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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).

      3.4.3 Dual Implicit Representation

      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.

      3.4.4 Dictionary between the Three Representations

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.

      3.5 The Lee Diagram and Expected Losses

       Graphical methods are widely used in mathematics and statistics to visually present ideas which would otherwise be abstruse.

      Lee, The Mathematics of Excess of Loss Coverages (1988)

      In

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