CIRPÉE Centre interuniversitaire sur le risque, les politiques économiques et l emploi

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1 CRPÉE Cenre ineruniversiaire sur le risque, les poliiques économiques e l emploi Cahier de recherche/working Paper Vehicle and Flee Random Eecs in a Model o nsurance Raing or Flees o Vehicles Jean-François Angers Denise Desardins Georges Dionne François Guerin Ocobre/Ocober 2004 Angers, Desardins, Guerin: CRT, Universiy o Monreal Dionne: HEC Monréal, CRT, CRPÉE, and THEMA (France)

2 Absrac: We are proposing a parameric model o rae insurance or vehicles belonging o a lee. The ables o premiums presened ake ino accoun pas vehicle accidens, observable characerisics o he vehicles and lees, and violaions o he road-saey code commied by drivers and carriers. The premiums are also adused according o accidens accumulaed by he lees over ime. The model proposed accouns direcly or explici changes in he various componens o he probabiliy o accidens. represens an exension o bonus-malus-ype auomobile insurance models or individual premiums (Lemaire, 985 ; Dionne and Vannase, 989 and 992 ; Pinque, 997 and 998 ; Frangos and Vronos, 200 ; Purcaru and Denui, 2003). The exension adds a lee eec o he vehicle eec so as o accoun or he impac ha he unobservable characerisics or acions o carriers can have on ruck acciden raes. This orm o raing makes i possible o visualize wha impac he behaviors o owners and drivers can have on he prediced rae o accidens and, consequenly, on premiums. Keywords: Flee o vehicles, random eecs, vehicle eec, lee eec, insurance pricing, behaviors o owners and drivers, Poisson, gamma, Dirichle Résumé: Nous proposons un modèle paramérique de ariicaion de l assurance de véhicules rouiers apparenan à une loe. Les ables de primes qui y son présenées iennen compe des accidens passés des véhicules, des caracérisiques observables des véhicules e des loes e des inracions au code de la sécurié rouière des conduceurs e des ransporeurs. De plus, les primes son ausées en oncion des accidens accumulés par les loes dans le emps. l s agi d un modèle qui prend en compe direcemen des changemens explicies des diérenes composanes des probabiliés d accidens. l représene une exension aux modèles d assurance auomobile de ype bonus-malus pour les primes individuelles (Lemaire, 985; Dionne e Vannase, 989 e 992; Pinque, 997 e 998; Frangos e Vronos, 200; Purcaru e Denui, 2003). L exension aoue un ee loe à l ee véhicule pour enir compe des caracérisiques ou des acions non observables des ransporeurs sur les aux d accidens des camions. Cee orme de ariicaion compore plusieurs avanages. Elle perme de visualiser l impac des comporemens des propriéaires des loes e des conduceurs des véhicules sur les aux d accidens prédis e, par conséquen, sur les primes. Elle mesure l inluence des inracions e des accidens accumulés sur les primes d assurance mais d une açon diérene. En ee, les ees des inracions son obenus via la composane de régression, alors que les ees des accidens proviennen des résidus non expliqués de la régression sur les accidens des camions via un modèle bayésien de ariicaion. Mos Clés: Tariicaion de l assurance, loe de véhicules, modèle bayésien, sécurié rouière, bonus-malus JEL Classiicaion: D8, G22

3 NTRODUCTON Very ew sudies have analyzed sysemaically he risks o accidens or vehicle lees. Marie- Jeanne (994) developed a raing model based on he size o he lee and Teugels and Sund (99) proposed raing based on he aggregaed loss o he lee. Oher researchers have conined hemselves o sudying he drivers o vehicles o obain a porrai o he risks posed by a carrier (Dionne e al., 200). This amouns o orgeing ha irms owners or managemen can also aec he acciden raes o heir vehicles. Decisions regarding driving hours, spending on vehicle mainenance, and guidelines or loading or securing cargo in vehicles can have repercussions on road saey. Dionne, Desardins, and Pinque (999 and 200) developed bonus-malus models ha use a semi-parameric approach o ake ino accoun he behaviors o boh he drivers and owners o vehicles. n his aricle, we propose a parameric model. Measuring he risks associaed wih lees o vehicles is diicul or a number o reasons. For one, he unis composing he lees mus be deined. Should we do his by observing drivers or vehicles? We answered ha quesion by oping or vehicles: For, wih inormaion readily available rom insurers, he link beween vehicles and carriers can be made coninuously. Linking inormaion on drivers o carriers is, in conras, very cosly, since he movemens o drivers rom one lee o anoher are no sysemaically recorded, whereas licensing and insurance conracing keep rack o vehicles as hey move among lees. The vehicles are aken o represen dieren individual risks. These risks are inluenced by he observable and unobservable characerisics o he vehicles, he drivers using hem, and he carriers who own or lease hem. is hus essenial o use care in modeling hese dieren sources o inormaion. 2

4 Anoher diiculy is weighing he inormaion obained on individuals and lees or raing purposes. An adequae model or raing he risks o lees mus inegrae he behaviors o drivers wih hose o owners so as o inroduce incenives or saey ailored o he various levels o decisions o be made when acing hierarchical moral hazard (Moses and Savage, 994, 996; Flue, 999; Winer, 2000). We are proposing a new raing model or vehicles belonging o a lee. The model is a parameric one which can accoun direcly or boh observable and unobservable characerisics o he vehicles, drivers, and owners associaed wih a paricular vehicle lee. The model proposed is a direc exension o bonus-malus-ype auomobile insurance models (Lemaire 985, 995; Dionne and Vanasse 989 and 992; Pinque 997, 998; Frangos and Vronos 200; Purcaru and Denui 2003) o individual premiums (see Pinque, 2000, or a review o he lieraure). The exension adds a random lee eec o he vehicle eec in he model, in order o ake ino accoun he unobservable eecs o carriers, vehicles, and heir drivers on ruck acciden raes in he Bayesian or a poseriori calculaion o premiums. Observable variables characerizing vehicles, lees, and he road-saey behavior o boh drivers and carriers are used in evaluaing he a priori risks o dieren vehicles. n he ollowing secion, we presen saisical models or esimaing acciden probabiliies or vehicles belonging o lees o various sizes. Secion 3 develops he opimal bonus-malus sysem inegraing boh lee and vehicle eecs. Secion 4 proposes dieren premium ables, while secion 5 discusses possible exensions o he model. 3

