General Iteration Algorithm for Classification Ratemaking

Transcription

1 General Iteraton Algorthm for Classfcaton Ratemakng by Luyang Fu and Cheng-sheng eter Wu ABSTRACT In ths study, we propose a flexble and comprehensve teraton algorthm called general teraton algorthm (GIA) to model nsurance ratemakng data. The teraton algorthm s a generalzaton of a decades-old teraton approach known as mnmum bas models. We wll demonstrate how to use GIA to solve all the multplcatve mnmum bas models publshed to date and the commonly used multplcatve generalzed lnear models (GLMs), such as gamma, osson, normal, and nverse Gaussan models. In addton, we wll demonstrate how to apply GIA to solve the broad range of GLM models, mxed addtve and multplcatve models, and constrant-optmzaton problems that prcng actuares often deal wth n ther practcal work. KEYWORDS GIA, GLM, classfcaton ratemakng, weghted average 193

2 Varance Advancng the Scence of Rsk 1. Introducton Insurance ratng for property and casualty lnes of busness went through a great expanson after World War II. The expanson lad down the foundaton for modern ratng plans, whch are farly complex n that they typcally consst of a wde range of ratng factors. However, t also created a sgnfcant challenge for the nsurance ndustry n how to determne the optmal values for each ratng varable n the plan. For example, a typcal personal automoble ratng plan contans garage terrtory, drver age, drver gender, drver martal status, vehcle usage, drvng dstance, vehcle model year, vehcle symbols, drver hstory (accdents and volatons), and a number of specal credts and debts such as mult-car dscounts, drvng school dscounts, and good student dscounts. To respond to the challenge of ratng plan expanson, Baley and Smon [2] and Baley [1] proposed a heurstc teraton approach called mnmum bas models, whch utlzes an teratve procedure n determnng smultaneously the optmal values for the ratng varables. Durng teraton, the procedure wll mnmze a target bas functon. Compared to the tradtonal one-way or two-way analyss, such multvarate procedures can reduce estmaton errors. Untl recent nterest n generalzed lnear models (GLMs), the mnmum bas approach was the maor technque used by property and casualty prcng actuares n determnng the rate relatvtes for a class plan wth multple ratng varables. We wll llustrate how the mnmum bas approach can be used to derve ndcated class plan factors. Because multplcatve models are more popular than addtve ones, we wll focus frst and prmarly on multplcatve models. Later, we wll also llustrate how to generalze the approach by developng addtve and mxed addtvemultplcatve models. Assume that we are conductng a two-varable (X and Y) ratng plan analyss based on loss cost. Varable X has a total of m categores of values, varable Y has a total of n categores of values, and the categores are represented by the subscrpt of (from 1,2,:::,m) and (from 1,2, :::,n). Defne r, as the observed loss cost relatvty, and w, as the earned exposures or weght for the classfcaton and for varables X and Y, respectvely, and let x and y be the relatvtes for classfcaton and classfcaton, respectvely. The multplcatve ratng plan proposed by Baley [1] s: E(r, )=x y, where =1,2,:::,m and =1,2,:::,n: Wth the above multplcatve formula, one type of error proposed by Baley s to measure the dfference between the estmated cost and the observed cost. The errors across the varable Y are n =1 w, (r, x y )for =1,2,:::,m. The errors across the varable X are m =1 w, (r, x y )for =1,2,:::,n. When the above errors are set to zero for every X and Y, t can be shown that the estmated relatvtes, ˆx, ŷ, can be derved teratvely as follows: Algorthm 1: ˆx = ŷ = w, r, w, y w, r, w, x : (1.1) Strctly speakng, t s somewhat msleadng to descrbe Baley s approach as mnmum bas models. Frst, what Baley proposed s essentally an teraton algorthm, not a set of statstcal models. The teratve procedure s a fxed pont teraton technque commonly employed n numerc analyss for root fndng. Second, the error functon above s not consstent wth the bas concept n statstcs. Bas generally refers to the dfference between the mean of an estmator and thetruevalueoftheparameterbengestmated. For example, suppose we are tryng to estmate 194

3 General Iteraton Algorthm for Classfcaton Ratemakng relatvty x usng an estmator ˆx (whch s some functon of observed data). Then the bas s defned as E(ˆx ) x.ife(ˆx ) x =0, ˆx s called an unbased estmator of x. Although Algorthm 1 does not measure the bas of the mean estmated relatvty from the true value, the approach has long been recognzed as the mnmum bas method by actuares. In fact, t s essentally a cross-classfcaton estmaton algorthm. In ths paper, we descrbe our new and generalzed approach, the general teraton algorthm (GIA), whch has greater statstcal rgor. Brown [3] was the frst one to ntroduce statstcal models and lnk Baley and Smon s mnmum bas approach to the maxmum lkelhood estmatons of the statstcal theores: L, = Br, = Bx y + ",, where L, s the observed loss cost, B s the base, ", s a random error, and L, follows a statstcal dstrbuton. Returnng to Algorthm 1, t can be proven that Algorthm 1 s equvalent to applyng the maxmum lkelhood (ML) method wth an assumpton that L, follows a osson dstrbuton. Therefore, the results from Algorthm 1 are the same as those from the ML osson model. Wth the ntroducton of statstcal theores and statstcal models to the mnmum bas approach, Brown further expanded the approach wth four more mnmum bas algorthms (three multplcatve and one addtve) by assumng dfferent dstrbutons for L, (or r, ): Algorthm 2: ˆx = 1 X r, : (1.2) n y Algorthm 2 assumes that L, follows an exponental dstrbuton. Algorthm 3: w, 2 ˆx = r,y : (1.3) w2, y2 Algorthm 3 s equvalent to an ML normal model. Algorthm 4: w, r, y ˆx = w : (1.4), y2 Algorthm 4 results from the least squares model. Another mnmum bas algorthm proposed by Baley and Smon [2] has a complcated format: Algorthm 5: Ã w, r 2! 1=2, ˆx = y 1 w, y : (1.5) Feldblum and Brosus [5] summarzed these mnmum bas algorthms nto four categores: balance prncple, maxmum lkelhood, least squares, and Â-squared. ² Algorthm 1 could be derved from the socalled balance prncple, that s, the sum of the ndcated relatvty = the sum of observed relatvty. Such a balance relatonshp can be formulated as: X w, r, = X w, x y : ² Algorthms 1, 2, and 3 can be derved from the assocated log lkelhood functons of observed pure premum relatvtes. ² Algorthm 4 can be derved by mnmzng the sum of the squared errors: Mn x,y X w, (r, x y ) 2 :, ² Algorthm 5 can be derved by mnmzng the Â-squared error, the squared error dvded by the ndcated relatvty: Mn x,y X, w, (r, x y ) 2 x y : In hs mlestone paper, Mldenhall [9] further demonstrated that classfcaton rates determned by varous lnear bas functons are essentally thesameasthosefromglms.onemanadvantage of usng statstcal models such as GLM 195

