Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 31

In the Hu/SCS Case Study, Ins Co. has catastrophe exposures from severe convective storms (SCS) and, independently, hurricanes (Hu). In practice, hurricane exposure is modeled using a catastrophe model. We proxy that using a very severe lognormal distribution in place of the gross catastrophe model event-level output. Both units are modeled by an aggregate distribution with a Poisson frequency and lognormal severity. The key stochastic parameters are shown in Table 2.4.

The Case includes a gross and net view. Net applies per occurrence reinsurance to the Hu unit with an attachment probability 0.05 (¤ 40) and detachment probability 0.005 (¤ 413). The reinsurance assumes an unlimited number of free reinstatements, to simplify computations. In practice, the net distribution would include inuring property per risk and quota share programs. It would be taken from the net catastrophe model event-level output.

Other attritional (i.e., noncatastrophe) losses are approximately constant in comparison, therefore we can exclude them from our calculations since reasonable risk measures are translation-invariant,. Loss statistics based on an FFT approximation are shown in Table 2.7. To improve convergence the FFT approximation limits all occurrence losses to 1.2×109, which accounts for the difference in means compared to the expected 100. All graphics are based on the FFT computation, but computations are shown based on FFT and simulation approaches. Table 4.8 shows various VaR, TVaR, and EPD levels.

Table 2.7 Hu/SCS Case Study estimated mean, CV, skewness, and kurtosis by line and in total, gross, and net. Per occurrence reinsurance applied to Hu with an attachment probability 0.05 (¤ 40) and detachment probability 0.005 (¤ 413)

	Gross			Net
Statistic	Hu	SCS	Total	Hu	SCS	Total
Mean	29.727	69.133	98.860	18.987	69.133	88.121
CV	10.923	0.736	3.324	16.246	0.736	3.548
Skewness	121	24.900	116	137	24.900	132
Kurtosis	27	9	26	32	9	31

Figure 2.6 shows the loss densities. These are more extreme versions of the Cat/Non-Cat Case. Cat is extremely skewed. The lognormal severity is thick-tailed for both units, though Cat is thicker. Figure 2.7 shows the bivariate plots. The right-hand plot shows the pinched behavior typical of thick-tailed distributions. A large loss combined with a small loss is the most likely nontrivial outcome. The distribution is clustered around the axes.