Introduction
A stop-loss treaty protects an insurer when losses over a defined period are unusually high in aggregate. Unlike a cover that responds separately to each claim or occurrence, stop-loss reinsurance compares the period’s aggregate eligible losses with a threshold. The reinsurer pays only for the amount above that threshold, subject to the contract’s terms.
The key calculation is simple, but two contract terms are easy to confuse. The attachment is the aggregate-loss level at which recovery begins. A coverage limit is the maximum amount the reinsurer will pay. If aggregate losses are above the attachment, that does not necessarily mean the whole excess is recoverable: a limit can cap the payment.
The formula assumes that the treaty responds to aggregate losses over one defined period, that the stated attachment and limit apply to the same loss basis, and that no other contract terms change the recovery. In particular, \(L\) must represent the losses the treaty actually counts. The wording may specify, for example, whether losses are gross or net of certain recoveries and which premium amount is used to calculate a loss-ratio attachment. Those definitions are part of the contract, not choices to make after calculating the losses.
Attachment and Coverage Limit
An attachment can be a fixed dollar amount or a percentage of a stated premium base. When the attachment is expressed as a proportion of premium, let \(P\) be the premium base specified by the treaty and \(q\) the attachment percentage, written as a decimal. The dollar attachment is
For example, an attachment of 80% of a premium base of $10 million corresponds to an attachment of $8 million. The reinsurer begins to pay only when eligible aggregate losses exceed $8 million. The percentage is applied to the premium base to find the attachment; it is not itself the recovery percentage.
For a limited contract, \(C\) is the maximum payment by the reinsurer, not the maximum amount of losses above the attachment that can be ignored. The largest aggregate loss level at which the contract can still increase its payment is \(A+C\). Above that level, the payment remains at \(C\), and any further eligible losses are retained by the insurer under this contract.
| Contract term | Meaning | Role in the calculation |
|---|---|---|
| Aggregate eligible losses, \(L\) | Losses counted for the period under the treaty wording | Compared with the attachment |
| Attachment, \(A\) | Loss level above which the cover responds | Subtract from \(L\), but not below zero |
| Coverage limit, \(C\) | Maximum amount the reinsurer pays | Caps the recovery |
| Premium base, \(P\) | Premium amount defined by the treaty | Used only when attachment is premium-based |
Why the Recovery Formula Works
The formula follows directly from the payment in each possible loss range. Assume the contract has a dollar attachment \(A\), a maximum payment \(C\), and no other adjustments. If losses do not exceed \(A\), there is no covered excess. If losses exceed \(A\) but do not exceed \(A+C\), the payment is the amount above \(A\). If losses exceed \(A+C\), the excess above the attachment is greater than \(C\), so the reinsurer pays only \(C\).
If \(L\leq A\), then \(L-A\leq 0\), so the recovery is zero.
If \(A<L<A+C\), the excess \(L-A\) is positive and is less than \(C\), so the recovery is \(L-A\).
If \(L\geq A+C\), then \(L-A\geq C\), so the maximum payment is \(C\).
These three cases combine into \(R=\min(\max(L-A,0),C)\). The inner maximum prevents a negative payment when losses are below attachment; the outer minimum prevents payment above the coverage limit. At the boundary \(L=A\), recovery is zero. At \(L=A+C\), recovery is exactly \(C\), so the formula agrees at both boundaries.
Worked Example: Fixed Attachment With a Payment Limit
A hypothetical insurer reports $12.40 million of eligible aggregate losses for a treaty period. The stop-loss contract has a $10.00 million attachment and a $3.00 million maximum payment. Calculate the recovery and the insurer’s retained losses under this contract. Treat the amounts as a deterministic settlement; no interest rate, mortality assumption, or probability distribution is needed because the aggregate loss is given.
Method. Apply the attachment first, then cap the resulting excess at the payment limit. The common mistake to avoid is treating the $3.00 million limit as a new attachment rather than as the maximum recovery.
Step 1: Calculate the loss excess above attachment.
Step 2: Compare the excess with the payment limit. Since $2.40 million is less than the $3.00 million limit, the limit does not reduce the payment:
Step 3: Calculate retained losses.
Check. The aggregate loss exceeds the attachment by $2.40 million, and the recovery is below the $3.00 million maximum. The insurer retains exactly the attachment amount.
Premium-Based Attachments
A stop-loss attachment is sometimes stated as a percentage of a premium base. This is often described as a loss-ratio attachment: the percentage specifies the loss level at which recovery starts relative to the treaty’s defined premium. The calculation still uses the same recovery formula; first convert the percentage attachment into dollars.
The calculation depends on using the correct premium base. A treaty might define that base by a particular period or type of premium. Do not substitute a convenient premium figure if the contract specifies another one. Nor should the stated attachment percentage be applied to losses to find the recovery: it determines \(A\), while the recovery is the excess of \(L\) above \(A\), subject to the limit.
Worked Example: Loss-Ratio Attachment With Uncapped Recovery
A hypothetical treaty attaches at 75% of a stated $18.00 million premium base and has a $2.40 million maximum payment. Eligible aggregate losses are $15.20 million. Find the dollar attachment, recovery, and retained losses. This is a deterministic settlement using the given premium and aggregate loss; no interest rate, mortality assumption, or probability distribution is used.
