Modeling Motorcycle Insurance Rate Reduction due to Mandatory Safety Courses

Transcription

1 Modelng Motorcycle Insurance Rate Reducton due to Mandatory Safety Courses Shujuan Huang 1 Vadveloo Jeyaraj 1 Valdez mlano 1 Garry D. Lapdus 2 1 Mathematc Department Unversty of Connectcut Storrs CT USA 2 Injury Preventon Center Connectcut Chldren s Medcal Center Unversty of Connectcut School of Medcne Hartford CT USA Abstract Alarmng statstc has ndcated that the rsk of fatalty assocated wth motorcycle crashes far exceeds that of automobles; hereby Connectcut ntroduced mandatory motorcycle safety tranng. Ths paper develops a unfyng framework to quantfy the effectveness of such mandatory programs and to translate ths n terms of a possble nsurance rate reducton. Overall Dscount Rate stmaton and Indvdual Dscount Rate Adjustment are acheved by nonlnear optmzaton and an Integer Valued Autoregressve (INAR) model respectvely. A heterogenety factor s njected nto the model to assess the mpact of the tranng programs. Fnally numercal Illustratons are gven wth data drawn from Connectcut. 1 Introducton 1.1 Research Purpose Intense call on Insurance Industry Involvement from Publc Transportaton and Health Department In U.S. motorcyclsts were klled and njured 12. Snce 1998 there has been a sgnfcant ncrease n deaths and non fatal njures. Per vehcle mle traveled motorcyclsts are about 37 tmes more lkely than passenger car occupants to de n a motor vehcle crash and 9 tmes more lkely to be njured 12. Motorcycle related deaths account for 14 percent of total traffc fataltes n 2008 although motorcycles only made up about 3 percent of regstered vehcles 12. More and more studes have ndcated that the rsk of drver and passenger fatalty assocated wth motorcycle crashes far exceeds that of automoble crashes. Ths alarmng statstc has caused ncreased attenton from state motor vehcle departments state health departments and the nsurance ndustry to pursue efforts to ntroduce programs to dramatcally reduce motorcycle accdents. In the "Natonal Agenda for Motorcycle Safety" whch s supported by Motorcycle Safety Foundaton (MSF) and Natonal Hghway Traffc Safety Admnstraton (NHTSA) of US Department of transportaton (DOT) t s strongly recommended that the nsurance ndustry should collect organze analyze and dstrbute motorcycle specfc loss data to better understand safety ssues and develop gudelnes to te approved tranng lcensng and safe rdng practces to premum reductons. However at present practce nsurers employ lmted avenues to enhance and encourage motorcycle safety. 1

2 Motorcycle nsurers are not currently requred to provde motorcycle specfc loss data for analyss or use a safety related database to gude nsurance polcy. It also should be noted that most research or projects whch focus on evaluatng the effectveness of motorcycle safety course systematcally and comprehensvely were completed before 2000 [ ]. These studes have shown mxed results on effectveness due to dfferent knds of methodologcal ssues. In addton there s no tranng ndcator varable ncluded n current natonal traffc accdent databases. Whle some nsurers have both related clam records and nformaton about the polcyholder and whether they have taken the tranng or not t would be extremely helpful to evaluate these the effcacy of motorcycle rder tranng programs f we could employ more resources n the nsurance ndustry Reduce Insurers Own Losses by Supportng Certan Responsble Rdng Practces wth Incentves. Currently n some states most motorcycle nsurance companes offer up to a 10% dscount wthn the last three years for the successful completon of the Basc or xperenced Motorcycle Safety Course. The courses are unformly desgned and guded by MSF and the content should be nearly consstent. However the dscount rate vares greatly from company to company and from state to state. From the perspectve of the nsurers own nterest developng a quanttatve methodology to evaluate the effectveness of the Motorcycle Safety Courses and the nsurance dscount ratewould enhance motorcycle safety and optmze nsurers rsk management as well. Table.1 Facts about current motorcycle nsurance dscount rate for safety course Insurance Dscount rate Detals 1 Company PROGRSSIV NA 2 Safety Course Completng an approved safety course could earn you a dscount. GICO 10% 10% dscount for completng a Motorcycle Safety Foundaton or Mltary Safety Course Allstate 5% Save 5% f you ve voluntarly passed a Motorcycle Safe Drvng n the past 36 months. USAA 5% approved safety course wthn the last three years FORMOST NA Motorcycle safety course dscount Natonwde up to 5% Save up to 5 percent on your motorcycle nsurance when you complete an approved safety course. MARKL NA Safety Course Dscount Daryland Cycle NA Motorcycle safety course completon Rder No dscount 1 From offcal webste of nsurance companes 2 NA means the specfc value for dscount rate s not dsclosed drectly on the webste. The customers need to consult the agent case by case. 2

