Evidence of Adverse Selection in Automobile Insurance Markets

Transcription

1 Evdence of Adverse Selecton n Automoble Insurance Markets by Georges Donne, Chrstan Gouréroux and Charles Vanasse Workng Paper Aprl 1998 ISSN : Ths research was fnanced by CREST and FFSA, n France, FCAR Quebec, CRSH Canada and the Rsk Management Char at HEC. We thank A. Snow for hs dscusson on dfferent ssues and for hs collaboraton n provdng complementary results of hs research and P.A. Chappor, P. Pcard and a referee for ther useful comments. Results of ths paper were presented at the France USA Conference on nsurance markets (Bordeaux, 1995), the Geneva Assocaton meetngs (Hannover, 1996) and the Delta Thema nsurance semnar (Pars, 1997).

2 Evdence of Adverse Selecton n Automoble Insurance Markets Georges Donne, Chrstan Gouréroux and Charles Vanasse Georges Donne holds the Rsk Management Char and s professor of fnance at École des HEC. Chrstan Gouréroux s Drector, Laboratory Fnance-Assurance CREST and Research Assocate, CEPREMAP. Charles Vanasse s research professonal, CRT, Unversté de Montréal. Copyrght École des Hautes Études Commercales (HEC) Montréal. All rghts reserved n all countres. Any translaton or reproducton n any form whatsoever s forbdden. The texts publshed n the seres Workng Papers are the sole responsblty of ther authors.

3 Evdence of Adverse Selecton n Automoble Insurance Markets Abstract In ths paper, we propose an emprcal analyss of the presence of adverse selecton n an nsurance market. We frst present a theoretcal model of a market wth adverse selecton and we ntroduce dfferent ssues related to transacton costs, accdent costs, rsk averson and moral hazard. We then dscuss an econometrc modelng based on latent varables and we derve ts relatonshp wth specfcaton tests that may be useful to solate the presence of adverse selecton n the portfolo of an nsurer. We dscuss n detal the relatonshp between our modelng and that of Puelz and Snow (199). Fnally, we present some emprcal results derved from a dfferent data set. We show that there s no resdual adverse selecton n the studed portfolo snce approprate rsk classfcaton s made by the nsurer. Consequently, the nsurer does not need a selfselecton mechansm such as the deductble choce to reduce adverse selecton. Keywords : Adverse selecton, emprcal test, rsk classfcaton, transacton costs. JEL classfcaton: D80. Résumé Dans cet artcle, nous proposons une analyse emprque sur la présence de l'antsélecton dans un marché d'assurance. Dans un premer temps, nous présentons un modèle théorque d'un marché avec antsélecton et nous ntrodusons dfférentes dscussons relées aux coûts de transacton, aux coûts des accdents, l'averson au rsque et le rsque moral. Pus, nous dscutons d'une modélsaton économétrque avec varables latentes et nous décrvons sa relaton avec des tests de spécfcaton qu peuvent être utles pour soler la présence de l'antsélecton dans le portefeulle d'un assureur. Nous dscutons en détal des lens entre notre modélsaton et celle de Puelz et Snow (197). Fnalement, nous présentons des résultats emprques obtenus d'une banque de données dfférente. Nous montrons qu'l n'exste pas d'antsélecton résduelle dans le portefeulle étudé parce qu'une classfcaton approprée des rsques est effectuée par l'assureur. Ce résultat mplque que l'assureur n'a pas beson d'utlser un mécansme d'autosélecton, comme un chox de franchse pour rédure l'antsélecton. Mots clés : Antsélecton, test emprque, classfcaton des rsques, coûts de transacton. Classfcaton JEL : D80.

4 Introducton Adverse selecton s potentally present n many markets. In automoble nsurance, t s often documented that nsured drvers have nformaton not avalable to the nsurer about ther ndvdual rsks. Ths explans the presence of many nstruments lke rsk classfcaton based on observable characterstcs (Hoy, 198 and Crocker and Snow, 1985, 1986), deductbles (Rothschld and Stgltz, 1976 and Wlson, 1977) and bonusmalus schemes (Donne and Lasserre, 1985, Donne and Vanasse, 199 and Pnquet, 1998). But the presence of deductbles can also be documented by moral hazard (Wnter, 199) or smply by transacton costs proportonal to the actuaral premum, and the bonus-malus scheme s often referred to moral hazard. It s then dffcult to solate a pure adverse selecton effect from the data. However, the presence of adverse selecton s necessary to obtan certan predctons that would not be obtaned wth only transacton costs and moral hazard. Ths dffculty of solatng a pure adverse selecton effect s emphaszed by the absence n the publshed lterature of theoretcal predctons when both problems of nformaton are present smultaneously. Very few models consder both nformaton problems (see however Donne and Lasserre, 1988 and Chassagnon and Chappor, 1996). The lteratures on moral hazard and adverse selecton were developed separately and tradtonally faced dfferent theoretcal ssues : n the adverse selecton lterature, the emphass was put on the exstence and effcency of compettve equlbra wth and wthout cross-subsdzaton between dfferent rsk classes whle n the moral hazard one the emphass was on the endogenous determnaton of contractual forms wth few dscusson on equlbrum ssues (see however Arnott, 199). The same remarks apply to mult-perod contractng. Moreover, both lteratures have neglected accdent cost dstrbutons : the dscusson was manly on the accdent frequences wth few exceptons (Wnter, 199; Donne and Doherty, 199 and Doherty and Schlesnger, 1995). 1

