Per-Occurrence And Aggregate Excess Structures

Why the Loss Base Matters

An excess cover does not respond to an amount called “loss” in the abstract. Its recovery depends on which losses are added together before the attachment and limit are applied. A per-occurrence cover applies its layer to each contract-defined occurrence. An annual aggregate cover applies its layer to the covered loss accumulated over the contract year. Because the calculation order differs, the same claims can produce different recoveries.

This distinction is central to describing a reinsurance or insurance layer accurately. It also prevents a frequent rating error: applying an annual attachment separately to each claim, or applying a per-occurrence attachment once to the annual total. The examples here are hypothetical and use deterministic losses to focus on contract mechanics, not on estimating expected layer cost.

Definition: A per-occurrence excess layer applies an attachment and limit separately to each contract-defined occurrence. An annual aggregate excess layer applies an attachment and limit to the total covered loss accumulated over the contract year. A claim is a reported or paid loss item; an occurrence is the event or grouping defined by the contract, and one occurrence may involve multiple claims.

Payment Functions and Loss Bases

For a layer attaching at $d$ with a limit of $u$ above the attachment, the payment on a single loss amount $x$ is $\min((x-d)_+,u)$, where $(z)_+=\max(z,0)$. The limit $u$ is the maximum amount paid by the layer, not the maximum covered ground-up loss. The layer exhausts at a ground-up loss of $d+u$.

For a per-occurrence cover, let $X_j$ be the covered loss assigned to occurrence $j$, after applying the contract’s stated loss basis. With attachment $d_O$ and limit $u_O$ per occurrence, total recovery for the year is the sum of the occurrence recoveries. For an annual aggregate cover, let $S$ be the covered annual subject loss; with annual attachment $d_A$ and limit $u_A$, recovery is calculated once on $S$.

R_{\mathrm{occ}}=\sum_{j=1}^{n}\min\bigl((X_j-d_O)_+,u_O\bigr), \qquad S=\sum_{j=1}^{n}X_j, \qquad R_{\mathrm{agg}}=\min\bigl((S-d_A)_+,u_A\bigr).

These formulas assume that the stated amounts are the losses subject to the relevant cover, that the same currency and valuation basis are used throughout, and that there are no additional provisions changing recovery. In particular, they omit reinstatements, annual aggregate deductibles or caps beyond those shown, exclusions, inuring recoveries, and adjustments for expenses or timing. Actual contract language determines whether a loss enters the subject-loss base and how occurrences are grouped.

Feature	Per-occurrence excess	Annual aggregate excess
Amount tested against attachment	Covered loss for each occurrence	Total covered loss for the contract year
How often the layer function is applied	Once per occurrence	Once per contract year
Limit in the basic structure	Maximum recovery per occurrence	Maximum recovery for the year
Key data requirement	Correct allocation of claims to occurrences	Correct annual subject-loss total and year boundary

Worked Examples

Worked Example: Applying a Per-Occurrence Layer to Several Losses

A hypothetical insurer has a per-occurrence layer attaching at $100,000 with a $100,000 limit per occurrence. Three separate covered occurrences have losses of $70,000, $160,000, and $250,000. Calculate the recovery for each occurrence and the total annual recovery. Assume no reinstatement or other contract adjustment.

Method. Apply the layer payment function separately to each occurrence, then add the payments. Work in thousands of dollars: $d_O=100$, $u_O=100$, and the occurrence losses are $70$, $160$, and $250$.

For the first occurrence, the loss does not reach the $100,000 attachment: $\min((70-100)_+,100)=\min(0,100)=0$.

For the second occurrence, the amount above attachment is $160-100=60$, below the $100,000 limit: $\min((160-100)_+,100)=\min(60,100)=60$.

For the third occurrence, the amount above attachment is $250-100=150$, but the layer pays at most $100,000: $\min((250-100)_+,100)=\min(150,100)=100$.

Adding the three occurrence payments gives $0+60+100=160$ thousand dollars, or $160,000. The annual ground-up loss is $70+160+250=480$ thousand dollars; that total is a useful check on the input, but it is not the loss to which this per-occurrence layer is applied.

Check and common mistake. The total recovery cannot exceed the sum of the three per-occurrence limits, $3(100)=300$ thousand dollars, and $160,000 is within that bound. A common error is to apply the attachment once to the $480,000 annual total, which would produce a different, aggregate-style calculation rather than the stated per-occurrence recovery.

Worked Example: Applying an Annual Aggregate Layer

Use the same hypothetical annual losses of $70,000, $160,000, and $250,000, but now consider an annual aggregate layer attaching at $200,000 with a $150,000 annual limit. Calculate the annual recovery. Assume all three amounts are covered subject losses in the same contract year.

Method. First sum the annual subject losses, then apply the annual attachment and annual limit once. In thousands of dollars, $S=70+160+250=480$, $d_A=200$, and $u_A=150$.

The annual loss above the attachment is $480-200=280$ thousand dollars. The layer is limited to $150,000, so: $\min((480-200)_+,150)=\min(280,150)=150$.

The annual aggregate recovery is therefore $150,000. The amount of annual subject loss retained below the attachment is $200,000; the amount above the attachment that exceeds the layer limit is $280-150=130$ thousand dollars. These amounts reconcile: $200+150+130=480$.

Check and common mistake. The recovery does not exceed the annual limit of $150,000. Applying this annual attachment separately to the three losses would incorrectly treat the contract as a set of per-occurrence layers.

Worked Example: Group Claims into Contract-Defined Occurrences

A hypothetical policy covers two contract-defined occurrences during one year. Occurrence A produces two covered claims of $50,000 and $70,000; occurrence B produces one covered claim of $200,000. A per-occurrence layer attaches at $80,000 and has a $100,000 limit per occurrence. Calculate the recovery, assuming claims from the same occurrence are combined before the layer is applied.

