Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 30

In the Cat/Non-Cat Case Study, Ins Co. has catastrophe and noncatastrophe exposures. The noncatastrophe unit proxies a small commercial lines portfolio. Balancing the relative benefits of units considered to be more stable against more volatile ones is a very common strategic problem for insurers and reinsurers. It arises in many different guises:

 Should a US Midwestern company expand to the East coast (and pick up hurricane exposure)?

 Should an auto insurer start writing homeowners?

 What is the appropriate mix between property catastrophe and noncatastrophe exposed business for a reinsurer?

This Case uses a stochastic model similar to the Tame Case. The two units are independent and have gamma and lognormal distributions with parameters shown in Table 2.4.

The Case includes a gross and net view. Net applies aggregate reinsurance to the Cat unit with an attachment probability 0.1 (¤ 41) and detachment probability 0.005 (¤ 121).

Loss statistics based on an FFT approximation are shown in Table 2.6. The FFT approximation is good, though not as exact as the first case, and it is used for all graphics and illustrations. Simulation can also be used. Table 4.7 shows various VaR, TVaR, and EPD levels.

Table 2.6 Cat/Non-Cat Case Study estimated mean, CV, skewness, and kurtosis by line and in total, gross, and net. Aggregate reinsurance applied to Cat with an attachment probability 0.1 (¤ 41) and detachment probability 0.005 (¤ 121)

	Gross			Net
Statistic	Cat	Non-Cat	Total	Cat	Non-Cat	Total
Mean	20.000	80.000	100.000	17.786	80.000	97.786
CV	1.000	0.150	0.233	0.737	0.150	0.182
Skewness	3.972	0.300	2.539	3.139	0.300	1.351
Kurtosis	35.933	0.135	19.173	55.220	0.135	16.336

Figure 2.4 shows the loss densities. The top left plot shows the different body behavior. Non-Cat is close to symmetric. Cat has a peaked distribution, with mode much lower than mean. The spike in the net plots is caused by claims generating a limit loss to the reinsurance on Cat. The different tail behavior of the two units is evident, especially on a log scale. noncat has a log concave density whereas Cat is eventually log convex. The impact of reinsurance is clear. It introduces a point mass at the aggregate attachment for Cat. The total distribution is still continuous. Figure 2.5 shows the bivariate plots. The left plot emphasizes the different tail behavior; Non-Cat remains localized close to its mean while Cat decays very slowly (vertically).