Statistical Modeling Methods in Automobile Insurance

Transcription

1 Statstcal Modelng Methods n Automoble Insurance Elena BURLACU PhD Student he Academy of Economc Studes of Moldova Republc of Moldova, elenaburlacu@gmal.com Abstract hs artcle descrbes the notons of the statstcal ndcators of automoble nsurance and also presents methods that descrbe nterdependence between them. Statstcal calculatons - the collecton, the pre-processng and the further data analyss s based on certan system of statstcal ndcators. hs system s composed of the ndcators, whch are supposed to be used n all frms n a gven sphere of actvty, n our case these are the nsurance companes whch operatng on the auto nsurance market. he nterdependence between nsurance ndcators was analyzed by correlaton analyss, multple regressons and testng the hypothess of normalty and model valdaton and ts coeffcents. Key words: nsurance compensatons, the amount of nsurance, correlaton, lnear regresson. JEL Classfcaton: G, C Introducton here s an obvous need of usng the calculatons and ratng n auto nsurance. akng nto account the market condtons for the nsurance company, the most mportant crteron for attractng clents s certanly the rate polcy, namely to ensure customer n analogous condtons to other companes, provdng the same amount but wth a lower prce for the nsured. Determnng ths value, the nsurer collate the need of calculus and lkelhood estmate of ts busness, wth need to buld mathematcal models, to analyze them and to choose the optmal model that best descrbes certan phenomena on auto nsurance market.. he statstcal modelng methods he nsurance amount of damages (Y s, probably, the man ndcator of the fnancal stablty of the nsurance company, therefore t represents nterest and the research of dependent factors has an nfluence on the formaton of ths ndcator.. he nterdependence of statstcal ndcators Analyzng whch s the nfluence that the sze of nsurance compensaton has, on whether t occur, can lead to the followng conclusons: - he overvew numbers of the nsurance payments depends on the total number of contracts that form the whole package of nsurance company. 58

2 - he total sze of nsurance compensatons, for the same reasons, depends on the number of nsurance requests to the entre nsurance package. - he type of contract. For example, statstcs shows that the number of requests nto coverng damages caused by means of transport s bgger than the number of shares on theft, but t s understood the fact, that the sze of the payment for the caused damage, on average s less than the sze payment for the theft. - he operatng regon of means of transportaton, drvng experence of the drver, car brand, the number of nsured events that occurred prevously wth the partcpaton of one or another drver etc, may also become nfluental factors - Rensurance. he amount of rensurance compensatons can be treated as a factor that reduces the total compensatons. It has to be mentoned that the nfluence of these factors on the whole or the degree of ther nfluence on the formaton of random value Y, descrbed above, s only at logcal reasonng level. herefore, t presents mathematcal research nterest of the nfluence of ndcators on the value Y, and the research of nterdependence of these ndcators.. Practcal approach From the strng of factors that may affect the amount of damages n an nsurance company, t s proposed to study the nterdependence of the three of them: the amount of nsurance payments (X, the number of nsurance contracts (polces (X, the number of cases provded (X3. As a bass for calculatng tme seres are gong to be used, for 4 months, for each of the ndcators selected from the reports out of an nsurance company. - the hypothess of normalty (crteron Pearson At the frst phase was tested the hypothess of normalty, for each of the three rows data of ndcators for whch has been used Pearson crteron. he result of calculatng the theoretcal frequency data were grouped nto sx ntervals, χ obs.. = 0,853, χ cr. (for degrees of varance v=3 and the mportance level 0,05 = 7,85. Snce the value of Pearson's statstcal analyss s less than the crtcal one, accordng to the crteron of Pearson, the normal hypothess dstrbutonal law s accepted the confdence level (Χ calculaton was based on Mcrosoft Excel 7.0 wth bult CHIDIS. he assumpton of normalty for the three ndcators was accepted at the confdence level of the lnear regresson he applcaton of lnear regresson, as a tool for the analyss of the data presented, derves from a compromse between the veracty of assumptons and smplcty the nterpretaton method. o determne the relatons between ndcators we wll use the correlaton analyss and the correlaton matrx: Revsta Română de Statstcă rm IV/03- Suplment 59

