to ver cosderable delays betwee occurrece ad paymet Frstly, the sured cotgecy tself may ot occur at a sgle stat ad may ot eve be recogzed as clamable evets utl may years after commecemet Secodly, the legal lablty of the surer may ot always be clear-cut, ad there may be cosderable delays before the surer (or the court) decdes that lablty exsts Thrdly, the quatum of damages may be mpossble to determe utl some perod of tme has elapsed sce occurrece of the evet Fourthly, there wll be processg delays, wth the surer's --
offce, the recordg of the clams, processg of the clams fle, authorzato of paymet ad drawg, despatch ad ecashmet of the clam paymet The predcto of outstadg clams amouts o-lfe surace wth short-term polces s, by ts very ature, hghly speculatve Specfc detals of methodologes for makg such predctos are cotaed a comprehesve ad hghly detaled survey coducted by Taylor (986) Oe feature commo to all of these methods s the utlzato of curret ad past records of clams amouts the form of ru-off tragle to calbrate the proposed predcto model before use Kremer (982) has show how the classcal cha ladder method for estmatg outstadg clams o geeral surace busess s strogly related to a two-way aalyss of varace Ths paper follow closely o the work o the statstcal treatmet of clams reservg Mack (99) whch oted the coecto betwee the methods of estmatg 'Icurred But Not Reported' (IBNR) clams reserves ad automoble ratg methods Ths parametrc model s ow mplemeted GLIM ad appled to clams data (see Weke, 2003) Our purpose s ot to add to the exstg plethora of methodologes but rather to retur to the grass roots of the subect by explorg more fully the statstcal settg for the basc cha-ladder ad related techques 2 LAIMS DATA 2 Data Presetato The methods for estmatg clams reserves that are dscussed requre data to be preseted the form of a ru-off tragle Ths presetato cross classfes the data accordg to the perod of org ad the perod of developmet The perod of org may be the year whe the clam was curred, or reported, or whe the polcy relatg to the clam was uderwrtte, whle the developmet perod refers to the legth of tme sce the perod of org whch the clams were curred, reported or pad By coveto, the developmet year relatg to the year of org s deoted as developmet year zero A clam cohort s defed depedg o the defto used for clams from each org perod ad developmet perod For example, we could have each etry the tragle as beg the value of the clam pad developmet year, the clam havg occurred year of org The geeral form of the ru-off tragle s gve by: Year of org 0 Year of developmet 0 2 0,0 0, 0, 2,0,, 2,0,, 2,0 0,,, 0, -2-
22 lams Data lams ru-off data are geerated whe delay s curred the settlemet of surace clams Typcally the format for such data s that a tragle whch the row deotes accdet years ad colum delay or developmet years The aalyss s based o a set of geeral surace data, show Table Ths data s take from a paper by Reshaw (989) ad cossts of clams from a portfolo of geeral surace polces wth exposure factors Table : Ru-off lams Data ad Exposure 35784 766940 60542 482940 527326 574398 46342 39950 227229 67948 3528 88422 933894 83289 445745 320996 527804 26672 280405 290507 00799 92629 06654 75086 46923 495992 2480405 30608 08250 77689 562400 272482 352053 206286 44360 69390 99983 769488 50484 470639 39632 937085 847498 805037 705960 440832 84763 3398 063269 359480 06648 443370 376686 986608 34404 Exposures 60 72 697 62 600 552 543 503 525 420 The exposures for each year of busess are dvded to the clams data before the aalyss s carred out 3 THE HAIN LADDER METHOD 3 The Basc ha Ladder Method The cha ladder method assumes that all exteral factors, for example, flato of clam costs, chage the mx of busess, chage the rate of settlemet of clams, ca effectvely be gored ad the model the assumes the form = x y + ε (3) where,, =,, deote the total amout of clams developmet year respect of accdet year, x s the ultmate total cost of clams accdet year, ad y s the proporto of total paymets made by the ed of developmet year Let S represets clams amout wrtte developmet year wth respect to accdet year the = S, + = + S, +, for all, ; + (32) -3-
