Estimating Risk Preferences from Deductible Choice. Labor Economics Seminar, UC Berkeley

Transcription

1 Estimating Risk Preferences from Deductible Choice Alma Cohen Tel-Aviv University Liran Einav Stanford University and NBER Labor Economics Seminar, UC Berkeley September 14, 2006

2 What is it about? Rich data on deductible choices in auto insurance in Israel. The average individual chooses between two options: regular : (p h,d h ) = ($911, $455) low : (p l,d l )=($966, $273) Or, in other words, pay extra $55 up-front towards saving $182 with (claim) probability λ. E(λ) = which is less than the ratio = 0.3. However, 18% of the sample pay for higher coverage. This can happen if either those people have higher risk ( adverse selection ) or higher risk aversion ( selection ), or both. We use ex-post information on claims to identify the joint distribution F (λ i,r i X). Note: throughout we restrict attention to rational expectedutility maximizers.

3 Two reasons to care The distribution of risk aversion (the marginal distribution of r i ): How risk averse is the average individual? How heterogeneous are risk attitudes? How do they relate to observables? Very important for macroeconomics, finance, and insurance. Large literature, but questions remain fairly open. Relationship between risk and risk aversion (the joint distribution of (λ i,r i )): How is risk aversion correlated with risk? What is the importance of unobserved risk exposure relative to unobserved risk attitudes in coverage choice? What are the implications for optimal insurance contracts? Closely related to a large recent empirical literature on insurance contracts. Results may guide theoretical work on multi-dimensional screening

4 Literature - Risk aversion Large body of literature that claims low degree of risk aversion (single-digit RRA). Underlying micro-level empirical evidence, however, is weak: Highly-selected populations: TV show participants (Gertner, 1993; Metrick, 1995; Beetsma and Schotman, 2001), horse race bettors (Jullien and Salanie, 2000) Hypothetical survey questions (Evans and Viscusi, 1990, 91; Barsky et al., 1997; Donkers et al., 2001; Hartog et al., 2002) Experiments (Holt and Laury, 2002; Choi et al., 2005) Large literature in finance in the context of the equity premium (driven by the built-in relationship between intertemporal substitution and static risk aversion) Backed out from labor supply (Chetty, forthcoming) Closest to ours: Cicchetti and Dubin (1994) on phone wire insurance (but stakes are small (55 cents/month), damage probability is tiny (0.005 per month), and room for other preference-based stories (next slide)). And, more recently, Sydnor (2005). All use representative agent models; many focus on testing expected utility theory, not on measurement. Note: as we will only estimate a point on the utility function rather than a function, the analysis is orthogonal to the recent debate about the empirical relevance of expected utility theory (Rabin, 2000; Rabin and Thaler, 2001).

5 Literature - Adverse selection Most of the literature uses reduced form tests for the existence of adverse selection: after controlling for observables, are outcomes correlated with coverage choices? Results are mixed: No: French auto insurance (Chiappori and Salanie, 2000), US long-term care insurance (Finkelstein and McGarry, forthcoming). Yes: UK annuities (Finkelstein and Poterba, 2004), our data of Israeli auto insurance (Cohen, 2005) We take a different approach: we impose a structure on adverse selection, and measure its relative importance. Cardon and Hendel (2001) use a structural approach which is somewhat similar to ours (see later). Finkelstein and McGarry (forthcoming) argue that adverse selection is not found due to negative correlation between risk and risk aversion, but their evidence is indirect. We will try to provide a more direct evidence on this correlation. It is not a coincident that none of the applications estimated risk preferences. All may involve other preferencebased explanations for the coverage choice. We think that our application is cleaner for this purpose, as other preference-based stories seem less relevant.

6 Data Entire records of an Israeli auto insurance company through its first five years of operation (11/ /1999). Roughly, 7% market share. Focus on new policies only: 105,800 records. Policy is similar to the US comprehensive insurance. Deductible is the only choice. The company is mostly the only one that offers direct insurance: Some market power Probable sample selection (later) We abstract from moral hazard (later).

