A markovian study of no claim discount system of Insurance Regulatory and Development Authority and its application

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1 Thailad Statisticia July 214; 12(2): htt://statassoc.or.th Cotributed aer A markovia study of o claim discout system of Isurace Regulatory ad Develomet Authority ad its alicatio Dili C. Nath* [a] ad Prasejit Siha [b] [a] Deartmet of Statistics, Gauhati Uiversity, Guwahati 78414, Idia. [b] Deartmet of Statistics, Triura Uiversity, Suryamaiagar 79922, Idia. * Corresodig author; dilic.ath@gmail.com Received: 8 Jauary 213 Acceted: 1 Jue 214 Abstract No claim discout (NCD) is oe of the more cotroversial areas of automobile isurace, beig a toic o which the motorist is liable to hold strog ad emotive views from time to time. I this aer we try to fid out the robabilities of claims by differet categories of olicyholders (motorists). The olicyholders are divided ito two grous viz. good drivers ad bad drivers accordig to their drivig exeriece as well as accidet records i last two years. Here, we use a trasitio robability matrix (TPM) for differet discout levels followig the Isurace Regulatory ad Develomet Authority (IRDA) rules of NCD usig Markov Chais. Usig this TPM we obtai the resective amout of remiums to be aid i the log ru by differet grous of olicyholders secially the drivers of the districts of Karimgaj, North Triura ad West Triura. The results of this study show that robability of claims ad differet NCD rates are ot arallel. Keywords: markov chais, statioary distributio, automobile isurace. 1. Itroductio I automobile isurace, amog other geeral isurace olicies, it is quite commo tedecy to reduce the remium by a factor i case the isured does ot make

2 224 Thailad Statisticia, 214; 12(2): ay claim i a give eriod. This is oularly kow as No Claim Discout (NCD). No claim discout systems (sometimes also called Bous-Malus systems) are exeriece ratig systems which are commoly used i motor isurace. NCD schemes rereset a attemt to categorize olicyholders ito relatively homogeeous risk grous who ay remiums relative to their claims exeriece. Those who have made few claims i recet years are rewarded with discouts o their iitial remium, ad hece are eticed to stay with the comay. Deedig o the rules i the scheme, ew olicyholders may be required to ay the full remium iitially ad the will obtai discouts i the future as a results of the claim free years [1]. A NCD ca sigificatly reduce the cost of your car isurace cover. A NCD system discourages small claims. This ricile is meat to reward olicy holders for ot makig claims durig a year; that is, to grat a bous to a careful driver. A bous ricile effects the olicy holder s decisio whether or ot to claim i a articular istace. No claim will be made for some of the accidets where there is oly slight damage. Philiso [2] called this heomeo huger for bous. It reduces claims costs to the isurer which offsets the decrease i remium icome from NCD systems. No claims discouts allow the driver to be more resosible about their vehicle ad whe drivig. If o claims are made, each year the remium reduces. The discout is calculated as a straightforward ercetage of the total cost of the isurace remium ad will be discouted each year a claim is ot made. Every car isurace comay has its ow method for determiig the exact amout to discout off the remium but the commoality is the maximum umber of years to accumulate the o claims discout is five years. Also, drivers ca isure their o claims discout to rotect it oce the five year discout maximum has bee reached. This charge is added to your isurace olicy. If ay claim is made or arises o his/her motor isurace olicy durig the eriod of cover, his/her NCD for that vehicle will be forfeited ad reverts to %. The olicyholder will have to begi to accumulate his/her discout i a ew cycle. So, we ca ow see how the NCD works for car isurace ad exactly how much moey a olicyholder could save. The differece i remiums ca be vast. I Idia NCD rewards the olicyholder with savigs o his/her car or motorcycle isurace for good drivig or ridig history. The NCD savigs start at 2% ad go as high as 5%, see Table 1. The isurer calculates the level of NCD based o the umber of years the olicyholder have bee drivig or ridig, his/her claims ad icidet history.

