Probability Weighting in Damage-Claiming Decisions

Transcription

1 Probabiliy Weighing in Damage-Claiming Decisions Yoram Eden and Doron Sonsino Absrac: We presen experimenal evidence suggesing ha insurance-holders ignore he possibiliy of damage-recurrence when deciding wheher o submi a claim for a curren small loss. The neglec resuls in successive claiming for curren small damage-levels. When he probabiliy of damage-recurrence is disclosed, subjecs increase he cuoff-damage for submiing a claim; he recurrence probabiliy is significanly overweighed in accordance wih prospec heory s principles. Our resuls sugges ha regulaory agencies and insurance companies should consider disclosure of saisical informaion on damage-recurrence o insurees. Keywords: claim decisions; damage recurrence; prospec heory; probabiliy weighing JEL classificaion: D8 G22 Graduae school of Business Adminisraion; College of Managemen; 7 Rabin Blvd; P.O.B. 907; Rishon Lezion 7590; Israel. s: yeden@colman.ac.il; sonsinod@colman.ac.il. Fax: We hank for he research auhoriy a he college of managemen for financial suppor. A draf of he paper was presened a he Wesern Risk and Insurance Associaion Meeings in Las Vegas 2004; he paper was presened a he World Risk and Insurance Economics Congress in Sal Lake Ciy We hank Ido Kallir, Reuven Horesh, Ernan Haruvy and conference paricipans for commens and suggesions. We also hank 2 anonymous referees and he ediors for consrucive suggesions.

2 . Inroducion Car-insurance raes in many counries follow a bonus-malus raing sysem where he annual premium raes depend on he number of claims made by he driver over several years. Drivers ha did no file a claim enjoy special "good-driver discouns", while drivers ha claimed damage once or more pay considerably higher raes. Empirical sudies (see Dionne and Ghali, 2005 and he references herein) sugges ha he bonus-malus sysem may effecively alleviae moral hazard and induce cauious driving by policy holders. The experimen discussed in his noe refers o he Israeli sysem where he annual premium rae depends on he number of claims made over an inerval of years. The raes paid by drivers wih one damage-claim are abou 50% higher han hose paid by drivers ha enjoy he no-claims discoun. The raes paid by drivers wih 2 damage-claims are more han wice larger han he minimal raes. The seep increase in premium-levels wih he number of preceding claims makes he decision wheher o file a claim for curren small damage, non-rivial. 2 On one hand, drivers would wan o file a claim in order o save he curren repair coss (above he deducible). On he oher hand, drivers should ake ino consideraion he increased premium schedule hey would face following he claim. In paricular, drivers should be aware of he fac ha claiming he insurance company for a curren small damage may double heir annual insurance raes in case a second damage occurs subsequenly. The formal lieraure in economics and insurance sudies he claiming-policy problem from he perspecive of raional decision under condiions of complee informaion. De Pril (979), Venezia and Levy (980), Venezia (984), Dellaer e al (99), in paricular, formulae he opimal claiming decision as a sochasic dynamic programming problem and characerize he minimal damage-level for which a raional driver should file a claim. These raional characerizaions however build on specific assumpions on he disribuion (and condiional disribuion) of damage and require Specifically, Dionne and Ghali (2005) demonsrae ha he inroducion of a bonus-malus sysem in Tunisia has significanly decreased he frequency of repored accidens among good risks bu did no significanly affec he frequency of claiming for bad risks (policy holders were classified good risks when he insurance period covered more han 5 years). 2 In he US drivers are obliged by law o repor all accidens where he propery damage exceeds some minimal amoun; in Israel and many counries in Europe reporing is no mandaory in accidens wih no injuries.

