Applatons of the Offset n Property-Casualty Predtve Modelng Jun Yan, Ph.D. James Guszza, FCAS, MAAA, Ph.D. Matthew Flynn, Ph.D. Cheng-Sheng Peter Wu, FCAS, ASA, MAAA Abstrat: Generalzed Lnear Model [GLM] theory s a ommonly aepted framework for buldng nsurane prng and sorng models. A helpful feature of the GLM framework s the offset opton. An offset s a model varable wth a known or pre-spefed oeffent. Ths paper presents several sample applatons of offsets n property-asualty modelng applatons. In addton, we wll onnet the offset opton wth more tradtonal atuaral tehnques suh as exposure and premum adjustments. A reurrng theme of the dsusson s that atuaral modelers have at ther dsposal several oneptually related tehnques that an be used to elmnate the mpat of varables that (for whatever reason are not ntended for nluson n a model, despte the fat that they mght be orrelated wth both the target varable and other predtve varables. Examples dsussed n ths paper nlude a lass plan analyss as well as a ter sorng applaton. Sample SAS ode for fttng GLMs wll be provded n the body of the paper. Key Words: Offset, Resdual, Generalzed Lnear Models, GLM, Predtve Modelng, Ratemakng, SAS Introduton In reent years, property-asualty nsurane ompanes have wdely embraed predtve modelng as a strateg tool for ompetng n the nsurane marketplae. Predtve modelng and n partular the use of Generalzed Lnear Models was orgnally ntrodued as a method for mprovng the preson of personal auto nsurane prng. The use of predtve modelng was subsequently extended to homeowners and ommeral lnes as well. Today, predtve modelng s a ore strateg apablty of many top nsurers and s appled n suh key operatons as marketng, underwrtng, prng, and lams management. Property-asualty nsurane s a omplex and dynam busness. As s often observed, t s unque n that the ultmate ost of ts bas produt s unknown at the tme of sale. A plethora of rsk fators affets the ost of provdng nsurane. Many of these are well understood and are refleted n the pre of nsurane. For example, a typal automoble nsurane ratng plan ontans more than 20 varables, nludng a wde range of drver, vehle, and terrtoral haratersts [1]. However, the ost of provdng nsurane s also greatly nfluened by suh dynam and exogenous fators as the underwrtng yle, medal nflaton, varatons n the sze of jury awards, and poorly understood exposures suh as asbestos and mold. It s therefore pratally mpossble for atuaral models to be omprehensve n the sense of nludng all relevant varables that affet the number and sze of lams. The non-deal nature of atuaral models s ompounded by the real-world fat that nsurane data s often nomplete, nonsstently oded, and generally drty. In addton, many relevant varables (suh as vehle symbol, ratng terrtory, or Workers Comp ndustry lassfatons are massvely ategoral, Casualty Atuaral Soety E-Forum, Wnter 2009 366
Applatons of the Offset n Property-Casualty Predtve Modelng leavng ndvdual nsurers wth nsuffently redble data to estmate ther own ratng fators as part of a ratng plan optmzaton exerse. For these and other reasons, atuaral modelers fae a gener problem: n many, f not most, modelng stuatons, they are fored to exlude varables that are relevant to predtng frequeny and sze of loss. If these omtted varables are orrelated wth both the target varables and one or more of the other modelng varables, they wll bas the estmates of the orrespondng model parameters [2]. Ths phenomenon s ommonly known as omtted varable bas [OVB]. In short, t wll never be possble to buld a sngle atuaral super model that aounts for every sngle determnant of loss. To avod the perl of OVB, atuares therefore must often adjust for or otherwse aommodate the effets of omtted varables as part of ther model desgn and model onstruton proess. Commonly known fators whh potentally bas property-asualty predtve modelng results nlude the underwrtng yle and external envronmental hanges (.e., tme, varaton n loss maturty, dstrbuton hannel, varaton n rate adequay aross states and through tme, and a hangng ompettve landsape, to name a few. A tradtonal atuaral response to the problem of OVB s to adjust the model s target varable (more presely, the exposure or premum omponent of the target varable. A oneptually smlar tehnque that has long been n the arsenal of atuaral modelers s runnng a prelmnary regresson model on the varables to be omtted (suh as poly year or state and then usng the resduals of ths model as the target varable gong forward. More reently, atuares have embraed the offset opton from Generalzed Lnear Model theory [3-7]. Eah of these tehnques offers a way of avodng OVB. That s, eah tehnque offers a way of aountng for the effet of omtted varables n a way that avods basng the model s parameters. Ths paper wll revew the bass of GLM theory and the GLM offset opton, provde varous sample applatons of the offset, and draw onnetons between the offset opton and tradtonal atuaral tehnques. Bakground: GLM Theory and the Offset Reall that a Generalzed Lnear Model [GLM] relates the expeted value of the target varable (µ E[Y] to a lnear ombnaton of predtve varables (β X va a lnk funton g( : g μ β β X β X... β X β X ( 0 1 1 2 2 In addton to the lnearty assumpton mplt n the above equaton, GLM theory assumes that the target varable s dstrbuted by the 2-parameter famly of dstrbutons known as the exponental famly. The exponental famly enompasses a wde range of dstrbutonal forms nludng Normal, Gamma, Bnomal, Posson, Negatve Bnomal, and many others. The exponental famly densty funton s expressed as: {( yθ b( θ / a( ϕ ( y, } f Y ( y; θ, ϕ exp ϕ The two parameters n ths famly, θ and φ, are known as the anonal parameter and dsperson parameter, respetvely. As we wll see, these are related to the mean and varane, respetvely, of Y. p p Casualty Atuaral Soety E-Forum, Wnter 2009 367