5 2 STATSTCAL MODELS Our mehodology is divided ino wo seps. n he irs sep, we use an economeric model o evaluae he acciden probabiliies or he vehicles o carriers. As a priori inormaion, we shall use esimaed parameers o calculae insurance premiums. These parameers ake ino accoun he inormaion available on he observable characerisics o vehicles and lees as well as on raic violaions by drivers and carriers. n order o ake unobservable characerisics and acions ino accoun or purposes o raing, we shall use he residuals o he economeric esimaions. One o he aricle s conribuions consiss in proposing a new model or esimaing acciden probabiliies, a model capable o explicily isolaing he lee eec rom he vehicle eec. n a second sep, we propose a bonus-malus sysem which can use boh he a priori inormaion obained rom he esimaed parameers and he a poseriori inormaion obained rom residuals o he esimaions o vehicle acciden disribuions. n order o show wha conribuion he dieren eecs make o insurance premiums, we shall disinguish beween one-vehicle and wovehicle carriers and hen generalize he model o carriers wih more han wo vehicles. 2. ECONOMETRC MODEL FOR ESTMATNG DSTRBUTONS OF VEHCLE ACCDENTS Mos economeric models applied o discree (or counable) variables are based on he Poisson disribuion, where probabiliy P ha a vehicle i belonging o lee will be involved in accidens a period can be represened by he ollowing expression: 4

6 λi i i i y ( ) i λi e P( y λ ) =. y! Wih he Poisson law, we obain ha he mahemaical expecaion o he number o accidens (E) is equal o he variance (Var), E( Y ) Var( Y ) ruck i belonging o lee a period and = =λ where i i i Y i is he number o accidens or λ i is he parameer o he Poisson disribuion. This modeling implicily supposes ha he disribuion o accidens can be enirely explained by observable heerogeneiy, which cancels any need or a bonus-malus sysem. Le us now suppose ha an unobservable heerogeneiy exiss owing o cerain characerisics or acions non observable by he insurer. Suppose ha λ i =iαθ i wih X i i = de β i where d i measures he number o days ha vehicle i o lee is auhorized o circulae during period, divided by he number o oal days in period. This measures he exposure o he risk o acciden in period. Using he exponenial o deine vecor X ( i x i,,xip ) i allows us o ensure he non-negaiviy o λ i. The = L conains he p characerisics o ruck i in lee observed a period ; his vecor conains speciic inormaion on he vehicle and oher characerisics on he lee. β is a vecor o parameers o be esimaed. Parameer α is he random eec associaed wih lee, ha is, he unobservable risk aribuable o he lee; whereas parameer θ i is he random eec o ruck i in lee. We suppose ha θ i = where is he oal number o vehicles in lee. n oher erms, θ i is he proporion o he risk or lee which can be aribued o vehicle i; he oal unobservable risk or vehicle i o lee is deined by αθ i. should be noed ha when 5

7 lee has only one vehicle such ha =, θ = by deiniion. This means ha he risk aribuable o vehicle corresponds o ha o he lee, rom which i ollows ha λ = α. We make he hypohesis ha θ i will ollow a Dirichle parameric disribuion wih parameers ( ν, ν, L, ν ) and ha 2. This α will ollow a gamma disribuion wih parameers (, ) parameizaion permis o obain a mean lee eec ha increases wih he number o vehicles in he lee. 2.. Size- carrier For period, he disribuion o he number o accidens or a lee wih one vehicle is given by: ( ) ( ) ( ) P y = P y α α dα, 0 which, assuming ha α ollows a gamma disribuion (, ), can be rewrien as ollows: ( ) P y ( y ) ( y ) ( ) + + Γ + = Γ + Γ y. () This disribuion has been used airly oen in he lieraure (Lemaire, 985; Dionne and Vanasse, 989; Hausman e al., 984; Gouriéroux, 999). is capable o modeling unobservable heerogeneiy and o inroducing a bonus-malus sysem or individual observaions. On he oher hand, i is no direcly applicable when esimaing he probabiliy o accidens or vehicles belonging o a lee, as i canno isolae he lee eec rom he vehicle eec. We now presen 6

8 our generalizaion o his basic model, saring wih he simple case o a lee composed o wo vehicles Carrier wih 2 vehicles The oin probabiliy o he number o accidens a period or he wo vehicles in lee is given by: where ( ) ( ) ( ) P y,y = P y,y θ θ dθ, θ =θ and 2 θ =θ. Condiionally on θ, he oin probabiliy o acciden is equal o: ( ) ( 2) ( ) ( ) ( y ) ( y2 ) ( 2 ) y y2 y y θ θ yi + 2 α( θ + ( θ ) 2+ ) P( y,y 2 θ ) = ( α ) i = e dα Γ + Γ + Γ. (2) 0 By inegraing (2) and subsiuing he value o P( y,y 2 θ ) in P( y,y 2) ( ) ( ( ) 2), we obain: 2 y y 2 y y 2 Γ 2 + yi ( 2 ) ( ) ( ) ( ) 2 θ θ i 2 = = 2 ( ) θ θ 0 Γ ( y + ) Γ ( y2 + ) Γ( 2 ) 2 + yi P y,y d +θ + θ. (3) n order o esimae he probabiliies o acciden wih a parameric approach, we mus now make he disribuion o θ more explici. As indicaed above, we suppose ha he vehicle eec will 7

9 ollow a Dirichle disribuion. By replacing he densiy uncion ( θ ) in equaion (3) wih he densiy o a parameric Dirichle ( ν,ν 2 ), we obain: 2 Γ ν ( θ ) = θ θ Γ ν i ν2 2 ν ( ) ( ) ( ) i, ( 2) P y,y = 2 2 y y 2 Γ 2 + yi y+ν y2+ν2 ( ) ( ) 2 Γ( ν +ν2) ( θ) ( θ) 2 d. θ Γ ( y + ) Γ ( y2 + ) Γ( 2 ) Γ( ν) Γ( ν2) yi ( ( ) ) i +θ + θ = 2 (4) To obain a value o he oin probabiliy in (4), we mus compue he inegral: ( θ ) ( θ ) y +ν y2+ν2 2 dθ yi ( +θ + ( θ ) 2) To do so, le s wrie he expression ( ) +θ + θ o he denominaor as ollows: θ, +2 ( ) which permis us o rewrie he inegral in (4): ( θ ) ( θ ) ( y ) ( y2 2) ( y y ) y +ν y2+ν2 Γ +ν Γ +ν 2 dθ = yi 2 yi + Γ + 2 +ν +ν2 ( ) ( ) θ +2 F y +ν ;2 + y + y ; ν +ν + y y 2;. +2 8

