Direct Analysis of Pre-Adjusted Loss Cost, Frequency or Severity in Tweedie Models

Transcription

1 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models Sheng G. Sh, Ph.D. Abstract Resonse data (loss cost, clam freqency or clam severty are often re-adsted wth known factors and drectly analyzed wth generalzed lnear models (GLM. Ths aer shows that the exosre weghts shold also be adsted f the Tweede dstrbton wth log lnk s sed n sch drect analyss. An advantage of the drect analyss over GLM offsettng s that the strctre of the orgnal dataset may be smlfed sgnfcantly after removng the known factors. Drect analyss s a convenent tool for drectly modelng loss rato and for removng known terrtory factors from the dataset. Imlementaton n EMBLEM and SAS s dscssed, and a comtatonally effcent SAS macro s rovded for Tweede models. Keywords. Predctve Modelng; GLM offset; Ratemakng; Tweede Model; EMBLEM; SAS. 1. INTRODUCTION In nsrance ratemakng, resonse data are often re-adsted wth known factors before redctve modelng. However, the effect of adstment on exosre weghts s sally ether gnored or not lnked to the resonse dstrbton. Ths s artclarly the case when the resonse varable s loss cost, whch s assmed to follow a Tweede dstrbton of ower (1<<. Alcaton of the GLM offset featre n roerty-casalty redctve modelng has been dscssed recently by Yan et. al.[7]. They translated the analyss on loss rato nto an analyss on loss cost wth remm offset. In ths aer, we wll show how loss rato, vewed as loss cost readsted wth remm rates, can be analyzed drectly. An advantage of the drect analyss s that the strctre of orgnal dataset may be smlfed sgnfcantly for sbseqent analyss. We frst show, n general, how the exosre weghts shold be modfed n Tweede models (ncldng the secal case of Posson and Gamma wth re-adsted loss cost, clam freqency or clam severty as the resonse.. CONNECTION BETWEEN OFFSETS AND PRE-ADJUSTMENT In ths secton, we gve two roostons that connect GLM offsets wth re-adstment. Prooston 1 blds a smle lnkage between the offsets and re-adstment. Prooston establshes a fondaton for data smlfcaton. Casalty Actaral Socety E-Form, Wnter 010 1

2 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models.1 Assmtons and Notatons Sose that there are two ratng factors U and V, where U has m categores and V has n categores. Denote as the relatvty of the th category ( 1,,..., m of U and v as the relatvty th of the category ( 1,,..., n of V. Let Y be a random varable for the rato of nterest n the ratng cell wth the th level of U and the th level of V sch that Y X / w. When the rato of nterest s loss cost, X L as loss amont and w e as earned exosre. When the rato of nterest s clam freqency, X c as clam cont and w e. When the rato of nterest s clam severty, X L and w c. Assme that the Y s are mtally ndeendent and Y follows a Tweede dstrbton wth ower arameter sch that E( Y v, (.1 Var ( Y ( v / w (. where s a constant dserson arameter [3]. To nclde the dsersed Posson and Gamma as secal cases of the Tweede dstrbton, we focs on the range of ower arameter, 1. As n a tycal analyss, we assme that the ower arameter and the constant dserson arameter are known or have been re-determned. We wll se log lnk n all the models.. Proostons..1 Smle lnk between GLM offsets and re-adstment Wth the Tweede model, an offset roblem can be translated nto a re-adstment roblem and vce versa as shown n the rooston below. Ths nterchangeablty also allows s to have a model wth both re-adstment and offsets. Prooston 1 Under the assmtons and notatons above, f s are known, then fttng the followng Tweede model (n Eq. (.3 of ower wth weghts w and log( as an offset, log E( Y log( v log(, (.3 where 1,,..., m and 1,,..., n, s eqvalent to fttng the Tweede model of ower below (n Eq. (.4 wth re-adsted resonse varable Y / and weghts w, where 1,,..., m and 1,,..., n. log E( log( v, (.4 In other words, can be vewed to follow Tweede dstrbton of the same ower and Casalty Actaral Socety E-Form, Wnter 010