3 1.2 Facts about Motorcycle Safety Course Nearly all the approved safety courses recognzed by dfferent nsurance companes rely on the Motorcycle Safety Foundaton (MSF) RderCourses whch are adopted by most states DMV. The Motorcycle Safety Foundaton s an nternatonally recognzed developer of the comprehensve research based Rder ducaton and Tranng System (MSF RTS) whch s a not for proft organzaton sponsored by BMW BRP Ducat etc. There are dfferent levels of Rder Courses for example: Basc RderCourse(BRC) Basc RderCourse 2( Lcense waver skll practce) Street RderCourse 1 Returnng Rder Basc Rder Course 3 Wheel Basc RderCourse (3WBRC) Scooter Basc RderCourse (SBRC) Street RderCourse 2 (SRC2) xperenced RderCourse (RC). Some of courses are just beng ntroduced or n development or desgn phase. Dfferent states and dfferent drvng schools approved by MSF may offer levels of nstructon or gve the courses dfferent names. For example n Waterbury CT the courses offered by the Rder ducaton Program nclude the Basc RderCourse Intermedate RderCourse and the xperenced RderCourse. In New York State they desgnate the ntermedate Rder Course as the Basc RderCourse 2. However the content of the Basc RderCourse 1 and 2 (sometmes called Intermedate RderCourse) are smlar. Both courses are for those motorcycle rders who do not have a lcense yet. Whle the advanced RderCourse are for those who have been rdng for some tme. In ths paper we wll only consder two categores: BRC (Basc RderCourse) and ARC (Advanced RderCourse). 1.3 Introducton to Automoble Insurance and Pror Ratng System In an nsurance portfolo the potental rsks exposed by polcyholders vary; specfcally for automoble nsurance the lkelhood of havng crashes vares among the nsured drvers. One of the man tasks of actuares s to farly allocate the burden of barng the potental losses among polcyholders whch s materalzed by quanttatve analyss to specfy ndvdual rsks and thus to determne the premums. Ths procedure s called prcng or rate makng. There are two man phases nvolved. A base premum s determned when the polcy s ssued and then the premum wll be adjusted by dscounts or surcharges as the polcy s carred out. A dscount for the motorcyclsts who have taken the safety course desgned by MSF would qualfy for a dscount. In theory and practce motorcycle rders should pay a premum correspondng to hs/her own rsk and evaluate ths dscount rate accordng to the effectveness on rsk reducton of motorcycle rder safety tranng program. 1.4 Why Research Studes about the ffectveness have Shown Mxed Results The results of research studes lookng at the effectveness of rder tranng have shown mxed results [ ]. Most of the studes revewed a tranng program that essentally conssted of a sngle course. Most state governments and nsurance company nvolvement n the U.S. are through the lcensng functon and therefore lmted prmarly to a basc novce course. In addton MSF Rder ducaton and Tranng System 3

4 (RTS) s expandng n breadth and depth to meet the growng needs of current and prospectve rders all the whle. For example the Street RderCourse was just ntroduced n the last year (2010) and there may be a tme lag to show program effectveness. Some courses are stll under the process of developng and have not been ntroduced yet. These may lead to a fallacy of a sngle tranng course servng as an n total countermeasure. 2 ffectveness of Dfferent Levels of Motorcycle Safety Courses 2.1 Defnton of effectveness ffectveness 3 s a measure of the extent to whch a specfc nterventon procedure regmen or servce when deployed n the feld n routne crcumstances does what t s ntended to do for a specfc populaton. In other words t means dong "rght" thngs.e. settng rght targets to acheve an overall goal (the effect). Then we need to defne our overall goal frst. To put t smply f the goal s to assure the mnmum rdng sklls for ntal entry nto the motorcyclng envronment then we can say MSF safety course has acheved at a 85 90% success rate n basc courses accordng to the records of tranng schools. However n most cases we need to consder a more comprehensve goal of safety courses whch s to determne f motorcycle rder safety tranng courses have any mpact on reducng the frequency of motorcycle crashes njures and nsurance clams. It may also nclude: qualty educaton and tranng knowledge sklls atttude habts values rsk management sklls self awareness and self assessment. Snce most motorcycle nsurance companes offer up to a 10% dscount for the successful completon of the Basc or xperenced Motorcycle Safety Course the effectveness of the safety course would drectly affect the costs of nsurance companes. From the pont of vew of Motorcycle nsurance prcng we could defne actuaraleffectveness as follows: Defnton: Actuaral ffectveness for Motorcycle Safety Tranng: The average reducton rato of ncurred loss per unt exposure (or average clam frequency) to nsurance company that clamed by a populaton of untraned motorcyclsts f all would have taken the safety tranng wthout any change else (e.g. weather motorcycle physcal condton transportaton envronment). Ths s essentally the same as the expected rsk reducton for a motorcyclst to attend the safety tranng from non attendng state. For example: Actuaral ffectveness of 10 percent means that an nsurance company can reduce ther ncurred loss for one polcy holder (or frequency of clams) by 10% smply by convncng that polcy holder to attend the safety tranng course. 3 Last JM. A Dctonary of pdemology (4th dton). New York: Oxford Unversty Press

