Per-Occurrence Excess of Loss Aggregation

Why the Order of Calculation Matters

A per-occurrence excess of loss contract responds to the covered losses assigned to one occurrence as a group. The attachment applies to the group total, not independently to each claim in that group. This distinction matters whenever several claims from the same occurrence are individually below the attachment but collectively exceed it.

The calculation therefore has an essential order: first identify and total the covered claims belonging to the occurrence; then apply the attachment and limit to that total. Applying the attachment to each claim before adding the claims changes the calculation and generally does not represent a per-occurrence cover.

Definition: Let $X_{oj}$ be the covered amount of claim $j$ assigned to occurrence $o$, after any applicable policy-level coverage adjustments. The covered loss total for occurrence $o$ is $S_o=\sum_j X_{oj}$. For a per-occurrence excess of loss layer with attachment $A$ and limit $L$, where $L$ is the maximum recovery for that occurrence, the recovery is $R_o=\min\{(S_o-A)^+,L\}$, where $z^+=\max(z,0)$. The net covered loss is $S_o-R_o$.

The attachment $A$ is the amount of covered loss retained below the layer. The limit $L$ is the layer’s maximum payment, so the top of the layer is $A+L$. The recovery can equivalently be written as the covered portion of the occurrence total lying between $A$ and $A+L$. If a contract uses “limit” differently, its wording must be interpreted before using this formula.

This tutorial assumes the claim amounts and occurrence assignments are already known. Each worked example uses hypothetical, deterministic dollar amounts valued at the same date. No probability distribution, interest rate, or mortality assumption is applicable. There are no additional coverage adjustments or aggregate caps in the examples unless stated.

Deriving the Occurrence Recovery Formula

The formula follows from the layer’s two payment boundaries. If the total occurrence loss does not reach the attachment, the layer pays nothing. Once the total exceeds the attachment, the layer pays the excess above $A$, but only until the recovery reaches $L$.

R_o = \begin{cases} 0, & S_o \le A,\\ S_o-A, & A < S_o < A+L,\\ L, & S_o \ge A+L. \end{cases}

For $S_o\le A$, the positive part $(S_o-A)^+$ is zero. For $A<S_o<A+L$, the positive part is $S_o-A$, which is below the limit. For $S_o\ge A+L$, the positive part is at least $L$, so taking the minimum with $L$ caps the payment. These three cases are exactly summarized by $R_o=\min\{(S_o-A)^+,L\}$.

Identify the occurrence and covered claims.
Include only the claims assigned to that occurrence and use their covered amounts as the inputs.

Aggregate before applying the layer.
Calculate $S_o=\sum_j X_{oj}$. Do not apply the occurrence attachment claim by claim.

Apply attachment and limit.
Calculate the amount above attachment, $(S_o-A)^+$, then cap it at $L$.

Check the net amount.
Subtract the recovery from the occurrence’s covered total. The recovery cannot exceed either the amount above attachment or the layer limit.

Worked Calculations

Worked Example: Several Claims Cross the Attachment Together

A hypothetical equipment insurer has three covered claims from one occurrence: $90,000, $160,000, and $210,000. The per-occurrence layer has an attachment of $250,000 and a limit of $150,000. Find the recovery and net covered loss. Treat the amounts as deterministic, with no interest or mortality assumption and no probability distribution.

Method: aggregate the covered claims first. The occurrence total is

S_o=\$90{,}000+\$160{,}000+\$210{,}000=\$460{,}000.

The amount above the attachment is

(S_o-A)^+=(\$460{,}000-\$250{,}000)^+=\$210{,}000.

The layer is limited to $150,000, so

R_o=\min(\$210{,}000,\$150{,}000)=\$150{,}000.

The net covered loss is

S_o-R_o=\$460{,}000-\$150{,}000=\$310{,}000.

Check: The layer’s top is $250,000 + $150,000 = $400,000. The occurrence total of $460,000 exceeds that top, so the full $150,000 limit is exhausted. The recovery and net loss reconcile to the gross covered total.

Worked Example: Comparing Occurrence and Claim-by-Claim Treatment

A hypothetical wind event produces covered claims of $180,000, $140,000, and $95,000. A per-occurrence layer attaches at $200,000 and has a limit of $150,000. Calculate its recovery. Then show what would result if the same attachment and limit were incorrectly applied separately to each claim. Amounts are deterministic and valued at the same date; no probability distribution, interest, or mortality assumption applies.

Correct method: occurrence aggregation. First total the three claims:

S_o=\$180{,}000+\$140{,}000+\$95{,}000=\$415{,}000.

The occurrence loss above the attachment is

(S_o-A)^+=(\$415{,}000-\$200{,}000)^+=\$215{,}000.

Capping that amount at the $150,000 limit gives

R_o=\min(\$215{,}000,\$150{,}000)=\$150{,}000.

The correct net covered loss is $415,000 − $150,000 = $265,000.

Incorrect claim-by-claim calculation: Applying the $200,000 attachment separately gives $(\$180{,}000-\$200{,}000)^+=0$, $(\$140{,}000-\$200{,}000)^+=0$, and $(\$95{,}000-\$200{,}000)^+=0$. The resulting sum of recoveries would be zero, rather than $150,000.

