Journal
of Public
ADVERSE
Economics
20 (1983) 121-130.
SELECTION
AND
North-Holland
Publishing
STATISTICAL
An analysis of Canadian automobile
Company
DISCRIMINATION
insurance
B.G. DAHLBY*
Received
Statistical
February
1980, revised version
received
October
1981
discrimination
occurs when a characteristic,
such as sex, is used as an indicator of the
The theory of adverse selection is used to explain the occurrence of
st;ill\tical discrimination.
A model of the market for collision insurance,
which is based on the
~hcorv of adverse selection, is estimated
on Canadian
data. The results suggest that adverse
selectjon occurs in this market. Simulations
of the effect of prohibiting
sexual discrimination
in
the 21-24 age group indicate that the premiums for single females would increase substantially
and that a significant proportion
would no longer purchase collision insurance.
I 1\1\youp of an individual.
1. Introduction
In 1978, a Board of Industry
in Alberta ruled that insurance
premiums
based on the sex of the insured
contravene
The Individual’s
Rights
Protection
Act (Statutes of Alberta, 1972). This act prohibits discrimination
services or facilities customarily
in the provision
of ‘. . . any accomodation,
available
to the public, because of the race, religious beliefs, colour, sex.
ancestry or place of origin of that person. ..‘. The Board of Inquiry
was
established because a number of males complained
that automobile
insurance
companies charge females, with similar characteristics,
lower premitims. This
practice by the insurance
industry is based on statistical evidence that the
accident rate for young females is lower than that for young males. However,
the Board of Inquiry ruled that ‘if a discrimination
prohibited by law exists it
is no less a prohibited discrimination
because it is supported by statistics’.’ A
policy of eliminating,
in stages, age, sex, marital
status, and geographic
location as factors in setting auto insurance
premiums has been initiated by
the publicly-owned
Insurance
Corporation
of British Columbia
under its
Fundamental
Auto Insurance
Rating (FAIR) program.
*I would like to thank A. Buse, N. Doherty, D. Gillen and two anonymous referees for their
comments
and suggestions,
A. Cooper and J. Lyndon of the Insurance
Bureau of Canada for
supplying
the data, and Y.K. Lee for his research assistance.
The financial
support
of the
Faculty of Graduate
Studies and Research at the University of Alberta in the form of a research
grant is gratefully acknowledged.
‘4lberta Human Rights Commission
(1978, p. 18).
0047-2727/83/000&0000/$03.00
0
1983 North-Holland
122
B.G. Dahlhy,
Adverse selection
tmd stutisticai
discrimination
Whether insurance
companies
should be able to levy premiums based on
the average risk of individuals
in different sex, marital status, or age groups
is a political
or ethical
question,
but the prohibition
of statistical
discrimination
has economic implications
because it will alter the premiums
that are levied in insurance markets. The objective of this paper is to analyze
statistical
discrimination
in insurance
markets using the theory of adverse
selection. In section 2 the theory of adverse selection is reviewed and the
nature
of the equilibrium
in a competitive
insurance
market
in which
statistical
discrimination
occurs is briefly described.
In section
3 some
statistics from the private automobile
insurance industry are presented which
suggest that there is an adverse selection problem in this market. A simple
model of the market for collision insurance, which is based on the theory of
adverse selection, is estimated and the results are consistent with the theory.
In the final section the model is used to simulate the effect of prohibiting
sexual discrimination
in collision
insurance
in the 21-24 age group. The
model
suggests
that the premiums
for single females
would
increase
substantially
and that there would be a significant decline in the proportion
of single females purchasing collision insurance.
2. Adverse selection and statistical
discrimination
The problem
of adverse selection arises in insurance
markets when the
purchaser of insurance has more information
about the probability
of a loss
than the insurance
company
and it has been analyzed
by Akerlof (1970),
Rothschild
and Stiglitz (1976), Wilson (1977) Miyazaki (1977), and Spence
(1977). In this literature it is usually assumed that there are two states of the
world, and in one state there is an accident which costs C dollars. The cost
and the probability
of an accident are exogenous variables, and thus there is
no problem
of moral hazard. There are only two risk groups, high-risk
individuals
and low-risk individuals,
with accident probabilities
of rrn and rrr,
respectively.