5 2.2 Formulaton of Actuaral ffectveness Note that only those Advanced Motorcycle Safety Course learners have prevous records and the novce tranng course learners don t whch means that they have no prevous clam record. Therefore there should be dfferent formula for dfferent courses effectveness. I wll begn by addressng the stuaton where the data s free from censorng or truncaton ffectveness for Basc Motorcycle Safety Course For the Basc Motorcycle Safety Course we need to compare the reducton on ncurred loss per unt of exposure after tranng wth those wthout tranng snce before tranng the BRC learners don t have a lcense yet. In partcular the estmated effectveness makes sense only when the chosen samples have smlar characterstcs such as age gender motorcycle models years rdng mles rdden per year and prmary purpose of rdng (commutng recreaton etc.). Hence matched par approach could be exploted here to calculate the effectveness for basc motorcycle safety course. Usually for nsurers we have the number of ncurred clams untes of exposures dollars of ncurred losses per year as well as whether they have taken the tranng or not (f yes the knd of course they have taken). The correspondng clam data about both clam frequency and severty of the polcyholder should be also avalable. For the matched par samples suppose we have obtaned the followng summary data about exposures clam count and ncurred losses. Here the tmelne should be based on the motorcyclsts tranng year. We denote the year they took tranng as t = 0 one year after they took the tranng as t = 1 and two years after they took tranng as t = 2. Whle for those who have not taken any tranng we don t need to consder such tme restrcton. Table 2 Matched par sample summary for BRC Types of polcyholders when loss occurs xposures for year t Total Clam Count Total Incurred Loss Have taken BRC 4 before e C L No tranng e Bt N If we defne the effectveness by the measure of reducton n ncurred loss the 2 2 L effectveness 5 Bt ebt t 0 t 0 for year t of BRC would be: ρ B = = = LN en C Bt N + L Bt N 4 For those have taken both BRC and RC n hstory only consder the most recent one n other words the RC. 5 When the value of ρbt s negatve we take t as 0 mean t s not effectve at all 5

6 If we defne the effectveness by the measure of reducton n clam frequency the effectveness for year t of BRC would be: 2 2 CBt ebt ρ t 0 t 0 B = = = CN en ffectveness for Advanced Motorcycle Safety Course For the Advanced Motorcycle Safety Course we can compare the clam data for the same polcyholder before and after they took the course. The tmelne should also be based on the motorcyclsts tranng year. For dfferent polcyholders however ther tranng years may be dfferent. Here let n denote the number of years for polcyholder s records before tranng and m denote the number of years after. Table3 Clam hstory statstcs for motorcyclsts who took the RC For those who has taken RC xposures Total Clam Count Total Incurred Loss Before taken RC L After taken RC e C 1 1 e C 2 L 2 If we defne the effectveness by the measure of reducton n loss the effectveness of RC would be: e 1 L 2 m ρ = e = 1 L1 n + If we defne the effectveness by the measure of reducton n clam frequency the effectveness for year t of BRC would be: e 1 C 2 m ρ = e = 1 C1 n 3 Motorcycle Insurance Rate Reducton Modelng In the followng sectons we wll use the effectveness by the measure of reducton n clam frequency that s the (frequency part of the) pure premum 6. Frstly we wll estmate the overall dscount rate usng nsurance clam data drawn from states where such mandatory programs have been ntroduced. Secondly smlar to the bonus malus scheme for each polcy holder when we need to determne the specfc dscount rate for each person we would consder both the polcy holder s clam hstory and overall dscount rate. 6 The complete pure premum ncludes also the cost of the clam. It s equal to the frequency part tmes the expected cost per clam when cost per clam and clam occurrence are ndependent