5 What are then the most nterestng predctons for emprcal research? If we lmt the dscusson to sngle-perod contractng 1 and adverse selecton, the presence of separatng contracts wth dfferent nsurance coverages to dfferent rsk classes remans the most nterestng one. Ths s the Rothschld-Stgltz result obtaned from a model descrbng a smple compettve nsurance market wth two dfferent rsk types and two states of nature : when the proporton of hgh rsk ndvduals s suffcently hgh, a separatng equlbrum exsts wth less nsurance coverage for the low rsk ndvduals. There s no subsdy between the dfferent rsk classes and prvate nformaton s revealed by contractng choces. Recently Puelz and Snow (199) obtaned results from the data of a sngle nsurer and concernng collson nsurance : they verfed that ndvduals of dfferent rsk type self-selected through ther deductble choce and no cross-subsdzaton between the classes was measured. In ths paper we focus our attenton on such an emprcal test. We wll frst present n Secton 1 a theoretcal dscusson on adverse selecton n nsurance markets by ntroducng dfferent ssues related to transacton costs, accdent costs and moral hazard. In Secton, we dscuss n detal the artcle of Puelz and Snow (199). Partcularly we analyze one mportant ssue related to ther emprcal fndngs : we queston ther methodology of usng the accdent varable to measure the presence of resdual adverse selecton n rsk classes. In Secton 3, we present an econometrc modelng based on latent varables and ts relatonshp wth the structural equatons whch may be useful to analyze the presence of adverse selecton n the portfolo of an nsurer. Fnally, we present our results derved from a new data set. We replcate on ths data set the analyss of Puelz and Snow, and then propose some extensons about the methodology used. We show that ther concluson s not robust and that resdual adverse selecton s not present when approprate rsk classfcaton s made. 1 But we know that the data may contan effects from long-term behavor.

6 1. Adverse selecton and optmal choce of nsurance 1.1 All accdents have the same cost Let us frst consder the economy descrbed by Rosthschld and Stgltz (1976) (see Akerlof, 1970, for an earler contrbuton). There are two types of ndvduals ( = H,L) representng dfferent probabltes of accdents wth p H > p L. We assume that at most one accdent may arrve durng the perod. Wthout nsurance ther level of welfare s gven by : V (p ) = (1 p ) U (W) + p U (W C), (1) where : p W C s the accdent probablty of ndvdual type, = H,L s ntal wealth s the cost of an accdent U s the von Neumann-Morgenstern utlty functon (U'( ) > 0, U''( ) 0) assumed, for the moment, to be the same for the two rsk categores (same rsk averson). Under publc nformaton about the probabltes of accdent, a compettve nsurer wll offer full nsurance coverage to each type f there s no proportonal transacton cost n the economy. In presence of proportonal transacton costs the premum can be of the form P = (1 + k)p l where l s nsurance coverage and k s loadng factor. Wth k > 0, less than full nsurance s optmal. However an ncrease n the probablty of accdent does not necessarly mply a lower deductble f we restrct the form of the optmal contracts to deductbles for reasons that wll become evdent later on. In fact we can show : Proposton 1 : In presence of a loadng factor (k > 0), suffcent condtons to obtan that the optmal level of deductble decreases when the probablty of accdent ncreases are constant rsk averson and p < ½ (1 + k). The suffcent condton s qute natural n automoble nsurance snce p s lower than 10% whle k s hgher than 10%. Ths means that ndvduals wth hgh probabltes of 3

7 accdents do not necessarly choose a low deductble under full nformaton and non actuaral nsurance. However, n general, dfferent rsk types have dfferent nsurance coverage even under perfect nformaton. Under prvate nformaton, many strateges have beng studed n the lterature (Donne and Doherty, 199, Hellwg, 1987 and Fombaron, 1997). The nature of equlbrum s functon of the nsurers' antcpatons of the behavor of rvals. Rothschld and Stgltz (1976) assume that each nsurer follows a Cournot-Nash strategy. Under ths assumpton, t can be shown that a separatng equlbrum exsts f the proporton of hgh rsk ndvduals n the market s suffcently hgh. Otherwse there s no equlbrum. The optmal contract s obtaned by maxmzng the expected utlty of the low rsk ndvdual under a zero-proft constrant for the nsurer and a bndng self-selecton constrant for the hgh rsk ndvdual who receve full nsurance. If we restrct our analyss to contracts wth a deductble, the optmal soluton for the lowrsk ndvdual s obtaned by maxmzng V (p L ) wth respect to D L under a zero proft constrant and a self-selecton constrant : Max D L p L U(W D L P L L L ) + (1 p )U(W P ) s.t. P L L = p (C D ) (1+ k) L () H H L U(W p C) = p U(W D L P ) + (1 p H )U(W P L ), where P L s the nsurance premum of the L type. The soluton of ths problem yelds D L* > 0 whle D H* = 0 when the loadng factor (k) s nul. If now we ntroduce a postve loadng fee (k > 0) proportonal to the net premum, the total premum for each rsk type becomes P = (1 + k) p (C D ) and we obtan, from the above problem wth the approprate defntons, that D L* > D H* > 0 whch mples that p H (C D H* ) > p L (C D L* ) or that P H* > P L*. We then have as second result :

8 Proposton : When we ntroduce a proportonal loadng factor (k > 0) to the basc Rothschld-Stgltz model, the optmal separatng contracts have the followng form : 0 < D H* < D L. Ths result ndcates that the tradtonal predcton of Rothschld-Stgltz s not affected when the same proportonal loadng factor apples to the dfferent classes of rsk. 1. Introducton of dfferent accdent costs If now we take nto account dfferent accdent costs n the basc Rothschld and Stgltz model, the optmal choce of deductble may be affected by the dstrbutons of costs condtonal to the rsk classes (or types). Fluet (199) and Fluet and Pannequn (199) obtaned that a constant deductble wll be optmal only when the condtonal lkelhood rato f f H L (C) (C) s constant for all C, where f (C) s the densty of costs for type whch mples that the two condtonal dstrbutons are dentcal and the observed amounts of loss do not provde any nformaton to the nsurer. By a constant (or a straght) deductble t s mean that the deductble s not functon of the accdent costs. We can show that the results of Fluet and Pannequn (199) are robust to the ntroducton of a proportonal loadng factor. We consder two costs levels C 1,C and we denote p the dstrbuton of the cost condtonal to the occurrence of an accdent n, p 1 class. In other words, the condtonal expected cost of accdent for ndvdual s equal to : E 1 1 (C) = p C + p C. (3) We also assume that p H > p L and p H (E H (C)) > p L (E L (C)). Under the assumpton that C 1 > D 1 and C > > p > L* L* p1 D, ( = H,L) t can be shown that D1 = D as =. < L L p < p H H 1 When k > 0, D = = > 0 whatever C j and the same relatve results are obtaned H* H* H* 1 D D for the low rsk ndvdual. In other words : 5

