Estimating Risk Preferences from Deductible Choicepublic.econ.duke.edu/.../socialinsurance/readings/Cohen_Einav07(2.1… · Estimating Risk Preferences from Deductible Choice By A

contrast the choice among different alternativesthat vary only in their financial parameters (thelevels of deductibles and premiums) is a case inwhich the effect of risk aversion can be moreplausibly isolated and estimated

The average deductible menu in our dataoffers an individual to pay an additional pre-mium of $55 (US) in order to save $182 indeductible payments in the event of a claim2 Arisk-neutral individual should choose a low de-ductible if and only if his claim propensity isgreater than the ratio between the premium($55) and the potential saving ($182) which is03 Although this pricing is actuarially unfairwith respect to the average claim rate of 024518 percent of the sample choose to purchase itAre these individuals exposed to greater riskthan the average individual are they more riskaverse or are they a combination of both Weanswer this question by developing a structuraleconometric model and estimating the joint dis-tribution of risk and risk aversion

Our benchmark specification uses expectedutility theory to model individualsrsquo deductiblechoices as a function of two utility parametersthe coefficient of absolute risk aversion and aclaim rate We allow both utility parameters todepend on individualsrsquo observable and unob-servable characteristics and assume that there isno moral hazard Two key assumptionsmdashthatclaims are generated by a Poisson process at theindividual level and that individuals have per-fect information about their Poisson claimratesmdashallow us to use data on (ex post) realizedclaims to estimate the distribution of (ex ante)claim rates Variation in the deductible menusacross individuals and their choices from thesemenus are then used to estimate the distributionof risk aversion in the sample and the correla-tion between risk aversion and claim risk Thuswe can estimate heterogeneous risk preferencesfrom deductible choices accounting for adverseselection (unobserved heterogeneity in claimrisk) which is an important confoundingfactor3

Our results suggest that heterogeneity inrisk preferences is rather large While themajority of the individuals are estimated to beclose to risk neutral with respect to lotteriesof $100 magnitude a significant fraction ofthe individuals in our sample exhibit signifi-cant levels of risk aversion even with respectto such relatively small bets Overall an in-dividual with the average risk aversion pa-rameter in our sample is indifferent aboutparticipating in a 50 ndash50 lottery in which hegains $100 or loses $56 We find that womenare more risk averse than men that risk pref-erences exhibit a U-shape with respect to ageand interestingly that most proxies for incomeand wealth are positively associated with abso-lute risk aversion

We perform an array of tests to verify thatthese qualitative results are robust to deviationfrom the modeling assumptions In particularwe explore alternative distributional assump-tions alternative restrictions on the von Neu-mannndashMorgenstern (vNM) utility function anda case in which individuals are allowed to makeldquomistakesrdquo in their coverage choices due to in-complete information about their own risk typesWe also show that the risk preferences we es-timate are stable over time and help predictother (but closely related) insurance decisionsFinally we justify our assumption to abstractfrom moral hazard and we discuss the way thisand other features of the setup (sample selectionand additional costs associated with an acci-dent) may affect the interpretation of the result

Throughout we focus primarily on absolute(rather than relative) risk aversion4 This allowsus to take a neutral position with respect to therecent debate over the empirical relevance ofexpected utility theory (Matthew Rabin 2000Rabin and Richard H Thaler 2001 Ariel Ru-binstein 2001 Richard Watt 2002 NicholasBarberis Ming Huang and Thaler 2006 Igna-

2 For ease of comparison we convert many of the re-ported figures from New Israeli Shekels to US dollars It isimportant to keep in mind however that GDP per capita inIsrael was 052ndash056 of that in the United States (067ndash070 when adjusted for PPP) during the observation period

3 Throughout the paper we use the term adverse selec-tion to denote selection on risk while selection on risk

aversion is just selection Some of the literature refers toboth selection mechanisms as adverse selection with thedistinction being between common values (selection on risk)and private values (selection on risk aversion)

4 Primarily as a way to compare our results with otherestimates in the literature Section III also provides esti-mates of relative risk aversion by following the literatureand multiplying our estimates for absolute risk aversion byannual income (in Israel) We obtain high double-digitestimates for the mean individual but below 05 for themedian

746 THE AMERICAN ECONOMIC REVIEW JUNE 2007

cio Palacios-Huerta and Roberto Serrano 2006)While the debate focuses on how the curvatureof the vNM utility function varies with wealthor across different settings we measure thiscurvature only at a particular wealth levelwhatever this wealth level may be By allowingunobserved heterogeneity in this curvatureacross individuals we place no conceptual re-strictions on the relationship between wealthand risk aversion Our estimated distribution ofrisk preferences can be thought of as a convo-lution of the distribution of (relevant) wealthand risk attitudes Avoiding this debate is also adrawback Without taking a stand on the wayabsolute risk preferences vary with sizes andcontexts we cannot discuss how relevant ourestimates are for other settings Obviously wethink they are But since statements about theirexternal relevance are mainly informed by whatwe think and less by what we do we defer thisdiscussion to the concluding section