3 soluion of complicaed mahemaical programming problems. In realiy, insuranceholders do no hold objecive informaion on he disribuion of fuure-damage. The experimenal economics lieraure moreover has long acknowledged ha even when such informaion is provided, subjecs consisenly deviae from he opimal plan (for few examples see, Hey and Dardanoni (988), Anderhub e al (2000) and he more recen discussion by Carbone and Hey (200)). In his noe we examine he acual claiming-consideraions of insurance holders in minimal-informaion condiions where policy-holders do no hold informaion on he disribuion of damage bu are aware of he bonus-malus premium schedule and may also respond o saisics published by insurance companies or he media. For his, we run an MBA survey-experimen where subjecs are asked o disclose he minimal damage level for which hey would file a claim in wo differen sylized scenarios: car insurance scenario and denal insurance case. We focus on he aspec of recurren-damage consideraion and check wheher disclosure of informaion on he probabiliy of recurrence affecs subjecs claiming-decisions in each case. The resuls of he experimen reveal ha subjecs end o ignore he possibiliy of damage-recurrence unless i is explicily disclosed in he insrucions. Our subjecs chose o file claims for minor damage-levels when he possibiliy of damage recurrence was no disclosed. When he probabiliy of recurrence was revealed (in par b of he survey), he subjecs have significanly increased he cuoff damage-level for submiing a claim, hereby choosing no o file a claim for such small damagelevels. Esimaion of he experimenal daa reveals ha he probabiliy of damagerecurrence is significanly overweighed in accordance wih prospec heory s principles. More han 2/ of he paricipans in he experimen held privae comprehensive car insurance policies a he ime of he survey (ohers drove a company s vehicle). The daa for he car-insurance scenario was based on marke premium-raes and empirical recurrence saisics. Our resuls hus sugges ha neglec of damage recurrence may characerize claiming decisions in realiy. Insurance markers may herefore benefi from regulaory disclosure of saisical informaion on he possibiliy of recurrence. 2

4 2. The Survey The survey was run on 74 MBA sudens a he College of Managemen, Israel. The average age of he paricipans was abou. Abou 66% of he subjecs held a privae comprehensive car insurance policy a he ime of he survey (mos of he ohers drove a company s vehicle ha was insured by he firm). More han 90% of he paricipans expressed familiariy wih he erm "deducible" in he insurance-conex; abou 80% marked familiariy wih he erm "no-claims discoun". The survey was divided ino wo pars (a) and (b). Each par referred o wo differen scenarios: a car-insurance case and a denal-insurance case. In par (a) subjecs were asked o assume ha an insurance-even has jus occurred and hey are waiing for a professional assessmen of he cos of damage. The insrucions described a simple bonus-malus premium sysem where policy-holders ha did no claim damage over years pay he lowes premium; hose ha claimed damage once over he las years pay a medium premium and hose ha claimed damage wice or more pay he highes premium. Subjecs were furher asked o assume ha hey did no claim he insurance company for years and have paid he corresponding (minimal) insurance-premium "yeserday" (a day before he insuranceeven). The quesionnaire also described he deducible sysem: a fixed claimindependen deducible of 200 NIS in he car-insurance case; a 20% deducible in he denal-insurance case. The basic radeoffs beween claiming and no-claiming for small damage-levels were briefly explained o he subjecs (see he ranslaed quesionnaire in he Appendix). Subjecs were asked o sae he minimal damageamoun for which hey would file a claim. Subjecs were also asked if hey would be ineresed in addiional daa in order o deermine heir cuoff damage-levels. Those ha have answered posiively were requesed o describe he addiional daa hey wish o examine. The specific daa presened o he subjecs in par (a) is summarized a he firs 4 rows of Table 2. The daa for he car-insurance case was based on he acual raes of a Premiums and damages were denominaed in New Israeli Shekel; he exchange rae a he ime of he experimen was abou 4.5 NIS for US dollar.

5 leading car-insurance company in Israel. 4 The daa for he denal-insurance case was hypoheical. 5 Table 2: Car-insurance and Denal-insurance Daa Car-insurance daa Denal-insurance daa Lowes premium Medium premium Highes premium Deducible 200 (fixed) 20% of damage Recurrence-probabiliy 25% 0% Expeced-damage in recurrence In par (b) of he survey (ha was disribued afer collecing par a), sudens received addiional informaion on he condiional probabiliy of damage-recurrence. The probabilisic informaion was ascribed o an independen exper and subjecs were asked o assume i is reliable and applies o heir individual case. In he car-insurance quesionnaire subjecs were old ha "The fac ha your vehicle was engaged in an acciden implies ha here is a 25% chance ha you would be engaged in anoher acciden during his year. The average damage-amoun in such recurren accidens is 5000" (see he boom lines on Table 2). In he denal-insurance case, subjecs were similarly old ha "Your denal condiions indicae ha here is a 0% chance ha you will need an addiional reamen (following he curren reamen) wihin he nex year. The cos of a second reamen is NIS on average". The basic daa on premiums and deducibles ha was provided in par (a) was represened again in small leers. Subjecs were asked if he addiional informaion changes he minimal 4 Car-insurance in Israel is divided o "compulsory" and "comprehensive". Compulsory insurancepremiums are fixed by law and inended o cover physical damage in case of accidens. The comprehensive (propery) insurance marke is compeiive. The daa quoed in our car-insurance scenario refers o comprehensive insurance raes for a car valued a 00,000 NIS (he recurrence-daa was colleced from company's officials; he saisics are no publicly available). 5 The marke for denal insurance policies in Israel is hin and diverse. Mos of he populaion eiher pays for privae denal services or uses denal services ha are provided by general healh insurance companies. Policies for specific denal insurance are diverse in erms and condiions. We have herefore buil a simple hypoheical policy for he experimen. 4

6 damage-level for which hey would file a claim and asked o wrie down heir updaed hreshold damage-level for claim submission (see Appendix).. Resuls The average Cuoff Damage Level (henceforh: CDL) provided by he subjecs for he car-insurance and he denal-insurance scenarios are provided in Table. In boh cases, he disclosure of informaion on he likelihood of damage-recurrence brough a significan increase in CDL. In he car-insurance case, for example, he average cuoff damage-level for filing a claim wih he insurance company increased by more han 20%, from o A Wilcoxon signed-ranks es (Siegel and Casellan, 988) confirms ha he increase is significan a p=0.00 (see he righ column on he able). An individual-level comparison reveals ha 8 of he 74 subjecs (5.%) have increased heir CDL in response o he addiional informaion; 27 subjecs (6.5%) choose he same CDL in boh pars of he quesionnaire; only 9 subjecs (2.2%) have decreased heir CDL in response o he informaion on damagerecurrence. 6 The increase in CDL is robus in he sense ha i appears for differen subgroups of he N=74 sample. In paricular, he increase appears for he subjecs ha held a privae insurance policy and for hose ha did no hold a privae policy. I also appears in each of he 5 classes from which he subjecs were recruied. Table : Average (median) Cuoff Damage Level Average (median) CDL in par (a) Average (median) CDL in par (b) Wilcoxon saisic (significance) Car-insurance case (N=74) (4650) (5500) z=.85 (p=0.00) Denal-insurance case (N=68) (400) (4450) z=2.629 (p=0.006) The daa for he denal-insurance case reveals similar rends. 7 The average CDL has increased by 2% following he disclosure of 0% recurrence-probabiliy (see he 6 Some of he subjecs ha decreased heir CDL explained ha he addiional informaion drove hem o "earlier" claiming for he firs damage since repeaed-claiming is 25%-likely. These subjecs seem o ignore he effec of repeaed claiming on annual premiums. 7 The sample size for he denal insurance case is 68 since 6 subjecs did no complee boh pars of he quesionnaire for his scenario. The car-insurance saisics for hese 68 subjecs are similar o hose repored on Table. 5

7 daa in Table ). Individual-level examinaion suggess ha 28 of he 68 subjecs (4.2%) increased heir CDL in par (b) of he survey; only 6 subjecs (8.8%) decreased heir claiming cuoff-levels. The proporion of subjecs ha did no revise heir CDL in response o he recurrence-informaion was larger in he denal-insurance scenario as 4 subjecs (50%) chose no o revise heir cuoff level in his case. Seveneen of hese subjecs admied ha hey did no ake ino consideraion a possible need for recurren-reamens in par (a) bu sill chose no o revise heir CDL in response o he addiional informaion. Some subjecs explained ha hey were "only concerned wih he immediae consequences of heir decisions" or ha hey would "raher ake he risk and ignore small probabiliies of recurrence". The large differences beween he CDL's provided in pars (a) and (b) of he survey sugges ha disclosure of informaion on he likelihood of damage-recurrence may significanly affec small-damage claiming decisions. I is hus ineresing o noe ha less han 20% of he paricipans asked for such addiional informaion in par (a) of he quesionnaire Esimaion We build on Kahneman and Tversky's (979) Prospec Theory o model he decision problems presened o he survey paricipans. The 5 specific assumpions of our model follow: (I) Subjecs do no ake ino accoun he possibiliy of damagerecurrence unless hey receive explici informaion on his possibiliy. 9 (II) The disuiliy from a loss (paymen) of X is independen of individual wealh-levels and represened by he CRRA funcion, L ( X ) = X. (III) Subjecs weigh he probabiliies of damage-recurrence provided in par (b) of he quesionnaire; w ( p) denoes he subjecive weigh of probabiliy p. (IV) In considering he possibiliy of damage-recurrence, subjecs pre-assume ha hey would claim he insurance company 8 Only 2 of 74 subjecs (6.2%) asked for informaion on he possibiliy of recurren-claiming in par (a) of he car insurance quesionnaire; while 4 of 68 subjecs (20.6%) asked for such informaion in par (a) of he denal-insurance quesionnaire. 9 The assumpion builds on he fac ha less han 20% of subjecs asked for addiional-informaion in par (a) of he survey. Noe however ha esimaion of generalized specificaions ha allow for subjecive consideraion of damage-recurrence before he disclosure of recurrence-probabiliy did no produce significan coefficiens (see discussion a he end of secion 4) 6

8 for he second damage. 0 (V) Subjecs discoun fuure paymens a a consan annual discoun rae δ. These assumpions are used o derive 4 equaions ha characerize he 4 CDL's colleced in he survey. Consider firs he car-insurance daa described in par (a). Le C > 200 denoe he acual amoun of damage. If he subjec does no claim he insurance company for he damage, hen she mus pay C now bu her annual premium would be a he lowes level, 000, in each of he nex years. If, on he oher hand, he subjec claims he insurance company for he damage, she would only pay 200 (he deducible) now bu her annual premium would rise o 4500 in he nex years. The cuoff-damage-level for his case would be he damage-amoun where he subjec is indifferen beween he wo alernaives. Using C o denoe he corresponding CDL we ge he equaion: () ( C) δ 000 = 200 = = δ 4500 In par (b) he subjecs learn ha here exiss a 25% probabiliy for damage recurrence a he same year. If he subjec does no claim he insurance company for he firs damage hen he second-acciden would increase he premium-level from he loweslevel (000) o he medium-level (4500). If, on he oher hand, he subjec claims damage for he firs acciden, hen he second even would increase he premium level from medium (4500) o high (7000). Using C2 o denoe he CDL for his case, we ge he equaion: 0 This assumpion is jusified by he large figures of expeced second-damage disclosed in par (b): 5000 for he car-insurance scenario; for he denal-insurance case (see Table 2). Equaions (2) and (4) assume ha subjecs cancel he recurren-damage deducible in comparing he claim/no-claim alernaives (see Kahneman and Tversky (979) for a discussion of he cancellaion heurisic). Esimaion of an alernaive formulaion ha akes ino accoun he deducible paid (in boh alernaives) in case of a recurren damage gave similar resuls. 7

9 (2) ( C2) 200 = = δ δ (( w(0.25)) 000 w(0.25) 4500 ) (( w(0.25)) 4500 w(0.25) 7000 ) = Similar consideraions give wo equaions for he denal-insurance scenario. In he following equaions we use C o denoe he CDL colleced in par (a) of he survey and C4 for he CDL colleced in par (b). () ( C) δ 600 = (0.2 C) = = δ 2000 (4) ( C4) (0.2 C4) = δ = (( w(0.)) 600 w(0.) 2000 ) δ (( w(0.)) 2000 w(0.) 5000 ) = For given values of C-C4, equaions ()-(4) define a sysem of 4 non-linear equaions in 4 unknowns:, δ, w (0.25) and w (0.). In he firs wo rows of Table 4, we provide he numeric soluions of he sysem for he average and median daa colleced in he experimen. 2 The soluion for is lower han which reflecs riskseeking in he loss-domain (Schoemaker and Kunreuher, 979). The annual discoun rae is abou 0%. The soluion for w (0.25) (abou 0.4) reflecs a considerable overweighing of he 25% loss-recurrence probabiliy. The soluion for w(0.0) on he average daa reflecs an overweighing of he 0% probabiliy o 2.8% weigh. The overweighing of he 0% probabiliy does no, however, appear in he soluion for he median daa (possibly, because of he large number of paricipans ha chose o ignore he 0% recurrence probabiliy in par (b); see he discussion in secion ). 2 The numeric soluions and he SUR esimaions were run on SAS vs. 9. using he MODEL procedure. The resuls repored on Table 4 are robus wih respec o he iniial parameer-values assumed in he esimaion. 8

10 Table 4: Soluion and Esimaion of Equaions Soluion/esimaion mehod δ γ w(0.25) w(0.0) Solving parameers on average daa NA Solving parameers on median daa NA SUR esimaion NA SUR esimaion / TK weighing* NA NA SUR esimaion / Prelec weighing NA NA *TK weighing refers o he Tversky and Kahneman (992) probabiliy weighing funcion. In he hird line of Table 4 we repor he resuls of SUR esimaion of he sysem of equaions ()-(4) on he experimenal daa. Zellner (962) seemingly unrelaed regression mehod is used o allow for correlaion of random errors in equaions ()- (4). The resuls of he esimaion are similar o he numeric soluion for he average daa. All parameers are saisically significan a p<0.00. A he 4-h line of he able, we assume ha probabiliy-weighing akes he commonly used Tversky and γ p Kahneman (992) funcional form: w( p) =. γ γ / γ ( p ( p) ) The esimaed parameer for γ (0.62) is saisically significan a p= (=.87) and similar o he ones obained in many preceding sudies (e.g., Camerer and Ho (994), Wu and Gonzalez (996), Abdellaoui (2000)). An esimaion of he Prelec (998) weighing γ funcion ( w( p) exp( ( ln p) )) = gives similar resuls (see he boom line of he able); he esimae for γ is significan a p=0.069 (=.85). To check he assumpion ha subjecs do no ake ino accoun he possibiliy of damage recurrence when i is no explicily inroduced (assumpion I), we generalized he equaions for par (a) (equaions and ) by adding parameers x and y ha represen he subjecive probabiliy of recurrence in each scenario. The revised form of equaion () for example is: The parameer γ deermines he shape of he probabiliy-weighing funcion; γ= represens he special case where w(p)=p for every probabiliy p. 9

11 ' ( ) ( C) 200 = = δ δ (( x) 000 x 4500 ) (( x) 4500 x 7000 ), where x denoes he subjecive assessmen of same-year recurrence (before he disclosure of 25% recurrence-probabiliy in par b). SUR esimaion of, δ, γ and x (wih TK probabiliy weighing) for he sysem ('), (2)-(4) did no produce a significan coefficien for x; he esimaed coefficiens =, δ and γ were close o hose obained for he consrained version wih x = 0. Similar resuls were obained in esimaion of oher specificaions and in corresponding esimaions for he denal insurance scenario Discussion We run an experimenal survey suggesing ha insurance-consumers neglec he possibiliy of damage recurrence in heir curren claiming decisions. The decision wheher o claim he insurance company for a curren loss is no affeced by he possibiliy ha addiional claiming may double he annual premiums for several years unless he recurrence probabiliy is specifically disclosed. The insurance markeplace is compeiive and esablished. Claiming-decisions however are made by individual policy-holders ha ypically do no posses saisical informaion on he disribuion of firs and recurren damage. Our survey hus provides insigh ino acual claiming decisions in realiy. 5 Our resuls imply ha he incomplee informaion under which claiming decisions are made, may induce inefficien claiming. The resuls hus sugges ha regulaory agencies and insurance companies may be 4 Noe also ha second/hird recurrence would have a similar affec on payoffs independenly of he decision wheher o file or no file a claim for he curren damage (in boh cases premiums would increase o he maximal-level). Thus, "subjecive" consideraion of such repeiions should no affec he cuoff-levels for curren-claiming. The cancellaion of hese secondary affecs is also implied by Kahneman and Tversky's (979) cancellaion heurisic. 5 To encourage subjecs paricipaion we have randomly seleced 0 paricipans ha received a check in he amoun of 00 NIS for heir paricipaion. Since he decisions ha are made in he survey affec payoffs in successive years, devising an incenive scheme where subjecs acually receive a payoff ha depends on heir decision is complicaed. Anderhub e al (200), for example, use differed checks o pospone payoffs o subjecs in a simpler discouning experimen. We do no believe ha such direc incenives would significanly affec our resuls. 0

12 ineresed in disclosure of probabilisic informaion on damage-recurrence o consumers filing a claim. Preceding experimenal research demonsraes ha subjecs violae expeced uiliy heory (Wa e al, 200) and follow principles of prospec heory (Schoemaker and Kunreuher, 979) in heir insurance-decisions. Hogarh and Kunreuher (985) demonsrae ha ambiguiy in probabiliies of risks may lead o large differences beween he minimum premiums ha firms were prepared o charge and he maximal premiums ha consumers were ready o pay. Di-Mauro and Maffiolei (200) show ha uncerainy abou he probabiliy of poenial loss has a weak affec on subjecs valuaion of an insurance conrac. These sudies however deal wih he insurance decision or he valuaion of insurance-conracs; we are no aware of preceding experimenal sudies of he insurance claiming-decision. The resuls colleced in our survey demonsrae ha general principles of prospec heory apply o claimingdecisions on exising insurance-policies. The consisency of our empirical esimaes of sandard probabiliy weighing funcions wih he esimaes obained in many oher sudies seems noeworhy. If probabiliy-weighing akes similar forms in various applicaions, hen he common curvaure may be used for opimal loery design (Quiggin, 99), opimal insurance-conracing (Ryan and Vaihianahan, 200) and oher applicaions. 6 In paricular, i seems challenging o examine he impac of probabiliy-weighing on he radeoff beween deducibles and premiums and on he effecive screening of insurance consumers. References Abdellaoui, M. (2000). "Parameer-free Eliciaion of Uiliies and Probabiliy Weighing Funcions," Managemen Science, 46, Anderhub, V. Güh, W., Müller, W. and Srobel, M. (2000). An Experimenal Analysis of Ineremporal Allocaion Behavior, Experimenal Economics, (2), See also he lieraure on he reflecion of probabiliy weighing in being or gambling decision; e.g. Ziegelmeyer e al (2004).

13 Anderhub, V., Gneezy, U., Güh, W. and Sonsino, D. (200). On he Ineracion of Risk and Uncerainy An Experimenal Sudy, German Economic Review, 2, Camerer, C. F. and Ho, T. H. (994). Violaions of he Beweeness Axiom and Nonlineariy in Probabiliy, Journal of Risk and Uncerainy, 8, Carbone, E. and Hey, J. D. (200). A Tes of he Principle of Opimaliy, Theory and Decision, 50(), De Pril, N. (979). Opimal Claim Decisions for a Bonus-Malus Sysem: A Coninuous Approach, The Asin Bullein, 0, Dellaer, N. P., Frenk, J. B. G. and van Rijsoor, L. P. (99). Opimal Claim Behavior for Vehicle Damage Insurance, Insurance: Mahemaics and Economics, 2(), Di-Mauro, C. and Maffiolei, A. (200). The Valuaion of Insurance under Uncerainy: Does Informaion abou Probabiliy Maer?, Geneva Papers on Risk and Insurance Theory, 26(), Dionne, G. and Ghali, O. (2005). The (992) Bonus-Malus Sysem in Tunisia: An Empirical Evaluaion. Journal of Risk and Insurance, 72(4), Hey, J. D. and Dardanoni, V. (988). A Large Scale Experimenal Invesigaion ino Opimal Consumpion under Uncerainy, The Economic Journal, 98(2), 05-6 Hogarh, R. M. and Kunreuher, H. C. (985). "Ambiguiy and Insurance decisions," American Economic Review, 75, Kahneman, D. and Tversky, A. (979). "Prospec Theory: an Analysis of Decision under Risk," Economerica, 47, 6-9. Prelec, D. (998). "The Probabiliy Weighing Funcion," Economerica, 66,

