Testing Adverse Selection Using Frank Copula Approach in Iran Insurance Markets

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1 Journal of mathematcs and computer Scence 5 (05) Testng Adverse Selecton Usng Frank Copula Approach n Iran Insurance Markets Had Safar Katesar,, Behrouz Fath Vajargah Departmet of Statstcs, Shahd Behesht Unverst, Iran, Department of Statstcs, Unverst of Gulan, Iran Artcle hstor: Receved November 04 Accepted Februar 05 Avalable onlne Februar 05 Abstract Exstence of adverse selecton n nsurance markets could have rreversble effects on enterprse decson-makng process and oblgatons of nsurance companes. In ths artcle, testng adverse selecton s done b jontl modelng the coverage selecton and accdents frequenc usng Frank's copula, where the dependence parameter states the exstence of relatonshp between coverage selecton and the frequenc of accdents. Our margns are modeled b ordered logstc regresson model for the coverage selecton and negatve bnomal regresson model for the accdents frequenc. The copula model s calbrated usng 59,547 one-ear cross-sectonal cases of collson nsurance coverage of Iran Insurance co. The results ndcate a sgnfcant postve coverage selecton-accdents frequenc relatonshp. Kewords: Adverse Selecton, Copula, Logstc Regresson, Collson Insurance.. Introducton Ever snce the semnal work of Sh and Valdez [6], statstcan theorsts have devoted much effort to usng Archemedan copula models for research on asmmetrc nformaton. Ths paper searchs on contrbute to examn the applcaton of these models on collson nsurance data. An ordnar approach to testng adverse selecton s correlaton testng between contract selecton and accdents frequenc that we conduct ths work wth usng of copula. In ths artcle we follow Sh and Valdez [6] approach and we use a bg sample data set that we catched from an nsurer operatng n the collson nsurance market n Iran where contaned 59,547 contracts. We use collson nsurance for modelng and rank them n three level. Lke the Sh and Valdez [6], we set accdents frequenc n levels 0,,, 3 and 4 and for margnals dstrbutons. Then we use ordered regresson model and negatve bnomal. For testng adverse selecton we emplo Frank copula that t can show both negatve and posstve assocaton between margns. Wth fttng to data, our result show posstve assocaton between polc selecton and accdent frequenc. 54

2 H. Safar Katehsar, B. Fath Vajargah / J. Math. Computer Sc. 6 (05) These results calculated wth dependence parameter.3 and for testng our result we appl Claton and Gumble copula. For the rest of the artcle, we have organzed t as follows. Secton ntroduces the bvarate copula model for testng adverse selecton. Secton 3 descrbes the emprcal data and calbrates the model usng ths data. In addton, the secton dscusses procedures used to examne model goodnessof-ft and ts mplcatons. A robust test of model mplcatons s provded n ths secton. Secton 4 concludes the paper wth a dscusson of addtonal further work.. Pror works on adverse selecton n nsurance market Akerlof [] demonstrated the problems arsng from nformaton asmmetr based on the used car market and referred to defectve used cars as lemons. Recent emprcal studes n the compettve automoble nsurance market show no sgns of adverse selecton n these markets. To take some of the outstandng studes, Chappor and Salan e [] fnd no sstematc relatonshp between rsk and coverage n the French automoble nsurance market. These studes use data from the compettve nsurance market and ther emprcal results suggest that nsurers are successfull managng adverse selecton or moral hazard, at least n the compettve nsurance market where the can freel set premums. For more nformaton see Sato [5]. 3. Model specfcaton to testng adverse selecton For the frst, we ntroduce the nature of collson nsurance n Iran nsurance co. Wth purchase a polc from Iran nsurance co everbod coverages hmself/herself from these man and overal clams: overal accdent, overal theft and overal fre, as well as everbod can purchase one or more plethora coverage n ths lst:. Damaged caused b flood, earthquck and hurrcanes,. Broken glass alone causes other than the man clams, 3. Instant stolen vehcle parts and accessores, 4. Damage caused b splls or splashes of pant, acd and chemcals, 5. Compensaton b not usng the vehcle n repar perod, 6. Slppage (onl n mnor damage). For the same argument then, we consder a multnomal measure and examne three tpes of coverage, ordnall rankng them from lowest to hghest: frst level: overal coverage of collson nsurance, second level: overal coverage of collson nsurance n addton one or two plethora coverage(s) n the lst, thrd level (comprehensve): overal coverage of collson nsurance n addton three or four or fve or all of plethora coverage(s) n the lst. Wth a cross-sectonal set of observatons, we begn b lettng and ndcate the choce of coverage and accdents frequenc, respectvel, for polcholder. Here, wth possble values of,, or 3, represents the choce of frst level (overal), second level, and thrd level (comprehensve) coverages, respectvel. Note that and are the observed varables whose values wll be determned accordng to the correspondng latent varables defned b and, respectvel. One could vew as the polcholder s preferred polc coverage and, as the nherent rsk level of the polcholder. We choose to model the observable varables and wth a parametrc copula to be denoted b C(.,. ). Then the jont probablt mass functon of and could be expressed as: f(, ) C ( F( ), F( )) C F F C ( F( ), F( )) C F F ( ( ), ( )) ( ( ), ( )) () where F and F are the cumulatve dstrbuton functons of and, respectvel. Due to the parametrc feature of the copula model, one needs specfcatons of the dstrbuton functon F and F for model dentfcaton. The coverage choce s measured on an ordnal scale. Thus, an ordered 55