10 2F is a hypergeomeric uncion whose value is equal o: ( ) [ ] 2 [ l] l l 2 2 y +ν 2 y + i ( ) [ ], l =! l l y i +νi wih [ l] ( ) ( ) h = h h L h l +, a decreasing acorial uncion. The disribuion o he number o accidens observed a period or he wo vehicles in lee is now given by: 2 2 y y2 Γ 2 + yi ( ) ( 2) Γ( ν +ν2) 2 = Γ ( y + ) Γ ( y2 + ) Γ( 2 ) Γ( ν) Γ( ν2) ( ) P y,y ( +2) yi i = ( y ) ( y ) ( y y2 2) Γ +ν Γ +ν +ν + + Γ + +ν +ν ; y y ; F y ;2 y y2 2 2 ν+ν We now generalizes he model o a lee o vehicles Carrier wih more han 2 vehicles The oin disribuion o he number o accidens a period or he vehicles in lee is given by: where ( ) ( ) ( ) P y, L,y = L P y, L,y θ, L, θ θ Lθ dθ L dθ (5) 0 0 9

11 i. θ = θ We can rewrie he condiional probabiliy in (5) as: ( L L ) ( ) ( ) L L P y,,y θ,, θ = P Y,,Y α, θ,, θ α dα 0 and by inegraing wih respec o α, we obain a negaive binomial disribuion whose oin condiional probabiliy o accidens is equal o: yi y i Γ + yi ( ) ( ) θ i θ θ = ( ) ( ) i yi = Γ + Γ + yi + θii ( L L ) P y,,y,,. (6) Thus, by replacing P( y,,y θ,, θ ) replacing he densiy uncion ( θ,, θ ) ( ν, ν, L, ν ) 2 ( L ) P y,,y L L in equaion (5) by is value given in (6) and by, we obain he ollowing expression: yi Γ + yi Γ νi ( i )... θ = Γ ( y ) i i + = Γ ( ) Γ( ν ) = L by he densiy o a parameric Dirichle yi +νi ( θi ) = dθ i + yi i i = i + θii Ldθ.(7) Once again, we mus esimae he mulidimensional inegral: 0

12 yi +νi ( θi )... d d θ = θ θ i i = + yi i = + θii L (8) o equaion (8) in order o esimae he model s parameers. Three possibiliies are now open. They are discussed in deail in Angers e al. (2004). Here we summarize he main resuls.. The irs possibiliy, which grealy simpliies he calculaions, is o suppose ha all he i o he vehicles are idenical. This irs scenario supposes implicily ha all he vehicles in he lee represen idenical a priori risks, which is probably a very srong hypohesis since, as we shall see, several variables disinguishing he vehicles and he behaviors o drivers are signiican in esimaing he probabiliies o accidens. Anoher possibiliy is o divide he vehicles ino dieren risk groups, as is done by insurers when classiying risks. 2. Under he second possibiliy, we can separae he vehicles ino wo groups, or example, and deine G =, L,g as all he vehicles in he irs group wih = g g g i and G2 = g+, L, as all he vehicles in he second group wih = g2 g+ i g. The inegral o equaion (8) can hus be approximaed by:

13 g ci ci ( θi ) ( θi ) g+ L dθ d L dθ (9) g 0 0 +g θ i +g2 θi g+ wih c = y +ν and i i i i d y = +. Taking ha = θ = L ; θi i ui i,,g g g v = θ i and θi wi = g+, L,, g θ i we can rewrie (9) and subsiue he new expression in equaion (7) o obain an approximaion or he disribuion o he number o accidens a period o he vehicles in lee : ( L ) P y,,y y i i + i g+ Γ +ν y ( ) Γ + yi Γ ν ( y i i )( i ) ( ) Γ g2 ( yi ) ( i ) + Γ + Γ ν Γ ( yi +νi ) g g2 g 2F ( y i+νi), + yi, ( y i +νi ),, +g2 (0) where 2 F is a hypergeomeric uncion as deined in secion This procedure in esimaing he inegral can be generalized o several homogeneous groups, bu i is no obvious ha he precision gained would be greaer han ha corresponding o a Mone Carlo approximaion o he mulivariae inegral o equaion (8). 2

14 3. We can esimae he inegral in (8) by he Mone Carlo mehod (see Angers e al., 2004, or deails). This esimaion could also be used o veriy he precision o he hypergeomeric approximaion. 2.2 ECONOMETRC ESTMATONS 2.2. Descripive saisics The daa come rom he iles o he Sociéé d assurance auomobiles du Québec (henceorh reerred o as he SAAQ), daing rom 997 o 998 (or a deailed descripion o he daa base see Dionne, Desardins, and Pinque, 999, 200). We had access o daa on he wo years rom 43,679 carriers o merchandise by ruck. More han wo hirds o he carriers have only one vehicle. A 3 December 997 and 3 December 998, hese small carriers owned abou 30% o he 03,848 heavy rucks wih auhorizaion o circulae a leas one day, so he economeric esimaion was made wih 73,328 rucks rom 3,59 carriers. We use he 998 daa or inormaion on accidens and characerisics o vehicles and lees and he 997 daa or raic violaions, so as o respec he SAAQ s raing policy. Moreover, his approach reduces he problem o simulaneiy beween he violaions and accidens variables. should be menioned ha a vehicle is no necessarily auhorized o circulae 365 days in 998. On average, a vehicle is auhorized o circulae 88.5% o 998. Depending on he size o he lee, his percenage will vary beween 86.7 and 93.9%. To obain an annual saisic, we calculaed he number o rucks in rucks-year, by adding he number o days each ruck was auhorized o circulae and hen dividing by 365 days. The average requency o oal accidens per ruck-year is 3

15 This average increases as he size o he lee increases, bu decreases when he lee conains more han 50 rucks Esimaion o parameers We used he maximum likelihood mehod o esimae he unknown parameers, ( ),( ν, ν, L, ν ), β= β, L, β. We used SAS/ML soware o apply he opimizaion 2 p algorihm. The resuls or all size o lees wih 2 vehicles and more are presened in Table in he Appendix. For his esimaion, he vehicles o lees wih more han wo rucks were divided ino wo risk groups, according o he average number o accidens per ruck prediced by he negaive binomial disribuion model. For lees wih wo vehicles, we used he exac model o Secion The variance-covariance marix was esimaed based on he SAS/NLFPDD subrouine. We used he 0% hreshold (p lower han or equal o 0.0) o consider a saisical coeicien dieren rom zero. We noe in Table ha he vehicles wih more experience (number o years as carrier) have ewer accidens. The resuls also indicae ha he acors explaining accidens include: he carrier s size and secor o aciviy; he ype o use o which he vehicle was pu; he ype o uel; he number o cylinders; and he number o axles. Vehicles wih lee violaions (violaions o rucking sandards) in 997 are more a risk or accidens in 998 han hose wihou hese ypes o oenses. Moreover, vehicles whose drivers have accumulaed demeri poins or violaions in 997 represen higher risks or accidens in 998 han hose wihou such poins. 4