3 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models dserson arameter as Y, bt wth dfferent weghts. Proof Note that the Tweede dstrbton belongs to the exonental dserson famly, whch s closed nder a scale transformaton (cf. [3] Formla 6 on. 7. Ths, follows a Tweede dstrbton wth ower arameter. Based on Eq. (.1 and Eq. (. above, and Var( Var( Y E( E( Y / v / v (.5 / ( v /( w v /( w. (.6 To show that two models are eqvalent, let l be the log-lkelhood fncton for Y y. Then, accordng to the roerty of the exonental dserson famly, we have l y V ( / w (.7 where the mean E( Y v and the varance fncton maxmm lkelhood estmate vˆ of v, we set for l v l v y 1,,..., n, v ( v / w 0 V (. To obtan the (.8 whch leads to the estmate for the model secfed n Eq. (.3, w y v w / ˆ. (.9 Now, let l * be the log-lkelhood fncton for l v * z v V ( v /( w z. Then, (.10 where the mean v E( and the varance fncton * lkelhood estmate v ˆ of v, we set for 1,,..., n, V ( v v. To obtan the maxmm l * v v z v /( w 0 (.11 Casalty Actaral Socety E-Form, Wnter 010 3

4 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models whch leads to the estmate for the model secfed n Eq. (.4, vˆ * w w z vˆ. (.1 * * l ( ˆ v l ( ˆ v It s easy to verfy that 0 and 0 v v for the maxma. Q.E.D. Note that the rght sde of Eq. (.4 s not related to the ndex. Ths, t may be smlfed by collasng over the ratng factor U as dscssed n Secton... Examle 1 Loss rato can be vewed as loss cost lan: L / e re-adsted wth the remm rates n a ratng Loss Rato = Losses/Earned Premms = Losses/(Exosres*Rates = (Losses/Exosres/Rates = (Loss Cost/Rates. Assme that the loss cost L / e follows Tweede of ower. Then, the loss rato L /( e can be analyzed wth the Tweede model of ower, bt the model weghts need to be adsted to Exosres*Rates^(- = e... Pre-adstment for data smlfcaton Aggregatng data redces the nmber of records n a dataset and smlfes the data strctre. Ths can be esecally benefcal when aggregatng across hgh-dmensonal varables, sch as terrtory. From a modelng ersectve, ths s acheved by collasng on the GLM offset varable, bt sbseqent analyses wll then need to be done wth re-adsted data as shown n the rooston below. Prooston Under the assmtons and notatons above, f s are known, then fttng the followng Tweede model (n Eq. (.13 of ower wth weghts w and log( as an offset log E( Y log( v log(, (.13 Casalty Actaral Socety E-Form, Wnter 010 4

5 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models where 1,,..., m and 1,,..., n, s eqvalent to fttng the smlfed Tweede model of ower below (n Eq. (.14 wth weghts w, log E( w ( Y / where w ; 1,,..., n. log( v ; 1,,..., n, (.14 (.15 In other words, can be vewed to follow the Tweede dstrbton of the same ower and dserson arameter as Y, bt wth dfferent weghts (cf. [4]. Proof Note that the Tweede dstrbton belongs to the exonental dserson famly, whch s closed nder a scale transformaton and follows the convolton formla (cf. [3] Formla 10 on. 74. Wrte Y /. We know from Prooston 1 that follows the Tweede dstrbton of the ower wth mean v, dserson arameter and ror weghts w. Therefore, for 1,,..., n, w w s stll Tweede dstrbted wth the ower arameter and E (.16 w E( Y / w ( v / ( v, (.17 w w Casalty Actaral Socety E-Form, Wnter 010 5