All individuals
have the same wealth in the absence of an
accident
and utility function
which displays
risk aversion.
An insurance
company cannot observe the risk group of an individual
when he purchases
an insurance
policy, but the individual
knows his risk group. Assuming that
there are no costs in writing insurance
policies and that firms are risk
neutral,
Rothschild
and Stiglitz (1976) have shown
that with a Nash
equilibrium
high-risk individuals
purchase full coverage at an actuarially
fair
premium, low-risk individuals
purchase partial coverage at an actuarially fair
premium, and high-risk individuals
are indifferent between the two types of
policies. Wilson (1977) has introduced
an alternative
concept of equilibrium
in which each firm correctly anticipates
which policies will be dropped by
other firms when it changes its menu of policies and his analysis has been
B.C. Dahlhy,
Advrrse
selection
und .stntistic.ul discrimincilion
123
extended
by Miyazaki
(1977) and Spence
(1977) to allow for crosssubsidization
of contracts purchased by high-risk and low-risk individuals.
The theory of adverse selection can explain the existence of statistical
discrimination
in a competitive
insurance
market. [See Dahlby (1980) and
Hoy (1982) for a detailed
analysis
of statistical
discrimination
based on
adverse selection.]
Suppose that an insurance
company
can observe, at no
cost, the sex of the policy-holder
and that the probability
of an accident for a
female is less than that for a male because the proportion
of females who are
high risk is lower than the proportion
of males who are high-risk. Then a
firm may find it profitable to offer policies with lower premiums to females.
The insurance
market will then be segmented with a Wilson equilibrium
in
each segment. High-risk
individuals
will purchase
full coverage,
low-risk
individuals
will purchase partial coverage and the policies purchased by lowrisk individuals
will subsidize the policies purchased by high-risk individuals
of the same sex.
This theory of discrimination
in insurance
markets
based on adverse
selection is empirically
testable and it may shed some light on the nature of
the equilibrium
in insurance markets. The model predicts that the proportion
of individuals
purchasing
full coverage in each segment of the market will
vary directly with the average probability
of an accident in that segment.
Cross-subsidization
will occur within
each segment
with low-coverage
Cross-subsidization
between
policies
subsidizing
high-coverage
policies.
different segments of the market will not occur. The model predicts that
discrimination
can occur under competitive
conditions,
but non-myopic
behaviour
on the part of firms is required. The Nash equilibrium
concept,
does not explain
the existence
of
which assumes
myopic
behaviour,
discrimination
in insurance
markets in the presence of an adverse selection
problem. Thus, although Rothschild
and Stiglitz (1976) have argued that the
Wilson equilibrium
concept is not relevant for competitive
markets, it does
explain discrimination
in competitive
insurance
markets
while the Nash
equilibrium
does not. Of course this is not conclusive evidence in favour of
the Wilson concept because discrimination
in insurance markets may be due
to a non-competitive
market structure.
In the following section we have attempted
to test for the presence of
adverse
selection
in
the
Canadian
insurance
market.
automobile
Unfortunately,
the available data do not indicate the premiums
levied and
the number of cars insured at different levels of coverage. Therefore, we have
been unable to perform a direct test of the model of adverse selection
described
above which emphasizes
differences
in the levels of coverage
purchased
by high-risk and low-risk drivers and the cross-subsidization
of
high- and low-coverage
policies within each segment of the market. Instead,
we have used Akerlof’s hypothesis
that low-risk individuals
are more prone
to drop out of an insurance
market
than are high-risk
individuals
in
developing a simple model of adverse selection which is empirically testable.
124
B.G. Dahlhy,
Adverse selection
and statistical
3. Adverse selection in Canadian automobile
discrimination
insurance: An empirical analysis
The characteristics
which are mainly used in setting premiums
by the
private automobile
insurance
industry
in Canada are the age, sex, marital
status,
and claim history
of the principal
driver
of the automobile.
