MODELLING ACCIDENT OCCURRENCE IN CAR INSURANCE IMPLEMENTATION ON TUNISIAN DATA

Modellng Asan-Afrcan Accdent Journal Occurrence of Economcs n Car and Insurance Econometrcs, Implementaton Vol. 12, No. on 2, Tunsan 2012: 395-406 Data 395 MODELLING ACCIDENT OCCURRENCE IN CAR INSURANCE IMPLEMENTATION ON TUNISIAN DATA Noureddne Benlagha *, Lanouar Charfeddne ** and Imen Karaa *** ABSTRACT Ths paper examnes the relatve sgnfcance of the varables explanng the number of responsble accdents reported by the nsured to hs nsurer n a Tunsan nsurance company durng the 2008 year. To nvestgate the effect of the null observatons (zero values) and the heterogenety of the correspondng populaton, we use tow count models, the standard Posson model and the Zero-nflated Posson model (noted ZIP). The man emprcal result was the absence of an adverse selecton n the Tunsan nsurance portfolo of drvers of four wheels vehcles. Keywords: Car Insurance, Accdent occurrences, Zero-Inflated Count Models. 1. INTRODUCTION Ths paper deals wth the econometrcs of car nsurance, that s to say, the estmaton of the relatve sgnfcance of the varables explanng the number of responsble accdents reported by the nsured to hs nsurer n a gven perod of tme. For nstance, the Posson and two forms of the negatve bnomal model overwhelmngly domnate the receved applcatons. Indeed due to the exstence n the nsurance portfolo of a bg number of polcyholders wthout accdent on one perod of tme (one year), we consder the problem of modellng count data wth excess. At the level of practce, the presence of a zero accdent record over a perod of tme may show ether that the polcyholder had no accdent or that the accdents are not reported. In the later case, the polcyholders prefer ndemnfyng drectly the opposng party. Ths non reported accdent can also be related to a ht-and-run. Ths phenomenon s commonly known as the hunger for bonus, Lemare (1977). To nvestgate the effect of the null observatons (zero values) and the heterogenety of the correspondng populaton, we use two count models, the standard Posson model and the Zero-nflated Posson model (noted ZIP). * IHEC-SFAX : Tunsa and Unversté Pars 2, ERMES-UMR7181-CNRS, 12 place du Panthéon, 75005 Pars, France, E-Mal: blnour2002@yahoo.fr ** Assstant Professor, College of Admnstratve scences, Najran Unversty. Adress : P.O.B. 1988 Alsawad, Arport RD, 11111 Najran, Araba Saud (KSA), E-mal: Lanouar_charf@yahoo.fr *** Insttut Supéreur de Geston de Gabès, Unversty of Gabès, Rue Jlan Habb 6000, Gabès. Tunsa, E-mal: karaa.men66@yahoo.com

396 Noureddne Benlagha, Lanouar Charfeddne and Imen Karaa The general problem of count data modellng has been consdered n the statstcal and econometrcs lterature by several authors, see for nstance Bohnng et al. (1999). An excellent revew of count data models can be found n Greene (2009). For a survey of count data applcatons, one can refer to Rose (1990) who have used ths knd of models to explan the number of arlne ncdents, Lee et al. (2002) to explan the number of accdents at young drvers of whose senorty of drvng lcence s lower than one year, they emprcally show on Australan data that the ZIP model s justfed because of the over dsperson of frequency loss. Melgar and al. (2005) used a ZINB model on Spansh company data and show that ths model s more sutable to the data. In ths paper, we examne the explanatory varables of loss frequences n the 2008 year usng two count models. Besdes, we examne, partcularly, f the choce of contracts affects the number of accdent. So, a postve correlaton between the premum pad by the polcyholders and the number of accdent explans the presence of adverse selecton n the nsurance portfolo. Ths emprcal study s new on two ponts: on the one hand, t concerns a recent Tunsan data, after the mplementaton of a new no-clams bonus system. On the other hand, we use the no-clams bonus rate whch translates the ndvdual s past drvng experence. The paper s structured as follows. Secton 2 presents the models. Secton 3 descrbes the emprcal methodology. Secton 4 dscusses the emprcal results and presents a comparson between the two count models. Secton 5 concludes. 2. MODELS The number of accdents s generally consdered n the emprcal lterature as a count data. It s also usually assumed to follow the Posson or the negatve bnomal dstrbuton and then can be modelled accordngly. However, ths count varable may contan zero excess above what s to be expected from the Posson model. Therefore, the ZIP seems o be more sutable to ths modellng. We can notce an abundant lterature on the use of these models: Greene (1997), Wooldrdge (2002), Cameron and Trved (1998), Wnkelmann (2000), Yau et al. (2003),Yang et al. (2007). Before presentng the emprcal fndngs let us brefly present the dfferent count models. 2.1. The Posson Standard Model The Posson regresson model specfes that each y (number of car accdents) s drawn from a Posson dstrbuton wth parameter, whch s related to the explanatory varables X (characterstcs of ndvdual drvers). The probablty that an ndvdual wll be nvolved n perod s then: y accdents over one year or one y e P( Y y / X ) (1) y!

