How a Per-Risk Layer Responds to a Loss
A per-risk excess of loss treaty responds to the loss on an individual insured risk once that loss exceeds a stated attachment point. The reinsurer pays only the part of the loss inside the layer, up to the layer limit. The cedent therefore retains losses below the attachment point and any amount above an exhausted layer.
The calculation has two boundaries. The attachment point is where the layer begins to pay. The exhaustion point is the attachment point plus the limit; losses above exhaustion do not increase the recovery. Getting these boundaries right is essential when translating treaty terms into ceded and retained amounts.
This definition treats \(X\) as the loss amount to which the treaty terms apply. In an actual contract, the covered amount may depend on policy limits, exclusions, or other wording. Those adjustments must be made before applying this formula; they are not part of the formula itself. Here, the examples specify the covered per-risk loss directly.
Deriving the Recovery Formula
The layer can pay only after the loss has passed the attachment point. The amount above attachment is \((X-A)^+\): it is zero when \(X\leq A\), and \(X-A\) when \(X>A\). But the reinsurer cannot pay more than the layer limit \(L\). Taking the smaller of the amount above attachment and \(L\) enforces both conditions.
The cases follow directly from the layer’s boundaries. At or below \(A\), there is no amount above attachment. Between \(A\) and \(A+L\), the amount above attachment is positive but less than \(L\), so the layer pays that amount. At or above \(A+L\), the amount above attachment is at least \(L\), so the limit caps the payment. The formula is continuous at both boundaries: the recovery is zero at \(X=A\), and it reaches \(L\) at \(X=A+L\).
For a per-risk treaty, use the covered loss for one risk, not a combined loss across several risks.
The layer begins at \(A\) and is exhausted at \(A+L\).
Use \((X-A)^+\), which cannot be negative.
The recovery is the smaller of the amount above attachment and \(L\).
Confirm that gross loss equals recovery plus retained loss.
Worked Applications
Worked Example: Loss Below the Attachment
A hypothetical property treaty covers an individual risk’s loss above an attachment point of $275,000, subject to a limit of $500,000. The covered loss on one property is $238,000. Calculate the recovery and the cedent’s retained loss. State the method and a mistake to avoid.
Method. Apply the per-risk layer formula, first comparing the loss with the attachment and then checking the limit.
The exhaustion point is \(A+L=\$275{,}000+\$500{,}000=\$775{,}000\). The loss is below attachment because \(\$238{,}000<\$275{,}000\).
The amount above attachment, floored at zero, is \((X-A)^+=\max(\$238{,}000-\$275{,}000,0)=\max(-\$37{,}000,0)=\$0\). Therefore, \(R(X)=\min(\$0,\$500{,}000)=\$0\).
The retained loss is \(X-R(X)=\$238{,}000-\$0=\$238{,}000\). The balance check gives \(\$238{,}000=\$0+\$238{,}000\), as required.
Common mistake avoided: Do not report a negative recovery of \(-\$37{,}000\). The reinsurer pays nothing when the loss is below attachment; the positive-part operation makes that explicit.
Worked Example: Loss Within the Layer
A hypothetical equipment portfolio has a per-risk layer attaching at $320,000 with a limit of $180,000. One covered equipment loss is $435,000. Calculate the recovery and retained loss, and verify whether the layer is exhausted.
Method. Find the exhaustion point, calculate the loss above attachment, and cap that amount at the limit.
The exhaustion point is \(A+L=\$320{,}000+\$180{,}000=\$500{,}000\). Because \(\$320{,}000<\$435{,}000<\$500{,}000\), the loss falls within the layer.
The amount above attachment is \(X-A=\$435{,}000-\$320{,}000=\$115{,}000\). Applying the limit gives \(R(X)=\min(\$115{,}000,\$180{,}000)=\$115{,}000\). The layer is not exhausted because the recovery is less than its $180,000 limit.
The retained loss is \(X-R(X)=\$435{,}000-\$115{,}000=\$320{,}000\). The balance check gives \(\$435{,}000=\$115{,}000+\$320{,}000\).
Common mistake avoided: The limit is not the recovery automatically. The $180,000 is the maximum the treaty can pay, not a fixed payment whenever a loss enters the layer.
