Why the Order of Inuring Covers Matters
When several reinsurance covers respond to the same gross claim, the target excess layer does not necessarily attach to the original gross loss. It attaches to the loss remaining after the underlying covers have responded, according to their contractual order and terms. A recovery that is incorrectly applied, omitted, or placed in the wrong sequence can materially change the target layer’s indicated loss cost.
The key calculation is therefore a sequence: start with the gross claim, determine each underlying recovery using the loss presented to that cover, subtract that recovery, and then calculate the target-layer payment from the residual loss. This tutorial focuses on that allocation and on the resulting pure loss cost. It does not add expenses, commissions, reinstatement premiums, or a risk load.
The order is not a universal mathematical convention: it is a feature of the contracts. A calculation must follow the stated order and specify what loss each cover insures. The formulas below assume covers respond to the same occurrence loss, recoveries are paid in dollars against that loss, and there are no aggregate deductibles, exclusions, or other terms that alter the stated response.
Layer and Residual-Loss Formulas
For an occurrence loss \(z\), a layer with attachment \(a\) and limit \(L\), written \(L\) excess of \(a\), pays the part of the loss above \(a\), up to \(L\). Its recovery is
The layer’s exhaustion point is \(a+L\). A loss at or below the attachment produces no recovery; a loss above the exhaustion point produces the full limit. The residual after the cover is the presented loss less the recovery.
If a quota-share cover cedes proportion \(q\) of the loss presented to it, its recovery is \(qz\) and the residual is \((1-q)z\). If it applies first to gross loss, the residual presented to the next cover is \((1-q)X\), where \(X\) is the gross claim. If it applies to a different defined base, use that base instead.
Why the Layer Formula Works
Think of a layer \(L\) excess of \(a\) as covering the vertical slice of loss amounts above \(a\), but no higher than \(a+L\). A loss of size \(z\) reaches that slice only if \(z>a\). When it does, the portion reached is \(z-a\), unless the loss exceeds the slice’s full height \(L\). Thus the payment is zero below attachment, \(z-a\) within the layer, and \(L\) above exhaustion. Those three cases are exactly represented by \(\min\{L,\max(0,z-a)\}\).
Worked Example: One Underlying Layer Before the Target
A hypothetical insurer reports three gross occurrence claims of $250,000, $600,000, and $1,000,000. An underlying cover pays $400,000 excess of $300,000 per occurrence. The target cover pays $200,000 excess of $500,000 per occurrence. Calculate the total target-layer loss across the three claims.
Assumptions. These are three specified occurrences, not a probability sample, so no probability distribution is assigned. There is no interest or mortality component. The underlying cover responds first, and each layer applies per occurrence.
Method. For each claim, calculate the underlying recovery, subtract it from gross loss, and apply the target-layer formula to that residual.
For the $250,000 claim, the underlying recovery is \(\min\{$400,000,\max(0,$250,000-$300,000)\}=$0\). The residual loss is $250,000. The target payment is \(\min\{$200,000,\max(0,$250,000-$500,000)\}=$0\).
For the $600,000 claim, the underlying recovery is \(\min\{$400,000,$600,000-$300,000\}=$300,000\). The residual loss is $600,000-$300,000=$300,000. The target payment is \(\min\{$200,000,\max(0,$300,000-$500,000)\}=$0\).
For the $1,000,000 claim, the underlying recovery is \(\min\{$400,000,$1,000,000-$300,000\}=$400,000\). The residual is $1,000,000-$400,000=$600,000. The target payment is \(\min\{$200,000,$600,000-$500,000\}=$100,000\).
Total target-layer loss: $0+$0+$100,000 = $100,000. The target cover’s limit does not bind on the third claim because its payment is only $100,000, below the $200,000 limit.
Check and common mistake avoided. The third claim leaves $600,000 after the underlying recovery, so it is $100,000 into the target layer. Do not subtract the underlying cover’s $300,000 attachment from gross loss as though it were a recovery: the actual underlying recovery is $400,000, and that is what reduces the gross claim.
