Deriving Stop-Loss Payments from Aggregate Loss

From Aggregate Loss to Stop-Loss Recovery

A stop-loss treaty responds to the total covered loss for a defined period, rather than separately applying a layer to each individual claim. Its recovery is therefore determined by comparing the period’s aggregate loss with the treaty’s attachment and payment limit. The key calculation is simple once the meanings of those two treaty terms are kept separate.

This tutorial considers a hypothetical treaty covering one accounting period. The calculation is deterministic: once the gross aggregate loss is known, the recovery follows from the contract terms. It does not calculate an expected recovery or a reinsurance premium.

Definition: Let $X$ be the insurer’s gross aggregate loss for the covered period, before this stop-loss recovery. Let $d$ be the aggregate attachment, and let $u$ be the maximum dollar amount the reinsurer will pay under the cover. Assume $d\geq 0$, $u\geq 0$, and $X\geq 0$. The recovery is zero below the attachment, increases dollar for dollar above it, and cannot exceed $u$.

Here, $u$ is the maximum recovery, not the aggregate-loss value at which the cover is exhausted. Exhaustion occurs when aggregate loss reaches $d+u$: at that point the loss above the attachment equals the maximum payment. This distinction determines the upper boundary of the covered range.

Deriving the Payment Formula

Before the limit is applied, the amount of aggregate loss above the attachment is $X-d$. If $X\leq d$, that amount is zero or negative, but a treaty does not make a negative recovery. The positive-part notation $z_+=\max(z,0)$ expresses this floor at zero. The treaty then caps that amount at the maximum payment $u$.

Y=\min\bigl((X-d)_+,u\bigr)

This expression is a direct translation of the contract terms: first find the positive amount above the attachment; then pay no more than the limit. To obtain a piecewise form, compare $X$ with the attachment $d$ and the exhaustion point $d+u$.

Below or at attachment.
If $X\leq d$, then $(X-d)_+=0$, so the recovery is zero.

Inside the covered range.
If $d<X<d+u$, then $0<X-d<u$. The recovery is the full amount above attachment, $X-d$.

At or above exhaustion.
If $X\geq d+u$, then $X-d\geq u$. The recovery is capped at $u$.

Y= \begin{cases} 0, & X\leq d,\\ X-d, & d<X<d+u,\\ u, & X\geq d+u. \end{cases}

The middle interval is sometimes called the covered range: each additional dollar of aggregate loss adds one dollar to the recovery until the limit is reached. The endpoint values are consistent with the adjoining cases. At $X=d$, the recovery is zero. At $X=d+u$, the amount above attachment is exactly $u$, so the recovery is $u$.

Why the Formula Works

The derivation follows from two contractual restrictions. First, the reinsurer pays nothing for the part of aggregate loss at or below the attachment. Second, the reinsurer’s total payment cannot exceed the stated limit. Starting with $X-d$, the positive-part operation enforces the first restriction and the minimum with $u$ enforces the second.

\begin{aligned} X\leq d &\implies (X-d)_+=0 \implies Y=0,\\ d<X<d+u &\implies 0<X-d<u \implies Y=X-d,\\ X\geq d+u &\implies X-d\geq u \implies Y=u. \end{aligned}

This also explains why the exhaustion point is $d+u$, not $u$. At exhaustion, the amount eligible for payment—aggregate loss above the attachment—has reached the payment limit. If $u=0$, the formula gives zero recovery for every $X$, as expected for a cover with no available payment.

Key result: For an aggregate attachment $d$ and a maximum recovery $u$, the recovery is $0$ through $d$, rises as $X-d$ between $d$ and $d+u$, and remains $u$ once $X$ reaches or exceeds $d+u$.

Worked Applications

Worked Example: Evaluating Losses Below, Within, and Above the Cover

A hypothetical insurer has a one-year stop-loss treaty with an aggregate attachment of $600,000 and a maximum recovery of $250,000. For each of three possible gross annual aggregate losses—$540,000, $725,000, and $910,000—calculate the recovery and the aggregate loss retained after this recovery.

Method. Use $Y=\min((X-d)_+,u)$, with $d=\$600{,}000$ and $u=\$250{,}000$. The exhaustion point is $d+u=\$600{,}000+\$250{,}000=\$850{,}000$.

Loss of $540,000. This is below the attachment: $(X-d)_+=\max(\$540{,}000-\$600{,}000,0)=\$0$. Thus $Y=\min(\$0,\$250{,}000)=\$0$, and retained aggregate loss is $\$540{,}000-\$0=\$540{,}000$.

Loss of $725,000. This is above the attachment but below exhaustion: $X-d=\$725{,}000-\$600{,}000=\$125{,}000$. Thus $Y=\min(\$125{,}000,\$250{,}000)=\$125{,}000$, and retained aggregate loss is $\$725{,}000-\$125{,}000=\$600{,}000$.

Loss of $910,000. This exceeds the exhaustion point: $X-d=\$910{,}000-\$600{,}000=\$310{,}000$. The amount above attachment is greater than the limit, so $Y=\min(\$310{,}000,\$250{,}000)=\$250{,}000$. Retained aggregate loss is $\$910{,}000-\$250{,}000=\$660{,}000$.

Check. Recoveries are respectively $0, $125,000, and $250,000; none is negative or greater than the $250,000 limit. In each case, gross aggregate loss equals recovery plus retained aggregate loss. The method avoids treating the $250,000 payment limit as the aggregate-loss exhaustion point.

Worked Example: Aggregating Claims Before Applying the Treaty

Under a hypothetical one-year treaty, the attachment is $800,000 and the maximum recovery is $350,000. The insurer’s four covered claims for the year are $310,000, $285,000, $190,000, and $245,000. Find the annual aggregate loss, the stop-loss recovery, and the retained aggregate loss.