7 Pricing For a given individual i, the company computes some deterministic number z i = f(x i ) where x i is a vector of observables. Everything else is derived from z i. Four possible deductible levels are: 0.6d i d i 1.8d i 2.6d i Priced at: 1.06z i z i 0.875z i 0.8z i d i and z i are related through: d i =min(0.5z i,cap t ) About third of the buyers are subject to the cap. The main identifying variation is due to: (i) variation in the cap over time (Figure 1); (ii) some experimentation with the weights during six months of the first year. We will focus throughout on the first two options.

8 Figure 1: Variation in the deductible cap over time 1,750 Uniform Deductible Cap (Current NIS) 1,700 1,650 1,600 1,550 1,500 1,450 1,400 10/30/94 10/30/95 10/30/96 10/30/97 10/30/98 10/30/99 Policy Starting Date This figure presents the variation in the deductible cap over time, which is one of the main sources of pricing variation in the data. We do not observe the cap directly, but it can be calculated from the observed menus. The graph plots the maximal regular deductible offered to anyone who bought insurance from the company over a moving seven-day window. The large jumps in the graph reflect changes in the deductible cap. There are three reasons why the graph is not perfectly smooth. First, in a few holiday periods (e.g., October 1995) there are not enough sales within a seven-day window, so none of those sales hits the cap. This gives rise to temporary jumps downwards. Second, the pricing rule applies at the date of the price quote given to the potential customer. Our recorded date is the first date the policy becomes effective. The price quote is held for a period of 2-4 weeks, so in periods in which the pricing changes, we may still see new policies sold using earlier quotes, made according to a previous pricing regime. Finally, even within periods of constant cap, the maximal deductible varies slightly (variation of less than 0.5 percent). We do not know the source of this variation. 52

9 Key descriptive figures (Tables 1,2) Menus: p d : (0.06; ) Policy Status: Expired (70.7%), Active (truncated) (15.0%), Canceled (14.3%). Average duration (0.28). Claims: 0(81.9%), 1 (15.7%), 2 (2.1%), 3 (0.28%), 4 (0.02%), 5(0.003%). Average claim rate: Choice (fraction chose,claim rate): Low (17.8%, 0.309) Regular (81.1%, 0.232) High (0.6%, 0.128) Very high (0.5%, 0.133)

10 Table 1: Summary statistics covariates Variable Mean Std. Dev. Min Max Demographics: Age Female Family Single Married Divorced Widower Refused to Say Education Elementary High School Technical Academic No Response Emigrant Car Attributes: Value (current NIS)¹ 66,958 37,377 4, ,000 Car Age Commercial Car Engine Size (cc) 1, ,000 Driving: License Years Good Driver Any Driver Secondary Car Business Use Estimated Mileage (km)² 14,031 5,891 1,000 32,200 History Length Claims History Young Driver: Young Gender Male Female Age > Experience < > Company Year: First Year Second Year Third Year Fourth Year Fifth Year The table is based on all 105,800 new customers in the data. 1 The average exchange rate throughout the sample period was approximately 1 US dollar per 3.5 NIS, starting at 1:3 in late 1994 and reaching 1:4 in late The estimated mileage is not reported by everyone. It is available for only 60,422 new customers. 44

