Occurrence excess

This is the operation we saw in step 3 of the CatXoL walkthrough — now isolated and formalized. The occurrence excess applies an attachment point $A$ and a limit $\ell$ to each occurrence independently:

$$L_{j,t,e}^{\text{gross}} = \min(\max(L_{j,t,e}^{\text{subject}} - A, 0), \ell)$$

In plain English:

1. Subtract the attachment: $L - A$
2. If negative (loss below attachment), gross loss is 0
3. If positive but above the limit, cap at the limit

Every occurrence is evaluated independently; the trial total is the sum of per-occurrence gross losses. The output preserves per-occurrence resolution — each row gets a gross_loss column.

Intuition

Think of the occurrence excess as a bandpass filter on the loss distribution. It blocks everything below the attachment, passes losses within the layer, and clips everything above the limit. The resulting gross distribution is concentrated between 0 and the limit, with a mass point at 0 (below-attachment trials) and potentially another at the limit (exhaustion trials).

Applied directly to Trial 9’s subject losses — all perils, no filter — at $15M xs $5M, every occurrence above the $5M attachment contributes. Eight occurrences do: the three large Florida hurricanes (each exhausting the $15M limit), plus an Arizona earthquake and four smaller Gulf and Atlantic hurricanes — events the larger $30M xs $10M Contract 1 layer would have ignored entirely. Everything below $5M still pays nothing:

Earthquake Hurricane

Occurrence excess (15M xs 5M) on all of Trial 9's subject occurrences. Dashed lines mark the attachment ($5M) and the top of the layer ($20M); the clear band between them is the layer, and the portion of each bar inside it is the gross. Eight occurrences reach the attachment — not just the Florida hurricanes.

Occurrence	Subject	Occurrence gross
FL hurricane — Jul 12	$29.7M	$15.0M
FL hurricane — Sep 6	$102.4M	$15.0M
FL hurricane — Oct 5	$80.8M	$15.0M
5 smaller events (AZ earthquake; GA, TX, LA hurricanes)	$34.2M	$9.2M
22 occurrences below the $5M attachment	$25.0M	$0.0M
Trial 9 total	$272.1M	$54.2M

helios_re/occurrence_excess.py Python

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Across all 20 trials, the occurrence excess clips each trial’s subject down to the sum of its per-occurrence layered losses:

Occurrence excess EP curves: SunCoast subject vs 15M xs 5M applied per occurrence (no filter). Each trial's gross is the sum of its per-occurrence layered losses.

The occurrence excess is the per-event layer at the heart of a CatXoL. Its sibling, the aggregate excess, applies the same attachment-and-limit shape to the trial total instead of each event.