Our analysis also provides two results regard-ing the relationship between the distribution ofrisk preferences and that of risk First we findthat unobserved heterogeneity in risk aversionis greater and more important (for profits andpricing) than unobserved heterogeneity in riskThis is consistent with the motivation for recenttheoretical work which emphasizes the impor-tance of allowing for preference heterogeneityin analyzing insurance markets (Michael Lands-berger and Isaac Meilijson 1999 Michael Smart2000 David de Meza and David C Webb 2001Bertrand Villeneuve 2003 Pierre-Andre Chiap-pori et al 2006 Bruno Jullien Bernard Salanieand Francois Salanie 2007) Second we findthat unobserved risk has a strong positive cor-relation with unobserved risk aversion and dis-cuss possible interpretations of it This isencouraging from a theoretical standpoint as itretains a single crossing property which weillustrate in the counterfactual exercise Thisfinding contrasts the negative correlation re-ported by Amy Finkelstein and Kathleen Mc-Garry (2006) for the long-term care insurancemarket and by Mark Israel (2005) for automo-bile insurance in Illinois We view these differ-ent results as cautioning against interpreting thiscorrelation parameter outside the context inwhich it is estimated Even if risk preferencesare stable across contexts risk is not and there-fore neither is the correlation structure

This study is related to two important strands

of literature The first shares our main goal ofmeasuring risk aversion Much of the existingevidence about risk preferences is based onintrospection laboratory experiments (Steven JKachelmeier and Mohamed Shehata 1992 Ver-non L Smith and James M Walker 1993Charles A Holt and Susan K Laury 2002) dataon bettors or television game show participants(Robert Gertner 1993 Andrew Metrick 1995Bruno Jullien and Salanie 2000 Roel MBeetsma and Peter C Schotman 2001 MatildeBombardini and Francesco Trebbi 2005) an-swers given by individuals to hypothetical sur-vey questions (W Kip Viscusi and William NEvans 1990 Evans and Viscusi 1991 Robert BBarsky et al 1997 Bas Donkers BertrandMelenberg and Arthur van Soest 2001 JoopHartog Ada Ferrer-i-Carbonell and NicoleJonker 2002) and estimates that are driven bythe imposed functional form relationship be-tween static risk-taking behavior and intertem-poral substitution5 We are aware of only a fewattempts to recover risk preferences from deci-sions of regular market participants Atanu Saha(1997) looks at firmsrsquo production decisionsand Raj Chetty (2006) recovers risk preferencesfrom labor supply In the context of insuranceCicchetti and Dubin (1994) look at individualsrsquodecisions whether to insure against failure ofinterior telephone wires Compared to theirsetting in our setting events are more frequentand commonly observed stakes are higher thepotential loss (the difference between the de-ductible amounts) is known and the deductiblechoice we analyze is more immune to alter-native preference-based explanations Finallyin a recent working paper Justin Sydnor (2006)uses data on deductible choices in homeownerrsquosinsurance to calibrate a bound for the impliedlevel of risk aversion6 An important differencebetween our paper and these papers is thatthey all rely on a representative individual

5 Much of the finance and macroeconomics literaturegoing back to Irwin Friend and Marshall E Blume (1975)relies on this assumption As noted by Narayana R Kocher-lakota (1996) in a review of this literature the level of staticrisk aversion is still a fairly open question

6 The possibility of using deductibles to make inferencesabout risk aversion was first pointed out by Jacques HDreze (1981) Dreze suggests however relying on theoptimality of the observed contracts (ldquosupply siderdquo infor-mation) while we rely on individualsrsquo choices of deduct-ibles (ldquodemand siderdquo information)

747VOL 97 NO 3 COHEN AND EINAV ESTIMATING RISK PREFERENCES FROM DEDUCTIBLE CHOICE

framework and therefore focus only on thelevel of risk aversion7 In contrast we explicitlymodel observed and unobserved heterogeneityin risk aversion as well as in risk We cantherefore provide results regarding the hetero-geneity in risk preferences and its relationshipwith risk which have potentially important im-plications for welfare and policy A representa-tive individual framework cannot address suchquestions

The second strand of related literature isthe recent empirical literature on adverse se-lection in insurance markets Much of thisliterature addresses the important question ofwhether adverse selection exists in differentmarkets As suggested by the influential workof Chiappori and Salanie (2000) it uses ldquore-duced formrdquo specifications to test whether aftercontrolling for observables accident outcomesand coverage choices are significantly corre-lated (Georges Dionne and Charles Vanasse1992 Robert Puelz and Arthur Snow 1994John Cawley and Tomas Philipson 1999Finkelstein and James Poterba 2004 Finkel-stein and McGarry 2006) Cohen (2005) appliesthis test to our data and finds evidence consis-tent with adverse selection As our main goal isquite different we take a more structural ap-proach By assuming a structure for the adverseselection mechanism we can account for itwhen estimating the distribution of risk prefer-ences While the structure of adverse selectionis assumed its relative importance is not im-posed The structural assumptions allow us toestimate the importance of adverse selectionrelative to the selection induced by unobservedheterogeneity in risk attitudes As we discussin Section IIC this approach is conceptuallysimilar to that of James H Cardon and IgalHendel (2001) who model health insurancechoices and also allow for two dimensions ofunobserved heterogeneity8

