Covariate-based pricing of automobile insurance

Transcription

1 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 José Antono Ordaz (Span), María del Carmen Melgar (Span) Covarate-based prcng of automoble nsurance Abstract Ths paper analyzes the most sgnfcant varables that explan the exstence of clams n the automoble nsurance sector. Ths queston s a key ssue for nsurers. Knowng all these factors, they could eventually fx more precsely ther premums and reach hgher levels of effcency. To acheve our target, a probt model specfcaton s provded usng a database from a Spansh prvate nsurance company. The results of our work pont out the sgnfcance of some varables such as the polcyholders drvng experence or ther regon of resdence. Addtonally, our research shows evdence of the exstence of relatonshps between the clams and the ncreasng polces levels of nsurance coverage, thus suggestng the presence of some usual problems n the nsurance markets such as the moral hazard and the adverse selecton. Keywords: automoble nsurance, clams, probt model. Introducton Automoble nsurance s one of the most mportant branches of the whole nsurance ndustry n all modern countres. In the case of Span, n 2008 the overall amount of the premums of ths sector represented 37.24% of total revenue from non-lfe nsurance, and % of all nsurance busness (DGSFP, 2009). These fgures justfy why automoble nsurance s the focus of much research. Its characterstcs make t also conducve to the mplementaton of econometrc models that seek to test the valdty of certan theoretcal results gven n the markets n the presence of asymmetrc nformaton. The works by Boyer and Donne (1989), Puelz and Snow (1994), Donne, Gouréroux and Vanasse (1999), and Chappor and Salane (2000) are some essental references on ths topc. Other studes have focused on analyzng the number of casualtes suffered by drvers, as well as dentfng the factors nfluencng t. In ths sense, Melgar, Ordaz and Guerrero (2004, 2005, 2006) have used count data models to determne the most mportant varables when estmatng the number of clams that polcyholders make to ther companes. Shankar, Mlton and Mannerng (1997), Rchaudeau (1999), and Lee, Stevenson, Wang and Yau (2002) are other papers whch deal wth ths ssue. The man objectve of the present analyss s to dentfy whch varables are the most relevant n the determnaton of the probablty process that the polcyholder makes or not clams. Characterstcs of the nsured vehcle, such as ts category and use, others relatng to the drver, such as age, gender, drvng experence and area of resdence, and those relatng to the polces, as ts annual premum and the chosen level of nsurance coverage, are some of the varables that are ordnarly taken nto consderaton by nsurance companes. To know the factors that may affect the clams, t s a matter of great nterest to nsurers. Fnally, the avalablty of a good rsk model would allow nsurance frms to establsh more precsely the premum that must be pad by ther polcyholders, whch would gve greater effcency to ths mportant ssue. To acheve the objectve, n ths study we take the nformaton on the varables above outlned from a Spansh nsurer, to whch we apply a probt dscrete bnary choce model to explan where the varable s defned so that t reflects the report of clams, by assgnng the values 1 and 0, respectvely. The paper s structured nto 5 sectons. After the ntroducton, secton 1 contans a descrpton of the man features of the data wth whch we have worked. In secton 2, we explan the econometrc model to be used. The results are then presented n secton 3. The fnal secton offers some bref conclusons. The paper fnshes wth the References and an Appendx contanng the lst of varables used n the present work. 1. Analyss of the sample In order to carry out the research that we ntend to perform, we have a database wth nformaton on a total of 130,000 polces, whch has been provded by a Spansh prvate nsurer. The tme nterval for ths dataset covers the perod from June 16, 2002 to June 