16 Table also repors he resuls on he parameers or random eecs disribuions. Regression indicaes ha he parameers o he negaive binomial are signiican, which means ha we can reec he Poisson disribuion and apply a bonus-malus insurance raing model o hese lees. is imporan o menion ha we esimaed seven parameers because hese parameers are aeced by lee size. The ν parameer is also signiican a he 99% hreshold. is no aeced by he lee size. These resuls signal ha boh he vehicle and lee eecs can be used in calculaing premiums. n conclusion, he β coeiciens will be very useul in esimaing a priori risks when calculaing insurance premiums, whereas coeiciens and ν will be useul in adusing premiums o i he pas acciden records o vehicles and lees in he bonus-malus model. 3. BONUS-MALUS 3. OPTMAL BONUS-MALUS SYSTEM To consruc an opimal bonus-malus sysem (Lemaire, 985; Dionne and Vanasse, 989, 992) based on he number o pas accidens recorded or a ruck as well as hose observed or is lee, we mus calculae he premium or a ruck o a given lee a period + using he ollowing mahemaical expecaion relaion: ( θα i ) E y,x + i E( θα i ). 5

17 The erm + corresponds o he par o he mahemaical expecaion obained rom he i economeric regressions. is equal o d + X i i e + β where d + i is he number o days ha vehicle i o lee is auhorized o circulae in period + divided by he oal number o days in period +. As already indicaed, his variable measures exposure o risk. The regression componen corresponds o X + i β where he vecor o coeiciens ( β ) is esimaed by means o economeric models and + ( + + X i x i,,xip ) = L represens he observable p characerisics o ruck i in lee a + + he beginning o period +. X ( X,,X,,X,,X ) = L L L gives he p characerisics o all he rucks in lee up o he + period. The vecor y ( y,,y,,y,,y ) = L L L represens he accidens o vehicles in lee up o period and E ( i y,x ) θα designaes he mahemaical expecaion o he lee and vehicle eecs aribuable o vehicle i, based on pas experience as measured by accidens accumulaed over he preceding periods. As we shall see, he modeling proposed will ake ino accoun boh he accidens o vehicle i and hose o is lee. These eecs accoun or he unobservable acors which can aec he accidens o rucks and lees: α is he eec associaed wih lee and θ i is he weigh ruck i in lee acually exers on his lee eec. Finally, ( ) E θα gives he mahemaical expecaion o he wo eecs i aribuable o ruck i no condiional on accidens. The las erm is used o normalize he bonusmalus acor a when he insurer has no experience wih a paricular vehicle. The preceding equaion comes rom a Bayesian analysis o he evoluion o accidens over ime. We are now going o show is explici orm under he hypoheses o saisical disribuion or he wo random eecs. We know ha he rue mahemaical expecaion o he number o accidens 6

18 + or ruck i o lee a period + is equal o i ( X,, i ) λ α θ. is a uncion o he vecor or he observable characerisics o he vehicle up o period and o he random acors or lee α and vehicle θ, which are supposed o be independen o ime. i Given he observaions obained up o period +, he opimal esimaor o his mahemaical expecaion a period +, ˆ + i ( y,x ) λ can be calculaed as ollows: ( αθ i ) ( ( L ) ) i E y,x Eθ E α θ,, θ,y,x y,x ˆ λ i ( y,x ) = i = i E( α ) E( θi) E( α ) E( θi). We know ha:... ( ( ) ) ( ) ( ) θ = Eθ E α θ, L, θ,y,x y,x = θ E α θ, L, θ,y,x θ, L, θ y,x dθ Ldθ i i i i = wih: ( L ) P( y θ, L, θ,x ) ( ) θlθ ( ) ( ) θ,, θ y,x =... P y,,,x d d θ = θ L θ θ Lθ θ L θ i i =. Similarly, we can calculae: wih: ( L ) ( ) L E α θ,, θ,y,x = α α y,x, θ,, θ dα 0 ( L ) α θ,, θ,y,x = 0 ( α θ L θ ) ( α ) P y,,,,x ( L ) ( ) P y α, θ,, θ,x α dα. 7

19 Now le s see how we can apply his Bayesian raing ormula o carriers o dieren sizes. 3.. Size- carrier n his siuaion, he condiional acciden probabiliy or he lee is given by: y ( α ) e ( ) y y α α = = P( y, L,y α,x ) = ( ) e = α = y! = y!. () Given pas accidens observed up o period, he mahemaical expecaion esimaor o he number o accidens or he ruck in lee a period + is equal o: + + y E ( α y, L,y, L,X, L,X ) + + = =. (2) E ( α ) + = Equaion (2) is he ormula used in he lieraure (Lemaire, 985; Dionne and Vanasse, 989, 992) or individual vehicles and does no have o accoun or he lee eec since he lee is he vehicle Carrier wih 2 vehicles n his siuaion, he condiional acciden probabiliy or he lee is given by: 8

20 y i ( ) α θ + θ 2 ( ) i y y 2 y i = = P( y θ, α,x ) = ( θ) = ( θ ) = ( α ) i = = e = i = Γ( yi ) (3). We know ha, given he pas accidens observed up o period and due o he values assigned o he random eecs o he 2 rucks in lee, he a poseriori densiy uncion or o a gamma densiy wih parameers: 2 yi + 2, +θ + ( θ ) 2. = = = α corresponds So: 2 yi 2 + i = = 2 ( ) y 2 2 i + α +θ + ( θ ) i = = 2 = = = = ( ) 2 Γ 2 + y i = +θ + θ ( α y,x, θ ) = α e. Given he pas accidens observed up o period and due o he values assigned o he random eecs o he 2 rucks in lee, he mahemaical expecaion o α is equal o: yi = +θ + ( θ ) 2 = = ( ) E α y,x, θ =. (4) Given he pas accidens observed up o period or he wo rucks o lee, he densiy uncion o θ is equal o: 9

21 ( ) θ y,x = D ν + y ν2 + y2 = = ( θ ) ( θ ) +θ + θ ( ) 2 = = yi i = = (5) where: 2 Γν i + yi = = = = 2 2 ν + + i ν i+ i 2 + y i = = = 2 i = = + 2 Γ ν i + yi + 2 = = = D F y ;2 y ; y ;. Given he pas accidens observed up o period or he wo rucks o lee, he mahemaical expecaion esimaor o he number o accidens or ruck i in lee a period + is hus equal o θ = θ i i = and θ i i 2 : i = θ α θ = + ( ) i i E i y,x i 2 yi E y,x = +θ + ( θ ) 2 = =. remains o calculae he expression: θ i E y,x +θ + ( θ ) 2 = =. By deiniion, 20

