Calculating Stop-Loss Recoveries from Annual Losses

Why Calculate at the Annual Aggregate Level?

A stop-loss reinsurance contract responds to the total loss over a specified period, such as one policy year. The settlement therefore depends on the annual aggregate loss, not on whether any single claim exceeds the attachment. After finding the year’s total, the calculation is governed by two contract terms: the point at which the cover begins to pay and the maximum amount the reinsurer will pay.

This tutorial focuses on deterministic settlement calculations for stated annual losses. No interest rate, mortality assumption, or probability distribution is needed: the claim amounts and the contract terms are given. The same method applies whether the annual total is supplied directly or must first be calculated by adding the claims for the year. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $S$ be the insurer’s total covered loss for one contract period, $A$ the stop-loss attachment, and $M$ the maximum recovery payable by the reinsurer. Assume $S\geq 0$, $A\geq 0$, and $M\geq 0$. The stop-loss recovery is $$ R(S)=\min\bigl(\max(S-A,0),M\bigr). $$ The insurer’s retained aggregate loss is $S-R(S)$. Here, $M$ means the maximum recovery, not the aggregate loss at which the cover is exhausted; that exhaustion point is $A+M$.

Calculate the Recovery and Retained Loss

If the year contains covered claims $X_1,X_2,\ldots,X_n$, first calculate the aggregate $S=X_1+X_2+\cdots+X_n$. Apply the attachment to this total. A total no greater than $A$ produces no recovery. A total above $A$ creates an amount above the attachment, but the recovery cannot exceed $M$. The insurer retains the part of the aggregate not recovered.

R(S)= \begin{cases} 0, & S\leq A,\\ S-A, & A<S<A+M,\\ M, & S\geq A+M, \end{cases} \qquad \text{Retained loss}=S-R(S).

At the boundary $S=A+M$, either adjoining expression gives a recovery of $M$. For example, if $A=100$ and $M=40$, the cover begins to respond above 100 and has paid its full 40 once the aggregate reaches 140. Above 140, the recovery stays at 40; any further increase in loss is retained by the insurer.

Why the Formula Works

The attachment determines how much of the aggregate is below the reinsurer’s layer. If $S\leq A$, none of the loss reaches the layer, so recovery is zero. If $S>A$, the amount above the attachment is $S-A$. That amount is the recovery only while it is no greater than the contract maximum $M$. Once it exceeds $M$, the contract pays no more than $M$. Taking the greater of $S-A$ and zero, followed by the lesser of that result and $M$, gives the stated formula.

Subtracting the recovery from the aggregate gives the insurer’s retained loss. In the middle region, the retained amount is exactly the attachment; after exhaustion, the insurer retains the amount above the recovery cap as well.

S-R(S)= \begin{cases} S, & S\leq A,\\ A, & A<S<A+M,\\ S-M, & S\geq A+M. \end{cases}

This retained-loss form is a useful independent check. Below attachment, retention equals the whole loss. Within the layer, retention remains at the attachment. After exhaustion, retention rises with the aggregate, because the recovery is capped.

A Reliable Settlement Procedure

Read the contract terms.
Identify the period, covered losses, attachment $A$, and maximum recovery $M$. Confirm whether a stated “limit” is the recovery cap or an exhaustion point.

Find the covered annual aggregate.
Add all covered claims for the period once. Do not apply an annual attachment separately to each claim.

Calculate the recovery.
Find the amount above attachment, $S-A$, replace negative results with zero, and cap the result at $M$.

Calculate and reconcile retention.
Subtract recovery from the aggregate. Confirm that recovery is between zero and $M$, and that recovery plus retention equals $S$.

Worked Example: Aggregate Below the Attachment

A hypothetical insurer reports three covered claims of $38,000, $27,000, and $16,000 during one policy year. The stop-loss cover has an attachment of $90,000 and a maximum recovery of $35,000. Calculate the recovery and the insurer’s retained aggregate loss. Treat the amounts as covered losses for this year; no interest, mortality, or probability model applies.

Method. Sum the claims first, then compare the annual total with the attachment. This avoids mistakenly treating the attachment as a per-claim deductible.

Step 1: Find the aggregate. $$ S=\$38{,}000+\$27{,}000+\$16{,}000=\$81{,}000. $$

Step 2: Apply the attachment. The total is below the attachment because $\$81{,}000<\$90{,}000$. Thus the amount above attachment, floored at zero, is $\max(\$81{,}000-\$90{,}000,0)=\$0$.

Step 3: Apply the recovery limit. $$ R(S)=\min(\$0,\$35{,}000)=\$0. $$

Step 4: Find retention and check. Retained loss is $\$81{,}000-\$0=\$81{,}000$. Recovery plus retention is $\$0+\$81{,}000=\$81{,}000$, equal to the annual aggregate. The result is consistent: a cover that has not attached pays nothing.

Worked Example: Recovery Within the Layer

For a hypothetical treaty year, an insurer has covered claims of $52,000, $31,000, and $29,000. The contract attaches at $90,000 and limits the reinsurer’s recovery to $35,000. Find the annual recovery and retained loss. These are deterministic amounts; no interest, mortality, or frequency model is assumed.

Method. Calculate the annual aggregate and then determine whether the amount above attachment reaches the recovery cap. The common error avoided here is assuming that every recovery above attachment is automatically capped.

