Attachment, Limit, and Exhaustion Points

Introduction

A per-claim excess-of-loss treaty covers a band of each claim’s loss. To calculate its payment, identify where the band begins, how wide it is, and the loss level at which it is fully used. The last of these levels is the exhaustion point. Confusing the layer limit with the exhaustion point can shift every recovery calculation.

This tutorial focuses on translating those treaty terms into loss amounts and allocating a claim between the insurer and reinsurer. The examples use hypothetical contracts and losses. The treatment is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: For a nonnegative claim loss $X$, let $a\geq 0$ be the attachment point and $L>0$ be the layer limit, meaning the maximum amount the reinsurer pays per claim. The exhaustion point is $a+L$. The reinsurer’s payment per loss is $Y^L=\min\{(X-a)_+,L\}$, where $z_+=\max\{z,0\}$. The insurer retains $X-Y^L$.

Three Loss Levels to Identify

The attachment point is the lower edge of the covered band. The reinsurer pays nothing when the claim is at or below $a$; a claim must exceed $a$ to produce a positive recovery. The layer limit $L$ is the maximum payment by the reinsurer on that claim. The exhaustion point $a+L$ is the loss level at which the full limit is reached.

For example, a treaty described as “$40,000 excess of $100,000 per claim” has attachment $a=\$100,000$, limit $L=\$40,000$, and exhaustion point $\$140,000$. The reinsurer pays the portion of a claim in the band above $100,000, capped at $40,000. A loss above $140,000 still produces only a $40,000 recovery; the insurer retains the amount above the exhaustion point as well as the first $100,000.

Claim loss level	Reinsurer payment	Insurer retention
$X\leq a$	0	$X$
$a<X<a+L$	$X-a$	$a$
$X=a+L$	$L$	$a$
$X>a+L$	$L$	$a+(X-a-L)$

At $X=a$, the covered band has not been entered and recovery is zero. At $X=a+L$, the layer has just been filled, so the recovery is exactly $L$. Above that level, the reinsurer remains capped at $L$; the insurer’s retained amount increases with the loss.

Payment Formula and Why It Works

The minimum formula is convenient because it handles losses below, within, and above the layer without requiring a separate calculation for each case. An equivalent expression uses positive parts:

$$ Y^L=(X-a)_+-(X-a-L)_+. $$

To see why, consider the possible positions of $X$ relative to the two layer boundaries. If $X\leq a$, both positive-part terms are zero, so the recovery is zero. If $a<X\leq a+L$, the first term is $X-a$ and the second is zero, so the recovery is $X-a$. If $X>a+L$, both terms are positive, and their difference is $(X-a)-(X-a-L)=L$. At the boundary $X=a+L$, the first term is $L$ and the second is $0$, so the difference is also $L$. These cases give exactly the capped payment in the definition.

The insurer’s retained loss can also be written in a useful piecewise form. It equals $X$ below attachment, $a$ while the claim is within the layer, and $a+(X-a-L)$ above exhaustion. Equivalently:

X-Y^L=\min\{X,a\}+(X-a-L)_+.

This allocation identity follows directly from the three regions. Below attachment, the minimum is $X$ and the positive-part term is zero. Within the layer, the minimum is $a$ and the positive-part term is zero. Above exhaustion, the terms are $a$ and $X-a-L$, whose sum is the insurer’s first $a$ plus the loss above the layer.

Key takeaway: For a layer attaching at $a$ with limit $L$, the exhaustion point is $a+L$, not $L$. The reinsurer pays at most $L$ per claim, while the insurer retains any amount above $a+L$.

Worked Examples

Worked Example: Applying a Layer to Claims on Both Boundaries

A hypothetical treaty pays up to $75,000 per claim excess of $25,000. For claim losses of $18,000, $25,000, $60,000, $100,000, and $130,000, find the exhaustion point, each recovery, and each insurer-retained amount.

Method. The attachment is $a=\$25,000$, and the limit is $L=\$75,000$. Thus the exhaustion point is $a+L=\$25,000+\$75,000=\$100,000$. For each claim, use $Y^L=\min\{(X-a)_+,L\}$, then subtract the recovery from the claim loss to find the retention.

For $X=\$18,000$, the loss is below attachment. The recovery is $\min\{0,\$75,000\}=\$0$, and the insurer retains $\$18,000-\$0=\$18,000$.

For $X=\$25,000$, the loss equals attachment. The recovery is $\min\{\$0,\$75,000\}=\$0$, and the insurer retains $\$25,000$.

For $X=\$60,000$, the amount above attachment is $\$60,000-\$25,000=\$35,000$. The recovery is $\min\{\$35,000,\$75,000\}=\$35,000$, and the insurer retains $\$60,000-\$35,000=\$25,000$.

For $X=\$100,000$, the amount above attachment is $\$100,000-\$25,000=\$75,000$. The recovery is $\min\{\$75,000,\$75,000\}=\$75,000$, and the insurer retains $\$100,000-\$75,000=\$25,000$. This loss is exactly at exhaustion.

For $X=\$130,000$, the amount above attachment is $\$130,000-\$25,000=\$105,000$, which exceeds the limit. The recovery is capped at $\$75,000$, and the insurer retains $\$130,000-\$75,000=\$55,000$.

Check. The recoveries are $0, $0, $35,000, $75,000, and $75,000. None exceeds the $75,000 limit. At exhaustion, the recovery equals the limit; above exhaustion, the additional $30,000 loss is retained by the insurer.

