Abstract
We introduce a model of the decision between precaution and insurance under an ambiguous probability of loss and employ a novel experimental design to test its predictions. Our experimental results show that the likelihood of insurance purchase increases with ambiguous increases in the probability of loss. When insurance is unavailable, individuals invest more in precaution when the probability of loss is known than when it is ambiguous. Our results suggest that sources of ambiguity surrounding liability losses may explain the documented tendency to overinsure against liability rather than meet a standard of care through precaution. The results provide support for our theoretical predictions related to risk management decisions under alternative probabilities of loss and information conditions, and have implications for liability, environmental, and catastrophe insurance markets.
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Notes
See Starmer (2000) for a review.
The behavioral literature also suggests that certain behavioral biases, such as overconfidence or optimism, as well as the tendency to overreact to recent events, may explain under- and overinsurance for certain types of losses. See, for example, Kunreuther et al. (2001).
See Jaspersen (2014) for a comprehensive review.
Many studies attempt to explain insurance markets by designing the experiments as auctions rather than choice tasks. See, for example, Camerer and Kunreuther (1989) and Hogarth and Kunreuther (1989). Although this design may work well as a mechanism for eliciting willingness to pay for insurance, and under a double auction, studying both sides of the insurance markets, the results are not necessarily generalizable to the insurance marketplace in which consumers face choice tasks rather than pricing tasks, as explained in Laury et al. (2009).
A driving quiz was chosen for the earnings task to ensure that all participants would be familiar with the subject matter and could successfully complete the quiz but still have some risk of making mistakes. The median quiz score was 75%. The median estimates for own score and others’ scores were 85 and 78% respectively. We required that participants had a driver’s license. The questions on the quiz were similar to those that would appear on a written state driving test. Although the risk of errors was therefore clearly related to auto accident risk, all instructions and the loss scenarios were framed in neutral language and not in the context of decisions over auto insurance per se.
Participants were presented with decisions, probabilities of loss, and total cost of precaution for each decision, separately for each scenario (presentation of a treatment). They were not presented with the expected loss. We did not use terms such as ‘precaution’ or ‘risk mitigation’, just the phrase ‘reduce your probability of loss.’
For example, a risk-neutral subject with an initial 10% probability of loss who estimates a score of 90% on the driving quiz is better off not purchasing insurance. But if the participant’s actual score on the driving quiz is 70% then the expected payoff is higher with insurance and such a subject is, therefore, penalized for error. Scoring rules are often applied in experiments to reward accurate reports of subjective probabilities, typically in the form of a fixed reward for the estimate plus a penalty for error. Yet in these cases subjects’ risk preferences can affect their reports. See, for example, Andersen et al. (2013) and Harrison et al. (2013).
Technically, because we don’t reward and penalize reported scores directly, participants could estimate one score, but report a different score. However, given the 20 different treatments and careful attention to detail required throughout the experiment we note this would be very cognitively costly. Combined with the lack of financial incentive to record a particular score different from a true estimate, we view this as highly unlikely.
The average probability of mistakes on the driving quiz was 25%, resulting in an expected probability of loss of 32.5% before risk mitigation.
In the other categories, full precaution is no longer a perfect substitute for insurance because of the risk of mistakes.
Analysis of subject-level data reveals only 12 reversals between the partial and full risk mitigation decisions across 120 decisions in the four no-mistakes treatments. There are four switches when the initial probability is 10% and eight under an initial probability of 32%. Separate McNemar tests by initial probability also show no significant difference in proportions.
These results are comparable to those found in the Laury et al. 2009 study.
We interpret this simply as a substitution away from insurance—insuring a realized loss has a relatively higher price increase under the low-loss event compared to the high-loss event. Given the loss occurs, then the insurance costs an additional $0.22 per dollar covered under the low loss event, but only an additional $0.02 per dollar covered under the high loss event.
We note that the 17 participants are distributed across all six sessions, with 1–4 instances in each session.
For example, participants may anticipate that choosing some risk-mitigation will lessen regret if a loss occurs, or may have an illusion of control resulting from taking a risk-mitigating action. See Jaspersen (2014) for discussion of entertainment value in hypothetical settings.
