4.2 Compound Doubly Stochastic Poisson Pricing Model

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 $ L_s$ and by the threshold loss $ D$. Set a probability space $ (\Omega,\mathcal{F},\mathcal{F}{}_t,\nu)$ and an increasing filtration $ \mathcal{F}{}_t\!\subset\!\mathcal{F}$, $ t\!\in\![0,T]$. This leads to the following assumptions:

Therefore, one has

$\displaystyle L_t=\sum_{t_i<t} X_i=\sum_{i=1}^{M_t} X_i.
$

The definition of the process implies that $ L$ is left-continuous and predictable. We assume that the threshold event is the time when the accumulated losses exceed the threshold level $ D$, that is $ \tau=\inf\{t:L_t\geq D\}$. Now define a new process $ N_t=I(L_t\geq D)$. Baryshnikov et al. (1998) show that this is also a doubly stochastic Poisson process with the intensity

$\displaystyle \lambda_s=m_s\left\{1-F(D-L_s)\right\}I(L_s<D).$ (4.1)

In Figure 4.2 we see a sample trajectory of the aggregate loss process $ L_t$ ( $ 0\leq t\leq T=$ 10 years) generated under the assumption of log-normal loss amounts with $ \mu= 18.3806$ and $ \sigma= 1.1052$ and a non-homogeneous Poisson process $ M_t$ with the intensity function $ m^1_s = 35.32 + 2.32\cdot 2\pi \sin\left\{2 \pi (s - 0.20)\right\}$, a real-life catastrophe loss trajectory (which will be analysed in detail in Section 4.3), the mean function of the process $ L_t$ and two sample 0.05- and 0.95-quantile lines based on $ 5000$ 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 $ 0.05$-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 $ D=60$ billion USD.

Figure 4.2: A sample trajectory of the aggregate loss process $ L_t$ (thin blue solid line), a real-life catastrophe loss trajectory (thick green solid line), the analytical mean of the process $ L_t$ (red dashed line) and two sample 0.05- and 0.95-quantile lines (brown dotted line). The black horizontal line represents the threshold level $ D=60$ billion USD.

\includegraphics[width=1.07\defpicwidth]{STFcat02.ps}