Stop-Loss Attachments and Aggregate Limits

What an Aggregate Stop-Loss Contract Protects

A stop-loss reinsurance contract responds to an insurer’s losses over a stated period, commonly one contract year. Its attachment point is a threshold for the eligible aggregate loss in that period, not a threshold applied separately to each claim. Once eligible losses exceed the attachment, the reinsurer may pay the excess, subject to the contract’s limit.

The first task is therefore to identify which losses count toward the annual aggregate. Gross losses and eligible losses may differ if the contract excludes specified items. The attachment and limit are then applied to the eligible aggregate, according to the treaty terms.

Definition: Let $Z$ be the eligible aggregate loss for one contract period, $B$ the stop-loss attachment point, and $M$ the maximum recovery, or aggregate limit. The attachment is the amount of eligible aggregate loss the insurer retains before the cover begins to respond. The exhaustion point is $B+M$: at that eligible aggregate, the maximum recovery has been reached. These definitions assume $B\geq 0$, $M\geq 0$, and that the limit is stated as a maximum recovery.

This convention matters: a “$600,000 limit” in a stop-loss treaty ordinarily means that the reinsurer can pay at most $600,000, not that eligible losses are capped at $600,000. The losses themselves can exceed the exhaustion point; it is the recovery that stops increasing.

Reading the Attachment and Limit Together

For eligible annual aggregate loss $Z$, the recovery $R(Z)$ is the portion above the attachment, but no more than the limit. For a nonnegative quantity $w$, write $w^+=\max(w,0)$. Then:

R(Z)=\min\bigl((Z-B)^+,\,M\bigr)

The same rule can be read in three ranges. Below the attachment, there is no recovery. Between the attachment and the exhaustion point, the recovery increases dollar for dollar with eligible aggregate loss above the attachment. At and above the exhaustion point, the recovery is fixed at the limit.

Eligible annual aggregate	Recovery	Interpretation
$Z\leq B$	0	The attachment has not been exceeded.
$B<Z<B+M$	$Z-B$	The layer is partly used.
$Z\geq B+M$	$M$	The limit is exhausted.

At exactly $Z=B$, the recovery is zero. At exactly $Z=B+M$, the recovery equals $M$. These boundary cases follow directly from the formula; no additional payment is triggered merely by reaching the attachment.

Why the Recovery Formula Works

First consider the amount of eligible loss above the attachment. If $Z\leq B$, then $Z-B\leq 0$, so its positive part is zero. If $Z>B$, the positive part is $Z-B$, the amount above the attachment. This gives $(Z-B)^+$ before applying the limit.

The treaty cannot pay more than $M$. Taking the minimum of the amount above the attachment and $M$ enforces that cap. Equivalently, below the attachment the payment is zero; between $B$ and $B+M$ it is $Z-B$; and once $Z\geq B+M$, the amount above the attachment is at least $M$, so the payment remains $M$. This establishes the piecewise interpretation.

R(Z)= \begin{cases} 0, & Z\leq B,\\ Z-B, & B<Z<B+M,\\ M, & Z\geq B+M. \end{cases}

If the insurer’s retained eligible loss is being measured after the recovery, it is $Z-R(Z)$. This amount includes both the loss below the attachment and any eligible loss above the exhaustion point. It is not necessarily equal to the attachment: the insurer retains more than the attachment when the eligible aggregate exceeds the exhaustion point.

Work from Eligibility to Recovery

A reliable calculation separates the treaty’s eligibility rule from its layer arithmetic. First determine the eligible annual aggregate. Then compare that aggregate with the attachment and exhaustion point. This prevents a common error: applying the correct stop-loss formula to the wrong total.

Determine the eligible aggregate.
Start with the losses in the treaty period and adjust for any stated exclusions or eligibility rules.

Find the exhaustion point.
Add the maximum recovery to the attachment: $B+M$.

Apply the layer.
Calculate the eligible amount above $B$, then cap that amount at $M$.

Check the result.
Confirm that recovery is between zero and $M$, and that retained eligible loss equals $Z-R(Z)$.

Worked Example: Locate the Layer and Its Exhaustion Point

A hypothetical treaty covers eligible losses for one contract year. The attachment is $3,000,000 and the maximum recovery is $1,000,000. Eligible annual aggregate losses are $4,250,000, with no exclusions. Find the exhaustion point, recovery, and eligible loss retained by the insurer. Treat the amounts as deterministic; no probability distribution is assumed.

Method. Identify the eligible aggregate, calculate the exhaustion point, and apply the capped excess formula. This avoids treating the limit as a cap on losses rather than on recovery.

Step 1: Exhaustion point.

B+M=\$3{,}000{,}000+\$1{,}000{,}000=\$4{,}000{,}000

Step 2: Amount above attachment.

Z-B=\$4{,}250{,}000-\$3{,}000{,}000=\$1{,}250{,}000

Step 3: Apply the limit.

R(Z)=\min(\$1{,}250{,}000,\$1{,}000{,}000)=\$1{,}000{,}000

Step 4: Retained eligible loss.