11 Table 2A: Summary statistics menus, choices, and outcomes Variable Obs Mean Std. Dev. Min Max Menu: Deductible (current NIS)¹ Low 105, , Regular 105,800 1, , High 105,800 2, , , Very High 105,800 3, , , Premium (current NIS)¹ Low 105,800 3, , , Regular 105,800 3, , , High 105,800 2, , , Very High 105,800 2, , p/ d 105, Realization: Choice Low 105, Regular 105, High 105, Very High 105, Policy Termination Active 105, Canceled 105, Expired 105, Policy Duration (years) 105, Claims All 105, Low 18, Regular 85, High Very High Claims per Year² All 105, Low 18, Regular 85, High Very High The average exchange rate throughout the sample period was approximately 1 US dollar per 3.5 NIS, starting at 1:3 in late 1994 and reaching 1:4 in late The mean and standard deviation of the claims per year are weighted by the observed policy duration to adjust for variation in the exposure period. These are the maximum likelihood estimates of a simple Poisson model with no covariates. Table 2B: Summary statistics - contract choices and realizations Claims Low Regular High Very High Total Share 0 11,929 (19.3%) 49,281 (79.6%) 412 (0.7%) 299 (0.5%) 61,921 (100%) 80.34% 1 3,124 (23.9%) 9,867 (75.5%) 47 (0.4%) 35 (0.3%) 13,073 (100%) 16.96% (30.8%) 1,261 (68.8%) 4 (0.2%) 2 (0.1%) 1,832 (100%) 2.38% 3 71 (31.4%) 154 (68.1%) 1 (0.4%) (100%) 0.29% 4 6 (35.3%) 11 (64.7%) (100%) 0.02% 5 1 (50.0%) 1 (50.0%) (100%) 0.00% Table 2B presents tabulation of choices and number of claims. For comparability, the figures are computed only for individuals whose policies lasted at least 0.9 years (about 73% of the data). The bottom rows of Table 2A provide descriptive figures for the full data. The percentages in parentheses in Table 2B present the distribution of deductible choices, conditional on the number of claims. The right-hand-side column presents the marginal distribution of the number of claims. 45

12 The choice structure Individual i is associated with two utility parameters: λ i (Poisson risk rate) and r i (coefficient of absolute risk aversion). λ i and r i are private information of the individual. She is faced with a particular menu of (p i h,di h )and(pi l,di l ). We analyze the static choice by thinking of a very shortterm policy duration t, overwhichtheriskisλ i t and premium is pt. (this is: (i) consistent with reality; (ii) allows to incorporate truncated policies in an internally consistent way; (iii) allows to abstract from other future risks; (iv) convenient).

13 The choice structure (cont.) vnm utility from (p, d) isgivenby v(p, d) =(1 λt)u(w pt)+(λt)u(w pt d) Taking limits of v(p h,d h )=v(p l,d l ), an indifferent individual satisfies u 0 (w) p = λ ³ u(w d l ) u(w d h ) Second-order expansion implies that i will choose a low deductible iff r i = u00 i (w Ã! i) 1 p u 0 i (w > i 1 1 i) λ i d i d l i +dh i 2 See Figures 2.

14 Figure 2: The individual s decision a graphical illustration Low deductible is optimal Regular deductible is optimal p i d i This graph illustrates the individual s decision problem. The solid line presents the indifference set equation (7) applied for the menu faced by the average individual in the sample. Individual are represented by points in the above two-dimensional space. In particular, the scattered points are 10,000 draws from the joint distribution of riskandriskaversionfortheaverageindividual(onobservables)inthedata,basedonthepointestimatesofthe benchmark model (Table 4). If an individual is either to the right of the line (high risk) or above the line (high risk aversion), the low deductible is optimal. Adverse selection is captured by the fact that the line is downward sloping: higher risk individuals require lower levels of risk aversion to choose the low deductible. Thus, in the absence of correlation between risk and risk aversion, higher risk individuals are more likely to choose higher levels of insurance. An individual with λ i > pi d i will choose a lower deductible even if he is risk neutral, i.e., with probability one (we do not allow individuals to be risk loving). This does not create an estimation problem because λ i is not observed, only a posterior distribution for it. Any such distribution will have a positive weight on values of λ i that are below p i Second, the indifference set is a function of the menu, and, in particular, of p i d i and d. An increase in p i d i d i. will shift the set up and to the right, and an increase in d willshiftthesetdownandtotheleft. Therefore, exogenous shifts of the menus that make both arguments change in the same direction can make the sets cross, thereby allowing us to nonparametrically identify the correlation between risk and risk aversion. With positive correlation (shown in the figure by the right-bending shape of the simulated draws), the marginal individuals are relatively high risk, therefore creating a stronger incentive for the insurer to raise the price of the low deductible. 53

15 The empirical model ln λ i = β 0 x i + ε i (1) ln r i = γ 0 x i + υ i (2) Ã εi υ i! iid N ÃÃ 0 0!, Ã σ 2 λ ρσ λ σ r ρσ λ σ r σ 2 r!! (3) claims i Poisson(λ i t i ) (4) Pr(choice i = low) =Pr(r i >ri (λ i)) (5) Ã! = Pr exp(γ 0 1 p x i + υ i ) > i 1 1 λ i d i d l i +dh i 2