3 Dili C. Nath 225 Table 1. Levels of NCD system of IRDA. Levels/ Age of Vehicle No Claim Discout Savig 5 5% 4 45% 3 35% 2 25% 1 2% % Each year at reewal, a olicyholder automatically moves u to the ext level of NCD if he/she have't made a claim for a accidet where he/she was at fault i that year. If olicyholder does make a claim for a accidet where he/she was at fault, the olicyholder will move dow zero level of NCD, uless he/she is o maximum NCD for life. If the olicyholder makes a claim for somethig that's ot his/her fault, for examle, his/her car or motorcycle is stole or damaged by a storm, or someoe scratches the aitwork, his/her NCD level will ot chage. There is a table fixed by Isurace Regulatory ad Develomet Authority (IRDA) for a NCD ad is give Table1. As is aaret therefore a NCD is a secial discout give for every claim-free year. This therefore reduces the remium i succeedig years. However a claim i the succeedig years would result i loadig, which is the iverse equivalet of NCD. The slabs for a o claim bous ow start at a 2% discout o remium for ow damage for o claims i the recedig year ad icreases to 25% for o claims i the recedig two years, 35% for three years, 45% for four ad a maximum of 5% for five years [3]. No Claim Discout (NCD) or Bous-malus systems (BMSs) are itroduced i Euroe i the early 196s, followig the semial works of Delaorte [4], Bichsel [5], ad Bühlma [6]. There exists a vast literature o BMSs i actuarial jourals, maily i the ASTIN Bulleti, the Scadavia Actuarial Joural ad the Swiss Actuarial Joural. Loimarata [7] develos formulas for some asymtotic roerties of bous systems, where bous systems are uderstood as Markov chais. Bous systems used i Demark, Norway, Swede, Filad, Switzerlad ad West Germay are studied by Vesäläie [8] o the basis of the method give by Loimarata. Lemaire [9] derives a algorithm for obtaiig the otimal strategy for a olicy holder. Lemaire [1] alies this algorithm to comare bous systems used i Demark, Norway, Swede, Filad, Switzerlad ad West Germay. Hastigs [11] resets a simle model based o a tyical British olicy, assumig that the umber of accidets is Poisso ad the amout of damage is egative exoetially distributed. The roblem is formulated as a Markov

4 226 Thailad Statisticia, 214; 12(2): decisio roblem ad is solved by dyamic rogrammig. Lamaire [12] comutes a merit-ratig system for automobile third arty liability isurace. The results are alied to the ortfolio of a Belgia comay ad comared to the remium system rovided by the exected value ricile. Kolderma ad Volgeat [13] reset a cotiuous model based o geeralized Markov rogrammig, alicable to bous-malus systems used by Dutch motor isurace comaies. Lemaire [14] obtais the data from the Actuarial Istitute of the Reublic of Chia of market wide observed loss severity distributios for roerty damage ad bodily ijury for accidet years 1987 to These distributios are very well rereseted by a logormal model. Lemaire ad Zi [15] aalyze 3 bousmalus systems (BMS) from aroud the world. All BMS are simulated, assumig that the umber of at-fault claims for a give olicyholder coforms to a Poisso distributio. Lemaire [16] studies the Markov chai theory for the desig, evaluatio, ad comariso the BMSs of the atios Brazil, Belgium, Jaa, Switzerlad ad Taiwa. The tools are the same, but the assumtios about the robability distributios for the umber of claims vary. Pitrebois et al [17] obtai the relativities of the Belgia Bous-Malus System, icludig the secial bous rule sedig the olicyholders i the malus zoe to iitial level after four claim-free years. The model allows for a riori ratemakig. All the above metioed studies are ot associated with the NCD of Isurace Regulatory ad Develomet Authority of Idia. This aer isects the desirability of this multi-layer remium system (NCD system); to start with this study works with a give umber of levels with fixed gas. The basic framework cosiders that of a discrete time arameter Markov chai, where the state sace cosists of the differet levels of the remium, ad the state of a articular isured shift radomly from a year to the ext. The radomess of the trasitio is govered by the trasitio robability of causig a accidet i a give year. This study models the robability to be varyig deedig o quality of the driver. For the most art, it would be cosiderig a fiitely may grous of olicyholders (drivers) characterized by resective robabilities of gettig ivolved i a accidet. A try has bee made to obtai the statioary distributio for each grou of olicyholders. This reflects the distributio of a articular grou over the various levels of remium i the log ru. For examle, oe ca obtai the ercetage of good drivers exected to receive the fully discouted rate i the log ru. A comarative study of these statioary distributios over the various grous cosidered, form the basis of aroriateess of the assumed NCD system. Briefly the objectives are as follows: a) To fid the robability distributio(s) of umber of accidets/claims i the study area.

