Master’s Thesis
Name: Drhimeur Ali Student ID: 11385715
Msc Track: Behavioral Economics and Game Theory Supervisor: Professor Theo Offerman
Thesis Title: The perception of rare events and the effects of bundling on
the take up of catastrophe insurance.
Abstract:
It is commonly known that individuals do under-insure against low probability-high consequence events. This paper is based on an online experiment that presents different scenarios to estimate the willingness to pay for catastrophe insurance and the probability threshold above which individuals do start considering purchasing insurance. In addition, the paper tests whether bundling coverage for catastrophic events fairs significantly better than offering independent contracts. It is found that most participants are willing to buy catastrophe insurance when the probability of the event to happen is higher than 1.5%. The paper also concludes that bundling is a successful strategy to nudge individuals towards more take up of coverage against disasters.
Statement of Originality
This document is written by Student Ali Drhimeur, who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and so sources other than those mentioned in the text and its references have been used to creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the content.
I- Introduction
In 2016, the total number of reported natural disasters1 was four times higher than in 1970 (290 compared to 68)2. The 21st century has seen an increase in the frequency and intensity of natural disasters and other devastating events, considered primarily as unlikely events (low probability) but with high costs of casualties. These can be floods, earthquakes, heat waves, nuclear leaking, etc. The effect of these events can be disastrous: for example, the Katrina Hurricane in 2005 had a cumulated loss of over $108 billion. To face this type of risks, the insurance industry plays an important role of covering for losses in order to help populations go through the catastrophic events and stand back up financially. After the Katrina Hurricane, insurers contributed in repaying $41.1 billion in claims for damages following the most devastating insured event in History (Insurance Information Institute, 2005). This represents only a coverage of 27% of total losses.
A striking remark is the lack of insurance purchase for this type of low probability-high consequences events. This may be due to many factors that blur individuals to decide rationally over buying insurance.
One explanation is the underestimation of small probabilities and optimism bias. Individuals tend to have unrealistic beliefs that they will not suffer a catastrophe, even if the risks of a disaster to happen are not zero.
Another plausible explanation is the expected government bailout. Many individuals do not purchase insurance for catastrophic events ex-ante if they think that the government will intervene and help cover the losses ex-post. However, this claim has been invalidated in an empirical analysis by Kunreuther et al (1978).
The concept of bounded rationality challenges the notion of homo oeconomicus. It presents individuals as seeking a satisfying solution rather than an optimal one. The difficulty to correctly process information assumes a simplified decision making process that leads to suboptimal decision making. Therefore, for low probability events, searching and processing information before making a decision can be considered as too costly for the consumer to engage in it.
According to the probability weighting function of Kahneman and Tversky, people do overestimate small probabilities. However, in insurance for catastrophic events, individuals
tend to underestimate the probabilities of it happening, and therefore do not feel the need to pay for insurance to cover against the losses (Slovic et al., 1977).
The basic idea is to try and alert people over the probabilities than these events could happen, and so the importance of covering losses.
A strategy to deal with the problem of under-purchase of insurance is to raise a policy’s loss probability above a certain threshold, so that search and thought become rational for the decision maker (Slovic et al., 1977). This can be done through bundling insurance policies. For example, floods and earthquakes are both low probability events that might individually fall below the presumed “threshold” probability for purchasing insurance. But, because they are independent events, bundling the two policies increases the overall risk of the policy coverage. Hence, this could warrant the consumers for this type of insurance products.
In 2004, Kunreuther and Pauly, develop a theoretical basis for this research paper. Their model shows that expected utility maximizing individuals “do face a cost to discovering the true probability of rare events”(H. Kunreuther & Pauly, 2004), and therefore are not willing to support that cost to buy insurance. As an extension to their work, they ask for future research to estimate the probability threshold that could strike individuals’ attention, and how can bundling insurance policies change the purchase behavior of coverage for rare events. This is what is investigated in this paper.
This research has three main objectives. The first one is to estimate the willingness to pay for insurance for low probability-high consequence catastrophic scenarios. The second goal of this research is to estimate the probability threshold above which most of the participants are willing to purchase insurance. The third objective is to experimentally test if bundling catastrophe insurance coverage leads to significantly more purchase compared to independent catastrophe insurance contracts.
In addition to the estimation of the willingness to pay for the pool of participants and the probability threshold above which they are willing to purchase insurance for rare events, the main hypothesis to be tested here is that the bundled contract will be more purchased than the separated individual contracts. This is because of the higher probability of the bundled contract, which groups the risks of the four independent catastrophic contracts.
To answer these questions and evaluate this hypothesis, an online experiment was designed on Qualtrics software. The questionnaire evenly and randomly distributes participants between the two treatments: one group decides over independent contracts and the other group decides over bundled contracts.
The first part asks the participants their willingness to pay for coverage regarding a catastrophic scenario. The second part tries to estimate the probability threshold above which most people do take up insurance regarding a catastrophic scenario. The last part consists of deciding to purchase or not an insurance contract, given the treatment assigned to.
The data collected is expected to reflect a statistically significant difference between the take up of independent contracts for rare events and the take up of bundled insurance contracts. It will also allow to see how far individuals’ willingness to pay can be from the actuarially fair premium for insurance contracts regarding the LP-HC casualty events, and finally estimate the probability threshold of insurance purchase for rare events.
Results show different risk attitudes from the willingness to pay for catastrophe insurance. It also displays that most participants are willing to buy catastrophe insurance when the probability of the event to happen is higher than 1.5%. The paper also concludes that bundling is a successful strategy to nudge individuals towards more take up of coverage against disasters. This paper will first present the related literature to insurance and to low-probability, high-consequence events. Then will follow a theoretical background of the paper from Kunreuther and Pauly before presenting the experimental design and the results derived from the dataset. The last parts will focus on discussing the results and the limitations of the paper.