Sequential Covers: Apply Each Recovery to Its Stated Base
When covers inure in sequence, the next cover generally responds to the loss remaining after the prior recovery, if that is what the contract specifies. It does not automatically respond to gross loss. This distinction matters especially when a quota share precedes an excess-of-loss cover: the excess cover then sees the retained portion, not the original gross claim.
Record the claim amount before any recoveries.
Use its attachment, limit, share, and contractual loss basis to calculate its recovery.
Subtract the recovery, then present that residual to the next cover if the contract says it inures after the first.
Apply the target’s own attachment and limit to the final residual, not automatically to gross loss.
Worked Example: Quota Share, Underlying Excess, and Expected Target Cost
A hypothetical portfolio has gross occurrence loss \(X\) with the following discrete distribution: $200,000 with probability 0.20, $800,000 with probability 0.50, and $1,500,000 with probability 0.30. A 20% quota share applies first to gross loss. A $300,000 limit excess of $400,000 then applies to the retained loss. The target layer is $250,000 excess of $600,000. Calculate the expected target payment per occurrence.
Assumptions. The listed outcomes and probabilities fully specify the loss distribution. The quota share applies to gross loss, the excess cover applies next to the quota-share residual, and the target applies last. There is no interest or mortality component.
For a gross loss \(x\), the quota-share recovery is \(0.20x\), so the retained loss is \(0.80x\). The underlying excess recovery is \(\min\{$300,000,\max(0,0.80x-$400,000)\}\). Subtract that recovery to obtain the loss presented to the target.
Outcome 1: \(X=$200,000\). Quota-share recovery = \(0.20($200,000)=$40,000\). Retained loss = $200,000-$40,000=$160,000. The underlying excess recovery is $0 because $160,000 is below its $400,000 attachment. Residual loss = $160,000. Target payment = $0 because $160,000 is below $600,000.
Outcome 2: \(X=$800,000\). Quota-share recovery = \(0.20($800,000)=$160,000\). Retained loss = $800,000-$160,000=$640,000. Underlying recovery = \(\min($300,000,$640,000-$400,000)=$240,000\). Residual loss = $640,000-$240,000=$400,000. Target payment = $0 because $400,000 is below $600,000.
Outcome 3: \(X=$1,500,000\). Quota-share recovery = \(0.20($1,500,000)=$300,000\). Retained loss = $1,500,000-$300,000=$1,200,000. The amount above the underlying attachment is $1,200,000-$400,000=$800,000, so the $300,000 limit is exhausted. Residual loss = $1,200,000-$300,000=$900,000. Target payment = \(\min($250,000,$900,000-$600,000)=$250,000\).
The expected target payment is \(0.20($0)+0.50($0)+0.30($250,000)=$75,000\) per occurrence. This is the expected payment for the specified discrete distribution, not the total cost for a year unless the number of occurrences is also incorporated.
Check and common mistake avoided. In the largest outcome, the underlying cover pays $300,000, leaving $900,000 from the $1,200,000 quota-share retention. The target therefore pays its full $250,000 limit. Applying the underlying cover to gross loss instead of to the stated retained-loss base would produce a different, incorrect result.
Multiple Inuring Excess Covers and Net Layer Cost
For multiple underlying covers, repeat the same calculation for each cover in order. For a sequence of per-occurrence covers, let \(z_0=X\). If cover \(j\) has recovery function \(R_j\) applied to the loss then presented, the residual after that cover is \(z_j=z_{j-1}-R_j(z_{j-1})\). The target payment is calculated from the final residual \(z_n\). This recursion is appropriate only when the contracts specify this sequence and loss basis.
For a set of observed occurrences, the total target-layer loss is the sum of their target payments. If instead a probability model for gross loss \(X\) is available, expected target cost per occurrence is \(E[Y_{\text{target}}]\), found by weighting each possible target payment by its probability or by integrating over the loss distribution. These are different quantities: one is a sum for a particular set of claims, while the other is a model-based average per occurrence.