7 3.1 Reflecton on Current Tranng Dscount Rate Polcy As shown n table 1 of secton 1.1 some nsurance companes use unfed dscount rate such as 5% n Allstate and 10% n GICO whle other nsurance companes use flexble polces for dfferent polcy holders such as up to 5% n Natonwde or negotate case by case n PROGRSSIV. It should also be noted that some nsurers requre the safety course should be taken wthn 3 years lke Allstate and USAA whle others not. Based on those dfferences from the pont of vew of rsk management and proft maxmzaton of nsurers we wll frst evaluate whether we need to add the constrants that the safety course should be taken wthn 3 years then determne what specfc value or upper bound we should use. 3.2 valuaton about the 3 year Constrants Ths could be easly carred out by check the effectveness of the BRC and RC. CN en CBt e Bt For BRC as we dscussed before ρ Bt = where we denote the year CN e N + they took tranng as t = 0 one year after they took the tranng as t = 1 two years after they took tranng as t = 2 and so on. Then we can compare ρ B4 wth ρ B1 ρ B2 and by a smple statstcal test f there are sgnfcant dfference between them we ρb3 could add the 3 year constrant otherwse t s unnecessary. Meanwhle we could evaluate other possble k year polcy usng smlar methods. e 1 C 2 m For the RC as we dscussed before ρ =. In order to evaluate the e = 1 C1 n + 3 year polcy we could fx n as or more. Then compare whether there s any sgnfcant statstcal dfference when n s larger than 3. If t s we suggest keep the 3 year polcy otherwse not. 3.3 Overall Dscount Rate stmaton Usng Past Insurance Clam Data Frstly we could use past nsurance clam data to estmate the overall dscount rate consderng both the tranng effect on pure premum and the demand. Then based on ths estmated overall dscount rate dscount for ndvdual motorcyclst could be further adjusted accordng to ther own clam hstory Incentve Adjustment on Dscount Rate Snce from January 1 st 2011 Connectcut requres motorcyclsts of all ages who want to get the lcense to take the BRC nsurance companes don t need to use dscount rate to attract the customers but should offer basc dscount based on the reduced rsk by tranng. Whle the RC s voluntary then t s possble that someone took the course just for the dscount and the rders vew on effectveness of the RC also counts. 7

8 Therefore the number of polcyholder who has taken RC (wthn three year) should be a functon of dscount rate r the effectveness of the course ρ and the total demand N. Ths paper denotes that functon as e f ( r ρ N ) t t t =. For most states where the BRC s not requred an nsurer could also consder the ncentve adjustment on BRC dscount rate to ncrease the number of polcyholder to reduce the total rsk and ncrease proft. There are three factors whch can affect total proft and rsk management of nsurance companes as follows: Table 4: Three factors that affect profts of nsurer Type varable Impact on nsurance company proft Reducton n ndvdual premum because Dscount rate r of the dscount offered to customers Possble Reducton n Clam Cost because the tranng effect n Rsk Reducton Possble ncreasng number of polcy holders because of the dscount ncentves or state government requrement on tranng ffectveness ρ B xposures et ebt e Nt ρ and Overall Dscount Rate for RC and BRC learners As we dscussed before for the RC learners we can compare ther clam hstory before and after the tranng. If we have derved (the frequency part of) pure premum before tranng PP 1 and after tranng PP 2 separately we could smply calculate the dscount rate as r = ( PP PP2) PP1. Meanwhle comprehensvely consderng the pure premum and demand we wll also consder ntroducng the adjustment factorκ. Suppose among all the polcyholders who have taken the safety course for year t the proporton for BRC s and the proporton for RC s. Once we have estmated r and r B based on the changes on pure premum the unfed dscount rate r could be r = κ r + r. We solve the optmzaton problem wth followng steps. ( ( ) B) Step 1: stmate r and A r B based on the changes on pure premum Actually here we could drectly use the value of ρ and ρb to ntally estmate r and r B because the formula for ρ and ρb exactly reflects the change on (severty/ frequency part of) pure premum usng the nsurance clam data. Step 2: stmate the total n force exposure at a certan tme t For current year s exposure we could use the basc methods n ratemakng to estmate the total n force exposures at a certan tme let s denote t as N. Then we could have e t + e Bt + e Nt = N t t + +/ 8