Modellng Accdent Occurrence n Car Insurance Implementaton on Tunsan Data 397 ' The most common formulaton for s the loglnear model, ln X,. It s easly shown that the expected number of events per perod s gven by Goureroux et al. (1984). X ', E( Y / X ) Var( Y / X ) e Hence s both the mean and varance of the dstrbuton. Moreover, usng a Posson dstrbuton to ft the observed dstrbuton of accdents for a group of ndvduals supposes mplctly that all ndvduals have the same probablty of beng nvolved n one or more accdents. 2.2. The Zero Inflated Posson Model Cragg (1971) developed dfferent models n the stuaton where for an endogenous varable, an event (as the purchase of a good or the reportng of accdents) can occur or no, as n the Tobt model (1958). If the event does not occur, the zero value s assgned to the response varable, whch s assumed to be contnuous. The process of decson s represented by a probt model and the second event (amount of the purchase or the dsaster) by a standard regresson model. Whle referrng to the realty of data, t s possble that the populaton of the polcyholders for whch Y= 0, s composed of two dfferent types of populatons: A populaton who takes the decson to partcpate n the event or the experence that accepts to report accdent when t happens. The value zero shows that the drver dd not have an accdent durng the consdered perod. A populaton who does not report a clam to ts nsurer. It s common knowledge that sometmes drvers do not report small collsons when they want to avod beng penalzed about the no-clams bonus or when the cost of the accdent s less than the excess. Here, the responsble drver ndemnfes the opposng party drectly. The other case concerns the drvers for example, hang a parked car, and do not stop to make a report (ht-and-run). Ths dstncton can be nterestng for the nsurer, because we can thnk that the no reportng of a responsble accdent (outsde of the cases of the ht-and-run) s more lnked to the non gravty of the accdent than to the objectve realty of the rsk that can be caused. Ths concept s known as the hunger for bonus. Accordng to Lemare (1995), the hanger for bonus nduces that the ntroducton of a no-clam bonus system creates a consdered vew of clam amount and frequency dstrbutons. Indeed some of the lowest clam amount wll not be reported to the nsurer. For the polcyholder, the natural queston s up to whch level of clam amount s t nterestng for me to bear the cost myself?. A standard Posson model does not allow to dstngush between these two populatons. In such case the Zero nflated Posson Model (ZIP) can be used to dstngush among the dfferent types of populaton. The dea of mxture of dstrbuton s the bases for the ZIP. Mullahy (1986), Helbron (1989, 1994) and Lambert (1992) poneered the use of regresson models based on the ZIP dstrbuton. These models do not assume that the null values and the strctly postve values are generated by the same process Greene (1994).