The rest of the paper is organized as followsSection I describes the environment the setup

and the data Section II lays out the theoreticalmodel and the related econometric model anddescribes its estimation and identification Sec-tion III describes the results We first provide aset of reduced-form estimates which motivatethe more structural approach We then presentestimates from the benchmark specification aswell as estimates from various extensions androbustness tests We discuss and justify some ofthe modeling assumptions and perform counter-factual analysis as a way to illustrate the impli-cations of the results to profits and pricingSection IV concludes by discussing the rele-vance of the results to other settings

I Data

A Economic Environment and Data Sources

We obtained data from a single insurancecompany that operates in the market for auto-mobile insurance in Israel The data containinformation about all 105800 new policyhold-ers who purchased (annual) policies from thecompany during the first five years of its oper-ation from November 1994 to October 1999Although many of these individuals stayed withthe insurer in subsequent years we focusthrough most of the paper on deductible choicesmade by individuals in their first contract withthe company This allows us to abstract fromthe selection implied by the endogenous choiceof individuals whether to remain with the com-pany or not (Cohen 2003 2005)

The company studied was the first company inthe Israeli auto insurance market that marketedinsurance to customers directly rather thanthrough insurance agents By the end of the stud-ied period the company sold about 7 percent ofthe automobile insurance policies issued in IsraelDirect insurers operate in many countries and ap-pear to have a significant cost advantage (J DavidCummins and Jack L Van Derhei 1979) Thestudied company estimated that selling insurancedirectly results in a cost advantage of roughly 25percent of the administrative costs involved inmarketing and handling policies Despite theircost advantage direct insurers generally have haddifficulty in making inroads beyond a part of themarket because the product does not provide theldquoamenityrdquo of having an agent to work with andturn to (Stephen P DrsquoArcy and Neil A Doherty1990) This aspect of the company clearly makes

7 An exception is Syngjoo Choi et al (2006) who use alaboratory experiment and similar to us find a high degreeof heterogeneity in risk attitudes across individuals

8 In an ongoing project Pierre-Andre Chiappori andBernard Salanie (2006) estimate an equilibrium model ofthe French auto insurance market where their model of thedemand side of the market is conceptually similar to the onewe estimate in this paper

748 THE AMERICAN ECONOMIC REVIEW JUNE 2007

the results of the paper applicable only to thoseconsumers who seriously consider buying directinsurance Section IIID discusses this selection inmore detail

While we focus primarily on the demand sideof the market by modeling the deductible choicethe supply side (pricing) will be relevant for anycounterfactual exercise as well as for understand-ing the viability of the outside option (which wedo not observe and do not model) During the firsttwo years of the companyrsquos operations the pricesit offered were lower by about 20 percent thanthose offered by other conventional insurersThus due to its differentiation and cost advantagethe company had market power with respect toindividuals who were more sensitive to price thanto the disamenity of not having an agent Thismakes monopolistic screening models apply morenaturally than competitive models of insurance(eg Michael Rothschild and Joseph E Stiglitz1976) During the companyrsquos third year of oper-ation (December 1996 to March 1998) it facedmore competitive conditions when the estab-lished companies trying to fight off the new en-trant lowered the premiums for policies withregular deductibles to the levels offered by thecompany In their remaining period included inthe data the established companies raisedtheir premiums back to previous levels leav-ing the company again with a substantialprice advantage9

For each policy our dataset includes all theinsurerrsquos information about the characteristicsof the policyholder demographic characteris-tics vehicle characteristics and details abouthis driving experience The Appendix providesa list of variables and their definitions andTable 1 provides summary statistics In addi-tion our data include the individual-specificmenu of four deductible and premium combi-nations that the individual was offered (see be-low) the individualrsquos choice from this menuand the realization of risks covered by the pol-icy the length of the period over which it was ineffect the number of claims submitted by the

policyholder and the amounts of the submittedclaims10 Finally we use the zip codes of thepolicyholdersrsquo home addresses11 to augment thedata with proxies for individualsrsquo wealth basedon the Israeli 1995 census12

The policies offered by the insurer (as all pol-icies offered in the studied market) are one-periodpolicies with no commitment on the part of eitherthe insurer or the policyholder13 The policy re-sembles the US version of ldquocomprehensiverdquo in-surance It is not mandatory but it is held by alarge fraction of Israeli car owners (above 70percent according to the companyrsquos executives)The policy does not cover death or injuries to thepolicyholder or to third parties which areinsured through a separate mandatory policyInsurance policies for car audio equipmentwindshield replacement car and towing ser-vices are structured and priced separately Cer-tain types of coverage do not carry a deductibleand are therefore not used in the analysis14

Throughout the paper we use and reportmonetary amounts in current (nominal) NewIsraeli Shekels (NIS) to avoid creating artificialvariation in the data Consequently the follow-ing facts may be useful for interpretation andcomparison with other papers in the literature

9 During this last period two other companies offeringinsurance directly were established Due to first-mover advan-tage (as viewed by the companyrsquos management) which helpedthe company maintain a strong position in the market thesetwo new companies did not affect pricing policies much untilthe end of our observation period Right in the end of thisperiod the studied company acquired one of these entrants

10 Throughout the analysis we make the assumption thatthe main policyholder is the individual who makes the deduct-ible choice Clearly to the extent that this is not always thecase the results should be interpreted accordingly

11 The company has the addresses on record for billingpurposes Although in principle the company could haveused these data for pricing they do not do so

