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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 20M, and layer 3: ¤50M excess 50M. The cedent retains losses in excess of ¤100M. (Note: ¤30M excess 20M refers to a layer with a limit of ¤30M that attaches at ¤20M.)

Example 25 Structured finance tranches asset-backed securities (MBS, CDO, etc.) in an analogous way. The tranches are generally determined to achieve a certain ratings defined by probability of default, meaning they have a dual implicit definition.

It is convenient to introduce the notation Ldd+l for the layer with limit l in excess of attachment d. The layer pays

(3.6)

for a subject loss x. The detachment or exhaustion point of the layer is d + l. The cover can be written succinctly as

The notation Ldd+l mimics integrals, with the attachment and detachment points as sub- and super-scripts, and makes it easy to add them: L0l1+Ll1l1+l2=L0l1+l2.

We use the two equivalent expressions L0l(X) and X∧l interchangeably for a ground-up cover.

The expected loss and premium for a layer divided by the layer’s limit are called loss on line and rate on line, respectively.

When applied to a random loss X, Ldd+l(X) becomes a random variable.

Exercise 26 Using multiple layers it is possible to create any continuous indemnity function that increases with subject loss. Describe in words and plot payments from the following towers as functions of the subject loss 0≤x≤1000.

1 L0500(x)

2 L250∞(x)

3 L2501000(x)

4 0.5L250500(x)+0.75L500750+L7501000

5 Which of (1)–(4) has the same payouts as a call option? What is its strike?

6 Write the payout function for a put option in terms of L functions.

Solution. (1)–(4) See Figure 3.9. Cover (4) includes co-participations or coinsurance clauses, where some layers are only partially placed. (5) L250∞ has the same payout as a call option with strike 250. (6) A put option with strike k has payout function equal to k−L0k.

Figure 3.9 Sample layering functions for Exercise 26.

Remark 27 Limits and deductibles can be applied per claimant, claim, occurrence, or in the aggregate. We assume the reader is familiar with these concepts. The exact meaning of a limit and deductible is defined by that of X. In this book X almost always represents aggregate loss on a portfolio.