15, For computatonal reasons, a random sample of 15,000 polces has been used, of whch we know certan characterstcs related to the type and use of the vehcle: age, gender, years of drvng experence and area of resdence of the polcyholder; and also the annual premum he/she pays and the level of nsurance coverage of the polcy. These varables, or a categorcal verson of them, are taken as explanatory varables. On the other hand, we have consdered a bnary varable (that we have labelled CLAIM) whch 1 and 0 values reflect f the nsured has made or not some knd of clam, respectvely 1. José Antono Ordaz, María del Carmen Melgar, As mentoned above, all varables we used are defned n the Appendx at the end of ths paper. 92

2 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 Snce our prmary objectve s to analyze whch factors are the most sgnfcant n determnng the report or non-report of any type of clam, we especally focus on the dfferences that arse n each of the avalable varables regardng ths ssue. Frst of all, we must emphasze the large number of zeros that exhbts the dependent varable: 11,558 polcyholders have not declared any loss, representng a rate of 77.1% from the total number of nsured drvers of our database. The vehcles have been classfed nto fve groups accordng to the type they belong: toursm or van, truck, coach, motorcycle and specal vehcle. The category that ncludes toursms and vans s the most common one, accountng for 80.5% of the total. After them, specal vehcles represent 10.3% and motorcycles appear wth 7.7%. Trucks and coaches only gve, jontly, the remanng 1.5%. As to the clams n each category, Table 1 shows 26.5% of cars or vans have regstered some clam n the reference perod, and the fgure for trucks s very smlar: 25.3%. In contrast, the behavor exhbted on the one hand, by coaches, and on the other hand, by motorcycles and specal vehcles, s very dfferent: 52.2% of coaches have reported some clam, but only 7% of motorcycles and 6.8% of specal vehcles regstered clams. Table 1. Clam rates by types of nsured vehcles Clams Types of vehcles No Yes Total Car or van 73.5% 26.5% 100.0% Truck 74.7% 25.3% 100.0% Coach 47.8% 52.2% 100.0% Motorcycle 93.0% 7.0% 100.0% Specal vehcle 93.2% 6.8% 100.0% All categores 77.1% 22.9% 100.0% The descrptve analyss of the fgures for the man use of the nsured vehcle ndcates that 79.8% of them are for personal use. Wth respect to professonal use (whch ncludes publc servce, ndustral uses, freght transport, school transport, passenger transport and general farmng), t accounts for 19.6% and fnally, the category of other (whch was rental concerns, drvng school, sale and wthdrawal of drvng lcenses) s only 0.6% of the total. Table 2 presents the detals of clams for each one of the uses we have ndcated. One can see the professonal and, ndeed, any other uses show lower clam rates, 16.3% and 12.0%, respectvely, than the ones whch are regstered n the case of personal use: 24.7%. Table 2. Clam rates by uses of nsured vehcles Clams Uses of vehcles No Yes Total Personal 75.3% 24.7% 100.0% Professonal 83.7% 16.3% 100.0% Other 88.0% 12.0% 100.0% All categores 77.1% 22.9% 100.0% Among the characterstcs of the nsured people, age s the frst varable we analyze. Four ntervals were consdered: years old, years old, years old and more than 70 years old. The majorty of consdered drvers belong to the mddle ntervals. In partcular, polcyholders between 26 and 45 years old represent 39.8% of the total and those 46 to 70 years old 51.8%. The remanng 8.4% s dstrbuted so that the younger group of 14 to 25 years old s accountng for 3.1% and that of the older ones, for 5.3%. In regard to the clams, Table 3 shows that the percentages of polcyholders who have some are for the frst three age groups around 22-24%. The category of polcyholders wth more than 70 years old, meanwhle, shows a remarkable lower fgure of clam rate: only 15.9%. Table 3. Clam rates by age of polcyholders Clams Groups of age No Yes Total [14-25] years old 76.6% 