Department of Economics University of Warsaw Warsaw, Poland Długa Str. 44/50.

Transcription

1 MIGRATIOS OF HETEROGEEOUS POPULATIO OF DRIVERS ACROSS CLASSES OF A BOUS-MALUS SYSTEM BY WOJCIECH OTTO Dearmen of Economcs Unversy of Warsaw Warsaw Poland Długa Sr. 44/50 woo@wne.uw.edu.l

2 . ITRODUCTIO I s usually assumed a mgraons of an ndvdual drver across classes of a bonus-malus BM sysem sasfy assumons of e omogeneous Marcov Can. Generalzaon of e model ono e case of a oulaon of drvers s sragforward rovded e oulaon s omogeneous. However n racce oulaons of drvers are eerogeneous. Moreover s eerogeney s e rason d êre of BM sysems as er am s o mae bad drvers ayng ger nsurance remum an good drvers do. Te am of e aer s o sudy mlcaons of e dearure from e assumon on omogeney of oulaon. 2

3 Some effecs of eerogeney of oulaon are well nown owever exlc model of e mgraon rocess across BM classes of e oulaon as a wole as robably neer been formulaed nor sysemacally analysed. Lac of an adequae model mg resul n msnerreaons esecally wen analysng emrcal daa on mgraons of a oulaon of drvers across classes of e BM sysem. Msnerreaons may come from nng n erms of roeres of a omogeneous Marov Can wereas n fac e dearure from e assumon a e oulaon s omogeneous mae several of ese roeres eavly dsored. 3

4 2. BASIC ASSUMPTIOS OF THE MIGRATIO S MODEL Te BM sysem consss of S classes. Transons of drvers beween classes aes lace once a year. A drver caracersed by e value λ of rs arameer Λ roduces losses accordng o e Posson rocess w nensy λ er year and clams all of em o e nsurer. Transon robables for a drver w Λ = λ form a marx: P λ = λ { } =2... S Transon rules are refleced by osons of non-zero elemens of P λ a corresond o evens of zero clams clam 2 clams... on-zero elemens of Pλ equal robables of ese evens for a drver caracersed by e value λ of e arameer Λ Transon rules ensure a P λ s ergodc for all λ > 0. 4

5 Abou e oulaon of drvers we assume: Te oulaon s closed. Poulaon consss of drvers caracersed by osve values λ λ... λ of rs arameer. 2 Mgraon rocesses of ndvdual drvers are ndeenden. In e aer e connuous varan of e model s also consdered were e dscree dsrbuon λ λ2... λ s relaced by a connuous densy f λ. For brevy e connuous varan s no resened ere. 5

6 3. MODEL WITH DISCRETE DISTRIBUTIO OF RISK PARAMETER Λ We consder a wo sage exermen: A e frs sage a drver s randomly drawn from e oulaon. Denong e resul number of drver beng drawn by K we ave: Pr K = = = 2 A e second sage e rocess of mgraon of e drver on e sace of BM classes aes lace. Condonally gven K = e rocess sasfy assumons of e omogeneous Marov Can. Essence: Te rocess concerns canges n me of e on dsrbuon over members of e oulaon and BM classes. Margnal dsrbuon over BM classes and condonal dsrbuons cange n a dfferen way. An exceon s a omogeneous oulaon case wen margnal and all condonal dsrbuons are dencal. 6

7 Basc condonal caracerscs of e rocess rovded e even K = as occurred a e frs sage are: { : = [... ] } = Sequence of vecors of robables of sayng n year n classes = 2...S. Transon robably marx P : = P λ w elemens 2 S Gven a sarng vecor we oban nex erms of e sequence: P = = wc under assumed ergodcy converges o e lm: : = lm for = Of course e lm sasfes e sysem of equaons: = P. 7

8 Basc margnal caracerscs of e rocess are: { } = [ ] : = 2... S Sequence of vecors of robables of sayng n year n classes = 2...S were: = = And sequence of marces { } P = w elemens: =... S = = reresenng robably of ranson from class n year o class n e nex year. Te resul confrm a marces P are n general me-deenden exce wen oulaon s omogeneous.e. wen λ = λ =... = λ. 2 8