5 Dili C. Nath 227 b) To study the log ru behavior of the claimig rocess i our study area usig IRDA rule. The article is orgaized as follows: I Sectio 2, we describe the NCD model as a discrete state Markov chai. Data aalysis ad coclusio are give i Sectio The Model A osteriori ratig is a very efficiet way of classifyig olicyholders ito cells accordig to their risk. Several studies have show that, if isurers are allowed to use oly oe ratig variable, it should be some form of merit-ratig. The best redictor of the umber of claims of a driver i the future is ot age, car, or the towshi of residece, but ast claims behavior. A isured eters the system, i the iitial class, whe he or she obtais a drivig licese. The, throughout the etire drivig lifetime, the trasitio rules are alied uo each reewal to determie the ew class as a fuctio of claims history. The recedig defiitio assumes that the NCD forms a Markov chai rocess. A (first-order) Markov chai is a stochastic rocess i which the future develomet deeds oly o the reset state but ot o the history of the rocess or the maer i which the reset state was reached. It is a rocess without memory, such that the states of the chai are the differet NCD classes. The kowledge of the reset class ad the umber of claims for the year suffice to determie ext year s class. It is ot ecessary to kow how the olicy reached the curret class. The Markov rocess, Markov chai, trasitio robability matrix (TPM) are described i brief i the followig sectios. A Markov rocess is a stochastic rocess that has a limited form of historical deedecy. Let {X(t): t } be defied o the arameter set ad assume that it reresets time. The values that X(t) ca obtai are called states, ad all together they defie the state sace S of the rocess. A stochastic rocess is a Markov rocess if it satisfies P[ X ( t P[ X ( t t ) 1 t ) 1 x X ( t ) x, X ( ), t ] x X ( t ) x ], t, t 1 (1) Let t be the reset time. (1) states that the evolutio of a Markov rocess at a future time, coditioed o its reset ad ast values, deeds oly o its reset value. The

6 228 Thailad Statisticia, 214; 12(2): coditio of (1) is also kow as the Markov roerty. Markov chais are classified as discrete or cotiuous. Cosider a Markov rocess as defied by (1) ad, without loss of geerality, let the state sace S be the set of oegative itegers. The Discrete Time Markov Chais (DTMCs) that characterizes the rocess catures its evolutio amog states of S over time t. The trasitio robability betwee state i to state j at time ( -1) is the robability P[ X j X 1 i]. A DTMC is time homogeeous if P[ X j X 1 i] P[ X m j X m1 1,2,3,.., m, i, j S i], (2) Further we defie ad the oe ste robability trasitio matrix P: P = i, 1,,,1 1,1 i,1, j 1, j i, j (3) where, ij are the robabilities of accidets ad hece claims. Each row of the robability trasitio matrix reresets the trasitio flow out of the corresodig state. Each colum of it reresets the trasitio flow ito the state. As the cumulative trasitio flow out of each state must be 1, the rows of matrix P must sum to 1 ad have all o-egative elemets (sice they are robabilities). A matrix that has o-egative elemets ad its rows sum to 1 is ofte called a stochastic matrix. If S is fiite, the P has fiite dimesio. The robability that the DTMC reaches, o the state i i ste is give by the Chama-Kolmogorov equatio: th ste, state j startig from m m i, j i, k. k, j, m. ks (4)

7 Dili C. Nath 229 Let π (π, π,.) i 1 be the robability vector whose elemet the robability that the DTMC is at state i at ste. Sice π i deotes 1 π π.p or π π.p, = 1, 2,.., the the robability of state i at ste is simly the sum of the robabilities alog all samle aths from j to i i stes weighted by the robability of startig at state j. A DTMC is irreducible if for each air of states iteger such that i, j ( S 2 i, j) there exist a N i i,. State i is ositive recurret if. A state is eriodic if i, i iff = k.d for some values of ad a fixed value of d > 1. If a state is ot eriodic (d=1), the it is aeriodic, where d is the GCD of. Clearly, state i is aeriodic if i, i. A state is called ergodic if it is ositive recurret ad aeriodic. If P is the robability trasitio matrix of a irreducible DTMC i a ergodic set of states, the limitig matrix statioary robability vector π lim P has idetical rows that are equal to the lim π. The statioary robability vector of a irreducible DTMC i a ergodic set of states is uique ad satisfies π π.p (5) Ad the ormalizatio coditio π.1 T 1 (6) This aer always refers to a ergodic ad irreducible Markov chai throughout the chater, ad iterested o comutig the statioary distributio vector π which characterizes the steady state robability distributio of the rocess. The steady state is reached after the rocess asses a arbitrary large umber of stes. I steady state, the total flow out of a state is equal to the total flow ito the state. This roerty is called flow balace ad is exressed i the form of a flow balace equatio. The collectio of all flow balace equatios for a DTMC is formally rereseted by (5).