16 Estimation Simple case: set ε i = 0 (no private information about λ i ): equation (2) becomes a straight Probit, and the level is identified by the structure. Full model: ε i 6= 0, so things are more complicated the two equations are related through (i) the deductible choice; and (ii) the correlation ρ; (iii) λ i becomes a latent variable. Likelihood can be constructed by integrating over the latent variables: L(claims i,choice i θ) =Pr(claims i,choice i λ i,r i )Pr(λ i,r i θ) This is time-consuming computationally, as it involves computing (numerically or using simulations) a two-dimensional integral for each consumer, repeatedly. Thus, we estimate the model using MCMC Gibbs sampler. With data augmentation (Tanner and Wong, 1987), this is a standard sampler conditional on (λ i,r i ). The posterior of r i is truncated Normal, so the only non-standard problem is sampling from the posterior of λ i, for which we use slice sampling (Damien et al., 1999). All results report the empirical mean and standard deviation from 90,000 draws from the joint posterior.

17 Intuition for identification Think about a group of individuals identical on observables. Suppose, for the moment, they also face an identical menu. Adatapointis(claims i,choice i ). If, say, max(claims i )= 2, we can represent the data by only five numbers. The key distributional assumption: claims are generated by a mixture of Poisson distributions (or any other oneparameter distribution). This allows us to back out the distribution of risk types (λ i s) only from claim data (this is the key conceptual difference from Cardon and Hendel, 2001). We can now construct a posterior risk distribution for each claim group f(λ i claims i = x) and integrate over it to obtain predicted choice probabilities as a function of (E(r),Var(r),ρ). We now have three equations (moments) in three unknowns, i.e. identification. For estimation, we do it all simultaneously.

18 Results Table 5: Implied levels and heterogeneity Positive correlation Covariates (Table 4): Demographics: U-shape in age, females 20% more risk averse, divorced less, higher education more risk averse Car: more expensive cars - higher risk, higher risk aversion, bigger cars - higher risk, lower risk aversion Driving: business use - higher risk, lower risk aversion, reported history - higher risk aversion, claim history - much higher risk Trend: strong trend downwards for both risk and risk aversion Estimates help predict related choices, and are stable over time. Robustness (Table 6): functional form and distributional assumptions, incomplete information story, sample selection.

19 Table 5: Risk aversion estimates Specification¹ Absolute Risk Aversion² Interpretation³ Relative Risk Aversion 4 Back-of-the-Envelope Benchmark model: Mean Individual th Percentile Median Individual th Percentile th Percentile th Percentile CARA Utility: Mean Individual Median Individual Learning Model: Mean Individual Median Individual Comparable Estimates: Gertner (1993) Metrick (1995) Holt and Laury (2002) Sydnor (2005) This table summarizes the results with respect to the level of risk aversion. Back-of-the-Envelope refers to the calculation we report in the beginning of Section 4, Benchmark Model refers to the results from the benchmark model(table4), CARAUtility referstoaspecification of a CARA utility function, and Learning Model refers toaspecification in which individuals do not know their risk types perfectly (see Section 4.4). The last four rows are the closest comparable results available in the literature. 2 The second column presents the point estimates for the coefficient of absolute risk aversion, converted to $US 1 units. For the comparable estimates, this is given by their estimate of a representative CARA utility maximizer. For all other specifications, this is given by computing the unconditional mean and median in the population using the figures we report in the Mean and Median columns of risk aversion in Table 6 (multiplied by 3.52, the average exchange rate during the period, to convert to U.S. dollars). 3 To interpret the absolute risk aversion estimates (ARA), we translate them into {x : u(w) = 1 2 u(w + 100)+ 1 2u(w x)}. That is, we report x such that an individual with the estimated ARA is indifferent about participating in a fifty-fifty lottery of gaining 100 U.S. dollars and losing x U.S. dollars. Note that since our estimate is of absolute risk aversion, the quantity x is independent of w. To be consistent with the specification, we use a quadratic utility function for the back-of-the-envelope, benchmark, and learning models, and use a CARA utility function for the others. 4 The last column attempts to translate the ARA estimates into relative risk aversion. We follow the literature, and do so by multiplying the ARA estimate by average annual income. We use the average annual income (after tax) in Israel in 1995 (51,168 NIS, from Israeli census) for all our specifications, and we use average disposable income in the US in 1987 (15,437 U.S. dollars) for Gertner (1993) and Metrick (1995). For Holt and Laury (2002) and Sydnor (2005) we use a similar figure for 2002 (26,974 U.S. dollars). 5 Holt and Laury (2002) do not report a comparable estimate. The estimate we provide above is based on estimating a CARA utility model for the 18 subjects in their experiment who participated in the 90 treatment, which involved stakes comparable to our setting. For these subjects, we assume a CARA utility function and a lognormal distribution of their coefficient of absolute risk aversion. The table reports the point estimate of the mean from this distribution. 49

