Catastrophe excess of loss

A CatXoL composes all five building blocks. The filter and coverage period select the covered events, the occurrence excess clips each one, the aggregate excess caps the annual total, and scaling applies participation. For a CatXoL with $n$ reinstatements and participation $p$:

$$\begin{aligned} \text{CatXoL}(A, \ell, n, p) = {} & \text{scale}(p) \circ \text{agg}\_{\text{excess}}(0,\; (1+n) \cdot \ell) \\ & \circ\; \text{occ}\_{\text{excess}}(A, \ell) \circ \text{period}(t_0, t_1) \circ \text{filter}(P) \end{aligned}$$

As with every composition, the formula reads right to left — the filter applies first — while data flows left to right. This is exactly the pipeline we dissected on Trial 9, now named term by term. For Contract 1:

Building block	Contract 1
Loss filter	peril = HU, geography = FL
Contract period	2024-01-01 to 2024-12-31
Occurrence excess	$30M xs $10M per event
Aggregate excess	cap at $60M (1 reinstatement)
Scaling	participation, 14.5%

The aggregate limit $(1 + n) \cdot \ell$ controls how many full occurrences the contract covers. Without reinstatements ($n = 0$), the aggregate limit equals the occurrence limit — one full event. Each reinstatement restores the per-occurrence limit once, typically in exchange for additional premium (covered in pricing).

On Trial 9, the four Florida hurricanes’ per-occurrence gross — $19.7M, $30M, $0, and $30M after the $30M xs $10M layer — accumulate against the $60M aggregate limit. The cumulative would reach $79.7M, but with one reinstatement (two full occurrences of cover) the limit caps it at $60M:

Hurricane Cumulative aggregate

Contract 1 on Trial 9: per-occurrence gross (after the $30M xs $10M layer) accumulating against the $60M aggregate limit. The cumulative is clipped at $60M — the standalone gross. (This is Step 4 of the dissection; the per-occurrence clipping is Step 3.)

helios_re/catxol.py Python

▶ Run Reset

Across all 20 trials, the standalone gross distribution — reported at 100%, before participation:

Contract 1 EP curves: Florida-hurricane subject vs the standalone CatXoL gross (30M xs 10M, 1 reinstatement). Trial 9 and Trial 19 are the years where the $60M aggregate limit binds.

This composition turns Trial 9’s $272.1M subject into a $60M standalone gross — Helios Re books its 14.5% share, $8.7M, at roll-up — and across all 20 trials produces a standalone expected loss of $17.4M.

Drop the per-occurrence layer and you have an AggXoL; gate the occurrence excess with a trigger and you have a franchise structure. The programs page covers the elaborations and how a CatXoL chains with other contracts.