I the absece of the exteral factors the dstrbuto of delays betwee the cdet gvg rse to a clam ad the paymets made respect of that clam rema relatvely stable over tme If t s assumed that the exogeeous flueces are small, the we may regard expected value of the rato of cumulatve clams amouts for year to year ( +) [ ] E, + as a estmate of the progresso from, to, + for complete rows, e for values for whch, s kow but s ot, + The method assumes that the factors y are costat for all years of accdet If b represets the rato of the cumulatve paymets made by the ed of year to the expected value of the cumulatve paymets made by the ed of year, the b may be estmated by + = = b +, = 2,, (33) =, The b factors are thus calculated by summg each colum the ru-off tragle (Table ) ad takg the rato to the prevous colum total excludg the last etry Let us deote the product of ( ) b 's by B, that s, z= + B = b z, =,, (34) the we ca estmate the amout of clams stll outstadg as at the ed of year ( ) + respect of accdet year by ( B ) If we represet the rato of outstadg lablty at the ed of developmet year for year of accdet to the cumulatve clams amout by b +, the b + s the estmate of the outstadg lablty as at the ed of developmet year (for year of org ) Ad Equato (34) ow becomes B = b z, =,, z= + B = b + (35) -4-
These estmates ca the be used to complete the ru-off of the later years of org up to the pot for whch past experece s avalable The flato adusted cha ladder method whch s based o adaptg the geeralzed model by troducg a assumed dex of clams cost ca also be cosdered Other methods, for example, the ch-square method (due to Baley ad Smo, 960), the method of margal totals ad the method of weghted squares ca be used Detals of these methods are avalable Weke (992, 2003) 32 Iflato Adusted ha Ladder Method Ths method adopts the geeral model the form: = S R X + e (36) + ad the parameters thus become: = s r + e (37) λ + where are the paymets made developmet year of year of org, (e, ocumulatve) s s the ultmate total cost real terms of clams curred the perod of org r s the proporto of total paymets real terms made developmet year λ + s a assumed dex of clams cost Uder the flato adusted method, the ru-off tragle has to be preseted as cremetal clams for each year of org ad developmet Usg a clams flato dex, the past values are brought to curret moetary values Icremetal clams alog the same dagoal (movg from bottom left to top rght) arse from the same year ad hece the same flato dex value s appled o them The adusted cremetal clams are the accumulated ad the ormal procedures of the basc cha ladder method are appled These estmated clams reserves are also curret moetary terms I order to estmate the cash value of future clam paymets, a assumpto has to be made about the lkely level of future clam flato Assumptos: - The clams developmet patter s stable - lams flato wll be at the assumed future rate 33 The Separato Techque It has the form of equato (36): = S R X + e wth parameters + -5-
λ + = r + e (38) where s the umber of clams curred the year of org ad λ + s related to the year of paymet I ths case λ + s derved from the data rather tha assumed from exteral sources The derved factors wll be related to creases clam costs but wll also be affected by other exteral factors ad by radom fluctuatos the clam sze As a result, they are lkely to correspod to ay assumed dex cosdered sutable for use wth the cha ladder method The method for aalysg the ru-off tragle s as follows I respect of each year of org, the clam paymet, made each developmet year are dvded by some exposure dex S, attrbutable to the perod of org These exposure measures may be vehcle years or eared premums However, they may ot accurately reflect the dffereces the rsks uderwrtte (a partcular problem usg eared premums) Furthermore, results wll be affected by chages clam frequecy To avod these problems, ormalzg factors are used, the most commo beg the umber of clams (Hossack et al, 999) The method of assessg the umber of clams should be cosstet from year to year As the latest year of org has developed least, the umber of clams assumed for earler years should be of the same durato For ths reaso, the clam umbers are usually related oly to the clams reported the frst year The r ad λ + parameters are the estmated as follows: Year of org 0 Year of developmet 0 2 r 0λ 0 r λ0 2λ0 r r 0λ r λ λ r λ 0 r 0 λ r λ0 r 2λ r λ r 2 λ rλ r r λ The ( r ) are the proporto of clams pad year so by defto r = The by summg the dagoal volvg λ of the ru-of tragle, we have: where d = λ ( r + r + + r ) = λ (39) 0 d s the total of the th dagoal -6-
Thus λ = d If ν s the sum of the colum of the tragle cotag r, the ν = ( λ + λ + + λ ) (30) r hece ν = rλ or r = ν / λ Summg the ext dagoal gves d = λ ( r + r + + r ) = λ ( r ) (3) 0 ad so λ = d /( ) r (32) ad therefore, recursvely: r = ν /( λ + λ + + λ ) (33) + λ d r+ r+ 2 r = /( ) (34) To estmate the clams reserves, t s ecessary to make a assumpto for future flato factors Ths s doe by examg the ratos of past λ + factors the lght of the kow flato rates as at the tme, ad takg to accout the expected levels of flato throughout the future ru-off perod Ths eables the future factors to be calculated whch, together wth the relevat r ad factors wll produce a estmate for each future developmet perod for the years of org ot fully ru-off Assumptos: The clams developmet patter s stable 34 The Borhuetter Fergusso Techque (The B F method) The B F method dffers from the basc cha ladder method that the ultmate clam, S, s replaced by a alteratve estmate, S BF, whch s based o exteral formato ad expert udgmet The model s thus of the form: = S BF R + e (35) wth parameters: = s BF r + e (36) where r s the proporto of total paymets made by the ed of developmet perod -7-
BF s could be a estmate foud by usg a smple loss rato o wrtte premums (or some other sutable measure of exposure) Assumptos: - the gve loss rato s correct - the clams developmet patter s stable - the past clams developmet does ot provde ay addtoal formato o the future developmet of clams If b s the rato of the expected amout of clams pad by the ed of perod ( ), the b ca be estmated by: b = = 0 = 0, whch s the same parameter as that of the cha ladder method ad defg B = b z z= + The estmated clams stll outstadg at the ed of year org year s gve by: S BF + wth respect to the ( B ) (37) The Borhuetter-Ferguso techque assumes that there s pror kowledge about the parameters of the model, makg t aalogous to a Bayesa approach The B-F method may also be appled o flato adusted clams data (as s the case wth the flato adusted cha ladder method), ad the future clams reserves estmated o a assumpto of the future rate of clams flato I the ext secto, the gamma dstrbuto model wth multplcatve errors s descrbed ad appled to clams data However, f we are dealg wth a stuato whch past flato rates chage wth tme (ad are kow) the a slght modfcato s ecessary 4 A STOHASTI APPROAH: A ASE OF THE GAMMA MODEL Let us suppose that a ru-off tragle we have total clams amout varable S (wth a realzato s ) of cells labeled (, ) each wth a kow measure of exposure, (depedet of ths partcular case of clams reservg) Let us also suppose that the total clams amout R k of each ut k =,, of cell (, ) has a gamma -8-
dstrbuto wth a costat shape parameter, α ad a mea value probablty desty fucto (pdf) of the total clams amout varable s α m The the s f s = α ( ) ( αs m ) Γ( α) m s exp (4) Ad the amout geeratg fucto (mgf) s M α mt ( t) = α (42) Assumg that all uts cell (, ) are depedet the s = R + R 2 + has mgf M S mt ( t ) = α ad hece ~ α m kow from the ru-off tragle ad ot those of dstrbuto of S S Gamma ( ) α α; Sce the realzato s of S are usually R k we therefore work wth Now cosder the multplcatve approach dsplayed the form m = x y, where x, y are ukow parameters for accdet year ad developmet year respectvely These ukow parameters ca be estmated by the maxmum lkelhood method Assumg that all S are depedet, the loglkelhood fucto s gve by l = log L = { s ( x y ) + α log( αs ) α log( x y ) log sγ( α) }, α (43) The mles are those values xˆ, ŷ of x, y, respectvely whch maxmze Equato (43) These lkelhood estmators are gve by xˆ, =,, = s y yˆ, =,, = s x (44) where = over The estmators Equato (44) are smlar to approxmatg xˆ by the -weghted mea of S y ad ŷ by the -weghted mea of S x ;, =,, The gamma dstrbuto model wth shape -9-
