Per-Claim Versus Aggregate Excess-of-Loss

Why the Treaty Basis Matters

An excess-of-loss treaty transfers losses above an attachment point, subject to a limit. The calculation changes depending on what loss the treaty applies to. A per-claim treaty applies its attachment and limit separately to each claim. An aggregate treaty applies them once to the total loss over the covered period. Consequently, identical claim amounts can produce different recoveries under the two bases.

The distinction matters when estimating how much a hypothetical portfolio retains or cedes. A per-claim layer responds to the distribution of claim sizes: it may make several payments if several claims penetrate its attachment. An aggregate layer responds to the total loss: many small claims can together exceed its attachment even when none would do so individually.

Definition: For a loss $x$, a per-claim excess-of-loss layer with attachment $a$ and limit $u$ pays $\min((x-a)^+,u)$, where $(z)^+=\max(z,0)$. The layer attaches at $a$, has a maximum payment of $u$, and is exhausted at $a+u$. For annual aggregate loss $S$, an aggregate layer with attachment $D$ and limit $U$ pays $\min((S-D)^+,U)$. These are separate treaty bases; the attachment and limit on one basis should not automatically be treated as the attachment and limit on the other.

If the portfolio has $N$ claims with losses $X_1,\ldots,X_N$, its annual aggregate loss is $S=\sum_{j=1}^{N}X_j$. The total per-claim recovery is the sum of the payments made on the individual claims:

C_{\mathrm{claim}}=\sum_{j=1}^{N}\min\big((X_j-a)^+,u\big).

By contrast, the aggregate recovery is calculated from the sum of the claims:

C_{\mathrm{aggregate}}=\min\big((S-D)^+,U\big).

The notation makes the order of operations explicit. For per-claim cover, apply the layer to each $X_j$, then add the payments. For aggregate cover, add the $X_j$ first, then apply the layer once to $S$. Neither formula is a shortcut for the other.

How the Two Calculations Differ

A per-claim layer can make no payment on a loss below its attachment even if that loss contributes to the annual total. An aggregate layer, however, counts every covered claim toward $S$, including losses below the aggregate attachment. Conversely, a large claim can exhaust a per-claim limit, while an aggregate treaty’s limit applies only to the total recovery for the period.

The following calculation shows why the order matters. Each layer is first applied to the amount named in its basis: an individual claim for per-claim cover, or the annual total for aggregate cover.

Identify the basis.
Determine whether the treaty applies separately to each claim or once to the sum of claims.

Apply the attachment.
For each relevant amount $z$, calculate the excess $(z-a)^+$, or $(z-D)^+$ for aggregate cover.

Apply the limit.
Cap each per-claim payment at $u$, or cap the single aggregate payment at $U$.

Find retained loss.
Subtract the treaty recovery from the original loss total for the same basis.

Worked Example: Comparing Recoveries for Four Claims

A hypothetical insurer has four claims of $18,000, $35,000, $52,000, and $90,000. Compare (a) a per-claim layer of $50,000 excess of $40,000 with (b) an annual aggregate layer of $50,000 excess of $40,000. Here “$50,000 excess of $40,000” means an attachment of $40,000 and a maximum recovery of $50,000. Find each recovery and the corresponding retained loss.

Method: calculate on the treaty’s stated basis. Work in thousands of dollars. For the per-claim layer, the recovery on claim $x$ is $\min((x-40)^+,50)$.

For the $18,000 claim, the amount above attachment is $(18-40)^+=0$, so recovery is $\min(0,50)=0$. For the $35,000 claim, it is $(35-40)^+=0$, so recovery is 0. For the $52,000 claim, it is $(52-40)^+=12$, so recovery is $\min(12,50)=12$. For the $90,000 claim, it is $(90-40)^+=50$, so recovery is $\min(50,50)=50$.

The total per-claim recovery is $0+0+12+50=62$, or $62,000. The original total is $18+35+52+90=195$, or $195,000. The insurer retains $195-62=133$, or $133,000.

