An experimental investigation of insurance decisions in low probability and high loss risk situations

Abstract

This experimental study investigates insurance decisions in low-probability, high-loss risk situations. Results indicate that subjects consider the probability of loss (loss size) when they make buying decisions (paying decisions). Most individuals are risk averse with no specific threshold probability.

1 Introduction

A low probability and high loss (LPHL hereafter) events can be expressed as risky situations where probability of occurrence is low, but the harmful effect can be very dreadful (e.g., bankruptcy, insolvency, terrorism, and natural disasters). Both individuals and firms implement different risk reduction mechanisms, such as life insurance, credit insurance, home insurance, and storm shelters, against these events. However, how people decide to insure specifically towards LPHL hazards is questionable. Theoretical frameworks concerning protective measures chosen by individuals against LPHL risk situations have been developed over the past 30 years (e.g., Arrow 1996; Cook and Graham 1975; Dong et al. 1996; Kunreuther and Slovic 1978). Most have consistently demonstrated that insurance markets for high probability events can be expressed by standard expected utility theory (EUT); however, the EUT is inadequate to explain decision making processes in low probability risk situations (Hershey and Schoemaker 1980). As Morgenstern (1979) 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. (Morgenstern 1979, p. 178).

Most survey studies indicate that some people perceive the risk as if no hazard exists, while others seem to react to the situation as if it is a frequent risk exposure condition (e.g. Camerer and Kunreuther 1989; McClelland et al. 1990; McDaniels et al. 1992; Slovic et al. 1980). The reasons why individuals behave in two opposite ways have not attracted enough attention in the literature. One possible reason might be that some people may take into account the low probability of occurrence and react to LPHL risks as if there is no such risk, while others may focus on the high loss and thus overreact (Etchart-Vincent 2004). In fact, Sjöberg (1999) finds that the demand for risk reduction is influenced by the severity of the hazard and not by the probability. Yet, some other scholars conclude that people make insurance decisions based on probability estimates (Slovic et al. 1977).¹ The lack of a definite answer to the question of whether individuals focus on low probability or high loss when they make insurance decisions in LPHL situations, stresses the necessity for further empirical research.

The present study contributes to this research stream by conducting an experiment to examine a dominant risk assessing consideration, namely, the probability of loss versus loss amount on individuals’ insurance decisions in LPHL risky situations. During the course of this, we investigate whether using different elicitation methods induces any change in these decisions. In addition, we check the effects of individuals’ risk attitudes, their self-determined threshold probabilities, and their demographic characteristics (such as age, income and gender) on these insurance valuations.

In our experiment, we use two different probabilities of loss (0.01 and 0.005)² and two loss amounts (all the income and half of the income) to test the dominance of probability and size of losses on the insurance decision. Furthermore, the design allows us to answer whether the two different elicitation mechanisms reported in the literature to detect the decision making process induce similar risk mitigation behaviour. Specifically, we try to examine the differences in individuals’ insurance decisions between the case when they are asked to decide whether they want to buy the insurance or not (asking dichotomous questions to elicit an individual’s choice), and the case when they are asked to state the amount of money they are willing to pay for the insurance (asking open-ended questions to elicit an individual’s valuation). The theoretical framework does not distinguish risk reduction mechanisms, such as insurance. However, according to the preference reversal phenomenon, individuals may consider different kinds of information when they make choice versus pricing/valuation decisions (Grether and Plott 1979; Holt 1986; Kagel and Roth 1995; Segal 1988; Tversky et al. 1988, 1990). More specifically, when people make choices they focus on the probability and when they assign values they look at the size of the outcome. Previous experimental studies about LPHL risks investigate either buying insurance decisions or paying for insurance decisions. To our knowledge there is no study that investigates both. According to the most recent experimental study that utilises buying decisions (through asking dichotomous questions) to examine insurance decisions in LPHL situations, the probability of occurrence of the risky event plays a dominant role in valuing insurance (Ganderton et al. 2000). However, another experimental study by McClelland et al. (1993) investigates insurance paying decisions (using open-ended questions), and concluded that some individuals are willing to pay zero while others are willing to pay much higher than the expected loss value. This contradiction necessitates further research.