parameter α (α s a costat) ad mea value m ca be ftted to a geeralzed lear model (GLM) by specfyg the radom compoet, systematc compoet ad the lk fucto (Mcullagh ad Nelder, 989) 5 STATISTIAL ANALYSIS OF THE MODEL I ths computer based statstcal model, goodess-of-ft statstc derved from the lkelhood rato (called the devace) s used The ftted values are set equal to the observed values ( µˆ = S ) Equato (43) to gve the maxmum achevable loglkelhood ad the devace therefore becomes { log L( s ; s ) log L( ˆ ; s )} Dev = µ 2 { log( s ˆ µ ) + ( s ˆ µ ) ˆ } = 2 α µ, (5) Ths test statstc depeds o a scale parameter estmate, α, ad whe o scale parameter has bee explctly set by the user, GLIM calculates the devace usg the above expresso wth α = ad uses the mea devace as a estmate of α for the purpose of calculatg stadard errors of the parameters The stadardzed Pearso resduals, r, are computed GLIM ad used to explore the adequacy of the ft of the model ( s ˆ µ ) { V ( ˆ )} 2 r = µ,, =,,, (52) where V ( ˆ µ ) deotes the varace fucto of the dstrbuto evaluated at the ftted value µˆ, ad also produce resdual plots for aalyss 6 IMPLEMENTATION AND APPLIATION Our ma am s to mplemet the model a GLIM statstcal package ad apply t to the ru-off clams data wth exposures Table to produce predcted clams amouts for the empty (, ) -th cell the southeast tragular rego We acheve ths obectve by user defed macros wth GLIM ad specfy four prmary macros (see Weke, 992 for further detals) The output of estmated clams, row totals of the estmated future clams ad total estmated future clams are sgfcatly mportat to the practtoers for forecastg purposes For ay geeral surace compay these values show the expected clams amouts arsg from evets whch have occurred but of whch o otfcato has yet bee receved IBNR), the predcted clams for each accdet year ad the overall total expected clams amout from the year of org to the preset tme The goodess-of-ft statstc of the model ca also be performed by usg the package sce both the devace ad degrees of freedom are gve the output -0-
7 RESULTS AND DISUSSION The results of the applcato of GLIM package to the o-cumulatve ru-off data wth exposures (Table ), for whch I am grateful to the aoymous suppler, are ow be dsplayed ad dscussed The parameter estmates for the grad mea, the accdet year parameters R () ad the developmet year parameters ( ) ;, =,, 0, wth ther respectve stadard errors were computed ad used to costruct Table 2 The GLIM system automatcally sets R ( ) = () = 0 The resduals ad the ftted values of the model for varous values of accdet year ad developmet year are computed ad ther compoets used to draw up the resdual plots Table 2 gves the observed clams amouts, S, ad estmated future clams S The system calculates the estmated future clams whch appear dow the `stars' from the observed clams amouts The S values are tur summed up columwse to produce the row totals of the estmated future clams Sce the clams amouts for accdet year s full we have that RT ( ) = 0 Ad Table 3 shows the row totals of the estmated future clams ad total estmated future clams Practtoers have kee terest the values Table 3 sce these values are estmates of the outstadg clams provso at the preset tme (eat the ed of accdet year 0) wth respect to year of org ad the total overall outstadg clams provso for the etre perod The estmates are of sgfcat use forecastg the IBNR clams provso ad geeral orgazato of busess It s usually ecessary to vestgate the extet of predctor stablty for ths model Geerally, t wll be observed that the predctor stablty weakes as data pots further to the spces of the ru-off tragle are vared A mprovemet of the model the case of kow clams umbers may also be approprate to cosder Table 2: Observed clams amouts ad estmated future clams 35784 766940 60542 482940 527326 574398 46342 39950 227229 67948 3528 88422 933894 83289 445745 320996 527804 26672 425046 93278 290507 00799 92629 06654 75086 46923 495992 2840405 356653 90296 30608 08250 77689 562400 272482 352053 206286 2475 36555 8044 44360 69390 99983 769488 50484 470639 339427 22857 337833 8553 39632 937085 847498 805037 705960 48038 35408 238439 35248 89223 440832 84763 3398 063269 6446 452746 383479 258236 38678 9663 359480 06648 443370 225467 685536 50553 427867 28827 425858 0787 376686 986608 98278 049747 587237 43279 36655 24683 364794 92357 34404 853339 87344 932640 52726 384446 325628 29279 324098 82054 Table 3: Row totals of the estmated future clams Row 2 3 4 5 6 7 8 9 0 RT 0 93278 446949 60874 99363 452200 28786 3665823 422963 456353 The total estmated future clams s: 8086988 --