23.4% 100.0% [26-45] years old 75.8% 24.2% 100.0% [46-70] years old 77.3% 22.7% 100.0% More than 70 years old 84.1% 15.9% 100.0% All categores 77.1% 22.9% 100.0% The drvng experence s another aspect to be taken nto consderaton. Ths was done through the varable referrng to the tme of possesson of a drvng lcense. Consderng all the nsured drvers, only 0.8% has less than 2 years of experence. However, ts clam rate accounts for 35.5%. Ths s a much hgher percentage than the one of the experenced drvers, namely 22.9%. We have consdered the gender of the polcyholders as well. The descrptve analyss of ths queston ndcates that 85.3% are men, and 22.3% of them made some clam. Regardng women, they show a slghtly hgher fgure, whch s 26.5%. The area of resdence s also a hghly relevant varable. Ths varable s normally taken as a proxy for the polcyholder usual drvng area. We have worked wth the dvson of the Spansh terrtory at the level of NUTS-1 Regons, accordng to the crteron of Eurostat. The exact defnton of each one of them can be seen n the Appendx of the present research. The Southern regon s the most represented one, brngng together 46.3% of the total nsured. We should then menton the followng regons: Central, whch accounts for 16.8%; North-western wth 15.4%; and Eastern, whch ncludes 12.1% of the whole of polcyholders. The other four regons share the remanng 9.4%. As to the clams found n each of the regons, Table 4 shows that resdents n the frst regon ( Southern ) presented clams n 24.0% of cases. Of the rest, t must be ponted out the sgnfcantly hgher fgure of Madrd, where the percentage of polcyholders wth clams 93

3 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 reaches 28.7%. At the other extreme, we see the Canaras and, especally, the Central regon, where the fgures of clams are 21.3% and 19.0%, respectvely. Table 4. Clam rates by areas of resdence of polcyholders Clams Areas of resdence No Yes Total Canaras 78.7% 21.3% 100.0% Central 81.0% 19.0% 100.0% Ceuta-Mellla 75.0% 25.0% 100.0% Eastern 75.6% 24.4% 100.0% Madrd 71.3% 28.7% 100.0% North-eastern 75.6% 24.4% 100.0% North-western 77.3% 22.7% 100.0% Southern 76.0% 24.0% 100.0% All categores 77.1% 22.9% 100.0% The last block of analyzed varables refers to features drectly related to the polces. In partcular, t has been taken nto consderaton the annual amount pad as premum and the level of nsurance coverage. Wth respect to the amount of the premum, t has been dvded nto four ntervals (n = euros): (0-300], ( ], ( ], and more than 600. The majorty of polcyholders belong to the nterval of cheapest premums, representng 32.2% of the nsured drvers of our database. The two mddle ntervals provde smlar fgures, representng 26.8% and 23.2%, respectvely. Fnally, the premums above 600 are only 17.8% of the total. As to the clam rates, the postve and growng relatonshp s very notceable between the amount of the premum and the report of clams shown n Table 5. Whle the percentage of clams of the polces of less than 300 s 11.8%, ths number s gradually rsng from fnally reachng the 36.9% n the case of polces wth premums n excess of 600. Table 5. Clam rates by amount of polces annual premums Clams Groups of annual premums (n ) No Yes Total (0-300] 88.2% 11.8% 100.0% ( ] 77.4% 22.6% 100.0% ( ] 71.9% 28.1% 100.0% More than % 36.9% 100.0% All categores 77.1% 22.9% 100.0% Regardng the coverage of the polcy, t has been dvded nto three levels based on the guarantees of the nsurance contract. The low level ncludes only the compulsory guarantees under the law; polces wth ths level of coverage account for 54.3% of the total. Those who want any addtonal optonal guarantee, such as that concernng the glass breakage, fre and/or theft, are ntegrated n the level of coverage that we have labelled as medum. Ths s the type chosen by 37.8% of nsured drvers of our database. Fnally, the hgh level also covers the own damage of the vehcles; here s the 7.9% of the total nsured. The analyss of clams for each one of the