Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 44

The implicit representation identifies an outcome with its value, creating an implicit event.

The implicit representation relabels the sample space by identifying the event {X=x} with the sample point x, creating a new random variable on the sample space of outcome values. It is easy to understand but hard to aggregate because there is no way to link outcomes. There is no easy way to specify dependence, which explains the interest in copulas, mathematical objects used to specify the dependence structure between two or more random variables. It is impossible to distinguish between implicit events that cause the same loss outcome.

The implicit representation defines a probability on the new outcome sample space by PX(A):=Pr(X∈A) for A⊂R. PX is uniquely determined by the distribution function F(x):=Pr(X≤x).

When risk is solely a function of loss outcome, rather than the cause of loss, it can be appropriate to work with implicit events.

Example 9 Ins Co. actuaries working with the homeowners line of business use the same catastrophe model as the commercial lines folks, but their spreadsheet has only the following columns:

 Hurricane/earthquake flag

 Homeowners portfolio loss

This information allows them to compute the probability distributions of losses, in total and also conditional on type of peril. However, it is insufficient to tie back to the commercial lines experience. The primary identifying feature for an event is the loss amount.

Example 10 Ins Co. wants to purchase California earthquake reinsurance. The reinsurer needs explicit (event ID, event loss) model output, so it can aggregate its exposure. But Ins Co. can evaluate its reinsurance programs using implicit events (loss amount, probability).