Discrimination
on the basis of sex and marital status is primarily used in the
under 25 age group. Table 1 shows some statistics for the 21-24 age group in
urban areas in Canada for the policy years 1975-78. There are two major
types of automobile
insurance,
namely bodily injury and property damage
(BIPD) insurance (which is also known as third party liability insurance), and
collision insurance.
BIPD insurance
was compulsory
in all of the provinces
from which these data were compiled (except Ontario
where most drivers
took out BIPD insurance
in any case because they were required to have
$100,000 minimum
coverage) and collision insurance
was optional. The first
row of the table shows that the average claim frequency for BIPD insurance
was substantially
higher for single males than for married males or females.
Table
Canadian
Driver
automobile
BIPD
statistics
1
for the 21-24 age group,
1975-78.
class
Age
Sex
Marital
insurance
status
21-22
23-24
Male
Single
Male
Single
21-24
Male
Married
21-24
Female
Single or
married
0.124
0.104
0.0980
0.0821
0.09 17
0.0783
0.0741
0.0659
0.175
0.311
0.312
0.289
0.109
0.0998
$162
$147
$96
$87
0.694
0.747
insurance
Claim frequency
entire
class (7~~)
-at least 5 years
claim-free (7r.J
Proportion
of
drivers with at
least 5 years of
claim-free driving
Collision insurance ($100 and $250 deductible)
Claim frequency
entire
class
0.123
0.142
(nC)
Average premium
in 1978 (P)
$264
$192
Expected loss
cost in 1978
$162
$137
(nC C)
Proportion
of
drivers purchasing
collision insurance
0.588
0.497
(Z)
Source:
Experience
1977
The Insurance
Bureau
of Canada,
and 1978 Automobile Insurance Experience.
Automobile
Insurance
B.C. Dahlby, Adverse selection
and statistical
discrimination
125
The second row shows that the average claim frequency for drivers with at
least live years of claim-free driving was about 25 percent lower than the
average claim frequency for all individuals
in that driver class. These figures
indicate that each class of drivers is not homogeneous
with respect to the
risk of an accident, but can be viewed as containing
different proportions
of
as
was
assumed
in
the
theoretical
model
high-risk
and low-risk drivers,
described in section 2. If there are members of the two risk groups in each
class of drivers,
then the average
probability
of an accident
among
individuals
with zero accidents for n years approaches
rrL as n approaches
infinity. In the empirical
analysis which follows, we have used the claim
frequency for BIPD insurance
among individuals
with at least live years of
claim-free driving as a proxy for the probability
of an accident for a low-risk
individual
in that driver class. The third row of the table shows that the
proportion
of drivers with at least live years of claim-free driving is about the
same for single males 23-24 years of age, married males, and females in spite
of the differences among these classes in the average claim frequencies and
the claim frequencies of individuals
with live years of claim-free driving. This
suggests that there may be differences in rrL between driver classes.
The average claim frequency
for individuals
purchasing
$100 or $250
deductible
collision insurance
is shown in the fourth row of the table. The
average claim frequency for collision insurance is higher than that for BIPD
insurance
in each driver class. One possible explanation
of higher claim
frequencies for collision insurance is that it may cover accidents where there
are no claims for third party damages, i.e. single car accidents where there is
no damage to the persons or property of others. However, these statistics
may also indicate the presence of adverse selection in the market for collision
insurance.
The theory of adverse selection predicts that low-risk individuals
will purchase
less than full coverage
for collision
damages
if insurance
companies cannot distinguish low-risk and high-risk individuals.
If some lowrisk individuals
do not purchase $100 or $250 deductible collision insurance,
then the average accident rate among those purchasing
collision insurance
will exceed the average accident rate for the entire population,
which in this
case may be approximated
by the claim frequency for BIPD insurance. Thus,
the higher claim frequency for collision insurance
may be explained by the
decisions of some low-risk individuals
not to purchase collision insurance
because it is not attractive
at the high premiums
that must be charged
because of the presence of high-risk individuals
in the market.