levels of nsurance coverage can be seen n Table 6. In ths, one can observe how the percentages wll grow as does the level of nsurance coverage. Thus, for the lowest level, the percentage of cases wth clams that s collected s 16.1%, for the ntermedate s 29.3%, and for the hghest one s 39.4%. Table 6. Clam rates by polces nsurance coverage levels Clams Levels of nsurance coverage No Yes Total Low 83.9% 16.1% 100.0% Medum 70.7% 29.3% 100.0% Hgh 60.6% 39.4% 100.0% All categores 77.1% 22.9% 100.0% Ths result s very nterestng. Even though ths should not necessarly mply that polcyholders wth dfferent levels of nsurance coverage dffer n rsk, t s true that from the perspectve of nsurers they really fnd these dfferences n the clam rates. On the one hand, the relatonshp of ths varable wth clams could ndcate a stuaton of moral hazard arsng from behavor by those excessve careless drvers who enjoy a wde coverage. Addtonally, on the other hand, t may also reflect the exstence of adverse selecton behavor as a drver aware of hs/her proneness to clams would generally contract a hgher coverage for reassurance. Both ssues are among the man problems that are seen n the nsurance market. 2. Model specfcaton A probt econometrc model s provded n ths research. The bnary dscrete choce models such as probt, are characterzed by the endogenous varable Y whch only takes two values, 1 and 0, correspondng to each of the two possble scenaros that are consdered 1. In ths study, the endogenous varable Y takes the ssue of whether the -th polcyholder made or not some type of clam to the nsurer such that: 1 f the -th polcyholder made some clam Y = 0 otherwse If we assume that the varable Y depends on a set of 1 There exst other bnary dscrete choce models, such as the lnear probablty model (LPM) and the logt model. As t s well known n the lterature, the frst one has some theoretcal lmtatons such as the assumpton of the constant margnal effects of the explanatory varables on the studed probablty. Wth respect to the logt model, ther results are usually qute smlar to those of the probt model. Our choce between both of them has been based on the better goodness-of-ft shown by the probt model n our subsequent emprcal analyss. 94

4 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 explanatory varables X, followng the general econometrc specfcaton: Y = F( β ) + ε, (1) X where ε represents the usual random dsturbance error, then takng nto account the assumpton that E ε =, we have: [ ] 0 X [ Y X ] E[ F( X β) X ] + E[ ε X ] F( X β) E = =. (2) Moreover, f we calculate the condtonal expectaton value of Y n terms of probabltes, then we wll obtan: E [ Y X ] = Y P( Y X ) = = P ( Y = 1 X ) + 0 P( Y = 0 X ) = 1 = P Y = 1 X ). (3) ( From ths t follows that: [ Y X ] F( X ) = P( Y 1 X ) E = β =. (4) Consderng that the varable Y can only take the values 1 and 0, meanng the model mples, therefore, t allocates a certan condtonal probablty that Y = 1, denoted by P,.e.: P ( Y = 1 X ) = P = F( X β ), (5) and consequently: P( Y = 0 X ) = 1 P = 1 F( X β ). (6) Fnally, the model estmates the probablty that the polcy of the -th ndvdual records any clam: Y ˆ = Pˆ = F( X ˆ) β. (7) From ths general approach, common to any bnary dscrete choce model, the probt model s characterzed by usng the dstrbuton functon for a standard normal: Φ. So, we wll have: Z F ( X β ) = Φ( X β ) = Φ( Z ) = φ( s) ds, (8) where: 2 s 2 1 φ ( s) = e (9) 1/ 2 (2π ) s the densty functon of normal dstrbuton and s s a latent ntegraton varable wth mean 0 and varance 1. Regardng the nterpretaton of the model, the estmated parameters do not drectly determne the margnal effect of changes n exogenous varables X j on the estmated probablty (as n the case of a lnear model). Its sgn and magntude, however, are ndcatve of the drecton of change and the relevance of these varatons. The margnal effect s then computed as a result of the product of the densty functon of standard normal dstrbuton at a determned pont (the polcy of the -th