6 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models Var( v v ( w * / ( w ( w ( ( w ( *( w w. w v * Var( Y /( w / w / (.18 To show the two models are eqvalent for estmatng v, we note from the roof of Prooston 1 that the maxmm lkelhood estmate for the model secfed by Eq. (.13 s gven n Eq. (.9. From Eq. (.17, t s rather trval that the maxmm lkelhood estmate for the model secfed by Eq. (.14 s the same as that n Eq. (.9, becase only a sngle Q.E.D. Examle s nvolved for estmatng v. In a loss rato analyss, a dataset wth nmeros remm rate levels may be smlfed by collasng over the remm varable. Note that a nqe remm rate level s defned by a nqe combnaton of all ratng varables n a ratng lan. The data sze can be redced drastcally n many cases by collasng over the remm varable. Before collasng, loss ratos L /( e are recorded for each exosre, where s are remm rates. We are nterested n fttng a Tweede model of ower wth other covarates that are combned nto v. After collasng, we can eqvalently 1 model weghted loss ratos ( L /( e wth adsted exosre weghts e. Note that the weghted loss ratos are not of the form ( L /( e. Examle 3 In a loss cost analyss, a dataset wth nmeros terrtores may be smlfed by collasng over the terrtory varable. Both loss cost and exosre weghts need to be adsted by known terrtory relatvtes for Tweede models. Casalty Actaral Socety E-Form, Wnter 010 6

7 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models 3. IMPLEMENTATION Sose that the EMBLEM (cf. [1] and [] data sorce s n a smmarzed table sch that each record has an observed level (ndexed by of a ratng factor (for examle, terrtory to be collased, an observed level (ndexed by of a olated combnaton of other ratng factors, along wth the nmber of clams ( c, the loss amont ( L and the exosre ( e at the level (,. Assme that the orgnal redctve models are as n Table 1 and log lnk s sed for all models. Wth the log lnk, s secfed as an offset n accordance wth the EMBLEM logc. The dserson arameter s ether secfed or estmated wherever arorate. Wth re-adstment, the resonse varable and the weght varable before collasng are gven n Table n accordance wth Prooston 1. After collasng, the resonse varable and the weght varable n Table 3 are ready for smlfed analyss n accordance wth Prooston. Table 1. Descrton of orgnal redctve models Model Dstrbton Resonse Varable Weght Varable Offset Freqency Posson Clam freqency, c / e Nmber of exosres, e Severty Gamma Clam severty, L / c Nmber of clams, c Loss cost Tweede( Loss cost, L / e Nmber of exosres, e Table. Descrton of re-adstment models before collasng Model Dstrbton Resonse Varable Weght Varable Freqency Posson Adsted clam freqency, Nmber of adsted c /( e exosres, e Severty Gamma Adsted clam severty, L /( c Loss cost Tweede( Adsted loss cost, L /( e Nmber of clams, c Nmber of adsted exosres, e Casalty Actaral Socety E-Form, Wnter 010 7

8 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models Table 3. Descrton of re-adstment models after collasng Model Dstrbton Resonse Varable Weght Varable Freqency Posson Weghted sm of adsted clam freqency, Total nmber of adsted exosres, c /( e e ( Severty Gamma Weghted sm of adsted clam severty, ( L / /( c Loss cost Tweede( Weghted sm of adsted loss amont ( 1 L /( e Total nmber of clams,c Total nmber of adsted exosres, e Imlementaton n SAS can be done smlarly. Wth known Tweede ower and dserson arameters, the GENMOD rocedre can be adoted wth ser defned dstrbton.[5] 4. REMARKS Throghot ths aer, we assmed that both the Tweede ower and dserson arameters are known. In ractce, the ower arameter s often taken from ror modelng exerence, whle the dserson arameter s estmated sng the Pearson, Devance or the lkelhood aroach [3]. Comared to the lkelhood aroach, an estmated dserson arameter sng ether the Pearson or Devance can be sgnfcantly dfferent for n the md-range of the nterval (1,. In the SAS envronment, PROC NLMIXED may be sed for smltaneos estmaton of all Tweede arameters sng the code wrtten by Flynn [7], bt convergence may become a roblem wth a large dataset and nmeros class varables. As an alternatve, the code n Aendx A may be aled. We assmed that the dserson arameter s a constant. However, t s often more arorate to allow to vary wth dfferent ratng cells sch that, esecally n a loss cost model [6]. In sch a case, f we nsst on fttng a model wth fxed, then a dfferent set of weghts may be necessary for an accrate solton. On the other hand, f s allowed to vary, we may t any adstment on weghts nto, leavng the orgnal weghts ntoched. The choce of weghts n Eq. (.4 and Eq. (.14 affects both the accracy of the model estmates Casalty Actaral Socety E-Form, Wnter 010 8

