Ambiguity, Risk and Earthquake Insurance Premiums: An Empirical Analysis. Toshio FUJIMI, Hirokazu TATANO

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1 京 都 大 学 防 災 研 究 所 年 報 第 49 号 C 平 成 8 年 4 月 Annuals 6 of Disas. Prev. Res. Inst., Kyoto Univ., No. 49 C, 6 Ambiguity, Risk and Earthquake Insurance Premiums: An Emirical Analysis Toshio FUJIMI, Hirokazu TATANO Synosis Ambiguity of insurance ayment in earthquake insurance can be one of main reasons of low urchase rate. We emirically investigate the influence of the ambiguity on the decision to buy hyothetical earthquake insurance and the relationshi with individual characteristics based on MEU model using questionnaire data. The main results of this aer are summarized as follows. First, resondent s references to the insurance with, 5, and % unreimbursement risk are generally inconsistent with exected utility theory. Second, resondents demanded more than 5% reduction in remium to offset each unreimbursement risk. Third, the ambiguity remium is larger in men who urchase earthquake insurance, have never received insurance ayment, and distrust insurance comanies than each corresondents, and increases with age, education level. Keywords: ambiguity, earthquake insurance, robabilistic insurance, Maximin Exected Utility. Introduction Individual disaster-revention efforts lay imortant roles to mitigate the damage and to romote the emergency restoration. As for earthquake, the earthquake insurance can be helful to reair or rebuild a damaged houses because the government can not sufficiently assist earthquake victims due to the financial budget constraint. However, urchase rate of earthquake insurance is low in Jaan (only 7.% households buy the earthquake insurance. There are several ossible reasons: adverse selection, ercetion bias, and ambiguity of insurance ayment. In the adverse selection mechanism, only eole who live in area with frequent earthquake urchase earthquake insurance and this raise the insurance fee and revent other eole from buying the earthquake insurance. Percetion bias is the underestimation of risk and anticiated damage from earthquake. Ambiguity of insurance ayment means the lack of knowledge on earthquake insurance olicy and insurance adjustment makes eole hesitate to urchase it. I focus the last reason, ambiguity of insurance ayment. In this study, we emirically investigate the influence of the ambiguity on the decision to buy hyothetical earthquake insurance and the relationshi with individual characteristics based on an econometric model using questionnaire data. Very few emirical studies on ambiguity are based on econometric models excet finance field where aggregated data are used. Our study is new in that we deal with the ambiguity of disaster and use individual data. This makes it ossible to examine the influence of individual characteristics to the ambiguity size that they erceive. The aer is structured as follows. We start with a exlanation why ambiguity of insurance ayment revents eole from urchasing the earthquake insurance in Section. Section resent the MEU model we will use, and Section 3 resents the estimation results.. Ambiguity Ambiguity of insurance ayment stems from the unclear criteria that the insurance adjuster will use to assess the damage from earthquake. Consumers feel ambiguity due to the following three reasons even if the criteria are clear on insurer s side. First one is asymmetric information. Earthquake insurance contract is too comlicated for consumers to understand. In

2 articular, exertise of house structure is required to assess the damage of house. Therefore, there is no way other than to believe the insurance adjuster s assessment. Second is the rarity of the ayment of the earthquake insurance comared to other insurances such as mobile, fire or injury insurance. Earthquake that damages houses rarely occurs so that we don t have a sufficient number of cases to roughly gras when and how mach earthquake insurance is aid. Thus, consumers feel concerned over the ossibility that the claim is not aid as they exect. In this case, they think of earthquake insurance as robabilistic insurance. Insurance is a contract in which an individual ays a fixed remium and is romised to be aid in the event that a secific hazard occurs. Kahneman and Tversky (979 introduced the notion of robabilistic insurance, namely an insurance olicy which, in the event that the hazard occurs, ay off with some robability strictly less than one. They showed, for a articular tye of robabilistic insurance, that while consumers find such olicy unattractive, an exected utility maximizer would actually erfect the robabilistic olicy (at an aroriate reduced remium to a olicy that ays off with certainty. we analyze a different tye of PI than the one originally addressed by Kahneman and Tversky. The original version of robabilistic insurance was selected for study because it was analytically tractable and led to the surrising result that a risk averse exected utility maximizer favors robabilistic over standard insurance. However, it has the secial feature that in the event that the claim was not aid, the remium would be refunded. This contingency dose not adequately cature the risk of default or fraud because in these instances a remium refund might be roblematic. In this article we investigate a more natural form of robabilistic insurance that dose not involve refunding of remia. Ambiguity is generally defined as indeterminacy of an unique subjective robability due to missing information about the decision roblem. Many emirical studies have found significant evidence of ambiguity affecting decision making. For examle, Kunreuther et. al (995, Einhorn and Hogarth (985, 986 studied lottery choice in exeriment; Heath and Tversky (99, Fox and Tversky (995 analyzed lottery choice in actual events; Maenhout (4 and Lin, Pan, and Wang (5 showed the effect of ambiguity on ortfolio selection in finance. We emirically investigate the influence of the ambiguity on the decision to buy hyothetical earthquake insurance and the relationshi with individual characteristics based on an econometric model using questionnaire data. 3. Survey data Questionnaires was sent out by mail to 3, households in Joyo city, Kyoto in the middle of January, 6. Samles are randomly selected from the NTT telehone book. 68 resonses have been collected (the resonse rate is 3.4%. Necessarily, this survey is hyothetical. It is imossible to have real incentives aid to the resondents. One could devise similar exeriments for real money. In earthquake insurance setting, however, the robability and loss of the relevant event have to be considerably lower and larger than the lottery choice in exeriment resectively. Therefore, the stake would have to be affordably low, which makes the exeriment comletely different from the earthquake insurance setting we want to consider. Hence, we believe that in this domain, thought exeriments for large sums can be more instructive than real exeriments for ennies. Fortunately, there is evidence indicating that there is no difference in resonse for resondents with and without real ayments. Beattie and Loomes (997 designed an exeriment to investigate the relevance of real incentives in decision roblems and concluded that in simle airwise choices, incentives aear to make very little difference to erformance. Further evidence is resented by, among others, Grether and Plott (979, and Conlisk (989, and is surveyed in Camerer (995. Binswanger (98 reorts absence of significant difference in his analytical results between individuals articiating in an exeriment with real money or only laying a hyothetical game. Similarly, Camerer and Hogarth (999 comare 74 exeriments and conclude that financial versus hyothetical incentives in exeriment occasionally imrove erformance although often do not. The questionnaire are structured as follows. First, the hyothetic situation is resented. Then the willingness to ay for full covered insurance and for robabilistic insurance are asked. (H Imagine that you have a house worth mil-

3 lion yen and the other asset (e.g. cash, stocks, or land worth million yen. Assume that earthquake with a seismic intensity 7 on the Jaanese scale will occur with robability of 5% in 5 years (or,.5% er year. If such earthquake haens, your house will be half destroyed ( 5 million loss with 5% robability and comletely destroyed ( million loss with 5% robability. (A. What is the most you would be willing to ay for an insurance olicy that will cover all damages due to earthquake? (B Imagine that you have been offered a different olicy that is identical to the revious one exect that there is about % (or 5%, % unreimbursement risk. That is, there is a ossibility with about % (or 5%, % that your claim will not be aid in case of half collase and only half of your claim will be aid in case of comlete collase. This risk is caused by the adjuster s too strict assessment. What is most you would be willing to ay for robabilistic fire insurance? These data indicate that robabilistic insurance is relatively unattractive. Figure shows the number of the resondent in each category of willingness to ay. (A is full covered insurance and (B is robabilistic insurance resectively. (A (B Fig. Willingness to ay for (A full insurance and (B robabilistic insurance From this figure, we can see that most resondents are located in uer left, which means the large reduction of the value by switching from the full covered insurance to the robabilistic insurance. It is noteworthy that while the majority of the resondents were willing to ay above the actuarially fair remium for full covered insurance, the majority of resondents were not willing to ay the actuarially fair remium for robabilistic insurance. 4. Model To examine the influence of ambiguity of insurance ayment, we analyze the data based on both the Exected Utility (EU model and Maximin Exected Utility (MEU model. EU is most widely used to model decision making under uncertainty. However it can not reresent observed individual choice under ambiguity. MEU is the generalized exected utility model to deal with the ambiguity develoed by Gilboa and Schmeidler (989. we exlain EU first, then describe MEU. 4. Exected Utility Model In the full covered insurance setting above, a decision maker has the rosect (, W Y;, W Y / ;, W where Y is the value of the house ( million yen and W is the value of the other assets ( million yen. is the robability of half collase (.5% er year and is the robability of half collase (.5% er year. Under EU, willingness to ay for the full covered insurance wt f is determined by the equation below. V ( wt f u( W Y wt f v ( where u( is a utility function and v is exected utility without any insurance. v ( u( W Y u( W Y / u( W In the robabilistic insurance setting, the objective rosect is written as Q q, W Y; q, W Y / ; q,, ( W ( ( where q, q, q and is unreimbursement risk (, 5, %. Q can be interreted as the objective robability distribution when the decision maker has the robabilistic insurance. Under EU, the willingness to ay for the robabilistic insurance wt is determined by the equation below.

4 V ( wt q u( W Y wt q u( W Y qu( W wt v / wt ( 4. Maximum Exected Utility Model Maximin Exected Utility (MEU model is generally written as below. V MEU f ( s dp( ( f min u s (3 PC S where s S is a state of world, f : S is an act that mas a state into an outcome, P is a subjective robability, C is the set of subjective robability distributions that the decision maker has. That is, C reresents the ambiguity that the decision maker may erceive in the decision roblem. This idea is suorted by the fact that we can naturally accet the rediction with some range such as "some event will occur with %5%". Each robability distribution P in C is a "ossible scenario" that the decision maker envisions. MEU imlies that a decision maker behaves in the worst case of his exected scenarios. In our earthquake insurance setting, MEU model is formulated as below. u( W Y wt V MEU ( wt min u( W Y / wt (4 P C u( W wt where P (, W Y;, W Y / ;, W is a subjective robability distribution. As for the full covered insurance, MEU is reduced to EU since no ambiguity exists. To estimate the model, the form of C is necessary to be secified. We aly robust control theory of Hansen and Sargent (. The right side of equation (4 can be seen as a constraint robust control roblem if C is secified as below. C { P : R( P, Q } where R( P, Q is relative entroy between P and Q. is a arameter that reresents the size of ambiguity. R( P, Q 3 K ln q k k Hansen and Sargent ( shows that the constraint robust control roblem has a same solution with a multilier robust control roblem as below. u( W Y wt u( W Y min P C u( W wt R( P, Q / wt (5 The arameter in the last roblem (5 can be interreted as an imlied Lagrange multilier on the constraint R ( P, Q. Since ( P, Q is convex in, R and, the first order condition gives the solution of (6. P (, W Y;, W Y / ;, W where q W Y wt u( W Y / wt / W Y wt q q e q e q q q q e u( W wt / W Y wt u( W Y / wt / qe W Y wt u( W Y / wt / W Y wt u( W wt / e qe W Y wt u( W wt / qe W Y wt u( W Y / wt / W Y wt u( W wt / qe Thus, the robabilistic insurance urchase decision can be modeled by MEU where wt is determined by the equation below. VMEU ( wt u( W Y wt u( W Y / wt (6 u( W wt v 5. Examination of Risk Aversion To examine the effect of risk aversion on the decision to urchase the earthquake insurance, we estimate the Pratt-Arrow coefficient of relative risk aversion. We assume that the relative risk aversion is constant in wealth, which derives the secific utility form, x u ( x. The relative risk aversion may vary across demograhic grous. Thus we connect it with resondent s social characteristics in linear x where is a intercet, x is a vector of resondent s characteristics variables, and is a arameter vector. Random utility model is alied to estimate the model. The value function consists of random art and non-random art. Resondent i choose B j if

5 Table The estimation results under EU Indicator Variables (N=56 mean Age Age (in yeas 6. Gender Dummy; if resondent is male.9 Married Dummy; if resondent is married.945 Childe Dummy; if resondent has a childe (under years old.77 Education Dummy; if resondent graduated an university or graduate school.379 Unemloyed Dummy; if resondent is unemloyed.79 Self-emloyed Dummy; if resondent is self-emloyed.3 Civil servant Dummy; if resondent is a civil servant.73 Exerience Dummy; if resondent has exerienced a economic loss from earthquake.74 Purchase Dummy; if resondent has urchased an earthquake insurance.3 Never_Paid Dummy; if resondent has never received any insurance ayment.337 Distrust Dummy; if resondent distrust insurance comanies.454 V where B j Vi ( wti Vi ( B, i ( j Bs are bids B B B B ( j j J shown to the resondent as insurance fee. Assume that follows normal distribution with mean and variance. The log likelihood can be written as N Vi ( B j v i Vi ( B j v i ln L ln ln. i where ( is the normal distribution function. This log likelihood is maximized to estimate arameters. Table rovides the exlanation of indeendent variables and these samle means. Table shows the estimation results for full covered insurance and robabilistic insurance. Estimated gamma reresents the estimate of relative risk aversion. Positive value of imlies risk aversion and negative value means risk rone. The of the full covered insurance is lausible value,.676 while that of the robabilistic insurance is -7.76, which is too low to accet. The unreasonably low value