Aggregate excess of loss

An Aggregate Excess of Loss contract (AggXoL) drops the per-occurrence layer: after the filter and coverage period, it applies the aggregate excess directly to the trial total, then scales by participation.

$$\text{AggXoL}(A, \ell, p) = \text{scale}(p) \circ \text{agg}\_{\text{excess}}(A, \ell) \circ \text{period}(t_0, t_1) \circ \text{filter}(P)$$

As with every composition, the formula reads right to left — the filter applies first — while data flows left to right.

Multiple occurrences accumulate against the attachment before the contract responds. Helios Re Contract 2 is the SunCoast Earthquake AggXoL — $10M xs $15M, filtering to SunCoast’s earthquake losses.

Applied to Trial 9’s earthquakes, no single quake is large — the biggest is $6.5M — but together they accumulate past the $15M aggregate attachment, and the contract pays the excess up to the $10M limit:

Earthquake Cumulative aggregate

Contract 2 (SunCoast EQ AggXoL, 10M xs 15M) on Trial 9. The 22 earthquakes are individually small, but the cumulative line crosses the $15M attachment and exhausts the layer at $25M. Aggregate subject $27.6M → gross $10M.

Trial 9 (earthquake)	Value
Aggregate subject (22 quakes)	$27.6M
Aggregate attachment	$15M
Gross (excess of attachment, capped at $10M)	$10M

helios_re/aggxol.py Python

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Order matters

The aggregate attachment and limit apply to the trial total, so the occurrences must be accumulated before the layer responds — here no single earthquake reaches the $15M attachment, but together they cross it. When a contract also carries a per-occurrence layer (as a CatXoL does), that layer is applied to each event first, and the aggregate then acts on the running total of those per-occurrence results. A robust implementation tracks the layer’s erosion event by event: each occurrence’s contribution to the aggregate gross is the increase in the aggregate-layered cumulative from the previous occurrence to this one — a per-occurrence audit trail, with the attachment and limit still applied to the aggregate.

Across all 20 trials, Contract 2 responds whenever SunCoast’s annual earthquake total clears $15M — which it does in most years, since the quakes are frequent and small:

Contract 2 EP curves: SunCoast earthquake aggregate vs the 10M xs 15M layer. Most trials pierce the $15M attachment; the heavier ones exhaust the $10M limit.

In Practice

Helios Re Contract 3 is also an AggXoL — $50M xs $300M on SunCoast’s entire US book — but it adds one more term: it operates on losses net of Contracts 1 and 2, so those recoveries inure to its benefit. That inuring step is a contract-chaining operation, covered in Contract chaining.