8 ONLUSIONS The ma focus of ths paper was to determe the best estmate of clams reserves for a partcular set of crcumstaces by comparg the reserve estmates produced by the dfferet methods The study revealed that for the partcular data, the separato techque geerally teded to gve the best ft to the observed clams experece It gave the lowest mea, meda, rage ad ter-quartle rage for the percetage resdual errors It gave the lowest total clams reserves The method would thus be sutable the case where t s mportat to have a far pcture of the reserves wthout beg pessmstc or optmstc Ths would geerally be so whe the reservg exercse s beg carred out for maagemet revew purposes ad whe determg premum rates The flato adusted cha ladder also gave a reasoably good ft o the observed clams experece However a tred for t to overestmate the clams later years of org was observed Ths may expla why the estmated clams reserves of ths method ted to be hgher tha those gve by the separato techque The basc cha ladder method gave smlar results but dd gve hgher total clams reserves The overestmato was however ot large both cases The methods would thus seem approprate where a coservatve approach s take the clams reservg exercse Determg clams reserves for the publshed accouts of the compay ad also for supervso of solvecy may be doe usg ether of these two methods Furthermore, the case that the surace compay s beg valued for a purchase, a coservatve value of the reserves s approprate ad ether of the two methods could be used The B-F methods gave poor fts to the observed clams data The flato-adusted method was observed to cosstetly overestmate the clams at all years of org It thus would ot be cosdered a approprate model to estmate clams reserves for ths class of busess The B-F method wthout flato adustmet would also ot be approprate for estmatg the clams reserves The mplemetato of ths model s GLIM provded eough statstcs the output for testg the ft of the model to the ru-off clams data ad also produced the estmated clams amouts for the empty (, ) -th cell the south east tragular rego The estmated clams amouts are tur used to produce the row totals of the estmated future clams ad the total estmated future clams whch are of terest to practtoers The estmated clams amouts, the row totals of the estmated future clams ad total estmated future clams reported Table 2 ad Table 3 are cosstetly lower value tha the correspodg estmates Reshaw (989) Ths slght varato the results s attrbuted to the fact that Reshaw (989) used a log-ormal dstrbuto model whch case the expected value of the estmated clams amout s a fucto of a addtoal term (e half the varace term) Thus the computato of predcted values s based o both the meas ad varaces of the parameter estmates the logrespose fucto -2-
The row totals of the estmated clams reported Table 3 le betwee the values calculated usg the dyamc estmato ad emprcal Bayes methods Verrall (989) We fd from the above comparsos that the results obtaed by the parametrc model are satsfactorly acceptable ad ca be used for practcal purposes Therefore ths model provdes a smple method whose applcato clams reservg s early as smple to execute as the cha-ladder method but has the advatage of provdg the goodess-of-ft test statstc ad the estmato error REFERENES Baley, R A ad Smo, L J (960) Two Studes Automoble Rate Makg ASTIN Bullet, pp 92-27 Kremer, E (982) IBNR-clams ad the two-way model of ANOVA, Scad, Act, J, Vol, 47-55 Mack, T (99) A Smple Parametrc Model for Ratg Automoble Isurace or Estmatg IBNR lams Reserves ASTIN Bullet, Vol 2, No, 93-09 Mcullagh, P ad Nelder, J R (989) Geeralzed Lear Models (2 d Edto), hapma ad Hall Reshaw, A E (989) ha Ladder ad Iteractve Modellg (lams Reservg ad GLIM) Joural of the Isttute of Actuares, Vol 6, Part 3, 559-587 Taylor, G (986) lams Reservg No-Lfe Isurace, North Hollad Verrall, R J (989) A State Space Represetato of the ha Ladder Lear Model Joural of the Isttute of Actuares, Vol 6, Part 3, 589-609 Weke, P G O (992) Statstcal Models for Estmatg IBNR lams Reserves, MSc Dssertato, The ty Uversty, Lodo Weke, P G O (2003) Estmatg IBNR lams Reserves usg Gamma model ad GLIM, The Ngera Joural of Rsk ad Isurace, 2003, Vol 4, No : Weke, P G O (2006) Determstc lams Reservg Short-Term Isurace otracts, EAJ of Statstcs, Vol, No 2: 98 23 (Wth A Mureth) Weke, P G O (2008) Estmato of IBNR lams Reserves usg Lear Models, KJST Seres A, 2008, Vol 3, No (accepted) -3-