9 For future year s total exposure we could use tme seres forecastng methods to estmate them. It should be noted that we need to consder the government mandatory polcy s affect on the number of motorcyclsts who have taken the course n the course of analyss. Step 3: Formulate e t and ebt e Bt should depend on the polces of dfferent States DMV. Some states requre all age motorcyclsts to take the BRC f they want to get the lcense (e.g. Connectcut Texas); whle others requre the motorcyclsts under 18 (or 16 21) to take the course. In ths paper we wll study the former stuaton. Snce we just ntroduced the polcy n 2011 we could estmate ebt by the entre new lcense ssung pattern n the past years by the method of tme seres analyss. As we dscussed before the number of polcyholder who have taken RC (wthn three years) should be a functon of dscount rate r the effectveness of the course ρ and the total demand N t that s et = f ( r ρ Nt ). Objectvely the functon f should be monotone non decreasng functon of both r and ρ when Nt s gven. We assume ρ to be a constant n a certan perod. ρ could be estmated by the data n secton 2.2. Here we treat r (or adjustment factorκ ) as a decson varable n our programmng. In fact the functon f could be treated as the utlty functon on the effectveness of the tranng and the nsurance dscount for customers. Smlar to the commonly used xponental Utlty 7 n nsurance ndustry we assume ( ρ ) ( ) ( r+ ) ( ) e = f r N = N e e α βρ (3.1) t t t Bt where the factor α β can be determned subjectvely or objectvely. To put t smply we could let both α and β be 0.5. Step 4: Fnal Programmng Objectve: St. 0 r 15% ; k > 0 ( ( 1 ) B) t 1 Nt 1 Arg Max ρ ρ ( ) r L L e + e r e + e P (3.2) t B Bt t Bt t et 1 ent 1 r = κ r + r (3.3) et + ebt + ent = Nt ( ) t t Bt ( ) ( 1 r+ ) e = N e e α βρ (3.4) 7 Utlty Theoretc Underwrtng 9

10 Where Nt ebtneed to be estmated or calculated by past data. LA t 1 ea t 1 LN t 1 en t 1 are based the prevous years data fle. ρ ρbare estmated n secton 2.2 the same for r and r B. Ths s a nonlnear programmng problem. We could use MATLAB to solve t. We could get the optmzed value for κ frstly then the value for r easly followed. 3.4 Indvdual Dscount Rate Adjustment by Personal Clam Hstory Snce we have estmated the overall dscount rate r n secton 3.3 now we am at determnng the specfc dscount rate for each person whch s smlar to the bonusmalus scheme for each polcy holder Dscount Rate for RC learners In ths paper the Integer Valued Autoregressve (INAR) method wll be used to model ndvdual annual clam count n consecutve polcy years. Al Osh and Alzad (1987) proposed what they have called an nteger valued frst order autoregressve (INAR(1)) model. Later Goureroux and Jasak (2004) appled t to car nsurance n bonus malus system desgn. They compared the advantages of INAR than the tradtonal negatve bnomal approach. Zhang(2009) generalzed the INAR to dynamc heterogenety wth applcatons n automoble nsurance. Here we wll use the smlar INAR(1) model but nterpret the heterogenety as tranng mpact factor. Let s consder the polcy holder and denote Nt the number of clams at year t submtted by ths ndvdual. We assume N 1... N t N t+ 1are ndependent condtonal on an unobservable heterogenety factor. We assume that the heterogenety s a tme ndependent random varable and follows a Gamma dstrbuton γ ( a ( 1 r) a). Here the parameters are desgn to ensure the expectaton of to be r ; where r s the overall dscount rate estmated n secton 3.3. N = B p o N + ε (3.5) Where { ε t } t values; B ( ) t = 1 ( ) ( ) t t t 1 t ε t = θ ~ Posson( θλ ) (3.6) r 8 ~ Gamma( a ) (3.7) a = r (3.8) [ ] 1 s a sequence of random varables takng nonnegatve nteger p s the so called Bnomal thnnng factor whch s ndependent of the error term and defned by ( ) N t 1 t o t 1 = j j= 0 B p N U U s a sequence of..d. Bernoull random j 8 Recall: For gamma dstrbuton Gamma( k θ ) where k s the shape parameter and θ s the scale parameter. Mean μ = kθ 10

