with sum
1 Create the model in a spreadsheet and confirm E[X]=28 and E[Xi]=14.
2 Plot X1, X2, and X as functions of ω=0,1,…,99.
3 Plot the survival functions, as functions of the outcome x.
4 Plot the Lee diagrams, as functions of probability p.
5 Are the random variables different? The survival functions? The Lee diagrams?
We return to this example in Chapter 15.
Solutions. Figures 3.5–3.7 show the random variables, the survival functions, and the Lee diagrams. The random variables are all distinct, but the survival function and Lee diagrams for each line are the same.
Figure 3.5 Random variables, functions of an explicit state.
Figure 3.6 Survival functions of the outcome.
Figure 3.7 Lee diagrams, function of a dual implicit state.
Remark 20 The relationships illustrated in Figure 3.3 are a discrete version of the formula for integration by parts. Consider approximating Riemann sums to the integral of the survival function. Let 0=x0<x1<⋯<xn<⋯, be a fine, but not necessarily equally spaced, dissection of the positive reals. We get two equivalent representations of E[X] by
using Taylor’s theorem to write S(xi−1)−S(xi)=S(xi−(xi−xi−1))−S(xi)=−S′(xi′)(xi−xi−1)=f(xi′)(xi−xi−1), for some xi−1≤xi′≤xi.
Exercise 21 Confirm the change in indexing between Eq. 3.2 and Eq. 3.3 is correct by looking at panels (d) and (e).
Technical Remark 22. In addition to the outcome-probability and survival function forms, there is a third, dual implicit outcome expression
by change of variable substitution F(x)=p, f(x)dx=dp. This view replaces the probability defined by X with the uniform probability dp on [0,1].
Technical Remark 23. Figure 3.8 relates integral expressions for the mean and the different ways of representing risk presented in Section 3.4. When F is absolutely continuous it has a density, giving the usual ∫xf(x)dx representation of the mean in the second to last line.
3.5.3 Layer Notation
It is common to use limits and deductibles to transform the insured loss. If X is a loss random variable, then applying a deductible d transforms it into
and applying a limit of l transforms it to
These notations are shorthand: for example, X∧l is the random variable with outcome (X∧l)(ω)=X(ω)∧l at sample point ω∈Ω.
When a policy has both a limit and a deductible, the limit is applied after the deductible. Applying a limit and a deductible creates what is called a limited excess of loss layer or simply a layer. Many reinsurance contracts and specialty lines policies are tranched into a coverage tower consisting of multiple layers, written by multiple insurers. (A tranche means a piece cut off or a slice.) In this context, a layer is sometimes identified with its limit and the deductible is called the attachment of the layer. A layer that attaches at 0 is called ground-up; all others are excess. Layers in a tower are typically arranged with no gaps.
Example 24 An aggregate reinsurance tower to ¤100M could be structured as a ¤10M retention, layer 1: ¤10M excess 10M, layer 2: ¤30M excess