of for the robabilistic insurance imlies that EU is unable to reresent the decision to buy that insurance. Ljungqvist and Sargent ( and Gollier ( said by the thought exeriment that the coefficient of relative risk aversion lies within the range from to 4. The emirical literature suort this. Friend and Blume (975 studied the demand for risky assets and conducted that generally exceeds unity and is robably greater than. Using exenditure data, Weber (975 estimated to lie within a range from.3 to.8, and Sziro (986 obtained a similar range using aggregate time-series data on roerty insurance. In a careful study of consumtion, Hansen and Singleton (98 found relative risk aversion arameters ranging from.68 to.97. In a subsequent study of investments, Table The estimation results of under EU Full covered insurance Probabilistic insurance Coeff -value Coeff -value Intercet Age Gender Married Childe Education Unemloyed Self-emloyed Civil servant Exerience sigma E E Estimated gamma N Log likelihood ratio The -value are based on a two-tailed test that true coefficient is zero.

6 Hansen and Singleton (983 found numerical estimates of, most of which ranged from.6 to.7, with outliers as low as and as high as Mankiw s study of consumtion sending obtained relative risk aversion estimates ranging from.44 to 5.6 for nondurable consumtion and from.79 to 3. for durable goods consumtion. As for full covered insurance, this estimation results suggest that EU works well. The obtained relationshi between risk aversion and resondent s characteristics is mostly consistent with the revious knowledge in the emirical literature. In revious studies, the relation between risk aversion and age is not unequivocal. Barskey et al. (997 reort a negative relation (u to age 6-64, while Riley and Chow (99 and Halek and Eisenhauer ( for over 65 age and Donkers et al. (999 for all age show a ositive relation. Our result suorts that risk aversion increase in ages. we found in our loss-gambling that men are statistically significantly more risk averse than women. It is also founded by Schubert et al. (999. Married resondents exhibit significantly risk aversion than unmarred ones. It is also founded by Halek and Eisenhauer (. We found that self-emloyed are less risk averse than emloyee. Praag (996, Cramer et al. (, and Barskey et al. (997 reort a lower risk aversion for the self emloyed. The effect of education on risk aversion is negative. The result is also reorted by Binswanger (98, 98 and by Donkers et al. ( Examination of Ambiguity Aversion The revious section shows EU can not exlain the decision to urchase the robabilistic earthquake insurance while it works well for full covered insurance. We now show that the aversion to robabilistic insurance is consistent with MEU. From the viewoint of risk (or objective robability distribution, full covered insurance and robabilistic insurance are very close. Therefore, ambiguity arameter will be estimated given that the coefficient of relative risk aversion for robabilistic insurance is same with that for full covered insurance. The ambiguity size may vary across demograhic grous. Hence we connect it with resondent s characteristics in linear, x..5. where.,. 5 and. 5 are dummy variables (= if unreimbursement risks are %, 5%, and % resectively, x is a vector of resondent s characteristics variables, and is a arameter vector. Table 3 shows that the estimation results for full covered insurance and robabilistic insurance. Estimated theta for.,.5, and. reresent the estimate of ambiguity arameter for unreimbursement risks %, 5%, and % resectively. Table 3 The estimation results of under MEU Coeff -value dummy_%.973e-3. dummy_5% 4.949E-3. dummy_% 6.48E-3. Age.6E-4.88 Gender 8.564E-4.3 Education -.47E-4.6 Exerience -.3E Purchase.87E-3.7 Never_Paid.695E-4. Distrust 4.387E-4.7 sigma 9.46E-5. Estimated theta for E-3 Estimated theta for E-3 Estimated theta for E-3 N 56 Log likelihood ratio.377 We cannot comare this result with revious knowledge in literature since this aer is first to estimate the ambiguity arameter with cross-section data. However, some emirical studies using aggregated data exist in finance. Maenhout (4 calibrated to be 7.48 E-3 and 4.94 E-3 resectively using a long annual samle from 89 to 994 and quarterly ostwar samle from 947. to These values are not so different from our results. The relationshi between risk aversion and resondent s characteristics is consistent with the common sense excet Purchase. Positive arameters means the corresondent variables increase ambiguity and enlarge the ga between objective and subjective robabilities. The ambiguity increase with age and is larger for male. High education (university or graduate school and exerience of a economic loss from earthquake reduce the ambiguity though these effects are not statistically significant. Resondents who have never received any insurance claim (including mobile, injury, fire, and life insurance or distrust the insurance comany erceive more ambiguous to insurance ay-