Table 1 also shows the average premium per car insured and the expected
loss per car insured in 1978, where the latter variable is the product of the
average claim frequency for collision insurance
and the average loss cost per
claim to the insurance
industry. Finally, the table shows that proportion
of
drivers purchasing
collision insurance was highest among females and lowest
among single males 21-22 years of age.
126
B.G. Dnhlhy,
Adverse selection
und statistical
discrimination
In attempting
to test for adverse selection in the market for collision
insurance,
we have utilized data from the Insurance
Bureau of Canada for
nine classes of drivers over the age of 20 for the years 1975-78. The figures
do not include Manitoba,
Saskatchewan,
and British Columbia
which have
public insurance
programs.
The proportion
of drivers purchasing
collision
insurance
in each driver class was obtained by dividing the number of cars
insured for collision by the numbers
of cars insured for bodily injury and
property damage. The data include all drivers in a given driver class and not
just those drivers with the same history of claims. We have treated the
average loss cost per claim in each driver class, C, as an exogenous variable
which proxies the average payout by the insurance industry. However, there
may be systematic
differences
in C between
driver classes because
of
variations
in coverage, and C does not equal the average payout because it
includes adjustment
costs. A more detailed description
of the data is given in
an appendix which is available from the author upon request.
In testing for adverse selection in the market for collision insurance,
we
have postulated
the following simple model which focuses on the decision to
purchase collision insurance:
lnZ=a,+cc,ln(P/(~,~C))+cc,lnh-t-
$
6jDj;
j=l
cc,<o, cc,>o,
Eq. (la) is based on the hypothesis
that the proportion
of individuals
in a
given driver class that purchase collision insurance,
Z, is inversely related to
the relative price of insurance for a low-risk individual
and is directly related
to the proportion
of high-risk individuals
in that class. The relative price of
insurance is the ratio of the average premium, P, to the expected payout for
a low-risk individual where the latter variable is the product of 7cL, the BIPD
claim frequency for drivers with live or more years of claim-free driving, and
C, the average loss cost per claim. A proxy for the proportion
of high-risk
drivers, h, is given by ((xb -xJ/Q
where nb is the average claim frequency
for BIPD insurance.’
A dummy
variable
for driver class j, Dj, is also
‘With
two
risk
+ In ((%,/RL) - 1)).
groups,
&’ is equal
to
(hn,,+(l
-h)n,),
and
h~((n~-~~)/nJ
equals
(In h
E.G. Dahlby, Adverse selection and statistical discrimination
127
included
in the equation
to reflect the effects of other variables,
such as
average net worth and attitudes
toward
risk, which may influence
the
decision to purchase collision insurance.
In eq. (2a) the premium
for collision
insurance
is determined
by the
expected loss cost per car insured,
rccI C, where nc is the average claim
frequency
for collision
insurance
in that driver class, a measure
of the
insurance risk for that driver class, R, and a time trend t. The elasticity of the
premium with respect to the expected loss cost per car should be positive but
less than one because the other costs associated
with providing
insurance,
such as administration
costs, probably
do not vary with the expected loss
cost. A measure of insurance
risk has been included in eq. (2a) because the
segmentation
of the insurance
market may limit the extent of risk-pooling
and because there may be unexpected
changes in the cost of automobile
repairs. The measure of insurance
risk that we have used is the ratio of the
variance in the loss cost per car insured to the average loss cost per car
insured for the period 1975-78 in each driver class. That is, R is equal to
(var (xc. C)/nTc),
where n”
is the average loss cost per car insured for the
period 1975-78. The data indicate that this measure of insurance
risk is
generally lower in the driver classes where the number
of cars insured is
relatively large. It should be noted that this measure of insurance
risk is
based on the variability
of the loss cost per car insured for the entire
industry and thus it is probably an underestimate
of the insurance risk for an
individual
firm. Finally, a time trend was included because a study of the
Canadian
insurance
industry
by Quirin et al. (1974) indicated
a significant
decline in operating costs over time.