ndvdual) and the correspondng parameter: P X j Φ( Xβ) = = φ( Xβ) βj. (10) X j The magntude of the varaton of probablty s based on the values of each and every one of the explanatory varables and ther respectve coeffcents n the partcular observaton we want to consder. Therefore, n order to obtan a representatve value of these margnal effects, they are usually evaluated for the mean values of the regressors. If X j s a dummy varable, whch s the case wth most of the explanatory varables n our model, the analyss of ther average effect s done through the dfference of the values provded by: E [ = 1] and [ = 0] Y X k E Y X k. (11) Wth respect to the estmaton of the model, t wll be done through the maxmum lkelhood method that provdes consstent and asymptotcally effcent estmators. To test the ndvdual sgnfcance of each parameter (and consequently of the correspondng explanatory varable) the Wald test s used, whose z-statstc, follows a standard normal dstrbuton. In such models, where the endogenous varable takes only values 1 or 0, the usual coeffcent of determnaton R 2 s not vald as a measure of goodness-of-ft. Instead, other alternatves have been developed, such as the McFadden R 2, rangng between 0 and 1 (although ts nterpretaton s not drectly comparable to the lnear R 2 ), the LR-statstc or lkelhood rato and pseudo-r 2 of predcton-evaluaton. Fnally, as for the detecton of the exstence of possble problems of endogenety n the model, one can use the so-called Hausman test (1978). 3. Estmated model and structural analyss of results Table 7 shows the probt model specfcaton of regster of clams whch has fnally been selected from among the varous tests that have been carred out 1. 1 As mentoned n secton 2, we have also used a logt model n our tests. We have fnally chosen a probt specfcaton due to goodness-of-ft crtera. 95

5 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 Dependent varable: CLAIM Table 7. Model estmaton output Model: Bnary probt Method: Maxmum lkelhood Included observatons: 15,000 Varable Coeffcents Margnal effects Standard error z-statstc P-value CONSTANT COACH MOTORCYC SP_VEH OTH_USE EXP<2Y CENTRAL MADRID NORTWEST COV_MED COV_HIGH Mean dependent varable LR statstc St. devaton dependent varable Degrees of freedom 10 Log lkelhood -7, Probablty of LR statstc Restrcted log lkelhood -8, McFadden R Expectaton-predcton evaluaton for bnary specfcaton (success cut-off: C = 0.5) Correct predctons for dependent varable = 0 11,530 Correct predctons for dependent varable = 1 18 Pseudo-R 2 (%) Ths choce s made based on the sgnfcance of the explanatory varables (at ρ < 0.05) and also to ensure goodness-of-ft and ts global sgnfcance. In ths regard t s noted that the value of the pseudo-r 2 of the predcton-evaluaton of the chosen specfcaton s 76.99%. Ths value, although not very hgh, s qute sgnfcant 1. Regardng the endogenety between the varables of the model, the Hausman test confrmed the presence of ths queston. Ths lmtaton s usual n ths type of research and s generally assumed. All varables ntroduced n the model are qualtatve, so ther entry s done through dummy varables 2. It should also be hghlghted that some of the ntally selected varables have not been sgnfcant enough n some specfcatons we have made, or have shown evdence of multcollnearty; for that reason, they have not been consdered n our fnal adjustment 3. The results of ths estmate, together wth the structural analyss that has been performed subsequently from them, allow the followng conclusons: The frst group of varables s devoted to the dfferent types of vehcles. We have taken, as the base category, 1 All the econometrc results shown n Table 7 have been carred out wth EVews v.6, except that references to margnal effects of the explanatory varables whch have externally been calculated accordng to equatons (11). 2 The ntroducton of dummy varables s performed addtvely, thus avodng problems that could arse when ncludng nteracton terms (Rons and Harrson, 1988). 