3 able. he results of correlaton analyss X Х ХЗ Y X Х 0, ХЗ 0, ,3344 Y 0,650 0,5037 0, he correlaton matrx talks about a strong connecton between the ndependent varables and the dependent varable Y and also sgnals that the ndependent varables do not correlate wth each other. he correlaton analyss shows that the tme seres have a lnear tendency n order to exclude the autocorrelaton between the varables, n the regresson model t s ntroduced an addtonal factor tme t. he fnal multple regresson pattern wll have the followng form: Yˆ = -944, Х+75,668 Х3+88,457 t (. - Fsher test In order to test the sgnfcance of the model, the Fsher test s used. For the gven model the tabular value F crt = 3,34 at level of sgnfcance α = 0,05, γ = 3, γ = 4, a value whch s much lower than Fobs = hus the model descrbes the real structure of nfluence of the analyzed factors on the amount of compensatons. - Student test It also showed an nterest, checkng the sgnfcance of regresson coeffcents. he sgnfcance of regresson coeffcents was checked by Student crteron (t-dstrbuton n a comparson of observaton t obs and t cr. t =.444, t =.68, t4 =.6045 values that are bgger than the tabular value t cr =,45 the number of thresholds of freedom d f = 4 and confdence level α = As observed, only two of the coeffcents for the varables X (amount of nsurance and ХЗ (number of clams can be consdered sgnfcant from the statstcally pont of vew. - Durbn Watson test o assess the exstence of autocorrelaton n regresson balances, whch creates dffcultes nto applyng the method of the smallest squares regresson equaton, we used the crteron of Durbn - Watson. Statstcs crtcsm of Durbn - Watson has the form: d n = = n where: ( ˆ ε ˆ ε = ˆ ε (. 60

4 ˆ = Y ε ( m0 + m * X mk X k - the dscrepancy method of the smallest squares. Frankly speakng, n the absence of autocorrelaton balances regresson, the sgnfcance of crtcal statstcs d, accordng to the crteron of Durbn - Watson, must not dvert more than. he sgnfcance statstcs d =.964 obtaned, allows us to make a concluson regardng the lack of autocorrelaton n the examned sequences. herefore the regresson coeffcent for the frst factor shows that the ncrease of the amount of nsurance payments by 000 u.c., the compensaton amount wll ncrease on average by 80 u.c. he coeffcent X3 shows that the ncreasng number of cases provded by one unt amount of compensaton wll ncrease on average by 75 u.c. may represent the average value of compensaton for nsured event. Regresson coeffcent of the tme factor shows that each month amount of compensaton payments wll ncrease by 88 u.c. Accordngly, to mprove the fnancal stablty of the nsurance company t s necessary when formng nsurance payment, to forecast for each clent, the probablty of occurrence of accdent.. Models wth qualtatve factors hs can be done through analyss of qualtatve factors. Upon a possble rsk of an accdent there are a number of qualtatve factors that mght nfluence: sex, age of drver, drvng experence, type of car (the techncal capabltes of the car and operatng regon. hese features can be descrbed by the vector: X = x, x,... ( x k (. If Y s a quanttatve varable, then y= f the ndvdual s nvolved n a car accdent y= 0 otherwse hen the vector: Y = ( y, y,... y n wll contan dchotomous varables. he analyss of dependence K Y = ( y, y,... y n of the characterstcs of ( X, X,... X wll be descrbed by the model: y = x + ε, =,..., n, (. = Where - number of observatons, (,,..., k - a set of ε unknown parameters, - random error. As y takes the value of 0 or and E( ε = 0 E ( y, then = P( y = + 0 P( y = 0 = P( y = = x. Accordngly the model (. can be expressed by a lnear probablty model: Revsta Română de Statstcă rm IV/03- Suplment 6

5 P ( y = = x (.3 he purpose of ths complex analyss conssts to form some groups from the data taken from a survey on drvers, and then to usng the regresson to detect the nterdependence between the probablty of occurrence of accdents accordng to age, gender, drver experence, techncal car specfcatons. Concluson In concluson we can terate that any statstcal calculatons should be based on the system of statstcal ndcators and use statstcal and mathematcal computng technques. o verfy the degree of nfluence of varous factors on the value of nsurance clams there were used correlatonal and regresson analyss: there were constructed matrces of correlaton coeffcents and regresson equatons and factor analyss was appled. It was examned the sgnfcance of regresson equaton obtaned and there were tested ts coeffcents. he researches allow to make a concluson about the fact that, the amount of compensaton s an amount dependent on the followng factors: the amount of nsurance and the number of nsured cases. Accordngly, to mprove the fnancal stablty of the nsurance company t s necessary when formng nsurance payment, to forecast for each clent, the probablty of occurrence of accdent, for ths purpose the analyss of nfluence of quanttatve and qualtatve factors on the occurrence of ths rsk, s mportant. Bblography Legea nr.44-xvi dn..006 Cu prvre la asgurarea oblgatore de răspundere cvlă pentru pagube produse de autovehcule publcată în MO nr.3-35/ dn Verejan O., Pârţach I. Statstca actuarală în asgurăr. Edtura Economca, Bucureşt, 004. Iacob A. I., anasou O. Modele econometrce vol.i, Edtura ASE, Bucureşt, 005 urturean C. I. Metode statstce de analză a serlor de tmp Edtura Sedcom, Iaş, 006 Лемер. Ж., Автомобильное страхование. Актуарные модели,. М.,Янус-Ко, 998 Frees, E. W. Regresson Modelng wth Actuaral and Fnancal Applcatons, Cambrdge Unversty Press, 00 6