8 23 Thailad Statisticia, 214; 12(2): A forecast of the future distributio of olicies amog the classes, say years from ow, ca be obtaied easily through simulatio or by comutig the -th ower of the trasitio matrix P. For may uroses, a asymtotic study is sufficiet to comare NCDs. A NCD forms a regular Markov chai: all its states are ergodic (it is ossible to go from every state to every other state), ad the chai is ot cyclic. 2.1 IRDA Trasitio Rule of NCD System The six levels of discout of IRDA are %, 2%, 25%, 35%, 45%, 5%. At the ed of each olicy year, olicyholders chage levels accordig to the followig rules: i) A olicyholder who has made o claim(s) durig a olicy year moves to the ext higher discout level or remai at 5 % if already at the highest level. ii) A olicyholder who has made at least oe claim durig a olicy year dros back to zero ercet level. 2.2 Trasitio Matrix The rules of a NCD system metioed i sectio 4 ca be summarized i a trasitio matrix showig robabilities of movemets amogst each level, see Figure 1, for the geeral otatio, where is the robability of o claim ad ( 1 ) is the robability of at least oe claim. Here, P = (Pij)5x5, Pi= 1-, i=,1,,5. Pi,i+1=, i=,,4. ad P55= Figure 1. Trasitio figure of discout levels of IRDA.

9 Dili C. Nath 231 Trasitio Probability Matrix: (7) Existig Discout Level Next Discout Level Geeral solutio of TPM (7) uder equilibrium coditio ), 1 ), 1 ), 1 ) ( ( 1 2 ( 3 ( 4 5, 4 ( 1 ), 5. (8) 2.4 The average yearly remium aid A(, m), i the steady state i terms of ad m A(,m) = m 5 i π i x ercetage of discout at differet levels. A(,m) = m (1 )[ /(1 )] (9) 1 where, m is the yearly amout of remium. 3. Data Aalysis ad Coclusio The rimary data have show i Tables 2 ad 3 are collected from the drivers of the districts of North Triura, West Triura ad Karimgaj durig Table 2 resets the frequecy of drivers who made, to 8 accidets durig their drivig life. Here, a maximum of eight accidets are made by drivers durig their drivig life. Agai,

10 232 Thailad Statisticia, 214; 12(2): Table 3 describes the two-ways cross-tabulatio of the umber of accidets made drivers i last 2 years agaist the drivig exeriece. Here, it is see that a maximum of two accidets made by drivers. The drivers are divided ito six differet grous viz. ( 5) yrs, (6 1) yrs, (11 15) yrs, (16 2) yrs, (21 25) yrs ad (25 +) yrs with resect to drivig exeriece. Table 2. Number of accidets made by drivers durig drivig life. No. of Accidets Frequecy Percet Cumulative Percet Total Table 3. Number of accidets i last 2 years vs. drivig exeriece. Drivig exeriece i years Total Number of accidets i last 2 years Zero Oe Two Total Uder the assumtio of umber of accidets idicatig umber of claims, it is see that the umber of accidets by automobile drivers, see Table 2, have a Poisso distributio with arameter λ =1.14 or Geometric distributio with arameter =.47 or Negative Biomial distributio with arameter r = 1 & =.48, see Figures 2, 3 ad 4. So the robability of o claim or accidet ( ) i the aforemetioed three cases are.32,.47 ad.48 resectively. Agai from the classical defiitio of robability, the

11 Dili C. Nath 233 robability of o claim or accidet ( ) be.45 which is very close to the results obtaied from Geometric distributio or Negative Biomial distributio. So, it may be cosidered that the umber of accidets by rofessioal automobile drivers of the study area follows geometric ( =.5) or egative biomial (r = 1, =.5) distributio. The steady state or equilibrium distributio of olicyholders usig geometric ( =.5) is π = (.5,.25,.125,.625,.3125,.3125). The exected amout of remium has to be aid yearly i the log ru for the risk is.87m where, m is the yearly remium i.e.13% less amout to be aid yearly. Figure 2. Fitted oisso distributio of umber of accidets (reseted i Table 2).

