Why the Treaty Basis Matters
A reinsurance contract can respond to losses grouped in different ways. Under an occurrence basis, claims arising from one covered event are combined before the attachment is applied. Under an annual aggregate basis, covered losses are combined across the policy year before the attachment is applied. The same claim amounts can therefore produce different recoveries depending on the treaty basis.
The key is to identify the unit to which the contract applies: one claim, one occurrence, or all covered losses during a policy year. This tutorial focuses on the distinction between the latter two. All examples are hypothetical and deterministic: the claim amounts and groupings are given, so no severity or frequency distribution is assumed. There is no discounting; equivalently, the interest rate is 0%. Mortality assumptions do not apply.
An occurrence is not automatically the same thing as one claim, one claimant, or one day. For example, a hypothetical contract might treat several claims arising from one storm as one occurrence, but the precise event definition must come from the contract. Do not infer the grouping solely from the number of claim records.
Layer Payment on the Specified Basis
For a loss amount \(z\), an attachment \(a\), and a reinsurance limit \(L\), define the layer payment function
The attachment is the amount that must be exceeded before the reinsurer pays. The limit \(L\) is the maximum recovery for the unit to which the contract applies. If a contract has no finite limit, the payment is \(\max(z-a,0)\). The formula assumes that the treaty limit is expressed as a maximum payment, not as an upper loss threshold.
Here, \(O_j\) is the sum of all covered claims assigned to occurrence \(j\), and \(T\) is the sum of all covered claims in the policy year. The two calculations differ because \(g\) is applied to different-sized groups. In general, summing payments calculated occurrence by occurrence is not the same operation as first summing all losses and calculating one annual payment.
Why Claims Must Be Grouped Before Applying the Attachment
Suppose two covered claims of amounts \(x_1\) and \(x_2\) arise from one occurrence. The occurrence treaty sees the combined amount \(O=x_1+x_2\), so its recovery is \(g(x_1+x_2;a,L)\). It does not independently apply the attachment to \(x_1\) and \(x_2\).
The reason follows directly from the treaty’s unit of coverage. Its attachment is defined on the occurrence loss, and that loss is the sum of claims assigned to the occurrence. Applying the attachment to each claim would instead calculate \(g(x_1;a,L)+g(x_2;a,L)\), which describes a per-claim basis. Those are different contract calculations.
The annual aggregate expression follows because the annual total is \(T=\sum_j O_j\). The attachment and limit are applied to that total once, not once to every occurrence. This distinction is important even when every individual claim is smaller than the attachment: claims can combine to exceed an occurrence or annual attachment.
Worked Calculations
Worked Example: Several Claims in One Occurrence
A hypothetical property treaty pays up to $45,000 in excess of a $50,000 attachment per occurrence. One covered event produces claims of $28,000, $47,000, and $36,000. A separate event produces claims of $22,000 and $41,000. Find the total recovery. The amounts are deterministic observations; no probability distribution is assumed. There is no discounting (interest rate 0%), and no mortality assumption applies.
Method: group claims by occurrence, then apply the layer once to each occurrence. For the first event, the occurrence total is
Its loss above the $50,000 attachment is \(111{,}000-50{,}000=61{,}000\). The recovery is limited to $45,000:
For the second event, the occurrence total is \(O_2=22{,}000+41{,}000=63{,}000\). The amount above attachment is \(63{,}000-50{,}000=13{,}000\), below the limit, so the recovery is $13,000. The total recovery is \(45{,}000+13{,}000=\textbf{$58,000}\).
Check. The two occurrence totals sum to $174,000. The insurer retains $111,000−$45,000=$66,000 on the first occurrence and $63,000−$13,000=$50,000 on the second, for total retained loss of $116,000. Retained loss plus recovery is $116,000+$58,000=$174,000. The common mistake avoided is applying the attachment separately to each claim rather than to each event total.
Worked Example: Three Occurrences Versus One Annual Aggregate
A hypothetical reinsurance arrangement is evaluated on the same three occurrence totals: $70,000, $85,000, and $110,000, all in one policy year. First calculate recovery under a per-occurrence treaty paying up to $75,000 in excess of $50,000 per occurrence. Then calculate recovery under an annual aggregate treaty with a $50,000 annual attachment and a $75,000 annual limit. The loss totals are deterministic; no distribution is assumed. There is no discounting (interest rate 0%), and no mortality assumption applies.
Method for the per-occurrence treaty. Apply the layer separately to each occurrence:
Thus, the per-occurrence recovery is \(20{,}000+35{,}000+60{,}000=\textbf{$115,000}\). Each uncapped amount is below the $75,000 per-occurrence limit.