Applatons of the Offset n Property-Casualty Predtve Modelng Two mathematal fats are helpful n nterpretng ths seemngly omplated expresson: and: E [ Y ] μ b ( θ Var ( Y b ( θ a( ϕ It s ommon to denote b (θ as V(µ and all t the varane funton. (N.B.: the varane funton, V(µ, s not the same thng as the varane of Y. Furthermore, the funton a(φ s often spefed to be φ/ω, where ω s a pror weght (suh as exposure or premum. Therefore, we have the followng expresson that relates the varane of Y to the mean of Y: ϕ Var ( Y V ( μ ω In the speal ase of un-weghted, ordnary least squares [OLS] regresson, we have: g(µ1 (dentty lnk, ω 1 (eah observaton s gven equal weght, a(φσ 2 (merely a dfferent namng onventon, b(θθ 2 /2, and (y,φ -½{y 2 /σ 2 log(2πσ}. The reader an verfy that these substtutons result n the famlar expressons for the Normal dstrbuton N(µ,σ 2 and homoskedastty (onstant varane: : 1 { ( μ 2 / 2 2 } 2 f Y ( y; μ, σ exp y σ 2 2πσ 2 Var ( Y σ In OLS regresson, the modeler selets the target varable, the approprate set of predtve varables, as well as the pror weghts ω, and must verfy that the assumptons of normalty (n partular homoskedastty and the lnearty on the addtve sale (.e., dentty lnk are satsfed. In the broader GLM framework, the normalty and lnearty assumptons are eah relaxed. The normalty assumpton s replaed wth the muh weaker assumpton that the dstrbuton of Y s from the exponental famly; and lnearty s replaed wth lnearty on the sale determned by the lnk funton. Commonly used dstrbuton/lnk funton ombnatons are dsplayed below: Dstrbuton V(µ Lnk Sample Applaton Normal 1 dentty General applatons Posson µ log Frequeny modelng Bnomal µ(1- µ logt Retenton, ross-sell Gamma µ 2 log Severty modelng Tweede µ p, pє(1,2 log Pure Premum modelng Note that what s often alled hoosng a dstrbuton for a GLM s tantamount to hoosng the varane funton V(µ that relates the varane of Y to the mean. Casualty Atuaral Soety E-Forum, Wnter 2009 368
Applatons of the Offset n Property-Casualty Predtve Modelng Wth the bas GLM framework n hand, we an turn to the offset feature. An offset s smply an addtonal model varable, ξ, whose oeffent s onstraned to be 1: g ( μ β X ξ In the ase of OLS regresson, ths amounts to subtratng ξ from the target varable pror to runnng the regresson. Therefore, offsets are not typally dsussed n the ontext of OLS regresson. Suppose that ξ s the predted value of Y from a prelmnary regresson model. Then, spefyng ξ as an offset s equvalent to usng the resdual from the prelmnary regresson as the target varable of the regresson of nterest. As mentoned above, ths s a well known method of removng the effets of a group of nusane varables from the target varable pror to runnng the model n order to avod omtted varable bas. In the remander of ths paper, we wll dsuss offsets n the ontext of multplatve models,.e., models onstruted usng the log lnk funton. As an asde, t s nterestng to note that the offset was orgnally an afterthought n the development of Generalzed Lnear Models theory by Nelder and Wedderburn n 1972 [3]. Quotng from the book by Hlbe [8, page 130]: Offsets were frst oneved by John Nelder as an afterthought to the [Iteratvely Reweghed Least Squares] algorthm he and Wedderburn desgned n 1972. The dea began as a method to put a onstant term dretly nto the lnear predtor wthout that term beng estmated. It affets the algorthm only dretly before and after regresson estmaton. Nelder only later dsovered that the noton of an offset ould be useful for modelng rate data. Offsets as a Measure of Exposure As the above quote suggests, the offset s most ommonly dsussed as a measure of exposure n the ontext of Posson regresson. For example, t shows up n essentally the same way n both atuaral and epdemologal work. In both ases, offsets are often nterpreted as a measure of exposure. In the latter settng, the exposure mght be the number of people exposed to a pathogen; and the response would be the number of people who ontrat the dsease. In the former settng, the exposure mght be the number of ar-years nsured; and the response would be the number of lams nurred. In both settngs, the value of the response s assumed to be roughly proportonal to the value of the exposure. As the fnal equaton n the prevous suggeston ndates, the offset must be on the same sale as the lnear predtor β X. Therefore, n the auto example above, log(exposure would be used as an offset. That s: ξlog(u where u ( unts denotes exposure: log( E [ C] β X log( u Casualty Atuaral Soety E-Forum, Wnter 2009 369