9 p H j Proposton 3 : Let p L j be condtonal lkelhood rato for accdent costs of type H relatve to type L and let H* D be the optmal deductbles of type H n the presence of a proportonal loadng factor k 0, then the optmal deductbles of ndvdual L have the followng property : D and D L* H* j = > D 0 for j 1, () L* 1 H H > p > L* p1 = D as =. (5) < L L p < p 1 The ntuton of the result s the followng one. The optmal contract of the low rsk ndvdual wll be a straght or constant deductble f the observed amount of loss does not provde nformaton to the nsurer. Otherwse, the level of coverage vary wth the sze of the loss. In the extreme case where the observed loss reveals all the nformaton, both rsk types wll buy the same deductble when k = 0 (Doherty and Jung, 1993). Snce n the above analyss t was assumed that both costs dstrbutons have the same support, all the nformaton cannot be revealed by the observaton of an accdent. For the analyss of other defntons of lkelhood ratos see Fluet (199). 1.3 Adverse selecton wth moral hazard The research on adverse selecton wth moral hazard s startng (see however Donne and Lasserre 1988). We know that a constant deductble may be optmal under moral hazard f the ndvdual can modfy the occurrence of accdents but not the severty (Wnter, 199). Here to keep matters smple we assume that an nsured can affect hs probablty of accdent wth acton a but not the severty. Moreover, p p H j L j s ndependent of the cost level j and k = 0. Under these assumptons, Chassagnon and Chappor (1996) have shown that some partculartes of the basc Rothschld-Stgltz model are preserved. Partcularly, a hgher premum s always assocated to better coverage and ndvduals wth a lower deductble are more lkely to have an accdent, whch permts to 6

10 test the assocaton between deductble and accdent occurrence. However, the presence of moral hazard may reduce dfferences between accdent probabltes. 1. Cross-subsdzaton between dfferent rsk types One dffculty wth the pure Cournot-Nash strategy les n the fact that a poolng equlbrum s not possble. Wlson (1977) proposed the antcpatory equlbrum concept that always results n an equlbrum (poolng or separaton). When the proporton of hgh rsk ndvduals s suffcently hgh, a Wlson equlbrum concdes wth a Rothschld- Stgltz equlbrum. Moreover, welfare of both rsk classes can be ncreased by allowng subsdzaton : low rsk ndvduals can buy more nsurance coverage by subsdzng the hgh rsks (see Crocker and Snow, 1985 and Fombaron, 1997, for more detals). 1.5 Dfferent rsk aversons The possblty that dfferent rsk types may also dffer n rsk averson was consdered n detal by Vlleneuve (1996). It s then necessary to control for rsk averson when we test for the presence of resdual adverse selecton. We wll see that the rsk classfcaton varables do, ndeed, capture some nformaton on rsk averson. In other words, we can also test for the presence of resdual rsk averson n rsk classes. 1.6 Rsk categorzaton In many nsurance markets, nsurers use observable characterstcs to categorze ndvdual rsks. It was shown by Crocker and Snow (1986) that such categorzaton s welfare mprovng f ts cost s not too hgh and f observable characterstcs are correlated wth hdden knowledge. The effect of rsk categorzaton s to reduce the gap 7

11 between the dfferent rsk types and to decrease the possbltes of separaton by the choce of dfferent deductbles. Ths result suggests that a test for the presence of adverse selecton should be appled nsde dfferent rsk classes or by ntroducng categorzaton varables n the model. It s known that the presence of adverse selecton s suffcent to justfy rsk classfcaton when rsk classfcaton varables are costless to observe. Now the emprcal queston becomes : Emprcal queston : Gven that an effcent rsk classfcaton s used n the market, should there reman resdual adverse selecton n the data? Another result of Crocker and Snow s to show that, wth approprate taxes and subsdes on contracts, no nsureds loose as a result of rsk categorzaton. Ths result can be obtaned for many types of equlbrum and partcularly for both Rothschld-Stgltz and Wlson (or Wlson-Myazak-Spence) equlbra. Snce rsk categorzaton facltates rsk separaton wthn the classes, t may reduce the need of cross-subsdzaton between rsk types of a gven class. However, there should be subsdzaton between the rsk classes accordng to the theory.. Emprcal measure of adverse selecton : some comments on the current lterature Dfferent tests can be used to verfy the presence of adverse selecton n a gven market and ther nature s functon of the avalable data. If we have access to ndvdual data from the portfolo of an nsurer and want to test that hgh rsk ndvduals n a gven class of rsk choose the lower deductble, the test wll be functon of the dfferent rsk classes used by the nsurer, and consequently of the explanatory varables ntroduced n the model. Intutvely, when the lst of explanatory varables s large and the classfcaton s approprate, the probablty to fnd resdual adverse selecton n a portfolo s low. 8

12 Very few artcles have analyzed the sgnfcance of resdual adverse selecton n nsurance markets. Dahlby (1983, 199) reported evdence of some adverse selecton n Canadan automoble nsurance markets and suggested that hs emprcal results were n accordance wth the Wlson-Myazak-Spence model that allows for crosssubsdzaton between ndvduals n each segment defned by a categorzaton varable. Hs analyss was done wth aggregate data. Untl recently, the only detaled study wth ndvdual data was that of Puelz and Snow (199) (see Chappor, 1998, for an overvew of the recent papers and Rchaudeau, 1997, for a thess on the subject). In ther analyss they consdered four dfferent adverse selecton models. They found evdence of adverse selecton wth market sgnalng and no-cross-subsdzaton between the contracts of dfferent rsk classes. In other words, they found evdence of separaton n the choce of deductble wth non-lnear nsurance prcng and no-cross-subsdzaton. To obtan ther results they estmated two structural equatons : a demand equaton for a deductble and a premum functon that relates dfferent tarfcaton varables to the observed prema. The demand equaton can be derved from the low rsk ndvdual maxmzaton problem n a pure adverse selecton model wth a postve loadng factor. Ths yelds D L* > D H* > 0 wth two types of rsk n a gven class (Proposton ). Unfortunately, t cannot be obtaned from the frst order condton () n Puelz and Snow whch corresponds to the frst order condton of the result presented n Proposton 1 above. Another crtcsm concerns the relatonshp on non-lnear nsurance prcng and Rothschld-Stgltz model. In fact from the dscusson above, the separaton result s due to the ntroducton of a self-selecton constrant n the low-rsk ndvdual problem and not from the fact that nsurance prcng s non-lnear. The two problems yeld dfferent emprcal tests. From Proposton, we do not need the non-lnearty of the premum schedule to verfy that a separatng contract s chosen. In Rothschld-Stgltz model ths s the self-selecton constrant that separates the rsk types. Therefore what we need to test s the fact that dfferent rsk types choose dfferent deductbles n the controlled classes of rsk and that the self-selecton constrant of the 9