9 4. WHERE THE COFUSIO COMES FROM For e margnal rocess e followng formula olds: + = P. Dese marces P are me-deenden convergence of condonal robably vecors for eac = 2... ensures a: Margnal robably vecors converge o er lmng value: As a resul P P as well And so e lmng values sasfy e sysem of equaons: = P. Te above resuls may sugges a errors made wen reang e rue mgraon rocess as a omogeneous Marcov Can: are sgnfcan for small rue bu for large could be negleced false 9

10 In e aer dsorons due o eerogeney of oulaon on roeres/caracerscs of e omogeneous Marcov Can are suded. Te sudy focus on e followng roeres/caracerscs: Te ably o measure e rae of convergence of vecors o er lm by e second-larges egenvalue of P Basc caracerscs of e dsrbuon of vecor of frequences of drvers sayng n classes 2... S n a gven year Basc caracerscs of e condonal dsrbuon of a vecor gven s value from year Bas and varance of esmaors of ranson robables based on emrcal daa on number of ransons beween classes a e end of year Some of resuls are llusraed by examles oers are resened n a more general form. 0

11 5. THE SECOD-LARGEST I ABSOLUTE TERMS EIGEVALUE OF P Le us consder a smle 3-class no clam dscoun sysem w a ranson robably marx: q 0 P = 0 q q 0 Were s a no-clam robably for a gven drver and q =. Te saonary dsrbuon on e sace of classes s smly gven by: 2 = [ q q ] as e curren oson of a drver deends only on wo las years. Egenvalues of marx P equal { ρ ρ2 ρ3} { 0 0} so e second larges egenvalue s zero wc confrms e fac a e sequence converge o n a fne number of ses.

12 Le us ae now a oulaon of drvers a dffer by no-clam 2 3 robably. Te lm of e sequence P P P... of ranson robably marces equals now: q 0 P = q q Were denoe robably of no-clam for a drver beng n class number n e saonary sae and q =. Egenvalues of s marx are e followng: = ρ ρ { } 2 23 = 3 ± Obvously e laer wo equal zero only wen = =

13 However can be formally sown a f ere s any dversy of arameer q n e oulaon en no-clam robables sasfy: < < 2 3. Te nerreaon s sragforward: srucure of class s based owards ger sare of bad drvers srucure of class 3 s based owards ger sare of good drvers class 2 s n s resec a medum one. On e oer and we now a all drvers reac er saonary sae n wo years so all e oulaon aan e saonary sae n wo years oo. Tus e concluson reads: Te second larges n absolue erms egenvalue of marx looses P e roery of reflecng e rae of convergence once we dear from e omogeney assumon. 3

14 6. OTATIOS FOR AALYSIS OF FREQUECIES Le us denoe by: - e number of drvers mgrang from class n year o class n e nex year S = : = - e number of drvers n class n year : = [... ] 2 s - e vecor of e above numbers Furer on we focus on e uncondonal dsrbuon of e vecor and condonal dsrbuon of s vecor gven. Terms uncondonal and condonal are convenonal as deends also on sarng condons. Te convenon s usfed by focussng on e case of large wen a deendence dsaears. 4

15 7. MOMETS OF THE UCODITIOAL DISTRIBUTIO OF Execed value of e counng vecor equals: = Ε. Due o assumed ndeendency of mgraon rocesses of ndvdual drvers e covarance marx s us a smle sum: cov = = 2... s 2... s s 2... s s s. Te above marx can be decomosed no a dfference of wo marces: 5

16 cov = { dag } = Were: Te frs erm aearng on e RHS s a covarance marx adequae for e case of omogeneous oulaon were = = 2 =... =. Second erm measures eerogeney wn e se of vecors I s n fac an -mulle of e covarance marx of e random vecor K beng a funcon of random varable K. For e second marx sablses. I reresens e endency realsed over a number of years owards concenraon of good drvers n bonus classes and bad drvers n malus classes. 6

17 8. THE CODITIOAL DISTRIBUTIO OF GIVE = P + ε were n e case of omogeneous oulaon we ave: Ε = P so a: Flucuaons of can be reresened by e sysem of equaons: Ε ε = 0 and cov ε ε deenden on. Dese e las comlcaon s a sysem of lnear regressons and so covarance marx of random erms can be easly calculaed: cov ε ε = cov P cov P. Te decomoson aes an esecally nce form for large : lm cov ε ε = dag P dag P. { } 7