Z-R(Z)=\$4{,}250{,}000-\$1{,}000{,}000=\$3{,}250{,}000

The recovery equals the limit because eligible losses exceed the $4,000,000 exhaustion point. The check is that the $3,250,000 retained amount plus the $1,000,000 recovery equals the $4,250,000 eligible aggregate.

Worked Example: Exclusions Change the Aggregate Tested

A hypothetical annual stop-loss treaty has an attachment of $2,400,000 and a maximum recovery of $600,000. Gross annual losses are $3,300,000, of which $200,000 is expressly excluded from the treaty’s aggregate. Find eligible aggregate losses, the exhaustion point, recovery, and eligible loss remaining after recovery. Assume no other exclusions or adjustments and no probability distribution.

Method. Remove the excluded amount before applying the attachment and limit. This avoids calculating the layer from gross losses when the treaty specifies an eligible aggregate.

Step 1: Eligible aggregate.

Z=\$3{,}300{,}000-\$200{,}000=\$3{,}100{,}000

Step 2: Exhaustion point.

B+M=\$2{,}400{,}000+\$600{,}000=\$3{,}000{,}000

Step 3: Amount above attachment and recovery. The amount above attachment is:

Z-B=\$3{,}100{,}000-\$2{,}400{,}000=\$700{,}000

The $700,000 excess is greater than the $600,000 limit, so:

R(Z)=\min(\$700{,}000,\$600{,}000)=\$600{,}000

Step 4: Eligible loss remaining.

Z-R(Z)=\$3{,}100{,}000-\$600{,}000=\$2{,}500{,}000

The eligible aggregate is $3,100,000, the exhaustion point is $3,000,000, the recovery is $600,000, and the eligible loss remaining is $2,500,000. The check is $\$2{,}500{,}000+\$600{,}000=\$3{,}100{,}000$.

Annual Boundaries and Common Traps

An annual attachment applies to the aggregate for the stated year. If a treaty has separate contract years and does not provide for carryover or pooling, apply the attachment and limit separately in each year. Do not combine two years’ losses first and apply one attachment to the combined total. Contract language controls how periods, adjustments, and eligibility are defined.

A frequent error is to subtract the attachment from the eligible aggregate incorrectly, or to stop once the recovery reaches the limit without checking the retained amount. For example, with eligible aggregate $3,100,000 and attachment $2,400,000, the amount above attachment is:

\$3{,}100{,}000-\$2{,}400{,}000=\$700{,}000

It is not $600,000. The $600,000 is the limit, which caps the $700,000 excess; the resulting recovery is $600,000. Subtracting the recovery from the eligible aggregate then gives $2,500,000 retained. Keeping these three quantities distinct—excess above attachment, recovery, and retained eligible loss—is an effective arithmetic check.

Do not apply the attachment per claim. It applies to the treaty-defined annual aggregate in this setting.
Do not assume gross loss equals eligible loss. Apply stated exclusions before testing the attachment.
Do not interpret the limit as an aggregate-loss ceiling. It caps the reinsurer’s payment.
Do not confuse the attachment with the exhaustion point. The latter is $B+M$.
Do not pool contract years without a stated carryover or pooling provision.

Key takeaway: For a stop-loss layer with attachment $B$ and maximum recovery $M$, test the eligible annual aggregate $Z$, not an individual claim. The attachment is where recovery begins, the exhaustion point is $B+M$, and recovery is $\min((Z-B)^+,M)$.

Check Your Understanding

For each hypothetical treaty, use the stated eligibility rules and treat the values as deterministic. The answers distinguish the aggregate tested from the maximum recovery.

A treaty has attachment $1,500,000 and maximum recovery $500,000. Eligible annual losses are $1,850,000. Find the exhaustion point and recovery. (Answer: Exhaustion point $2,000,000; recovery $350,000.)
A treaty has attachment $2,200,000 and maximum recovery $700,000. Gross annual losses are $3,050,000, including $150,000 excluded by the treaty. Find eligible aggregate losses, exhaustion point, and recovery. (Answer: Eligible aggregate $2,900,000; exhaustion point $2,900,000; recovery $700,000.)
A treaty has attachment $4,000,000 and maximum recovery $1,250,000. Eligible annual losses are $3,760,000. Find recovery and eligible loss retained. (Answer: Recovery $0; eligible loss retained $3,760,000.)
A treaty has attachment $900,000 and maximum recovery $400,000. Eligible aggregate losses are $1,550,000. Find the amount above attachment, recovery, and eligible loss retained. (Answer: Amount above attachment $650,000; recovery $400,000; eligible loss retained $1,150,000.)
A treaty has an attachment of $1,600,000 and maximum recovery of $600,000 for each contract year, with no carryover between years. Eligible losses are $2,050,000 in Year 1 and $2,300,000 in Year 2. Find each year’s recovery and the total recovery over both years. (Answer: Year 1 recovery $450,000; Year 2 recovery $600,000; total recovery $1,050,000.)

Eligible annual aggregate	Recovery	Interpretation
\(Z\leq B\)	0	The attachment has not been exceeded.
\(B<Z<B+M\)	\(Z-B\)	The layer is partly used.
\(Z\geq B+M\)	\(M\)	The limit is exhausted.

What You'll Learn