13 hgh rsk ndvduals s bndng. In that perspectve, the estmaton of both equatons (6) and (7) n Puelz and Snow (199) reman useful f we do not have access to the tarfcaton book of the company. Otherwse, the estmaton of (6) s not useful. For dscusson we reproduce here ther equatons (6) and (7) : P = β β1 SYM + β T + = 7 = 11 = = 11 + β β 5 1 D + β T D D 1 = 11 β + β 6 3 A + β T D 10 β 3 SYM + β A D β D 1 + MALE + β A D 5 10 = 7 8 β + β 6 SYM MR D PERAGE + ε 1 (6) D = α 0 + α 1 RT + α ĝ d + α 3 W 1 + α W + α 5 W 3 + α 6 MALE (7) + α 7 PERAGE + ε, where A s the age of the automoble; MR = 1 for a multrsk contract and 0 otherwse; SYM s the symbol of the automoble; T s the terrtory; D = 0 for D = $100, D = 1 for D = $00, and D = for D = $50; W 1, W, W 3 = wealth dummy varables; MALE = 1 for a male and 0 for a female; PERAGE s the age of the ndvdual; RT s for rsk type measured by the number of accdents; and ĝ d s the deductble prce on whch we wll come back. The dependent varable of equaton (6) s the gross premum pad by the nsured and both D 1 and D are dummy varables for deductble choce. Puelz and Snow used equaton (6) to generate a margnal prce varable and to test for the non-lnearty of the premum equaton. Equaton (6) yelds the values of deductble prces and equaton (7) ndcates f dfferent rsks choose dfferent deductbles gven that we have controlled for the dfferent prces and other characterstcs that may nfluence that choce. They also estmated a prce equaton to determne ther prce varable ĝ d n the demand equaton for a deductble (7) and used the number of accdents (RT) at the end of the current perod to approxmate the ndvdual rsks. Both varables have sgnfcant parameters wth rght sgns. But t s not clear that they had to estmate ĝ d. It would have been 10

14 easer to use drectly the values obtaned from equaton (6). Fnally, very few varables are used n (7): the age and the symbol of the automoble are not present. 3. A new evaluaton of adverse selecton n automoble nsurance In ths secton we present an econometrc model and emprcal results on the presence of adverse selecton n an automoble nsurance market. The data come from a large prvate nsurer n Canada and concern collson nsurance snce the nsured has the choce for a deductble for that type of nsurance only. There s no bodly njures n the data and lablty nsurance for property damages s compulsory. In that respect we are close to Puelz and Snow (199). 3.1 Latent model Pure adverse selecton model In order to perform carefully the analyss of adverse selecton n ths portfolo from a structural model, t s mportant to desgn a basc latent model. The dscusson presupposes that two deductbles D 1 < D are avalable. The latent varables of nterest are for the ndvdual : the tarfcaton varables from the nsurer : P 1 the premum for the contract wth the deductble D 1 P the premum for the contract wth the deductble D. Snce D 1 < D, t s clear that P 1 > P. the ndvdual rsk varables : Ths rsk can be measured by accdent occurrences and costs. For the moment, we lmt the number of potental accdents n a gven perod to one : 11

15 Y = 1, f ndvdual has an accdent, 0, otherwse; C = potental cost of accdent for ndvdual. the deductble choce varable : Fnally, we must analyze the deductble choce by ndvdual. Snce we have only two possble choces, ths yelds a bnary varable : Z = 1, f the ndvdual chooses deductble D 1, 0, otherwse. A latent model may correspond to : p 1 = log P 1 = g 1 (x,θ) + ε 1, p = log P = g (x,θ) + ε, *, Y * > Y = wth Y = g (x, θ) + ε, c = log C = g (x,θ) + ε *, z * > Z = wth Z = g (x, θ) + ε, where denotes the ndcator functon. The latent model would be very smplfed f the dfferent error terms are uncorrelated ε = (ε 1, ε, ε 3, ε, ε 5 ) ~ N(0,Ω). However these correlatons may be dfferent from zero and have to be analyzed. In fact, they wll become very mportant n the dscusson of the test for the presence of adverse selecton n the nsurer portfolo. Moreover, the above dependent varables are not necessarly observable. At least two dependent sources of bas have to be consdered : 1) Accdent declaratons The nsurer observes only the accdents for whch a payment has to be made, that s only the accdents that generate a cost hgher than the chosen deductble. Moreover, the 1

16 nsured may also take nto account of the ntertemporal varaton of hs prema when he fles a clam and declares only the accdents that wll not ncrease to much hs future prema. For example, n our data set, we observe very few rembursements below $50 for the nsured ndvduals wth a deductble of $50 whch means that they do not fle clams between $50 and $500 systematcally. The same remark apples for those who choose the $500 deductble. Therefore, lmtng ourselves to a statc scheme, the observed accdents are the clams fled : 1, f ndvdual wth deductble D = 0, otherwse; Ŷ 1 1 1, f ndvdual wth deductble D = 0, otherwse. Ŷ had an accdent and fled a clam, had an accdent and fled a clam, Smlarly, accdent costs faced by the nsurer correspond to ther true values C 1, C only when Ŷ 1 = 1 and Ŷ = 1 respectvely. Therefore, when approprate precautons are not taken, we should obtan an undervaluaton of the accdent probabltes and an overvaluaton of the accdent costs. ) Avalable prema When the tarfcaton book of the nsurer s avalable, all prema P 1, P consdered by each ndvdual are observable for the determnaton of the two functons g 1 (x 1,θ) and g (x,θ). In practce, we may often be lmted to the chosen premum Pˆ P1, f Z = 1, P, f Z = Introducng moral hazard Under moral hazard, the agent effort s not observable. The nsurer can ntroduce ncentve schemes to reduce the negatve moral hazard effects on accdent and costs dstrbutons, but does not elmnate all of them n general. Ths s the standard trade-off 13

