17.4 Examples

Let us assume that claims appear in good and bad periods. According to (17.9) we are able to approximate the risk process by:

$\displaystyle R_H(t)=u+(c-\lambda\mu)t+\lambda^{H}B_H(t),
$

where $ B_H(t)$ is a fractional Brownian motion, $ c$ is the premium rate, $ \mu$ is the expected value of claims, $ \sigma^2=\textrm{E}B^2(1)$ is their variance, $ \lambda $ is the claim intensity, and $ u$ is the initial capital.

We can compute finite and infinite time ruin probabilities for different levels of the initial capital, premium, intensity of claims, expectation of claims and their variance (see Tables 17.1 and 17.2). We approximate the finite time ruin probabilities by formula (17.12) and the infinite time ruin probabilities using the estimator given in (17.23). Sample paths of the process $ R_H$ are depicted in Figure 17.1.

Figure 17.1: Sample paths of the process $ R_H$ for $ H=0.7$, $ u=40$, $ c=100$, $ \mu =20$, $ \sigma =10$, and $ \lambda =3$.

\includegraphics[width=1.04\defpicwidth]{STFgood01.ps}

The results in the tables show the effects of dependence structures between claims on the crucial parameter for insurance companies - the ruin probability. Numerical simulations are performed for different values of the parameter of self-similarity $ H$ which defines the level of the dependence between claims. It is clearly visible that an increase of $ H$ increases the ruin probability. The tables also illustrate the relationship between the ruin probability and the initial capital $ u$, premium $ c$, intensity of claims $ \lambda $, expectation of claims $ \mu$ and their variance $ \sigma$. It is shown that for dependent damage occurrences the ruin probability is considerably higher than for independent events. Thus ignoring possible dependence (existence of good and bad periods) and its level might lead to wrong choices of the premium $ c$.