Moreover, our experimental design also allows us to elicit a threshold probability in individuals’ minds and gauge their risk attitudes towards the effects of these insurance decisions. A “threshold probability” is presented as the minimum probability in an individual’s mind for a given amount of loss for which he/she buys insurance. We also determine subjects’ risk attitudes by following the calculation used by McClelland et al. (1993). This enables us to test the consistency of our results with the well-known fourfold patterns of risk attitudes (Mauro and Maffioletti 2004). 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. Finally, we examine how an individual’s endowment in the experiment, gender, age, and income influences his/her insurance buying and paying decisions.

The results of the current study are important for academicians and practitioners in the insurance market in many aspects. It contributes to the literature by answering whether using different methods to elicit individuals’ insurance valuations changes their decisions and affects the dominant consideration of probability of loss and amount of loss. In other words, the study answers the question “do individuals consider different kinds of information when they make choice versus pricing insurance decisions?”. Further, it explores individuals’ threshold probabilities in low-probability and high-loss risk context. The practitioners in the insurance industry may use information about (1) the dominant consideration of probability of loss versus size of loss on consumers’ decisions, (2) the minimum probability of loss in their mind necessary to start considering insuring, (3) their risk attitudes and (4) the role of gender, income and age on decisions for motivating society to mitigate LPHL risks (such as natural disasters).

2 Experimental method

Ninety six subjects take part in the experiment. All of them first receive a questionnaire asking their demographic characteristics such as their age, gender, and income before receiving written instructions on the experiment.³ The experiment has two phases. The purpose of the first phase is to give subjects a “hard earned income”, which they then have to protect in the second phase of the experiment. This raises the salience of the incentive scheme, since we believe subjects perceive this income as their income and not as manna from heaven.⁴ More specifically, in Phase 1, subjects are presented with a set of pairwise choice questions; each pairwise choice is composed of two lotteries, labelled “Lottery A” and “Lottery B”, of the kind depicted in Fig. 1. Each subject has to report his/her preference between the two lotteries.

Fig. 1

Sample presentation of lotteries

The same 10 pairwise choice lotteries (reported in Table 1) are given to all subjects to make their choice decisions. The time taken to complete each session varies across subjects, since subjects are explicitly encouraged to proceed at their own pace.⁵ The incentive mechanism is that, among the ten pairwise choice lotteries, one pairwise choice question is selected at random for each subject and his/her lottery choice for that pair is played for real by the computer. This is how payments to each subject are determined in Phase 1. All the lotteries used in Phase 1 were composed of two of the four possible consequences 200 ECU, 300 ECU, 400 ECU, and 500 ECU.

Table 1 The 10 pairwise choice questions

Out of 96 subjects that answer pairwise choice in Phase 1, 9 of them earned 200 ECU, 25 of them earned 300 ECU, 11 of them earned 400 ECU and 51 of them earned 500ECU. Note that we use Experimental Currency Units (ECU) and convert it to Euros at the end of the experiment. The money earned in Phase 1 determines the initial endowment for Phase 2.