3 H. Safar Katehsar, B. Fath Vajargah / J. Math. Computer Sc. 6 (05) multnomal model s used to descrbe the relatonshp between the response and the latent varable : f f () 3 f where and are unknown thresholds to be addtonall estmated. We consdered an ordered logstc regresson model n the estmaton. Henceforth, we use, ' exp( ( x)) F( ) P( Y ), ' (3) exp( ( x)), 3 accdents frequenc s specfed usng a negatve bnomal regresson model. More specfcall, ts probablt mass functon s expressed as: f ( ) P ( Y ) (4) wth the dsperson parameter for polcholder. The model specfed n ths secton b ts nature s full parametrc and can therefore be easl estmated usng lkelhood-based methods. To accommodate the fact that the choce of coverage and the frequenc of accdents could possbl be ether postvel or negatvel assocated, we consder the Frank copula whch permts such flexblt: u u ( e )( e ) C ( u, u ; ) lo g[ ], (5) e where s the dependence parameter that captures the assocaton between the two responses. The flexblt of allowng for ether drecton of assocaton has been one of the prmar reason for ts populart n applcatons n nsurance, fnance and medcal statstcs. Addtonal statstcal propertes of the Frank s faml of copulas n (5) have been explored n Genest [3] and Nelsen [4]. 4. Calbratng the model Data used to calbrate the model specfed n Secton 3 was drawn from a portfolo of automoble nsurance polces of a major nsurer n Iran. In partcular, we use the observatons n calendar ear for ths nsurer where we have a total of 59,547 polces that were recorded n the collson nsurance portfolo. Smlar to several jursdctons worldwde, Iran requres drvers to have, at the mnmum, a thrd part lablt coverage to be able to drve a vehcle on the road, and at the same tme, drvers have the lbert to choose beond ths mnmum level of coverage. Our data set ndeed comes from a subsample of the Iran nsurance co portfolo. Table provdes a summar of the frequenc statstcs for our two prmar varables of nterest. 56