9 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models and the valdty of hyothess tests even f an estmate of v s nbased. Both roostons 1 and can be generalzed to the case wth more than two ratng factors. Note that mltle adstments can be combned and seqenced wth ndex and other covarates may be combned and seqenced wth ndex. Acknowledgment The athor thanks Trevor Handley, FCAS, MAAA for revewng ths aer. Aendx A The SAS macro rovded n ths aendx may be sed exermentally for Tweede models. The macro s based on the orthogonal roerty between the mean arameter and the ower/dserson arameter (, [3], whch allows ther searate otmzatons. It terates ntl convergence between the -ste wth PROC GENMOD and the (, -ste wth PROC NLMIXED. The GENMOD rocedre s easy to converge and has a handy CLASS statement, whch s stable for Tweede models wth known (, and hgh dmenson of. Ths aroach redces the brden on PROC NLMIXED so that t s sed only to estmate (, wth assmed known. **************************************************************** * MACRO FOR TWEEDIE MODEL * * * * Athor: Sheng G. Sh * * Paramters: * * dn -- dataset name * * vformat -- lst of formats * * vclass -- class varables * * wght -- weght varable * * res -- resonse varable (mst be non-negatve * * red -- redctors * * clmcnt -- clam conts * * offset -- offset varable * * Warnng: * * Check ott for convergence of GENMOD and NLMIXED; * * Check log for reslts; * * Ttle3 wll be over-wrtten. * ****************************************************************; %macro tweede(dn=,vformat=,vclass=,wght=,res=,red=,clmcnt=,offset=; /* Intalzaton */ ttle3; data Est_save_; format _ h_ sgma lower _er h_lower h_er sgma_lower sgma_er 15.4 _change sgma_change 15.4; _ = 1.5; _lower =.; Casalty Actaral Socety E-Form, Wnter 010 9

10 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models _er =.; h_ = 1; h_lower =.; h_er =.; sgma_ = 1; sgma_lower =.; sgma_er =.; _change =.; sgma_change =.; call symt('',trm(left(t(_,15.4; call symt('h',trm(left(t(h_,15.4; call symt('sgma',trm(left(t(sgma_,15.4; call symt('_lower',trm(left(t(_,15.4; call symt('_er',trm(left(t(_,15.4; call symt('h_lower',trm(left(t(h_,15.4; call symt('h_er',trm(left(t(h_,15.4; call symt('sgma_lower',trm(left(t(sgma_,15.4; call symt('sgma_er',trm(left(t(sgma_,15.4; rn; /* Maxmm lkelhood estmaton */ %let converge = 0; %let =1; %do %ntl ((&converge eq 1 or (& gt 10; ttle3 "Otmzaton Ste &"; %otmze(&dn,&vformat,&vclass,&wght,&res,&red,&clmcnt,&offset,0; %let = %eval(&+1; data Est_save_(dro=_old sgma_old; set Est_save_ end=last; _old = _; sgma_old = sgma_; retan _old sgma_old; ott; f last then do; _ = &; _lower = &_lower; _er = &_er; h_ = &h; h_lower = &h_lower; h_er = &h_er; sgma_ = &sgma; sgma_lower = &sgma_lower; sgma_er = &sgma_er; _change = abs(_-_old; sgma_change = abs(sgma_-sgma_old; _old = _; sgma_old = sgma_; f (_change le 1e-5 and (_change ne. and (sgma_change le 1e-5 and (sgma_change ne. then call symt('converge','1'; ott; end; rn; %end; Casalty Actaral Socety E-Form, Wnter