22 i i E θ θ y,x = ( θ y,x ) dθ. +θ + ( θ ) 2 +θ + ( θ ) 2 = = = = By replacing ( y,x ) θ by is value given in (5), we obain ha: ν + ( ) y ν2 + y2 i ( ) = = θ θ θ i E θ y,x = D 2 dθ. 2 + yi + ( ) +θ + θ 2 i = = = = +θ + ( θ ) 2 = = Calculaing he inegral, we obain: θ i E y,x +θ + ( θ ) 2 = = = = 2F Ι+ν + y ;+ 2 + y i;+ ν i+ y i; = = i = i = = ν+ i y i + 2 = = = = = = ν+ i yi 2F ν+ y ;2 + y i; ν+ i y i; i = = = = i = i = = + (6) 2 = wih θ = θ i i = and θ i i 2 and he indicaive uncion = i i = and = 0 i i = 2. i = Thus, he opimal esimaor o vehicle i is equal o: 2

23 E ( θα 2 i y,x ) + + ν +ν2 θi i = i yi + 2 E y,x E( ) E( i) α θ = i = 2 νi +θ i + ( θ ) 2 = = + ν+ν 2 = i 2 ν i = = 2 2F y +ν +Ι ; yi+ 2 + ; ν i+ yi+ ; = = yi +νi yi 2 = = = = yi +νi = = = = 2F y +ν ; yi+ 2 ; ν i+ yi; = = i = i = = + 2 =. We noe ha or each vehicle i, he opimal esimaor or accidens a period + is a uncion o he ollowing acors: he parameers observable when he insurance policy is being renewed a period +; he accidens accumulaed by vehicle i over he preceding periods; he oal accidens o he lee over he same periods; he observable characerisics o he wo vehicles over he preceding periods; and he gamma and Dirichle parameers. We shall apply his ormula o our daa in secion 4. Bu le s now see how i is possible o generalize his insurance raing ormula o a lee o vehicles Carrier wih more han 2 vehicles This secion is divided ino hree subsecions corresponding o he hree approximaion hypoheses or he muliple inegral discussed in secion All he i or he vehicles are idenical 22

24 n his siuaion, he condiional acciden probabiliy or he lee is given by: ( L ) P y θ, θ, α,x = ( ) ( θα i ), = i yi y! e θiα y i y y i i α i = = = = = ( θi ) ( α ) e = y i!. (7) The opimal esimaor ˆ + i λ is hus equal o: + yi +ν i yi + i i E ( θiα y,x ) ν + = = = i E( θα i ) νi + yi + νi = =. (8) This ormula compares raher well wih he one presened in equaion (2) or a carrier wih a single vehicle. Here, as all he vehicles are idenical in erms o he observable variables, diereniaion o he wo ormulas will be principally he work o he experience variables. On he one hand, all he accidens o he lee come ino play and, on he oher hand, he weigh o pas accidens akes ino accoun he parameers o he Dirichle disribuion, on an individual basis ν i or each vehicle and on an aggregaed basis νi or all he vehicles. Divide he vehicles ino 2 groups 23

25 we now have dieren vehicles, we can orm groups wih homogeneous characerisics or risks o obain an explici ormula. n ac, insurers orm more or less homogeneous risk classes by using dieren classiicaion variables such as he ype o car, he erriory Pas experience serves o pinpoin he dierences which are no observable a priori. we limi ourselves o wo groups, he condiional acciden probabiliy or he lee is given by: wih ( L ) P y θ,, θ, α,x = ( θα i i ) = i yi y! e iθiα g y y i i α i g θ i+ g 2 θ i = = = g g+ ( i ) ( ) e i y i! = θ α g i i g+ = e = g g 2 g g or he wo groups respecively. ˆ + i The opimal esimaor λ is hus equal o: 24

26 E ( θα i i y,x ) ν + + θi i = i yi + E y g,x E( ) E( i) α θ = ν i + g θ m + g 2 θm m= m= g+ g2 g g = = 2F y i+ν i + Ι ; yi+ + ; yi+ ν i + ; i i ν y i i +νi + yi = = = = = + + g2 = = = = i ν i g2 + yi +νi g2 g g = = 2F y i +ν i; yi + ; y i +νi; = = = = = + g2 = (9) where he indicaive uncion: = i he ruck belongs o group 0 i he ruck belongs o group 2. This ormula is very diicul o generalize o more han wo groups. he lee has several more or less homogeneous groups o vehicles, i may be more advanageous o rely on a Mone Carlo simulaion approach. Mone Carlo simulaion approach n he general case, he condiional acciden probabiliy or he lee is given by: y ( i ) ( yi ) i y y i i α i θi = = = i i = = P( y, θ, L, θ ) ( ) ( ), α,x = θ i α e = Γ +. (20) We also obain ha: 25

27 + yi = + θi i = ( ) E α y,x, θ, L, θ = (2) and ( L ) θ,, θ y,x =... θ = i y i i +ν = + θ i i = + yi i = = + θ i i = y i i +ν = + θ i i = dθ Ldθ + yi i = = + θ i i = (22) We can esimae he muliple inegral: yi i ( i ) +ν θ = L dθ L dθ i θ i = + yi i = = o equaion (22) wih he Mone Carlo mehod by using he imporance uncion (weighing) (Lange, 999) ( ) wih h θ where θ=θ, L, θ % % such ha: ( θ) % ( θ) N g... g( θ) dθ= h θ dθ= w θ h θ dθ w θ % % h % % % % % N % % ( ) ( ) ( ) ( ) l θ = i θ = i θ = i l= ( θ) ( θ % ) g w ( θ ) =. % h % 26

28 By aking: h Γ yi +ν = ( θ ) = ( θ ) % Γ y +ν i i = i i y +νi = i, he opimal esimaor ˆ + i λ is approximaely equal o: N θli l= + yi + i = = + ν + mθ m θα l m= = N l= + yi i = = + m θ m l m= = yi E i ( i y,x ) + + = i i E( θα i ) νi (23) wih θ = li a li a li where he a l i are values o a Gamma: G y, i +νi or i =,, and l =,, N. = 27

29 4. APPLCATON OF THE BONUS-MALUS SYSTEM n his secion, we propose premium ables over several years, represening exensions o hose proposed in he lieraure on auomobile insurance or individual vehicles. Given ha we did no model he condiional disribuion or he cos o claims, we suppose ha he average cos o claims is $0,000, seemingly a reasonable value or accidens involving rucks in Norh America (Dionne, Laberge-Nadeau e al., 999). 4. FLEET OF 2 TRUCKS Table 2 presens an example o premiums calculaed or a ruck belonging o a lee o wo rucks. The irs line o he able (Flee accidens) gives he sum o he accidens or he lee over hree years. The maximum indicaed is 2 accidens bu i could be higher. The second line (Truck accidens) gives he sum o accidens or he ruck in quesion. For example, in he hird column where he lee accumulaes wo accidens, he ruck concerned may have had 0, or 2 accidens. Thus each corresponding scenario o premiums depends on he ruck s and he lee s own experience. we use he resul o Table showing ha ν =ν 2 =ν, a ruck has a bonusmalus acor () equal o: 28