II- Literature review
Catastrophic events do have disastrous consequences on the society and its main actors. First, citizens could lose their properties and suffer the financial burden of rebuilding their lives. Second, insurance companies do endure the repayment costs of the claims from insured customers, which can cause them financial distress. Finally, the government also has to deal with the rebuilding of the community destroyed from the disaster, and bailouts uninsured damaged households. That is why promoting insurance purchase is essential to society in order to minimize the ruinous effects of a catastrophe.
Theoretically, there is no consensus in the literature on which model best describes the decision process of individuals that face risky decisions like buying insurance policies. A majority of the literature considers the Expected Utility Theory as the baseline model for decisions made under uncertainty (Friedman & Savage, 1948; Arrow, 1996; Kunreuther, 1996; Dong, Shah, & Wong, 1996). Many papers try to estimate utility function by deriving the factors and determinants that drive insurance decisions. However, McClelland, Schulze, & Coursey
agents’ behavior. Mainly, rank-dependent utility theory (Quiggin, 1982) and Cumulative Prospect Theory (Tversky & Kahneman, 1992) try to explain decisions in an insurance context by including a probability weighting function. This feature disentangles the actual probability of an event to happen with the subjective weight placed on each objective probability. Results of both models show that decision makers do overestimate low probabilities but underestimate large ones (Machina, 1982; Viscusi & Evans, 1990).
However, because theoretical models cannot settle on deterministic factors for insurance decisions, literature developed empirical and experimental studies to try and explain the decision process of consumers towards this type of uncertainty protection products.
Krutilla (1966) suggests that imposing a national flood insurance program can improve the efficiency of coverage and help population cover against flooding risks. However, Chivers & Flores (2001) survey populations and claim that information is a capital factor for the consumer to engage in purchasing insurance, no matter the obligation to purchase insurance. Despite information, literature suggests that households under-insure disastrous events and therefore expose themselves to large financial losses (Camerer & Kunreuther, 1989). According to the literature, many reasons can explain the low take up of insurance for low probability-high consequence (LP-HC) events. First, because of the unfamiliar risk faced with catastrophic events, individuals do not have an accurate assessment of the situation that they might be facing (Cutler and Zeckhauser, 2004). This is caused by a lacking salience of a disaster which causes consumers to overlook the risk (Tversky & Kahneman, 1973). Also, experimental evidence from Ganderton et al (2000) shows that individuals do not compute expected loss, but consider either the probability of the disaster to happen or the magnitude of the loss suffered in order to decide about the purchase of insurance. Hence, when assessing risks, bimodal behaviours are found in the literature. Some individuals perceive low risks as zero, while others do overbid for insurance policies on low probability-high consequence events (Kunreuther, Desvousges, & Slovic, 1988; McClelland et al., 1993). One additional cause for low take up of catastrophe insurance is the agents’ belief that the government will intervene ex-post a disaster. The government bailout does decrease the incentive of individuals to pay for insurance against high loss events ex-ante (Birot & Gollier, 2001; Harrington, 2000). Yet, other empirical data using the National Flood Insurance Program in the USA shows a positive relationship between flood insurance purchase and government involvement in financially supporting households after a catastrophe (Browne & Hoyt, 2000)
better information about probabilities and more transparency from the insurer about their profits and loading factor could help individuals decide better about their insurance decisions. As a solution, Slovic et al. (1977) and Kunreuther & Pauly (2004) do propose to bundle several policies into one so that the probability threshold of individual is reached and their attention is drawn to the risks.
This paper offers to the literature two main additions. First, it tries to infer a probability threshold above which individuals are willing to purchase insurance products for low probability-high consequence events, which has not been treated experimentally before. Second, this paper tests whether bundling disaster coverage into one single contract does increase insurance take up compared to offering several independent contracts, which also have not been studied in previous literature, as far as the author knows.
III- Theoretical background
This experimental research is inspired from a theoretical model developed by H. Kunreuther & Pauly (2004) in their paper “Neglecting Disaster: Why Don't People Insure Against Large Losses”.
The two authors consider that the under-purchase of insurance for low probability-high consequence events is due to the implicit costs of “thinking seriously about the purchase of insurance”. According the them, the under-purchase of insurance for disaster risks depends mainly on the availability of information and the costs related to acquiring it. The decision of not buying insurance is therefore solely dependent on the current beliefs about the risk, which is generally not in phase with the actual risk.
In their model, it is assumed that the decision maker is utility maximizer under uncertainty, has full information about the losses and the premium charged. However, the individual has no information on the probability of the event to occur or the loading charges that reflect the administrative expenses and profits of the insurance company.
The model then studies the expected utility of searching for additional information against the expected utility of not searching for additional information on the probability that a catastrophe happens to make a well-rounded decision.
In case the agent wants additional information on the risks faced, he can bear a search cost (S) which will explicit the probability of the catastrophe to happen in order to help him make a decision.
long hours of prospection. Hence search cost becomes more expensive, which deters any incentive to purchase catastrophe insurance, especially if individuals had no previous experience with disasters which creates an availability bias.
The paper suggests possible solutions to be implement by the supply side to make the consumer more aware about the risks encountered. The authors consider bundling coverage as an interesting feature so that “search and thought become rational”, and ask for further research to estimate the probability threshold above which individuals start considering buying insurance and test if bundling can be a valid suggestion to implement.
IV- Methodology
1- Experimental design and procedures:
In order to answer the questions targeted, an online experiment has been prepared to derive meaningful insights about the perception of small probabilities in a catastrophic scenario by individuals and the take up of bundled insurance products.