4 Varance Advancng the Scence of Rsk s that the characterstcs of the models, such as the parameters confdence ntervals and hypothess testng, can be thoroughly studed and determned by statstcal theores. Also, the contrbuton and sgnfcance of the varables n the models can be statstcally evaluated. Another advantage s that GLMs are more effcent because they do not requre actuares to program the teratve process n determnng the parameters. 1 However, ths advantage can be dscounted somewhat due to the powerful calculaton capablty of modern computers. Due to these advantages, GLMs have become more popular n recent years. Of course, actuares need to acqure the necessary statstcal knowledge n understandng and applyng the GLMs and rely on specfc statstcal modelng tools or software. On the other hand, we beleve that the formats and the procedures for the mnmum bas types of teraton algorthms are smple and straghtforward. The approach s based on a target or error functon along wth an teratve procedure to mnmze the functon wthout dstrbuton assumptons. Actuares have been usng the approach for many decades. So, compared to GLM, some advantages of the teraton approach are that t s easy to understand; easy to use; easy to program usng many dfferent software tools (for example, an Excel spreadsheet); and does not requre advanced statstcal knowledge, such as maxmum lkelhood estmatons and devance functons of GLM. One ssue assocated wth most prevous work on the mnmum bas approach and GLM s the model-selecton lmtaton. GLMs assume the underlyng dstrbutons are from the exponental famly. Also, commonly used statstcal software typcally provdes lmted selecton of GLM dstrbutons, such as osson, Gamma, normal, neg- atve bnomal, and nverse Gaussan. On the other hand, only fve types of multplcatve models and four types of addtve models are avalable from prevous mnmum bas work. 2 These lmtatons, we beleve, may reduce estmaton accuracy n practce snce nsurance and actuaral data are rarely perfect and may not ft well the exponental famly of dstrbutons or exstng bas models. In addton, there are two other common and practcal ssues that actuares have to deal wth n ther daly prcng exercses. Frst, many realworldratngplansareessentallyamxedaddtve and multplcatve model. For example, for personal auto prcng, the prmary class plan factor for age, gender, martal status, and vehcle use s often added wth the secondary class plan factor for past accdent and volaton ponts, and then the result s multpled wth other factors. The commonly used GLM software, to our knowledge, does not provde optons that can solve such mxed models because the dentty lnk functon mples an addtve model, whle the log lnk functon mples a multplcatve model. Second, t s possble that usng ether a GLM or a prevous mnmum bas teraton approach wll result n parameters for some varables whch are questonable or unacceptable by the marketplace. One practcal way to deal wth ths ssue s to select or constran the factors for the varables based on busness and compettve reasons whle leavng other factors to be determned by multvarate modelng technques. Snce all the varables are connected n the multvarate analyss, any constraned factors should flow through the analyss, and the constrant wll mpact the results for the other unconstraned factors. We can call ths ssue a constrant optmzaton problem. 1 GLMs may also nvolve an teratve approach. The most commonly used numercal method to solve the GLM s the teratve reweghted least squares algorthm. 2 Feldblum and Brosus [5] lst sx multplcatve mnmum bas models n ther summary table. However, the balance prncple model s the same as the maxmum lkelhood osson model. 196

5 General Iteraton Algorthm for Classfcaton Ratemakng In ths study, we propose a more flexble and comprehensve approach wthn the mnmum bas framework, called a general teraton algorthm (GIA). The key features of GIA are: ² It wll sgnfcantly broaden the assumptons for dstrbutons n use, and, to a certan degree, t totally relaxes any specfc form for the dstrbutons. Therefore, GIA wll be able to provde a much wder array of models from whch actuares may choose. Ths wll ncrease the model-selecton flexblty. ² Its flexblty wll mprove the accuracy and the goodness of ft of classfcaton rates. We wll demonstrate ths result through a case study later. ² Smlar to past mnmum bas approaches, t s easy to understand and does not requre advanced statstcal knowledge. For practcal purposes, GIA users only need to select the target functons and the teraton procedure because the approach s dstrbuton free. ² Whle GIA stll requres the teratve process n determnng the parameters, we beleve that the effort s not sgnfcant wth today s powerful computers. ² It can solve the mxed addtve-multplcatve models and the constrant optmzaton problems. In the followng sectons, we wll frst prove that all fve exstng multplcatve mnmum bas algorthms are specal cases of GIA. We wll also propose several more multplcatve algorthms that actuares may consder for ratemakng based on nsurance data. Then, we wll demonstrate how to apply GIAs to solve the mxed models and constrant optmzaton problems. The numercal analyss of multplcatve and addtve models gven later s based on severty data for prvate passenger auto collson n Mldenhall [9] and McCullagh and Nelder [8]. The results from selected algorthms wll be compared to those from the GLM models. Followng Baley and Smon [2], the weghted absolute bas and the earson ch-square statstc are used to measure the goodness of ft. We also calculate the weghted absolute percentage bas, whch ndcates the magntude of the errors relatve to the predcted values. The remander of ths paper s organzed as follows: ² Secton 2 dscusses the detals of 2-parameter multplcatve, 3-parameter multplcatve, constrant, addtve, and mxed GIA. ² Secton 3 addresses the resdual dagnoss of GIA. ² Secton 4 nvestgates the calculaton effcency of GIA. It shows that GIA could converge rapdly and s not necessarly neffcent n numercal calculatons. ² Secton 5 revews numercal results for two case studes usng multplcatve and mxed models. ² Secton 6 outlnes our conclusons. ² The appendx reports the numercal results for theexamplesdscussednsecton5wthseveral selected multplcatve GIAs. It also shows the teratve convergences of selected multplcatve, addtve, and mxed GIAs. 2. General teraton algorthm (GIA) 2.1. Two-parameter GIAs Followng the notaton used prevously, n the multplcatve framework for two ratng factors, the expected relatvty for cell (, ) should be equal to the product of x and y : E(r, )=¹, = x y : (2.1) By (2.1), there are a total of n alternatve estmates for x and a total of m estmates for y : ˆx, = r, =y, ŷ, = r, =x, =1,2,:::,n =1,2,:::,m: (2.2) 197

6 Varance Advancng the Scence of Rsk Followng actuaral conventon, the fnal estmates of x and y could be calculated by the weghted average of ˆx, and ŷ,.ifweusethe straght average to estmate the relatvty: ˆx = X 1 n ˆx, = 1 n X r, y : (2.3) Smlarly, ŷ = (1=m)ŷ, =(1=m) (r, =x ). Ths s Algorthm 2, the ML exponental model ntroduced by Brown [3]. If the relatvty-adusted exposure, w, ¹,,s used as the weght n determnng the estmates: ˆx = X = Smlarly, w, ¹, w, ¹ ˆx, = X, w, y r, w, y y w, r, w, y : (2.4) ŷ = X w, ¹, w, ¹ ŷ, = X, = w, r, : w, x w, x r, w, x x The resultng model s the same as Algorthm 1, the balance prncple or ML osson model. If the square of the relatvty-adusted exposure, w, 2 ¹2,, s used as the weght: ˆx = X w, 2 ¹2, w, 2 ˆx, = r,y : w2, ¹2, w2, y2 (2.5) The resultng model s the same as Algorthm 3, the ML normal model. If the exposure adusted by the square of relatvty, w, ¹ 2,, s used as the weght: ˆx = X w, ¹ 2, w ˆx, = w, r, y, ¹2, w :, y2 (2.6) The resultng model s the same as Algorthm 4, the least-squares model. From the above results, we propose the 2-parameter GIA approach by usng w p, ¹q, as the weghts for the bas functon: 2-arameter GIA: When ˆx = X w p, ¹q, ˆx, = wp, ¹q, wp, r, yq 1 : wp, yq (2.7) ² p = q = 0, t s the ML exponental model, Algorthm 2; ² p = q = 1, t s the ML osson model, Algorthm 1; ² p = q = 2, t s the ML normal model, Algorthm 3 ² p =1 and q = 2, t s the least-squares model, Algorthm 4. In addton, there are two more models that correspond to GLM wth the exponental famly of gamma and nverse Gaussan dstrbutons. 3 When the exposure s used as the weghts, that s, p =1andq = 0, the GIA wll lead to a GLM gamma model and becomes: Algorthm 6: ˆx = X w, w ˆx, =, w, r, y 1 w : (2.8), When p =1 and q = 1, the GIA leads to a GLM nverse Gaussan model and becomes: Algorthm 7: ˆx = X w, y 1 w, r, =y 2 w ˆx, =, y 1 w, =y : (2.9) Equaton (2.7) suggests that n theory there s no lmtaton for the values of p and q that can be used and they can take on any real values. It s wth ths feature that GIA should greatly 3 For detaled nformaton, please refer to Secton 7 of Mldenhall [9]. 198

7 General Iteraton Algorthm for Classfcaton Ratemakng enhance the flexblty for actuares when they apply the algorthm to ft ther data. Of course, n realty we do not expect that extreme values for p and q wll be found useful. In ratemakng applcatons, earned premum could be used f exposure s not avalable. Normalzed premum (premum dvded by relatvty) s a reasonable opton for the weght. Ths suggests that q could be negatve. In general, p should be postve: the more exposure/clams/premum, the more weght assgned Three-parameter GIAs So far, we have used the 2-parameter GIA n Equaton (2.7) to represent several commonly used models, Algorthms 1 to 4, but not Algorthm 5, the Â-squared multplcatve model. In order to represent Algorthm 5, we further expand the 2-parameter GIA to a 3-parameter GIA usng the lnk functon concept from GLM. One generalzaton of GLMs as compared to a more basc lnear model s done by ntroducng a lnk functon to lnk the lnear predctor to the response varable. Smlarly, we ntroduce a relatvty lnk functon to lnk the GIA estmate to the relatvty. The proposed relatvty lnk functon s dfferent n several aspects from the lnk functon n GLMs. In GLMs, the lnk functon determnes the type of model: log lnk mples a multplcatve model and dentty lnk mples an addtve model. Ths s not the case for GIA. Multplcatve GIA, for example, could have a log, power, or exponental relatvty lnk functon. For a 3-parameter GIA, nstead of usng (2.2), we estmate the relatvty lnk functons of f(ˆx ) and f(ŷ ) from f(ˆx, )andf(ŷ, ) frst; and then calculate ˆx and ŷ by nvertng the relatvty lnk functon, f 1 (f(ˆx )) and f 1 (f(ŷ )). The functons f(ˆx, )andf(ŷ, ) can be estmated by: Takng the weghted average usng parameters p and q: Ã! r, wp, yq f f(ˆx )= X f(ŷ )= X Thus, w p, ¹q, w p f(ˆx, )=, ¹q, w p, ¹q, f(ŷ, )= wp, ¹q, 0 Ã! 1 r, ˆx = f 1 wp, yq f y w p, yq C A y w p, yq (2.11) wp, xq f μ r, 0 μ 1 w p r, ŷ = f 1, xq f x B A : wp, xq wp, xq x : (2.12) One possble selecton of the relatvty lnk functon s the power functon, f(ˆx )=ˆx k and f(ŷ )=ŷ k. In ths case, equaton (2.12) leads to a 3-parameter GIA: 0 11=k ˆx wp, rk, yq k w p A : (2.13) 4, yq When k =2,p =1,andq = 1, Equaton (2.13) s equvalent to: Ã ˆx = w! 1=2, r2, y 1, (2.14) w, y and ths s Algorthm 5, the Â-squared multplcatve model. Another example of a new teratve algorthm occurs when k =1=2, p =1,andq =1: Algorthm 8: 0 w ˆx r 1=2 12, y1=2 A : (2.15) w, y f(ˆx, )=f(r, =y ), f(ŷ, )=f(r, =x ), =1,2,:::,n =1,2,:::,m: (2.10) 4 There s no unque soluton for these equatons. For one group of solutons, we can dvde each x by a factor and multply each y by the same factor to obtan another group of solutons. To guarantee a unque soluton, we can add a constrant to force the average of the x s to be one. 199