11 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models /* Reslts */ %f (&converge eq 1 %then %do; ttle3 'Tweede Model wth Converged Parameter Estmates'; %otmze(&dn,&vformat,&vclass,&wght,&res,&red,&clmcnt,&offset,1; %t Converged; %t Power arameter = & wth 95% C.I. (&_lower, &_er; %t Dserson arameter = &h wth 95% C.I. (&h_lower, &h_er; %t SAS scale arameter = &sgma wth 95% C.I. (&sgma_lower, &sgma_er; %end; %else %do; %t Not converged after 10 teratons: ; %t Power arameter = &; %t Dserson arameter = &h; %t SAS scale arameter = &sgma; %t at the end of 10th teraton.; %end; ttle3; %mend tweede; %macro otmze(dn,vformat,vclass,wght,res,red,clmcnt,offset,flag; roc genmod data=&dn; format &vformat; class &vclass /aram=glm; _ = &; m_ = _MEAN_; y_ = _RESP_; v_ = m_**_; f y_ gt 0 then d_ = *(y_*(y_**(1-_-m_**(1-_/(1-_-(y_**(-_-m_**(-_/(- _; else d_ = *(m_**(-_/(-_; varance var = v_; devance dev = d_; weght &wght; model &res = &red /lnk=log noscale scale=&sgma %f %length(&offset eq 0 %then ; %else offset=&offset;; ott ot=ot_m_ red=yhat_; rn; %f &flag ne 1 %then %do; ods trace on; ods ott ParameterEstmates=Est_; roc nlmxed data=ot_m_; format _ 15.4 h_ 15.4; arms _=& h_=&h; bonds 1<_<, h_>0; n_ = &clmcnt; w_ = &wght; y_ = &res; m_ = yhat_; t_ = y_*m_**(1-_/(1-_-m_**(-_/(-_; Casalty Actaral Socety E-Form, Wnter

12 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models a_ = (-_/(_-1; f (n_ eq 0 then rll_ = (w_/h_*t_; else rll_ = n_*((a_+1*log(w_/h_+a_*log(y_-a_*log(_-1-log(-_ -lgamma(n_+1-lgamma(n_*a_-log(y_+(w_/h_*t_; /* log lkelhood of (_,h_ wth m_ known */ model y_ ~ general(rll_; rn; ods trace off; data _nll_; set Est_; f Parameter eq '_' then do; call symt('',trm(left(t(estmate,15.4; call symt('_lower',trm(left(t(lower,15.4; call symt('_er',trm(left(t(uer,15.4; end; else f Parameter eq 'h_' then do; call symt('h',trm(left(t(estmate,15.4; call symt('h_lower',trm(left(t(lower,15.4; call symt('h_er',trm(left(t(uer,15.4; call symt('sgma',trm(left(t(sqrt(estmate,15.4; call symt('sgma_lower',trm(left(t(sqrt(lower,15.4; call symt('sgma_er',trm(left(t(sqrt(uer,15.4; end; rn; %end; %mend otmze; **************************************************************; Here s an examle that calls the %tweede macro: %tweede(dn=cardata, vformat=modelyr 4. SYM $symfmt., vclass=modelyr SYM, wght=eex, res=losscost, red=modelyr SYM, clmcnt=clamcnt, offset=logep ; Casalty Actaral Socety E-Form, Wnter 010 1

13 Drect Analyss of Pre-Adsted Loss Cost, Freqency or Severty n Tweede Models 5. REFERENCES [1] EMBLEM Gettng Started Gde, EMB Software Lmted, [] EMBLEM User s Gde, EMB Software Lmted, [3] Jørgensen, B., and M. C. P. de Soza, Fttng Tweede's comond Posson model to nsrance clams data, Scandnavan Actaral Jornal, 1994, [4] Ohlsson, E. and B. Johansson, Credblty theory and GLM revsed, Research Reort 003:15, Mathematcal Statstcs Stockholm Unversty, 003. [5] SAS/STAT User s Gde, Cary, NC: SAS Insttte Inc., [6] Smyth, G. K. and B. Jørgensen, Fttng Tweede's comond Posson model to nsrance clams data: dserson modellng, ASTIN Blletn, 00, 3 (1, [7] Yan, J., and J. Gszcza, M. Flynn, C. P. W, Alcatons of the Offset n Proerty-Casalty Predctve Modelng, Casalty Actaral Socety E-Form, Wnter 009, Bograhy of the Athor Sheng G. Sh has ffteen years of exerence n analyzng nsrance data. He has a Ph.D. n statstcs from the Unversty of Texas at Astn. He s crrently wth Safeco Insrance, a member of Lberty Mtal Gro. Casalty Actaral Socety E-Form, Wnter