Читать книгу Pricing Insurance Risk - Stephen J. Mildenhall - Страница 46

All three representations are used in insurance. The explicit representation could be a specific claim file, as a claim adjuster sees it. Its implicit representation identifies it with its loss amount and ignores other details. And its dual implicit representation identifies it with its nonexceedance probability level. Many explicit events can have the same loss amount or probability level. The probability depends on the reference portfolio. Table 3.1 summarizes the three ways of representing a risk. The last row shows how to compute expected values in each representation, the subject of the next section.

Table 3.1 Representing risks: event definitions and natural sample spaces

Facet	Random variable	Distribution function	Quantile function
Notation	X	F	F −1
Events	Explicit	Implicit	Dual implicit
Sample space	Ω	R=(−∞,∞)	[0,1]⊂R
Sample point	ω∈Ω	Outcome, x	Probability or rank, p
Probability	Pr	PX(a,b]=P(X∈(a,b])=F(b)−F(a)	Uniform
Mean E[X]	∫ΩX(ω)Pr(dω)	∫0∞(1−F(x))dx=∫0∞xdF(x)	∫01F−1(p)dp

Example 15 Lloyd’s uses an explicit event approach to measure risk because of its intended use of the results. Each syndicate at Lloyd’s reports to the Corporation of Lloyd’s their estimated losses from 16 specific Realistic Disaster Scenarios (RDS), such as a Miami-Dade Florida Windstorm or Japanese Typhoon. The Corporation then aggregates them over all syndicates to determine its total exposure. If Lloyd’s merely asked each syndicate for its 100 year or 250 year loss, it would be unable to determine its aggregate exposure. Rating agencies and regulators evaluate companies on a standalone basis. They do not need to aggregate exposure in this way, and so a dual implicit event, such as a VaR or TVaR loss amount at an extreme return period, fits their intended purpose. The RDS system is described in Lloyd’s (2021). We discuss it further in Sections 5.1 and 5.A.