Risk Management Decisions in Low Probability and High Loss Risk Situations: Experimental Evidence

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1 Risk Management Decisions in Low Probability and High Loss Risk Situations: Experimental Evidence Ozlem Ozdemir Associate Professor Middle East Technical University (METU) Department of Business Administration, Ankara, Turkey Phone: Andrea Morone (Corresponding Author) Assistant Professor Università degli studi di Bari, Dipartimento di Scienze Economiche, Italy Phone: Fax:

2 Risk Management Decisions in Low Probability and High Loss Risk Situations: Experimental Evidence Abstract Most theoretical and empirical studies have consistently demonstrated that insurance markets for high probability events can be expressed by standard expected utility; however, the EUT is inadequate to explain decision making processes in low probability risk situations. The current study aims to investigate the way people make their insurance decisions in low probability and high loss contexts such as bankruptcy, insolvency, natural disasters etc. More specifically, we try to examine whether elicitation method (buying or paying) induces similar risk management decisions or not, and whether risk attitude and/or threshold probabilities affect the valuations. We test whether preference reversal exists and we elicit individuals threshold probabilities to explore prospective reference theory. The results indicate that subjects considered the probability of loss (the size of the loss) when they face buy or not decision (willingness to pay decision). In addition, the second-order stochastic dominance is supported, however, no evidence is found to conclude that a threshold probability is present in subjects minds. Implications for the insurance market are derived. Keywords: risk management, insurance, low probability. JEL Classification: C91, D81, G 22, G 32 Acknowledgement: The authors would like to gratefully acknowledge the helpful comments of Werner Güth, Jamie Brown Kruse, and the contributions of the participants at the International Association of Research in Economic Psychology and Society for the Advancement of Behavioral Economics Joint Meeting, This paper has been made possible through the financial support of Max Planck Institute of Economics, Jena, Germany. 2

3 I. INTRODUCTION The low probability and high loss (LPHL from now on) events can be expressed as risky situations: the probability of occurrence is low, but the harmful effect can be very dreadful (e.g., bankruptcy, insolvency, natural disasters, terrorism, and some environmental hazards). Both individuals and firms use different risk reduction mechanisms (protective measures) such as credit life insurance, credit insurance, home insurance, storm shelters etc.) in LPHL events. The risky situations have been investigated based on studies conducted at Decision Research (an organization founded in 1976 by Paul Slovic). The different behavioral finance theories and concepts that influence an individual's perception of risk for different types of financial services and investment products are heuristics, overconfidence, prospect theory, loss aversion, representativeness, framing, anchoring, familiarity bias, perceived control, expert knowledge, affect (feelings), and worry (Ricciardi, 2008). Theoretical framework concerning risk and the protective measures chosen by individuals against LPHL risk situations have also been developed over the past thirty years (e.g., Kunreuther, 1979, 1996; Cook and Graham, 1975; Dong et al., 1996; Arrow, 1996). A majority of these models are developed within the framework of Expected Utility Theory (EUT) (von Neumann and Morgenstern, 1944) and most of them attempt to explain the mechanism through which insurance is purchased, specific decision making processes individuals go through, and the factors at play during this incidence (e.g., Friedman and Savage, 1948). However, as Morgenstern (1979, p.178)) mentioned: [T]he domain of our axioms on utility theory is also restricted.for example, the probabilities used must be within certain plausible ranges and not go to 0.01 or even less to 0.001, then be compared to other equally tiny numbers such as 0.02, etc. 3