The questionnaire has been designed using the Qualtrics software and includes a total of eight questions divided in four parts. The first three parts consist of three independent scenarios to reveal the participant’s perception of small risks in a disastrous environment. The last part includes five questions that aim to collect personal information on the participants.
In each scenario, participants are given a hypothetical initial wealth value so that their thinking about the insurance contracts are within a framework of property, as if they were to decide over covering their own wealth.
There a fairly large literature that discusses the pros and cons of using hypothetical stakes in an experiment, whether it can change the decision making process of individuals or not. Edwards (1953) finds an increase in the willingness to take risks while Slovic (1969) found different strategies when comparing real to hypothetical payoffs. On the other hand, Tversky & Kahneman (1992) confirm that none of their conclusions using hypothetical stakes were contradicted when using real ones. More specifically, when the decisions involve large stakes like purchasing insurance against large losses, Laury & Holt (2008) advise to use hypothetical scenarios as it is not feasible to compensate experimental subjects over large sums of money. Therefore, the use of hypothetical stakes in this experiment is in line with what the literature suggests and can be considered as valid given the high stakes involved in the decisions to make in the scenarios.
a table that explicit the chance and value of each potential loss (Table1). The question then asks for the participants’ willingness to pay for insurance coverage against the disaster described. Participants could choose among six premiums to pay for insurance coverage ranging from $0 to $10.000, with the actuarially fair premium being $1.100. The latter has been calculated by multiplying the probability of the event to happen (1%) with the expected losses from the catastrophe ($110.000) derived from table 1.
Value of wealth destroyed Chance of destruction
$50 000 50%
$100 000 25%
$200 000 15
$300 000 10%
Table 1: Potential losses and chances of happening in Scenario1
The second part of the experiment tries to estimate the probability threshold above which most of the participants are willing to buy the insurance contract.
Respondents are presented with another scenario: they own a property valued at $400.000. There is a chance that a storm destroys up to $300.000, and they can insure their property for a yearly premium of $1.500. They are then asked to choose for which probability of the catastrophe to happen they would be willing to buy coverage at that price against the damage of the catastrophe, among eight choices of probabilities ranging from 0.01% to 15%. The premium chosen in this scenario is associated with a risk of a catastrophe occurring of 0.5%, but it is unknown to the respondent.
The third part consists of deciding whether to purchase or not an insurance contract based on another independent scenario where the total value of the wealth could be destroyed if a catastrophe happens. This is equivalent to a hypothetical $400.000 loss. This part of the experiment aims to contrast the take up of insurance for independent contracts and bundled contracts, which constitute the two treatments analysed in this experiment.
Hence, after responding to parts 1 and 2, the program evenly and randomly distributes participants between the two treatments: one group will decide over the take up of four independent contracts and the other group will decide over the take up of four coverages bundled in one contract.
The first treatment group is presented with four independent contracts covering different damages: earthquake, flooding, storm and industrial accidents. In addition to that is explicitly stated the likelihood of each event to happen and the premium to pay to subscribe to the corresponding coverage. Those probabilities have been randomly chosen, with respect to the actual likelihood magnitude and the actual ranking between the chances of these events to happen. Hence, the respondents need to choose whether to purchase or not each contract. On the other hand, the second treatment group is presented with one contract only to choose to purchase or not. However, this single contract is a bundle including coverage against the same four damages presented in the first treatment. The probability presented have been carefully chosen as being the probability of at least one of the catastrophes to happen. Assuming that the catastrophic events are completely unrelated, the probability used in the bundle is:
𝑷 𝑨𝒕 𝒍𝒆𝒂𝒔𝒕 𝒐𝒏𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒂𝒕𝒂𝒔𝒕𝒓𝒐𝒑𝒉𝒆𝒔 𝒕𝒐 𝒉𝒂𝒑𝒑𝒆𝒏 = 𝟏 − 𝑷(𝑵𝒐𝒏𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒂𝒕𝒂𝒔𝒕𝒓𝒐𝒑𝒉𝒆𝒔 𝒕𝒐 𝒉𝒂𝒑𝒑𝒆𝒏)
The last part of the questionnaire collects personal data on the participant in order to be used for analysis. Questions concerning the previous experience of a catastrophe or the ownership of a property can be relevant. Indeed, an individual who has experienced a catastrophe could be more likely to be aware of the risks and the drawbacks and therefore more likely to purchase insurance, at a higher price, or starting from a lower probability of occurrence. On the other hand, similarly, an owner of a property which constitutes part of his own wealth could be more entitled to purchase insurance at a higher price if the losses are large. The distribution of the questionnaire has been done through different social medias in order to reach the largest pool of participants, students and professionals from different age ranges.
2- Hypotheses:
The perception of risks for every individual is mostly intuitive and based on the personal experiences, or the so called “availability bias” (Tversky & Kahneman, 1973). Hence, when faced with the decision of judging probabilities, individuals use heuristics and tend to reduce the decision process to simpler ones. In insurance, the same logic is valid. Decision makers will be more likely to buy a policy insurance for the type of events that have been experienced first hand, either in person or through an acquaintance. The issue arises regarding unfamiliar risks and small chances of happening, like natural disasters or man-made catastrophes (provoked fires, industrial accidents, etc.). Therefore, the rational judging of the likelihood of these events to happen is difficult given the lack of salience of the experience for this type of events.
Consequently, the purchase of insurance is expected to be relatively low for very low probability events, even if the damages can be very high.
Because coverage against catastrophes is necessary, this research aims to explicit three main behaviours regarding the decision making process for the low probability-high casualty events.
