Quota share

A quota share is the simplest contract structure: a fixed cession percentage $q$ applied to every covered loss. As a composition, it is the scaling building block behind a filter and a coverage period:

$$\text{QS}(q) = \text{scale}(q) \circ \text{period}(t_0, t_1) \circ \text{filter}(P)$$

As with every composition, the formula reads right to left — the filter applies first — while data flows left to right. The gross loss is simply the scaled subject loss:

$$L_j^{\text{gross}} = q \cdot L_j^{\text{subject}}$$

Helios Re Contract 5 is the Baltica European Windstorm Quota Share at $q = 0.25$. Baltica writes only EU windstorm, so the filter ($\text{peril} = \text{WS}$) selects its whole book and the full-year period passes every event — both terms are identities here, but they are still part of the composition and would bite for a multi-peril cedent or a mid-year inception.

Applied to Baltica’s Trial 9 — a windstorm season clustered in October–December — the quota share cedes a quarter of every loss:

Windstorm

Baltica Trial 9 ceded at 25%: every windstorm loss becomes a quarter of its subject value. Subject $103.1M → gross $25.8M. Note the European windstorm season (autumn–winter), unlike SunCoast's summer hurricanes.

Baltica Trial 9	Subject	Gross (×0.25)
Nov 19 windstorm (DE/FR/UK)	$23.7M	$5.9M
Nov 21 windstorm (FR/UK)	$15.6M	$3.9M
Trial 9 total (11 occurrences)	$103.1M	$25.8M

helios_re/quota_share.py Python

▶ Run Reset

Across all 20 trials, the gross curve is an exact vertical scaling of the subject curve, and the cedent’s net curve mirrors it — together they sum to the subject at every return period:

Contract 5 EP curves: Baltica EU Windstorm Quota Share (25%). The gross curve is exactly 25% of the subject curve at every return period.

Key Insight

A quota share’s value is not analytical complexity but business utility — quota shares are widely used for capital management, risk sharing within large enterprise groups, and accessing different capital sources (e.g., a parent company ceding to a subsidiary to move risk to a different balance sheet).

With event limit

Real quota share contracts often cap the reinsurer’s exposure per occurrence. An event limit is an occurrence excess with zero attachment, placed before the cession scaling:

$$\text{QS}\_{\text{event limit}}(q, \ell_e) = \text{scale}(q) \circ \text{occ}\_{\text{excess}}(0, \ell_e) \circ \text{period}(t_0, t_1) \circ \text{filter}(P)$$

The order matters: the limit is applied before the cession percentage. Each occurrence is capped at $\ell_e$, then the capped amount is shared at ratio $q$. With a $15M event limit, Baltica’s Trial 9 has two windstorms above the cap — the $23.7M November storm and the $15.6M one just after — both clipped to $15M before cession:

Windstorm

QS $15M event limit on Baltica Trial 9 (subject losses, before the 25% cession). Each windstorm is capped at $15M; the Nov 19 ($23.7M) and Nov 21 ($15.6M) storms are clipped, the rest pass through unchanged.

helios_re/quota_share_event_limit.py Python

▶ Run Reset

Across all 20 trials, the ceded gross falls below a plain 25% cession wherever an individual windstorm pierces the $15M limit — thirteen of the twenty trials:

QS with $15M event limit (Baltica EU WS, 25% cession): subject vs ceded gross. The gross sits below a flat 25% of subject in the trials where a windstorm exceeds the event limit.

With annual limit

Adding an aggregate excess on top creates an annual limit on the ceded losses. The aggregate excess goes between the occurrence excess and the cession:

$$\text{QS}\_{\text{capped}}(q, \ell_e, \ell_a) = \text{scale}(q) \circ \text{agg}\_{\text{excess}}(0, \ell_a) \circ \text{occ}\_{\text{excess}}(0, \ell_e) \circ \text{period}(t_0, t_1) \circ \text{filter}(P)$$

The aggregate excess introduces a granularity change: its output is per-trial, not per-occurrence. All downstream components — including the cession — operate on the aggregated result. On Trial 9, the event-limited windstorms accumulate to $93.8M, and an $80M annual limit clips the total before cession:

Windstorm Cumulative aggregate

QS $80M annual limit on Baltica Trial 9 (after the $15M event limit, before cession). The capped windstorms accumulate to $93.8M; the cumulative line is clipped at the $80M annual limit, so $80M × 25% = $20.0M is ceded.

helios_re/quota_share_annual_limit.py Python

▶ Run Reset

Across all 20 trials, the annual limit flattens the ceded gross at $20M (= $80M × 25%) in the heaviest windstorm years — Trials 5, 9, and 10:

QS with $15M event limit + $80M annual limit (Baltica EU WS, 25% cession): subject vs ceded gross. The annual limit caps the ceded loss at $20M in the worst years.

These variants show the catalog’s compositional power directly: an event limit is one extra term, an annual limit two — the same blocks that build a CatXoL, rewired.