4 Theories other than the expected utility (e.g. rank-dependent utility theory, generalized expected utility, and prospect theory) have a tendency to overweight low-probability or to be oversensitive towards it (Kahneman and Tversky, 1979; Tversky and Kahneman, 1981; Machina, 1982; Karmarkar, 1978). With little help from theory, researchers conducted survey and laboratory studies that have mixed conclusions for explaining how individuals value insurance against LPHL risk situations. In numerous survey studies, people are found to perceive the risk as if no hazard exists, while others rate low-probability risk equal to more frequent risk exposure (e.g. McClelland et al., 1990, Brookshire et al., 1985; McDaniels et al., 1992; Slovic et al., 1980; Camerer and Kunreuther, 1989; Kunreuther, 1996). Most laboratory studies have consistently demonstrated that insurance markets for high probability events can be expressed by standard expected utility; however, the EUT is inadequate to explain decision making processes in low probability risk situations 1 (e.g., Schoemaker and Kunreuther, 1979; Hershey and Schoemaker, 1980). Most experiments that took into account the size of loss 2 in addition to the probability of loss conclude that individuals have insurance preference for high probability, low loss risks over low probability, high loss events, when expected losses are the same (Slovic et al., 1977; Schoemaker and Kunreuther, 1979; Kunreuther and Slovic, 1978; Kunreuther and Pauly, 2004; Etchart-Vincent, 2004). While there are many studies investigating insurance decisions under various probabilities of loss events, to the authors knowledge, there exist two main studies on the insurance purchase decisions specifically for LPHL risk situations. McClelland et al. (1993) found bimodal distribution for willingness-to-pay for insuring, the ratio of insurance bid to expected value, at probability of 0.01 and loss of $40 out of $50. However, Ganderton et al. 4

5 (2000) concluded that the insurance bid-expected value ratio values are not showing bimodality (for probabilities between 0.01 and 0.36). Further, they found that probability of the occurrence of the risky event has a dominant role in valuing insurance for low probability and high loss situations. In sum, McClelland et al. (1993) investigated paying decisions while Ganderton et al. (2000) examined buying decisions. The general purpose of our study is to investigate the way people make their insurance decisions in LPHL contexts. Specifically, we try to examine whether elicitation method (buying or paying) induces similar decisions or not, and whether risk attitude and/or threshold probabilities affect the valuations. For the first objective of the current study, we design an experiment that includes both open-ended question to elicit individual willingness to pay for protective measure i.e. insurance (WTP hereafter) and dichotomous question to individual buying the insurance or not decision in order to examine whether Ganderton et al. (2000) conclusion about the relative importance of probability on individual insurance buying decision holds for WTP decision. In fact, according to the preference reversal phenomenon, individuals may consider different kinds of information when they make choice versus pricing decision (Kagel and Roth 1995, Slovic and Lichtenstein, 1968, Grether and Plott, 1979; Holt, 1986; Segal, 1988). The second purpose of our experiment is to determine the subjects risk attitudes. For that, we calculated the ratio of willingness to pay values to expected values and compare the distribution of our data with McClelland et al. (1993). It is important to note that we particularly use the probability of 0.01 since it is the common low probability used in both McClelland et al. (1993) and Ganderton et al. (2000). Third, we aim to elicit threshold probability in individuals minds that make them start to consider having insurance. According to the prospective reference theory, people 5

6 overestimate the risk if the probability is below a particular threshold and underestimate when it is above the threshold probability, clarifying the difference between the actual versus perceived risk (Viscusi and Evans, 1990). The paper is organized as follows: section II presents a model that explains the empirical framework along with the hypotheses to be tested. Section III presents the experimental design and sections IV and V respectively results and conclusion of the paper. II. THE RESEARCH MODEL Suppose the loss event occurs with a probability of P and the amount of loss is L, the value of the willingness to pay to reduce risk is B, hence given the amount of endowment, W; the individual expected utility would be: 1. without the insurance: (1) 2. with the insurance: (2) where 0 < P < 1, L 0, B 0, and W > 0. Individual maximum WTP can be found by setting Equation (1) equal to Equation (2) and solving for B i.e. certainty equivalent. We determine the certainty equivalents for four different risk situations in our experiment, depending upon whether the loss size is partial, L= W/2 or full, L=W, and upon the probability of loss equals 0.01 or Note that the expected values for full loss size with probability and partial loss size with probability 0.01 are equal. With the framework explained above, we aim to show whether B is mainly affected by P or L. We believe it depends on how B is elicited from the subjects in the experimental setting. If dichotomous question (whether they would buy insurance or not) is asked, then B has a binary value and subjects have to make a choice decision, but if open ended question 6