30 ˆ ˆ = = 2 2F Ι+ν+ ˆ y ;+ 2 ˆ + y i;+ 2ν+ ˆ y i; i i ˆ = = = = = ν+ y ˆ ˆ i 2 ˆ + yi + 2 = = = yi 2 2 ˆ2 ˆ = = = = ˆ ˆ ˆ 2F ν+ y ;2 + y i;2ν+ y i; = = = ˆ ˆ + 2 = = ˆ ˆ ˆ + ν+, where he indicaive uncion is deined as beore. The esimaed values o he parameers are equal o ˆ = and ν ˆ = (Table ). Le s ake he column No acciden or he lee and he ruck. We noe ha he premium or he ruck decreases over ime. The ollowing column gives he variaions in he premiums i he lee does have an acciden and depending on wheher or no he ruck has an acciden. We noe ha he premium or he ruck increases in comparison o he irs column even i he ruck did no have an acciden, or i is penalized by he lee eec. Bu he increase is less han he one corresponding o he case where i did incur an acciden. 29

31 Table 2: Table o insurance premium or vehicles belonging o a size-2 lee Flee 2 accidens i 2 = i 2 = y i i = = i = = i = = = 2 Truck y accidens i = 0 yi = 0 yi = yi = 0 yi = yi = = = = = = = i i i i i i i $0,000 $0,000 $0,000 $0,000 $0,000 $0, $, $ $, $2, $, $2, $3, $ $, $, $, $2, $2, $ $, $,57.48 $, $, $2, $ $ $, $, $, $2, $ $877.5 $, $, $, $2, $ $80.09 $, $ $, $, $ $ $, $ $, $, $ $ $ $ $, $, $ $ $ $ $, $, FLEET OF SEVERAL TRUCKS All vehicles in lee have he same risk characerisics n his siuaion, he insurance premium esimaed or ruck i belonging o carrier is given by: ˆ + yi ν+ ˆ yi + = = + i i ˆ ˆ ˆ ν+ yi + i = i = = ˆ =ˆ which is (8) when ν i = ˆ ν. Table 3 presens his example or a lee o 0 idenical rucks wih ˆν = and ˆ = (see Table ). Suppose ha he carrier accumulaes 2 accidens over he nex period, wih 6 rucks incurring no acciden nor speeding violaion; 2 rucks incurring no acciden bu one speeding 30

32 violaion; ruck incurring an acciden bu no speeding violaion; and ruck incurring an acciden as well as a speeding violaion. Sill supposing ha he average cos o claims is $0,000, he a priori insurance premium or a vehicle when no accoun is aken o pas experience is esablished a $,850 (0. 85 $0,000). Since all he vehicles o he lee are idenical in erms o observable risk, hey all have he same i = 0.85 and a equal o a he sar o he insurance conrac. The oal premium or he lee is esablished a $8,500 (0 $,850). n he ollowing period (+), he insurance premiums or each o he records o he vehicles in he lee are given in Table 3. Table 3: Table o insurance premiums or vehicles belonging o a size-0 lee when he lee accumulaes 2 accidens a year + Accumulaion Speeding + i Number o i o accidens violaion i $0,000 rucks $4,507 $4, $2,573 $2, $3,0 2 $6, $,770 6 $0,620 Toal $23,902 We noe ha accidens aec he bonus-malus acor () o all he vehicles (lee eec), whereas speeding violaions aec he a priori risk via he regression componen or he vehicles which accumulae hem. The deailed calculaion o he or he acciden accumulaion o a ruck involved in he acciden corresponds o: = =

33 We noe ha he is higher or vehicles having had an acciden han or hose which did no. We also noice ha he a priori risk measuremen increases signiicanly or vehicles which + i have accumulaed a speeding violaion. none o he 0 vehicles in he lee had been involved in an acciden nor had been charged wih speeding, he oal premium would have decreased rom $8,500 o $8,440 (0 $844), or he would be equal o and he individual ruck premium o $8,440 ( $0,000 = $844). However, in our example, he oal premium goes rom $8,500 o $23,902 based on he accumulaed experience o he 0 vehicles. Now, i he carrier has accumulaed 3 pas accidens and has 9 rucks wih no acciden and no speeding violaion and ruck wih 3 accidens bu no speeding violaion, he oal premium is $24,776. The insurance premiums o he lee or each o he experiences are given in Table 4. Table 4: Table o insurance premiums or vehicles belonging o a size-0 lee when he lee has accumulaed 3 accidens + Accumulaion o + i Number o i accidens i $0,000 rucks $2,8 9 $9, $5,47 $5,47 Toal 3 0 $24,776 The deailed calculaion o he or he accumulaed 3 accidens is equal o: = = We noe ha he premium or a vehicle wih no acciden nor speeding violaion is $2,8 when i belongs o a lee having accumulaed 3 accidens and drops o $,770 i i belongs o a lee 32

34 having accumulaed 2 accidens, while reaining he same characerisics (Table 3). This resul is explained by he ac ha he s o all he vehicles are aeced by he lee s accumulaion o accidens. We also noe ha, when he vehicle comes rom a lee having accumulaed 2 accidens, accumulaing 3 accidens increases he insurance premium more ($5,47) han accumulaing one acciden and one speeding violaion. ($4,507) Dividing he vehicles ino 2 groups n his siuaion, he esimaed insurance premium o a ruck i belonging o a carrier is given by (9) wih ν = νˆ. i Suppose ha he accidens accumulaed by he carrier over he nex period is 0, wih 4 rucks belonging o group (a priori expeced number o accidens below or equal o ) and 6 rucks belonging o group 2 (a priori expeced number o accidens above ). By supposing ha he average cos o claims is $0,000, he insurance premiums or he hisory o each o he lee s vehicles in he ollowing period are given in Table 5. Table 5: Table o insurance premiums or vehicles belonging o a 0-ruck lee when he lee has no accumulaed a single acciden Group + Accumulaion + ˆ i Number o gi o accidens i $0,000 rucks $594 4 $2, $,026 6 $6,56 Toal 0 0 $8,532 33