11 varables wth autoregressve parameter p.the mxture operaton o s called bnomal thnnng. Note: () Intutvely Nt s ntroduced nto two parts one s the lagged clam counts from prevous years Bt( p) o N t 1 the second part s the newly arrved clam counts ε t. Lagged clam counts n the frst part are ntroduced because durng the loss settlement perod whch can be many years n duraton addtonal facts regardng ndvdual clams and trends often wll become known (ncludng unpad and often unreported losses to ther ultmate settlement values) () For expostory purpose we focus on the autoregressve process of order 1 but the approach s easy to extend to hgher autoregressve orders. () Nt s the average annual clam frequency for polcy holder durng the year t (We denote the year before they took RC as t = 1 the year they took tranng as t = 0 one year after they took the tranng as t = 1 two years after they took tranng as t = 2 etc.) Therefore here the ndex t s not ndcatng the exact year but a relatve tme compare to the tme when the motorcyclsts took the tranng. Then Nt s the correspondng statstcs. Here we only consder one year before the tranng because there should be no tranng mpact for consecutve years before the tranng hence msmatchng wth formula (3.5) and (3.6). At yeart the (frequency part of ) pure premum s Pt = Nt + 1 N 1 N0... N t = Nt + 1 N 1 N0... Nt N 1 N0... N t = pnt λ N 1 N0... N + t = pnt + λ N 1 N0... N t Where λ s predetermned by the basc ratemakng category of such polcy holder.the rsk on the count varable N t + 1 can be measured by Rt = V Nt + 1 N 1 N0... N t = V Nt + 1 N 1 N0... Nt N 1 N0... N t + V Nt + 1 N 1 N0... Nt N 1 N0... N t = p(1 p) Nt λ N 1 N0... N + t + V pnt λ N 1 N0... N + t 2 = p(1 p) N t + λ N 1 N0... N t λ V N 1 N0... N + t 11

12 As suggested by many nvestgatons the perod that the safety courses play a key role to reduce rsk s wthn 3 years after the course taken tme (Of course t s easy to generalze 3 to any other number k f necessary) here we only need to consder the short clam hstory t = that pertans to a customer wth a senorty for up to 3 years or new customers wth smlar hstory. Proposton 3.1 Condtonal Dstrbuton of Where () For t = 1 the condtonal dstrbuton of () For t = 0 the condtonal dstrbuton of r s γ ( a ) a gven N 1s 1/ a λ γ a+ N 1 + r p () For t = 1 the condtonal dstrbuton of gven N 1 N0 s ( 2 1 0) mn( N 1 N0) z2 = 0 mn( N 1 N0) z2 = 0 a λ π ( z2 N 1 N0 ) γ a+ N 1 + N0 z21/ + λ + 1 r 1 p π ( z2 N 1 N0) a λ τ ( a+ N 1 + N 0 z2) + λ p 1! z2 z p r 2 = N 1 z2 p λ N z π z N N C ( 0 2 ) a+ N 1 + N0 z2 (v) For t = 2 the condtonal dstrbuton of gven N 1 N 0 N1 s a π ( z2 z3 N 1 N0 N1 ) γ a+ N 1 + N0 + Nz2 z31/ + 2λ + = 0 z = 0 r 1 mn( N1 N0 ) mn( N 1 N0 ) π z z N N N mn( N N ) mn( N N ) z3 2 Where π ( z2 z3 N 1 N0 N1) = z3= 0 z2= 0 ( ) a λ z2+ z τ 3 ( a+ N 1 + N0 + Nz2 z3) + 2λ + z z r p p 2 CN C 0 N 1 z2+ z3 1 p λ ( N0 z2 )!( N1 z3 )! Proof: Case t = 0 The jont dstrbuton of N 1and s ( ) ( ) ln ( ) = l N l 1 1 a+ N 1 + N0 + Nz2 z3 12 λ p

13 N 1 λ 1 a 1 a 1 a λ a = exp( ) exp 1 p 1 p N! 1 r τ a r ( )( ) N 1 a 1 a λ + ~ exp + r p Where INAR(1) defnes a count process whch has a margnal Posson dstrbuton wth a modfed parameter P λ /1 ( p) Therefore the condtonal dstrbuton of gven N 1s 1/ a λ γ a+ N 1 + r p The proof for other cases s smlar to ths and easy to generalzed. a Proposton 3.2 Predcton of the Course Impact Factor and (Frequency Part of) Pure Premum () For 1 () For 0 () t = [ ] 1 = r P 1 = a λ 1 = + 1 / + r p λ t = N ( a N ) a 1 P0 = pn 1 + λ N 1 = pn 1 + ( a+ N 1) / + λ ( r) p For mn( N N ) ( )( ) 1 0 π z2 N 1 z2 0 N0 a+ N 1 + N0 z = 2 t = 1 N 1 N 0 = mn( N 1 N0) a λ π ( z2 N 1 0) z2 0 N λ = + + r p P = pn + λ N N (v) For t = 2 N 1 N0 N 1 = mn( N1 N0 ) mn( N 1 N0 ) π z3= 0 z2= 0 mn( N1 N0 ) mn( N 1 N0 ) z3= 0 z2= 0 ( z2 z3 N 1 N0 N1 )( a+ N 1 + N0 + Nz2 z3) π a λ + 2λ + r p ( z2 z3 N 1 N 0 N 1) P = pn + λ N N N