18 All e above roeres are desroyed once we adm eerogeney n e oulaon as P dffers en from s lm P. However we can sll ose e queson weer e dynamcs of e vecor can be descrbed analogously as n e case of omogeneous oulaon a leas for large. Te answer s negave as urns ou a Ε s a nonlnear funcon of e vecor. 8

19 Ts can be easly sown n e smle case wen e BM sysem assumes a a drver can ener class only from class. We can focus en on e condonal execaon Ε. Drec calculaons for exreme cases = and = render followng resuls: = Ε = = wc generally dffers from Ε and: = = = = wc dffers from as well. 9

20 I s farly dffcul o derve more general resuls. However examles suded srongly suor e followng nerreaon: For large execed srucure of e bonus malus class s based owards larger sare of good bad drvers However e nformaon a n year ere are more drvers an execed n e class mles a e bas s weaer e srucure of e class s more ale e srucure of e oulaon To e conrary unexecedly small number of drvers n e class mles sronger selecon and so sronger bas Hence f for some large e robably of no-clam n class s greaer smaller an average n e oulaon en s robably condonal on number of drvers n s class s a decreasng ncreasng funcon of. 20

21 Te vecor of random erms of e sysem of equaons: = P + ε can be decomosed no wo effecs: ε = Ε + Ε P. { } { } Te frs comonen reresens flucuaons around e condonal execed value wereas e second one reresens lnearzaon error. I urns ou a e sare of covarance marx of e second comonen n e oal covarance marx of e vecor ε does no dsaear even wen bo and end o nfny. Te concluson s no derved n full generaly bu raer based on examles suded. 2

22 9. ESTIMATIG TRASITIO PROBABILITIES WHE DATA O TRASITIOS ARE AVAILABLE Transon robables could be esmaed on e bass of emrcal daa on number of ransons. A naural esmaor of e ranson robably can be defned en as: ˆ : = w some redefned consan for e case wen = 0. Usng formulas for momens of e quoen of samle means well nown on e ground of samlng eory we oban e followng aroxmaons: 22

23 Ε var ˆ 2 ˆ + 2 and: = 2 2 = Bo formulas are exac n resec of erms of order / or larger 2 wereas erms of order and smaller are negleced. In bo cases e frs comonen reresens e classcal esmaor of e condonal robably of some even wen e condon s sasfed for an ndvdual observaon w robably. Heerogeney resuls n bas of order /. Heerogeney resuls also n effcency gan a s of order / as well as e wole varance. Tus e relave effcency gan does no dsaear wen oulaon sze ncreases

24 Aroxmaons for covarances are as follows: = m m n m 2 ˆ ˆ cov n m n m m wen.e. ransons are from wo dfferen classes and: 2 n n n 2 ˆ ˆ cov n n = are from e same class. wen ransons o dfferen classes A searae effec ndeenden of eerogeney concerns only ransons from e same class o wo dfferen classes. Under eerogeney all ransons are n a way deenden evens. Larger an execed gven mly larger fracon of bad 24 good drvers an execed among ose movng from class o and so smaller fracon of em n e res of oulaon.

25 0. SUMMARY Maor mlcaons of e eerogeney of e oulaon: Transon robables vary n me for small no really new Dfferen roeres of e rocess even for large : o Probably of loss deendng on class well nown bu ofen nerreed as beng due o oer facors an eerogeney o Loss of nerreaon of e second larges egenvalue of robably ranson marx P new resul o on-lneary of funcon Ε new facs and nerreaons o Effcency gan wen esmang ranson robables new resul 25

26 Lac of awareness of e mlcaons may lead o msnerreaons of varous anomales a mg be observed n emrcal researc. Tese anomales are que common and may come from: e unger for bonus enomenon varous facors a cange arameers of e mgraon rocess n subsequen calendar years. REFERECES [3] Lee T. C. Judge G. G. Zellner A. Esmang e arameers of e Marov robably model from aggregae me seres daa or-holland Amserdam 970. [4] Lemare J. Bonus-malus sysems n auomoble nsurance Kluwer Boson 995. [5] McCall J. J. A Marovan model of ncome dynamcs Journal of e Amercan Sascal Assocaon vol