3 That s the case of the age and the gender of the nsured drvers, and the annual premum of the polcy, as we wll dscuss later. the cars and vans. In comparson, all categores have proved statstcally sgnfcant except for the trucks. The ncdence of these categores n the regster of clams s unequal both quanttatvely and n the sgn. So, whle the drvers of coaches show a greater propensty for clams as the set of categores that do not appear explctly, motorcycles and specal vehcles have less chance of clams. Fgure 1 shows the results of structural analyss performed on ths varable. It can be seen that the estmated average probablty of clams 4 for coaches s Meanwhle, for cars or vans, jontly wth trucks, t s Motorcycles and specal vehcles, however, offer substantally lower fgures, specfcally, and 0.097, respectvely Car or van Coach Motorcyle Specal vehcle Types of vehcles Fg. 1. Estmated average probablty of clams by types of nsured vehcles 4 These values are obtaned by always takng the mean values of the other explanatory varables. 5 It s noted that the result of motorcycles, n prncple, could be strkng. However, ths may be due to the hard demands the nsurance company may be mposng to the polcyholders of such vehcles. 96

6 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 Another relevant varable s the use of vehcles. In partcular, t has been sgnfcant through the category related to other uses (OTH_USE), whch ncludes all other uses dfferent from personal and professonal ones, as defned n the descrptve analyss of the data gven n secton 1. Compared to these two uses, the other category shows a negatve relatonshp to the clams; n partcular, the probablty of makng a clam n ths case s 14.1% lower. The experence of drvers s revealed as one of the most mportant varables n explanng the clams n the sector. As expected, the lack of experence s a decsve factor n the occurrence of accdents. Structural analyss of results leads us to verfy that polcyholders wth ther lcences less than 2 years old have an average probablty of sufferng a loss equal to 0.494, whle ths probablty for those who possess a drvng lcence for more than 2 years s (Fgure 2) Up to 2 years More than 2 years Polcyholders' drvng experence Fg. 2. Estmated average probablty of clams by polcyholders drvng experence The area of resdence of the nsured drver and therefore, ther usual traffc area s another sgnfcant varable to explan clams n automoble nsurance. Of the eght great regons n to whch the Spansh terrtory s dvded, three have behaved sgnfcantly dfferent from the rest: the Central, Madrd and North-western. The frst one refers to the Autonomous Communtes of Castlla y León, Castlla-La Mancha and Extremadura, the second one corresponds to the Autonomous Communty of Madrd and the thrd one concerns Cantabra, Galca and Asturas. Whle the nfluence of the Central and Northwestern regons s negatve n the clams, the Madrd regon shows a postve relatonshp whch s also greater n quanttatve terms than others. Fgure 3 gves the numbers of the structural analyss from the modellng results on ths varable. It can be seen that the estmated average probablty of clams s consderably greater n Madrd (0.351) than n the rest of the Spansh State (0.273). However, the other two regons that we have hghlghted appear wth lower numbers Central Madrd North-w estern Rest of Span Polcyholders' resdence areas Fg. 3. Estmated average probablty of clams by polcyholders resdence areas The last varable that has shown ts relevance s the extent of polces nsurance coverage. Startng from the lowest level as base category, the other two categores we have consdered,.e. the ntermedate (COV_MED) and the hghest levels (COV_HIGH), play an mportant role n the model. The nfluence of ths varable on clam rates s clearly postve and ncreasng. As can be seen n Fgure 4, the estmated average probablty of clams for each one of the possble levels of nsurance coverage, from lowest to hghest, s 0.197, and Our econometrc analyss confrms the results we saw n our prevous