In Phase 2, the subjects have to make insurance decisions to protect the money they earned during Phase 1. In this phase (willingness to pay, WTP, treatment), they are randomly divided into two groups. The first group of 48 subjects states their willingness to pay (their valuations) for insurance that reduces their losses 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 the willingness to pay in a loss situation where subjects may lose all the money earned during Phase 1 with a probability of 0.01; WTP2 is the willingness to pay in a loss situation where subjects may lose all the money earned during Phase 1 with a probability of 0.005; WTP3 is the willingness to pay in a loss situation where subjects may lose half of the money earned during Phase 1 with a probability of 0.01; WTP4 is the willingness to pay in a loss situation where subjects may lose half of the money earned during Phase 1 with a probability of 0.005.⁶ The Becker et al. (1964) mechanism (BDM hereafter) is used to elicit the willingness to pay (WTP hereafter) values: whether subjects buy the insurance or not depends on whether their stated WTP is greater or equal to the random price determined by the computer. This random price is between 0 and the amount of money earned during Phase 1. It is important to note that we use BDM mechanism to elicit the threshold probabilities as it is a very well- established method in the literature to elicit the true values of prices, preferences, and choices (See Kagel and Roth 1995, pp. 19, 20, 70, 73, 74, and 79). There are many studies that compare different elicitation mechanisms, including BDM, in terms of their advantages and disadvantages of using one or another in different circumstances (e.g., Noussair et al. 2004; Schade et al. 2012; Hey et al. 2009; Morone 2010; Safra et al. 1990), however, it is not in the scope of this study to find the best mechanism to elicit threshold probabilities.

The other group of 48 subjects states whether they want to buy insurance or not (buy or not treatments, BON1, BON2, BON3, and BON 4). The subjects make their choice for the same 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). The prices of the insurances are given to subjects as equal to the expected losses. BDM mechanism is used to elicit subjects’ choices about whether they want to buy or not to buy (BON) the insurance.

In Table 2 we report the money subjects earned (endowments) in Phase 1 and the number of subjects make WTP and BON decisions.

Table 2 Number of subjects according to endowment and decision type

After both groups complete their buying and paying decisions,⁷ they are asked to state the minimum probability necessary to incite them to buy insurance in the four different loss situations, i.e. their threshold probabilities. The prices of the insurances are set equal to the expected losses. Whether the subject buys the insurance depends on whether he/she stated his/her minimum probability as smaller or equal to the random number selected by the computer (the random number being between 0 and 1). After all the decisions are made, one of the loss situations is randomly selected by the computer and played for real to determine subjects’ money balances at the end of the experiment. As noted previously, we use ECU and convert it to Euros at the end of the experiment. Three randomly selected subjects have their ECUs converted 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 500 ECU = €10).⁸

3 Results

The experiment is run at the Max Planck Institute of Economics laboratory, in Jena, Germany. The computerized experiment software is developed in z-Tree (Fischbacher 2007). Students from Jena University are recruited to participate in the experiment using the ORSEE software (Greiner 2004). 45 % of subjects are male and the average age is 23 (minimum 19 and maximum 39). The average monthly income earned is 348 Euro (minimum 0 and maximum 1,100 Euro). It is important to note that the data is available from the authors upon request.

Table 3 represents the statistics for 48 individuals, stating their maximum willingness to pay for the insurance (paying decisions) in four loss situations:

WTP1::

is a treatment where subjects state their willingness to pay, to protect their endowment, when the probability of the loss is equal to 0.01, and the loss is equal to all the endowment;

WTP2::

is a treatment where subjects state their willingness to pay, to protect their endowment, when the probability of the loss is equal to 0.005, and the loss is equal to all the endowment;

WTP3::

is a treatment where subjects state their willingness to pay, to protect their endowment, when the probability of the loss is equal to 0.01, and the loss is equal to half of the endowment

WTP4::

is a treatment where subjects state their willingness to pay, to protect their endowment, when the probability of the loss is equal to 0.005, and the loss is equal to half of the endowment.

Accordingly, individuals give very close values for insurance against risk situations with the same amount of loss. Specifically, the average willingness to pay in scenarios WTP1 and WTP2 are very close. Similarly, in WTP3 and in WTP4, both averages are also very close. This may indicate the dominant influence of potential amount lost rather than the probability of loss on subjects’ insurance valuations. Furthermore, it is important to note that subjects state higher mean values for WTP2 than WTP3, even though both have the same expected losses.