Step 1: Aggregate the claims. $$ S=\$52{,}000+\$31{,}000+\$29{,}000=\$112{,}000. $$

Step 2: Calculate the amount above attachment. $$ S-A=\$112{,}000-\$90{,}000=\$22{,}000. $$

Step 3: Compare with the maximum recovery. Since $\$22{,}000<\$35{,}000$, the cap does not reduce the recovery: $$ R(S)=\min(\$22{,}000,\$35{,}000)=\$22{,}000. $$

Step 4: Calculate retention. $$ \text{Retained loss}=\$112{,}000-\$22{,}000=\$90{,}000. $$

Check. The aggregate is between the attachment of $90,000 and the exhaustion point of $\$90{,}000+\$35{,}000=\$125{,}000$. Retention should therefore equal the attachment, and it does. The reconciliation is $\$90{,}000+\$22{,}000=\$112{,}000$.

Worked Example: Recovery Limit Reached

A hypothetical insurer records four covered claims of $44,000, $36,000, $28,000, and $41,000 in a year. Its stop-loss treaty attaches at $75,000 and has a maximum recovery of $45,000. Calculate the recovery and retained annual loss. Assume all four claims fall in the same treaty period, with no additional contract adjustments.

Method. Aggregate all claims for the year, calculate the amount above attachment, and then apply the maximum recovery. This prevents the common mistake of applying the attachment claim by claim.

Step 1: Calculate the annual aggregate. $$ S=\$44{,}000+\$36{,}000+\$28{,}000+\$41{,}000=\$149{,}000. $$

Step 2: Calculate the amount above attachment. $$ S-A=\$149{,}000-\$75{,}000=\$74{,}000. $$

Step 3: Cap the recovery. The amount above attachment exceeds the maximum recovery of $45,000, so the treaty is exhausted: $$ R(S)=\min(\$74{,}000,\$45{,}000)=\$45{,}000. $$

Step 4: Calculate retained loss. $$ \text{Retained loss}=\$149{,}000-\$45{,}000=\$104{,}000. $$

Check. The exhaustion point is $\$75{,}000+\$45{,}000=\$120{,}000$, and the aggregate of $149,000 exceeds it. The retained amount also agrees with $S-M=\$149{,}000-\$45{,}000=\$104{,}000$. Recovery plus retention equals $\$45{,}000+\$104{,}000=\$149{,}000$.

Common Mistakes and Checks

A frequent error is to apply an annual stop-loss attachment separately to each claim. For the four claims in the preceding example, subtracting $75,000 from each claim and flooring at zero would produce no recovery, because every individual claim is below $75,000. That is the wrong calculation: the contract responds to the annual aggregate, which is $149,000, not to each claim individually. The correct recovery is $\min(\$149{,}000-\$75{,}000,\$45{,}000)=\$45{,}000$.

Another error is to confuse the recovery limit with the aggregate exhaustion point. With attachment $A$ and maximum recovery $M$, exhaustion occurs at $A+M$. A contract that attaches at $75,000 and allows a maximum recovery of $45,000 is exhausted at an aggregate loss of $120,000, not $45,000 or $75,000. If contract wording instead gives the aggregate amount at which the layer ends, subtract the attachment to find the maximum recovery.

Before finalizing a result, check each of the following:

The aggregate includes only losses covered by the stated period and contract.
The attachment is applied once to the annual total.
The recovery is not negative and does not exceed the maximum recovery.
The retained amount is nonnegative, and recovery plus retention equals the aggregate.
The attachment, recovery limit, and claims are all expressed in the same currency and units.

These checks catch most arithmetic and interpretation errors without changing the contract calculation. Keep the settlement tied to the actual contract wording: exclusions, coinsurance, reinstatements, or special adjustment provisions would require terms beyond the simple cover defined here.

Key takeaway: For a stop-loss cover with attachment $A$ and maximum recovery $M$, calculate the annual aggregate $S$ first, then use $R(S)=\min(\max(S-A,0),M)$. The insurer retains $S-R(S)$.

Check Your Understanding

For each hypothetical annual settlement, calculate the recovery and retained aggregate loss. Assume the claims listed are all covered, and use the stated maximum recovery as the treaty limit.

An insurer has annual claims of $22,000, $31,000, and $18,000. The attachment is $60,000 and the maximum recovery is $20,000. Find the recovery and retention. (Answer: Recovery $11,000; retained loss $60,000.)
Claims for the year are $45,000, $35,000, and $28,000. The attachment is $80,000 and maximum recovery is $25,000. Find the recovery and retention. (Answer: Recovery $25,000; retained loss $83,000.)
The annual aggregate is $74,000, the attachment is $75,000, and the maximum recovery is $20,000. Determine the recovery and retained loss. (Answer: Recovery $0; retained loss $74,000.)
A year has claims of $16,000, $39,000, $24,000, and $51,000. The attachment is $90,000, with a maximum recovery of $30,000. Find the annual aggregate, recovery, and retained loss. (Answer: Aggregate $130,000; recovery $30,000; retained loss $100,000.)
In Year 1, covered claims total $33,000, $29,000, and $22,000. In Year 2, they total $50,000, $43,000, and $37,000. Each year has its own attachment of $100,000 and maximum recovery of $25,000. Find the recovery and retained loss in each year. (Answer: Year 1 recovery $0 and retained loss $84,000; Year 2 recovery $25,000 and retained loss $105,000.)

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