Catastrophe excess of loss

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

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

As with every composition, the formula reads right to left — the filter applies first — while data flows left to right. The composition defines the contract at 100% — the whole layer. Helios Re’s 14.5% participation is not one of the contract’s terms: participation is a scaling applied at portfolio roll-up, where the reinsurer’s share of each contract is booked. 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)

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.

Watch Out

The aggregate limit $(1 + n) \cdot \ell$ is a claims-side encoding: it reproduces the recoveries exactly and drops everything else about the reinstatement provision. A real provision is multi-dimensional — each reinstatement can be free or paid, and a paid one carries an additional premium computed pro rata to the amount of limit restored, pro rata to the time remaining in the contract period, or both.¹ Those premium terms are commercially material: two contracts with the same layer and the same reinstatement count can cost the cedent very different amounts. A fuller treatment of reinstatement provisions is planned for this catalog.

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 of the demo tier, 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.

---

Footnotes

1. Richard R. Anderson and Weimin Dong, “Pricing Catastrophe Reinsurance with Reinstatement Provisions Using a Catastrophe Model”, CAS Forum, Summer 1998 — prices layers whose reinstatement premiums are pro rata to the amount of limit reinstated and, optionally, to the time remaining. On the free-versus-paid distinction and the variations in practice, see IRMI, “More Porridge Please: Reinstatements in Reinsurance”, and IRMI, “Reinstatement Premium”. ↩