What a Per-Risk Recovery Measures
A per-risk excess-of-loss contract protects the insurer against covered losses on individual risks above a specified retention. For each risk, the reinsurer pays only the part of the covered loss that falls within the contract’s layer. The insurer remains responsible for the loss below the layer and for any amount above the layer’s limit.
The key is to apply the contract separately to each risk. A recovery on one risk does not use up the limit available for another risk under a per-risk cover. This differs from a contract whose limit applies to losses combined across an occurrence; the unit to which the contract applies must be read from the stated terms.
The notation is read in standard order: the limit comes first, then the retention. Thus \(40{,}000\) xs \(100{,}000\) means the reinsurer can pay at most \(40{,}000\) on a risk, and pays only above the insurer’s first \(100{,}000\) of covered loss. The layer runs from \(100{,}000\) to \(140{,}000\); it is not a \(100{,}000\) limit attaching above \(40{,}000\).
Calculate the Recovery and the Retained Loss
For an individual risk, compare its covered loss with two thresholds: the retention \(R\) and the exhaustion point \(R+L\). Below the retention, there is no recovery. Inside the layer, the recovery is the amount above the retention. At or above exhaustion, the recovery is capped at the limit.
The insurer retains the part not paid by the reinsurer. Equivalently, the retained loss is the portion up to the retention plus any loss above the top of the layer.
These expressions follow directly from the layer’s boundaries. If \(X\le R\), none of the loss is above the attachment point, so the recovery is zero. If \(R<X<R+L\), the amount above the retention is \(X-R\), and it is still below the limit. If \(X\ge R+L\), the full layer of width \(L\) has been reached, so the recovery cannot exceed \(L\). In every case, the insurer’s retained amount is the original loss minus that recovery.
Identify the recovery limit \(L\) and retention \(R\), in that order. Find the exhaustion point \(R+L\).
For each covered loss \(X\), calculate the amount above \(R\), then cap that amount at \(L\). A convenient calculation is \(C(X)=\min(\max(X-R,0),L)\).
Use \(I(X)=X-C(X)\). Check that the recovery and retained amount add back to the covered loss for that risk.
For a portfolio under a per-risk cover, add the individual recoveries and retained losses after the contract has been applied separately to every risk.
Worked Applications
Worked Example: A Loss Below, At, and Within the Layer
A hypothetical insurer has a per-risk cover of \(40{,}000\) xs \(100{,}000\). A covered loss on one risk is \(125{,}000\). Calculate the recovery and the insurer’s retained loss. State how the answer would change if the loss were exactly \(100{,}000\).
Method. The limit is \(L=40{,}000\), the retention is \(R=100{,}000\), and the layer is exhausted at \(R+L=140{,}000\). The common mistake to avoid is treating the first number in the notation as the retention.
For the \(125{,}000\) loss, the amount above the retention is \(125{,}000-100{,}000=25{,}000\). This is below the \(40{,}000\) limit, so the recovery is \(25{,}000\). The insurer retains \(125{,}000-25{,}000=100{,}000\).
At exactly the \(100{,}000\) retention, there is no loss above the attachment point. The recovery is \(0\), and the insurer retains the full \(100{,}000\). For the original loss, the check is \(25{,}000+100{,}000=125{,}000\), as required.
Worked Example: Recoveries for Several Separate Risks
A hypothetical insurer has a per-risk treaty of \(40{,}000\) xs \(100{,}000\). Four separate risks have covered losses of \(70{,}000\), \(100{,}000\), \(125{,}000\), and \(175{,}000\). Find the recovery and retained loss for each risk, then find the portfolio totals.
Method. Apply the same \(100{,}000\) retention and \(40{,}000\) limit independently to each risk. The exhaustion point is \(100{,}000+40{,}000=140{,}000\). The common mistake to avoid is pooling the losses first and applying one layer to their total.
For the \(70{,}000\) loss, the loss does not reach the retention: recovery \(=0\), retained loss \(=70{,}000\).
For the \(100{,}000\) loss, the loss is exactly at the retention: recovery \(=0\), retained loss \(=100{,}000\).
For the \(125{,}000\) loss, the amount above the retention is \(125{,}000-100{,}000=25{,}000\). It is below the limit, so recovery \(=25{,}000\), and retained loss \(=125{,}000-25{,}000=100{,}000\).
For the \(175{,}000\) loss, the amount above the retention is \(75{,}000\), which exceeds the \(40{,}000\) limit. Recovery \(=40{,}000\), and retained loss \(=175{,}000-40{,}000=135{,}000\).
| Covered loss | Recovery | Retained loss |
|---|---|---|
| \(70{,}000\) | \(0\) | \(70{,}000\) |
| \(100{,}000\) | \(0\) | \(100{,}000\) |
| \(125{,}000\) | \(25{,}000\) | \(100{,}000\) |
| \(175{,}000\) | \(40{,}000\) | \(135{,}000\) |
| Total | \(65{,}000\) | \(405{,}000\) |
The total covered loss is \(70{,}000+100{,}000+125{,}000+175{,}000=470{,}000\). The portfolio check is \(65{,}000+405{,}000=470{,}000\). Each individual row also reconciles to its own covered loss.