Table 3 Descriptive statistics for willingness-to-pay values

For further investigation, we calculate the “relative WTP” values through dividing subjects’ WTPs by their endowment (Table 4). Notice that individuals are willing to give up considerable portions of their endowments to reduce the risk in four different risk situations (27.1, 27.8, 22.7, and 21.2 % respectively, where n = 48). Further, the relative WTPs are getting larger with higher endowments. In addition, we see that relative WTP2 is not significantly larger than WTP3. The relative WTPs in median endowments (300 and 400) do decrease a lot when the amount of loss decrease, but the relative WTPs in low and high endowments (200 and 500) do not decrease as much as one might expect. However, before concluding, these results should be strengthened by further statistical analyses.

Table 4 Descriptive statistics for relative willingness-to-pay values

We test the statistical significance of the differences of means of WTP values through a nonparametric test (Table 5). Accordingly, we conduct Wilcoxon Signed Rank test, yet we use different procedures of the test for small sample case, where the sample size (the number of subjects) is less than or equal to 15 (Black 2008).

Table 5 Wilcoxon signed rank test between each pair of WTP (valuation/paying decision)

The results show that, in overall, the mean values of the WTP with different loss amounts are not drawn from the same parental distribution, with the probabilities of loss remaining the same. More specifically, according to the valuations of all 48 subjects, the difference between the means of WTP1 and WTP3 is statistically significant (at 95 % confidence level, $p$ value = 0.043). This means that when the loss amount changes from half of the endowment to all of the endowment and the probability of loss staying at 0.01, individual insurance valuations differ. Further, the difference between the means of WTP2 and WTP4 is also statistically significant (at 90 % significant level, $p$ value = 0.083). This means that when the amount lost changes from half of the endowment to all of the endowment and the probability of loss staying at 0.005, the individual insurance valuations also differ. However, the means of WTP values are not found to be significantly different when the probabilities of loss change, with amount lost being the same (WTP1 and WTP2, $p$ value = 0.162, WTP3 and WTP4, $p$ value = 0.281). This result suggests that it is the amount of loss that influenced individuals’ insurance valuation (i.e. their paying decisions). It is important to note that this result does not change when the amounts of endowments (and with respect to that the sample sizes/the number of subjects) change. Note further that according to this statistical test, the mean differences between WTP2 and WTP3 is not statistically significant ($p$ value = 0.777). Reducing the loss probability, and increasing its size in such a way that preserves the mean is a “mean-preserving spread” (also known as second-degree stochastic dominance) in the standard terminology (Rothschild and Stiglitz 1970; Machina and Pratt 1997; Müller 1998; Borling and Keiding 2002). In sum, WTP2 is not larger than WTP3 because they share the same expected loss.⁹

Table 6 represents frequencies and percentages for 48 individuals that state their choice in four buying situations:

BON1::

is a treatment where subjects state their buying decision when the probability of the occurrence of the loss event is 0.01, and the loss amount is the whole endowment;

BON2::

is a treatment where subjects state their buying decision when the probability of the occurrence of the loss event is 0.005, and the loss amount is the whole endowment;

BON3::

is a treatment where subjects state their buying decision when the probability of the occurrence of the loss event is 0.01, and the loss amount is half of the endowment;

BON4::

is a treatment where subjects state their buying decision when the probability of the loss event is 0.005, and the loss amount is half of the endowment.