In eq. (3a), rcc is hypothesized
to vary directly with h and rc,_ and inversely
with Z. The latter relation arises because low-risk individuals
have a greater
tendency to drop out of the insurance
market than do high-risk individuals.
Thus. the lower the proportion
of individuals
in a given driver class
purchasing
collision insurance, the higher the ratio of high-risk individuals
to
all individuals
purchasing
collision
insurance
and hence the higher the
average claim frequency
for collision
insurance.
Therefore,
in this simple
model of adverse selection, Z, P, and 7~’ are endogenous
variables, and h, rcL,
C, R, t, and the Dts are exogenous variables.
This system of equations
was estimated using two-stage least-squares
on
pooled cross-section
data for nine driver classes for the policy years 19755
78.” The results are shown below with the absolute values of the r-statistics
3The observation
for driver class 06 (occasional
male
included in the regressions because the value for ((nb-~JzL)
as large as it was in other years.
driver under 25) in 1978 was not
in that year was about three times
B.C. Dahlby, Adverse selection
128
and statistical
discrimination
in parentheses:
lnZ=0.3160-0.2893
(1.83)
ln(P/(rrr,.
C))+O.2441
(2.64)
lnh;
(4.29)
R2 = 0.8475,
In P= 1.3376+0.8793
(5.54)
(lb)
ln(n”. C)+O.O2193 In
(16.67)
R-0.07772t;
(2.05)
(5.32)
R= = 0.9482,
(2b)
In 7cc=0.6051 +0.2175 In h+0.9916
(6.41)
(10.52)
(40.29)
In rcL-0.07639
In Z;
(1.22)
R2 = 0.9826.
(3b)
All of the coefficients in the demand relation in eq. (lb) have the predicted
signs and are significant
by the t-test. (The estimated
coefftcients of the
dummy variables are not reported here, but are shown in an appendix which
is available from the author upon request.) One problem with the log-linear
specification
of eq. (lb) is that Z is not restricted to be less than one. A logit
specification
of the demand relation was also estimated and is shown below:
=4.568 - 2.609 In (P/(X,. C)) + 0.8792 In h,
(5.57)
R2 = 0.9205.
(4)
The logit specification
also indicates
that the proportion
of individuals
purchasing
collision insurance in a given driver class varies inversely with the
relative price of insurance
for a low-risk individual
and directly with the
proportion
of high-risk individuals
in that driver class. Eq. (2b) indicates that
the elasticity of the premium with respect to the expected loss cost per car
insured is positive, as expected, and the null hypothesis that p2 equals one is
rejected by the t-test at the 95 percent confidence
level. The measure of
insurance
risk has a significant
positive
effect on premiums
and the
coefficient of the time trend indicates
that, in the absence of increases in
expected loss costs, the average premium would have declined at an average
annual rate of 8 percent, which is somewhat higher than anticipated.
Eq. (3b)
indicates that h and zL have significant positive effects on 7~’ and that Z has
B.G. Dahlhy, Adverse selection and statistical
discrimination
129
a negative effect, but the null hypothesis that y4 is zero cannot be rejected at
the 95 percent confidence level.
The within sample predictions
of Z, P, and rc’, based on reduced form
equations
using the estimates of the structural
parameters
in eqs. (lb)(3b),
were calculated
and the statistics in table 2 provide some measures of the
predictive
power of the model. The correlations
between the actual and
predicted values of the variables are high and the mean absolute prediction
error is less than 5 percent of the mean value of Z and rcc and 8 percent of
the mean value of P. Therefore we conclude that the simple model estimated
in eqs. (lbH3b)
is consistent
with the observed behaviour
of the market for
collision insurance, and it supports the view that there is an adverse selection
problem in this market.