For the aggregate treaty, first calculate $S=195$ thousand. Its recovery is $\min((195-40)^+,50)=\min(155,50)=50$, or $50,000. Aggregate-basis retained loss is $195-50=145$, or $145,000.

Check. In each calculation, recovery plus retained loss equals the $195,000 total. The per-claim recovery exceeds the aggregate recovery by $12,000 in this example; the two layers need not pay the same amount, even when their attachment and limit happen to have the same numerical values.

Common mistake avoided: Applying the $50,000 limit once to the sum of the per-claim recoveries would incorrectly replace the per-claim calculation with an aggregate calculation. Here the per-claim sum is $62,000, which is greater than $50,000, but the treaty permits up to $50,000 on each claim, not just $50,000 across the year.

Why the Formulas Have Different Orders

The per-claim formula follows from the contract being evaluated claim by claim. For a particular claim $X_j$, nothing is ceded until it exceeds $a$. The amount above $a$ is then capped at $u$, giving $\min((X_j-a)^+,u)$. Summing those payments over the $N$ claims gives the total recovery under that basis.

For aggregate cover, the treaty’s covered amount is the annual total $S$, not any one $X_j$. The amount above the annual attachment is therefore $(S-D)^+$; applying the annual limit $U$ gives the aggregate formula. This is also why adding claim-level amounts above an aggregate attachment would be the wrong operation: the attachment is deducted once from $S$, not once from each claim.

\text{Per-claim basis: }\sum_{j=1}^{N}\min\big((X_j-a)^+,u\big) \qquad \text{Aggregate basis: }\min\bigg(\Big(\sum_{j=1}^{N}X_j-D\Big)^+,U\bigg).

These expressions generally differ because the positive-part operation and the limit are applied at different stages. No general rule says that one recovery must always be larger than the other: the result depends on the claims and on both treaties’ attachments and limits.

Worked Example: Aggregate Recovery When Individual Claims Are Mostly Below Attachment

A hypothetical portfolio has claims of $12,000, $24,000, $45,000, $31,000, and $28,000. Compare a per-claim layer of $60,000 excess of $40,000 with an aggregate layer of $60,000 excess of $40,000. Find the recoveries and retained totals.

Per-claim calculation. In thousands, apply $\min((x-40)^+,60)$ to each claim. The $12,000, $24,000, $31,000, and $28,000 claims are each below $40,000, so each has recovery 0. The $45,000 claim has excess $45-40=5$, which is below the $60,000 limit, so its recovery is 5. Total per-claim recovery is $0+0+5+0+0=5$, or $5,000.

The total loss is $12+24+45+31+28=140$ thousand. Per-claim-basis retained loss is $140-5=135$, or $135,000.

Aggregate calculation. Apply the aggregate layer once to $S=140$: $\min((140-40)^+,60)=\min(100,60)=60$. Aggregate recovery is $60,000, and aggregate-basis retained loss is $140-60=80$, or $80,000.

Check. Each recovery plus its corresponding retained loss equals $140,000. The four claims below the per-claim attachment still contribute to the aggregate total, which is why the aggregate layer pays its full $60,000 limit here.

Common mistake avoided: The aggregate attachment is not applied separately to each claim. Doing that would produce the per-claim result of $5,000 rather than the correct aggregate recovery of $60,000.

Same Aggregate, Different Claim Patterns

An aggregate treaty depends on the annual total, so two portfolios with the same total have the same aggregate recovery when their aggregate treaty terms are the same. A per-claim treaty can distinguish between those portfolios, because it responds to the size of each individual claim. The next example holds the total fixed to isolate that difference.

Worked Example: Equal Annual Losses but Different Per-Claim Recoveries

Consider two hypothetical portfolios, each with three claims and total loss of $60,000. Portfolio A has claims of $5,000, $5,000, and $50,000. Portfolio B has claims of $20,000, $20,000, and $20,000. For each portfolio, compare a per-claim layer of $20,000 excess of $30,000 with an aggregate layer of $30,000 excess of $50,000.