20 Table 4: The benchmark model Variable Ln(λ ) Equation Ln(r ) Equation Additional Quantities Demographics: Constant ^ (0.0073) ^ (0.1032) Var-Covar Matrix (Σ): Age (0.0026) ^ (0.0213) σ λ (0.0097) Age^ ( ) ^ ( ) σ r (0.0773) Female (0.0086) ^ (0.0643) ρ (0.0265) Family Single omitted omitted Married (0.0115) ^ (0.0974) Unconditional Statistics: 1 Divorced ^ (0.0155) (0.1495) Mean λ (0.0013) Widower (0.0281) (0.2288) Median λ (0.0017) Other (NA) (0.0968) (0.7397) Std. Dev. λ (0.0019) Education Elementary (0.0333) (0.2156) Mean r (0.0002) High School omitted omitted Median r ( ) Technical (0.0189) (0.1341) Std. Dev. r (0.0015) Academic ^ (0.0124) ^ (0.0840) Corr(r,λ ) (0.0085) Other (NA) (0.0107) (0.0819) Emigrant (0.0090) (0.0651) Obs. 105,800 Car Attributes: Log(Value) ^ (0.0177) ^ (0.1272) Car Age ^ (0.0023) ^ (0.0176) Commercial Car ^ (0.0187) (0.1239) Log(Engine Size) ^ (0.0235) (0.1847) Driving: License Years (0.0017) (0.0137) License Years^ ( ) ( ) Good Driver ^ (0.0112) (0.0822) Any Driver ^ (0.0105) ^ (0.0722) Secondary Car ^ (0.0141) (0.0875) Business Use ^ (0.0134) ^ (0.1124) History Length (0.0052) ^ (0.0518) Claims History ^ (0.0154) (0.1670) Young Driver: Young driver ^ (0.0253) (0.2290) Gender Male omitted - Female ^ (0.0061) - Age omitted ^ (0.0121) ^ (0.0124) - > (0.0119) - Experience <1 omitted (0.0104) - > ^ (0.0121) - Company Year: First Year omitted omitted Second Year ^ (0.0122) ^ (0.0853) Third Year ^ (0.0137) ^ (0.1191) Fourth Year ^ (0.0160) ^ (0.1343) Fifth Year ^ (0.0249) ^ (0.1368) Standard deviations based on the draws from the posterior distribution in parentheses. ˆ Significant at the five-percent confidence level. 1 Unconditional statistics represent implied quantities for the sample population as a whole, i.e., integrating over the distribution of covariates in the sample (as well as over the unobserved components). 48