12 The Israeli Central Bureau of Statistics (CBS) associateseach census respondent with a unique ldquostatistical areardquo eachincluding between 1000 and 10000 residents We matchedthese census tracts with zip codes based on street addressesand constructed variables at the zip code level These con-structed variables are available for more than 80 percent of thepolicyholders As a proxy for wealth we use (gross) monthlyincome which is based on self-reported income by censusrespondents augmented (by the CBS) with Social Security data

13 There is a substantial literature that studies the optimaldesign of policies that commit customers to a multiperiodcontract or that include a one-sided commitment of theinsurer to offer the policyholder certain terms in subsequentperiods (Georges Dionne and Pierre Lasserre 1985 RussellCooper and Beth Hayes 1987 Georges Dionne and Neil ADoherty 1994 Igal Hendel and Alessandro Lizzeri 2003)Although such policies are observed in certain countries(Dionne and Vanasse 1992) many insurance markets includ-ing the one we study use only one-period no-commitmentpolicies (Howard Kunreuther and Mark V Pauly 1985)

14 These include auto theft total loss accidents and notldquoat faultrdquo accidents

749VOL 97 NO 3 COHEN AND EINAV ESTIMATING RISK PREFERENCES FROM DEDUCTIBLE CHOICE

The exchange rate between NIS and US dollarsmonotonically increased from 301 in 1995 to414 in 1999 (on average it was 352)15 Annualinflation was about 8 percent on average andcumulative inflation over the observation periodwas 48 percent We will account for these effects

as well as other general trends by using yeardummy variables throughout the analysis

B The Menu of Deductibles and Premiums

Let xi be the vector of characteristics individuali reports to the insurance company After learningxi the insurer offered individual i a menu of fourcontract choices One option offered a ldquoregularrdquodeductible and a ldquoregularrdquo premium The term

15 PPP figures were about 10 percent lower than the nom-inal exchange rates running from 260 in 1995 to 374 in 1999

TABLE 1mdashSUMMARY STATISTICSmdashCOVARIATES

Variable Mean Std dev Min Max

Demographics Age 41137 1237 1806 8943Female 0316 047 0 1Family Single 0143 035 0 1

Married 0779 042 0 1Divorced 0057 023 0 1Widower 0020 014 0 1Refused to say 0001 004 0 1

Education Elementary 0016 012 0 1High school 0230 042 0 1Technical 0053 022 0 1College 0233 042 0 1No response 0468 050 0 1

Emigrant 0335 047 0 1

Car attributes Value (current NIS)a 66958 37377 4000 617000Car age 3952 287 0 14Commercial car 0083 028 0 1Engine size (cc) 1568 385 700 5000

Driving License years 18178 1007 0 63Good driver 0548 050 0 1Any driver 0743 044 0 1Secondary car 0151 036 0 1Business use 0082 027 0 1Estimated mileage (km)b 14031 5891 1000 32200History length 2847 061 0 3Claims history 0060 015 0 2

Young driver Young 0192 039 0 1Gender Male 0113 032 0 1

Female 0079 027 0 1Age 17ndash19 0029 017 0 1

19ndash21 0051 022 0 121ndash24 0089 029 0 124 0022 015 0 1

Experience 1 0042 020 0 11ndash3 0071 026 0 13 0079 027 0 1

Company year First year 0207 041 0 1Second year 0225 042 0 1Third year 0194 040 0 1Fourth year 0178 038 0 1Fifth year 0195 040 0 1

Note The table is based on all 105800 new customers in the dataa The average exchange rate throughout the sample period was approximately $1 per 35 NIS starting at 13 in late 1994

and reaching 14 in late 1999b The estimated mileage is not reported by everyone It is available for only 60422 new customers

750 THE AMERICAN ECONOMIC REVIEW JUNE 2007

regular was used for this deductible level bothbecause it was relatively similar to the deductiblelevels offered by other insurers and because mostpolicyholders chose it The regular premium var-ied across individuals according to some deter-ministic function (unknown to us) pit ft(xi)which was quite stable over time The regulardeductible level was directly linked to the regularpremium according to

(1) dit min12

pit capt

That is it was set to one-half of the regular pre-mium subject to a deductible cap capt whichvaried over time but not across individuals Thepremiums associated with the other options werecomputed by multiplying pit by three differentconstants 106 for ldquolowrdquo deductible 0875 forldquohighrdquo deductible and 08 for ldquovery highrdquo de-ductible The regular deductible dit was con-verted to the other three deductible levels in asimilar way using multipliers of 06 for low 18for high and 26 for very high

There are two main sources of exogenous vari-ation in prices The first arises from companyexperimentation The multipliers described abovewere fixed across individuals and over time formost of the observation period but there was asix-month period during the insurerrsquos first year ofoperation (May 1995 to October 1995) in whichthe insurer experimented with slightly modifiedmultipliers16 This modified formula covers al-most 10 percent of the sample The secondsource of variation arises from discrete ad-justments to the uniform cap The cap variedover time due to inflation competitive condi-tions and as the company gained more expe-rience (Figure 1) The cap was binding for

16 For individuals with low levels of regular premiumsduring the specified period the regular deductible was set at53 percent (instead of 50 percent) of the regular premiumthe low deductible was set at 33 percent (instead of 30percent) of the regular premium and so on