7 (their WTP) is asked, then B has a continuous value and subjects have to make a payment decision. Hence, Hypothesis 1: When B has a binary value, B is mainly affected by the probability of loss, while it is affected by the loss size when it has a continuous value (consistent with preference reversal) We think it can be interesting to test whether second-degree stochastic dominance (SDSD hereafter) is satisfied for our data. The stochastic dominance, which was introduced by Rothschild and Stiglitz (1970) using a mean-preserving spread concept, has had many theoretical and empirical applications in the risk and insurance literature (e.g., Machina and Pratt, 1997; Müller, 1998). It can be briefly explained as: a probability distribution dominates another probability distribution if the expected utility of G is at least large as the expected utility of F for every concave and non-decreasing utility function (Borglin and Keiding, 2002). If we define B1 as the certainty equivalent in the risk situation with full loss size, probability of loss being 0.01, B2 as the certainty equivalent for the risk with full loss size, probability of loss being 0.005, B3 as the certainty equivalent for the risk with partial loss size, probability of loss being 0.01, B4 as the certainty equivalent for the risk with partial loss size, probability of loss being 0.005, then: Hypothesis 2: The second-degree stochastic dominance is satisfied, meaning that B1>B2>B3>B4. The relative risk attitude is measured through calculating the value of B/EV (EV is the expected value) based on McClelland et al. (1993). Values of B/EV higher than 1 indicates 7

8 risk aversion lower than 1 indicates risk seeking, and the equality to 1 indicates risk neutrality. 3 According to the well-known fourfold patterns of risk attitudes as suggested by Prospect Theory (Kahneman and Tversky, 1979), people are risk averse for gains and risk seekers for losses in high probability events, and risk averse for losses and risk seekers for gains in low probability events. Hypothesis 3: Consistent with well-known fourfold patterns of risk attitudes as suggested by Prospect Theory (Kahneman and Tversky, 1979), B/EV values are greater than 1 for individuals that make an insurance decision under low probability of loss situations. In addition, individual threshold probability of loss i.e. the minimum probability in individual s mind for a given amount of loss for which s/he buys insurance is also elicited in the experiment. Thus, taking into account the equations (1) and (2), we try to determine minimum P for a given B. Further, B is stated to be equal to the expected values of four different risk situations mentioned above. Hypothesis 4: Consistent with Prospective Reference Theory, P values (threshold probabilities) stated by individuals should be equal to or smaller than probability 0.01 for the risk situation with probability 0.01 and should be equal to or smaller than probability 0.05 for the risk situation with probability III. THE EXPERIMENTAL DESIGN The experiment was run in December 2005 at the lab of the Max Planck Institute of Economics, Jena, Germany. The software of the computerized experiment has been developed in z-tree (Fischbacher, 2007). 96 students from Jena University, 32 in each of the 3 8

