An actuarial risk model is a mathematical description of the behavior of a collection of risks generated by an insurance portfolio. It is not intended to replace sound actuarial judgment. In fact, a well formulated model is consistent with and adds to intuition, but cannot and should not replace experience and insight (Willmot; 2001). Even though we cannot hope to identify all influential factors relevant to future claims, we can try to specify the most important.
A typical model for insurance risk, the so-called collective risk model, has two main components: one characterizing the frequency (or incidence) of events and another describing the severity (or size or amount) of gain or loss resulting from the occurrence of an event, see also Chapter 18. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk (Embrechts, Kaufmann, and Samorodnitsky; 2004). In the former, for example, the main risk components are the number of credit events (either defaults or downgrades), and the amount lost as a result of the credit event.
The stochastic nature of both the incidence and severity of claims are fundamental components of a realistic model. Hence, in its classical form the
model for insurance risk is defined as follows (Embrechts, Klüppelberg, and Mikosch; 1997; Grandell; 1991). If
is a probability space carrying
(i) a point process
, i.e. an integer valued stochastic process with
a.s.,
for each
and
nondecreasing realizations, and (ii) an independent sequence
of positive independent and identically distributed
(i.i.d.) random variables, then the risk process
is given by
The modeling of the aggregate loss process consists of modeling the point process and the claim size sequence
. Both processes are usually assumed to be independent, hence can be treated independently of each other. The modeling of claim severities was covered in detail in Chapter 13. The focus of this chapter is therefore on modeling the claim arrival point process
.
The simplicity of the risk process (14.1) is only illusionary. In most cases no analytical conclusions regarding the time evolution of the process can be drawn. However, it is this evolution that is important for practitioners, who have to calculate functionals of the risk process like the expected time to ruin and the ruin probability, see Chapter 15. All this calls for numerical simulation schemes (Burnecki, Härdle, and Weron; 2004).
In Section 14.2 we present efficient algorithms for five classes of the claim arrival point processes. Next, in Section 14.3 we apply some of them to modeling real-world risk processes. The analysis is conducted for the same two datasets as in Chapter 13: (i) the PCS (Property Claim Services) dataset covering losses resulting from catastrophic events in USA that occurred between 1990 and 1999 and (ii) the Danish fire losses dataset, which concerns major fire losses of profits that occurred between 1980 and 1990 and were recorded by Copenhagen Re. It is important to note that the choice of the model has influence on both the ruin probability (see Chapter 15) and the reinsurance strategy of the company (see Chapter 20), hence the selection has to be made with great care.