8 Varance Advancng the Scence of Rsk Mldenhall [10] ndcated that the 3-parameter GIA s equvalent to a GLM wth the parameters x k and y k, the weght w p, and the response varable r k followng a dstrbuton wth varance functon Var(¹)=¹ 2 q=k.whenk =1andp =1, we can conclude that: ² when q = 2, the normal GLM model s the same as the GIA Algorthm 4 n Equaton (1.4); ² when q = 1, the osson GLM model s the same as the GIA Algorthm 1 n Equaton (1.1); ² when q = 0, the gamma GLM model s the same as the GIA Algorthm 6 n Equaton (2.8); ² when q = 1, the nverse Gaussan GLM model s the same as the GIA Algorthm 7 n Equaton (2.9). Also, for Â-squared mnmum bas model wth k =2,p =1,andq = 1, the GIA theory ndcates that r 2 follows a Tweede dstrbuton wth a varance functon Var(¹)=¹ 1:5. In actuaral exercses, we often exclude the extremely hgh and low values from the weghted average to yeld more robust results. In the case of several ratng varables, there may be thousands of alternatve estmates. Actuares have the flexblty to use the weghted average wthn selected ranges (e.g., the average wthout the hghest and the lowest 1% percentle). Ths s smlar to the concept of trmmed regresson used wth GLMs whereby observatons wth undue nfluence on a ftted value are removed. Fnally, we would lke to extend GIA to reserve applcatons. Mack [7] dscussed the connecton between ratemakng models of auto nsurance and IBNR reserve calculaton because reserves can be estmated by a ratemakng model wth two ratng varables, accdent year and development year. He showed that the mnmum bas method produces the same result as the chan ladder loss development method. Recently, actuares have appled GLMs to estmate reserves usng the ncremental loss as the response varable. Let, be the ncremental pad loss n accdent year and development year, thats,, s the cell (,) of the ncremental payment trangle. England and Verrall [4] used the followng GLM wth log lnk functon and osson dstrbuton to estmate the expected values of future payments: E(, )=m, and Var(, )=Ám,, log(m, )=C = 1 =0: Several other models were also proposed for reserve estmates. For example, Renshaw and Verrall [11] appled the GLM wth a gamma dstrbuton. The only dfference between the gamma and osson models s that the gamma model s varance functon s Var(, )=Ám, 2. Let ¹, =(m, =m 1,1 ), x = e, and y = e ; then the above GLM reserve models can be smlarly transferred to the GIA multplcatve algorthm by settng ¹, = x y. So GIA can also be used to estmate reserves based on the trangles of ncremental pad loss. When k =1,p =1 and q = 1, GIA yelds the same result as a osson GLM reserve model; when k =1,p =1,and q = 0, GIA produces the same result as a gamma GLM model Constrant GIA In real-world ratemakng applcatons, some factors need to be selected or capped wthn a certan range for busness or compettve reasons. Snce n a multvarate analyss, all the varables are related, other factors should be adusted to reflect the mpact of the subectve selectons. When ths ssue arses, the standard GLM or other approaches may have lmtatons f the selected factors are outsde of the ftted confdence nterval. For example, the mult-car dscount used for prvate passenger auto prcng s typcally be- 200