descrptve analyss. Ths can nvolve nherent behavors of the nsurance market such as moral hazard and/or adverse selecton. We fnd that ths s one of our most mportant results. Fnally, t s necessary to note that the varables related to age and gender of the nsured drvers and the polcy premums, ntally consdered n the descrptve analyss, have not been retaned n the econometrc estmaton of the model. In the case of age, ther categores have not been sgnfcant enough; ts effect, perhaps, s most lkely felt ndrectly through the varable experence of the drver. Regardng gender, t has not been sgnfcant ether. And as the premums are concerned, because of problems of endogenety n the extent of polcy coverage, we decded aganst ts entry nto the fnal specfcaton of the model Low Medum Hgh Polces' coverage levels Fg. 4. Estmated average probablty of clams by polces coverage levels 97

7 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, 2010 References 98 Conclusons The weght that automoble nsurance ndustry nowadays represents n the whole nsurance busness n the developed economes, and the mportance for companes of knowng anythng related to ts actvty, are the fundamental reasons that have motvated ths work. Thus, the focus of analyss we have carred out has been the determnaton of the most sgnfcant varables n the regster of clams. To ths end, we have worked wth data relatng to 15,000 polces provded by a Spansh prvate nsurance company to whch we have appled a probt bnary model, snce we consder an endogenous varable takng only 1 and 0 values, dependng on whether the polcy has or has not recorded some clam. After developng an ntal exploratory descrptve analyss, the most mportant varables contaned n the database n relaton to clams report have been carred out to perform the econometrc estmaton of a probt model. Our fnal selected specfcaton has allowed us to pont out what s the nfluence of each of these varables wth respect to the clams and to estmate ther margnal effects, conductng a structural analyss of the results and estmatng the average odds of clams for each category consdered. Hghlghts of ths structural analyss are of mportance n clams of the type of vehcle (for example, coach), as well as of the polcyholders drvng experence. Thus, a coach can be up to 28.8% more lkely to clam than most vehcles. Regardng drvng lcense, people whose experence s less than 2 years can ncrease ther probablty of clams up to a 23.4% aganst those who are more expert. Also, notable results have been obtaned from the use of vehcle and the area of resdence of the nsured drver. Whle other uses than personal and professonal ones have exhbted less proneness to regster a clam (partcularly up to 14.1%), to lve n regons such as the Autonomous Communty of Madrd makes the probablty of a clam to be 7.8% hgher than n most of the Spansh terrtory. Fnally, what deserves a specal menton s the postve relatonshp that has been observed between the clams and the varable measurng the levels of nsurance coverage. It was found that there s an ncreased regster of clams wth ncreasng level of nsurance coverage of the polcy. The rsk of clams s 18.6% hgher n cases n whch the nsured enjoys the greatest level of coverage aganst the lowest level, the mnmum allowed legally. Ths may provde evdence of moral hazard and/or adverse selecton stuatons. Both aspects are closely lnked to nsurance markets wth asymmetrc nformaton and our analyss appears to ndcate them. To conclude, we must notce that ths last result leads us to thnk that t could be probably preferable to establsh separate models for each level of nsurance coverage. In the same way, t could be approprate to defne dfferent models for each type of vehcles, ther uses or, even, the drvng experence or the area of resdence of the polcyholders. In ths paper, we have preferred to consder jontly all the varables and ther respectve categores n order to clarfy ther partcular sgnfcances n the analyss as a whole. Our conclusons can be now a good startng pont