RISK PREMIUM IN MOTOR VEHICLE INSURANCE. Banu ÖZGÜREL * ABSTRACT

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1 D.E.Ü.İİ.B.F. Dergs Clt: 0 Sayı:, Yıl: 005, ss:47-60 RISK PREMIUM IN MOTOR VEHICLE INSURANCE Banu ÖZGÜREL * ABSTRACT The pure premum or rsk premum s the premum that would exactly meet the expected cost of the rsk covered gnorng management expenses, commssons, contngency loadng, etc. Clam frequency rate and mean clam sze are requred for estmaton n calculatng rsk premums. In ths study, we dscussed to estmate clam frequency rate and mean clam sze wth several methods and calculated rsk premums. Data, whch supported our study, s provded by nsurance company nvolvng wth motor vehcle nsurance. Key words: Number of Clams, Clam Szes, Clam Frequency Rate, Rsk Premum. 1. Introducton Insurance s an mportant part of rsk management programs for organsatons and ndvduals. Insurance s a rsk fnancng transfer under whch an nsurer agrees to accept fnancal burdens arsng from loss. More formally, nsurance can be defned as a contractual agreement between two partes: the nsurer and the nsured. Insurance may be classfed nto three groups nsurance of person, nsurance of property and pecunary nterest, and nsurance of lablty. General nsurance, or Non-Lfe nsurance or Property/Casualty nsurance as t s sometmes called, normally relates to the nsurance of property and lablty. In ths study, we are nterested n motor vehcle nsurance. Motor vehcle nsurance contracts are schedule contracts that the permt the nsured to purchase both property and lablty nsurance under one polcy. The contract can be dvded, however, n two separate contracts one provdng nsurance aganst physcal damage to automobles, and the other protectng aganst potental lablty arsng out of the ownershp, mantenance, or use of an automoble. Some automoble nsurance Öğr. Gör., Dokuz Eylül Ünverstes Fen Edebyat Fakültes, İstatstk Bölümü, Tınaztepe Kampüsü, Buca-İZMİR, 47

2 Banu ÖZGÜREL contracts, notably those ssued by nsurers assocated wth automoble fnance companes, cover only physcal damage nsurance.. The Rsk Premum The pure premum or rsk premum s the premum that would exactly meet the expected cost of the rsk covered gnorng management expenses, commssons, contngency loadng, etc. To compute t, we need to estmate the clam frequency rate and mean clam sze. Multplyng clam frequency rate by mean clam sze yelds the rsk premum. A clam s an asserton of a rght to payment, as when a customer notfes a manufacturer of an njury from a defectve product and expenses a belef that the njury justfes compensaton. Usually, the payment s to occur at some future pont n tme. It should not be assumed that all accdents become clams, nor that the amount of the clam wll always be the same as the amount of the damage. The clam sze s the sum, whch the nsurer has to pay on the occurrence of a fre, an accdent, death or some other nsured event. The sum of the ndvdual clams consttutes the aggregate clam amount, whch s one of the key concerns, both n the practcal management of an nsurance company and n theoretcal consderaton. The clam frequency rate s a rate whch can be estmated as the number of clams dvded by the number of unts of exposure. It s mportant that the two components, clam frequency rate and clam sze be consdered separately. Only n exceptonal crcumstances can the use of rate of payment of clams be consdered satsfactory for the assessment of a rsk premum. 3. Method of Study Analyss s based on smple formulatons as follows (Hossack, 1989: 13-15): (1) Dependng on yearly total clam szes and number of clams, mean clam szes for each year can be calculated by the formula 48

3 Rsk Premum In Motor Vehcle Insurance TCS MCS = c (1) where the notaton s MCS TCS C : mean clam sze for year : total clam sze for year : number of clams n year () Dependng on yearly total amount of premums and number of polces, average offce premum per polcy for each year can be calculated by the formula TOP AOP = N () where the notaton s AOP : average offce premum per polces for year. TOP : total amount of premums for year. N : number of polces n year. (3) Dependng on yearly total clam szes and number of polces, rsk premum for each year can be calculated by the formula TCS RP = N (3) where RP denotes rsk premum for year. (4) Dependng on yearly offce premum and rsk premum, the rato can be calculated by the formula 49