Because the ranking of outcomes remains constant across the design, we are unable to rule out rank-dependent expected utility (see Quiggin 1982), even if we find support for another representation.
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Acknowledgments
The authors would like to thank the anonymous referee, the Editor Kip Viscusi, Glenn Harrison, James Sundali, Bill Rankin, and seminar participants at Colorado State University, Ludwig-Maximilian University, University of Münster, and at a Behavioral Insurance Workshop sponsored by the Georgia State University Center for the Economic Analysis of Risk for helpful comments on earlier drafts of this paper. They are grateful for financial support from the Colorado State University College of Business and the Nevada Insurance Education Foundation.
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Appendices
Appendices
1.1 Appendix 1: Ambiguity aversion increases willingness to pay
In this Appendix we show that ambiguity aversion increases the willingness to pay to avoid risk when individuals can exercise care.
The probability of a loss is π(c, ε) where ε is a random variable with distribution F. We do not restrict the dependence of π on ε, nor do we require that beliefs be unbiased. Let
the argument is still valid if utility is separable in effort. The individual has the second order expected utility function
Let c* denote the optimal value of care. The willingness to pay to avoid the risk, P, is
where A is an ambiguity premium. Then we have
For an ambiguity neutral individual, the ambiguity premium is zero and the optimal level of care, c 0, maximizes E F {U(c, ε)}. Then
For an ambiguity averse individual the ambiguity premium is positive and the optimal level of care, c 1, maximizes E F {Φ(U(c, ε))} Willingness to pay is given by
Since E F {U(c 0, ε)} > E F {U(c 1, ε)} and A > 0, we have P 1 > P 0, ambiguity aversion increases the willingness to pay to avoid risk when an individual’s ability to take care affects the probability of a loss.
Now suppose that π is free of c, so that c 0 = c 1 = 0. Then E F {U(c 0, ε)} = E F {U(c 1, ε)}. Then A > 0 implies that P 1 > P 0. The results in Alary et al. (2010) and Snow (2011) are special cases of the result here.
1.2 Appendix 2: Examples of precaution-only and precaution and insurance treatments
Menu of choices in a precaution-only treatment:
Choose ONE of the following options below.
Decision | Up-front Cost to Replace Orange Balls | New # of Orange Balls | New # of White Balls | Probability Orange Ball is Drawn |
A | $0.00 | 10 | 90 | 10% |
B | $1.50 | 9 | 91 | 9% |
C | $3.00 | 8 | 92 | 8% |
D | $4.50 | 7 | 93 | 7% |
E | $6.00 | 6 | 94 | 6% |
F | $7.50 | 5 | 95 | 5% |
G | $9.00 | 4 | 96 | 4% |
H | $10.50 | 3 | 97 | 3% |
I | $12.00 | 2 | 98 | 2% |
J | $13.50 | 1 | 99 | 1% |
K | $15.00 | 0 | 100 | 0 |
Your decision in Scenario 1
Menu of choices in a precaution/insurance treatment:
Choose ONE of the following options below.
Decision | Up-front Cost to Replace Orange Balls | New # of Orange Balls | New # of White Balls | Probability orange ball is drawn |
A | $0.00 | 10 | 90 | 10% |
B | $1.50 | 9 | 91 | 9% |
C | $3.00 | 8 | 92 | 8% |
D | $4.50 | 7 | 93 | 7% |
E | $6.00 | 6 | 94 | 6% |
F | $7.50 | 5 | 95 | 5% |
G | $9.00 | 4 | 96 | 4% |
H | $10.50 | 3 | 97 | 3% |
I | $12.00 | 2 | 98 | 2% |
J | $13.50 | 1 | 99 | 1% |
K | $15.00 | 0 | 100 | 0 |
L (Insurance) | $14.50 | 10 | 90 | N/A |
Your decision in Scenario 7
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Bajtelsmit, V., Coats, J.C. & Thistle, P. The effect of ambiguity on risk management choices: An experimental study. J Risk Uncertain 50, 249–280 (2015). https://doi.org/10.1007/s11166-015-9218-3
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DOI: https://doi.org/10.1007/s11166-015-9218-3