14 Proposton 3.3 Indvdual Dscount Rate P0 P 1 () For t = 0 r 0 = P 1 + P1 P 0 () For t = 1 r 1 = P 0 + P2 P 1 () For t = 2 r 2 = P 1 + Where rt s the dscount rate for the th polcy holder n the year t (t s a relatve tme to the tranng). P 1 P 0 P1 P 2 are estmated through Proposton Indvdual Dscount Rate for BRC learners For the BRC learners we don t have the BRC learners prevous clam hstory record. One straghtforward soluton s to use the overall dscount rate estmated n secton 3.3 whch already comprehensvely consdered the effectveness of BRC program and other factors. 4 Numercal Illustratons 4.1 stmaton of ffectveness If we could cooperate wth nsurance company to obtan the ndvdual level nsurance clam data as well as the characterstcs of those polcy holders (to show whether they have taken any safety course or not) we could use the method n secton 2 to estmate the effectveness of BRC and RC respectvely. Because the data s unavalable we wll try to use the roughly estmated result from prevous research to show our analytcal framework. In James and Barbara(1989) Does Motorcycle Tranng Reduce Crashes? They showed overall the untraned group had 32 % more crashes than the traned group. John W Bllhemer (1998) valuaton of Calforna motorcyclst safety program showed that analyses of statewde crashes trend ndcate that fatal motorcycle crashes have dropped 69 percent snce the ntroducton of the CMSP. Whle many other researchers showed that the program s not effectve at all. For the llustraton purposes we assume the effectveness of BRC s 10% and the effectveness for RC s 6%. 4.2 stmaton of Overall Dscount Rate for Motorcycle Safety Course Year Table 5 CT total student regstraton BRC regstraton RC regstraton Total Regstered Motorcycles 9 9 Source: Hghway Statstcs 14

15 Total Percentage 96% 4% Based on the data n table 5 we could estmateα = 4% α = 96% and r and r as B r = r + r ) the data 2% and 10% respectvely. Usng the rough value of r ( ( ) e t e Bt and Nt from 2000 to 2010 we estmate the values of parameter α and β n ( A ) e = N e e αr+ βρ as 0.5 and 0.5 respectvely. ( ) t t Bt ( ) In addton snce from January 2011 all applcants must successfully complete a novce safety course before obtanng a motorcycle endorsement. It can be antcpated that there would be a sharp rse on the 2011 BRC regstraton. Based on the pattern n table 5 let s suppose t return to the level n 2008 say Whle the number of total regstered motorcycles seems to be relatvely stable these years let s forecast we have total regstered motorcycles n From the webste of Rocky Mountan Insurance Informaton Assocaton we got the Cost of Auto Insurance by State 10 as follows. Here we assume the rate of motorcycle nsurance s on the same level of other autos. Then base on the pattern n table 6 we estmate the average expendtures for and 2011 are and 915 respectvely by the method of Two Movng Averages. Table 6 CT average auto nsurance expendture State 2008 Average Average Average 2006 xpendture 11 Rank xpendture Rank xpendture Rank Connectcut $950 9 $ $ B 10 Cost of auto nsurance 11 The average nsurance expendture s calculated by addng all auto nsurance premum collected for lablty comprehensve and collson coverage and dvdng by the number of nsured cars for the year. Ths average s based on all polces ncludng lablty only and polces wth optonal comprehensve and collson coverage. Lmts on polces vary wdely and are based on state requrements as well as consumer choce. 15