4 Banu ÖZGÜREL k = AOP RP (4) year. where k denotes the rato of offce premum to rsk premum for (5) A fxed clam frequency rate can be calculated dependng on the total number of polces and the total number of clams. In fact, clam frequency rate s the rato of the number of clams n a perod to the exposure to rsk for ths perod. The man problem n calculatng clam frequency rate s to fnd the exposure to rsk can be calculated by varous methods. One of them s the Census Method. Let N(t) denote the number of polces booked by the nsurance company at tme t. Durng the short nterval of tme δt mmedately followng tme t, the change n the number of polcyholders a premum wll be neglgble. Each of the N(t) premum polcyholders currently on the books wll contrbute δ t years of exposure durng that small tme nterval so that the total number of years exposure durng the nterval of tme δ t followng tme t wll be N(t) δ t. The total exposure durng wll, therefore, be approxmately t = 0 to 1 n steps of δt N( t) δt the calculaton becomes more and more accurate the smaller δ t becomes, and exact when δ t becomes too small. Then n mathematcal terms dt s wrtten nstead of δ t, and becomes 1 0 N( t) dt 50

5 Rsk Premum In Motor Vehcle Insurance.e. the area under the N(t) curve between t=0 and t=1. For example, when data gven n Table 4. s consdered Fgure 4.1 s the approxmate graph of N(t). Fgure 1. Avalable number of polces durng October 1,1997 to October 1, Number of Polces ,5 1 1,5 t The area under N(t) curve s 3 t = 0 1 t t + 1 N + N 4 4 and s calculated to be 481, snce the number of clams durng ths perod s reported to be 183, the clam frequency rate s sad to be 183/481 and equals When the perod s extended to k years and total number of clams booked each year are assumed to be exposure to rsk for that year, then total exposure to rsk becomes N 1 +N + +N k.. The total number of clams gven by

6 Banu ÖZGÜREL C 1 +C + +C k. therefore yelds the clam frequency rate to be calculated by the formula m k = 1 = k = 1 C N (5) where m s the clam frequency rate whch s assumed to be constant durng k years. (6) On the mplcatons of the mean clam szes and total number of clams booked by the company the growth rate can be calculated. In the growth rates geometrc progresson can be assumed or both geometrc and arthmetc progressons can be assumed. 4. Applcaton The nsurance company s establshed n 199 and the avalable data comprses of the number of polces booked each year total amount of the premums collected for each year yearly number of clams total clam sze for each year begnnng from 1993 to These data are gven n Table 1. 5

7 Rsk Premum In Motor Vehcle Insurance Table 1. Yearly Income and Outgo Years Number of Polces Total Amount of Premum ($) Number of Clams Total Clam Sze ($) Total Besdes, the numbers of polces booked by the company October 1, 1997 to October 1, 1998 are also avalable. Ths data s presented n Table. Table. Avalable number of polces durng October 1, 1997 to October 1, 1998 Dates Number of Polces October 1, January 1, Aprl 1, July 1, October 1, Number of clams durng the tme between October 1, 1997 to October 1, 1998 s 183. In ths secton the results wll be obtaned by applyng each step explaned n secton are presented. (1) Mean clam szes for each year are presented n Table 3. 53

8 Banu ÖZGÜREL Table 3. Mean Clam Szes Years Number of Clams Total Clam Szes ($) Average Clam Szes ($) () Average offce premum for each year are presented n Table 4. Table 4. Average Offce Premums Years Total Offce Premums Number of Polces Average Offce Premums (3) Rsk premum for each year are presented n Table 5. Table 5. Rsk Premums Years Total Clam Szes ($) Number of Polces Rsk Premums