FIGURE 1 VARIATION IN THE DEDUCTIBLE CAP OVER TIME

Notes This figure presents the variation in the deductible cap over time which is one of the main sources of pricing variation inthe data We do not observe the cap directly but it can be calculated from the observed menus The graph plots the maximalregular deductible offered to anyone who bought insurance from the company over a moving seven-day window Thelarge jumps in the graph reflect changes in the deductible cap There are three reasons why the graph is not perfectlysmooth First in a few holiday periods (eg October 1995) there are not enough sales within a seven-day window sonone of those sales hits the cap This gives rise to temporary jumps downward Second the pricing rule applies at thedate of the price quote given to the potential customer Our recorded date is the first date the policy becomes effectiveThe price quote is held for a period of two to four weeks so in periods in which the pricing changes we may still seenew policies sold using earlier quotes made according to a previous pricing regime Finally even within periods of constantcap the maximal deductible varies slightly (variation of less than 05 percent) We do not know the source of this variation

751VOL 97 NO 3 COHEN AND EINAV ESTIMATING RISK PREFERENCES FROM DEDUCTIBLE CHOICE

about a third of the policyholders in our dataAll these individuals would be affected by achange in the cap Much of the variation ofmenus in the data is driven by the exogenousshifts in the uniform deductible cap The un-derlying assumption is that conditional onobservables these sources of variation pri-marily affect the deductible choice of newcustomers but they do not have a significantimpact on the probability of purchasing insurancefrom the company Indeed this assumption holdsin the data with respect to observables there is nodistinguishable difference in the distribution ofobservable characteristics of consumers who buyinsurance just before and those who buy just aftera change in the deductible cap

C Summary Statistics

The top of Table 2A summarizes the deduct-ible menus all are calculated according to theformula described earlier Only 1 percent of thepolicyholders chose the high or very high de-ductible options Therefore for the rest of theanalysis we focus only on the choice betweenregular and low deductible options (chosen by811 and 178 percent of the individuals respec-tively)17 Focusing only on these options doesnot create any selection bias because we do notomit individuals who chose high or very highdeductibles For these individuals we assumethat they chose a regular deductible This as-sumption is consistent with the structural modelwe develop in the next section which impliesthat conditional on choosing high or very highdeductibles an individual would almost alwaysprefer the regular over the low deductible

The bottom of Table 2A as well as Table2B present summary statistics for the policy re-alizations We focus only on claim rates and noton the amounts of the claims This is because anyamount above the higher deductible level is cov-

ered irrespective of the deductible choice and thevast majority of the claims fit in this category (seeSection IIIE) For all these claims the gain fromchoosing a low deductible is the same in the eventof a claim and is equal to the difference betweenthe two deductible levels Therefore the claimamount is rarely relevant for the deductible choice(and likewise for the companyrsquos pricing decisionwe analyze in Section IIIF)

Averaging over all individuals the annualclaim rate was 0245 One can clearly observesome initial evidence of adverse selection Onaverage individuals who chose a low deductiblehad higher claim rates (0309) than those whochose the regular deductible (0232) Those whochose high and very high deductibles had muchlower claim rates (0128 and 0133 respectively)These figures can be interpreted in the context ofthe pricing formula A risk-neutral individual willchoose the low deductible if and only if her claimrate is higher than (pd) (plow pregular)(dregular dlow) When the deductible cap is notbinding which is the case for about two-thirds ofthe sample this ratio is given directly by thepricing formula and is equal to 03 Thus anyindividual with a claim rate higher than 03 willbenefit from buying the additional coverage pro-vided by a low deductible even without any riskaversion The claim data suggest that the offeredmenu is cheaper than an actuarially fair contractfor a nonnegligible part of the population (13percent according to the benchmark estimates re-ported below) This observation is in sharp con-trast to other types of insurance contracts such asappliance warranties which are much more ex-pensive than the actuarially fair price (Rabin andThaler 2001)

II The Empirical Model

A A Model of Deductible Choice

Let wi be individual irsquos wealth (pih di

h) theinsurance contract (premium and deductible re-spectively) with high deductible (pi

l dil) the

insurance contract with low deductible ti theduration of the policy and ui(w) individual irsquosvNM utility function We assume that the num-ber of insurance claims is drawn from a Poissondistribution with an annual claim rate iThrough most of the paper we assume that i isknown to the individual We also assume that iis independent of the deductible choice ie that

17 The small frequency of ldquohighrdquo and ldquovery highrdquochoices provides important information about the lowerends of the risk and risk aversion distributions but (for thatsame reason) makes the analysis sensitive to functionalform Considering these options or the option of not buyinginsurance creates a sharp lower bound on risk aversion forthe majority of the observations making the estimates muchhigher Given that these options are rarely selected how-ever it is not clear to us whether they were regularlymentioned during the insurance sales process renderingtheir use somewhat inappropriate

752 THE AMERICAN ECONOMIC REVIEW JUNE 2007

there is no moral hazard Finally we assumethat in the event of an accident the value of theclaim is greater than di

h We revisit all theseassumptions in Sections IIID and IIIE For therest of this section i subscripts are suppressedfor convenience

In the market we study insurance policies are

typically held for a full year after which theycan be automatically renewed with no commit-ment by either the company or the individualMoreover all auto-insurance policies sold inIsrael can be canceled without prior notice bythe policyholder with premium payments beinglinearly prorated Both the premium and the