9 treatments, were recruited to participate in the experiment using the ORSEE software (Greiner, 2004). 45 percent of the subjects were male and 55 percent was female. The average age was 23 (minimum 19 and maximum 39), average monthly income earned by the individual was 348 Euro (minimum 0 and maximum 1100 Euro). Subjects received written instructions after being seated at a computer terminal. 4 As instructions stated, only three out of 96 subjects earned a big payoff. In each of the 3 treatments we had 32 subjects divided in 2 groups in order to test for order effect. The experiment had two phases. In the first phase, we used pair-wise choice (PC), willingness-to-pay (WTP) and willingness-to-accept (WTA) mechanisms. The purpose of this first part was not only to make subjects practice Becker, DeGroot, and Marschak (1964) (hereafter BDM) 5 procedure but also to give them a hard earned income that they had to protect in the second part of the experiment, this would raise the salience of the incentive scheme since we believed subjects would perceive this income as their income and not as manna. The procedure of Phase 1 differed for each treatment. In the first session/treatment, 32 subjects were asked to choose their preferred lottery from 10 (pair-wise choice) lotteries. In the second session, 32 subjects were given 500 ECU (Experimental Currency Unit) in the beginning of the experiment to buy the same 10 lotteries. They were asked to state their buying price for each lottery. We determined whether they bought the lotteries or not through BDM procedure. In the last session, 32 subjects were given 10 lotteries and stated their selling price for each lottery. The selling process was determined through BDM mechanism. A randomly chosen decision was played to determine the initial endowment for Phase 2. All the lotteries used in Phase 1 were composed by two of the four consequences 200 ECU, 300 ECU, 400 ECU, and 500 ECU. The probabilities of these consequences were 9

10 recorded in Table 1. In the experiment the lotteries were presented as segmented circles on the computer screen. Insert Table 1 The procedure of Phase 2 was the same for all three sessions. The subjects completed the first phase of the experiment were randomly divided into two groups. Thus, the first group of 16 subjects stated willingness to pay for buying full insurance that reduced all of their loss in four different loss situations: two different probabilities of loss (p=0.01 and 0.005) and two loss amounts (all the endowment and half of the endowment). For example, WTP1 is WTP 6 in the loss situation where subjects can lose all their money they earned from Phase 1 with a probability of 0.01, WTP2 is WTP in the loss situation where subjects can lose all their money they earned from Phase 1 with a probability of 0.005, WTP3 is WTP in the loss situation where subjects can lose half of their money they earned from Phase 1 with a probability of 0.01, WTP4 is WTP in the loss situation where subjects can lose half of their money they earned from Phase 1 with a probability of BDM mechanism 7 was used to elicit the WTP values: whether subjects got the insurance or not depends on whether their stated WTP was greater or equal to the random price determined by the computer. This random price was between 0 and the amount of money earned from Phase 1. The other 16 subjects stated whether they would buy the insurance (p=0.01 and 0.005, L= all the endowment and half of the endowment)or not. The prices of the insurances were equal to the expected values, ten times the expected values, twenty times the expected values, and finally fifty times the expected values. 8 After all the decisions were made, one of them was selected randomly and played for real to determine subjects money balances at the end of the experiment. 10

11 Then, all of the subjects were asked to state their threshold probabilities after they made buying and paying decisions 9 for the same four different loss situations with prices that were equal to the expected values. Threshold probability was the smallest probability of the occurrence of the loss situation for which the subject would buy the insurance. Whether the subjects got the insurance depended on whether their probability number was smaller or equal to the random number selected by the computer and the random number was between 0 and 1. After subjects completed all 8 situations, one of the loss situations was chosen randomly by the computer and played for real to determine subjects money balances in the experiment. ECU was converted to Euros at the end of the experiment for each session separately. One person in the entire group of respondents participating in each session (about 32 people) converted ECU to Euros at the following exchange rate: 1ECU = 1, and for the others the rate is: 1 ECU = 0.02 (for example, for one person 500 ECU= 500, for others 500ECU = 10). To select that person, we drew a number (1 to 32), and if the number selected matches with the subject s seat number, that person actually received a good amount of money. 10 It is important to note that our data support the second-degree stochastic dominance hypothesis (Hypothesis 2). Indeed, WTP4 dominates WTP3 which dominates WTP2, which dominates WTP1 (WTP1 > WTP2 > WTP3 >WTP4). Further, reducing the probability of loss and increasing the size of loss in a way which preserves the mean (as done in risk situations to determine WTP2 and 3) is called mean-preserving spread (Rothschild and Stiglitz, 1970). Note the WTP 2 is riskier than WTP3. IV. RESULTS Table 2 represents the statistics for 48 individuals that stated their maximum willingness to pay for the insurance in four loss situations. 11

