Allocating Catastrophe Recoveries Across Layers

Why Layer Allocation Requires a Clear Loss Basis

A catastrophe program may contain several layers, each responding to a different range of loss. To allocate an event loss correctly, first identify the loss amount the contracts recognize, then apply each layer’s attachment point, limit, and participation terms to the appropriate subject loss. A recovery calculation can be arithmetically correct yet wrong for the contract if it starts from the wrong loss basis or treats a prior recovery as inuring when it does not.

The calculations here concern one hypothetical catastrophe event and deterministic loss amounts. They do not involve discounting, mortality, or a probability distribution. The methods are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: A layer with attachment point $A$ and limit $L$ responds to the portion of its subject loss above $A$, up to a maximum covered-loss width of $L$. Its exhaustion point is $A+L$. Under the convention used here, $L$ is the width of the layer before any reinsurer participation share is applied.

The subject loss is not necessarily the insurer’s gross claim total. It is the loss amount defined by the contract: for example, eligible covered loss after policy deductibles and exclusions, and after any prior recoveries that the layer’s wording requires to inure. Earlier material on treaty definitions of loss and reinsurance recoveries establishes why contract wording controls this basis.

Layer Formula and Allocation Logic

Let $S$ be the loss subject to a particular layer, $A$ its attachment, and $L$ its limit. Define $x^+=\max(x,0)$. For a 100% participating layer, the recovery is:

R=\min\bigl((S-A)^+,L\bigr).

If the reinsurer participates at share $q$, with $0\le q\le1$, and the contract defines the limit as the width of the subject-loss layer before that share, then:

R=q\min\bigl((S-A)^+,L\bigr).

Always verify how the contract defines a limit when a participation share applies. Some wording may instead specify a maximum payment after applying the share. In that case, the stated payment cap, rather than the convention above, governs.

Derivation: Think of the loss as occupying the interval from zero to $S$, and the layer as covering the interval from $A$ to $A+L$. The covered amount is the length of their overlap: $\max(0,\min(S,A+L)-A)$. If $S\le A$, that length is zero; if $A<S<A+L$, it is $S-A$; and if $S\ge A+L$, it is $L$. These three cases are exactly $\min((S-A)^+,L)$.

For nonoverlapping layers that apply to the same subject loss, calculate each recovery against that same $S$. Do not subtract a lower layer’s recovery before testing the next vertical layer. A prior recovery is different: subtract it from the loss basis only when the relevant wording makes it inure before the layer being calculated.

Layer description	Attachment $A$	Limit $L$	Exhaustion point
1 million excess of 1 million	1 million	1 million	2 million
3 million excess of 2 million	2 million	3 million	5 million

The first layer covers losses in the range from 1 million to 2 million; the second covers the range from 2 million to 5 million. The phrase “excess of” gives the attachment, while the amount before it gives the limit under the usual “amount excess of amount” convention.

Worked Allocations

Worked Example: Allocating a Modeled Event Loss Across Three Layers

A hypothetical catastrophe model produces an eligible event loss of $8.70 million. A program has three 100% layers: $2 million excess of $1 million, $4 million excess of $3 million, and $5 million excess of $7 million. Determine each recovery and the retained loss. Assume no prior inuring recovery, no other contractual adjustment, no discounting, and no participation shares. The loss is a fixed modeled amount, not a random variable.

Method. Apply the layer formula to the same subject loss, $S=\$8.70$ million, for each layer.

Layer 1. Here $A=1$ and $L=2$, in millions. The loss above attachment is $8.70-1=7.70$. The layer is capped at 2, so $R_1=\min(7.70,2)=\$2.00$ million.

Layer 2. Here $A=3$ and $L=4$. The loss above attachment is $8.70-3=5.70$. Thus $R_2=\min(5.70,4)=\$4.00$ million.

Layer 3. Here $A=7$ and $L=5$. The loss above attachment is $8.70-7=1.70$, below the limit. Thus $R_3=\min(1.70,5)=\$1.70$ million.

Total recovery is $2.00+4.00+1.70=\$7.70$ million. Retained loss is $8.70-7.70=\$1.00$ million. As a check, that retained amount is the loss below the first attachment, and all loss above $1 million is covered by the three contiguous layers through their respective exhaustion points. The method avoids treating each layer as if it had a different subject loss.

Worked Example: Applying the Covered-Loss Definition and Prior Inuring Recovery

For a hypothetical event, an insurer’s initial property-loss estimate is $6.40 million. The treaty’s definition excludes $0.40 million, and policy terms leave $5.60 million of eligible covered loss. A separate $0.60 million recovery is explicitly stated to inure before the catastrophe layers. The layers are $2.50 million excess of $1 million and $4 million excess of $3.50 million, both at 100%. Calculate the layer recoveries and the final retained covered loss. Assume no other adjustments or discounting; all amounts are deterministic.

Method. Use the contract-defined covered loss, not the initial estimate, and then deduct the recovery that the wording explicitly makes inuring.

First, the eligible covered loss is $5.60 million. The subject loss after the inuring recovery is $S=5.60-0.60=\$5.00$ million.

For the first layer, $A=1$ and $L=2.50$, so $R_1=\min(5.00-1,2.50)=\$2.50$ million.

For the second layer, $A=3.50$ and $L=4$, so $R_2=\min(5.00-3.50,4)=\$1.50$ million.

The catastrophe-layer recoveries total $2.50+1.50=\$4.00$ million. Including the prior $0.60 million recovery, total recoveries are $4.60 million. The remaining covered loss is $5.60-4.60=\$1.00$ million, which also equals $S-(R_1+R_2)=5.00-4.00$. The two checks agree. The method avoids applying the layers to the initial $6.40 million estimate or ignoring the explicit inuring provision.