17 between nsurance coverage and effort effcency. Ths means that there may reman a resdual moral hazard effect n the data that s not taken nto account even by an extended latent model wth moral hazard. Resdual moral hazard can affect accdent occurrences and costs jontly wth deductble choce : non observable low effort levels mply hgh accdent probabltes and hgh accdent costs. Moreover, resdual moral hazard can explan why, for example, predcted low rsk ndvduals n an adverse selecton model wth moral hazard may choose the lowest deductble D 1, when they antcpate low effort actvtes n the contract perod. In order to take nto account of the moral hazard effect, we extend the above model by ntroducng a non observable varable a that summarzes all the efforts of ndvdual not already taken nto account explctly. Ths varable can be affected by non observable costs and ncentve schemes. But some of them are observable. Partcularly, the bonusmalus scheme of the nsurer may nfluence the prema, both accdents numbers and effort costs dstrbutons and deductble choce. An nsured that s not well classfed accordng to hs past accdents record (hgh malus) at the begnnng of the perod, may want to mprove hs record by ncreasng hs safety actvtes (less speed, no alcohol whle drvng, ) durng the current perod. These actvtes should reduce accdent occurrences and accdent costs. They may also nfluence the deductble choce f the antcpated actons affect partcularly low cost accdents. The explct ntroducton of moral hazard goes as follows : let a a contnuous varable measurng non observable ndvdual's acton be a functon of a vector of dfferent observable explanatory varables x~ and of non observable varables. The former are called explct moral hazard varables whle the second take nto account of the resdual moral hazard. One can extend the latent model n the followng way : premum functons are naturally affected by the observable explanatory varables for the explct moral hazard whle the two dstrbutons for cost and accdents and the deductble choce are functon of two ngredents: the explct and the resdual moral hazard. Introducng the relaton a = x~ δ +, the three relatonshps can be rewrtten as follows : ε * Y = wth Y = g (x, θ) + γ x~ δ + ε + γ ε, Y * > 0,

18 c = g (x θ) + γ x~ δ + ε + γ ε, * Z = wth Z = g (x, θ) + γ x~ δ + ε + γ ε. Z * > 0, For the premum functon we just have to ntroduce the x~ varables n the regresson component. We must say that ths form of moral hazard may ntroduce some autocorrelaton between the dfferent equatons (same ε ) and some lnk between the parameters ( γ3, 5 δ, γ δ γ δ ). 3. Some specfcaton tests Comparson of the observed and the theoretcal prema The observed prema P 1 and P can be compared to the ndvdual underlyng rsks, for nstance through the pure prema. The pure prema may be taken equal to the expected clams, contract by contract,.e. deductble by deductble. For the contract wth deductble D 1 the correspondng pure premum s gven by : 1 = E ( Y D ( C ) Π Z = 1). C> D1 / Equvalently, we have : = E ( Y D ( C ) Π Z = 0). C> D / If we assume that there s no correlaton between Z, Y and C when the explanatory varables are taken nto account, we obtan : Π Π = P ( 1 C D 1 ). = 1 E ( C D ). ( Y = 1) E ( C D ) 1 > = P ( Y ) ( ) C > D 15

19 Then usng the cost equaton we deduce : E ( ), (( C D) > D ) = E ( exp( g + σ u) D ) exp ( g +σ u) C > D where u s a normal varable N(0,1). We then have : E (( C D) C > D) = E ( exp g exp σ u D) = log D g σ = exp g = exp g = exp g ( exp g exp σ u D) log D g σ σ + exp log D g σ σ g + φ ( σ u) u > ϕ(u) du ϕ(u) du D ϕ(u σ log D g σ log D g σ ) du D + σ log D g D φ σ ϕ(u) du log D g σ ϕ(u) du log D. σ Ths last expresson s lke a Black-Scholes prce equaton for an European call opton. In fact, we obtan E (( C D) C > D) = E ( C D) +. Ths s an opton on the rembursement cost (C) where the deductble (D) s the exercse prce. For the nsured, the contract valuaton ncludes a prvate opton of non declaraton. From the above expresson and the correspondng expresson P (Y = 1) we obtan : Π 1 ( θ) g3 = φ σ ( x θ) 3 x exp g ( x, θ) and a correspondng expresson ( θ) Π σ + g φ ( x, θ) + σ log D g ( x, θ) σ by replacng D 1 by D. 1 φ D1 log D σ 1 After the estmaton of the dfferent parameters of the model, pure and observed prema θ ˆ ˆ whch can be compared by usng a regresson model of the type gk ( x, ) = αk Π k ( θ) + βk 16

20 wll measure the lnks between prema and ndvdual rsks and the estmated coeffcents wll provde nformaton on margnal profts or fx costs. We can also compare margnal profts for dfferent deductbles by comparng ( α 1, β 1) to ( α ), β. We may also verfy whether the nsurance tarfcaton s set manly from accdent frequences or f the pure g prema s sgnfcant by dong a regresson of g 1 (x,θ) on φ σ ( x θ) 3, 3 and then testng the sgnfcance of the effect on average cost. Fnally, we may also consder some aspects related to the rsk averson by consderng f V(C D) + premum. nfluences also the Comparson of the observed and theoretcal deductble choces Another mportant structural aspect s the ndvdual choce of deductble. Suppose there are only two possbltes D 1 < D and let us assume rsk neutralty for the moment. When ndvdual chooses the premum k, hs payments are equal to : ( C C < D k Dk C ) ( ) > D = P k k + YC Y C Dk C Dk P k + Y > In expected value we obtan : P k + E + ( Y C / x ) E(Y (C D ) / x ). k D 1 s prefered to D by ndvdual f : P P 1 E P + ( Y ( C D1) / x ) < P E Y ( C D ) 1 E + ( / x ) + ( Y ( C D ) / x ) + E Y ( C D ) + ( / x ) > 0. 1 Therefore t s possble to check ths knd of behavor by comparng the observed choces Z to the one modelng (as soon as P 1 and P are known). Z * 1 = + + P P1 E( Y ( C D ) / x ) + E ( Y ( C D1) / x ) correspondng to ths > 0 17

