Demand for Repeated Insurance Contracts
with Unknown Loss Probability
Emilio Venezian
Venezian Associates
Chu-Shiu Li
Feng Chia University
Chwen-Chi Liu
Feng Chia University
1
Agenda
 Introduction
 Purpose
 The basic assumptions
 Dynamics of self-selection for compulsory
coverage
 Dynamics of self-selection for voluntary
coverage
 Conclusion
2
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.
3
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.
4
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.
5
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
6
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 noncompulsory coverage.
7
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.
8
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.
9
The General Model
Updating priors on the accident frequency
The gamma distribution of an accident rate at time 0 is given as :
1
f ( ) 
k   1e  k
( )
  an accident rate per period
The parameters can be related to the mean and variance of
the random variable by :
 

k
 
2

k2
10
The General Model
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
Expected value
Variance
 
 n
 
2
k 1
 n
(k  1) 2
11
The General Model
If the individual has experienced n accidents,
the accident probability will be
p(n  )  e

n
n!
The priors of individuals follow a gamma distribution, thus
the probability of n accidents during the next period is:
 

(  n)  k   1 
p(n)   p(n  ) f ( ) d 

 

n!( )  k  1   k  1 
 o
This is a negative binomial distribution
12
n
The General Model
Max Eu(W )
W (n)  W  P( D)  nD
P( D)  C  D(1   )
1  e  rW
u (W ) 
r
Thus the optimal deductible for the individual is :
  (1   )( k  1) 
1
D  Ln 

r



(
1


)


*
13
Under Compulsory Coverage System
--- Selection of a Deductible
14
Dynamics of Self-Selection for Compulsory
Coverage Given Two Deductibles
The choice of a low deductible , D1 , implies that
Eu(W ( D1 ))  Eu(W ( D2 ))
or, equivalently :
e  rD1 (1 ) (k  1  erD1 )   e  rD2 (1 ) (k  1  erD2 ) 
r ( D2  D1 )(1   )
 
 k  1  e rD1 
Ln
 k  1  e rD2 



  z (k )
where
z (k ) 
(30)
r ( D2  D1 )(1   )
 k  1  e rD1 

Ln
rD2 
 k 1 e 
15
Dynamics of Self-Selection for Compulsory
Coverage Given Two Deductibles
After one period, n accidents have been experienced,
then individuals will switch from Low deductible to
High Deductible if
  z (k )
n  z (k  1)  
    z (k )
16
Under Voluntary Coverage System
--- Selection of a Deductible
Or no insurance
17
Dynamics of Self-Selection for Voluntary
Coverage Given Two Deductibles
The condition for selecting no insurance can be expressed as
r (C  D2 )(1   )
 
 k  1  e rD2 
Ln
 k  1  e rC 



where
D*  C
In the next period, the individual with no insurance to switch
to insurance with a deductible Di we have
r (C  Di )(1   )
 n
 k  2  e rDi 
Ln
 k  2  e rC 



18
Dynamics of Self-Selection
for Voluntary Coverage
The condition for switching is, therefore
r (C  Di )(1   )
r (C  Di )(1   ) r (C  Di )(1   )
n
 

0
rDi
rDi
rD2
 k 2e 
 k 2e 
 k 1 e 



Ln
Ln
Ln
rC 
rC 
rC 
 k 1 e 
 k 2e 
 k 2e 
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
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