Applatons of the Offset n Property-Casualty Predtve Modelng In the Posson ase, ths s mathematally equvalent to replang lam ount wth lam frequeny (lams dvded by exposures: FC/u as the target varable; usng exposure as the weght; and dspensng wth the offset: log( E[ F] β These two models spefatons are summarzed n the table below: Opton 1 Opton 2 GLM famly: Posson Posson Target: C F Weght: (none u Offset: log(u (none Opton 2 s the more ommonly adopted model spefaton. The equvalene of these two spefatons s demonstrated n Appendx A. X Exposure Adjustments and the Offset To avod omtted varable bas, atuares ommonly perform as a prelmnary step varous exposure or premum adjustments to remove the effets of varables not nluded n the model. Suh adjustments are ommonly used n prng plan analyses for reasons nludng prng struture omplexty, data avalablty, data redblty, busness or regulatory onsderatons, ompettve onsderatons, and the desre to mtgate polyholder mpats. For example, suppose we wsh to model lam frequeny n terms of the followng varables: Mult-ar ndator Drver age Vehle use Symbol Terrtory Beause of the large number of Terrtory and Symbol ategores, the analyst mght wsh to estmate Symbol and Terrtory fators n a separate analyss. Merely droppng these varables from the model wth no further aton would rase the problem of OVB. Suppose, for example, that a ertan terrtory has a dsproportonately large number of young drvers. If Terrtory were smply exluded from the model wth no further adjustment, the Drver Age varable would at partly as a proxy for terrtory. The fnal ratng plan, nludng both Terrtory and Drver Age, would overharge young drvers n ths hypothetal terrtory. Ths problem s sometmes dealt wth by adjustng the exposure feld. Assumng a ompletely multplatve ratng plan, adjustng the exposures means smply multplyng exposures ether by the exstng terrtory and symbol relatvtes, or by a set of relatvtes that have been estmated n a separate modelng exerse. As above, let u denote the exposure measure and let τ σ j denote the produt of the Terrtory and Symbol relatvtes for Terrtory and Symbol j. We ompute f adj Casualty Atuaral Soety E-Forum, Wnter 2009 370
Applatons of the Offset n Property-Casualty Predtve Modelng /(u*τ σ j. We use ths adjusted frequeny (lams dvded by adjusted exposure quantty rather than unadjusted frequeny (f/u as the target varable. Gven the above dsusson and the result of Appendx A, t should be lear that one ould equvalently use the un-adjusted frequeny feld f as the target varable and also nlude log(τ σ j as an offset term n the model: log ( E [ F] β β β log( τσ mult drverage vehleuse In other words, the tradtonal atuaral response to the OVB problem s equvalent to usng a strategally seleted offset term, that s, adjustng exposures s equvalent to nludng the prespefed ratng fators as an offset n the model and allowng the remanng fators to onform to ths offset. Loss Rato Modelng and the Offset The dsusson n the prevous seton s analogous to the dstnton between Loss Rato and Pure Premum models. Suppose we wsh to onstrut a redt sorng model, for eventual use n target marketng, ompany plaement, and prng refnement. Suppose also that the urrent ratng plan s up-to-date, wth no base rate or ratng relatvty hanges needed. Examples of the varables used to onstrut the redt sorng model mght be number of late payments n the past x days, balane-tolmt rato, and number of derogatory publ reords n the past y years. Usng Pure Premum as the target varable n suh a model would obvously ntrodue the possblty of omtted varable bas. It s possble that some of the parameters n the resultng redt sorng model would double ount a penalty or redt gven n one of the exstng ratng fators [9]. The tradtonal atuaral response to ths problem s to use Loss Rato rather than Pure Premum as the target varable [10]. Ths s analogous to the above dsusson of adjusted exposures n Pure Premum modelng: we replae loss/u (Pure Premum wth loss/(u*τσ υloss/prem (Loss Rato as the target varable. Ths s oneptually equvalent to usng dollars of loss as the target varable, and nludng log(prem as the offset term n the model: log ( E[ loss] β β β... log( prem latepay baltolm derog In ths way, Loss Rato modelng as an alternatve to Pure Premum modelng an be vewed as yet another nstane of strategally usng the offset feature to avod the problem of omtted varable bas. Please note that our pont s not to reommend that atuares abandon the use of Loss Rato as a target varable n favor of usng Pure Premum or dollars of loss wth an offset. We only wsh to make the pont that modelng Loss Rato rather than Pure Premum s oneptually yet another nstane of usng an offset to ntegrate pror onstrant nto one s model. In loss rato modelng, the goal s to buld a sorng model to be layered on top of the exstng ratng plan. The pror onstrant s therefore the urrent ratng plan n ts entrety, properly adjusted and on-leveled. Casualty Atuaral Soety E-Forum, Wnter 2009 371
Applatons of the Offset n Property-Casualty Predtve Modelng Usng the Offset to Constran Seleted Ratng Fators Another useful applaton of the offset s onstranng ertan ratng fators to take on pre-spefed values. Constrants suh as these are often motvated by regulatory and marketng onsderatons [7,11]. For example: The nsurane marketplae mght demand that that the dsount for mult-ar or home-auto pakage poles be no greater than 15%, regardless of the ndaton of a statstal analyss. Calforna s Proposton 103 requres that a good drver dsount be at least 20% below the rate the nsured would otherwse be harged. In both ases, we must onstran the values of ertan ratng fators n advane, and allow the remanng ratng fators to optmally onform to these onstrants. The offset allows one to easly ntegrate onstrants suh as these nto one s model. Ths s only a short step from the exposure adjustment example dsussed above. We wll gve an example wth the added omplexty that we wsh to onstran some, but not all, of the levels of a ertan ratng varable. In passng, we should note that offsets should not be appled blndly or n a mehanal fashon. Werner and Guven [12] provde a helpful example of a ase n whh one would not want the other fators n a ratng plan to help make up for a pror onstrant to a ratng fator. In general, one should be mndful of the aveat that no modelng desons (modelng tehnque, target varable desgn, hoe of predtve varables and offsets, modelng dataset desgn, and so on should be made wthout due regard for the busness ontext of one s work. Suppose we wsh to optmze two fators of a multplatve ratng plan: drver age group (wth values {1,2,3,4} and mult-ar ndator. We have already multpled the exposures by all other ratng varables as desrbed n the exposure adjustment seton above. Our target varable s adjusted frequeny: lam ount dvded by adjusted exposure. Detals of the dataset used n ths and the followng examples an be found n Appendx B. Let us further assume that (ether for ompettve or regulatory reasons the relatvtes for DRIVER_AGE_GROUP 3 and 4 must be onstraned to take on the values 1.05 and 1.25, respetvely. The followng SAS ode shows how to buld a model that norporates ths onstrant. Model 1 data freq_data; set nput; FREQ_ADJ CLAIM_COUNT / EXPOSURE_ADJ; offset_fator 1; f DRIVER_AGE 3 then offset_fator1.05; f DRIVER_AGE 4 then offset_fator1.25; logoffsetlog(offset_fator; run; f DRIVER_AGE_NEW n (1,2 then DRIVER_AGE_NEW DRIVER_AGE; else DRIVER_AGE_NEW 99; pro genmod datafreq_data; Casualty Atuaral Soety E-Forum, Wnter 2009 372
Applatons of the Offset n Property-Casualty Predtve Modelng run; lass DRIVER_AGE_NEW; weght EXPOSURE_ADJ; model FREQ_ADJ DRIVER_AGE_NEW MULTICAR / dstposson lnklog offset logoffset; Table 1 Model 1 Output Varable varable value beta e beta DRIVER_AGE_NEW 1 0.75 2.11 DRIVER_AGE_NEW 2 0.65 1.91 DRIVER_AGE_NEW 99 0.00 1.00 MULTICAR 1-0.27 0.77 MULTICAR 0 0.00 1.00 In ths example, we onstran DRIVER_AGE by lettng the offset take on the onstraned values for age groups 3 and 4; and the 1.0 for the other age groups. At the same tme, we re-ode the age group values 3 and 4 to the value 99 to ensure that the model parameters for these levels wll be 0. (Ths s a SAS trk: SAS treats the hghest value of a ategoral value as the base ategory. Therefore, the model estmates beta parameters for age groups 1 and 2, as well as the mult-ar ndator, subjet to the onstrant that age groups 3 and 4 must have relatvtes of 1.05 and 1.25, respetvely. The fnal relatvty for eah level of DRVER_AGE and MULTICAR wll by exp(beta log_offset e beta *offset. The fnal ratng relatvtes are dsplayed below. Table 2 Combnng Model 1 Parameters wth Offset Values Varable varable value model beta e beta offset fnal relatvty DRIVER_AGE 1 0.75 2.11 1 2.11 DRIVER_AGE 2 0.65 1.91 1 1.91 DRIVER_AGE 3 0.00 1.00 1.05 1.05 DRIVER_AGE 4 0.00 1.00 1.25 1.25 MULTICAR 1-0.27 0.77 1 0.77 MULTICAR 0 0.00 1.00 1 1.00 Construton of a Cross-overage Ter Sore In many nsurane ratng plans, a ter struture s a ratng omponent that s layered on top of a lass plan. In most ases, ter prng s appled on a poly level aross overages. The purpose of ratng ters s to nlude n the prng proess further varables suh as personal redt sore or not-at-fault adents whh are not part of standard lass plans. Ratng ters an also be used to apture nteraton effets between the lass plan varables, suh as the nteraton between drvng reord and drver age, whh are not fully refleted due to the lmtaton of prng strutures. In the next example we llustrate how an offset tehnque an be used to reate a ross-overage ter struture. Suppose we wsh to add a ter struture to an exstng standard personal auto lass plan wth two overages: property damage lablty (PD and omprehensve (Comp. The ters are to be Casualty Atuaral Soety E-Forum, Wnter 2009 373
Applatons of the Offset n Property-Casualty Predtve Modelng omprsed of two fators: number of poly-level not-at-fault adents n the past 3 years (NAF and redt sore (CREDIT. The ter struture and the ter sore are requred to be the same aross the two overages. Suppose we start wth two separate data fles, one for PD lablty and one for Comp. Table 3 shows some sample reords of the two data fles on the exposure/vehle level. Note that the PD and Comp adjusted exposures were alulated usng the log desrbed n the seton of Exposure Adjustments and Offset. Spefally, the PD lablty adjusted exposure s the unadjusted exposure multpled by the orrespondng terrtory fators; and the Comp adjusted exposure s the unadjusted exposure multpled by the orrespondng fators for terrtory, vehle symbol and dedutble. Table 3 Sample Reords from PD Dataset Credt Sore Group Poly Level NAF Count Current Plan Ratng Fator Adjusted Pure Premum Poly Number Vehle Number Adjusted Exposure Inurred Loss 00003 1 2 1 0.41 0.73 0 0 00004 1 0 0 1.63 1.46 1664 1143 00005 1 0 0 0.58 1.25 0 0 00006 1 2 0 0.61 0.52 0 0 00007 1 1 0 1.12 1.25 1344 1077 Sample Reords from Comp Dataset Credt Sore Group Poly Level NAF Count Current Plan Ratng Fator Adjusted Pure Premum Poly Number Vehle Number Adjusted Exposure Inurred Loss 00003 1 2 1 1.38 1.07 0 0 00004 1 0 0 0.79 1.60 495 309 00005 1 0 0 1.05 1.51 566 375 Our frst step s to smply stak these two datasets together, addng a overage ndator to dentfy whether the reord s PD vs. Comp. Table 4 PD and Comp Combned Dataset Credt Sore Group Poly Level NAF Count Current Plan Ratng Fator Adjusted Pure Premum Poly Number Vehle Number PD_IND Adjusted Exposure Inurred Loss 00003 1 1 2 1 0.41 0.73 0 0 00004 1 1 0 0 1.63 1.46 1664 1143 00005 1 1 0 0 0.58 1.25 0 0 00006 1 1 2 0 0.61 0.52 0 0 00007 1 1 1 0 1.12 1.25 1344 1077 00003 1 0 2 1 1.38 1.07 0 0 00004 1 0 0 0 0.79 1.60 495 309 00005 1 0 0 0 1.05 1.51 566 375 Casualty Atuaral Soety E-Forum, Wnter 2009 374
Applatons of the Offset n Property-Casualty Predtve Modelng The followng GLM an be used to estmate the parameters for NAF and Credt: Model 2 Input Dataset: Target Varable: Predtve Varables: Dstrbuton: Lnk: Offset: Weght: Staked dataset Pure Premum (loss / exposure_adj; Credt, NAF Tweede Log PD_Relatvty*β 1 β 2 β p (produt of exstng ratng plan fators exposure_adj; In the above model spefaton, we are usng the offset to reflet the exstng ratng plan fators. We must also aount for the varaton n Pure Premum between the two overages: learly we expet a hgher Pure Premum for PD reords than Comp reords. Not nludng a PD ndator n the model desgn would lead to a partularly egregous example of omtted varable bas. In the above model desgn, we hoose to nlude the PD relatvty as an offset fator along wth the ratng plan fators other than redt sore and not-at-fault adent ount. Note that other model desgns are possble. For example, t would also be possble to nlude the PD relatvty as part of the exposure adjustment step. Ether way, we must perform a prelmnary analyss to estmate the Pure Premum relatvty for PD vs. Comp, and nlude ths relatvty ether as part of the offset or the exposure adjustment step. Beause our target varable n ths example s Pure Premum, the Tweede s an approprate hoe of dstrbutons. Ths has been dsussed extensvely n the atuaral lterature [4,6], so we wll revew ths top only brefly. For lam ount (or frequeny modelng, t s ustomary to assume that the varane of the target varable s proportonal to the mean: V(µφµ. Ths s the Posson model desgn used n the prevous examples. For severty modelng, t s ustomary to assume that the varane s proportonal to the square of the mean: V(µφµ 2. Ths s known as a Gamma model desgn. Pure Premum s the sum of a (Posson dstrbuted random number of (Gamma dstrbuted szes of loss. It s a onvenent mathematal fat that the varane of ths target varable s proportonal to the mean rased to a power between 1 and 2, pє(1,2: V(µφµ p. Ths model desgn s also exponental famly, and s known as the Tweede. Unfortunately, the ommonly used SAS statstal pakage does not automatally support the Tweede model n the GENMOD GLM modelng proedure. One alternatve to GENMOD s to ft Tweede models usng the NLMIXED proedure. Detals of ths are gven n Appendx C. Table 5 shows the parameter estmates from the above Tweede model. The PD relatvty used n the offset s 3.44. Casualty Atuaral Soety E-Forum, Wnter 2009 375
Applatons of the Offset n Property-Casualty Predtve Modelng Table 5 GLM Output and Pure Premum Relatvtes Parameter Estmate Pure Premum Relatvty redt_grp_0 1.09 2.96 redt_grp_1 1.23 3.44 redt_grp_2 0.74 2.10 redt_grp_3-0.14 1.15 redt_grp_4 0.00 1.00 naf_pol_0-0.15 0.86 naf_pol_1-0.03 0.97 naf_pol_2 0.00 1.00 Thus the ter sore for a poly wth NAF1 and Credt2, for example, s exp(0.74-0.032.03. Please note that the PD ndator s not used to alulate the ter fator. Sequental Modelng The prevous two examples, buldng a redt sore and a terng struture on top of an exstng ratng plan, may be thought of as exerses n sequental modelng. By sequental modelng we mean buldng a model to aount for varaton not already explaned by a pre-spefed model. The pre-spefed model (the exstng ratng plan n the above examples n other words serves an as offset when buldng the seond model. Sequental modelng tehnques have a wde range of applatons. As noted above, the frst two examples estmatng ratng plan fators after Terrtory and Symbol fators have been determned n a separate analyss; and buldng a redt sorng model on top of an exstng ratng plan are examples of sequental modelng. Sequental modelng an also be useful for regulatory omplane. For example, Calforna s Proposton 103 requres that safety/drvng reord and mleage be the greatest determnants of auto premums. Insurers typally use sequental methods when developng ther rates n Calforna. We wll gve one fnal example of sequental modelng before losng the paper. In ths example, we wll estmate frst the man effets of a ratng plan and then an nteraton term n sequental fashon. There an be many motvatons for sequental modelng strateges suh as the one exemplfed here. For example, perhaps the nteraton fators wll be used only n ertan states; but the man effet fators are desred to be ommon aross all states. Sequental modelng usng an offset would be a pratal way to approah suh a stuaton. Another motvaton mght be that one wshes to keep the man effets model smple, wthout the omplaton of estmatng an nteraton term n the same step. In ths fnal example, we suppose we are modelng PD pure premum usng the three ratng varables: drver age group, multar ndator, and pleasure use ndator. We wll buld an ntal GLM model for these three man effets. We wll next buld a seond model usng the frst model sore as an offset to estmate the fators for a drver age/pleasure use nteraton term. As dsussed above, the man effets ratng plan mght be used natonally; the addtonal nteraton fators mght be mplemented n seleted states. Casualty Atuaral Soety E-Forum, Wnter 2009 376
Applatons of the Offset n Property-Casualty Predtve Modelng Model 3 Target Varable: Predtve Varables: Dstrbuton: Lnk: Offset: Weght: PD Pure Premum (pd loss / pd exposure_adj; DRIVER_AGE, MULTICAR, PLEASURE_USAGE Tweede Log (none exposure_adj; The ratng fators resultng from ths model are dsplayed n the table below. Table 6 Model 3 Parameter Estmates and Pure Premum Relatvtes Varable Value Beta e beta MULTICAR 1-0.26 0.77 MULTICAR 0 0.00 1.00 DRIVER_AGE 1 0.37 1.45 DRIVER_AGE 2 0.04 1.04 DRIVER_AGE 3-0.83 0.44 DRIVER_AGE 4 0.00 1.00 PLEASURE 1-0.36 0.70 PLEASURE 0 0.00 1.00 Let η denote the lnear omponent of the sorng formula orrespondng to the table above: η β DRIVER_AGE β MULTICAR β PLEASURE. We wll use η as the offset n the model for Step II of the sequene. Model 4 Target Varable: Predtve Varables: Dstrbuton: Lnk: Offset: Weght: PD Pure Premum (pd loss / pd exposure_adj; DRIVER_AGE * PLEASURE Tweede Log η exposure_adj; Model 4 dffers from Model 3 only n the hoe of predtve varables; and the fat that we re usng the lnear omponent of the Model 3 sorng formula (ETA as the offset. Note although exp(η s Model 3 s estmate of PD Pure Premum, we are usng η, not exp(η, as the offset n Model 4 (below. Ths s beause we are buldng a multplatve model (usng the log lnk funton. Therefore the offset must be on the log sale. Table 7 dsplays the ratng fators resultng from Model 4. Casualty Atuaral Soety E-Forum, Wnter 2009 377
Applatons of the Offset n Property-Casualty Predtve Modelng Table 7 Model 4 Parameter Estmates and Pure Premum Relatvtes DRIVER_AGE PLEASURE Model 3 Estmates Pure Premum Relatvty 1 1 0.54 1.72 1 0 0.63 1.88 2 1 0.45 1.57 2 0 0.78 2.18 3 1 0.65 1.92 3 0 0.75 2.12 4 1-0.05 0.95 4 0 0.00 1.00 In states for whh Model 4 s nteraton fators are not used, the fators n Table 6 onsttute the ratng plan. In states for whh the nteraton s ntended to be used, we must ntegrate the results of tables 6 and 7. Ths s done n tables 8 and 9: Table 8 Pure Premum Relatvtes for Type Varable Value Relatvty MULTICAR 1 0.77 MULTICAR 0 1.00 Table 9 Pure Premum Relatvtes for DRIVER_AGE and PLEASURE PLEASURE DRIVER AGE PLEASURE1 PLEASURE0 1 1.74 2.71 2 1.63 2.26 3 0.71 0.97 4 0.90 1.00 Conluson The GLM offset feature s a pratal and versatle tool for dealng wth a varety of ssues suh as: data onstrants, redblty ssues (as n Symbol fator development, regulatory onsderatons (e.g. Calforna s Proposton 103, the desre to layer a further ratng, sorng, or ter model on top of an exstng ratng plan (redt sorng, ter fator development, and the need to add state-spef varatons to a bas ountrywde ratng plan (sequental modelng. Generally speakng, the offset opton s helpful when omtted varable bas [OVB] threatens to dstort one or more model parameters. The lass use of an offset s to norporate a measure of exposure when modelng rates. For example f some reords n a personal auto dataset orrespond to 6-month poles whle other reords orrespond to 12-month poles, then t s approprate to use (log of months of exposure as an offset. Falure to do so would rase the speter of OVB: model varables orrelated wth months of exposure mght possbly pk up some of the varaton that should be explaned by months of exposure. Ths would result n based parameter estmates. Beyond ths lassal use, the offset opton s helpful n a number of atuaral applatons. For example, we have desrbed how the offset opton an be used to buld GLM models subjet to Casualty Atuaral Soety E-Forum, Wnter 2009 378
Applatons of the Offset n Property-Casualty Predtve Modelng ertan ratng fator onstrants; to optmze the ratng fators of some, but not all, of the varables n a ratng plan; and to buld predtve or ratng models n sequental fashon. We dsussed redt sorng, ter varable reaton, and state-exepton sub-models as examples of atuaral prng models bult n sequental fashon. The offset opton provdes atuares wth a unfyng framework enompassng suh tradtonal tehnques as exposure adjustments and loss rato modelng as an alternatve to pure premum modelng for avodng omtted varable bas. It s therefore approprate to onsder usng an offset when performng a multvarate analyss subjet to varable exlusons or other a pror onstrants. Casualty Atuaral Soety E-Forum, Wnter 2009 379
Applatons of the Offset n Property-Casualty Predtve Modelng Referenes [1] MClenahan, C. L., Ratemakng, Foundatons of Casualty Atuaral Sene, Casualty Atuaral Soety, 1990, 25-90. [2] Stenmark, A. J., and C. P. Wu, Smpson s Paradox, Confoundng Varables, and Insurane Ratemakng, Proeedngs of Casualty Atuaral Soety, 2004, Vol. XCI, 133-198 [3] Nelder, John A., and R. W. M. Wedderburn, Generalzed Lnear Models, Journal of the Royal Statstal Soety, Seres A, 1972, Vol. 135, No. 3, 370-384. [4] Mldenhall, S. J., A Systemat Relatonshp Between Mnmum Bas and Generalzed Lnear Models, Proeedngs of Casualty Atuaral Soety, 1999, Vol. LXXXVI, 393-487. [5] Brokman, M. J., and T. S. Wrght, Statstal Motor Ratng: Makng Effetve Use of Your Data, Journal of the Insttute of Atuares, 1992, Vol. 119, Part III, 457-526. [6] Anderson, D., S. Feldblum, C. Modln, D. Shrmaher, E. Shrmaher, and N. Thand, "A Prattoner's Gude to Generalzed Lnear Models", Casualty Atuaral Soety Dsusson Paper Program, 2004, 1-116. [7] Fu, L., and C. P. Wu, Generalzed Mnmum Based Models, Casualty Atuaral Soety Forum, Wnter 2005, 72-121. [8] Hlbe, J., Generalzed Lnear Models and Extensons, College Staton, TX: Stata Press, 2001. [9] Monaghan, J. E., The Impat of Personal Credt Hstory on Loss Performane n Personal Lnes, Casualty Atuaral Soety Forum, Wnter 2000, 79-105. [10] Credt Reports and Insurane Underwrtng, NAIC Whte Paper, 1997, Kansas Cty, MO: Natonal Assoaton of Insurane Commssoners. [11] Murphy, K.P., M. J. Brokman, and P. K. W. Lee, Usng Generalzed Lnear Models to Buld Dynam Prng Systems, Casualty Atuaral Soety Forum, Wnter 2000, 107-139. [12] Werner, G., and S. Guven, GLM Bas Modelng: Avodng Common Ptfalls, Casualty Atuaral Soety Forum, Wnter 2007, 257-272. [13] Dunn, Peter K., Ourrene and Quantty of Preptaton an be Modeled Smultaneously, Internatonal Journal of Clmatology, 2004, Vol. 24, No. 10, 1234-1239, http://www.s.usq.edu.au/staff/dunn/researh.html [14] Smyth, Gordon K., and Bent Jørgensen, Fttng Tweede's Compound Posson Model to Insurane Clams Data:Dsperson Modellng, ASTIN Bulletn, 2002, Vol. 32, No. 1, 143-157, http://www.stats.org/smyth/pubs/nsuran.pdf Casualty Atuaral Soety E-Forum, Wnter 2009 380
Applatons of the Offset n Property-Casualty Predtve Modelng Casualty Atuaral Soety E-Forum, Wnter 2009 381 Appendx A: Two Equvalent Ways of Modelng Frequeny wth Posson Regresson Suppose we wsh to model lam frequeny F as a generalzed lnear funton of several ovarates {X 1, X 2,, X N }. Let C denote the number of lams for a gven poly, and u (for unts denote number of exposures. Then: FC/u. We wll demonstrate that the followng two ways of modelng F are equvalent: Opton 1 Opton 2 GLM famly: Posson Posson Target: C F Weght: (none u Offset: log(u (none Let us start wth Opton 1 and demonstrate that t s equvalent to Opton 2. Let denote the observaton number. The Posson regresson assumpton s that:! ( e C P λ λ Where... exp(log( 2 2 1 1 N N X X X u β β β α λ Note that f we let:... exp( 2 2 1 1 N N X X X β β β α μ Then we have the relatonshp: λ u*μ. The log-lkelhood funton for the Opton 1 Posson regresson model s: κ λ λ λ β α λ log(! log(, ( e l (For smplty we are assumng that over-dsperson does not exst n the data. That s, φ1. We an reast the above expresson n terms of F and μ: ' ' log( ' log( ( ' log( / ( ' log( log(, ( κ μ κ μ μ κ μ μ κ μ μ κ μ μ β α μ f e u f u u u u u u l In the above expressons, {κ, κ, κ } denote onstants that do not depend on the model parameters. Ths last expresson s the log-lkelhood funton for the Posson regresson, ast n the terms Opton 2.
Applatons of the Offset n Property-Casualty Predtve Modelng Appendx B: Detals of the Dataset Used n Examples The data used n ths paper was smulated by Delotte Consultng usng a typal prvate passenger auto (PPA ratng struture. The data onssts of 50,000 vehle-level reords orrespondng to 24,993 sngle-ar poles and 11,038 mult-ar poles. Two overages, Property Damage lablty (PD and Comprehensve (Comp, were smulated for eah vehle. By onstruton, 50% of the vehles have exposures n both overages, whle the other 50% of the vehles have PD exposure. The followng ratng varables were smulated for eah vehle reord: Multar ndator {0,1} 0 sngle ar 1 mult Car Poly age {0,1,2,,15} Drver age group {1,2,3,4} Pleasure use ndator {0,1} 1 Pleasure Use 0 Not Pleasure Use Credt sore group {0,1,2,3,4} Terrtory {T1, T2, T3, T4} Vehle symbol {1,2,3,4,5} Poly-level at fault adents {0,1,2} Poly-level not at fault adents {0,1,2} All of these varables are treated as ategoral varables n the examples desrbed n the body of ths paper. The followng target felds were also smulated for eah vehle reord: PD nurred loss, PD lam ount (7,414 lams, or 14%, PD exposure, Comp nurred loss, Comp lam ount (6,143 lams, or 12.2% and Comp exposure. Casualty Atuaral Soety E-Forum, Wnter 2009 382
Applatons of the Offset n Property-Casualty Predtve Modelng Appendx C: The Tweede Compound Posson Model and Correspondng SAS Code Matthew Flynn, Ph.D. Followng Smyth & Jorgenson [14], seton 4.1, page 11, the Tweede Compound Posson jont lkelhood funton as: wth ω f, ϕ a ( n, y; ϕ w, ρ a( n, y; ϕ w, p exp t( y, μ p ( n, y; ϕ w, p α 1 n α ( w ϕ y 1 α ( p 1 ( 2 p n! Γ( nα y where α (2-p/(p-1 and. t ( y, μ, p 1 p 2 p μ μ y 1 p 2 p 1 p 2 p μ μ t( y, μ, p y 1 p 2 p. The SAS odes usng Pro NLMIXED to ft the above lkelhood funton for the ross overage terng sore example n the paper s gven as follows: pro nlmxed datathe_appended_dataset; parms p1.5; bounds 1<p<2; eta_mu b0 1*(redt_grp12*(redt_grp23*(redt_grp34*(redt_grp4 naf1*(naf_pol1naf2*(naf_pol2 overage_comp*(overage COMP ; mu exp(eta_mu urrent_fator; eta_ph ph0 ph_1*(redt_grp1 ph_2*(redt_grp2 ph_3*(redt_grp3 ph_4*(redt_grp4 ph_naf1*(naf_pol1 ph_naf2*(naf_pol2 ph_overage_comp*(overage COMP ; ph exp(eta_ph; n lams; w nsured; y pp; t ((y*mu**(1 - p/(1 - p - ((mu**(2 - p/(2 - p; a (2 - p/(p - 1; f (n 0 then loglke (w/ph*t; else loglke n*((a 1*log(w/ph a*log(y - a*log(p - 1 - log(2 - p - lgamma(n 1 - lgamma(n*a - log(y (w/ph*t; model y ~ general(loglke; replate adjexp; estmate 'p' p; run; Casualty Atuaral Soety E-Forum, Wnter 2009 383
Applatons of the Offset n Property-Casualty Predtve Modelng The above odes an be broken down nto the followng major setons: Frst we all the Pro NlMIXED, addressng the desred nput dataset: pro nlmxed datathe_appended_dataset; The PARMS statement provdes a startng value for the algorthm s parameter searh. Multple startng values are allowed, as well as nput from datasets (from pror model runs, for example. Wth some doman knowledge we antpate ths parameter to be n the neghborhood of 1.5. parms p1.5; Parameters an also be easly restrted to ranges, suh as to be postve, and here we requre the estmated Tweede power parameter to fall between one and two. bounds 1<p<2; Next we spefy the lnear model/predtor for the mean response. Pro NLMIXED does not have the onvenent CLASS statement of some of the other regresson routnes, lke Pro GENMOD or Pro LOGISTIC. However, the desgn matrx an be reated on-thefly, so to speak, by effetvely nludng programmng statements n the Pro NLMIXED ode. Here, we reate dummy varables by odng the lnear model wth logal statements. For example, the phrase, (redt_grp1 resolves to ether true (1 or false (0 at runtme, reatng our desred ndator varables to test dsrete levels of rght-hand sde varables. As a remnder, for a GLM, the lnear predtor s requred to be lnear n the estmated parameters, so non-lnear effets suh as hgh powers of ovarates or splnes an be aommodated. eta_mu b0 1*(redt_grp12*(redt_grp23*(redt_grp34*(redt_grp4 naf1*(naf_pol1naf2*(naf_pol2 overage_comp*(overage COMP ; Next we reate a log lnk that maps the lnear predtor to the mean response. That log lnk on the left hand sde, beomes an exponental as the nverse lnk (on the rght-hand sde. mu exp(eta_mu urrent_fator; A great feature of usng Pro NLMIXED s ts flexblty. Here we are spefyng what Smyth & Jorgenson [13] refer to as a double GLM. Instead of a sngle onstant dsperson onstant, we an ft an entre seond lnear model wth log lnk for the dsperson fator. eta_ph ph0 ph_1*(redt_grp1 ph_2*(redt_grp2 ph_3*(redt_grp3 ph_4*(redt_grp4 ph_naf1*(naf_pol1 ph_naf2*(naf_pol2 ph_overage_comp*(overage COMP ; ph exp(eta_ph; Casualty Atuaral Soety E-Forum, Wnter 2009 384
Applatons of the Offset n Property-Casualty Predtve Modelng Pro NLMIXED allows a number of datastep style programmng statements. Here we are assgnng nput dataset varables lams, nsured, and pp as new varables (n, w and y to be used subsequently n buldng out our lkelhood equaton. That way, one an easly adapt pre-exstng ode to a partular nput dataset, wthout requrng modfatons to the guts of the log-lkelhood equaton (t s omplated enough already. n lams; w nsured; y pp; Now one an begn to spefy the loglkelhood. Here, for larty, we buld t out n several steps. Smply refer to the Tweede Compound Posson lkelhood desrbed above from Smyth & Jorgensen [13], and lay t out. t ((y*mu**(1 - p/(1 - p - ((mu**(2 - p/(2 - p; a (2 - p/(p - 1; f (n 0 then loglke (w/ph*t; else loglke n*((a 1*log(w/ph a*log(y - a*log(p - 1 - log(2 - p - lgamma(n 1 - lgamma(n*a - log(y (w/ph*t; Pro NLMIXED nludes several pre-spefed lkelhoods, for example, Posson and Gamma, the GENERAL spefaton allows the great flexblty to spefy one s desred model spefaton. model y ~ general(loglke; Weghts an ether be nluded dretly n the loglkelhood above, or wth the handy REPLICATE statement. Eah nput reord n the dataset represents an amount represented by the nput varable adjexp. replate adjexp; The ESTIMATE statement an easly alulate and report a varety of desred statsts from one s model estmaton. Here, we are nterested n the Tweede Power parameter. estmate 'p' p; Wthout usng any of the addtonal Mxed modelng power, Pro NLMIXED performs as a great Maxmum Lkelhood Estmator usng a varety of numer ntegraton tehnques. Casualty Atuaral Soety E-Forum, Wnter 2009 385