21 It s clear that, f the tarfcaton s based on pure prema only, the nsured would be ndfferent between the two deductbles. It becomes also evdent that we must study jontly the two structural aspects related to the nsurance tarfcaton and the deductbles choce to verfy the presence of some adverse selecton effects. Ths s the topc of the next secton. 3.3 Econometrc results We now present econometrc results from two structural equatons lke those proposed n Puelz and Snow and dfferent extensons. At ths pont we have not yet analyzed the accdent costs and not taken nto account moral hazard explctly. However, we wll use some tarfcaton varables of the nsurers that take nto account accdent costs ndrectly and moral hazard. These varables are: 1) the tarfcaton group varable for dfferent automoble characterstcs; ) the age of the car; and 3) the bonus-malus varables. Dfferent contracts correspondng to varous levels for a straght deductble are proposed by the nsurer. From the data, we observe that the deductble choce does matter for only two deductble levels $50 and $500 and n fact the choce of $500 s done only by about % of the overall portfolo, whle t s made by nearly 18% of the young drvers. The next fgure shows how the choce of the $500 does matter for rsk classes hgher than 3. We wll then concentrate our analyss to these classes. (See Appendx I for formal defntons of classfcaton varables.) 18

22 100% 90% 80% 70% 60% 50% 0% 30% 0% 10% 0% Fgure 1 Observed Deductble Choces Accordng to Classes Classes $500 Deductble $50 Deductble A prelmnary analyss of the data showed that the choce of the $500 deductble was sgnfcant only for groups of vehcles 8 to 15 and for drvers n drvng classes to 19 or for,77 polcy holders of the entre portfolo : n these classes, 13.5% of potental permt holders choose the $500 deductble whle 86.5% choose the $50 deductble. The correspondng accdent frequences are for the $500 deductble and for the $50 deductble. Many factors can explan these observatons. The most mportant one s the type of car. We wll control for ths pattern by usng the "group of vehcle" varable. Another factor may be rsk averson. As n Puelz and Snow (199), we use the "chosen lmt of lablty nsurance" varable to approxmate ndvduals' wealth. The rebate assocated to a larger deductble can also nfluence the choces snce ths s a prce varable. Ths margnal prce varable wll also be consdered and the nformaton comes from the tarff book of the nsurer. It s mportant to notce here that snce we do have access to ths prce varable drectly, we do not have to estmate (as n Puelz and Snow, 199) ths prce nformaton. However, for matter of comparson, we wll compare results obtaned from both methods. The whole lst of varables s presented n Appendx 1. Let us frst consder the choce of the deductble. As dscussed n the prevous secton, f we want to test the predcton of Rothschld and Stgltz (1976) that low resdual rsk ndvduals choose the hgher deductble, we must use a measure of ndvdual's rsk. That measure of ndvdual rsk has to represent some asymmetrcal nformaton between the nsurer and the nsured n the sense that, at the date of contract choce, the 19

23 nsured has more nformaton than the nsurer about hs ndvdual (resdual) rsk durng the contractual perod. A frst rsk varable s the expected number of accdents. Snce we have access to all clams we can estmate the ex-ante probablty of accdent the nsured knew at the begnnng of the perod. In that sense we may have more nformaton than the nsurer but probably less than the nsured snce we have access to only part of hs prvate nformaton. However, snce the estmated probablty of accdent s obtaned by usng observable characterstcs, ts value does not contan asymmetrcal nformaton. We may also use the number of accdents as n Puelz and Snow (199), but precautons have to be made on ts nterpretaton. To obtan the ndvdual probabltes of accdent we estmated the regresson coeffcents for the equatons assocated wth the ndvdual's rsks n the latent model and we used the predcton of ths regresson to construct the ndvdual expected number of accdents. In ths secton we do not take nto account of the accdent costs but we allow for more than one accdent durng the perod. Results are presented n Table 1. They come from the estmaton of an Ordered Probt Model where the dependent varable consders three categores : no accdent (wth a clam hgher than $500) durng the perod, one accdent and and more accdents (see Appendx for a descrpton). Snce only one ndvdual had three accdents, ths last category was grouped wth that of two accdents. (See Donne et al, 1997, for results wth the Negatve bnomal model. The results are dentcal.) Clams between $50 and $500 were not used to elmnate potental selecton bas assocated to the fact that these clams are not observable for those who have the $500 deductble. Table 1 Ordered Probt on Clams (0, 1, and more) Varable Coeffcent T-rato Intercept (7.01) Intercept µ 1.10 (17.30) SEXF (.18) MARRIED (1.08) AGE (0.885) NEW (.96) Group of vehcles G (0.189) G (0.80) 0

24 G (0.8) G (0.98) G (.00) G (0.001) G (0.185) Terrtory T 0.79 (0.958) T (0.963) T 0.7 (.555) T (0.99) T (1.509) T (1.91) T (.00) T (0.787) T (0.06) T (3.333) T (1.53) T (0.59) T (.90) T (0.50) T (1.577) T (0.71) T (1.859) T (3.56) T (.887) T (1.837) T (.016) Number of observatons,77 Log-Lkelhood 1, Observed Frequences 0, In order to ntroduce a prce n the deductble equaton, we used two dfferent approaches. The frst one was to calculate the prema varatons from the nsurer's book of prema for dfferent deductbles where the rsk classes are dentfed by the control varables n the regresson. Ths yelded the GD varable. In the second approach we estmate a premum equaton and calculate the prema varatons by usng the deductble coeffcent whch yelded the Ĝ D varable. We have to emphasze here that the ĜD varable n the deductble equaton s dfferent from the ĝ D varable n Puelz and Snow. Ther ĝ D varable was obtaned from a regresson, where a Ĝ D varable lke ours was 1