Table 2
Measures
of the within sample
predictive
P
Z
Correlation
coefficient
between the actual and
predicted values
Regression coeffkient
of actual on predicted
Root-mean-squared
error
Mean absolute error
Ratio of the mean
absolute error to the
mean values
power of the model
71’
0.901
0.970
1.028
0.0409
0.03 1
1.044
15.60
12.85
0.995
0.00438
0.00368
0.047
0.080
0.0381
4. Simulating the effect of prohibiting
collision insurance
discrimination
0.988
on the basis of sex in
The model
has been
used to simulate
the effect of prohibiting
discrimination
on the basis of sex in collision insurance
in the 21-24 age
group in 1977. This year was chosen for the simulation
because the average
absolute prediction
error of the premiums
with discrimination
was smallest
in this year. If discrimination
on the basis of sex had been prohibited
in
1977, it is assumed that uniform premiums would have been charged in three
classes of drivers, namely single individuals
21-22 years, single individuals
23-24 years, and married individuals
21-24 years. The first row in table 3
shows the predicted premiums
in 1977 with discrimination4
The predicted
premiums in 1977 without discrimination
are $221 for a single individual
21I
22 years, $187 for a single individual
23-24 years, and $153 for a married
individual
21-24 years. As anticipated,
the premiums
paid by males decline
4The differences betweyl the actual premiums and the predicted premiums are $21 for single
males 21-22 years, -$6 for single males 23-24 years, $7 for married males 21-24 years. and $16
for females 21 24 years.
B.G. Dahlby, Adverse selection and statistical discrimination
130
Table 3
A simulation
of the effect of prohibiting
on the basis of sex in collision
23-24
Single
21-22
Single
Age
Marital
Sex
discrimination
in 1977.
status
Male
Predicted premium
With discrimination
Without discrimination
Change
Percentage
change
$250
$221
- $29
-11.6
Predicted proportion
of drivers purchasing
collision insurance
With discrimination
Without discrimination
Change
0.47
0.49
0.02
Female
$137
$221
$84
61.3
0.73
0.63
-0.10
Male
$205
$187
-$18
- 8.8
0.56
0.57
0.01
Female
$137
$187
$50
36.5
0.73
0.67
-0.06
insurance
21-24
Married
Female
Male
$160
$153
-$7
-4.4
0.69
0.70
0.01
$137
$153
$16
11.8
0.73
0.71
- 0.02
and the premiums paid by females increase when discrimination
on the basis
of sex is prohibited.
The increases in the premiums paid by single females are
62.3 and 36.5 percent in the 21-22 and 23-24 age groups, respectively, Table
3 also shows that the percentage
of males purchasing
collision insurance
would increase
slightly and that the percentage
of females purchasing
collision
insurance
would decline by 10 percentage
points among single
females 21-22 years, six percentage points among single females 23-24 years,
and two percentage
points among married females 21-24 years. Thus, the
model predicts that the prohibition
of sexual discrimination
in collision
insurance would lead to substantial
increases in the premiums paid by single
females and a significant
decline in the proportion
purchasing
collision
insurance.
References
Akerlof, G., 1970, The market for lemons: Qualitative
uncertainty
and the market mechanism,
Quarterly
Journal of Economics 84,488-500.
Alberta Human Rights Commission,
1978, Report of a board of inquiry (Edmonton).
Dahlby,
B.G., 1980, The welfare effects of prohibiting
statistical
discrimination
in insurance
markets, Research paper 80-5, Department
of Economics, University of Alberta.
Hoy, M., 1982, Categorizing
risks in the insurance industry, Quarterly
Journal of Economics 97,
321-336.
Insurance Bureau of Canada, 1977 and 1978, Automobile insurance experience (Toronto).
Miyazaki, H., 1977, The rat race and internal labour markets, Bell Journal of Economics 8, 394
418.
Quirin, G., et al., Competition,
economic efficiency and profitability
in the Canadian
property
and casualty insurance industry (Insurance Bureau of Canada, Toronto).
Rothschild,
M. and J. Stiglitz, 1976, Equilibrium
in competitive
insurance
markets, Quarterly
Journal of Economics 90, 629-649.
Spence, M., 1977, Product
diflerentiation
and performance
in insurance
markets, Journal
of
Public Economics 10, 427447.
Wilson, C., 1977, A model of insurance
markets
with incomplete
information,
Journal
of
Economic Theory 16, 167-207.