First, according to the expected value theory, it is hypothesized that participants should all choose the actuarially fair premium at $1.100. Therefore, the research tries to estimate the willingness to pay for an insurance policy in a catastrophic scenario, and see to what degree it does contrast with the actuarially fair premium. On the second part of the experiment, assuming rationality and the expected utility theory, all participants should consider buying insurance when the probability of happening exceeds 0.5%. However, because perception of risk differs in between respondents, and because of the underestimation of risk and the availability bias, it is expected that the probability threshold above which individuals become willing the purchase coverage against the risks is higher than the actual probability related to the actuarially fair premium.
Lastly, the aim of the third part of the experiment is to test how good can the bundling of insurance coverage in one contract fair against the take up of four independent insurance contracts. Because the probability of occurrence shown in the bundled is higher overall than the probabilities shown in each single independent contracts, it is hypothesized that the purchase of bundled contracts will be significantly higher than the purchase of independent contracts.
V- Data Insights 1- Summary statistics
The questionnaire designed on Qualtrics software was shared through social medias from the 27th of June 2017 and data was collected on the 10th July 2017.
The number of recorded answers was 108. Among those, 64 were complete answers ready to be used in the analysis, and the rest was incomplete answers from the respondents. Table 2 provides an overview of the complete answers recorded.
Recorded answers Total Complete Incomplete
108 64 44
Gender Males Females
36 (56.25%) 28 (43.75%)
Age range 18-25 25-35 35+
47 (73.44%) 14 (21.88%) 3 (4.69%)
Catastrophe experience Yes Some
Acquaintance No
8 (12.5%) 13 (20.31%) 43 (67.19%)
Property owner Yes No
10 (15.62%) 54 (84.38%)
Table 2: Overview of the pool of participants
2- Scenario 1: Willingness to pay for coverage of a catastrophic risk
In the first scenario presented, the participants are facing a maximum loss of $300.000 over their property if a catastrophe were to happen with a probability of 1%. They are offered a table that explicit the distribution of losses such that the expected value of losses is $110.000. Then they are asked to choose their willingness to pay for an insurance policy that will cover their losses, ranging from $0 to $10.000. In this scenario, the actuarially fair premium is $1.100. The results show that 46.15% of participants were willing to pay less than the actuarially fair premium ($0 or $500) and 24.08% were willing to pay more than $1.100.
Willingness to pay Frequency Percentage Cumulative Percent. $0 12 18.46% 18.46% $500 18 27.69% 46.15% $1.100 20 30.77% 76.92% $2.500 11 16.92% 93.85% $5.000 1 1.54% 95.38% $10.000 3 4.62% 100.00% Total 65 100.00%
Table 3: Distribution of choices over the willingness to pay for catastrophe insurance
Assuming that the willingness to pay is a proxy to the aversion to risks, then the majority (46.15%) of participants were underestimating the risk of the catastrophe to happen.
On the other hand, 24.08% overestimated the risk of the catastrophe to happen, and therefore were willing to pay more to cover themselves against the catastrophe.
These results are partially in line with the underlying theory of underestimation of small probabilities in insurance.
This result can be explained by the fact that the participants that are willing to pay higher than the actuarially fair premium may be balancing between the small probability of the catastrophe happening and the big losses that might be generated. The description of the scenario and the associated losses should have added clarity for the decision maker because it offered all needed information to rationally infer the fair premium, but it could also have played a confusing role given the availability bias or the common misunderstanding of insurance policies.
2.1- The perception of probabilities and losses
The following tables try to explicit the perceived probability and the perceived losses by the respondent for each premium to be chosen.
In insurance theory, the premium is computed using the formula:
𝑃𝑟𝑒𝑚𝑖𝑢𝑚 = 𝑃 𝐸𝑣𝑒𝑛𝑡 𝐻𝑎𝑝𝑝𝑒𝑛𝑖𝑛𝑔 ×𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑙𝑜𝑠𝑠𝑒𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 (Eq.1)
In table 4, the formula considered is
Table 4 shows the probability that is perceived by the participant instead of the actual 1% of the catastrophe to happen.
Therefore, for each premium chosen is displayed the perceived equivalence to the 1% risk of the catastrophe to happen. This is calculated from (Eq.2) by dividing the chosen premium over the expected losses from the catastrophe (here $110.000). For instance, if the participant’s willingness to pay for the insurance policy is $500, then the 1% chance of a catastrophic event is perceived as being 0.45%, confirming the underestimation of chances for the disaster to happen. In the other extreme, when the chosen willingness to pay for the insurance policy is $10.000, a 1% chance is perceived as 9%. Hence the overestimation of the chances of the disaster happening.
Premium Perceived probability
equivalent to 1% Premium
Perceived losses equivalent to $110.000 $0 0% $0 $0 $500 0.45% $500 $50.000 $1.100 1% $1.100 $110.000 $2.500 2.27% $2.500 $250.000 $5.000 4.5% $5.000 $500.000 $10.000 9% $10.000 $1.000.000
Table 4: Perceived probability Table 5: Perceived losses
In table 5, the formula considered is
𝐶ℎ𝑜𝑠𝑒𝑛 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 = 𝑃 𝐸𝑣𝑒𝑛𝑡 𝐻𝑎𝑝𝑝𝑒𝑛𝑖𝑛𝑔 ×𝑃𝑒𝑟𝑐𝑒𝑖𝑣𝑒𝑑 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑙𝑜𝑠𝑠𝑒𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 (Eq.3) Therefore, for each chosen premium is opposed the perceived expected loss from the event which is equivalent to the true expected loss of $110.000. This is calculated by dividing the chosen premium by the probability of the catastrophe to happen (here 1%). For instance, the perceived expected loss from the catastrophe when a participant is willing to pay $500 is $50.000, and the perceived expected loss when the willingness to pay is $10.000 is $1.000.000. In theory, a rational decision maker should always choose the premium $1.100, which is the actuarially fair premium, but only 30.77% of the participants made this choice. Hence it is clear that most of them perceived the risk of the magnitude of losses in an irrational way.