16 Fatal Injury Table 7 CT motorcycle crash number and nvolved person number Incapacta Not tng njury njured Nonncapacta tng evdent njury Possble njury(cla m of nonevdent njury) Total Involved Person Total Crash Number Year K A B C N Based on the data n table 7 we get the estmated values of Lt 1and LN t 1 as Lt 1 = LNt 1 = Then e + e = N e = = t Nt t Bt ( αr+ βρ ) ( ) ( ) 1 r+ t = t Bt = ( ) ( ) e N e e e et 1 = 106 ent 1 = P t = 915 Then the objectve functon turns nto (0.5r ) (0.5r ) ( ) ( ( ) ) MAX e r e where 0 r 15% It s easy to solve ths nonlnear programmng and we get the value for r r = % and κ = stmaton of Indvdual Dscount Rate The values of other parameters are chosen to be p = 0.3and λ = 0.3 for comparson between dfferent clam patterns. r = 8% [ ] = r. Then usng the result n Proposton 3.1 to 3.3 we could obtan the estmated result n table 8 to table 10. Table 8 xpected value of the tranng mpact factor gven the clam hstory Clam Hstory N 1 N 1 N 0 N 1 N0 N 1 (001) (010)

17 (120) (101) (110) (011) (200) (210) (300) (100) Table 9 Predcton of ( the frequency part of ) Pure Premum Clam Hstory P 1 P 0 P 1 P 2 (001) (010) (120) (101) (110) (011) (200) (210) (300) (100) Table 10 Indvdual Tranng Dscount for Motorcycle Insurance Clam Hstory r 0 r 1 r 2 (001) (010) (120) (101) (110) (011) (200) (210) (300) (100) We should be notce that there may be other factor would affect the clam hstory of customers not just because of the tranng. Therefore for those values of dscount rate hgher than 10% we wll use 10% f lower than 10% we wll use the estmated value. r = mn( r 10%) t t 17

18 5 Concluson The geness of ths paper comes from the debates about ntroducng of Mandatory Motorcycle Safety Course n Connectcut n January However most research or projects whch focus on evaluatng the effectveness of motorcycle safety course systematcally and comprehensvely are before 2000 and have shown mxed results on effectveness. We also notced that the safety courses are unformly desgned and guded by MSF but the dscount rate offered by dfferent nsurance companes vares greatly. Therefore ths paper has developed a unfyng framework to quantfy the effectveness of such mandatory programs and to translate ths n terms of a possble nsurance rate reducton. Frst the defnton of effectveness s gven and formulated for both Basc and Advanced Motorcycle Safety Course by the measure of reducton n ncurred loss or clam frequency. Then evaluaton about current most wdely adopted 3 year constrants s carred out. Next our research s dvded nto two major steps: Overall Dscount Rate stmaton and Indvdual Dscount Rate Adjustment by Personal Clam Hstory. For the Overall Dscount Rate we use past nsurance clam data to estmate the overall dscount rate n nonlnear optmzaton programmng taken nto consderaton both the tranng effect on pure premum and the demand. For the Indvdual Dscount Rate Adjustment the Integer Valued Autoregressve (INAR) method s used to model ndvdual annual clam count n consecutve polcy years and nterpret the heterogenety as tranng mpact factor. Through the strct theoretcal dervaton condtonal dstrbuton of predcton of the Course Impact Factor and (Frequency Part of) Pure Premum as well as Indvdual Dscount Rate are derved. To llustrate our analytcal framework numercal Illustratons are gven n the last part of ths party. Snce the ndvdual level nsurance clam data as well as the characterstcs of those polcy holders (to show whether they took the safety tranng or not) are temporarly unavalable we made some assumptons about the effectveness for BRC and ARC respectvely; but regstered motorcycles Average Auto Insurance xpendture Motorcycle Crash Number and Involved Person Number are based on Connectcut data. Based on our assumpton the overall dscount rate for motorcycle safety course should be 8% whch may not reflect the real stuaton but can serve as an llustraton. If nsurance clam data were avalable we could update the assumptons n our model and determne the estmaton for overall dscount rate. For the Indvdual Dscount Rate stage we consdered 10 dfferent knds typcal clam hstores n the past three years and generated the Indvdual Tranng Dscount Adjustment table. Possble future work could be the collecton and combnaton of the data both from the transportaton health department and nsurers to conduct the analyss presented n ths paper; or ntroduce more stochastc factors n the calbraton of effectveness. 18

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20 [16] Tm Buche Sherry Wllams and Ray Ochs(2010)Motorcycle Safety Foundaton MSF RTS: A System Desgned To Succeed presented n support of the Vulnerable Road Users Conference n Jerusalem May 30 June [17] Walhn J. F. and J. Pars (2000). The true clam amount and frequency dstrbutons wthn a bonus malus system. ASTIN Bulletn