21 Table 6: Robustness Specification¹ Sample Obs. Claim Risk (λ ) Absolute Risk Aversion (r ) Corr(r,λ ) ρ Mean Median Std. Dev. Mean Median Std. Dev. Baseline Estimates: Benchmark Model All New Customers 105, The vnm utility function: CARA Utility All New Customers 105, @ @ The claim generating process: Benchmark Model No Multiple Claims 103, Thinner-Tail Risk Distribution All New Customers 105, # # # The distribution of risk aversion: Lower Bound Procedure All New Customers 105, Incomplete information about risk: Benchmark Model Experienced Drivers 82, Benchmark Model Inexperienced Drivers 22, Learning Model All New Customers 105, Sample Selection: Benchmark Model First Two Years 45, Benchmark Model Last Three Years 60, Benchmark Model Referred by a Friend 26, Benchmark Model Referred by Advertising 79, Benchmark Model Non-Stayers 48, Benchmark Model Stayers, 1st Choice 57, Benchmark Model Stayers, 2nd Choice 57, This table presents the key figures from various specifications and subsamples, tracing the order they are presented in Section 4.4. Full results (in the format of Table 4) from all these specifications are available in the online appendix (and at 1 Benchmark Model refers to the benchmark specification, estimated on various subsamples (the first row replicates the estimates from Table 4). The other specifications are slightly different, and are all described in more detail in the corresponding parts of Section The interpretation of r in the CARA model takes a slightly different quantitative meaning when applied to non-infinitesimal lotteries (such as the approximately 100 dollar stakes we analyze). This is due to the positive third derivative of the CARA utility function, compared to the benchmark model, in which we assume a small third derivative. Thus, these numbers are not fully comparable to the corresponding figures in the other specifications. # The interpretation of λ in the thinner-tail distribution we estimate is slightly different from the standard Poisson rate, which is assumed in the other specifications. Thus, these numbers are not fully comparable to the corresponding figures in the other specifications. 50

22 Counterfactuals We hold fixed the distribution of (λ i,r i ). To be meaningful, counterfactuals should be unlikely to have significant effects on inflow/outflow of customers. Underlying assumption: sequential decision making by individuals. First, decide about the company by observing the regular premium-deductible combination. Then choose the best contract. Thus, we analyze company s (additional) profits from the choice of the low combination. These are given by: n π0 +Pr(low; d, p)) h p d eλ(low; d, p) io max d, p All figures use the mean individual in the data (and her corresponding menu). See Figures 4,5.

23 Figure 4: Counterfactuals profits 8 Additional Profit per Policy (NIS) Benchmark model Negative rho No risk-aversion heterogeneity No risk heterogeneity ,000 1,125 1,250 1,375 1, Deductible Difference (Regular Deductible - Low Deductible) (NIS) This figure illustrates the results from the counterfactual exercise (see Section 4.6). We plot the additional profits from offering a low deductible as a function of the attractiveness of a low deductible, d = d h d l, fixing its price at the observed level. The thick solid line presents the counterfactual profits as implied by the estimated benchmark model. The other three curves illustrate how the profits change in response to changes in the assumptions: when the correlation between risk and risk aversion is negative (thin solid line), when there is no heterogeneity in risk aversion (dot-dashed line), and when there is no heterogeneity in risk (dashed line). The maxima (argmax) of the four curves, respectively, are 6.59 (355), 7.14 (583), 0, and 7.04 (500). The dotted vertical line represents the observed level of d (638), which implies that the additional profits from the observed low deductible are 3.68 NIS per policy. 55

24 Figure 5: Counterfactuals selection Expected Risk (Lambda) for Low Deductible Buyers Benchmark model Negative rho No risk-aversion heterogeneity No risk heterogeneity ,000 1,125 1,250 1,375 1,500 Deductible Difference (Regular Deductible - Low Deductible) (NIS) 1.00 Benchmark model Share of Buyers Choosing Low Deductible Negative rho No risk-aversion heterogeneity No risk heterogeneity ,000 1,125 1,250 1,375 1,500 Deductible Difference (Regular Deductible - Low Deductible) (NIS) This figure illustrates the results from the counterfactual exercise (see Section 4.6). Parallel to Figure 4, we break down the effects on profits to the share of consumer who choose a low deductible (bottom panel) and to the expected risk of this group (top panel). This is presented by the thick solid line for the estimates of the benchmark model. As in Figure 4, we also present these effects for three additional cases: when the correlation between risk and risk aversion is negative (thin solid line), when there is no heterogeneity in risk aversion (dot-dashed line), and when there is no heterogeneity in risk (dashed line). The dotted vertical lines represent the observed level of d (638), for which the share of low deductible is 16 percent and their expected risk is This may be compared with the corresponding figures in Table 2A of 17.8 and 0.309, respectively. Note, however, that the figure presents estimated quantities for the average individual in the data, while Table 2A presents the average quantities in the data, so one should not expect the numbers to fit perfectly. 56