2.2- Insights from the characteristics of the respondents:
The noticed misperception of the scenario can be due to a confusion in understanding the insurance policy, or eventually to the decision maker himself. Indeed, if the decision maker or some of his acquaintance did experience a catastrophe before, then it can be supposed that his willingness to pay would be higher than the actuarially fair premium. It is similar for the owners of a property as they might be more inclined to insure and be interested in covering their properties from catastrophic losses.
However, there is no significant correlations between the willingness to pay and the ownership of a property or with the previous experience of a catastrophe even at the 10% significance level.
This might be due to the small pool of participants (only 10 property owners and 22 individuals experienced or know an acquaintance that experienced a catastrophe), so the effect of the two characteristics could be more insightful with a larger number of participants.
According to Hsee, Loewenstein, Blount, & Bazerman (1999), the notion of evaluability of an attribute, here the probabilities, plays an important role in interpreting and understanding the risks faced. It is defined as the difficulty to evaluate independently a certain attribute (here the probabilities of occurrence), which depends on the amount of information the individual has on the attribute. Hence, when an individual has a reference point prior to his decision making, it will affect his decision and represent an anchor in the decision process of the participant. In this case, it is assumed that respondents who do not have any experience with catastrophes cannot evaluate and estimate the probability of the event to happen from prior experience. Therefore, they do not have any referential probability and their evaluability function is flat such that the probabilities that the catastrophe might happen are perceived by the respondents to be all similar whether it is 0% or 9% (see table 4), because of the difficulty to understand the probability (H. Kunreuther, Novemsky, & Kahneman, 2001). There is no rational distinction between the different probabilities offered. Therefore, their perception of the risk that the catastrophe occurs is random or based on heuristics, so is the decision making process.
On the other hand, for the experienced individuals, the memories from the catastrophic experience lived before can play the role of an anchor and so the probability reference that the experienced respondents use is higher because the memory is still fresh. For instance, an individual who recently experienced a catastrophe might consider the chances of it happening as higher than 1%. Hence, when he is faced with a risk of 1%, it is perceived as very low because of the availability bias. Indeed, according to the data, out of the 21 participants that had some experience with disasters, all but one have chosen a premium to pay less or equal to $2.500, and only three of them chose to pay $2.500 for that coverage, all the other 17 were willing to pay less ($0, $500, or $1.100), as shown in table 6. Therefore, their reference threshold of the perceived risk could be higher than the 1% chance for the catastrophe to happen stated in the scenario, and therefore their willingness to pay for this specific probability of 1% is lower.
+
Evaluation0
_
-
0% 9%Figure 1. Evaluability function when the participant has no referential probability. This figure shows the evaluability function of an individual with no prior
experience or knowledge of catastrophe probabilities. In his perception, there is no distinction between the probabilities of a catastrophe up to a certain level.
Willingness to
pay Frequency Mann-Whitney U test
3
No experience of a catastrophe
Experienced a
catastrophe Critical Value = 1
$0 5 7 U-Value = 2
$500 13 4 U-Value > Critical Value
$1.100 14 6 Do not reject H0
>=$2.500 11 4 Populations are not
significantly different
Total 43 21
Table 6. Frequency of willingness to pay condition on the experience of a catastrophe
In the same paper by Hsee et al., (1999), the evaluability function of the experienced respondents is S shaped (Figure 2). Because the catastrophe experienced is still relatively recent, the referential probability of a catastrophe happening could be higher than 1%. In this case the 1% probability of the scenario presented would lie on the left hand side of the reference point in Figure 2. That is eventually why most respondents that experienced themselves or an acquaintance a catastrophe do tend to be willing to pay relatively low premiums compared to the actuarially fair one, because at 1%, they do not value a risk of a disaster as important.
+
Evaluation 0_
-
1% Reference point Figure 2: Evaluability function when there is a reference point information This figure shows the evaluability function of an individual with prior experience of catastrophe probabilities. In his perception, because of the fairly recent event, the subjective probability of a catastrophe is higher than 1%.The data collected also shows a significant difference in the decision process between males and females.
First, females are more willing to pay for insurance coverage against small probability catastrophes than males. A correlation analysis shows a negative relationship between being a male and the willingness to pay for insurance, where the correlation coefficient is -0.2471, statistically significant at the 5% significant level. Therefore, in the pool of participants that responded to the questionnaire, females are more risk averse and may overestimate the risk of a catastrophe to happen.
This inference is emphasized by the fact that among the experienced4 respondents, most females were willing to pay $1.100 or more, while all men were willing to pay $1.100 or less only (Table 7).
Experienced respondents Mann-Whitney U test5
Willingness to
pay Females Males Critical Value = 1
$0 2 3 U-Value = 7
$500 0 2 U-Value = Critical Value
$1 100 2 1 Do Not Reject H0
>= $2 500 3 0 Populations are not significantly
different
Total 7 6
Table 7. Willingness to pay for experienced individuals by gender
In this case, if the reference point is higher than the 1% chance of occurrence for males, as shown in Figure 2, then the reference point for females is lower than 1%, and this risk would on the right hand side of the reference point, given the fact that females are willing to pay more for the same risk in order to purchase insurance.
4 Experienced participants include participants who experienced a catastrophe and those who know an acquaintance who suffered a catastrophe.