Portfolio A, per-claim basis. In thousands, the $5,000 claims each produce $\min((5-30)^+,20)=0$. The $50,000 claim produces $\min(50-30,20)=20$. Total recovery is $0+0+20=20$, or $20,000. Retained loss is $60-20=40$, or $40,000.

Portfolio B, per-claim basis. Each $20,000 claim is below the $30,000 attachment, so each recovery is 0. Total recovery is $0, and retained loss is $60,000.

Aggregate basis for either portfolio. Both have $S=60$ thousand. The aggregate recovery is $\min((60-50)^+,30)=10$, or $10,000. The aggregate retained loss is $60-10=50$, or $50,000, for each portfolio.

Check. Each portfolio totals $60,000, so their aggregate recoveries match. Their per-claim recoveries differ because Portfolio A has one claim above the per-claim attachment, while Portfolio B has none.

Common mistake avoided: Treating the same annual total as enough information to determine per-claim recovery. It is not: the claim-by-claim amounts are needed.

Exam Traps and Practical Interpretation

The most reliable way to avoid a basis error is to write down the treaty’s unit of application before calculating. For per-claim cover, calculate one payment for each claim and add those payments. For aggregate cover, calculate the total loss first and make one capped payment. Keep the retained loss on the same basis as the recovery being discussed.

A common incorrect calculation is to take each claim’s excess above the aggregate attachment and then sum those excesses. For example, if an aggregate attachment is $D$, this is generally not $\sum_j(X_j-D)^+$. The treaty applies $D$ once to $S$, so the appropriate pre-limit amount is $(\sum_jX_j-D)^+$. Similarly, a per-claim limit is applied to each claim’s payment, not once to the annual sum.

A concise comparison table can help keep the operations separate:

Feature	Per-claim excess of loss	Aggregate excess of loss
Amount tested against attachment	Each individual claim	Total loss for the covered period
How often the attachment applies	Once per claim	Once to the aggregate
How the limit applies	Capped separately for each claim	Capped once for the aggregate recovery
Information needed	Claim amounts	Aggregate loss and treaty terms

This comparison is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. In a calculation, explicitly stating the basis and showing the order of operations makes the result easier to check and helps prevent an attachment or limit from being applied the wrong number of times.

Key takeaway: Per-claim recovery is the sum of separately attached and limited claim payments. Aggregate recovery is one attached and limited payment on the total loss. The two values can differ even when the numerical attachment and limit are the same.

Check Your Understanding

For each problem, identify the treaty basis before calculating. Amounts are in thousands of dollars unless stated otherwise.

A portfolio has claims of 8, 22, and 47. Find the per-claim recovery under a layer of 30 excess of 25 and the aggregate recovery under a layer of 30 excess of 25. (Answer: Per-claim recovery = $22,000; aggregate recovery = $30,000.)
A portfolio has claims of 15, 28, 34, and 39. Find the total recovery under a per-claim layer of 25 excess of 30, then find the recovery under an aggregate layer of 40 excess of 50. (Answer: Per-claim recovery = $13,000; aggregate recovery = $40,000.)
A portfolio has claims of 14, 26, 40, and 50. Calculate the per-claim recovery for a layer of 15 excess of 20 and the aggregate recovery for a layer of 30 excess of 45. (Answer: Per-claim recovery = $36,000; aggregate recovery = $30,000.)
Two portfolios have the same aggregate loss of $90,000: one has claims of $10,000 and $80,000, and the other has claims of $45,000 and $45,000. Find each portfolio’s per-claim recovery for a layer of $25,000 excess of $40,000, and the common aggregate recovery for a layer of $30,000 excess of $60,000. (Answer: Per-claim recoveries are $25,000 and $10,000, respectively; aggregate recovery is $30,000 for each.)
A portfolio has claims of $18,000, $32,000, $44,000, and $76,000. Calculate the per-claim recovery for a layer of $25,000 excess of $35,000 and the aggregate recovery for a layer of $50,000 excess of $55,000. (Answer: Per-claim recovery = $34,000; aggregate recovery = $50,000.)

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