Women’s Travel Issues
Proceedings from the Second National Conference
740
modeling by industry actuaries that treats traffic accidents as a random process, however, shows that
this popular idea is erroneous (Industry Advisory Committee 1979, Butler and Butler, 1989, Butler,
1993b). A simple thought-experiment explains why. Imagine a jar containing 100 black balls
representing individual cars. Draw out one ball at random to represent an accident involvement, and
then replace it in the jar and stir before the next drawing. To keep track of the accident record of
individual balls, change the color of a drawn ball before replacing it from black to white (first
accident), then from white to green if drawn a second time, then from green to red for a third draw of
the same ball. Since 100 insured cars typically produce 5 claims a year and since insurers use the
records of the past three years to determine surcharges, draw and replace 15 balls. Then count the
balls by color. Poisson probability predicts that about 86 of the balls will still be black (accident
free), 13 white (1 accident), 1 green (2 accidents), and 0.05 red. If the experiment were scaled up to
10,000 balls—approximately the number of cars actuaries require for a credible risk class—five of
them would be red, indicating 3 accidents apiece in a period of three years. Is this proof that some
balls are more likely to be drawn than others? Not at all. By design, all of the balls had an equal
chance of being picked in each draw.
In defense of classifying cars by accident record, insurance companies point to the well-established
fact that the subclass of drivers who had accidents in a three-year period subsequently averages more
accidents in the following (fourth) year than accident-free drivers in the same class. This result,
however, can be modeled by specifying that not all of the balls spend the same amount of time in the
jar and thus have different chances of being drawn (Butler and Butler, 1989, pp. 206-208, Butler,
1993b, pp. 58-60). Rather than appealing to compound Poisson models of balls with different
exposures, we can instead think of an accident-record subclass as a random sample of a class of cars
on the road.
A three-year’s sample of cars picked at random by accident involvement from a class of cars would
include cars driven many miles and also cars driven few miles. The cars driven more than the class-
average, however, would be over-represented in the accident sample because they were more
exposed to risk of accident, while the cars driven less than average would be under-represented in
this sample. In the coming (fourth) year, therefore, the subclass of cars whose drivers have had
accidents in the last three years would average a higher mileage and more accidents than the large
class of cars with accident-free drivers. An example emphasizes important consequences.
Typically a subclass of cars defined by having had accidents in the past three years, taken from a
class with 5 claims a year per 100 cars, subsequently averages about 7.5 claims a year per 100 cars,
which is a fifty percent increase. This apparently large increase in accident rate would simply mean
that there has been a similar increase in annual mileage, say from a class average of 10,000 miles to
an accident subclass average of 15,000 miles. Finally, it is important to realize that despite this large
difference in accidents per year between the main accident-free class and the recent-accident
subclass, a very large majority of the cars in each would have identical accident-free records in this
fourth year: about 93% in the recent-accident subclass compared with slightly more than 95% in the
rest of the class.
In discussing a paper of mine that made the analogy between accident record classes and random
samples biased to higher average mileages, the chief actuary of the Automobile Insurers Bureau in
Boston noted that the effect of differences in miles of exposure to risk on the road "is one that I