9 General Iteraton Algorthm for Classfcaton Ratemakng tween 5% and 25%. Any factor outsde ths range s not lkely to be accepted by the market, no matter what the ftted value s for the ndcated dscount. In the followng we wll demonstrate how to apply GIA to solve the ssue. For example, let x 1 and x 2 be the sngle and mult-car factors, respectvely, and we wll cap the mult-car dscount to be between 5% and 25%. The constrant can be represented by 0:75x 1 x 2 0:95x 1. Addng ths constrant to (2.13), we can solve the problem by: ˆx 1 = à ˆx 2 =max wp 1, rk 1, yq k wp 1, yq à 0:75ˆx 1,mn! 1=k, à 0:95ˆx 1, à wp 2, rk 2, yq k wp 2, yq (2.16)!!! Wth the constrant, we can contnue the teraton process untl the values for all other ratng factors converge. Ths flexblty 5 assocated wth GIA wll provde actuares another beneft n dealng wth ther practcal problems Addtve GIA Followng the same notatons as above, the expected cost for classfcaton cell (, ) wth an addtve model should be equal to the sum of x and y : Thus, ˆx, = r, y, ŷ, = r, x, 1=k E(r, )=¹, = x + y : (2.17) =1,2,:::,n =1,2,:::,m: : (2.18) In the multplcatve models, we use the relatvty-adusted exposure, w p, ¹q,, as the weghtng functon and ntroduce the power relatvty lnk functon. However, the weghtng functons and the relatvty lnk functons cannot be appled n an addtve process. For the addtve GIA, we are lmted to the followng one-parameter model usng w p, as the weght: ˆx = X w p, w p, ˆx, = (r, y ) : wp, wp, (2.19) When p = 1, t leads to the model ntroduced by Baley [1] or the Balance rncple model n Feldblum and Brosus [5]. Mldenhall [9] also proved that t s equvalent to an addtve normal GLM model. When p = 2, t leads to the ML addtve normal model ntroduced by Brown [3]. When p = 0, t leads to the least squares model by Feldblum and Brosus [5]. There s no further generalzaton for the addtve GIAs wth addtonal parameters or lnk functons. Except for the exponental famly of dstrbutons, the lognormal dstrbuton s probably the most wdely used dstrbuton n actuaral ractce. If r, follows a lognormal dstrbuton, log(r, ) wll follow a normal dstrbuton and the multplcatve ratng plan can be transformed to log(r, ) = log(x ) + log(y )+",. The addtve GIA algorthms can be used to derve the parameters for the lognormal dstrbuton assumpton Mxed addtve and multplcatve GIAs A smplfed mxed addtve and multplcatve model 7 can be llustrated as follows: r,,h =(x + y ) z h + ",,h, (2.20) where =1,2,:::,m; =1,2,:::,n; andh =1,2, :::,l. Therearen l alternatve estmates for x : ˆx,,h = r,,h z h y : 5 Usng the offset term, GLMs can solve fxed factor constrants, such as x 2 =0:8. GIA s more flexble n ts capablty of solvng almost all formats of constrants. 6 E(r, )=x y exp(0:5 Var(", )). 7 The models wth more complex addtve-multplcatve structures can be derved smlarly. 201

10 Varance Advancng the Scence of Rsk There are m l alternatve estmates for y : ŷ,,h = r,,h x z : h There are m n alternatve estmates for z h : ẑ,,h = Usng w p,,h as the weght: ẑ h = = ˆx = r,,h x + y : wp,,h ẑ,,h wp,,h à r,,h wp,,h x + y w p,,h h w p,,h ˆx,,h h wp,,h! μ r,,h h wp,,h y z = h ; h wp,,h ŷ = h wp,,h ŷ,,h h wp,,h μ r,,h h wp,,h x z = h h w p :,,h, (2.21) There s no unque soluton for (2.21). For example, f a, b, andc areasolutontoestmatethe factors x, y, andz, then2a, 2b, and0:5c are another possble soluton. In order to facltate the teraton convergence, we need to add some constrants n the procedure. If we use the sample mean as the base, the weghted average of multplcatve factors from one ratng varable should be close to one. So n each teraton we can adust all the z s proportonally so that the average s reset to one. Mathematcally, the constrant s: h wp,,hẑh h w p,,h =1: (2.22) 3. Resdual dagnoss For a statstcal data-fttng exercse, t s mportant to conduct a dagnostc test to valdate the dstrbuton assumpton n use. Such dagnostc tests typcally consst of a resdual plot n whch the resduals are the dfference between the ftted values and the actual values. In ths secton, we wll descrbe how to conduct such resdual analyss for GIA, and the resdual plot results for the case study are gven n the next secton. We have dscussed that a 3-parameter GIA s equvalent to a GLM, assumng the response varable r k follows a dstrbuton wth varance functon Var(¹)=¹ 2 q=k. The raw resduals (r, k ˆx kŷk ) from GIA do not asymptotcally follow an ndependent and dentcal normal dstrbuton because the varances of resduals are postvely correlated to the predcted values. 8 As n GLM, we defne the scaled earson resdual of GIA as e, = rk, ˆr k p Var(¹) = rk, ˆx k ŷk q (ˆx k ŷk )2 q=k = rk, ˆx k ŷk qˆx 2k q ŷ 2k q (3.1) where e, s approxmately ndependent and dentcally dstrbuted snce 0 1 Var(e, )=Var@ rk, ˆx kŷk q A (ˆx k ŷk )2 q=k = Var(rk, ) (ˆx k =1: (3.2) ŷk )2 q=k We can use the scaled earson resduals to conduct the resdual dagnoss for GIA, such as developng a scattered resduals plot and a quantleto-quantle (Q-Q) plot. If the GIA algorthms ft the data well, scaled earson resduals are randomly scattered and the Q-Q plot s close to a straght lne. 8 The addtve models are equvalent to GLM normal models, so that the raw resduals from addtve and mxed models can be used drectly for dagnoss tests., 202