14 Quiggin, J. (99). On he Opimal Design of Loeries, Economica, 58, -6. Ryan, M. J. and Vaihianahan, R. (200). Medical Insurance wih Rank Dependen Uiliy, Economic Theory, 22, Schoemaker, P. J. H. and Kunreuher, H. C. (979). "An Experimenal Sudy of Insurance Decisions," Journal of Risk and Insurance, 46, Siegel, S. and Casellan, N. J. (988). Non Parameric Saisics for he Behavioral Sciences. McGraw-Hill Inernaional Ediions. Tversky and Kahneman (992). Advances in Prospec Theory: Cumulaive Represenaion of Uncerainy, Journal of Risk and Uncerainy, 5, Venezia, I. and Levy, H. (980). "Opimal Claims in Auomobile Insurance," Review of Economic Sudies, 47, Venezia, I (984). "Aspecs of Opimal Auomobile Insurance," Journal of Risk and Insurance, 5, Wa R., Vazquez F. J. and Moreno, I. (200). "An Experimen on Raional Insurance Decisions", Theory and Decision, 5, Wu, G. and Gonzalez, R. (996). "Curvaure of he Probabiliy Weighing Funcion," Managemen Science, 42, Zellner, A. (962). "An Efficien Mehod for Esimaing Seemingly Unrelaed Regressions and Tess for Aggregaion Bias," Journal of he American Saisical Associaion, 57, Ziegelmeyer, A., Broihanne, M. H. and Koessler, F. (2004). Sequenial Parimuuel Being in he Laboraory, Journal of Risk and Uncerainy,

15 Appendix: Car-Insurance Quesionnaire 7 Par (a) Assume he following daa on your car-insurance policy: -The annual premium is paid cash a he beginning of each year -The amoun of annual premium depends on he number of claims made over he las years: *If you did no claim, you pay 000 NIS *If you claimed once, you pay 4500 NIS *If you claimed wice or more, you pay 7000 NIS - There is a fixed deducible of 200 NIS for each claim (e.g., if you claim he insurance company for a damage of 0000 NIS you would only receive 8800 NIS) Addiional Assumpions: *Your vehicle was involved in an acciden his morning *You are currenly waiing for a professional damage-esimae *Your annual premium was paid yeserday *Since you did no claim damage over he las years, you paid 000 NIS *You plan o renew he curren policy in each of he nex years * No changes are anicipaed in he policy-erms Remember ha you are currenly waiing for he professional damage-esimae. Afer receiving he esimae you would have o decide wheher o claim he insurance company. We now ask you o sae he minimal damage amoun for which you would claim he insurance company. Noe: If he damage is lower han he deducible, here's clearly no poin o claim damage. Moreover, when claiming damage, you increase he insurance-premiums for he nex years. Thus, you should no claim-damage when he amoun of damage is "relaively low". However, you should obviously claim damage when he level of damage is "high enough". Final commen: Your response depends on your personal preferences. Please give us your independen answer wihou consuling your colleagues. The minimal damage for which I will claim he insurance company is: Would you wish o receive addiional informaion before deermining he minimal claim-level? YES/NO (if your answer is posiive please ouline he addiional informaion ha you would ask for) Par (b) Recall he car-insurance daa provided in par (a) (The daa was represened in smaller leers) 7 The quesionnaire for he denal insurance case was similarly formulaed; i is available a hp://www2.colman.ac.il/business/doron 4

16 Addiional Informaion We now wish o draw your aenion o addiional informaion ha was provided by an independen appraiser. Here is he addiional informaion as provided by he appraiser: "The fac ha your vehicle was involved in an acciden his morning suggess ha here is a 25% chance ha i will be involved in anoher acciden his year. Tha is, of each 4 vehicles (similar o he one a your possession) ha were involved in an acciden, would be involved in anoher acciden a he same year. The average damage-level in such recurren accidens is 5000 NIS" *Assume ha he informaion described above is reliable and applies o your case *Remember ha you are waiing for a professional esimae of he damage made in he curren acciden. Afer receiving he esimae you will have o decide wheher o claim damage from he insurance company. Does he "addiional informaion" provided above change your decision abou he minimal damage level for which you will claim he insurance company? YES/NO Given he addiional informaion above, wha is he minimal damage level for which you will claim he insurance company? Did you ake ino consideraion he possibiliy of addiional accidens (ha may increase your annual premium o 7000 if you claim damage wice) in answering he parallel quesion in par (a)? YES/NO 5