The CAT bond we are interested in is described by specifying the region, type of events, type of insured properties, etc. More abstractly, it is
described by the aggregate loss process and by the threshold loss
. Set a probability space
and
an increasing filtration
,
. This leads to the following assumptions:
Therefore, one has
The definition of the process implies that is left-continuous and predictable. We assume that the threshold event is the time when the
accumulated losses exceed the threshold level
, that is
. Now define a new process
. Baryshnikov et
al. (1998) show that this is also a doubly stochastic Poisson process with the intensity
In Figure 4.2 we see a sample trajectory of the aggregate loss process (
10 years) generated under the assumption of
log-normal loss amounts with
and
and a non-homogeneous Poisson process
with
the intensity function
, a real-life catastrophe loss trajectory (which will be analysed in detail in Section
4.3), the mean function of the process
and two sample 0.05- and 0.95-quantile lines based on
trajectories of the aggregated
loss process, see Chapter 14 and Burnecki, Härdle, and Weron (2004). It is evident that in the studied log-normal case, the historical trajectory falls
outside even the
-quantile line. This may suggest that ``more heavy-tailed'' distributions such as the Pareto or Burr distributions would be
better for modelling the``real'' aggregate loss process. In Figure 4.2 the black horizontal line represents a threshold level of
billion USD.
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