11 General Iteraton Algorthm for Classfcaton Ratemakng 4. Calculaton effcency One ssue assocated wth GIA s the calculaton effcency. Mldenhall [9] dscussed that one advantage of GLMs compared to the mnmum bas models s the calculaton effcency because GLMs do not requre an teratve process n estmatng the parameters. He showed that the addtve mnmum bas model by Baley [1], or GIA wth p = 1, does not converge even after 50 teratons usng the well-nvestgated data gven by McCullagh and Nelder [8]. However, wth several adustments to the teraton methodology, we can show that GIA can converge very quckly. Usng the same data, the addtve GIA can complete the convergence n fve teratons. One adustment s to nclude as much updated nformaton as possble that s, the latest y s should be used to estmate the next x s and vce versa. In GLMs and prevous mnmum bas models, a specfc class s usually selected as the base (e.g., age 60+ and pleasure). For GIA, we suggest usng the average as the base, because, when usng a specfc class as the base, the numercal value of the base wll vary from one teraton to next, requrng addtonal teratons to force the factor for the base class to be one. Another well-known ssue for the teraton procedure concerns how to set the startng pont for the frst teraton. The closer the startng pont to the fnal results, the faster the convergence. Usng average frequency=severty=pure premum as the base, the average factor of a ratng varable s one for multplcatve models and the average dscount s zero for the addtve models. Therefore, n ths study, we chose the startng values of x,0 and y,0 to be 1 for the multplcatve models and 0 for the addtve models. 5. Numercal analyss The numercal analyss of testng varous multplcatve GIAs s based on the severty data for prvate passenger auto collson gven n Mldenhall [9] and McCullagh and Nelder [8]. Usng ths well-researched data wll help us to compare the emprcal results of ths paper wth prevous studes. The data ncludes 32 severty observatons for two classfcaton varables: eght age groups and four types of vehcle use. In ths severty case study, the weght w, s the number of clams. Table 1 n the Appendx lsts the data. In order to test mxed addtve and multplcatve GIAs, we need at least three varables n the data. The data n Mldenhall [9] and McCullagh and Nelder [8] contan only two varables. Therefore, we wll use another collson pure premum dataset to demonstrate the mxed algorthm. In addton to age and vehcle use, ths data ncludes credt score as a thrd varable, wth four classfcatons from low to hgh. In ths pure premum case study, the weght w, s the earned exposure. Table 2 n the Appendx dsplays the data. Four crtera are used to evaluate the performance of these GIAs: the absolute bas, the absolute percentage bas, the earson ch-squared statstc, and the combnaton of absolute bas and the ch-squared statstc: ² The weghted absolute bas (wab) crteron s proposed by Baley and Smon [2]. It s the weghted average of absolute dollar dfference between the observatons and ftted values: w, Br, Bx y wab = : w, ² The second one, weghted absolute percentage bas (wapb), measures the absolute bas relatve to the predcted values: wapb = w, Br, Bx y Bx y w, : ² The weghted earson ch-squared (wch) statstc s also proposed by Baley and Smon [2] and t s approprate to test f dfferences between the raw data and the estmated relatvtes should be small enough to be caused 203

12 Varance Advancng the Scence of Rsk by chance : wch = w, (Br, Bx y ) 2 Bx y w, : ² Lastly, we combne the absolute bas and earson ch-squared statstc, p wab wch, tobe the fourth crteron for the model selecton. Table 3 lsts the relatvtes for Algorthms 1 8 and Table 4 dsplays the four performance statstcs of those models, wab, wapb, wch, and p wab wch. 9 In all the cases, class age 60+ and pleasure are used as the base. To llustrate the resdual dagnoss of GIA, we show the resdual plots for GIA wth k =1,p = 1, and q = 0:5. Fgure 1 n the Appendx reports the scattered resduals by observatons; Fgures 2 and 3 show the scattered resduals by age and by vehcle use, respectvely; Fgure 4 s the Q-Q plot. It s clear that the classfcaton of age and busness use s an outler. 10 Ths s not surprsng because of the small sample sze n the cell (fve clams). A practcal way to solve the problem s to cap the severty. As stated before, we fnd that GLMs wth common exponental famly dstrbuton assumptons are specal cases of GIA (k =1andp =1).Comparng the GIA factors n Table 3 wth those from GLMs wth normal, osson, gamma, and nverse Gaussan dstrbutons, we confrm: ² when k =1, p =1, and q = 2, the least squares GIA has the same results as GLM wth a normal dstrbuton; 11 ² when k =1,p =1,andq =1,GIAsthesame as a osson GLM; ² when k =1,p =1,andq = 0, GIA s the same as a gamma GLM; and ² when k =1,p =1andq = 1, GIA s the same as a GLM wth nverse Gaussan dstrbuton. As dscussed n Secton 2, a GIA wth k =1 and p =1 0 w, r, y q 1 = w A, yq s equvalent to the multplcatve GLMs wth the varance functon of Var(¹)=¹ 2 q for an assumed exponental famly dstrbuton. It s well known that nsurance and actuaral data s generally postvely skewed. The skewness for the symmetrc normal dstrbuton s zero, and s ncreasngly postve from osson, to gamma, and to nverse Gaussan. For the multplcatve GIA algorthms, the skewness can be represented by q. Whenq = 2, the GIA s the same as a normal GLM. When q = 1, t s the same as a osson GLM.Itsthesameasagammawhenq =0 and the same as nverse Gaussan when q = 1. Thus, smaller q values should be selected when thegiasappledtomoreskeweddata. The authors also attempted to fnd the global mnmum error ponts. 12 In ths case study, f wab s used to measure the model performance, when k =1:95, p =3:15, and q = 14:06, the weghted absolute error s mnmzed wth wab = 10:0765. If wapb s used to measure the model performance, when k =1:98, p =3:15, and q = 14:04, the weghted absolute percentage error s mnmzed wth wapb =3:461%. The result suggests that the best-ft model, n ths example, does not occur wth any of the commonly 9 We tested hundreds of 3-parameter algorthms. For the detaled reports on all the tested models, please refer to Fu and Wu [6]. 10 The severty of age and busness use does not ft any tested GLMs and GIAs well. 11 The underlyng assumpton of least squares regresson s that the resduals follow a normal dstrbuton. So the least squares method s the same as a normal GLM. 12 Resolvng such global mnmum error ssues requres addtonal n-depth research and s beyond the scope of ths paper. Snce the error measures are easy to calculate explctly n a spreadsheet, an Excel bult-n tool lke Solver can be used to fnd the optmzaton solutons. If the data s larger than the spreadsheet s capacty, nterested readers can apply Newton s method to obtan the mnmums usng SAS, Splus, or Matlab. The frst and second dervatves can be estmated usng a fnte dfference method. 204

13 General Iteraton Algorthm for Classfcaton Ratemakng Fgure 1. Scattered resdual plot of GIA wth k =1, p =1,andq = 0:5. Fgure 2. Scattered resdual plot by age of GIA wth k =1, p =1,andq = 0:5. used mnmum bas models and generalzed lnear models. It clearly demonstrates the fact that nsurance data may not be perfect for predetermned dstrbutons. On the other hand, f wch s used, the Âsquared model (k =2, p =1, and q =1) provdes the best soluton. Ths s expected because the Â-squared model s calculated by mnmzng the earson ch-squared statstc. If we use the crteron of p wab wch to select models, when k =2:45, p =1:16, and q = 0:06, the combned error s mnmzed wth 205

14 Varance Advancng the Scence of Rsk Fgure 3. Scattered resdual plot by vehcle use of GIA wth k =1, p =1,andq = 0:5. Fgure 4. Q-Q plot of GIA wth k =1, p =1,andq = 0:5. p wab wch =3:3061. Agan, the fve commonly used mnmum bas algorthms are not the best soluton when absolute bas and the chsquared statstc are consdered smultaneously. Based on the results of ths research and our experence, we suggest for actuaral applcatons the followng ranges of values for k, p, q: ² 1 k 3. ² p q, 0:5 p 4, and q 1. ² The hgher the skewness of the data, the smaller the value of q should be. 206

15 General Iteraton Algorthm for Classfcaton Ratemakng Fnally, we use another collson pure premum dataset to demonstrate the results for the mxed algorthm. Table 11 reports the fnal factors of the model. For the purpose of llustraton, we wll only calculate the model wth p =1. To show that GIAs can converge rapdly, n the Appendx we report the teraton processes of selected GIAs: ² Table 5 shows the multplcatve factors for the gamma GIA usng average severty as the base. ² Table 6 translates those factors usng the classfcaton age 60+ and pleasure as the base. ² Table 7 reports the teratve process for the coeffcents of a GLM wth the gamma dstrbuton and log lnk. ² Table 8 translates those coeffcents to the multplcatve factors of a gamma GLM. ² Table 9 lsts the addtve factors for the GIA wth p =1. ² Table 10 shows the addtve dollar values for the GIA wth p = 1 and uses the classfcaton age 60+ and pleasure as the base. ² Table 11 reports the convergence process of the mxed model wth p =1. From Tables 5 8, the multplcatve gamma GIA converges n four teratons. Ths s as fast as the correspondng GLM model. As expected, the numercal solutons between the two models are dentcal, and the solutons are also dentcal to the prevous results of Algorthm 6 gven n Table 3 for k =1,p =1,andq =0. Tables 9 and 10 report the teratve process for the GIA addtve algorthm wth p = 1. Mldenhall [9] used ths model as an example to show that the mnmum bas approach s not effcent. He showed that the mnmum bas model converges slowly to the GLM results, and that the dollar values at the 50th teraton are about two cents dfferent from those by GLM. However, usng our numercal algorthm, the GIA calculaton converges completely n fve teratons wth solutons dentcal to GLM results. Table 11 shows the teratve process for the GIA mxed model. Even though the algorthm s more complcated than the multplcatve and addtve models, the convergence takes only sx teratons. The above example llustrates an optmzaton case wth two and three varables. However, n typcal ratng plans, we need to optmze more than two varables. Our experence ndcates that the mproved numercal approach for GIA wll converge farly quckly for typcal actuaral ratng exercses wth fve to 15 varables. 4. Conclusons In ths research, we propose a general teraton algorthm by ncludng dfferent weghtng functons and relatvty lnk functons n the approach. As ndcated by the severty example gven prevously, nsurance and actuaral data are rarely perfect, so we expect that the best ftted results typcally wll not be based on a predetermned dstrbuton, such as those n the exponental famly of dstrbutons. Therefore, GIA can provde actuares a great deal of flexblty n data fttng and model selecton. The case studes gven n the paper ndcate that the best ftted results occur when the underlyng dstrbuton assumptons are not commonly used dstrbutons. In theory, the parameters n GIA can take on any real values and there s no lmtaton on the relatvty lnk functons when GIA s appled to a dataset. Therefore, GIA wll provde actuares many more optons than prevous mnmum bas algorthms or GLMs. However, due to the fact that nsurance and actuaral data s postvely skewed n nature, we do not expect that a very wde range of weghtng or relatvty lnk functonsneedstobeusednpractce. For the severty example used n the study, we searched and dentfed the best models wth the mnmum ftted errors. One ssue may exst: GIA uses an teratve process n determnng the pa- 207