to further researches n the future. Acknowledgements Ths work has receved support from the Spansh Mnstry of Scence and Innovaton and FEDER grant ECO /ECON. 1. Boyer M., G. Donne. An Emprcal Analyss of Moral Hazard and Experence Ratng // Revew of Economcs and Statstcs, 1989, No 71, pp Chappor P.A., B. Salané. Testng for Asymmetrc Informaton n Insurance Markets // Journal of Poltcal Economy, 2000, No 108 (1), pp DGFSP. Seguros y Fondos de Pensones. Informe 2008, Madrd, Dreccón General de Seguros y Fondos de Pensones (DGSFP), Mnstero de Economía y Hacenda, 2009, 300 pp. 4. Donne G., C. Gouréroux, C. Vanasse. Evdence of Adverse Selecton n Automoble Insurance Markets // Automoble Insurance: Road Safety, New Drvers, Rsks, Insurance Fraud and Regulaton, Donne G., C. Laberge- Nadeau (eds.), 1999, pp Hausman J.A. Specfcaton Tests n Econometrcs // Econometrca, 1978, No 46, pp Lee A.H., M.R. Stevenson, K. Wang, K.K.W. Yau. Modelng Young Drver Motor Vehcle Crashes: Data wth Extra Zeros // Accdent Analyss and Preventon, 2002, No 34, pp Melgar M.C., J.A. Ordaz, F.M. Guerrero. The Man Determnants of the Number of Accdents n the Automoble Insurance: An Emprcal Analyss // Études et Dossers, Nº 286. pp Melgar M.C., J.A. Ordaz, F.M. Guerrero. Dverses Alternatves pour Détermner les Facteurs Sgnfcatfs de la Fréquence d Accdents dans l Assurance Automoble // Assurances et Geston des Rsques Insurance and Rsk Management, 2005, No 73 (1), pp Melgar M.C., J.A. Ordaz, F.M. Guerrero. Une étude économétrque du nombre d accdents dans le secteur de l assurance automoble // Brussels Economc Revew Cahers Economques de Bruxelles, 2006, No 49 (2), pp

8 Insurance Markets and Companes: Analyses and Actuaral Computatons, Volume 1, Issue 2, Puelz R., A. Snow. Evdence on Adverse Selecton: Equlbrum Sgnallng and Cross-Subsdzaton n the Insurance Market // Journal of Poltcal Economy, 1994, No 102 (2), pp Rchaudeau D. Automoble Insurance Contracts and Rsk of Accdent: An Emprcal Test Usng French Indvdual Data // Geneva Papers on Rsk and Insurance Theory, 1999, No 24, pp Rons D.L., K.A. Harrson. Statstcal Interactons n Studes of Physcan Utlzaton // Medcal Care, 1988, No 26 (4), pp Shankar V., J. Mlton, F. Mannerng. Modelng Accdent Frequences as Zero-Altered Probablty Processes: an Emprcal Inqury // Accdent Analyss and Preventon, 1997, No 29 (6), pp Appendx 1. Defnton of the used varables n the econometrc analyss Dependent varable CLAIM Explanatory varables VEH_TYPE VEH_USES AGE FEMALE EXP<2Y NUTS-1 PREMIUM LEV_COV Bnary varable: 1 for any made clam; 0 otherwse Types of vehcles: Dummy varables: TRUCK (truck), COACH (coach), MOTORCYC (motorcycle), SP_VEH (specal vehcle: t ncludes overall ndustral and agrcultural vehcles). Excluded category: car or van. Uses of vehcles: Dummy varables: PROF_USE (professonal use) and OTH_USE (other uses). Excluded category: personal use. Age of polcyholders (years old): Dummy varables: AG26_45 (between 26 and 45 years old), AG46_70 (between 46 and 70 years old) and AG71_ (more than 70 years old). Excluded category: between 14 and 25 years old. Gender of polcyholders: 1 for female; 0 otherwse. Drvng experence: 1 for less than two years drvng experence; 0 otherwse. Large regons or areas (NUTS-1) of polcyholders resdence: Dummy varables: CANARIAS (Islas Canaras), CENTRAL (Central regon: Castlla-La Mancha, Castlla y León and Extremadura), CEU_MEL (Ceuta and Mellla), EASTERN (Eastern regon: Cataluña, Comundad Valencana and Islas Baleares), MADRID (Madrd), NORTEAST (North-eastern regon: Aragón, Euskad, La Roja and Navarra), NORTWEST (North-western regon: Asturas, Cantabra and Galca). Excluded category: Southern regon (Andalucía and Murca). Annual premums ( ): Dummy varables: P301_400 (between 301 and 400 ), P401_600 (between 401 and 600 ) and P601_ (more than 600 ). Excluded category: less than 301. Levels of nsurance coverage: Dummy varables: COV_MED (medum coverage level) and COV_HIGH (hgh coverage level). Excluded category: low coverage level. 99