Table 6 Individual buying (choice) decisions

As for the analysis of the 48 subjects that are asked to decide whether they want to buy insurance or not (buying decision) in Phase 2 (Table 6), 79.2 % say “yes” when we ask them if they want to buy the insurance when the probability of the occurrence of the loss event is 0.01 (BON1 and BON3 risk situations), no matter what the loss amount is (all of the endowment or half of the endowment). Furthermore, more subjects decide to buy the insurance when the probability of the occurrence of the loss event is higher rather than when the loss amount is higher. The percentage of subjects that say “yes” for BON3 is 79.2 % and the percentage of subjects that say “yes” for BON2 is 70.8 %, given that expected losses remain the same. Given the potential amounts lost remaining the same, a higher percentage of subjects want to buy insurance against BON1 (the risk situation with the probability of loss being 0.01, and the loss amount being all of the endowment) than BON2 (the risk situation with the probability of loss being 0.005, and the loss amount being all of the endowment) and similarly more subjects want to buy BON3 (the risk situation with the probability of loss being 0.01, and the loss amount being half of the endowment) than BON4 (the risk situation with the probability of loss being 0.005, and the loss amount being half of the endowment). In sum, the frequency of the buying decisions indicates that the probability of loss rather than the loss amount most influence subjects’ buying decisions. This conclusion, however, needs support from further statistical analyses.

In order to test this result further, we conduct a nonparametric McNemar test¹⁰ (Table 7). Note that the values in the table are the $p$ values of the corresponding analysis. Accordingly, we find that subjects more frequently want to buy insurance in the loss situation when the probability of loss is 0.01 (BON1) and when the probability of loss is 0.005 (BON2) (with 90 % confidence, $p$ value = 0.10). Furthermore, for the risk situation with the amount of loss being half of the endowment, the insurance buying decisions are different between the cases with the probability of the occurrence of the loss being 0.01 (BON3) and with the probability of the loss being 0.005 (BON4) (with 95 % confidence, $p$ value = 0.025). These results, in sum, suggest that subjects change their insurance buying decisions according to the probability of occurrence of the LPHL event.

Table 7 McNemar test between each pair of buying (choice decision)

For 48 subjects that participate in the WTP part of the experiment, the individual risk attitudes are calculated by taking the ratios of the WTP values to their expected values (WTP/EV). A WTP/EV greater than 1 indicates risk aversion, equal to 1 indicates risk neutrality and smaller than 1 indicates a risk seeking attitude (see McClelland et al. 1993 for further details on calculations). Accordingly, when each individual’s WTP/EV ratio is calculated, we find that these subjects are risk averse, thus they all have WTP/EV ratios greater than 1, which is consistent with well-known fourfold patterns of risk attitudes as suggested by Prospect Theory (Kahneman and Tversky 1979; Tversky and Kahneman 1981; Harbaugh et al. 2002). Intuitively, a risk averse individual would have been more likely to state a threshold probability that is not much higher than the probabilities used to calculate the expected values: 0.01 and 0.005. However, the mean values for the threshold probabilities are found to be 9, 3.7, 5.5, and 4.5 % to buy insurance in the four loss situations, BON1 to BON4, respectively, used in our experiment.¹¹ These subjects state way higher threshold probabilities and thus, unfortunately, are not adequate for reasonable interpretation.

In addition to the analyses explained above, we examine the effects of endowment, gender, age, income, and threshold probability on individuals’ paying decisions and on their buying decisions. We decide to use the risk situation with a probability of loss equal to 0.01 and loss amount of all the endowment (WTP1) because these have been the most frequently used probability and loss amount values in LPHL contexts (e.g., McClelland et al. 1993; Ganderton et al. 2000). As a result of the regression analysis (Table 8), it can be observed that the higher the subject’s endowment generated during phase 1, the higher the value he/she is willing to pay to buy the insurance ($p$ values = 0.000). In addition, women are found to state higher values for their willingness to pay to buy insurance than men ($p$ value = 0.041).

Table 8 Factors that affect WTP1

Moreover, according to the binary logistic regression used to analyze factors (endowment, gender, age, income, and threshold probability) affecting individuals’ buying decisions (against loss situation with the probability of loss being 0.01 and loss amount being all of the endowment, BON1), gender (women are more inclined to buy the insurance than men are) is found to be the only statistically significant factor ($p$ value = 0.019) that influence the buying decision (Table 9).