16 Varance Advancng the Scence of Rsk rameters, so when t further ncorporates multple dstrbuton assumptons n the searchng process, the approach may become even more tme-consumng and neffcent. However, we do not beleve ths ssue s sgnfcant because of the powerful computatonal capablty of modern computers. Mldenhall [10] ndcates, n hs comments on our pror work, that GLM can be extended to replcate the comprehensve GIA proposed n ths study. However, snce commonly used GLM software has lmted selectons for the statstcal dstrbuton assumptons, t s dffcult to perform Mldenhall s extenson. In addton, we demonstrate how to extend GIA to solve mxed addtvemultplcatve models and constrant optmzaton problems. To our knowledge, at ths stage, there s no soluton provded by GLM users to deal wth these ssues. Wth the fast development of nformaton technology, actuares can analyze data n ways they could not magne a decade ago. Currently there s a strong nterest n data mnng and predctve modelng n the nsurance ndustry, and ths calls for more powerful data analytcal tools for actuares. Whle some new tools, such as GLM, neural networks, decson trees, and MARS, have emerged recently and have receved a great deal of attenton, we beleve that the decades-old mnmum bas algorthms stll have several advantages over other technques, ncludng beng easy to understand and easy to use. We hope that our work n mprovng the flexblty and comprehensveness of the mnmum bas teraton approach s a tmely effort and that ths approach wll contnue to be a useful tool for actuares n the future. Acknowledgments The authors thank Steve Mldenhall for hs valuable comments. References [1] Baley, R. A., Insurance Rates wth Mnmum Bas, roceedngs of the Casualty Actuaral Socety 50, 1963, pp [2] Baley, R. A., and L. J. Smon, Two Studes n Automoble Insurance Ratemakng, roceedngs of the Casualty Actuaral Socety 47, 1960, pp [3] Brown, R. L., Mnmum Bas wth Generalzed Lnear Models, roceedngs of the Casualty Actuaral Socety 75, 1988, pp [4] England,. D., and R. J. Verrall, Analytc and Bootstrap Estmates of redcton Errors n Clams Reservng, Insurance: Mathematcs and Economcs 25:3, 1999, pp [5] Feldblum, S., and J. E. Brosus, The Mnmum Bas rocedure A racttoner s Gude, roceedngs of the Casualty Actuaral Socety 90, 2003, pp [6] Fu, L., and. C. Wu, Generalzed Mnmum Bas Models, Casualty Actuaral Socety Forum, Wnter 2005, pp [7] Mack, T., A Smple arametrc Model for Ratng Automoble Insurance or Estmatng IBNR Clams Reserves, ASTIN Bulletn 21:1, 1991, pp [8] McCullagh,., and J. A. Nelder, Generalzed Lnear Models (2nd edton), London: Chapman and Hall, [9] Mldenhall, S. J., A Systematc Relatonshp between Mnmum Bas and Generalzed Lnear Models, roceedngs of the Casualty Actuaral Socety 86, 1999, pp [10] Mldenhall, S. J., Dscusson of General Mnmum Bas Models, Casualty Actuaral Socety Forum, Wnter 2005, pp [11] Renshaw, A. E., and R. J. Verrall, A Stochastc Model Underlyng the Chan Ladder Technque, Brtsh Actuaral Journal, 1998, pp

17 General Iteraton Algorthm for Classfcaton Ratemakng Appendx. Data and numercal results Table 1. A collson severty data for multplcatve and addtve algorthms Age VUSE Severty Clam leasure DrveShort DrveLong Busness leasure DrveShort DrveLong Busness leasure DrveShort DrveLong Busness leasure DrveShort DrveLong Busness leasure DrveShort DrveLong Busness leasure DrveShort DrveLong Busness leasure DrveShort DrveLong Busness leasure DrveShort DrveLong Busness Table 2. A collson pure premum data for the mxed algorthm Age VUSE Credt Exposure Loss ure rem Busness Busness Busness Busness DrveLong , DrveLong , DrveLong , DrveLong , DrveShort 1 1, , DrveShort , DrveShort 3 2, , DrveShort 4 1, , leasure , leasure , leasure 3 2, ,

18 Varance Advancng the Scence of Rsk Table 2. (Contnued) Age VUSE Credt Exposure Loss ure rem leasure 4 1, , Busness , , Busness Busness , Busness DrveLong , DrveLong , DrveLong , DrveLong , DrveShort 1 1, , DrveShort 2 1, , DrveShort 3 2, , DrveShort 4 1, , leasure , leasure , leasure 3 1, , leasure , Busness , , Busness , Busness , Busness DrveLong , DrveLong , DrveLong , DrveLong , DrveShort 1 2, , DrveShort 2 2, , DrveShort 3 4, , DrveShort , leasure 1 1, , leasure 2 1, , leasure 3 1, , leasure , Busness , Busness , Busness , Busness DrveLong , DrveLong , DrveLong , DrveLong , DrveShort 1 3, , DrveShort 2 2, , DrveShort 3 6, , DrveShort 4 1, , leasure 1 1, , leasure 2 1, , leasure 3 3, , leasure , Busness Busness Busness , Busness , DrveLong , DrveLong , DrveLong 3 1, , DrveLong ,

19 General Iteraton Algorthm for Classfcaton Ratemakng Table 2. (Contnued) Age VUSE Credt Exposure Loss ure rem DrveShort 1 2, , DrveShort 2 2, , DrveShort 3 7, , DrveShort 4 2, , leasure 1 1, , leasure 2 1, , leasure 3 5, , leasure 4 1, , Busness , Busness , Busness , Busness , DrveLong , DrveLong , DrveLong 3 2, , DrveLong 4 1, , DrveShort 1 6, , DrveShort 2 5, , DrveShort 3 17, ,922, DrveShort 4 8, , leasure 1 4, , leasure 2 4, , leasure 3 15, ,842, leasure 4 7, , Busness , Busness , Busness , Busness , DrveLong , DrveLong , DrveLong 3 1, , DrveLong 4 1, , DrveShort 1 4, , DrveShort 2 4, , DrveShort 3 15, ,512, DrveShort 4 9, , leasure 1 4, , leasure 2 4, , leasure 3 15, ,653, leasure 4 11, , Busness Busness , Busness , Busness , DrveLong , DrveLong , DrveLong , DrveLong , DrveShort 1 1, , DrveShort 2 1, , DrveShort 3 6, , DrveShort 4 6, , leasure 1 5, , leasure 2 4, , leasure 3 22, ,129, leasure 4 31, ,601,

20 Varance Advancng the Scence of Rsk Table 3. The age and vehcle-use relatvtes for Algorthms 1 8 Algorthm Busness DTW Long DTW Short leasure Table 4. The performance measurements for Algorthms 1 8 Algorthm wab wapb wch p wab wch % % % % % % % % Table 5. Numercal teratons for multplcatve gamma GIA factors usng average severty as the base DTW DTW Iteraton Base Busness Long Short leasure Table 6. Numercal teratons for multplcatve gamma GIA factors usng 60+ and pleasure as the base DTW DTW Iteraton Base Busness Long Short leasure Table 7. Numercal teratons for multplcatve gamma GLM coeffcents usng 60+ and pleasure as the base DTW DTW Iteraton Base Busness Long Short leasure : : : :