12 Insert Table 2 Subjects stated higher mean value for WTP2 than WTP3 even though both had the same expected values, however Wilcoxon sign rank test (Z=-0.368, p-value= 0.713) concluded that these two values were coming from the same parental distribution which was consistent with EUT 11. To test the statistical significance between willingness-to pay values, we used a Sign test (Table 3). We rejected the hypothesis that the two samples with different loss amounts were drawn for the same parental distribution (p-value for WTP1-WTP3 and p-value for WTP2 - WTP4 are below 0.05). This result might point out that it was the amount of loss that actually determined individual protection valuation when subjects were asked to state their WTP to protect themselves i.e. when they made payment decisions (consistent with Hypothesis 1). Insert Table 3 As for the analysis of 48 subjects that stated whether they would buy the insurance or not in Phase 2, as it can be seen from Table 4, 20.8% of the individuals said yes when asked if they would buy the insurance when the probability of the occurrence of the loss event was 0.01, no matter what the loss amount was (all of the endowment or half of it). The frequency of the buy or not binary responses indicated the dominance of the probability change rather than the loss amount change on the individual decision-making (supporting Hypothesis 1). Insert Table 4 As a result, subjects changed their decisions of buying the insurance or not when probability of occurrence of the event changed; this result was also supported by McNemar test (Table 5). Having analyzed the connection between the effect of the cost of insurance on individual buy- or- not decision and their risk attitudes, we observe that the higher the cost of the insurance was the lower was the percentage of individuals buying it. More specifically, 12

13 when the cost was lower than the twenty times the expected value, the percentage of individuals who decided to buy was higher (around 80%); when the cost of the insurance was the twenty times the expected value, the percentage of subjects who decided to buy and the ones who decided not to buy was almost equal (around 50%). Finally, when the cost was fifty times the EV, the percentage of the individuals that did not buy the insurance (around 75%) became higher than the rate of the ones who bought it. Insert Table 5 The relative risk aversion was calculated as WTP/EV by using willingness to pay for insurance (see McClelland et al. (1993) for details). Higher values of WTP/EV indicated higher level of relative risk aversion. For all individuals, the WTP/EV was above one, indicating risk aversion, which was consistent with the well known fourfold pattern of risk attitudes i.e. Hypothesis 3 (Tversky and Kahneman, 1992; Harbaugh, Krause, and Vesterlund, 2002). A bimodal distribution was found for the ratio of the willingness to pay for protection in situations of probability 0.01 and loss of all endowment (Figure 1). This can be interpreted as evidence to the fact that while some individuals stated 0 as their WTP values others stated values higher than the expected value. Insert Figure 1 Intuitively, a risk-averse individual should more likely to state a threshold probability that is not too much higher than, the probabilities used to calculate the expected values: 0.01 and However, the mean values for the threshold probabilities were 9%, 3.7%, 5.5%, and 4.5% to buy insurance in four loss situations (rejecting the hypothesis 3). In addition, we checked the effects of endowment, gender, age, income 12, and threshold probability on individuals willingness to buy the insurance for a probability of loss that was equal to 0.01 and loss amount of all of the endowment by using a linear regression 13