3- Scenario 2: The probability threshold of considering insurance purchase:
The second scenario presents the risk for a storm to destroy up to $300.000 of a $400.000 hypothetical property. The participants are offered to buy an insurance policy at the price of $1.500. This premium is actuarially fair if and only if the chances of the storm to hit is 0.5%. They are then asked to choose at which probability of the catastrophe to occur were they willing to purchase insurance at the price of $1.500. The probabilities ranged from 0.01% and 15%. According to the expected utility theory, given that the probability for which the price of the insurance contract offered is actuarially fair is 0.5%, all risk averse and risk neutral participants should be willing to buy this coverage when the probability of the event exceeds or equals 0.5%. However, every individual has a different probability threshold above which considering to buy insurance becomes rational, independently of the actual chances of the event to occur (H. Kunreuther & Pauly, 2004). This is shown in data at the extremes, where at the smallest probability presented some individuals do take up insurance, while at the highest probability some individuals are still insensitive to the catastrophic risk.
The probability threshold above which the participants are willing to purchase insurance is defined here as the minimum probability at which the majority of respondents are willing to purchase the contract offered in the scenario.
Figure 3 shows that the majority of participants start considering buying insurance only if the probability of the catastrophe happening is at least 1.5%.
At 10% or more, more than 94% of respondents were willing to buy insurance at that price. The frequency of participants that are willing to buy insurance increases with the probability of the storm to happen (see scenario in appendix). This is rationally consistent. One anomaly in the results is between the 0.01% and 0.1% thresholds for which one single individual switched behavior.
In real life, the chances of an independent disaster to happen are lower than 5%. Therefore, based on the threshold derived here, most individuals will not consider purchasing coverage. But when bundled with other catastrophe coverages, it is more likely that the overall contract likelihood exceeds the 5% chance of occurrence. This makes it more attractive to buy for the decision maker.
Figure 3. Distribution of the willingness to purchase for each probability
3.1- Insights from the characteristics of the respondents:
By looking at the characteristics of the respondents and the minimum probability above which an insurance contract becomes attractive, one major insights can be derived.
For the owners of a property and the respondents who experienced a catastrophe, the probability threshold at which they are willing to purchase insurance is 0.50%, and not 1.5%.
Figure 4A. Willingness to purchase for each probability for experienced respondents
3 2 6 12 12 17 20 21 18 19 15 9 9 4 1 0 0 , 0 1 %0 , 1 0 %0 , 5 0 % 1 % 1 , 5 0 % 5 % 1 0 % 1 5 %
THRESHOLD FOR
EXPERIENCED
Yes No
Figure 4B. Willingness to purchase for each probability for property owners
Indeed, for both groups, the majority of respondents were willing to pay for insurance for a chance of occurrence higher than 0.5%. For experienced individuals, it might be due to the availability bias, while for property owners, this low threshold might be due to the fear of having their lifetime savings destroyed.
4- Scenario 3: Independent vs Bundled contracts.
In the third part of the online experiment, participants are randomly distributed over the two treatment groups. One group will decide over independent contracts and the second will decide over bundled coverages in a single contract.
The scenario provided is also independent of the two previous ones. Participants are provided with a hypothetical property valued at $400.000. This property is facing four potential catastrophes that can destroy it: an earthquake, a storm, a flood and an industrial accident, each with a different probability of occurrence.
Within the participants, 31 were assigned to the independent contracts treatment, and 33 were assigned to the bundled contract treatment.
2 1 2 7 7 8 10 10 8 9 8 3 3 2 0 0 0 , 0 1 % 0 , 1 0 % 0 , 5 0 % 1 % 1 , 5 0 % 5 % 1 0 % 1 5 %
THRESHOLD FOR
OWNERS
Yes No4.1- Insights from independent contracts treatment:
The analysis of the pool of participants that were assigned to the independent contracts treatment do hand some interesting insights.
One of the main questions was to look for the determinants of insurance take up for disaster insurance. Hence what could increase the probability that a participant subscribes to all four contracts?
A probit model was designed to estimate the marginal effects that explicative variables can have on the probability of purchasing all four available contracts at the same time.
The independent variables used in the model are the individual threshold probability of insurance purchase derived from part 2 of the experiment, whether the participant did experience a catastrophe previously, whether he is owner of a property and the gender to account for any possible gender gaps.
The three models displayed in table 8 all show a significant effect of threshold on the probability of purchasing all four insurance contracts. However, according to the two information criteria, AIC (Akaike Information Criteria) and BIC (Bayesian Information Criteria), the fittest model to express the data available is Model 3, that includes only the threshold variable (see Table 8).
Model 1 Model 2 Model 3
Variable Coefficient (Std. Error) Significance (P-Value) Coefficient (Std. Error) Significance (P-Value) Coefficient (Std. Error) Significance (P-Value) Threshold -0.234 (0.0963) 0.015* -0.205 (0.0902) 0.023* -0.155 (0.0740) 0.036* Cat. Experience -1.112 (0.7762) 0.152 0.839 (0.5895) 0.155 - - Ownership 0.404 (0.6462) 0.531 - - - - Gender -0.611 (0.6832) 0.371 - - - - Constant 0.733 (0.8261) 0.375 0.256 (0.4606) 0.579 -0.159 (0.3347) 0.636 AIC 37.75341 35.1724 35.19083 BIC 44.92334 39.47436 38.0588 Table 8: Probit regression results for the determinants of purchase of all four independent contract in the the treatment using robust standard errors. *Significance level at 5%.
The decision to use the Bayesian Information Criterion as the main one is because the BIC is more parsimonious and therefore more robust.
In this analysis, the only component that is statistically significant is the probability threshold derived in the second part of the experiment (Table 8), above which participants are willing to acquire insurance.
A one-unit increase in the threshold decreases the probability of subscription to all four contracts by 0.044. Hence the threshold of acceptance does play a role in considering disaster insurance take up. If participants have a relatively high probability threshold for considering disastrous events, then their take up for coverage against it is less likely.