9 Rsk Premum In Motor Vehcle Insurance (4) k for each year are presented n Table 6. Table 6. Yearly ratos of offce premums to rsk premums Years Average Offce Premums Rsk Premums k (5) Clam frequency rate dependng on data gven Table 1 and calculated by formula (5) s 146 m = = ,35 and ths can be assumed to be a crude estmate. (6) The most reasonable yearly mean clam sze fgures consttute a geometrc progresson. The frst mean clam sze n Table 3 s $88 and four year after t becomes $135. Let r denote the growth rate n mean clam sze and wth the assumpton of geometrc growth MCS + 1 = (1 + r) MCS Therefore, r can be calculated from and s found to be 11.9%. 135 = (1 + r)

10 Banu ÖZGÜREL The number of polces booked by the nsurance company whch are gven n Table 1 on yearly bass mples not only a geometrc growth but also an arthmetc growth. The relaton between the successve number of total polces s found to be as follows by tral and error. n =1.785n 1 n 3 =( )n n 4 =( )n 3 n 5 =( )n 4 or expressed by a general formula as follows [ ( 1) 0. ] n n = We begn the work wth the followng assumptons: (1) In 1993, 614 polcyholders are to be booked, and mean clam sze s to be $88. () Clam frequency rate durng the followng 5 years s to be constant and equal to 35%. (3) Mean clam sze ncreases by a constant rate whch s equal to 11.9%. (4) Number of polcyholders ncreases by the relaton [ ( 1) 0. ] n n = (5) Only half of the clams reported n a year are pad n that year, and the other half of the clams are pad n the succeedng year. Wth respect to the above assumptons mean clam szes, number of polces, number of clams and total clam sze pad each year can be calculated. The calculated values are showed n Table 7. 56

11 Rsk Premum In Motor Vehcle Insurance Table 7. Number of polces and clam amount under the assumptons Years Average Clam Szes ($) Number of Polces Number of Clams Total Clam Szes($) ( ) = ( ) = ( ) = ( ) = ( ) = Calculaton of clam frequency rate as a rato of number of clams to the exposure to rsk yelds a relaton whch s very useful to estmate the number of clams for each year. The number of clams whch are denoted by c n Table 7 are calculated by c = m n (6) Total clam sze for each year s to be calculated by consderng the number of clams produced that year and half of them to be pad mean clam sze of that year, and the other half to be pad mean clam sze of the succeedng year wth respect to assumpton (5) stated above: TCS c ( MCS + MCS ) = + 1 (7) The numercal fgures presented n Table 7 can be used n order to calculate the expected mean clam szes by applyng formula (1), and rsk premums by formula (3). 57

12 Banu ÖZGÜREL 8. The rsk premums obtaned by these calculatons are presented n Table Table 8. Rsk premums under the assumptons I Years Number of Clams Number of Polces Total Clam Szes Rsk Premum The total row n Table 8 contans the most mportant numercal fgures n our analyss. As a result of the analyss total number of polces booked n 5 years s to be 6101, total number of clams n ths perod s to be 137, and the total amount of clam szes to be pad n ths perod s $ If the calculated rsk premums n Table 8 are to be appled durng the years , the number of polces realzed (data gven n Table 1) would gve rse to collecton of total amount of net premums as presented n Table 9. Table 9. Total amount of net premums Years Realzed Number of Polces Planned Rsk Premum Total amount of premum Total The total rows both n Table 1 and 9 show that 6093 polcy are booked n fve years, total number of clams n ths perod n 146 (Table 1) and the total amount of rsk premums to be collected by the proposed plan s $ (Table 9). 5. Concluson 58