14 analysis (Table 6). As a result, the higher the endowment of the individual was and the more relatively risk averse the individual was, the higher was value he or she stated as the willingness to pay to buy the insurance (p-values are 0.000). In addition, women were found to state higher values as their willingness to pay to buy the insurance (p-value is 0.041). Insert Table 6 Further, the binary logistic regression was used to analyze the buy- the- insurance or not binary decision (against a loss situation where the probability of loss was 0.01 and loss amount was all of the endowment) as the dependent variable and endowment, gender, age, and income as the independent variables. Gender was found to be the only statistically significant factor (p-value is 0.019) that influenced the buy- or- not decision as women were more inclined to buy the insurance (Table 7). Insert Table 7 V. CONCLUSION AND DISCUSSIONS The LPHL events have not been fully explained by the EUT. While there are many empirical studies investigating insurance decisions under various probabilities of loss events, only two previous works (McClelland et al., 1993; Ganderton et al., 2000) with contradictory results examined insurance decisions in LPHL situations. The current study explores individual insurance valuations for LPHL risks, particularly, it tests whether the valuations differ because some people focus on probability, while others focus on loss or because each individual has his/her own threshold probability (that is consistent with the prospective reference theory) or it is individual risk attitudes that dominates people s decisions. For these purposes, we design an experiment that uses four loss situations with probabilities and 0.01 and loss sizes all income and half income to elicit individuals insurance valuations. To our knowledge, our experiment is the first attempt that 14

15 determines insurance decisions in LPHL situations using both open-ended and dichotomous questions. That not only allows us to compare our results with previous research, but also to test preference reversal phenomenon and to determine individual risk attitudes using the ratios of willingness-to-pay values to the expected values. Further, there is no empirical work that tries to elicit threshold probabilities in individuals minds. In our experiment, we ask subjects to write down the likelihood of the monetary loss that will make them think buying insurance. Our results show that when individuals are asked to state their willingness-to-pay to buy the insurance (payment decision), they perceive the risk to be higher in the case of higher amount of loss rather than higher probability of loss (it is interesting to note that our WTP values support the stochastic dominance introduced by Rothschild and Stiglitz (1970)). However, when individuals are asked whether they will buy the insurance or not (choice decision), the frequencies of the buy or not binary responses seem to support the dominance of the probability of loss rather than the loss amount on the individual decision making. Note that Ganderton et al. (2000) through using a buy or not question (rather than WTP question) concluded the relative importance of probability on individual insurance buying decision. In sum, one important contribution of this study is to reveal the distinction between the action (or choice) decision and the valuation (or payment) decision embodied in the different descriptions of WTP. Economic theory does not distinguish the WTP measures along these lines. However, this result seems supportive of the preference reversal phenomenon (individuals may consider different kinds of information when they make choice versus pricing decision) of which significance remains even in experiments with different structures (Pommerehne et al., 1982). In fact, regulatory agencies even distinguish between probabilistic (concerned with likelihood) and deterministic (concerned with magnitude) risk assessments (Kuhn and Budescu, 1996). 15

16 When people decide whether to buy insurance or not against LPHL events, they focus on probability that is low and this may be a possible reason of why people do not prefer to insure themselves against for example bankruptcy. However, when people pass the buying decision stage and come to the payment decision, they primarily take into account the high loss amount. For that, it is very important to convince individuals to decide to buy credit insurance. For all subjects, the ratio of the willingness to pay to the expected value is above one, indicating risk aversion, which is consistent with well-known fourfold patterns of risk attitudes as suggested by Prospect Theory: risk averse for gains and risk seeking for losses in high probability events, and risk averse for losses and risk seeking for gains in low probability events. As an additional investigation for the ratio of WTP to expected value, McClelland et al. (1993) bimodal distribution conclusion for the probability 0.01 is supported in our study. One possible reason may be we used BDM mechanism that enables us to see zero bids. However, as it is mentioned by Ganderton et al. (2000), dichotomous buying insurance or not question might not allow to see zero bids and that is why Ganderton et al. (2000) results contradicts with McClelland et al. (1993) results. This interpretation might have been generalized if we could estimate WTP values from buy or not questions. The threshold probabilities, in average, are much higher than the probabilities used to calculate the expected values and seem to have no significant correlation with any variable, providing no support for the prospective reference theory. Consistent with intuition, initial endowment of the individuals is found to have a positive impact on the value of their willingness to pay. Finally, being a woman has a positive impact on both individual insurance buying decision and willingness to pay for buying the insurance. 16