On the other hand, the insignificance of catastrophe experience on the take up of all four insurance contracts can be due to the negative experience participants had with the insurance companies. Therefore, the lack of trust could have a negative effect on insurance take up, as found in Cai & Song, (2017).
4.2- Insights from bundled contract treatment:
The answers collected from participants that were assigned to the bundled contracts treatment did not display the same determinants of take up as the population that took the independent contracts treatment.
In an attempt to estimate determinants of bundled disaster insurance purchase, a probit model have been designed. The dependent variable is the probability that a participant does purchase the bundled contract. The independent variables are different characteristics of the respondents, as in the previous section.
None of the suggested determinants do offer a significant relationship with the probability of bundled insurance take up.
This result might be due to the low pool of participants and therefore inferences can be hardly drawn from the data. But this result can also be due to the relatively high probability of the occurrence of the contract offered that includes four different coverage (8.7%). Therefore, the bundled contract could have a positive effect on the perception of risks and catastrophe insurance take up.
Model 1
Variable Coefficient Significance
(P-Value) Threshold 0.034 (0.0625) 0.591 Cat. Experience 1.112 (0.6624) 0.064 Ownership -1.171 (0.7195) 0.104 Gender 0.439 (0.5103) 0.390 Constant 0.307 (0.4543) 0.499 Table 9: Probit regression results for the determinants of purchase in the bundled treatment using robust standard errors
4.3- Effect of Independent versus Bundled Contracts:
In order to know if the bundling of catastrophe insurance coverage succeeded in attracting the attention of the consumers, a difference in proportion test from two different samples was conducted. First has been derived the proportion of take up of bundled insurance contracts, equal to 75.8%. Then, within the treatment group with independent contracts, the proportion of individuals who were willing to buy all four contracts at the same time, equal to 25.8%. It is important to choose the population who was willing to buy all four independent contracts because then there exist a common ground for comparison, i.e. all individuals will have the same coverage for the four catastrophic risks.
The hypothesis tested here is if the proportion of individuals who were willing to buy bundled insurance (𝑝R) is significantly higher than the proportion of individuals who were willing to buy all four independent insurance contracts separately (𝑝S). In other words, 𝐻N = 𝑝R− 𝑝S > 0.
Hypothesis tested Proportion 𝒑𝑩 (Sample size) Proportion 𝒑𝑰 (Sample size) Z-Score p-value 𝑯𝟎= 𝒑𝑩− 𝒑𝑰 > 𝟎 0.758 (33) 0.258 (31) 3.9961 0.000* Table10: Difference in proportion test. *Significance level at 5%.
From Table 10 it is clear that there is significant difference in the proportion of insurance purchasers between the bundle treatment and those who bought all 4 independent insurance contract. Therefore, the initial hypothesis of the research is verified.
Bundling insurance coverage have been significantly more attractive to the purchasers than independent contracts. This could be due to the fact that the probability shown on the insurance contract (8.7%) was higher than the “threshold” above which those individuals do consider a risk as important, and therefore start thinking about purchasing insurance.
Low probability events tend to have a low expected return from seeking information, even if the consequences can be very damaging. In that case, consumers do not bother to bear the costs of research when their perception of the risk incurred is lower than their personal threshold (H. Kunreuther & Pauly, 2004).
VI- Shortcomings and improvements in the design of the experiment:
This paper suffers some shortcomings that can be corrected for better results and some improvements can be proposed to add value to the topic of this paper.
First, an internal validity check shows a potential lack of control over the degree of focus and the understanding of participants of the scenarios that were proposed to them. This might be due to the lack of clarity in the terminology used to describe insurance contracts or to the unfamiliarity of the pool of participants with the decision situations that they were put in. Because most of them are under 25 years old, many might not have had to decide over buying insurance or not at all in their lifetime. Additionally, the use of hypothetical stakes could have induced a lack of interest from this pool of participants, even if literature does not strictly defend this idea. Therefore, their answers would be not in line with the decisions they could take when
However, this does not guarantee more involvement from the experimental subjects, and also, earnings cannot be as high as the potential losses from a catastrophe. Another suggestion for better data is to include several rounds for the same decision, to see how consistently behave individuals across scenarios.
Finally, in an attempt to improve the dataset collected, the experiment could have included another treatment offering individuals the choice between the independent contracts and the bundled. This can be more relevant for real life situations, as consumers can always search for alternatives in the insurance policies they want to purchase.
VII- Discussion and conclusive remarks:
There is a consensus that there exists a lack of insurance take up for low probability-high consequence events among customers. Even if risks are increasing, people fail to realize how important coverage against catastrophic events is important to avoid financial bankruptcy, especially in some critical areas. Therefore, it is necessary to understand the decision making process of individuals and experiment new remedies that might incentivize them to think more about the take up of catastrophe insurance. This paper uses an online experiment to try and estimate the willingness to pay of individuals in facing a catastrophic scenario, estimate the probability threshold above which individual do consider purchasing catastrophe insurance, and finally tests whether bundling disaster coverage into one contract can increase insurance take up.
The results of this paper offer several insights on the decision making of individuals for low probability-high consequences insurance contracts.
First, when facing a catastrophic risk, the majority of respondents are not willing to pay the actuarially fair premium, which denotes a misperception of the scenario presented and the magnitude of potential losses related to the small risk of occurrence. Second, most of the experimental subjects are willing to purchase a catastrophe insurance contract only if the probability of the event to happen is higher than 1.5%. Third, it was shown that bundling did indeed incentivize participants to purchase more catastrophe insurance, compared to offering independent insurance contracts.
There is no contradiction between the results obtained from the dataset and the literature and theory about catastrophe insurance purchase: without external assistance or nudging, purchasing catastrophe insurance is not a common fact.