25 Caveats Imperfect information about risk types. Moral hazard: Affecting driving/care behavior. Affecting the propensity to file a claim. (see Figure 3) Selection: Sample selection Choice-based selection

26 Figure 3: Claim distributions Regular Deductibles Density % % Low Deductible Claim Amount / Regular Deductible Level This figure plots kernel densities of the claim amounts, estimated separately, depending on the deductible choice. For ease of comparison, we normalize the claim amounts by the level of the regular deductible (i.e., the normalization is invariant to the deductible choice), and truncate the distribution at 10 (the truncated part, which includes a fat tail outside of the figure, accounts for about 25% of the distribution, and is roughly similar for both deductible choices). The thick line presents the distribution of the claim amounts for individuals who chose a low deductible, while the thin line does the same for those who chose a regular deductible. Clearly, both distributions are truncated from below at the deductible level. The figure shows that the distributions are fairly similar. Assuming that the claim amount distribution is the same, the area below the thicker line between 0.6 and 1 is the fraction of claims that would fall between the two deductible levels, and therefore (absent dynamic incentives) would be filed only if a low deductible was chosen. This area (between the two dotted vertical lines) amounts to 1.3 percent, implying that the potential bias arising from restricting attention to claim rate (and abstracting from the claim distribution) is quite limited. As we discuss in the text, dynamic incentives due to experience rating may increase the costs of filing a claim, shifting the region in which the deductible choice matters to the right; an upper bound to these costs is about 70 percent of the regular deductible, covering an area (between the two dashed vertical lines) that integrates to more than seven percent. Note, however, that these dynamic incentives are a very conservative upper bound; they apply to less than fifteen percent of the individuals, and do not account for the exit option, which significantly reduces these dynamic costs. 54

27 Table 7: Representativeness Variable Sample² Population³ Car Owners 4 Age¹ (12.37) (18.01) (14.13) Female Family Single Married Divorced Widower Education Elementary High School Technical Academic Emigrant Obs. 105, , ,435 1 For the age variable, the only continuous variable in the table, we provide both the mean and the standard deviation (in parentheses). 2 The figures are derived from Table 1. The family and education variables are renormalized so they add up to one; we ignore those individuals for which we do not have family status or education level. This is particularly relevant for the education variable, which is not available for about half of the sample; it seems likely that unreported education levels are not random, but associated with lower levels of education. This may help in explaining at least some of the gap in education levels across the columns. 3 This column is based on a random sample of the Israeli population as of We use only adult population, i.e., individuals who are 18 years old or more. 4 This column is based on a subsample of the population sample. The data only provide information about car ownership at the household level, not at the individual level. Thus, we define an individual as a car owner if (i) the household owns at least one car and the individual is the head of the household, or (ii) the household owns at least two cars and the individual is the spouse of the head of the household. 51

28 Discussion Two contributions: Measurement: measures of the distribution of risk preferences from micro-level choice data. Methodology: Conceptual framework for estimating demand system in the presence of adverse selection. Need individual-level menus and ex-post risk data. Can be applied to other insurance and credit markets. The marginal of r i (extrapolatable?): Higher average level of risk aversion than previous studies. Significant unobserved heterogeneity. Selection on unobservables may be important in voluntary markets. Interesting variation with covariates. In particular, positive wealth/income relationship. Caution against inference based on representative consumer models.

29 Discussion (cont.) The joint of (λ i,r i ) (more context specific): Positive correlation (driving may be different from other risks; omitted variables) Unobserved heterogeneity in risk aversion more important than adverse selection Interesting implications for pricing Multi-dimensional screening models: two dimensions, but only one instrument, leading to bunching (Armstrong, 1999): How do optimal contracts look like? How do they vary with the correlation structure? Canthishelpinexplainingdifferences in types of contracts across markets?