Worked Example: A Loss at and Above Layer Exhaustion
A hypothetical per-risk cover is \(75{,}000\) xs \(125{,}000\). Three separate risks have covered losses of \(150{,}000\), \(200{,}000\), and \(260{,}000\). Calculate each recovery and retained amount, and reconcile the portfolio totals.
Method. Here \(R=125{,}000\), \(L=75{,}000\), and exhaustion occurs at \(125{,}000+75{,}000=200{,}000\). The common mistake to avoid is paying the full amount above retention even when it exceeds the limit.
For the \(150{,}000\) loss, the excess above retention is \(150{,}000-125{,}000=25{,}000\), less than the limit. Recovery \(=25{,}000\); retained loss \(=150{,}000-25{,}000=125{,}000\).
For the \(200{,}000\) loss, the excess above retention is \(200{,}000-125{,}000=75{,}000\), exactly the limit. Recovery \(=75{,}000\); retained loss \(=200{,}000-75{,}000=125{,}000\).
For the \(260{,}000\) loss, the excess above retention is \(260{,}000-125{,}000=135{,}000\), but recovery is capped at \(75{,}000\). Retained loss \(=260{,}000-75{,}000=185{,}000\). The amount above the layer remains with the insurer.
Total covered loss is \(150{,}000+200{,}000+260{,}000=610{,}000\). Total recovery is \(25{,}000+75{,}000+75{,}000=175{,}000\), and total retained loss is \(125{,}000+125{,}000+185{,}000=435{,}000\). The check is \(175{,}000+435{,}000=610{,}000\).
Common Mistakes and Checks
An answer should reflect both the layer boundaries and the per-risk basis. A plausible-looking total can still be wrong if the contract has been applied to the wrong unit or its limit has been interpreted in the wrong order.
- Reversing the notation. In \(L\) xs \(R\), \(L\) is the limit and \(R\) is the retention. For example, \(30{,}000\) xs \(70{,}000\) attaches at \(70{,}000\), not at \(30{,}000\).
- Ignoring the limit. The recovery is not always \(X-R\). That expression applies only while the loss remains within the layer; above exhaustion, the recovery stays at \(L\).
- Giving a recovery at the retention. If \(X=R\), the amount above retention is zero, so there is no recovery.
- Pooling risks before applying a per-risk cover. Calculate each risk’s recovery using its own covered loss. Add the resulting recoveries only afterward.
- Forgetting the amount above the layer. When \(X>R+L\), the insurer retains \(X-L\), not merely \(R\).
A reliable final check is \(X=C(X)+I(X)\) for every risk, and the equivalent portfolio check that total covered losses equal total recoveries plus total retained losses. This calculation assumes the losses provided are covered amounts to which the stated per-risk terms apply; exclusions, deductibles, reinstatements, or other contract provisions would need to be handled if specified.
Check Your Understanding
For each hypothetical contract, calculate the recoveries and retained losses using the stated per-risk basis. Check that the two amounts reconcile to the covered loss or portfolio total.
- A cover is \(25{,}000\) xs \(60{,}000\). Two separate risks have losses of \(45{,}000\) and \(72{,}000\). Find each recovery and retained loss. (Answer: Recoveries \(0\) and \(12{,}000\); retained losses \(45{,}000\) and \(60{,}000\).)
- A cover is \(50{,}000\) xs \(80{,}000\). Separate risks have losses of \(110{,}000\), \(130{,}000\), and \(150{,}000\). Find the total recovery and total retained loss. (Answer: Total recovery \(130{,}000\); total retained loss \(260{,}000\).)
- A cover is \(35{,}000\) xs \(90{,}000\). Three risks have losses of \(125{,}000\), \(140{,}000\), and \(200{,}000\). Find the recovery for each risk and the portfolio totals. (Answer: Recoveries \(35{,}000\), \(35{,}000\), and \(35{,}000\); total recovery \(105{,}000\); total retained loss \(360{,}000\).)
- A cover is \(60{,}000\) xs \(140{,}000\). Risk A has two covered loss components, \(95{,}000\) and \(80{,}000\), which the stated contract terms require to be combined into one covered loss for that risk. Risk B has a covered loss of \(230{,}000\), and risk C has a covered loss of \(300{,}000\). Find the recovery and retained amount for each risk, then the totals. (Answer: Recoveries \(35{,}000\), \(60{,}000\), and \(60{,}000\); total recovery \(155{,}000\); total retained loss \(550{,}000\).)
- A cover is \(45{,}000\) xs \(105{,}000\). Five separate risks have losses of \(104{,}000\), \(105{,}000\), \(149{,}000\), \(151{,}000\), and \(220{,}000\). Find each recovery and the total retained loss. (Answer: Recoveries \(0\), \(0\), \(44{,}000\), \(45{,}000\), and \(45{,}000\); total retained loss \(595{,}000\).)