Also, given the results, it can be said that decision makers do balance between the probability of the catastrophe to happen and the magnitude of losses. In insurance specifically, because the risk involves a large loss, individuals might not focus on the chances of the disaster to happen, but only on their potential loss. Huber, Wider, & Huber (1997) confirm that only a small proportion of subjects search for probability information or use the information furnished to them when facing uncertainty.
There is a fundamental need to increase insurance purchase for low probability-high consequence events, so that agents do not suffer a large financial distress in case it occurs. Therefore, how can the results derived in this research help reach this objective?
It is necessary to have more transparency concerning the probabilities of catastrophes and the loading factors of the insurance company, i.e. how much they price above the actuarially fair premiums for the contracts they offer. This will allow individuals to judge the probabilities by themselves and realize the actual risk they are ignoring by not buying coverage.
There is also a need from insurers to offer bundled coverage for customers in an attempt to increase their warrant about catastrophe insurance, especially in a world where the threat of environmental disasters has increased since the past. Some insurance companies do offer bundled coverage, but it is a mix of a rare event and a more common event, the point being to force their customers to be covered against a specific rare event.
One additional suggestion for insurance take up improvement is to try to gauge the effect of a bundle containing coverage for a catastrophic risk and a lottery ticket. The goal of this manoeuvre is to switch the context of decision making from a context of potential losses to a
involved in a lottery that can earn him a fairly large amount of money, comparable to the potential amount lost in case of a catastrophe. However, the probability to win the lottery should be as small as the probability that the event insured happens.
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Appendix
Introduction
Welcome dear participants,
Many thanks for participating in this survey as part of my master's thesis project about insurance.
Please note that all answers will be recorded anonymously, treated as such, and used only for the purpose of this research.
First some definitions.
An insurance contract is a signed accord between the insurer and the client that guarantees financial coverages when certain events occur and the client suffers a specified loss, damage, illness or death, in return for payment of a specified premium.
Low Probability-High consequence events (LPHC) are events that happen with a small probability but cause a large damage and have expensive cost of casualties like hurricanes, earthquakes, heatwaves, industrial accidents, etc.
A premium is the price of the insurance contract, or how much the client pays the insurer in order to cover his risks.
You are asked to answer some questions on the purchase of insurance products that covers those LPHC events. The three scenarii that you will be dealing with are totally independent from each other. There are no correct or incorrect answers, just choose the option that suits you best!
Willingness to buy
Q2. Assume now that you have a furnished house worth 300.000 dollars. You are facing the following risk:
There is a chance that a natural catastrophe happens in your area, and it can destroy up to 300.000 dollars of your wealth (i.e. everything).
Authorities and insurers think that the catastrophe will happen with a probability of 1% (or once in 100 years).
The table below uncovers the value of wealth destroyed if the catastrophe happens. On the insurance contract, the information is presented as follow:
This means that, if the catastrophe happens, you have 50% chance of suffering $50.000 losses on your house, 25% chance of suffering $100.000 losses, etc.
How much would you be willing to pay yearly for covering your wealth against this catastrophe if it happens?
o
$ 0 (1)o
$ 500 (2)o
$ 1100 (3)o
$ 2500 (4)o
$ 5000 (5)o
$ 10000 (6)Q3. The following scenario is independent of the previous one. Now assume that you face the following problematic:
You have a furnished property valued at 400.000 dollars.
There is a chance that a storm hits your area and might destroy up to 300.000 dollars of your wealth.
The premium (price) to pay to be covered next year against this risk and buy insurance is 1.500 dollars.
When would you be willing to buy insurance?
Remember that you would suffer a loss up to 300 000 dollars. Please answer Yes or No for each proposition
Yes (1) No (2) If chance of catastrophe happening: 0.01% (1)
o
o
If chance of catastrophe happening: 0.1% (2)o
o
If chance of catastrophe happening: 0.5% (3)o
o
If chance of catastrophe happening: 1% (4)o
o
If chance of catastrophe happening: 1.5% (5)o
o
If chance of catastrophe happening: 5% (6)o
o
If chance of catastrophe happening: 10% (7)o
o
If chance of catastrophe happening: 15% (8)o
o
Independent Contracts
Q4. The following scenario is independent of the previous ones.
Assume you have an estate property of total value of 400.000 dollars.
There is a risk that a catastrophe happens and therefore destroys all of your property.
You will be offered four insurance contracts against low probability - high consequence events.
You will choose whether or not you want to purchase any of them. Please read the contracts carefully and answer the question below.
Would you be willing to purchase any of these insurance contracts in order to cover yourself? Yes (1) No (2) Contract A (1)
o
o
Contract B (2)o
o
Contract C (3)o
o
Contract D (4)o
o
Bundled Contracts
Q5. The following scenario is independent of the previous ones.
Assume you have an estate property of total value of 400.000 dollars.
There is a risk that a catastrophe happens and therefore destroys all of your property.
You will be offered one insurance contract covering against four low probability - high consequence events.
Please read the following carefully and answer the question below.
Would you be willing to buy this insurance contract?
o
Yes (1)o
No (2)
Generalities
Q6. What is your age range?
o
18 - 25 (1)o
25 - 35 (2)o
35 - 45 (3)o
45- 55 (4)o
55 + (5)Q7. What is your gender?
o
Male (1)o
Female (2)o
Prefer not to say (3)Q8. Which continent were you born in?
o
Africa (1)o
North America (2)o
South America (3)o
Asia (4)o
Europe (5)o
Oceania (6)Q9. Are you owner of a real estate property?
o
Yes (1)o
No (2)Q10. Have you ever experienced a catastrophe? (Earthquake, flooding, disastrous heatwave, hurricane, etc)