398 Noureddne Benlagha, Lanouar Charfeddne and Imen Karaa Wth reference to the prevous standard models, we can suppose therefore that the observed stochastc varable s the mxture of a bnary dstrbuton and a standard Posson dstrbuton or negatve bnomal dstrbuton: Y = Z Y * (4) The unobservable varable Z s modelled usng a logstc regresson to estmate the probablty that y = 0. The dependent varable z s dchotomous: For an nsured, z = 0 f the nsured dd not declare any accdents and z = 1 n the contrary case. The random varable Y * corresponds to the standard Posson model and s used to predct the value of Y for the polcyholders who reported an accdent (z = 1). Ths equaton estmates the mean of y. The ZIP model ncludes two parts, therefore: the one relatve to the count model (for Y *, to estmate the number of accdents when the nsured s n the stuaton of declaraton) and the one relatve to the zeros nflaton (Logt) that explans the probablty of non declaraton. At ths stage, for a ZIP model, we denote q the probablty of z = 0 (no reported accdent) and the Posson parameter for the accdent s frequences that depends, as prevously, of the explanatory varables (3), then the probablty dstrbuton Y s: and, for non Zero y, one has ' exp( X P( Y 0 / X ) q (1 q ) e ) Where q ' (5) 1 exp( X ) y P( Y y / X ) (1 q ) e (6) y! The probablty s number of accdents condtonally to z = 1 s equal to the uncondtonal probablty of the unobserved varable y. 3. EMPIRICAL STUDY * The paper s based on data that we receved from an nsurer who operates n the market for automoble nsurance n Tunsa. The data conssts of a cross secton of 19715 holders of drvng lcence n the year 2008. It must be noted that one regstraton corresponds to one rsk/accdent covered (f the polcyholder has a two cars, then he wll do two regstratons). Two data sets are merged. The frst data set contans the nformaton about the contracts (nputs) and the second s made up of the clams varables (outputs). 3.1. Data The contaned varables can be classfed nto three groups. 3.1.1. The Polcyholder s Characterstcs Sex: a dchotomous varable (1 f male and 0 f female). Age: consdered as a contnuous varable. Classfcaton: an ordnal varable.

Modellng Accdent Occurrence n Car Insurance Implementaton on Tunsan Data 399 Table 1 Classfcaton of Polcyholders by the Age Drvers Age Code Novce 18 age 21 0 Young 22 age 30 1 Expermented 31 age 55 2 Senor 56 age 75 3 Aged > 75 4 Occupaton: Occupaton of the drver (lberal, offcal, senor executve, mddle manager, craft, employee, unemployed, retred). Place of resdence: Tunsa s made up of 24 Governorates. So, we defne 4 regons (North, South, mddle and Coast). We used the same classfcaton as Sato s approach (2006). No clam bonus rate: s the dscount that the nsurer gves for a clams-free year. In Tunsa a new method of accordng a bonus was promoted n 2007. It conssts of 8 classes. For example; a novce drver s classfed n the 8 th and a drver wth two clams-free years s reclassfed n the 5 th class. The dataset contans also, nformaton about the car s characterstcs: Car Usage: personal or commercal car. The make of car: We classfed cars by the countres of orgn (example: French cars: Peugeot, Renault, Ctroen Italan cars: Fat ). 3.1.2. Contracts The premum pad by the polcyholder nforms about the contracts suppled by the nsurer. We can dstngush between dfferent polces on the bass of the level of the premums. In Tunsa all cars must be nsured at the Thrd party lablty nsurance (TPL). Ths nsurance covers damage nflected to other conductors or ther cars. The second type of nsurance contract s the comprehensve one. Ths nsurance covers dfferent rsks as theft, collson... Comprehensve nsurance polces are dfferentated by the value of the deductble. 3.1.3. Accdent Characterstcs Number of accdents: ths count data s treated as a dependent varable. Cumulatve payment: When ths amount s mentoned for an accdent, t represents the cumulatve expenses from the reportng date of the accdent 3.2. Explanatory Data Analyss Table 2 Dstrbuton of Accdent Occurrence by the Sex Male Female Total Y = 0 81.22% 18.78% 100% Y = 1 80.86% 19.14% 100%

400 Noureddne Benlagha, Lanouar Charfeddne and Imen Karaa The table 2 shows that 81.22% of polcyholders wth no accdent are male and 18.78% are females. Whle 80.86% of the polcyholders, wth at least one accdent, are males and 19.14% females. Table 3 Dstrbuton for Clam Number Number Frequency Percentage Cumulatve frequences 0 12650 64.16 64.16 1 2775 14.08 78.24 2 2490 12.63 90.87 3 1064 5.40 96.27 4 430 2.18 98.45 5 176 0.89 99.34 6 89 0.45 99.79 7 22 0.11 99.90 8 15 0.08 99.98 9 3 0.02 99.99 10 1 0.01 100.00 Total 19715 100.00 Fgure 1: Plot of Clam Number