17 For further studies, insurance can be assumed to reduce the possible monetary loss to a certain level rather than to zero. Various elicitation mechanisms (see Holt, 1986 for the drawbacks of paying only one round) and loss situations with different probabilities and loss amounts can be used to investigate the distinction between insurance payment and choice decisions to generalize the results. The dichotomous versus open-ended questions used to elicit individual protection valuation can be designed in a way that enables the researcher to compare the risk attitude measurements for both. The current experiment takes the expected values as the prices of the insurances to get the threshold probabilities in individuals minds, different price levels may help for further contributions. Finally, an extended theoretical investigation is necessary to support the findings of the current experiment s results. 17

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23 Table 1. The probabilities & gains of the lotteries in part ECU 300 ECU 400 ECU 500 ECU

24 Table 2. Descriptive Statistics for Willingness-to-pay Values* Mean Standard deviation Minimum Maximum WTP WTP WTP WTP *WTP1 the willingness to pay amount when probability of loss= 0.01 and loss amount=all of the endowment, WTP2 is the willingness to pay amount when probability of loss= and loss amount= all of the endowment, WTP3 is the willingness to pay amount when probability of loss= 0.01 and loss amount= half of the endowment, WTP4 is the willingness to pay amount when probability of loss= and loss amount= half of the endowment. 24

25 Table 3. Sign Test between Each Pair of WTP Values (Payment Decision) WTP3- WTP2 WTP2- WTP1 WTP3- WTP1* WTP4- WTP3 WTP4- WTP2* WTP4- WTP1* Z values p-values * Significant at 95% level (2-tailed) in that these values are not derived from the same parental distribution. WTP3-WTP2 = the difference in the mean values of WTP when expected values are the same. WTP2- WTP1= the difference in the mean values of WTP when probability changes from to 0.01, loss amount being the same (L= all of the endowment). WTP3-WTP1 = the difference in the mean values of WTP when loss amount changes from half of the endowment to all of the endowment, probability being the same (p= 0.01). WTP4-WTP3 = the difference in the mean values of WTP when probability changes from to 0.01, loss amount being the same (L= half of the endowment). WTP4-WTP2 = the difference in the mean values of WTP when loss amount changes from half of the endowment to all of the endowment, probability being the same (p= 0.005). 25

26 Table 4. Frequencies of Individual Buy or Not (Choice) Decisions* Frequency Percent BON BON BON BON *BON1= buy or not decision when the probability of the occurrence of the loss event is 0.01, loss amount is all of the endowment, BON2= buy or not decision when the probability of the occurrence of the loss event is 0.005, loss amount is all of the endowment, BON3= probability of the occurrence of the loss event is 0.01, loss amount is half of the endowment, BON4= probability of the loss event is 0.005, loss amount is half of the endowment. 0 refers to not buying the insurance and 1 refers to buying the insurance. 26

27 Table 5. McNemar Test between Each Pair of Buy or not Values (Choice Decision) BON1& BON2** BON1& BON3 BON2& BON4 BON1& BON4* BON2& BON3 BON3& BON4* N p-values BON1= buy or not decision when the probability of the occurrence of the loss event is 0.01, loss amount is all of the endowment, BON2= buy or not decision when the probability of the occurrence of the loss event is 0.005, loss amount is all of the endowment, BON3= probability of the occurrence of the loss event is 0.01, loss amount is half of the endowment, BON4= probability of the loss event is 0.005, loss amount is half of the endowment. 0 refers to not buying the insurance and 1 refers to buying the insurance. *Significant at 95% level (2-tailed) in that these values are not derived from the same parental distribution. ** Significant at 90% level (2-tailed) in that these values are not derived from the same parental distribution. 27