Check and lesson: Each claim is below the attachment, but their combined occurrence total is $415,000. The claim-by-claim method incorrectly treats an occurrence attachment as though it were a separate attachment for every claim. The net amount under the actual occurrence calculation also checks: $265,000 retained plus $150,000 recovered equals $415,000.

Worked Example: Applying the Layer Separately to Two Occurrences

A hypothetical insurer has two distinct occurrences. The first has covered claims of $70,000, $60,000, and $40,000; the second has covered claims of $160,000 and $90,000. Each occurrence is subject to a $100,000 attachment and a $100,000 limit. Assume the full limit is available for each occurrence and there is no other aggregate cap. Find the recovery and net loss for each occurrence and in total. The amounts are deterministic; no probability distribution, interest rate, or mortality assumption applies.

First occurrence. Its total is

S_1=\$70{,}000+\$60{,}000+\$40{,}000=\$170{,}000.

Its amount above attachment is $170,000 − $100,000 = $70,000, below the $100,000 limit. Therefore $R_1=\$70{,}000$, and its net loss is $170,000 − $70,000 = $100,000.

Second occurrence. Its total is

S_2=\$160{,}000+\$90{,}000=\$250{,}000.

Its amount above attachment is $250,000 − $100,000 = $150,000. The limit caps the recovery at $R_2=\$100{,}000$, leaving a net loss of $250,000 − $100,000 = $150,000.

Across the two occurrences, the gross total is $170,000 + $250,000 = $420,000, and the total recovery is $70,000 + $100,000 = $170,000. The total net loss is $100,000 + $150,000 = $250,000. Check: $170,000 + $250,000 = $420,000, so the net losses plus recoveries reconcile to the gross total. Each occurrence receives its own attachment calculation and its own available limit.

Common Exam Trap: Attaching Before Aggregating

The most consequential error is to calculate an excess amount for each claim, add those amounts, and call the result the per-occurrence recovery. That calculation substitutes a different structure for the contract described. In particular, it can miss a recovery when no individual claim exceeds the attachment but the total occurrence loss does.

For example, with claims of $80,000 and $90,000 from one occurrence and an attachment of $150,000, the correct attachment calculation is based on $170,000 − $150,000 = $20,000 before applying the layer limit. Applying the $150,000 attachment separately to $80,000 and $90,000 instead produces zero for both claims. The correction is to combine the covered claims first, then apply the attachment once to the occurrence total.

A related error is to combine claims from distinct occurrences before applying the layer. The per-occurrence formula uses $S_o$ for one occurrence, not the total across unrelated occurrences. If several occurrences are covered, calculate each occurrence’s recovery using its own total and then sum those recoveries, subject to any additional contract terms that apply.

Interpreting the Result

A useful reasonableness check is to compare the recovery with both the excess above attachment and the layer limit:

0\le R_o\le (S_o-A)^+ \quad\text{and}\quad R_o\le L.

If $S_o\le A$, recovery must be zero. If $A<S_o<A+L$, recovery equals $S_o-A$. If $S_o\ge A+L$, the full limit is paid. These checks catch arithmetic and interpretation errors before the result is used in a gross-to-net loss analysis.

The distinction between occurrence and claim units is not merely a calculation convention: it determines how the attachment is triggered. A candidate should establish the covered amounts and occurrence grouping from the stated facts, then use the contract’s attachment and limit at that same unit. This tutorial assumes the occurrence grouping is given; interpreting disputed occurrence definitions is a separate contract question.

Key takeaway: For a per-occurrence excess of loss layer, sum the covered claims assigned to one occurrence before applying the attachment. Calculate $R_o=\min\{(S_o-A)^+,L\}$, then subtract that recovery from the occurrence total to obtain net covered loss.

Check Your Understanding

For each problem, aggregate the covered claims by occurrence before applying the stated attachment and limit.

A single occurrence produces covered claims of $45,000, $70,000, and $105,000. The layer attaches at $150,000 and has a $80,000 limit. Find the recovery and net covered loss. (Answer: Recovery $70,000; net loss $150,000.)
One occurrence has claims of $80,000, $95,000, and $60,000. The attachment is $175,000 and the limit is $100,000. Find the occurrence recovery and explain whether every individual claim exceeds the attachment. (Answer: Recovery $60,000; none of the individual claims exceeds the attachment.)
Occurrence 1 has claims of $130,000 and $90,000; occurrence 2 has claims of $55,000 and $40,000. Each occurrence has a $100,000 attachment and a $75,000 limit, with the full limit available for each. Find the total recovery and total net loss. (Answer: Total recovery $75,000; total net loss $240,000.)
A single occurrence has covered claims of $210,000, $180,000, and $160,000. Its layer attaches at $300,000 and has a $125,000 limit. Calculate the recovery and net covered loss. (Answer: Recovery $125,000; net loss $425,000.)
A single occurrence produces claims of $30,000, $55,000, and $50,000. The layer attaches at $80,000 and has a $60,000 limit. Compare the occurrence recovery with the result of incorrectly applying the $80,000 attachment separately to each claim. (Answer: Correct occurrence recovery $55,000; incorrect claim-by-claim result $0.)

What You'll Learn