Modellng Accdent Occurrence n Car Insurance Implementaton on Tunsan Data 401 The Fgure 1 shows that the null values are preponderant as well as a break n the declne of populaton at the number of accdents. Table 4 Number of Accdents by Regon Regon Number North South Mddle Coast 1 3066 1195 5715 2674 2 656 236 1294 589 3 592 245 1107 546 4 269 100 463 232 5 104 46 192 88 6 38 17 78 43 7 20 10 41 18 8 3 1 13 5 9 3 0 7 5 10 1 1 The dentfcaton of the varable geographcal zone s one of the most mportant aspects of accdent studes. The table 4 shows that the number of accdents on the mddle of Tunsa s hgher than those n the other locatons. Ths can be explaned by the fact that the larger area consdered n ths study s the mddle Zone. Furthermore a poor road nfrastructure s responsble for the hgh rate of road accdents n ths regon. The followng table descrbes dfferent quanttatve varables: Table 5 Descrptve Statstcs of Quanttatve Varables Mean SD MIN MAX No-clam bonus rate 5.813 2.305 1 8 Age of polcyholders 45.84 12.45 18 98 Premum pad 326687.1 237455.9 0 7308659 The hgh mean value of the no-clam bonus rate ndcates that the portfolo nsurance can be consdered rsky. The premums pad by the polcyholders are not very hgh relatvely to the car s value n Tunsa. The age of polcyholder s a factor of rsk n car nsurance; t ndcates that n average the nsurers of ths portfolo are nexperenced ones. 4. EMPIRICAL RESULTS In subsecton 4.1 we estmate the Posson model and the nflated Posson model to dentfy the relatve sgnfcance of the varables explanng the number of responsble accdents. Fnally, we use a statstcal test and crterons to compare between the estmated models.

402 Noureddne Benlagha, Lanouar Charfeddne and Imen Karaa 4.1. Posson and Zero Inflated Posson Models We recall that the adverse selecton occurs when an nformed ndvdual s tradng decson depends on her unobservable characterstcs n a manner that adversely affects the unnformed agents n the market. Partcularly, n car nsurance market, the adverse selecton occurs when rsky drvers tend to buy comprehensve nsurance polces wth low deductbles. Whle the low rsk drvers buy a thrd party lablty nsurance wth hgh deductble (Rothschld and Stgltz (1976), Wlson (1977), Spence (1978)). In order to nvestgate the hypothess of adverse selecton we estmate the Posson and Zero nflated Posson models. Table 6 below reports the emprcal results of the estmaton of these models. It shows that the no clam bonus rate and the premum are sgnfcant and then explan the probablty of accdent when usng a Posson model. The negatve and sgnfcant relaton between the premum pad by the polcyholders and the number of accdent s nterestng nsofar as t makes t possble to check the presence of adverse selecton on the nsurance portfolo. But ths result s not satsfactory enough snce t can be explaned by the penalty n terms of no-clams bonus rate granted to polcyholders followng the occurrence of accdent. However, the results show that the varable level of premum s also sgnfcant. Ths varable s negatvely and sgnfcantly related to the number of accdent whch confrms the absence of the adverse selecton phenomenon, see for nstance Chappor and Salané (2000), Cohen (2005) and Vasechko et al. (2009). In other words, when choosng ther automoble nsurance contracts, ndvduals behave as though they had no better knowledge of ther rsk than nsurance companes. Table 6 The Estmaton Results of the Posson and Zero Inflated Posson Models Posson model Zero nflated Posson model Varables Coeff Std.Err Z Coeff Std.Err Z No-clam bonus 0.039 0.005 6.48 0.000 0.022 0.006 3.75 0.000 Premum -1.58e -07 7.80e -08-2.03 0.042 - - - - Age -0.006 0.001-4.82 0.000-0.004 0.001-2.98 0.003 Sex_male -0.084 0.023-3.69 0.000 - - - - Commercal car 0.0518 0.029 1.76 0.078 - - - - Low-prem -0.303 0.112-2.69 0.007-0.245 0.131-1.86 0.062 Med-prem -0.250 0.098-2.54 0.011 - - - - Hgh-prem -0.166 0.090-1.84 0.066 - - - - Senor_execu 0.154 0.051 3.00 0.003 0.124 0.059 2.08 0.038 D_offcal - - - - 0.069 0.042 1.65 0.099 D_experement - - - - -0.176 0.104-1.69 0.090 D-Young - - - - -0.223 0.120-1.85 0.064 Inflaton model Logstc regresson Rsk Exposure - - - - 2.743 0.096 28.49 0.000 Constant - - - - 0.185 0.035 5.27 0.000 LR Ch2 140.13 52.71 Prob > Ch2 0.000 0.0022 Log lkelhood -25678.419-22607.05