12 234 Thailad Statisticia, 214; 12(2): Figure 3. Fitted geometric distributio of umber of accidets (reseted i Table 2). Figure 4. Fitted egative biomial distributio of umber of accidets (reseted i Table 2).

13 Dili C. Nath 235 Agai, the robability of makig a accidet by good driver (6 to 2 years exerieced) ad bad driver ( to 5 years ad 21 (+) years old i.e. either very youg or too old) are.2 ad.45 resectively. So, the robabilities of o claim ( ) of the two categories of claimat are.8 ad.55 resectively (Table 3). For a good driver, the average remiums aid i the log term is.7m ad for a bad driver it is.85m i.e. they ay 3% ad 15% less yearly remiums resectively. So, the bad drivers are almost twice as likely to claim as good drivers but the remium is oly, o average margially higher. The umber of accidets by rofessioal automobile drivers of the study area follows geometric or egative biomial distributio. The bad drivers are almost twice as likely to claim as good drivers but the remium is oly, o average margially higher. Therefore, oe could adjust discouts levels of NCD slabs of IRDA so that bad drivers ay a remium doubled tha that of good drivers. Alteratively, itroductio of differet tyes of discout system like exadig the umber of categories of discout levels will be helful to ecourage the good drivers. A loger ad more gradual scale might differetiate betwee the differet risk grous more efficietly. Refereces [1] Bolad, P.J., Statistical methods i geeral isurace, ICOTS-7, 26. [2] Philiso, C., The Swedish system of bous, AUSTIN Bulleti, 196; 1: [3] Isurace Regulatory ad Develomet Authority, Available Source: htt://blog.easyisuraceidia.com/idex.h/21/7/8/title ad [4] Delaorte, P., Tarificatio du risque idividuel d accidets ar la rime modelil ée sur le risqué, ASTIN Bulleti, 1965; 3: [5] Bichsel, F., Erfahrugs-Tarifierug i der Motorfahrzeug-halflichtversicherug, Milleiluge der Vereiigug Schweizerischer Versicherugsmathematiker, 1964; [6] Bühlma, H., Otimale Prämiestufe systeme, Milleiluge der Vereiigug Schweizerischer Versicherugsmathematiker, 1964; [7] Loimarata, K., Some asymtotic roerties of bous systems, The ASTIN Bulleti, 1972; 6: [8] Vesäläie, S., Alicatios to a theory of bous systems, The ASTIN Bulleti, o. 1972; 6:

14 236 Thailad Statisticia, 214; 12(2): [9] Lemaire, J., Driver versus comay, otima1 behaviour of the olicy holder, Scadiavia Actuarial Joural, 1976; 59: [1] Lemaire, J., La soif du bous, The ASTIN Bulleti, 1977; 9: [11] Hastigs, N.A.J., Otimal claimig o vehicle isurace, Oeratioal Research Quarterly, 1976; 27: [12] Lemaire, J., How to defie a Bous-Malus system with a exoetial utility fuctio, The ASTIN Bulleti, 1979; 1: [13] Kolderma, J. ad Volgeat, A., Otimal claimig i a automobile isurace system with bous-malus structure, Joural of the Oeratioal Research Society, 1985; 36: [14] Lemaire, J., Selectig a fittig distributio for Taiwaese automobile losses, Uublised Mauscri, [15] Lemaire, J. ad Zi, H.M., A comarative aalysis of 3 Bous-Malus systems, ASTIN Bulleti, 1994; 24: [16] Lemaire, J., Bous-Malus systems: The Euroea ad Asia Aroach to Merit- Ratig, NAAJ, 1998; 2(1): [17] Pitrebois, S., Deuit, M. ad Walhi, J.F., Fittig the belgia Bous-Malus system, Belgia Actuarial Bulleti, 23; 3(1):