Worked Example: Applying a Participation Share and Checking Exhaustion

A hypothetical event produces $5.80 million of covered subject loss. The first layer is $2 million excess of $1 million, with the reinsurer taking an 80% share of the layer. The second is $3 million excess of $3 million, with 100% participation. The stated limits are layer widths before participation. There are no prior recoveries, no discounting, and no other adjustments. Find the recovery from each layer and the amount retained.

Method. Calculate the full layer amount first, then apply the stated share to the first layer.

For the first layer, loss above attachment is $5.80-1=4.80$ million, but the layer width is only $2 million. Its full layer amount is $\min(4.80,2)=\$2.00$ million. The reinsurer’s recovery is $0.80(2.00)=\$1.60$ million.

For the second layer, loss above attachment is $5.80-3=2.80$ million, which is below its $3 million limit. Its recovery is therefore $2.80 million.

Total ceded recovery is $1.60+2.80=\$4.40$ million. Retained loss is $5.80-4.40=\$1.40$ million. This result includes the 20% of the first layer retained by the insurer: $0.40 million, plus the $1.00 million below the first attachment. The components sum to $1.40 million, confirming the calculation. The method avoids applying 80% to the entire catastrophe loss.

Prior Recoveries, Layer Order, and Reconciliation

A recovery from another contract affects a layer’s calculation only if the terms establish that it is inuring to that layer. If the layer responds to gross covered loss, use that gross covered loss as $S$. If it responds to covered loss net of specified prior recoveries $I$, a simple net basis is:

S=\max(C-I,0),

where $C$ is the covered loss defined for the layer. This subtraction is appropriate only when the recoveries included in $I$ are eligible and the contract requires them to inure. If a treaty uses a different definition of ultimate net loss, includes expenses, or specifies another order, follow that wording instead. Do not subtract a recovery twice.

Establish the eligible loss.
Start with the event loss and apply the relevant covered-loss definition, including specified exclusions, deductibles, and other adjustments.

Set the subject-loss basis.
Identify whether the layer responds to gross eligible loss or to loss net of named inuring recoveries. Make only the adjustments required by its wording.

Calculate each layer.
For each attachment and limit, find the loss above attachment, cap it at the layer width, and apply the participation convention.

Reconcile.
Compare total recoveries with the relevant covered-loss basis and calculate the remaining covered loss. Investigate any result that exceeds that basis or violates the stated limits.

For full, contiguous layers on a common subject-loss basis, the layer intervals divide the covered ranges without overlap or gaps. Their recoveries therefore add to the covered portion of that loss. Gaps, partial shares, separate definitions of loss, or prior inuring recoveries can leave additional retained amounts; they should be identified rather than forced into a simple “total limit” calculation.

Common Exam Trap: Subtracting One Layer’s Recovery Before Applying the Next

Suppose a $6.00 million subject loss is covered by two contiguous, 100% layers: $2 million excess of $1 million and $3 million excess of $3 million. A tempting but incorrect approach is to subtract the first recovery before calculating the second:

R_1=2,\qquad R_{2,\text{wrong}}=\min\bigl((6-2)-3,3\bigr)=1.

This gives total recovery of $3 million. The correction is to apply both vertical layers to the same $6 million subject loss. The first recovery is $\min(6-1,2)=2$ million; the second is $\min(6-3,3)=3$ million. Total recovery is $5 million and retained loss is $1 million, the amount below the first attachment. Subtracting a layer recovery first is justified only when contract wording makes that recovery inure to the next layer; ordinary vertical layer boundaries do not do so.

Key takeaway: A layer recovery depends on the covered-loss definition, the layer’s subject-loss basis, its attachment and limit, and any participation terms. Apply every vertical layer to its contractually defined basis, then reconcile all recoveries and retained loss.

Check Your Understanding

For each hypothetical single-event problem, assume deterministic dollar losses, no discounting, and no adjustments other than those stated. Calculate the requested recovery or retained amount.

A $9.20 million subject loss is covered by a $3 million excess of $1 million layer and a $5 million excess of $4 million layer, both at 100%. Find each recovery and retained loss. (Answer: Recoveries are $3.00 million and $5.00 million; retained loss is $1.20 million.)
Eligible covered loss is $7.30 million. A $1.10 million recovery explicitly inures before two layers: $2.50 million excess of $1 million and $4 million excess of $3.50 million. Find the two layer recoveries and retained covered loss, including the inuring recovery. (Answer: Layer recoveries are $2.50 million and $2.70 million; retained covered loss is $1.00 million.)
A $4.60 million subject loss enters a $2.50 million excess of $1 million layer with 60% participation and a $2 million excess of $3.50 million layer with 100% participation. Limits are widths before shares. Find total recovery and retained loss. (Answer: Recoveries are $1.50 million and $1.10 million; total recovery is $2.60 million and retained loss is $2.00 million.)
A $10.40 million covered loss has a $2.00 million recovery that explicitly inures before layers of $2 million excess of $1 million, $3 million excess of $4 million, and $2 million excess of $8 million. Find the layer recoveries and the insurer’s remaining covered loss after all recoveries. (Answer: Layer recoveries are $2.00 million, $3.00 million, and $0.40 million; remaining covered loss is $3.00 million.)
A $6.20 million subject loss is covered by two 100% layers: $2 million excess of $1 million and $3 million excess of $4 million. Neither layer inures to the other. Find the recoveries and retained loss, and state whether the second layer exhausts. (Answer: Recoveries are $2.00 million and $2.20 million; retained loss is $2.00 million; the second layer does not exhaust.)

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