Modellng Accdent Occurrence n Car Insurance Implementaton on Tunsan Data 403 It s also nterestng to note that age, gender and car usage are sgnfcant. These three last varables are dummes. To analyze the effect of a dummy varable on the number of accdent we consder the gender of polcyholder as an example. The exact nterpretaton of the coeffcents nvolves the calculaton of the odds rato. Wth a dummy varable s coeffcent d, the odds rato s smply exp ( d ). The result shows that the coeffcent relatve to the varable sex_male s equal to -0.084, so the odds rato s exp(-0.084) whch equals to 0.91. One can conclude that the otherwse average male polcyholder (sex_male = 1) has a probablty of accdent 0.91 percentage pont lower than the otherwse average female polcyholder (sex_male = 0). The second part of the table 6 shows the results of the Zero nflated Posson model. One note that the Zero nflated Posson s a two part model. The frst part relates to the count model of the number of accdents and the second part (nflaton of zeros) corresponds to the logstc regresson. The frst part whch corresponds to the Posson regresson ndcates that the accdent occurrence ncreases wth the no-clams bonus rate and for the drvers wth occupaton as senor executve and offcal. Ths probablty (accdent occurrence) decreases wth age of the drvers. Partcularly, the accdent occurrence s negatvely related wth the expermented and young drvers. In the second part, for the logstc regresson, the postve sgn ndcates that the probablty of the number of clams ncreases wth the polcyholder exposure degree,.e. n our data set we have many polcyholders do not declare ther clams. In the next paragraph, we use the nformaton crtera and a statstcal test to compare between models estmated. 4.3. Comparson of the Models 4.2.1. Usng the Informaton Crtera A standard method of comparng count models s to use the nformaton crtera, such as the Akake Informaton Crtera or Bayesan Informaton Crtera. One can use also the log lkelhood statstc. All these nformaton crtera are functon of the number of parameters of the model and the number of observatons. Table 7 shows, that the Zero nflated model s better than the standard Posson model. Table 7 Comparson Crtera s of the Posson and ZIP Models Crtera Posson ZIP AIC 51634.7 45274.1 BIC 51413.8 45510.78 Log-lkelhood -25678.9-22607.05 4.2.2. Usng a Vuong Test Several authors have devsed tests for overdsperson wthn the context of the Posson model (See Goureroux et al. 1984, Cameron et Trved 1990). One can also consult the Greene work (2002) for more of detals on these dfferent tests. In partcular, Vuong (1989) has proposed a

404 Noureddne Benlagha, Lanouar Charfeddne and Imen Karaa test statstc whch s well suted to ths applcaton. The Vuong test statstc s smply the average log-lkelhood rato sutably normalsed. The test statstc s V nm, where s respectvely the mean and varance of the varable m, m 1 n m n 1 m and 1 s m m are n 2 2 m ( ) n 1 1 The hypotheses of the Vuong test are: f ( y ) Pr( y _ ZIP) m log log 1 f2 ( y ) Pr( y _ Posson). H 0 1 : E m 0 H : E m 0 The null hypothess of the test s that the two models are equvalent. Vuong shows that asymptotcally, V has a standard normal dstrbuton. As Vuong notes, the test s drectonal. If V s less the predetermned crtcal value, then the test does not favour one model or the other. A large postve value of V shows that the ZIP model s statstcally better than the Posson model. Large negatve value of V shows that the Posson model s statstcally better than the ZIP model. The test requres estmaton of both models and computaton of the sample of predcted probabltes. The Vuong test, (wth a Z value equal to 36.61 whch sgnfcant at the 0.1% level), shows that the ZIP model s better than the standard Posson model. It should be noted that, alpha corresponds to the overdsperson parameter n the count models; the null hypothess s that In ( ) = 0, f ths null hypothess wll be rejected then the Posson model wll be not sutable. 5. CONCLUSIONS In ths paper we have estmated two count models of accdent occurrence n order to test for the adverse selecton hypothess. The selecton of the sutable econometrc model s an mportant step to mplement nsurance premum that reduces asymmetrc nformaton, partcularly the adverse selecton effects. The use of an nflated count model s also mportant to dstngush between the no accdent and the non reportng one. The man emprcal result was the absence of an adverse selecton problem n the Tunsan nsurance portfolo of drvers of four wheels vehcles. The lmt of ths work s that n practce, count data are often overdspersed relatve to the Posson and Zero Inflated Posson dstrbuton, then a possble extenson of ths work s to

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