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Demand for Repeated Insurance Contractswith Unknown Loss Probability

Emilio Venezian Venezian Associates

Chwen-Chi LiuFeng Chia University

Chu-Shiu LiFeng Chia University

2

Agenda

Introduction Purpose The basic assumptions Dynamics of self-selection for compulsory

coverage Dynamics of self-selection for voluntary

coverage Conclusion

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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.

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Introduction-2

Mossin (1968) assumes that insurer’s estimate of the probability of lo

ss 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.

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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 t

hat 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 in

surance policy or may it be evidence of rational behavior.

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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

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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.

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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.

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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.

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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 :

kekf

1

)(

1)(

k

22

k

periodper rateaccident an

The General Model

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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

1

k

n

22

)1(

k

n

The General Model

Expected value

12

n

o kk

k

n

ndfnpnp

1

1

1)(!

)( )( )()(

This is a negative binomial distribution

!)(

nenp

n

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:

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Thus the optimal deductible for the individual is :

)( WuEMax

)1(

)1)(1(1*

k

Lnr

D

)1()( DCDP

r

eWu

rW

1)(

nDDPWnW )()(

The General Model

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Under Compulsory Coverage System

--- Selection of a Deductible

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Dynamics of Self-Selection for Compulsory Coverage Given Two Deductibles

The choice of a low deductible , D1 , implies that

or, equivalently : ))(())(( 21 DWuEDWuE

)1()1( 2211 )1()1( rDDrrDDr ekeeke

(30)

11

)1)((

2

1

12

rD

rD

ekek

Ln

DDr

2

1

11

)1)(()( 12

rD

rD

ekek

Ln

DDrkz

)(kz where

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Dynamics of Self-Selection

After one period, n accidents have been experienced, then individuals will switch from Low deductible to High Deductible if

)(kz

)1(kzn

)(kz

)(kz

Dynamics of Self-Selection for Compulsory Coverage Given Two Deductibles

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Under Voluntary Coverage System --- Selection of a Deductible

Or no insurance

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rC

rD

ekek

Ln

DCr

11

)1)((2

2

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

rC

rDi

ekek

Ln

DCrn

i

22

)1)((

iD

C*Dwhere

Dynamics of Self-Selection for Voluntary Coverage Given Two Deductibles

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Dynamics of Self-Selection

for Voluntary Coverage

0

11

)1)((

22

)1)((

22

)1)((2

rC

rDi

rC

rDi

rC

rDi

ekek

Ln

DCr

ekek

Ln

DCr

ekek

Ln

DCrn

ii

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.

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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.

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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.

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Thank you