Method for the annual aggregate treaty. First add the three occurrences:
The amount above the annual attachment is \(265{,}000-50{,}000=215{,}000\). The annual limit caps the recovery at \(\min(215{,}000,75{,}000)=\textbf{$75,000}\). The two recoveries differ because the per-occurrence treaty has a separate limit for each occurrence, whereas the annual aggregate treaty has one limit for the entire year.
Check. The per-occurrence payments total $115,000, which is less than the $225,000 total of three separate $75,000 maximum payments. The annual aggregate payment cannot exceed its single $75,000 annual limit. A common mistake is to add the annual total and then allow three separate annual limits; that would not follow the stated annual aggregate terms.
Worked Example: Annual Attachment Across Multiple Events
A hypothetical annual aggregate treaty pays up to $90,000 in excess of a $120,000 attachment for one policy year. Four occurrences produce covered claims as follows: occurrence 1 has claims of $20,000 and $28,000; occurrence 2 has claims of $15,000 and $24,000; occurrence 3 has claims of $25,000 and $32,000; occurrence 4 has one claim of $30,000. Find the annual aggregate recovery. Treat the listed values as deterministic losses, with no probability distribution, no discounting (interest rate 0%), and no mortality assumption.
Method: total covered claims for the year before applying the annual layer. The occurrence totals are \(20{,}000+28{,}000=48{,}000\), \(15{,}000+24{,}000=39{,}000\), \(25{,}000+32{,}000=57{,}000\), and $30,000. Therefore,
The annual loss above the attachment is \(174{,}000-120{,}000=54{,}000\). This is less than the annual limit of $90,000, so the recovery is \(\min(54{,}000,90{,}000)=\textbf{$54,000}\).
Check. The insurer retains $120,000, and the reinsurer pays $54,000; together they equal the $174,000 annual loss. The common mistake avoided is charging the $120,000 attachment separately against each occurrence. The contract specifies one annual attachment, so the annual total—not four separate totals—determines the recovery.
Comparison and Exam Traps
| Feature | Occurrence basis | Annual aggregate basis |
|---|---|---|
| Loss grouped before applying attachment | Claims assigned to one occurrence | All covered losses in one policy year |
| Number of attachment applications | One per occurrence | One per policy year |
| Limit applies to | Each occurrence, if the treaty says so | The annual recovery, if the treaty says so |
A particularly tempting error is to apply an occurrence attachment claim by claim. For instance, suppose two claims of $30,000 and $45,000 arise from one occurrence, under a treaty with a $50,000 attachment and a $40,000 per-occurrence limit.
That calculation uses a per-claim basis. On the stated occurrence basis, first combine the claims: \(30{,}000+45{,}000=75{,}000\). The amount above attachment is $25,000, below the $40,000 limit, so the correct recovery is \(g(75{,}000;50{,}000,40{,}000)=\textbf{$25,000}\).
Before calculating, check the contract basis and the event grouping, then apply the attachment and limit the correct number of times. The stated limit must also be interpreted on that same basis: a per-occurrence limit is not an annual limit, and an annual aggregate limit is not multiplied by the number of events.
Check Your Understanding
For each problem, identify the treaty basis before applying the attachment and limit.
- A hypothetical occurrence treaty pays up to $40,000 in excess of $45,000 per occurrence. One event produces claims of $42,000 and $31,000. Find the recovery. (Answer: $28,000.)
- A hypothetical annual aggregate treaty has an attachment of $90,000 and a limit of $50,000. Annual losses are $24,000, $38,000, and $52,000. Find the recovery. (Answer: $24,000.)
- Three separate occurrences have totals of $68,000, $44,000, and $95,000. A hypothetical treaty pays up to $40,000 in excess of $50,000 per occurrence. Find total recovery. (Answer: $58,000.)
- In one policy year, two occurrences have claims of $30,000 and $25,000, and $60,000 and $35,000, respectively. A third occurrence has a claim of $20,000. A hypothetical annual aggregate treaty has a $100,000 attachment and a $60,000 limit. Find recovery. (Answer: $60,000.)
- The occurrence totals in a hypothetical policy year are $55,000, $25,000, and $80,000. Compare recovery under a per-occurrence treaty with a $40,000 attachment and $35,000 limit per occurrence against an annual aggregate treaty with an $80,000 attachment and a $70,000 annual limit. (Answer: Per-occurrence recovery $50,000; annual aggregate recovery $70,000.)