TABLE 2AmdashSUMMARY STATISTICSmdashMENUS CHOICES AND OUTCOMES

Variable Obs Mean Std dev Min Max

Menu Deductible (current NIS)a Low 105800 87548 12101 37492 103911Regular 105800 145299 19779 62486 171543High 105800 260802 35291 112475 308778Very high 105800 376305 50853 162464 446013

Premium (current NIS)a Low 105800 338057 91404 132471 1923962Regular 105800 318922 8623 124972 1815058High 105800 279057 75451 109351 1588176Very high 105800 255137 68984 99978 1452046

pd 105800 0328 006 03 18

Realization Choice Low 105800 0178 038 0 1Regular 105800 0811 039 0 1High 105800 0006 008 0 1Very high 105800 0005 007 0 1

Policy termination Active 105800 0150 036 0 1Canceled 105800 0143 035 0 1Expired 105800 0707 046 0 1

Policy duration (years) 105800 0848 028 0005 108Claims All 105800 0208 048 0 5

Low 18799 0280 055 0 5Regular 85840 0194 046 0 5High 654 0109 034 0 3Very high 507 0107 032 0 2

Claims per yearb All 105800 0245 066 0 19882Low 18799 0309 066 0 9264Regular 85840 0232 066 0 19882High 654 0128 062 0 12636Very high 507 0133 050 0 3326

a The average exchange rate throughout the sample period was approximately $1 per 35 NIS starting at 13 in late 1994and reaching 14 in late 1999

b The mean and standard deviation of the claims per year are weighted by the observed policy duration to adjust forvariation in the exposure period These are the maximum likelihood estimates of a simple Poisson model with no covariates

TABLE 2BmdashSUMMARY STATISTICSmdashCONTRACT CHOICES AND REALIZATIONS

Claims Low Regular High Very high Total Share

0 11929 (0193) 49281 (0796) 412 (0007) 299 (0005) 61921 (100) 080341 3124 (0239) 9867 (0755) 47 (0004) 35 (0003) 13073 (100) 016962 565 (0308) 1261 (0688) 4 (0002) 2 (0001) 1832 (100) 002383 71 (0314) 154 (0681) 1 (0004) 0 (0000) 226 (100) 000294 6 (0353) 11 (0647) 0 (0000) 0 (0000) 17 (100) 000025 1 (0500) 1 (0500) 0 (0000) 0 (0000) 2 (100) 000003

Notes The table presents tabulation of choices and number of claims For comparability the figures are computed only forindividuals whose policies lasted at least 09 years (about 73 percent of the data) The bottom rows of Table 2A providedescriptive figures for the full dataset The numbers in parentheses in each cell represent percentages within each row Theright-hand-side column presents the marginal distribution of the number of claims

753VOL 97 NO 3 COHEN AND EINAV ESTIMATING RISK PREFERENCES FROM DEDUCTIBLE CHOICE

probability of a claim are proportional to thelength of the time interval taken into account soit is convenient to think of the contract choice asa commitment for only a short amount of timeThis approach has several advantages First ithelps to account for early cancellations andtruncated policies which together constitute 30percent of the policies in the data18 Second itmakes the deductible choice independent ofother longer-term uncertainties faced by the in-dividual so we can focus on static risk-takingbehavior Third this formulation helps to obtaina simple framework for analysis which is attrac-tive both analytically and computationally19

The expected utility that the individual obtainsfrom the choice of a contract (p d) is given by

(2) v p d

1 tuw pt tuw pt d

We characterize the set of parameters that willmake the individual indifferent between the twooffered contracts This set provides a lower (up-per) bound on the level of risk aversion for indi-viduals who choose the low (high) deductible (fora given ) Thus we analyze the equation v(phdh) v(pl dl) By taking limits with respect to t(and applying LrsquoHopitalrsquos rule) we obtain

(3)

limt30

1

t(u(w pht) u(w plt))

(u(w pht) u(w pht dh)) (u(w plt) u(w plt dl))

pl phuw

uw dl uw dh

or

(4)

pl phuw uw dl uw dh

The last expression has a simple intuition Theright-hand side is the expected gain (in utils) perunit of time from choosing a low deductibleThe left-hand side is the cost of such a choiceper unit of time For the individual to be indif-ferent the expected gains must equal the costs

In our benchmark specification we assumethat the third derivative of the vNM utilityfunction is not too large A Taylor expansion forboth terms on the right-hand side of equation(4) ie u(w d) u(w) du(w) (d22)u13(w) implies that

(5)pl ph

uw dh dluw

1

2dh dldh dlu13w

Let d dh dl 0 p pl ph 0 andd 1frasl2 (dh dl) to obtain

(6)p

duw uw d u13w

or

(7) r u13w

uw

p

d 1

d

where r is the coefficient of absolute risk aver-sion at wealth level w

18 As can be seen in Table 2A 70 percent of the policiesare observed through their full duration (one year) About15 percent are truncated by the end of our observationperiod and the remaining 15 percent are canceled for var-ious reasons such as change in car ownership total-lossaccident or a unilateral decision of the policyholder tochange insurance providers