Table 9 Factors that affect BON1

4 Discussions and conclusion

Individual insurance decisions in low-probability and high-loss risky events have not been fully explained in theory. In survey studies, some people are found not to buy the insurance at all, while giving very high values that are above the expected losses. The contradiction of these individual level insurance valuations becomes even more apparent on studying the results of the two previous experimental works (McClelland et al. 1993; Ganderton et al. 2000) about insurance decisions in low-probability and high-loss situations. The current experimental study contributes to the literature in many ways. First, it explores this inconsistency through examining the dominance of the probability of loss and amount of loss on individuals’ decisions. Furthermore, it uses both a dichotomous question (buying decision) and an open-ended question (paying decision) to gauge subjects’ insurance valuations. To our knowledge, our experiment is the first attempt to determine insurance decisions in low-probability and high-loss risk situations, eliciting both valuation/paying decisions and choice/buying decisions. This not only allows comparison of our results with previous research, but also allows testing the preference reversal phenomenon. Second, there is no empirical work to date 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 would make them consider buying insurance. The experimental design also allows us to determine individuals’ risk attitudes. Further, it enables us to see the effects of the threshold probabilities, endowments, and some demographic characteristics of subjects (such as income, gender, and age) on their buying and paying insurance decisions.

Our results show that when individuals are asked to state their willingness-to-pay for insurance (paying decision), they perceive the risk to be higher in the case of a higher amount of loss rather than a higher probability of loss. However, when individuals are asked whether they want to buy the insurance or not (buying decision), the outcome of the buy or not responses change according to the probability of loss rather than the loss amount on the individual’s decision making. This finding supports Ganderton’s et al. (2000) conclusion.

In summary, an important contribution of this study is revealing the distinction between the buying (choice) decision and the paying (valuation/pricing) decision embodied in the different descriptions of willingness to reduce low-probability and high-loss risk. This result seems supportive of the preference reversal phenomenon (individuals may consider different kinds of information when they make choice versus pricing decisions) of which the 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). However, economic theory does not distinguish the risk reduction measures along these lines. When people decide whether to buy insurance or not against events, they focus on lower probability, thus, intention to buy without any concern about price of insurance is affected by the probability of loss. This may be a possible reason why people do not prefer to insure themselves against low-probability, high-loss events (e.g., natural disasters and bankruptcy). However, when people pass the intention stage and decide to buy, they primarily take into account the high loss amount for pricing insurance. Hence, once insurance companies convince individuals to buy insurance by explaining the effectiveness or usefulness of insurance itself, rather than focusing on the risk (because people take into account low probability of loss rather than high loss amount and thus do not really worry about risk), they can persuade people to pay high amounts of money to buy insurance (because people focus on high loss amount when making paying decisions). In addition, most subjects are found to be risk averse, 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. The threshold probabilities, on average, are much higher than the probabilities used to calculate the expected values and seem to have no significant correlation with any variable. In fact, the results of the regression analysis indicate that threshold probability is an insignificant factor in insurance valuations for both paying and buying decisions. The analyses also show that individuals’ initial endowment has a positive impact on the value of their willingness to pay. Finally, it is important to note that being female is found to be an important positive factor that influences individual insurance buying decisions and paying decisions. This finding offers guidelines for businesses in the insurance industry in motivating low-probability and high-loss risk mitigation.

The current study suggests that the inconsistent findings in prior research may be due to differences in elicitation methods. For example, the studies that use dichotomous questions to ask individuals’ insurance purchase decisions mostly find the probability of loss affecting the decision (e.g., Ganderton et al. 2000), which does not support the findings of McClelland et al. (1993)’s study, which uses open-ended willingness to pay questions. 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 payments and choice decisions to generalize the results. The dichotomous versus open-ended questions approach used to elicit individual protection valuation can be designed to enable the researcher to compare both risk attitude measurements. The current experiment takes the expected values as the prices of the insurances in order to gauge individuals’ self-determined threshold probabilities and different price levels may contribute further information. Finally, an extended theoretical investigation is necessary to support the findings of the current experiment’s results.