28 Figure 1. The frequency distribution for WTP1/EV* *WTP1/EV refers to the ratio of the willingness to pay value for the insurance when probability of the occurrence of the loss event is 0.01 and the loss amount is all of the endowment to the expected value (EV= 0.01*all of the endowment earned in Phase 1). 28

29 Table 6. Regression Analysis Results of WTP1* Coefficient t-statistics p-value s Intercept Endowment** Gender** Age Income Threshold Risk attitude** Adj.R-square=0.77, degrees of freedom=6, F-statistics= significance p-value=0.000 * Dependent variable: WTP1= WTP value in the loss situation where subjects can lose all their money they earned from Phase 1 with a probability of the occurence of loss event **Significant at the 95% level in that being women, having larger amount of endowment, being relatively more risk averse person have positive impact on WTP1. Endowment is the amount of money that subjects earn form Phase 1 of the experiment. Data for age and income (the monthly income of the subjects) are obtained through a questionnaire asked before the experiment. Threshold is the threshold probability that is stated by the subjects during the experiment. Risk attitude is the four WTP/expected values that are used as a multiple-indicant measure of the participants risk attitudes. The internal consistency measure (Crobach s alpha) across these four indicants is estimated as 0.88, suggesting that the risk attitude measure is highly reliable. 29

30 Table 7. Regression Analyses for Buying Insurance Decisions Coefficient p-value Endowment Gender * Age Income Wald= , degrees of freedom=1, p-value = R-square=0.46, degrees of freedom=5, F-statistics=7.258, p-value=0.000 Dependent variable: BON1= Buying the insurance or not binary values, 0 refers to not buying and 1 refers to buying, in the loss situation where subjects can lose all their endowment they earned from Phase 1 with the probability of the occurence of loss event being *Significant at the 95% level in that being women has a positive impact on buying the insurance. Date for age and income (the monthly income of the subjects) is obtained through a questionnaire asked before the experiment. Endowment is the amount of money subjects earned from Phase 1 of the experiment. 30

31 1 The low probability is taken below 0.2 in Camerer (Kagel and Roth, Ch 8, pg. 641, 1995), 0.01 in McClelland et al. (1993), and between and 0.36 (mostly below 0.01) in Ganderton et al. (2000). 2 The frequency of EU violations appears to depend on the size of gamble payoffs. [ ] There are more violations when payoffs are large in magnitude Camerer (1995). 3 The proof is provided by McClelland et al. (1993, pg. 98). 4 The original instructions were in German. Instructions in English are available upon request. 5 We chose to use BDM, because we wanted the subjects to get used to the mechanism used in the second phase. 6 WTP refers to B value in our research model. 7 We intentionally choose to use BDM mechanism to elicit willingness to pay values (Kagel and Roth, 1995), because we wanted the mechanism to be noncompetitive to compare our results with the results of McClelland et al. (1993) competitive auction study (see Ganderton et al. (2000) for more discussions). 8 The reason why these prices are chosen is that expected values are already very low as for the price of the insurance that would eliminate all the loss (for example for the probability of with the endowment of 500 ECU, the expected value would be only 5 ECU and this is the highest value for the price that can be stated), that, most people are expected to be willing to pay that price. However, it is interesting to detect how far they can pay for this insurance that will enable us to determine the degree of their risk aversion. 9 The reason of not asking some subjects threshold questions first and then the willingness to pay questions is that the main aim of the paper is to relate somehow the bids with the threshold probabilities. By asking the threshold questions initially, most probably subjects would have stated extremely high threshold probability values that would be out of the scope of our research objective. 10 We wanted subjects to make their decisions considering a high loss by not giving them any information about who is going to be selected for the big payment. We had to go through this procedure of selecting only one person because we did not have enough budget to give every subject good amount of money. 11 In fact this result is supported by t-statistics test. 12 The information about subjects age, gender, and income are gathered from a questionnaire asked before the experiment. 31