25 the dependent varable! The estmaton results are gven n Tables and 3 for GD whle those for Ĝ D are n Tables A1 and A n the Appendx. Our results for the frequences of accdents goes n the expected drecton. The observed statstcs ndcated that the ndvduals who choose the larger deductble have an average frequency of accdent (0.081) lower than the average one (0.098) of those who choose the smaller deductble. In fact, from Table, we observe n Model that the predcted probablty of accdent E(acc) (whch should be the rght varable to measure the ndvdual observable rsk f we do not take care of the accdent costs) s sgnfcant and has a negatve coeffcent ( 5.30) to explan the choce of the hgher deductble. However, ths varable may take nto account of some non-lneartes that are not modelzed yet. Table Probt on Deductble Choce wth GD (Z = 1 f $500 deductble) Model 1 Model Model 3 Varable Condtonal on the number of clams Condtonal on the expected number of clams Condtonal on the number of clams and expected number of clams Coeffcent T-rato Coeffcent T-rato Coeffcent T-rato Intercept (5.006) (3.13) (3.111) Acc (1.983) (1.57) E(acc) (6.17) (6.78) GD (5.75) (7.13) (7.13) SEXF (8.96) (9.33) (9.338) AGE (7.975) 0.00 (7.78) 0.05 (7.79) Lablty lmt W (0.177) (0.65) (0.87) W (1.87) (1.88) (1.860) W (0.17) (0.597) (0.58) W (.990) (.156) (.166) Group of vehcles G (.683) (.9) (.9) G (3.359) (3.685) (3.698) G (3.67) (3.769) 0.9 (3.770) G (.36) (5.6) (5.70) G (.85) (6.80) (6.783) G (.937) (.795) (.813) G (3.51) (3.667) (3.680) YMALE (0.73) (0.01) (0.9) Number of observatons,77,77,77 Log-lkelhood 1, , ,71.961

26 For comparson we dd also estmate the same equaton by usng the numbers of accdents as n Puelz and Snow (RT). The varable "accdent" (Acc) yelded a smlar result but ts coeffcent s less mportant n absolute value ( 0.16) than that of E(acc) n Model. However, f we compare the log lkelhood values of the two regressons ( compared to ), any test wll choose the regresson wth the expected number of clams. Another possblty s to nclude both varables n the same equaton whch s a natural method for ntroducng a correcton for msspecfcaton problems (see Donne et al, 1997, for more detals). As shown n Table, only the E(acc) varable s sgnfcant when both varables are ntroduced n the same regresson (Model 3). Ths result s very mportant for our man purpose. It ndcates that when we control for the ndvduals' observable rsk by usng the E(acc) varable, there s no resdual adverse selecton n the portfolo snce the Acc varable s no more sgnfcant. It also ndcates that a concluson on the presence of resdual adverse selecton obtaned from a regresson wthout the E(acc) varable s msleadng : the coeffcent of the accdent varable s sgnfcant because there s a msspecfcaton problem. By ntroducng the E(acc) varable, we ntroduce a natural correcton to ths problem (see Donne, Gouréroux, Vanasse, 1997, for more detals). Results n Table 3 ntroduce a further step by addng more rsk classfcaton varables n the model. We observe that when suffcent classfcaton varables are present, both Acc and the E (acc) varables are not sgnfcant. In other words, an nsurer that uses approprate rsk classfcaton varables can elmnate the presence of resdual adverse selecton and can take nto account the non lneartes. Our results ndcate clearly that there s no resdual adverse selecton n the portfolo studed. 3

27 Varable Table 3 Probt on Deductble Choce wth GD and More Rsk Classfcaton Varables (Z = 1 f $500 deductble) Model 1' Condtonal on the number of clams Model ' Condtonal on the number of clams and expected number of clams Coeffcent T-rato Coeffcent T-rato Intercept 1.10 (.57) (.90) Acc (1.76) (1.80) E(acc) (0.188) GD (0.55) (0.550) W (0.859) (0.858) W (1.001) (1.000) W (1.77) (1.78) W (0.77) 0.03 (0.78) G (3.05) (3.058) G (.01) (3.933) G (3.318) (3.11) G (.1) (3.97) G (.61) (.09) G (.763) (.76) G (.30) (.18) YMALE (1.588) (1.58) Terrtory T (0.105) (0.009) T (1.56) (1.398) T (1.71) (0.831) T (1.093) (0.971) T (0.798) (0.55) T (.983) (1.90) T (0.15) (0.79) T (3.93) (.96) T (1.36) (1.356) T (0.78) (0.393) T (0.011) (0.1) T (0.391) (0.93) T (1.697) (0.86) T (1.753) (1.75) T (1.157) (0.699) T (0.88) (0.886) T (0.569) (0.30) T (0.) (0.058) T (0.373) (0.09)

28 T (1.097) (0.586) T (.05) (1.05) Drver's class CL (7.38) (7.376) CL (1.91) (1.58) CL (0.8) (0.89) CL (0.09) (0.09) CL (5.70) (5.50) CL1 0.7 (3.35) 0.63 (3.1) CL (.6) (.9) CL (1.859) (1.63) CL (6.869) (5.78) NEW (.0) (.30) AGECAR (3.7) (3.5) Number of observatons,77,77 Log-lkelhood 1,66.1 1,66.39 In Appendx, we reproduce smlar results (Tables A1, A) when Ĝ D (nstead of GD) s used. Its value s obtaned from the regresson of the premum equaton presented n Table A3. The same conclusons on the absence of resdual adverse selecton are obtaned. In the premum equaton we verfy that the average effect of havng a $500 deductble (deductble varable and nteractons wth age, sex, martal status, use of the car, terrtores ) on the prema s negatve and sgnfcant ( $). Ths s the sum of the drect and nteracton effects. Table summarzes the dfferent results. Agan we observe that the use of GD nstead of Ĝ D does not affect the conclusons of the paper. 5

29 Table Summary of econometrc results D H = $ 50 E H (acc) = D L = $ 500 E L (acc) = Coeffcent of E(acc) n a regresson of the deductble choce wth GD 5.30 (Table, Model ) Coeffcent of GD (n the same regresson) taken from the nsurer book 0.01 (Table, Model ) Coeffcent of Acc n a regresson of the deductble choce wth GD 0.16 (Table, Model 1) Coeffcent of GD (n the same regresson) taken from the nsurer book 0.01 (Table, Model 1) Coeffcent of Acc n a regresson on the deductble choce wth GD and E(acc) (Table, Model 3) (no resdual adverse selecton) Not sgnfcant Coeffcents of E(acc) and Acc n Table 3 (Models 1' and 3') Not sgnfcant Coeffcent of E(acc) n the regresson of the deductble choce wth (Table A1, Model 5) ĜD 3.80 Coeffcent of Ĝ D (n the same regresson) obtaned from results n table A (Table A1, Model 5) 6