Worked Example: Loss Above Exhaustion
A hypothetical commercial property treaty provides $300,000 of per-risk cover excess of $450,000. A covered loss on one insured property is $1,020,000. Calculate the exhaustion point, recovery, and retained loss, including the part of the loss above the layer.
Method. Compare the loss with the exhaustion point, then apply the limit. A loss above exhaustion uses the entire limit, not the entire amount above attachment.
Here \(A=\$450{,}000\) and \(L=\$300{,}000\), so \(A+L=\$450{,}000+\$300{,}000=\$750{,}000\). The loss exceeds exhaustion by \(\$1{,}020{,}000-\$750{,}000=\$270{,}000\).
The amount above attachment is \(X-A=\$1{,}020{,}000-\$450{,}000=\$570{,}000\). The recovery is therefore \(R(X)=\min(\$570{,}000,\$300{,}000)=\$300{,}000\).
The retained loss is \(X-R(X)=\$1{,}020{,}000-\$300{,}000=\$720{,}000\). This retained amount consists of the first $450,000 below attachment plus the $270,000 above exhaustion: \(\$450{,}000+\$270{,}000=\$720{,}000\). The final check is \(\$1{,}020{,}000=\$300{,}000+\$720{,}000\).
Common mistake avoided: Paying the full $570,000 above attachment would ignore the $300,000 limit. The amount above attachment must always be capped at \(L\).
Interpreting Recovery and Retention
A useful way to check a calculation is to express the retained loss by region. Below attachment, the cedent retains the entire loss. Within the layer, the cedent retains the amount up to attachment. Above exhaustion, the cedent retains the attachment amount plus the loss above the layer.
The middle case is especially useful: while the loss is within the layer, each additional dollar of loss is recovered, so the retained amount remains \(A\). Once the layer is exhausted, the recovery cannot increase; further loss is retained by the cedent. These checks also reveal why it is important to distinguish the attachment point from the limit. The attachment determines where payment begins, while the limit determines the maximum payment and, together with attachment, where payment stops.
| Loss position | Recovery | Retained loss |
|---|---|---|
| At or below attachment \(A\) | Zero | Entire loss \(X\) |
| Between \(A\) and \(A+L\) | \(X-A\) | \(A\) |
| At or above exhaustion \(A+L\) | Limit \(L\) | \(X-L\) |
Common Exam Traps
- Confusing the limit with the exhaustion point. A layer attaching at \(A\) with limit \(L\) exhausts at \(A+L\), not at \(L\).
- Allowing a negative recovery. If \(X<A\), use \((X-A)^+=0\), not \(X-A\) as a payment.
- Ignoring the cap. The recovery cannot exceed \(L\), even when the loss is far above attachment.
- Using the wrong unit of loss. A per-risk calculation uses the loss on one risk. Combining losses before applying the layer can change the calculation and does not follow the stated per-risk basis.
- Failing to check the accounting identity. For a fully specified covered loss, the calculation should satisfy \(X=R(X)+(X-R(X))\).
Check Your Understanding
For each problem, treat the stated loss as the covered loss on the specified risk. Apply the per-risk formula and check whether the recovery respects both layer boundaries.
- A layer attaches at $125,000 and has a $225,000 limit. A risk has a covered loss of $110,000. What is the recovery? (Answer: $0.)
- A layer attaches at $210,000 and has a $160,000 limit. A risk has a covered loss of $325,000. Calculate the recovery and retained loss. (Answer: Recovery $115,000; retained loss $210,000.)
- A layer attaches at $400,000 with a $250,000 limit. A risk has a covered loss of $730,000. Find the exhaustion point, recovery, and retained loss. (Answer: Exhaustion point $650,000; recovery $250,000; retained loss $480,000.)
- A hypothetical per-risk treaty attaches at $175,000 and has a $325,000 limit. Three separate risks have covered losses of $150,000, $280,000, and $610,000. Apply the layer to each risk separately and find the sum of the three recoveries. (Answer: Recoveries $0, $105,000, and $325,000; sum $430,000.)
- A layer attaches at $260,000 and has a $190,000 limit. A risk’s recovery is $145,000. Assuming the stated recovery is correct and the loss is covered, determine the loss and verify that it lies within the layer. (Answer: Loss $405,000; exhaustion point $450,000, so the loss is within the layer.)