13 Rsk Premum In Motor Vehcle Insurance We see that the realzed values of the fve years totals of number of clams and number of polces are very close to the values obtaned under the plannng assumptons. Besdes, the total amount of rsk premums whch would be collected under the proposed plan would cover the realzed total amount of clam szes. In ths paper, we studed the data provded by an nsurance company nvolvng wth motor vehcle nsurance n the lght of theory of nsurance rsk model and rsk premums. Frst of all, avalable data s analysed. After words, reasonable assumptons on the growth of number of polces, ncrease of clam frequency rate, ncrease of mean clam sze by a constant rate, a rsk premum payment plan for fve years s proposed. In concluson, the proposed plan s found to gve rse to very close values the realzed ones, and be capable to cover the realzed expenses. Özet Rsk prm, br sgorta şrketnde yönetm masrafları, komsyonlar göz önüne alınmadan rskn beklenen malyetn karşılayan prmlerdr. Rsk prmlernn hesaplanablmes çn hasar sıklık oranı ve ortalama hasar büyüklüğünün tahmn edlmes gerekr. Bu çalışmada, hasar sıklık oranı ve ortalama hasar büyüklüğünün çeştl yöntemlerle tahmn edlmes tartışılmış ve rsk prmler hesaplanmıştır. Çalışmamızı destekleyen verler kasko sgortalarını çeren sgorta şrket tarafından elde edlmştr. prm. Anahtar Kelmeler: hasar sayısı, hasar büyüklüğü, hasar sıklık oranı, rsk REFERENCES BOWERS, N.L., H.U. GERBER, J.C. HICKMAN, D.A. JONES ve C.J. NESBITT, (1986), Actuaral Mathematcs, USA: Socety of Actuares. BÜHLMAN, H. (1996), Mathematcal Methods n Rsk Theory, New York: Sprnger Verlag. DAYKIN, C. D., T.PENTIKAINEN T. and M. PEASONEN (1994), Practcal Rsk Theory for Actuares, London: Chapman Hall. 59

14 Banu ÖZGÜREL DEGROOT, M.H. (1975), Probablty and Statstcs, Canada: Addson Wesley. FREUND, J.E. (1973), Introducton to Probablty, New York: Dover Publcaton. GERBER, H.U. (1979), An Introducton to Mathematcal Rsk Theory, Phladelpha: Unversty of Pennsylvana. GRANDELL, J. (1991), Aspects of Rsk Theory, New York: Sprnger Verlag. HART, D.G., R.A BUCHANAN and B.A. HOWE (1996), The Actuaral Practce of General Insurance, Sydney: Insttute of Actuares of Australa. HOGG, R.V. and A.T. CRAIG (1995), Introducton to Mathematcal Statstcs, New Jersey: Prentce Hall. HOSSAK I.B., POLLARD J.H. and ZEHNWIRTH (1989), Introductory Statstcs wth Applcatons n General Insurance, Cambrdge Unversty Press. LARSON H.J. (198), Introducton to Probablty Theory and Statstcal Inference, Sngapore: John Wley & Sons. LEMAIRE, J. (1985), Automoble Insurance Actuaral Models, USA: Kluwer Njhoff. PANJER, H.H. and G.E. WILMOTT (199), Insurance Rsk Models, USA:Socety of Actuares. REJDA, G.E. (1994), Socal Insurance and Economc Securty, New Jersey: Prentnce- Hall. ROSS, S. (1998), A Frst Course n Probablty, USA: Prentce Hall. SCHEAFFER, R. L. and W. MENDENHALL (1975), Introducton to Probablty: Theory and Applcatons, Massachusetts: Duxbury Press. SKIPPER, H.D. Jr. (1998), Internatonal Rsk and Insurance: An Envronmental Manageral Approach, Sngapore: Irwn/McGraw- Hll. 60

15 Rsk Premum In Motor Vehcle Insurance STRAUB, E. (1980), Non-Lfe Insurance Mathematcs, New York: Sprnger- Verlag. Webster s New Collgate Dctonary (1991). WILLIAM, C.A. JR., SMITH, M.L. and YOUNG, P.C. (1998), Rsk Management and Insurance, Sngapore: Irwn/McGraw-Hll. 61

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