Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

20 Actuarial Modelling of Claim Counts
(i) The process has independent increments
(ii) The number of events in any interval of length t follows a Poisson distribution with
mean t (therefore it has stationary increments), i.e.
PrNt + s − Ns = k = exp−t tk k= 0 1 2
k!
Exposure-to-Risk
The Poisson process setting is useful when one wants to analyse policyholders that have
been observed during periods of unequal lengths. Assume that the claims occur according to
a Poisson process with rate . If the policyholder is covered by the company for a period of
length d then the number N of claims reported to the company has probability mass function
PrN = k = exp−d dk k= 0 1
k!
that is, N ∼ oid. In actuarial studies, d is referred to as the exposure-to-risk. We see
that d simply multiplies the annual expected claim frequency in the Poisson model.
Time Between Accidents
The Poisson distribution arises for events occurring randomly and independently in time.
Indeed, denote as T 1 T 2 the times between two consecutive accidents. Assume further
that these accidents occur according to a Poisson process with rate . Then, the T k s are
independent and identically distributed and
PrT k >t= PrT 1 >t= PrN t = 0 = exp−t
so that T 1 T 2 have a common Negative Exponential distribution.
Note that in this case, the equality
PrT k >s+ tT k >s= PrT k >s+ t
PrT k >s
= PrT k >t
holds for any s and t ≥ 0. It is not difficult to see that this memoryless property is related
to the fact that the increments of the process Nt t ≥ 0 are independent and stationary.
Assuming that the claims occur according to a Poisson process is thus equivalent to assuming
that the time between two consecutive claims has a Negative Exponential distribution.
Nonhomogeneous Poisson Process
A generalization of the Poisson process is obtained by letting the rate of the process
vary with time. We then replace the constant rate by a function t ↦→ t of time t
and we define the nonhomogeneous Poisson process with rate ·. The Poisson process
defined above (with a constant rate) is then termed as the homogeneous Poisson process.
A counting process Nt t ≥ 0 starting from N 0 = 0 is said to be a nonhomogeneous
Poisson process with rate ·, where t ≥ 0 for all t ≥ 0, if it satisfies the following
conditions: