Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 45

The implicit representation is a useful simplification. But there are situations where we can simplify it further. If we don’t care about the monetary outcome but care only about the rank of the outcomes, we can use the dual implicit representation and identify an outcome X = x with its nonexceedance probability p=F(x)=Pr(X≤x). Equivalently, we can relate x with s=S(x)=Pr(X>x), its exceedance probability. We work with the former and call it a dual implicit event.

Identifying an outcome with its probability is scale invariant and straightforward. It is easy to make comparisons: no matter the range of X, F(X) lies between 0 and 1. In fact, for continuous X, F(X) is uniformly distributed between 0 and 1—the basis of the inversion method for simulating random variables.

The dual implicit representation has the same disadvantage as the implicit: it is hard to aggregate. In addition, it suffers from being relative to an often unspecified reference portfolio. After Hurricane Katrina, news reports described it from a 25 year (relative to all US landfalling hurricanes) to a 400-year event (storms along a similar track).

Example 11 Investors and rating agencies assess bonds by their estimated probability of default. They are using a dual implicit representation of the bond.

Example 12 Although catastrophe models generally simulate explicit events, the results are often summarized by exceedance probability. Rating agency and regulatory capital models charge for catastrophe risk using events defined by exceedance probability, rather than objective event.

Example 13 The homeowners reinsurance actuaries in Ins Co. get a spreadsheet similar to the commercial lines simulation output, with losses arranged in ascending order. Their concern is identifying where various proposed reinsurance programs attach and exhaust (Section 3.5.3). They flag various rows with notations such as program 1 attaches, program 2 exhausts, etc. Since the rows are in ascending loss order, they can count rows above or below a particular notation and calculate the probability of attachment and exhaustion. They prepare a report that focuses on these probabilities without mentioning the actual dollar loss amounts.

Remark 14 The mapping from p back to x is defined by the inverse of the distribution function. Spreadsheet programs provide several, including: BETA.INV, CHISQ.INV, F.INV, GAMMA.INV, LOGNORM.INV, NORM.INV, and T.INV.