19 This specification ignores the option value associatedwith not canceling a policy This is not very restrictiveSince experience rating is small and menus do not changeby much this option value is likely to be close to zero Asimple alternative is to assume that individuals behave as ifthey commit for a full year of coverage In such a case themodel will be similar to the one we estimate but willdepend on the functional form of the vNM utility functionand would generally require taking infinite sums (over thepotential realizations for the number of claims within the year)In the special case of quadratic expected utility maximizerswho care only about the mean and variance of the number ofclaims this is easy to solve The result is almost identical to theexpression we subsequently derive in equation (7)

754 THE AMERICAN ECONOMIC REVIEW JUNE 2007

Equation (7) defines an indifference set in thespace of risk and risk aversion which we willrefer to by (r() ) and ((r) r) interchange-ably Both r() and (r) have a closed-formrepresentation a property that will be computa-tionally attractive for estimation20 Both termsare individual specific as they depend on thedeductible menu which varies across individu-als For the rest of the paper we regard eachindividual as associated with a two-dimensionaltype (ri i) An individual with a risk parameter

i who is offered a menu (pih di

h) (pil di

l)will choose the low-deductible contract if andonly if his coefficient of absolute risk aversionsatisfies ri ri(i) Figure 2 presents a graphi-cal illustration

B The Benchmark Econometric Model

The Econometric ModelmdashThe econometricmodel we estimate is fully described by the fiveequations presented in this section Our ob-jective is to estimate the joint distribution of(i ri)mdashthe claim rate and coefficient of abso-lute risk aversionmdashin the population of policy-holders conditional on observables xi The

20 For example estimating the CARA version of themodel (Section IIID) for which r() does not have aclosed-form representation takes almost ten times longer

FIGURE 2 THE INDIVIDUALrsquoS DECISIONmdashA GRAPHICAL ILLUSTRATION

Notes This graph illustrates the individualrsquos decision problem The solid line presents the indifference setmdashequation (7)mdashappliedfor the menu faced by the average individual in the sample Individuals are represented by points in the two-dimensional spaceabove In particular the scattered points are 10000 draws from the joint distribution of risk and risk aversion for the averageindividual (on observables) in the data based on the point estimates of the benchmark model (Table 4) If an individual is eitherto the right of the line (high risk) or above the line (high risk aversion) the low deductible is optimal Adverse selection is capturedby the fact that the line is downward sloping higher-risk individuals require lower levels of risk aversion to choose the lowdeductible Thus in the absence of correlation between risk and risk aversion higher-risk individuals are more likely to choosehigher levels of insurance An individual with i (pidi) will choose a lower deductible even if he is risk neutral ie withprobability one (we do not allow individuals to be risk loving) This does not create an estimation problem because i is notobserved only a posterior distribution for it Any such distribution will have a positive weight on values of i that are below(pidi) Second the indifference set is a function of the menu and in particular of (pidi) and d An increase in (pidi) willshift the set up and to the right and an increase in d will shift the set down and to the left Therefore exogenous shiftsof the menus that make both arguments change in the same direction can make the sets ldquocrossrdquo thereby allowing us toidentify the correlation between risk and risk aversion nonparametrically With positive correlation (shown in the figureby the ldquoright-bendingrdquo shape of the simulated draws) the marginal individuals are relatively high risk therefore creatinga stronger incentive for the insurer to raise the price of the low deductible

755VOL 97 NO 3 COHEN AND EINAV ESTIMATING RISK PREFERENCES FROM DEDUCTIBLE CHOICE

benchmark formulation assumes that (i ri) fol-lows a bivariate lognormal distribution Thuswe can write the model as

(8) ln i xi i

(9) ln ri xi vi

with

(10) i

vi

iid

N00

2 r

r r2

Neither i nor ri is directly observed Thereforewe treat both as latent variables Loosely speak-ing they can be thought of as random effects Weobserve two variables the number of claims andthe deductible choice which are related to thesetwo unobserved components Thus to completeour econometric model we have to specify therelationship between the observed variables andthe latent ones This is done by making two struc-tural assumptions First we assume that the num-ber of claims is drawn from a Poisson distributionnamely

(11) claimsi Poisson(i ti )

where ti is the observed duration of the policySecond we assume that when individuals maketheir deductible choices they follow the theoreti-cal model described in the previous section Themodel implies that individual i chooses the lowdeductible (choicei 1) if and only if ri ri(i)where ri is defined in equation (7) Thus theempirical model for deductible choice is given by

(12) Prchoicei 1 Prri

pi

idi 1

d i

Prexp(xi vi)

pi

exp(xi i)di 1

d i

With no unobserved heterogeneity in risk (i 0) equation (12) reduces to a simple probit In

such a case one can perfectly predict risk from thedata denote it by (xi) and construct an addi-tional covariate ln([pi((xi)di) 1]d i) Giventhe assumption that risk aversion is distributedlognormally running the probit regressionabove and renormalizing the coefficient onthe constructed covariate to 1 (instead ofthe typical normalization of the variance of theerror term to 1) has a structural interpretation withln (ri) as the dependent variable However Cohen(2005) provides evidence of adverse selection inthe data implying the existence of unobservedheterogeneity in risk This makes the simple probitregression misspecified Estimation of the fullmodel is more complicated Once we allow forunobserved heterogeneity in both unobserved riskaversion (vi) and claim rate (i) we have to inte-grate over the two-dimensional region depicted inFigure 2 for estimation