Method. Sum claims within each occurrence first. For occurrence A, subject loss is $50+70=120$ thousand dollars. Its payment is $\min((120-80)_+,100)=\min(40,100)=40$ thousand dollars.

Occurrence B has subject loss of $200,000. Its payment is $\min((200-80)_+,100)=\min(120,100)=100$ thousand dollars. Total recovery is $40+100=140$ thousand dollars, or $140,000.

The total ground-up loss is $50+70+200=320$ thousand dollars. The layer recovery of $140,000 is below the maximum across two occurrences, $2(100)=200$ thousand dollars.

Check and common mistake. The recovery is calculated on two occurrence totals, not three separate claim amounts. If each claim were incorrectly treated as its own occurrence, the payments would be $0$, $0$, and $100$ thousand dollars, for a mistaken total of $100,000. The contract’s occurrence definition—not the number of claim records—controls the grouping.

Why the Two Calculations Differ

The per-occurrence formula applies the layer function to each occurrence loss and adds the results. The aggregate formula first adds the losses and then applies the layer function to that annual sum. In general, applying a capped, attached layer function before summing is not equivalent to applying it after summing. The order matters because the function has both a threshold and a cap.

For a simple justification, consider the same three losses from the first example. The per-occurrence layer uses $d_O=100$ and $u_O=100$, so its separate payments are $0$, $60$, and $100$, totaling $160$. The annual total is $480$. If an annual aggregate layer instead uses $d_A=200$ and $u_A=150$, its payment is $150$. Although both structures see the same underlying losses, their attachments, limits, and order of application differ; there is no reason for the recoveries to match.

\sum_j \min\bigl((X_j-d_O)_+,u_O\bigr) \quad\text{and}\quad \min\bigl((\sum_j X_j-d_A)_+,u_A\bigr)

These expressions are not interchangeable. They need not be equal, and neither structure is universally guaranteed to pay more: that depends on the losses, occurrence grouping, attachment, and limit provisions. For example, a high per-occurrence attachment may leave several moderate occurrences unrecovered, while an aggregate attachment may be crossed by their combined annual total. Conversely, an occurrence that reaches its per-occurrence limit can produce substantial recovery even if the aggregate arrangement has a high annual attachment.

Identify the covered loss basis.
Determine which amounts enter the calculation and whether losses are gross, net of another recovery, or otherwise adjusted under the stated contract terms.

Determine the aggregation unit.
For a per-occurrence layer, identify contract-defined occurrences and combine their claims. For an annual aggregate layer, combine covered losses over the contract year.

Apply the attachment and limit at that unit.
Calculate the excess above the relevant attachment, floor negative amounts at zero, and cap the recovery at the applicable limit.

Reconcile and label the result.
Check whether the recovery respects the per-occurrence or annual cap, and report whether the result is per occurrence or for the year.

Common Traps and Practical Checks

Confusing a claim with an occurrence. A single event can generate multiple claims. Follow the occurrence definition before applying a per-occurrence attachment.
Applying the annual attachment repeatedly. An annual aggregate layer has one annual loss base in the basic structure shown here; do not reset its attachment for each claim unless contract wording explicitly provides a different mechanism.
Misreading the limit. A $100,000 limit above a $100,000 attachment exhausts at a ground-up loss of $200,000. It does not mean the layer covers all ground-up loss up to $100,000.
Adding losses on inconsistent bases. An annual sum is meaningful only if the losses share the stated valuation basis, currency, coverage terms, and contract year. A gross amount and a net amount should not be combined without a defined basis.
Assuming the structures have a fixed ranking. Per-occurrence and aggregate covers can respond differently. Compare the actual attachment, limit, and loss grouping rather than relying on a general claim that one always pays more.

These mechanics are a necessary foundation for pricing excess covers: before estimating a layer’s loss cost, the actuary must know what loss enters the layer and at what level the attachment and limit operate. Expected-loss calculations require additional distributional assumptions; the deterministic payment calculations here do not require a frequency or severity model.

Check Your Understanding

For each hypothetical problem, apply the contract’s stated loss basis and identify whether the calculation is per occurrence or annual aggregate.

A per-occurrence layer attaches at $40,000 and has a $60,000 limit per occurrence. Three separate occurrences have covered losses of $25,000, $75,000, and $130,000. Find each recovery and the annual total. (Answer: $0, $35,000, and $60,000; total $95,000.)
An annual aggregate layer attaches at $180,000 and has a $90,000 annual limit. Covered annual losses are $55,000, $85,000, and $160,000. Find the recovery and the annual loss above the attachment that remains unpaid by the layer. (Answer: recovery $90,000; $30,000 above the attachment remains unpaid.)
A per-occurrence layer attaches at $100,000 with a $100,000 limit per occurrence. One occurrence has claims of $60,000 and $90,000; a second occurrence has a claim of $240,000. Claims within an occurrence are combined. Find total recovery. (Answer: $150,000.)
An annual aggregate layer attaches at $250,000 and has a $120,000 annual limit. In one year, four covered losses are $90,000 each. Calculate annual recovery and the portion of total loss retained below the attachment. (Answer: recovery $110,000; $250,000 is retained below the attachment.)
Two occurrences have covered losses of $80,000 and $220,000. Compare total recovery under a per-occurrence layer attaching at $100,000 with a $100,000 limit per occurrence, and under an annual aggregate layer attaching at $150,000 with a $100,000 annual limit. (Answer: per-occurrence recovery $100,000; annual aggregate recovery $100,000.)

What You'll Learn