35 The deailed calculaion o he or a ruck belonging o group corresponds o: = [.434] = and ha or a ruck belonging o group 2 is given by = [.063] = Now, i he lee has accumulaed acciden and i he vehicle involved in he acciden belongs o group 2, he insurance premiums or he lee s vehicles are given in Table 6. Table 6: Table o insurance premiums or vehicles belonging o a 0-ruck lee when he lee has accumulaed acciden (in group 2) Group + Accumulaion + ˆ i Number o gi o accidens i $0,000 rucks $940 4 $3, $,606 5 $8, $2,333 $2,333 Toal 0 $4,23 The deailed calculaion o he or a ruck belonging o group corresponds o: = [.466] = Tha o a ruck belonging o group 2 and no having had any acciden is given by: = [.0974] = , whereas ha o a ruck in group 2 having had acciden is equal o: 34

36 = [.0974] = n conras, i he vehicle involved in he acciden belongs o group, we obain he values shown in Table 7. Table 7: Table o insurance premiums or vehicles belonging o a 0-ruck lee when he lee has accumulaed acciden (in group ) Group + Accumulaion + ˆ i Number o gi o accidens i $0,000 rucks $950 3 $2, $,38 $, $,625 6 $9,750 Toal 0 $3, 98 The deailed calculaion o he or a ruck belonging o group corresponds o: = [.598] = and ha o a ruck in group having had acciden is equal o: = [.598] = Finally, he or a ruck belonging o group 2 is given by: = [.095] = Table 8 sums up all he cases (numbers no in parenheses). 35

37 Table 8: Table o insurance premiums or vehicles belonging o a 0-ruck lee separaed ino wo risk groups Flee accidens Group accidens + i 0 = i = 4 = i = Group yi = (0.456) = y = 0 i y = 0 i + i $0,000 $594 ($594) 4 = i = (0.720) y = 0 i 0 = i = + i BM F $0,000 $940 ($940) y = i 4 = i = (0.728) yi =.058 = (.059) y = i + i $0,000 $950 ($950) $,38 ($,382) 4 = i = y = 0 i + i BM F $0,000 0 = i = 4 = i = y = 2 i y = i + i BM F $0,000 4 = i = y = 2 i + i $0, $, $, $, $, $,867 yi = $2,45 = Groupe 2 0,233 yi = (0.44) = $,026 ($,026) (0.689) yi =.00 = (.000) $,606 ($,606) $2,333 ($2,33) (0.697) $,625 ($,625) yi = $4,06 = 0.94 $2, $2, $2, $3, $3,26 should be denoed ha he Mone Carlo compuaions o premiums are idenical o hose wih he hypergeomeric approximaion when we assume ha all rucks are idenical inside he wo groups. They correspond o he numbers in parenhesis in Table 8. One advanage o Mone Carlo simulaions is ha we can consider all rucks as dieren in a given lee. We now presen resuls or en dieren rucks using Mone Carlo simulaions o make a poseriori compuaions. We sill use he economeric resuls o Table or a priori evaluaions. The simulaions are repeaed 500,000 imes; his akes abou 0 minues or a scenario like he ones presened in Table 9 in he Appendix, whereas he hypergeomeric approximaions are insananeous. 36

38 Table 9 presens he premium evoluion over ive years or hree scenarios. The a priori expeced number o accidens or hese hree scenarios is Scenario is or a lee ha accumulaes many accidens over ime. n he irs column, we observe he en dieren a priori values. n he hird column, we have he corresponding saring premiums or he hree scenarios which amoun o a oal premium o $9,206 or he lee. Accumulaing eigheen accidens over ive years yields a oal premium o $33,90 or he nex period. n scenario 2, he lee accumulaes only ive accidens over he ive years and he oal premium drops o $,96. Finally, in scenario 3, he lee has wo accidens each year (is average), resuling in an almos consan premium over ime. n Figure, we graphically represen he hree scenarios wih solid lines. The doed lines correspond o he case where he lee eec is no compued in boh he regression and he premium compuaions (see he numbers in Table 0 in he Appendix). The dierences are signiican. nroducing he lee eec increases he lucuaions in he premiums and should inroduce more incenives or road saey. (Figure here) 5. CONCLUSON n his aricle, we have developed a parameric model or raing insurance premiums or lees o vehicles. We have shown how aking ino accoun boh lee and vehicle eecs can aec he Bayesian calculaion o insurance premiums over ime. The model proposed was esimaed using 37

39 daa over a single period. An imporan exension would be o model a panel eec which would ake ino accoun he repeiions o inormaion on lees and vehicles over ime (see Abowd e al., 999, or a irs analysis o his ype o model). The raing ormula developed presupposes a decenralized managemen o road saey as regards carriers. n eec, charging dieren premiums or each o he vehicles in a lee based on he experience o boh he lee and is rucks will promp road-saey managers hemselves o keep a close eye on road-saey policy and o se up insiuional incenives moivaing drivers and carriers o adop pruden behaviors. ndeed, knowing which drivers and carriers are risky, hese managers can hen assign sliding premiums according o he risk levels o he dieren drivers and rucks. A irs version o he paper vas presened a he 2004 Risk Theory Seminar in New York Ciy. The research was inanced by he Programme d acion concerée en sécurié rouière FCAR- SAAQ-MTQ and by he Canada Research Chair in Risk Managemen. We hank Claire Boisver or her conribuion in he preparaion o he manuscrip. 6. REFERENCES Abowd, J-M., Kramarz, F., and Margolis, D.N. (999) High Wage Workers and High Wage Firms. Economerica 67, Angers, J.F., Desardins, D., Dionne, G., and Guerin R. (2004) ndividual and Firms Random Eecs in he Esimaion o Even Disribuions. Working Paper, Canada Research Chair in Risk Managemen, HEC Monréal, and CRT, Universiy o Monreal. Dionne, G., Desardins, D., and Pinque, J. (999) L évaluaion du risque d acciden des ransporeurs en oncion de leur seceur d acivié, de la aille de leur loe e de leur dossier d inracions. Research repor 00-28, Laboraoire sur la sécurié des ranspors du Cenre de recherche sur les ranspors, Universiy o Monreal, 54 p. 38