30 Average effect of $500 deductble on the prema (Sum of the nteracton varables and deductble varable) $ Coeffcent of Acc n the regressons of the deductble choce wth (Table A1, Model ) ĜD 0.16 Coeffcent of Ĝ D (n the same regresson) obtaned from results n Table A (Table A1, Model ) Coeffcent of Acc n a regresson on the deductble choce wth E(acc) (Table A1, Model 6) (No resdual adverse selecton) Ĝ D and Not sgnfcant Coeffcents of E(acc) and Acc n Table A (Models ' and 6') Not sgnfcant. Concluson In ths paper we have proposed a new emprcal analyss on the presence of adverse selecton n an nsurance market. We have presented a theoretcal dscusson on how to test such presence n a market wth transacton costs where moral hazard may be present and where accdent costs may dffer between the nsurance polces. Our econometrc results were derved, however, from a model wthout dfferent accdent costs. They show that ndvduals who choose the larger deductble have an average frequency of accdent lower than the average one of those who choose the smaller one. However, snce the expected numbers of accdents were obtaned from observable varables, ths result does not mean that there s adverse selecton n the portfolo. Further analyses show that, n fact, there s no resdual adverse selecton n the portfolo studed. The nsurer s able to control for adverse selecton by usng an approprate rsk classfcaton procedure. In ths portfolo, no other selfselecton mechansm (as the choce of deductble) s necessary for adverse selecton. Deductble choces may be explaned by proportonal transacton costs as suggested by Proposton 1. 7

31 REFERENCES Akerlof, G.A., "The Market for 'Lemons':Qualty Uncertanty and the Market Mechansm", Q.J.E. 8 (August 1970): Arnott, R., "Moral Hazard n Compettve Insurance Markets", n Contrbutons to Insurance Economcs, G. Donne (ed.), Kluwer Academc Press, 199. Chassagnon, A. and P.A. Chappor, "Insurance Under Moral Hazard and Adverse Selecton: The Case of Pure Competton", Paper presented at the nternatonal conference on nsurance economcs, Bordeaux, Chappor, P.A., "Asymmetrc Informaton n Automoble Insurance: an Overvew", Workng paper, Economcs Department, Unversty of Chcago (publshed n ths volume), Chappor, P.A., and B. Salané, "Asymmetrc Informaton n Automoble Insurance Markets: An Emprcal Investgaton", Mmeo, Delta, Crocker, K.J. and A. Snow, "The Effcency Effects of Categorcal Dscrmnaton n the Insurance Industry", J.P.E. 9 (Aprl 1986): 31-. Crocker, K.J. and A. Snow, "The Effcency of Compettve Equlbra n Insurance Markets wth Asymmetrc Informaton", J. Publc Econ. 6 (March 1985): Dahlby, B.G., "Testng for Asymmetrc Informaton n Canadan Automoble Insurance", n Contrbutons to Insurance Economcs, edted by Georges Donne, Boston: Kluwer, 199. Dahlby, B.G., "Adverse Selecton and Statstcal Dscrmnaton: An Analyss of Canadan Automoble Insurance", J. Publc Econ. (February 1983): Donne, G. and N. Doherty, "Adverse Selecton n Insurance Markets: A Selectve Survey", n Contrbutons to Insurance Economcs, edted by Georges Donne, Boston: Kluwer, 199. Donne, G. and N. Doherty, "Adverse Selecton, Commtment and Renegotaton : Extenson to and Evdence from Insurance Markets", J.P.E. (Aprl 199): Donne, G., N. Doherty and N. Fombaron, "Adverse Selecton n Insurance Markets" n Handbook of Insurance, edted by Georges Donne, Boston: Kluwer, 1998 (forthcomng). Donne, G., C. Gouréroux and C. Vanasse, "The Informatonal Content of Household Decsons wth Applcatons to Insurance Under Adverse Selecton", Dscusson paper, CREST and Rsk Management Char, HEC-Montréal,

32 Donne, G. and P. Lasserre, "Adverse Selecton, Repeated Insurance Contracts and Announcement Strategy", Rev. Econ. Studes 50 (October 1985): Donne, G. and P. Lasserre, "Dealng wth Moral Hazard and Adverse Selecton Smultaneously", Workng Paper, Economcs Department, Unversté de Montréal, Donne, G. and C. Vanasse, "Automoble Insurance Ratemakng n the Presence of Asymmetrcal Informaton", Journal of Appled Econometrcs, 7 (199): Doherty, N. and H.N. Jung, "Adverse Selecton When Loss Severtes Dffer: Frst-Best and Costly Equlbra", Geneva Papers on Rsk and Insurance Theory, 18 (1993): Doherty, N. and H. Schlesnger, "Severty Rsk and the Adverse Selecton of Frequency Rsk", Journal of Rsk and Insurance, 6, (December 1995): Fluet, C., "Second-Best Insurance Contracts Under Adverse Selecton", Mmeo, Unversté du Québec à Montréal, 199. Fluet, C. and F. Pannequn, "Insurance Contracts Under Adverse Selecton wth Random Loss Severty", Mmeo, Unversté du Québec à Montréal, 199. Fombaron, N., "Contrats d'assurance dynamques en présence d'antsélecton : les effets d'engagement sur les marchés concurrentels", thèse de doctorat, Unversté de Pars X-Nanterre, 1997, 305 pages. Hellwg, M.F., "Some Recent Developments n the Theory of Competton n Markets wth Adverse Selecton", European Economc Revew 31 (March 1987): Hoy, M., "Categorzng Rsks n the Insurance Industry", Q.J.E. 97 (May 198): Pnquet, J., "Experence Ratng for Heterogeneous Models", forthcomng n Handbook of Insurance, G. Donne (ed.), Kluwer Academc Press (1998). Puelz, R. and A. Snow, "Evdence on Adverse Selecton: Equlbrum Sgnalng and Cross-Subsdzaton n the Insurance Market", Journal of Poltcal Economy 10, (199), Rchaudeau, D., "Contrat d'assurance automoble et rsque router : analyse théorque et emprque sur données ndvduelles françases ", thèse de doctorat, Unversté de Pars I Pantheon-Sorbonne, 1997, 331 pages. Rothschld, M. and J. Stgltz, "Equlbrum n Compettve Insurance Markets: An Essay on the Economcs of Imperfect Informaton", Q.J.E. 90 (November 1976): Vlleneuve, B., "Essas en économe de l'assurance", Ph.D. thess, EHESS, 69 pages (1996). 9