EstimationmdashA natural way to proceed is toestimate the model by maximum likelihoodwhere the likelihood of the data as a function ofthe parameters can be written by integrating outthe latent variables namely

(13) Lclaimsi choicei

Prclaimsi choiceii riPri ri

where is a vector of parameters to be esti-mated While formulating the empirical modelusing likelihood may help our thinking aboutthe data-generating process using maximumlikelihood (or generalized method of moments(GMM)) for estimation is computationally cum-bersome This is because in each iteration itrequires evaluating a separate complex integralfor each individual in the data In contrastMarkov Chain Monte Carlo (MCMC) Gibbssampling is quite attractive in such a case Us-ing data augmentation of latent variables (Mar-tin A Tanner and Wing Hung Wong 1987)according to which we simulate (i ri) and latertreat those simulations as if they are part ofthe data one can avoid evaluating the complexintegrals by sampling from truncated normaldistributions which is significantly less compu-tationally demanding (eg Luc Devroye 1986)This feature combined with the idea of aldquosliced samplerrdquo (Paul Damien John Wake-field and Stephen Walker 1999) to sample from

756 THE AMERICAN ECONOMIC REVIEW JUNE 2007

an unfamiliar posterior distribution makes theuse of a Gibbs sampler quite efficient for ourpurposes Finally the lognormality assumptionimplies that F(ln(i)ri) and F(ln(ri)i) follow a(conditional) normal distribution allowing us torestrict attention to univariate draws furtherreducing the computational burden

The Appendix provides a full description ofthe Gibbs sampler including the conditionaldistributions and the (flat) prior distributions weuse The basic intuition is that conditional onobserving (i ri) for each individual we have asimple linear regression model with two equa-tions The less standard part is to generate drawsfor (i ri) We do this iteratively Conditionalon i the posterior distribution for ln(ri) fol-lows a truncated normal distribution where thetruncation point depends on the menu individ-ual i faces and its direction (from above orbelow) depends on individual irsquos deductiblechoice The final step is to sample from theposterior distribution of ln(i) conditional on riThis is more complicated as we have bothtruncation which arises from adverse selection(just as we do when sampling for ri) and thenumber of claims which provides additionalinformation about the posterior of i Thus theposterior for i takes an unfamiliar form forwhich we use a ldquosliced samplerrdquo

We use 100000 iterations of the Gibbs sam-pler It seems to converge to the stationarydistribution after about 5000 iterations There-fore we drop the first 10000 draws and use thelast 90000 draws of each variable to report ourresults The results are based on the posteriormean and posterior standard deviation fromthese 90000 draws Note that each iterationinvolves generating separate draws of (i ri) foreach individual Performing 100000 iterationsof the benchmark specification (coded in Mat-lab) takes about 60 hours on a Dell Precision530 workstation

C Identification

The parametric version of the model is iden-tified mechanically There are more equationsthan unknowns and no linear dependenciesamong them so (as also verified using MonteCarlo exercises) the model parameters can bebacked out from simulated data Our goal in thissection is not to provide a formal identificationproof Rather we want to provide intuition for

which features of the data allow us to identifyparticular parameters of the model The discus-sion also highlights the assumptions that areessential for identification vis-a-vis those thatare made only for computational convenience(making them in principle testable)

Discussion of Nonparametric Identifica-tionmdashThe main difficulty in identifying themodel arises from the gap between the (exante) risk type i which individuals usewhen choosing a deductible and the (ex post)realization of the number of claims we ob-serve We identify between the variation inrisk types and the variation in risk realizationsusing our parametric distributional assump-tions The key is that the distribution of risktypes can be uniquely backed out from theclaim data alone This allows us to use thedeductible choice as an additional layer ofinformation which identifies unobserved het-erogeneity in risk aversion21 Any distribu-tional assumption that allows us to uniquelyback out the distribution of risk types fromclaim data would be sufficient to identify thedistribution of risk aversion As is customaryin the analysis of count processes we make aparametric assumption that claims are gener-ated by a lognormal mixture of Poisson dis-tributions (Section IIID discusses this furtherand explores an alternative) Using a mixtureenables us to account for adverse selectionthrough unobserved heterogeneity in risk Italso allows us to better fit the tails of theclaim distribution In principle a more flexi-ble mixture or a more flexible claim-generat-ing process could be identified as long as theclaims data are sufficiently rich22

21 Cardon and Hendel (2001) face a similar identificationproblem in the context of health insurance They use vari-ation in coverage choice (analogous to deductible choice) toidentify the variation in health-status signals (analogous torisk types) from the variation in health expenditure (analo-gous to number of claims) They can rely on the coveragechoice to identify this because they make an assumptionregarding unobserved heterogeneity in preferences (iidlogit) We take a different approach as our main goal is toestimate rather than assume the distribution of preferences

22 Although it may seem that the claim data are limited(as they take only integer values between 0 and 5 in ourdata) variation in policy duration generates continuousvariation in the observed claim propensity Of course thisvariation also introduces an additional selection into themodel due to policy cancellations which are potentially

757VOL 97 NO 3 COHEN AND EINAV ESTIMATING RISK PREFERENCES FROM DEDUCTIBLE CHOICE