40 Dionne, G., Desardins, D., and Pinque, J. (200) Experience Raing Schemes or Flees o Vehicles. ASTN Bullein 3, Dionne, G., Desardins, D., ngabire, M.G., and Akdim, R. (200) La percepion du risque d êre arrêé chez les camionneurs e ransporeurs rouiers. Research repor , Laboraoire sur la sécurié des ranspors du Cenre de recherche sur les ranspors, Universiy o Monreal, 39 p. Dionne, G., Laberge-Nadeau, C., Desardins, D., Messier, S., and Maag, U. (999) Analysis o he Economic mpac o Medical and Opomeric Driving Sandards on Coss ncurred by Trucking Firms and on he Social Cos o Traic Accidens. n Auomobile nsurance: Road Saey, New Drivers, Risks, nsurance Fraud and Regulaion (ed. G. Dionne and C. Laberge- Nadeau), pp , Kluwer Academic Publishers, Boson. Dionne, G. and Vanasse, C. (989) A Generalizaion o Auomobile nsurance Raing Models: The Negaive Binomial Disribuion wih a Regression Componen. ASTN Bullein 9, Dionne, G. and Vanasse, C. (992) Auomobile nsurance Raemarking in he Presence o Asymmerical normaion. Journal o Applied Economerics 7, Frangos, N. and Vronos, S.D. (200) Design o Opimal Bonus-Malus Sysems wih a Frequency and a Severiy Componen on an ndividual Basis in Auomobile nsurance. ASTN Bullein 3, -22. Flue, C. (999) Commercial Vehicle nsurance: Should Flee Policies Dier rom Single Vehicle Plans? n Auomobile nsurance: Road Saey, New Drivers, Risks nsurance Fraud and Regulaion (ed. G. Dionne and C. Laberge-Nadeau), pp. 0-7, Kluwer Academic Publishers, Boson. Gouriéroux, C. (999) Saisiques de l assurance. Economica, Paris, 297 p. Hausman, J.A., Hall, B.H., and Griliches, Z. (984) Economeric Models or Coun Daa wih an Applicaion o he Paens R&D Relaionship. Economerica 52, Lange, K. (999) Numerical Analysis or Saisicians. Springer: New York, secion 2.2, Lemaire, J. (985) Auomobile nsurance: Acuarial Models. Huebner nernaional Series on Risk, nsurance and Economic Securiy, Kluwer Academic Publishers, Boson, 248 p. Lemaire, J. (995) Bonus-malus Sysems in Auomobile nsurance. Kluwer Academic Publishers, Boson, 283 p. Marie-Jeanne, P. (994) Problèmes spéciiques des loes auomobiles. Proceedings o he SUP conerence Cours Avancés sur l Assurance Auomobile. 39

41 Moses, L.N., Savage,. (994) The Eec o Firm Characerisics on Truck Accidens. Acciden Analysis & Prevenion 26, Moses L.N., Savage,. (996) deniying Dangerous Trucking Firms. Risk Analysis 6, Pinque, J. (997) Allowance or Cos o Claims in Bonus-Malus Sysems. ASTN Bullein 27, Pinque, J. (998) Designing Opimal Bonus-Malus Sysems rom Dieren Types o Claims. ASTN Bullein 28, Pinque, J. (2000) Experience Raing hrough Heerogeneous Models. n Handbook o nsurance, (ed. G. Dionne), Kluwer Academic Publishers, Boson, Purcaru, O., Denui, M. (2003) Dependence in Dynamic Claim Frequency Credibiliy Models. ASTN Bullein 33, Teugels, J.L. and Sund, B. (99) A Sop-Loss Experience Raing Scheme or Flees o Cars. nsurance: Mahemaics and Economics, Norh-Holland, Winer, R. (2000) Opimal nsurance under Moral Hazard. n Handbook o nsurance (ed, G. Dionne), Kluwer Academic Publishers, Boson, Corresponding addresses: Georges Dionne, Canada Research Chair in Risk Managemen, HEC Monréal, 3000, Chemin de la Côe-Saine-Caherine, Monreal (Qc) Canada, H3T 2A7. Telephone: (54) Fax: (54) georges.dionne@hec.ca. Jean-François Angers, Denise Desardins, and François Guerin, Cener or Research on Transporaion, Universiy o Monreal, C.P. 628, Succ. Cenre-Ville, Monreal, Canada, H3C 3J7. 40

42 APPENDX Table : Esimaion o parameers o predic he number o accidens or rucks in all he lees by dividing he rucks ino wo groups or lees wih more han 2 rucks Explanaory variables Coeicien Saisic P Consan < Number o years as carrier as o 3 December < Secor o aciviy in 998 Oher secors General public rucking Bulk public rucking Reerence group Privae rucking Shor-erm renal irm < Size o lee 2 Reerence group o < o < o < o < More han < Number o days auhorized o circulae in < Number o violaions o rucking sandards in 997 For overload < For excessive size For poorly secured cargo For ailure o respec service hours For ailure o pass mechanical inspecion For oher reasons Type o vehicle use Commercial use including ranspor o goods wihou C.T.Q. permi Transpor o oher han "bulk" goods Transpor o "bulk" maerials Reerence group Type o uel Diesel Reerence group Gas < Ohers Number o cylindres o 5 cylindres o 7 cylindres < or more han 0 cylindres Reerence group Number o axles 2 axles (3,000 o 4,000 kg) < axles (more han 4,000 kg) < axles < axles axles < axles or more Reerence group Number o violaions wih demeri poins in 997 For speeding < For driving under suspension For running a red ligh < For ignoring sop sign or raic agen < For no wearing a sea bel For oher oenses < ν < (lees o 2 rucks) < (lees o 3 rucks) < (lees o 4 o 5 rucks) < (lees o 6 o 9 rucks) < (lees o 0 o 20 rucks) < (lees o 2 o 50 rucks) < (lees o more han 50 rucks) < Log-likelihood -30,494 Number o carriers 3,59 Number o vehicles 73,328 4

43 Scenario (( ν= ; = ) Table 9: Mone Carlo Simulaions i Premium $ Premium $ Premium $ Premium $ Premium $ Premium $ ,90.23, , * 2, * 2, * 3, , * 2, * 3, , * 3, , ,408.99, ** 3, , , , ,45.97,693.32,857.26, * 2, ** 3, ,633.79, ,08.70* 2, , , ,28.652* 3, * 5, , , , ,30.646* 3, , , , * 4, , , , ,494.45* 3, , ,633. 2, , * 3, , , , , , , , ,309 Toal 9,206 Toal 25,083 Toal 30,079 Toal 29,99 Toal 3,906 Toal 33,90 Scenario i Premium $ Premium $ Premium $ Premium $ Premium $ Premium $ , *, 0.866, , , , , , *, , , , *, , *, ,28.004* 2, , , , , , , , , , , , , , , , , , , , , , , , , ,44 0.4, ,220 Toal 9,206 Toal 4,06 Toal 2,565 Toal,870 Toal 9,409 Toal,96 Scenario i Premium $ Premium $ Premium $ Premium $ Premium $ Premium $ , , , , , *, , , , , , , , , , , , , , *, * 2, * 2, , , , , * 2,04.62,898.04, ,28.330* 3, , , , , , , , , , , ,42.325* 3, , , , * 3, , ,373.87* 3, , * 3, , , , , , , ,725 Toal 9,206 Toal 9,734 Toal 9,808 Toal 9,770 Toal 9,83 Toal 9,820 One * indicaes ha he ruck had one acciden during he previous period while wo * is or wo accidens during he previous period. 42