Method. Convert the percentage to a dollar attachment, then calculate and limit the excess. The common mistake to avoid is multiplying aggregate losses by 75% and calling that amount the attachment.
Step 1: Convert the attachment percentage to a decimal and calculate the attachment.
Step 2: Calculate the amount above attachment.
Step 3: Apply the coverage limit. The $1.70 million excess is below the $2.40 million maximum:
Step 4: Calculate retained losses.
Check. The recovery brings retained losses back to the $13.50 million attachment because the payment limit does not bind.
Aggregate the Losses Before Applying the Attachment
The attachment applies to the aggregate eligible losses for the contract period, not separately to each claim. The calculation therefore has two conceptually distinct stages: determine \(L\) from the losses counted under the treaty, then apply the stop-loss formula once to that aggregate. Applying the attachment separately to each claim would change the contract’s response.
The following example also shows why a limited stop-loss recovery need not leave the insurer with exactly the attachment amount. Once the excess above attachment is greater than the maximum payment, the difference remains with the insurer.
Worked Example: Claim Aggregation and a Binding Payment Limit
Under a hypothetical treaty, the insurer has four eligible claims of $2.75 million, $3.10 million, $1.85 million, and $4.05 million. The attachment is 85% of a specified $12.00 million premium base, and the maximum payment is $1.20 million. Calculate aggregate losses, recovery, and retained losses. Claim amounts and premium are given for a deterministic settlement; no interest rate, mortality assumption, or probability distribution is needed.
Method. Sum all eligible claims first, convert the premium-based attachment to dollars, and then cap the aggregate excess. The common mistake to avoid is applying the attachment separately to each claim.
Step 1: Sum the eligible claims.
Step 2: Calculate the attachment.
Step 3: Find the excess and apply the payment limit.
Step 4: Calculate retained losses.
Check. The loss level at which the payment reaches its cap is \(A+C=\$10.20+\$1.20=\$11.40\) million. The actual aggregate loss is $11.75 million, or $0.35 million above that level. The insurer retains the $10.20 million attachment plus that $0.35 million excess: $10.55 million, as calculated.
Common Mistakes and Practical Checks
Stop-loss calculations are easier to audit when the attachment, excess, limit, recovery, and retained loss are shown separately. A useful check is that recovery cannot be negative or greater than the stated limit. Retained losses under this treaty must equal aggregate eligible losses minus the recovery.
- Using the limit as the attachment. If \(C\) is the maximum payment, it caps \(L-A\); it does not replace \(A\). For example, with \(A=\$10\) million, \(C=\$2\) million, and \(L=\$11.50\) million, recovery is \(\$1.50\) million, not zero.
- Applying the attachment to each claim. A stop-loss attachment is applied to the treaty-period aggregate. Sum the eligible losses first unless the wording explicitly defines a different basis.
- Treating a premium percentage as a recovery share. For an attachment of \(q\) of premium \(P\), calculate \(A=qP\). Do not multiply the excess or the claim amounts by \(q\) to determine the payment.
- Ignoring the treaty’s measurement basis. The aggregate \(L\) and premium \(P\) must follow the contract definitions. A calculation using the wrong loss or premium basis can be arithmetically correct and still give the wrong contractual result.
Stop-loss reinsurance transfers part of the insurer’s adverse aggregate-loss outcome once losses cross the agreed threshold. The attachment determines when that transfer starts; the coverage limit determines how much can be transferred under the stated cover. Keeping these roles separate makes the calculation transparent and helps reconcile the recovery to the retained loss.
Check Your Understanding
For each hypothetical treaty, calculate the requested recovery using the stated treaty basis. Amounts are deterministic, so no interest, mortality, or probability distribution is assumed. Treat dollar amounts as millions where stated.
- A treaty has an $8.60 million attachment, a $2.00 million maximum payment, and $7.90 million of eligible aggregate losses. What is the recovery? (Answer: $0.00 million.)
- A treaty has a $9.25 million attachment, a $1.50 million maximum payment, and $10.10 million of eligible aggregate losses. Calculate the recovery and retained losses. (Answer: Recovery $0.85 million; retained losses $9.25 million.)
- A treaty attaches at 70% of a $16.00 million premium base, has a $2.00 million maximum payment, and covers $12.40 million of eligible aggregate losses. Calculate the dollar attachment and recovery. (Answer: Attachment $11.20 million; recovery $1.20 million.)
- A treaty attaches at 80% of a $11.50 million premium base and has a $1.50 million maximum payment. Eligible claims for the period are $3.20 million, $2.80 million, and $4.10 million. Calculate aggregate losses and recovery. (Answer: Aggregate losses $10.10 million; recovery $0.90 million.)
- A treaty attaches at 90% of a $25.00 million premium base, has a $2.50 million maximum payment, and has $26.40 million in eligible aggregate losses. Calculate the recovery and the insurer’s retained losses. (Answer: Recovery $2.50 million; retained losses $23.90 million.)