Common mistake avoided: Treating $75,000 as the exhaustion point. It is the layer limit; the exhaustion point is $25,000+$75,000=$100,000.

Worked Example: Translating “Excess of” Wording

A hypothetical property treaty is stated as “$110,000 excess of $40,000 per claim.” Find the attachment, limit, and exhaustion point. Then calculate the recovery and retention for losses of $39,999, $92,000, $150,000, and $174,000.

Method. In “$110,000 excess of $40,000” wording, $40,000 is the attachment and $110,000 is the layer limit. Therefore $a=\$40,000$, $L=\$110,000$, and the exhaustion point is $\$40,000+\$110,000=\$150,000$.

At $X=\$39,999$, the loss is below attachment, so $Y^L=\$0$ and retention is $\$39,999$.

At $X=\$92,000$, the amount above attachment is $\$92,000-\$40,000=\$52,000$. Since this is below the limit, recovery is $52,000 and retention is $\$92,000-\$52,000=\$40,000$.

At $X=\$150,000$, the amount above attachment is $\$150,000-\$40,000=\$110,000$. Recovery is the full $110,000 limit, leaving retention of $\$150,000-\$110,000=\$40,000$.

At $X=\$174,000$, the amount above attachment is $\$174,000-\$40,000=\$134,000$, which exceeds the limit. Recovery is capped at $110,000, and retention is $\$174,000-\$110,000=\$64,000$. This is the initial $40,000 plus $24,000 above exhaustion.

Check. The exhaustion point is exactly $150,000, the recovery there is the full $110,000 limit, and the $174,000 loss leaves $24,000 above the layer in addition to the initial $40,000.

Common mistake avoided: Adding the attachment to a stated amount that is already the exhaustion point. Here the wording gives the layer limit, so it must be added to the attachment to find exhaustion.

Worked Example: Separating the Insurer’s Retention Above Exhaustion

A hypothetical equipment treaty is written as “$35,000 excess of $65,000 per claim.” A claim is adjusted to $112,000. Calculate the layer recovery, the insurer’s total retention, and the part of that retention above the layer.

Method. The attachment is $a=\$65,000$, and the limit is $L=\$35,000$. The exhaustion point is $\$65,000+\$35,000=\$100,000$.

The loss exceeds attachment by $\$112,000-\$65,000=\$47,000$. The reinsurer’s payment is the smaller of $47,000 and the $35,000 limit, so the recovery is $35,000.

The insurer’s total retention is $\$112,000-\$35,000=\$77,000$. Of this, $65,000 is the amount up to attachment. The amount above exhaustion is $\$112,000-\$100,000=\$12,000$. Thus the retention also checks as $\$65,000+\$12,000=\$77,000$.

Check. The insurer’s retention plus the reinsurer’s recovery is $\$77,000+\$35,000=\$112,000$, the original claim loss.

Common mistake avoided: Calling $77,000 the amount above the layer. It is the insurer’s total retention; only $12,000 lies above exhaustion.

Common Wording and Calculation Traps

Before calculating, identify what each stated amount represents. “$L excess of $a” ordinarily gives the layer limit $L$ and attachment $a$; the exhaustion point is their sum. If a problem instead directly states the loss level at which coverage is exhausted, that level is already $a+L$, so do not add the attachment again. State the interpretation explicitly when wording could be ambiguous.

A second trap is to apply the cap to the entire claim loss rather than to the amount above attachment. For a claim $X$ inside the layer, recovery is $X-a$, not $X$. A third is to stop the insurer’s retention at $a$ for every claim. That is true only up to exhaustion; for $X>a+L$, the insurer also retains $X-a-L$ above the layer.

Exam check: For every claim, verify $0\leq Y^L\leq L$, verify that $Y^L=0$ when $X\leq a$, and check that insurer retention plus reinsurer recovery equals $X$.

Check Your Understanding

For each hypothetical per-claim treaty, identify the relevant layer amounts and calculate the requested allocation. Treat all losses as nonnegative.

A treaty pays $28,000 excess of $52,000. Find the exhaustion point and the recovery on a $70,000 claim. (Answer: exhaustion point $80,000; recovery $18,000.)
A treaty pays $45,000 excess of $30,000. A claim is $90,000. Find the recovery, total insurer retention, and amount retained above exhaustion. (Answer: exhaustion point $75,000; recovery $45,000; total retention $45,000, consisting of $30,000 below the layer and $15,000 above it.)
A treaty is described as “$60,000 excess of $90,000.” For a $145,000 loss, calculate the recovery and retention. (Answer: exhaustion point $150,000; recovery $55,000; retention $90,000.)
A treaty attaches at $125,000 and has an exhaustion point of $200,000. Find its limit, then calculate the recovery and insurer retention on a $230,000 claim. (Answer: limit $75,000; recovery $75,000; retention $155,000, including $30,000 above exhaustion.)
A treaty pays $80,000 excess of $40,000. Three separate claims are $35,000, $90,000, and $150,000. Find the exhaustion point, each recovery, total recovery, and total insurer retention. (Answer: exhaustion point $120,000; recoveries $0, $50,000, and $80,000; total recovery $130,000; total retention $145,000.)

Claim loss level	Reinsurer payment	Insurer retention
\(X\leq a\)	0	\(X\)
\(a<X<a+L\)	\(X-a\)	\(a\)
\(X=a+L\)	\(L\)	\(a\)
\(X>a+L\)	\(L\)	\(a+(X-a-L)\)

What You'll Learn