Method. Because this is a stop-loss cover, first add the covered claims to obtain the annual aggregate. Do not apply the attachment separately to each claim.

The aggregate loss is $\$310{,}000+\$285{,}000+\$190{,}000+\$245{,}000=\$1{,}030{,}000$. The exhaustion point is $\$800{,}000+\$350{,}000=\$1{,}150{,}000$. Since $\$800{,}000<\$1{,}030{,}000<\$1{,}150{,}000$, the aggregate lies within the covered range.

The amount above attachment is $\$1{,}030{,}000-\$800{,}000=\$230{,}000$. Therefore, the recovery is $\min(\$230{,}000,\$350{,}000)=\$230{,}000$. The retained aggregate loss is $\$1{,}030{,}000-\$230{,}000=\$800{,}000$.

Check. The recovery is less than the $350,000 limit, and recovery plus retention is $\$230{,}000+\$800{,}000=\$1{,}030{,}000$, equal to the aggregate loss. The method avoids incorrectly applying the $800,000 annual attachment to each claim separately.

Worked Example: Recognizing a Fully Exhausted Limit

A hypothetical stop-loss cover has an aggregate attachment of $1,100,000 and a maximum payment of $450,000. The covered annual aggregate loss is $1,720,000. Calculate the recovery and retention, and explain how much of the loss above attachment is not reimbursed.

Method. Find the exhaustion point and compare it with the aggregate loss before calculating the recovery.

The exhaustion point is $\$1{,}100{,}000+\$450{,}000=\$1{,}550{,}000$. Because $\$1{,}720{,}000>\$1{,}550{,}000$, the limit is fully exhausted. Equivalently, the amount above attachment is $\$1{,}720{,}000-\$1{,}100{,}000=\$620{,}000$, which exceeds the $450,000 maximum payment.

Thus the recovery is $\min(\$620{,}000,\$450{,}000)=\$450{,}000$. The retained aggregate loss is $\$1{,}720{,}000-\$450{,}000=\$1{,}270{,}000$. The part of the loss above attachment that is not reimbursed is $\$620{,}000-\$450{,}000=\$170{,}000$.

Check. Retention can be written as the $1,100,000 below the attachment plus the $170,000 above the exhausted layer: $\$1{,}100{,}000+\$170{,}000=\$1{,}270{,}000$. Adding the $450,000 recovery gives the original $1,720,000 aggregate. The method avoids assuming that a fully exhausted cover reimburses all loss above attachment.

Recovery and Retained Aggregate Loss

If the insurer has no other recovery or adjustment affecting the same aggregate, its retained aggregate loss after this treaty is gross aggregate loss minus the stop-loss payment. This identity provides a useful arithmetic check, but it does not change the treaty’s recovery formula.

\text{Retained aggregate loss}=X-Y

In the covered range, the recovery brings retained aggregate loss to the attachment. Once the limit is exhausted, additional aggregate loss remains with the insurer. In particular, for $X\geq d+u$, retention is $X-u$, which can be greater than $d$.

X-Y= \begin{cases} X, & X\leq d,\\ d, & d<X<d+u,\\ X-u, & X\geq d+u. \end{cases}

This piecewise retention result follows by subtracting each recovery case from $X$. For example, within the covered range, $X-(X-d)=d$; above exhaustion, $X-u$ continues to increase with aggregate loss. It is therefore incorrect to assume that the insurer’s retained loss can never exceed the attachment.

Common Interpretation Errors

Using $u$ as the exhaustion point. If $u$ is the maximum payment, exhaustion occurs at $d+u$. Compare $X$ with that sum, not with $u$ alone.
Paying below the attachment. When $X\leq d$, recovery is zero. Do not report $X-d$ as a negative payment; the positive-part operation sets it to zero.
Forgetting the cap. If $X-d>u$, the recovery remains $u$, not $X-d$. The excess above the capped layer is retained.
Applying the cover claim by claim. The treaty in this tutorial responds to the aggregate for the covered period. Add the covered losses first, then apply the aggregate attachment and payment limit.
Confusing recovery with retention. Recovery is $Y$; retained aggregate loss is $X-Y$. State which quantity is requested before substituting values.

Key takeaway: For a stop-loss cover with aggregate attachment $d$ and maximum recovery $u$, calculate $Y=\min((X-d)_+,u)$. The covered range ends at $d+u$; the retained aggregate loss is $X-Y$.

Check Your Understanding

For each hypothetical treaty, determine the requested amount from the stated aggregate loss and terms.

A cover has attachment $420,000 and maximum recovery $180,000. If aggregate loss is $395,000, calculate the recovery and retention. (Answer: recovery $0; retention $395,000.)
A cover has attachment $700,000 and maximum recovery $260,000. If aggregate loss is $835,000, calculate the recovery and retention. (Answer: recovery $135,000; retention $700,000.)
A cover has attachment $950,000 and maximum recovery $300,000. If aggregate loss is $1,310,000, calculate the recovery and retention. (Answer: recovery $300,000; retention $1,010,000.)
A hypothetical annual aggregate consists of claims of $240,000, $175,000, and $310,000. The stop-loss attachment is $600,000 and maximum recovery is $200,000. Find the aggregate loss and recovery. (Answer: aggregate loss $725,000; recovery $125,000.)
A cover has attachment $1,250,000 and maximum recovery $400,000. Aggregate loss is $1,710,000. Find the recovery and the amount by which loss above attachment exceeds the available payment limit. (Answer: recovery $400,000; loss above attachment exceeds the limit by $60,000.)

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