7 Table 4 Subjective robability under MEU = % = 5% = % Objective robability.5%.5%.5% Subjective robability.4%.77%.3% Ratio (subjective/objective ment. The ositive coefficient of Purchase means that urchase of the actual earthquake insurance raise the ambiguity. This seems against our exectation. However, we can interret this result as meaning that only eole with high ambiguity tolerance buy the actual earthquake insurance because it has some unreimbursement risk. Table 4 shows the subjective and objective robability that the resondent s hyothetical wealth is 5 million less insurance fee. In our setting, the case of million wealth level has never haened if he buy the robabilistic insurance. Thus, we can focus on the robability million wealth level since the robability of million can be calculated by it. This shows that the subjective robability become times bigger than the objective robability due to the ambiguity aversion. We can notice that the subjective robability dose not so vary across the unreimbursement risks while the objective robability roortionally changes with them. Table 5 shows the risk and ambiguity remium that are additional ayments to buy earthquake insurance because of risk and ambiguity, resectively. Here, a willingness to ay consists of exected loss, risk remium, and ambiguity remium. than each corresondents. And it increases with age, education level. Wlllingness to ay In the classical economic analysis, insurance is exlained by concavity of utility. In MEU theory, insur- Exected loss Risk remium Ambiguity remium ance is exlained by ambiguity aversion. The observed Table 5 Risk and ambiguity remium (Yen aversion to robabilistic insurance suggests that the = % = 5% = % Exected loss 5,73 4,863 4,35 Risk remium 5,75 5,66 5,55 Ambiguity remium Willingness to ay A % unreimbursement risk reduces more than half value of earthquake insurance. This result is similar with the outcome of the exeriment survey conducted by Wakker, Thaler, and Tversky (997 where resondent exhibit more than a % reduction in the willingness to ay in order to comensate for % unreimbursement risk in life insurance setting. The difference between % and 5% is larger than that between 5% and %, which suggest that the marginal effect of ambiguity aversion with resect to unreimbursement risk declines. 7. Conclusion In this aer we used data from the survey where a set of questions on hyothetical earthquake insurance is resent. We emirically investigated the effect of ambiguity on the decision to buy hyothetical earthquake insurance and the relationshi with individual characteristics. The main results of this aer may be summarized as follows. First, we have observed that eole dislike robabilistic insurance: Most resondents demanded more than 5% reduction in remium to offset a % unreimbursement risk. Second, we have demonstrated that such references are generally inconsistent with exected utility theory. Third, we have shown that the reluctant to buy robabilistic insurance is redicted by the Maximin Exected Utility model. Forth, the ambiguity that resondents erceive is larger in men who urchase earthquake insurance, have never received insurance ayment, and distrust insurance comanies urchase of insurance is driven rimarily by the robust-rone to ambiguity rather than by diminishing marginal utility. Although this aer dealt with the earthquake insurance, there are many other decision roblems in which one erceive the ambiguity for the outcome from the investment to reduce the robability of some hazard. Examles are earthquake retrofit, medical check-us and the installation of burglar alarm. Our result suggests that guarantee of their erformance or comlete recomense in case of failing may dramatically increase their value.

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