4 H. Safar Katehsar, B. Fath Vajargah / J. Math. Computer Sc. 6 (05) eable. Number and percentage of polc choce and reported accdents Polc Choce 3 Clam Count Total Number Percent Total Number Percent Estmaton results and dscusson The resultng (maxmum lkelhood) estmates for the copula model are presented n table. In examnng the effect of vehcle characterstcs, we fnd that the age of the vehcle exhbts sgnfcant effect on both polc choce and accdent occurrence. Another explanator varable that s worth makng an observaton s the NCD (No Clam Dscount) factor. Frst, there s a sgnfcant effect of NCD on polc choce n the sense that a polcholder wth a hgh NCD tends to purchase better nsurance coverage on ts vehcle. Consstentl, a drver wth a lower NCD tends to have more accdents. Table. Estmates of Frank copula model for all reported accdents Choce - Cumulatve Logt Rsk - Negatve Bnomal Estmate StdErr Estmate StdErr Choce Choce Rsk-ntercept Choce-sex (woman) Rsk -sex (woman) Choce-vehcle age Rsk -vehcle age Choce-(NCD=) Rsk -(NCD=) Choce-(NCD=3) Rsk -(NCD=3) Choce-(NCD=4) Rsk -(NCD=4) Choce-vehcle applcaton () Rsk -vehcle applcaton () Choce-vehcle tpe () Rsk -vehcle tpe () Dsperson Dependence parameter Loglkelhood Qualt of ft tests Goodness-of-ft tests are performed for the margnals as well as for the copula. For margnal dstrbutons, we exhbt the observed and ftted frequences for both the polc choce and accdents frequenc n table 3. The consstenc between the actual and ftted frequences suggests ver satsfactor ft for both margnals. Accordng to table, the estmaton of dependence parameter n the Frank copula s roughl.3, whch translates to a Spearman s rho coeffcent of roughl percent. Ths provdes an evdence of the postve assocaton between the polc choce and level of rsk of the polcholder. We fnd that drvers wth better coverage tend to be more prone to make clams. Ths observaton would be explaned b the presence of adverse selecton. A fnal examnaton of the copula model s a robustness test. We re- 57

5 H. Safar Katehsar, B. Fath Vajargah / J. Math. Computer Sc. 6 (05) calbrated the copula model under two other customarl used Archmedean-tpe copulas, the Gumbel copula and the Claton copula. The estmated postve relatonshp that we alread observed based on the Frank copula between polc choce and rsk n Secton 3, suggests both copulas are elgble to test for possble robustness. The dependence parameter of the Gumbel copula s. that translates to a Spearman s rho of 0.5. The dependence parameter of the Claton copula s 0.6 that corresponds to a Spearman s rho of 0.0. Both models suggest a postve assocaton between the polc selecton and rsk level of the polcholder. The results ndcate a sgnfcant postve coverage selecton-accdents frequenc relatonshp. Table 3. Goodness-of-ft tests of the margnals Choce Rsk Value Observed Ftted Value Observed Ftted Concluson In ths paper we used a bvarate copula regresson method to jontl examne the polcholder s coverage choce and the level of rsk. To test for the presence of adverse selecton, the polcholder s coverage selecton was measured b an ordnal categorcal varable and the degree of rsk was approxmated b an expost rsk measure, the number of tmes a polcholder has clamed n a calendar ear. To calbrate the copula model, we used a cross-sectonal emprcal observaton of an nsurance portfolo from a major automoble nsurer n Iran. After controllng for the rsk factors (polcholder and vehcle characterstcs) observed b the nsurer, we found evdence of a strong postve coverage-rsk assocaton, whch suggests the possble exstence of prvate nformaton b the polcholders. References [] G.A. Akerlof, The market for lemons : qualt uncertant and the market mechansm, Quarterl Journal of Economcs, 84 (970) [] P-A. Chappor, B. Salan e, Testng for asmmetrc nformaton n nsurance market, Poltcal Econom. 08 (000) [3] C. Genest, "Frank s faml of bvarate dstrbutons, Bometrka. 73 (987) [4] R.B. Nelsen, An ntroducton to Copulas, Portland, Sprnger, New York, (006). [5] K. Sato, "Testng for asmmetrc nformaton n the automoble nsurance market under rate regulaton, The Journal of Rsk and Insurance. 73 (006) [6] P. Sh, E.A. Valdez, A Copula approach to test asmmetrc nformaton wth applcatons to predctve modelng, Insurance Mathematcs and Economcs. 49 (0)

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