1
Introduction
1.1 Rating of Risks and Claims Experience in Insurance
The basic idea underlying insurance is that individuals, who have the “same”
exposure to a particular risk, join together to form together a “community-
at-risk” in order to bear this perceived risk. In modern society, this idea is
most often realized in the form of insurance. On payment of a premium,
the individuals in the “community-at-risk” transfer their risk to an insurance
company.
We consider an insurance company with a portfolio consisting of
I
insured
risks numbered
i
=1
,
2
,...,I
.Inawe
l
l
-de
f
ned insurance period, the risk
i
produces
•
a number of claims
N
i
,
•
with claim sizes
Y
(
:
)
i
(
:
,
2
,...,N
i
)
,
•
which together give the aggregate claim amount
X
i
=
P
N
i
:
=1
Y
(
:
)
i
.
We will refer to the premium payable by the insured to the insurer, for
the bearing of the risk, as the gross premium. The premium volume is the
sum, over the whole portfolio, of all gross premiums in the insurance pe-
riod. The basic task underlying the rating of a risk is the determination of
the so-called pure risk premium
P
i
=
E
[
X
i
]
.O
f
t
en
,w
eu
s
eju
s
tth
et
e
rm
“risk premium”. The classical point of view assumes that, on the basis of
some objectively quanti
f
able characteristics, the risks can be classi
f
ed into
homogeneous groups (risk classes), and that statistical data and theory (in
particular, the Law of Large Numbers) then allow one to determine the risk
premium to a high degree of accuracy.
In reality, of course, it is clear that a huge number of factors contribute to
the size of the risk premium. In order to have reasonably homogeneous risk
classes, one would have to subdivide the portfolio into a very large number
of classes. For example, if we were to use four characteristics to rate the risk,
then assuming that each characteristic had 10 possible values, this would lead