Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2008
Part 5 – Aggregate Loss Models
Roughly speaking, an aggregate loss model is used to describe the total loss of an insur-
ance portfolio in a ﬁxed time period.
1.
Individual Risk Model:
There are
n
policyholders in an insurance portfolio. Assume
that policyholder
i
will produce a loss/claim of
X
i
, i
= 1
,
2
,...,n.
Then, the total or
aggregate loss of the insurance is
S
n
=
X
1
+
···
+
X
n
.
Such an aggregate loss model is called the individual risk model.
2.
Collective Risk Model:
The number of claims in an insurance portfolio is a counting
random variable
N
. The amount of the
i
th claim is
X
i
, i
= 1
,
2
,...
. Then the aggregate
loss/claim of the insurance is
S
=
X
1
+
···
+
X
N
with
S
= 0 if
N
= 0. Such an aggregate loss model is called the collective risk model.
Note that unless stated otherwise, in a collective risk model, we assume that
N,X
1
,X
2
,...
are independent and
X
1
,X
2
,...
have the same distribution function
F
(
x
) as
X
. Fur-
thermore, we denote the probability function of
N
by
p
n
= Pr
{
N
=
n
}
,n
= 0
,
1
,
2
,....
(a) The distribution function of
S
is given by, for any
x
,
F
S
(
x
) = Pr
{
S
≤
x
}
=
∞
X
n
=0
Pr
{
X
1
+
···
+
X
n
≤
x
}
p
n
=
∞
X
n
=0
p
n
F
*
n