1 Demand for Repeated Insurance Contracts with Unknown Loss Probability Emilio Venezian Venezian Associates Chwen-Chi Liu Feng Chia University Chu-Shiu Li Feng Chia University

2 Agenda Introduction Purpose The basic assumptions Dynamics of self-selection for compulsory coverage Dynamics of self-selection for voluntary coverage Conclusion

3 Introduction -1 Under repeated contracting for automobile insurance, the insured might stay with the same insurer and the same policy or switch to other policy, switch to another insurer or even buy no insurance for the next year. Thus, this paper tries to build a simple theoretical model to examine the buying behavior of multi-period contract.

4 Introduction -2 Mossin (1968) assumes that insurer’s estimate of the probability of loss is the same as the insured. Venezian (1980) the first to examine a model in which the probability is not knowable, but this has never been used in a framework of choice of insurance coverage. Eisenhaier(1993) assumes that insurers and the insured hold different estimates of the probability of loss.

5 Introduction -3 Jeleva and Villeneuve (2004) assume that consumers whose beliefs and objective probability differ. Venezian (2005) argues that the relevant utility function is not the one that applies at the time that the decision is made, it is the one that applies when uncertainty is resolved. Li, et al.(2007) find out that decision makers tend to stick with prior insurance policy or may it be evidence of rational behavior.

6 Introduction -4 Several papers explore multi-period insurance contracts such as Palfrey et al.(1995), Cooper and Hayes(1987) Dionne and Doherty (1994), Nilssen (2000) Reynolds(2001) However, none of these papers take into account the role of unknown loss probability

7 Purpose To explore the choice of deductibles by individuals and the effect on sequential decisions of assuming that the decision makers are uncertain about the accident frequency that will be observed in the policy year. To examine how likely decision makers who chose high deductible and experience one accident are likely to switch to low deductible. To analyze a theoretical model under the cases in which insurance is compulsory coverage and non- compulsory coverage.

8 The General Model Assumptions-1 We assume that a population is actually homogeneous with respect to accident rate and has accidents that follow a Poisson distribution. Individuals differ with respect to their priors on their own accident rates, each having a gamma distribution as the form of the prior, but with parameters that may differ from those of their peers.

9 The General Model Assumptions-2 Individuals are utility maximizers with constant absolute risk aversion that is known to the individuals, but the risk aversion may differ among individuals. To enquire on the conditions under which Bayesian incorporation of information about accidents into the prior distribution of the accident rate of individuals might account for observations of switching behavior.

10 Updating priors on the accident frequency The gamma distribution of an accident rate at time 0 is given as : The parameters can be related to the mean and variance of the random variable by : The General Model

11 If the individual experience n accidents, so the posterior, which is the prior at the beginning of the next period (at time 1), is a gamma distribution with Variance The General Model Expected value

12 This is a negative binomial distribution The General Model If the individual has experienced n accidents, the accident probability will be The priors of individuals follow a gamma distribution, thus the probability of n accidents during the next period is:

13 Thus the optimal deductible for the individual is : The General Model

14 Under Compulsory Coverage System --- Selection of a Deductible

15 Dynamics of Self-Selection for Compulsory Coverage Given Two Deductibles The choice of a low deductible, D 1, implies that or, equivalently : where

16 Dynamics of Self-Selection After one period, n accidents have been experienced, then individuals will s witch from Low deductible to High Deductible if Dynamics of Self-Selection for Compulsory Coverage Given Two Deductibles

17 Under Voluntary Coverage System --- Selection of a Deductible Or no insurance

18 The condition for selecting no insurance can be expressed as In the next period, the individual with no insurance to switch to insurance with a deductible we have where Dynamics of Self-Selection for Voluntary Coverage Given Two Deductibles

19 Dynamics of Self-Selection for Voluntary Coverage The condition for switching is, therefore Thus at least one accident is necessary for a switch from no insurance to insurance with some deductible, but one accident might not be sufficient.

20 Conclusion -1 A simple model of uncertainty in accident frequency with Bayesian updating of the prior distribution can explain the main features of switching behavior in insurance purchases. The model implies that a single accident is NOT sufficient to motivate a switch from high to low deductible and a single accident free period is NOT enough to motivate a switch from low to high deductible.

21 Conclusion -2 Absolute certainty in the value of accident frequency implies that experience will not affect the change in the selection of a deductible. Some uncertainty in the estimate will lead to Bayesian updating and the possibility of switches based on past history. We suggest that the failure to switch from high to low deductible after one accident, or from low to high deductible after one accident free period may just be a maximization of expected utility under uncertainty.

22 Thank you