Worked Example: Two Sequential Underlying Covers
A hypothetical annual claim listing contains three gross occurrence losses: $450,000, $900,000, and $1,400,000. Cover A pays $300,000 excess of $200,000. Cover B pays $250,000 excess of $500,000 and inures after Cover A, applying to the residual left by Cover A. The target is $300,000 excess of $700,000. Find the total target-layer loss for the three occurrences.
Assumptions. The three claims are fixed observations with no probabilities assigned. Both underlying covers and the target are per occurrence and respond in the stated order. There is no interest or mortality component.
Claim 1: $450,000. Cover A pays \(\min($300,000,$450,000-$200,000)=$250,000\). Residual after A = $450,000-$250,000=$200,000. Cover B pays $0 because the residual is below its $500,000 attachment. Residual after B = $200,000. The target pays $0 because this is below $700,000.
Claim 2: $900,000. Cover A’s amount above attachment is $900,000-$200,000=$700,000, so its recovery is capped at $300,000. Residual after A = $900,000-$300,000=$600,000. Cover B pays \(\min($250,000,$600,000-$500,000)=$100,000\). Residual after B = $600,000-$100,000=$500,000. The target pays $0.
Claim 3: $1,400,000. Cover A’s amount above attachment is $1,400,000-$200,000=$1,200,000, so it pays its full $300,000 limit. Residual after A = $1,400,000-$300,000=$1,100,000. Cover B’s amount above attachment is $1,100,000-$500,000=$600,000, so it pays its full $250,000 limit. Residual after B = $1,100,000-$250,000=$850,000. The target payment is \(\min($300,000,$850,000-$700,000)=$150,000\).
Total target-layer loss: $0+$0+$150,000 = $150,000. The third claim reaches $150,000 into the target layer, which is below its $300,000 limit.
Check and common mistake avoided. Cover B attaches at $500,000 after Cover A’s recovery; it is not applied to the original $1,400,000 gross loss. Applying it to gross loss would ignore the specified inuring order and overstate Cover B’s recovery.
Common Calculation Traps
- Subtracting an attachment as if it were a recovery. An attachment determines when a cover starts paying; it is not itself a payment. Calculate the recovery with the layer formula, then subtract that recovery from the presented loss.
- Applying a later cover to gross loss. If the contract says it applies after an underlying recovery, use the residual loss as its input.
- Forgetting the layer limit. The amount above attachment can exceed the limit. The recovery is capped at the stated limit.
- Using the wrong loss base for a quota share. A share of gross loss and a share of the loss retained after another cover are not interchangeable. Follow the stated terms.
- Confusing expected cost with observed total loss. A probability-weighted expected payment is an average per occurrence. A claim listing’s total is the sum for those particular occurrences.
Actual treaty wording can include aggregate limits, reinstatements, special event definitions, or other terms that make a simple occurrence-by-occurrence calculation insufficient. The formulas here apply when the stated per-occurrence terms define the response. The calculation is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, follow the stated recovery order and apply each limit to the correct loss base.
- A gross claim is $720,000. An underlying cover is $250,000 excess of $300,000, followed by a target of $200,000 excess of $500,000. What is the target payment? (Answer: $0)
- Three gross claims are $420,000, $760,000, and $1,100,000. An underlying cover is $350,000 excess of $250,000, and the target is $250,000 excess of $600,000. Find the total target-layer loss. (Answer: $150,000)
- Gross loss \(X\) is $500,000 with probability 0.20, $900,000 with probability 0.50, or $1,400,000 with probability 0.30. A 25% quota share applies first; a $200,000 limit excess of $350,000 applies to the retained loss; and the target is $150,000 excess of $500,000. Find expected target payment per occurrence. (Answer: $45,000)
- A gross claim is $800,000. Cover A pays $200,000 excess of $100,000. Cover B pays $150,000 excess of $350,000 and applies to the residual after A. The target is $250,000 excess of $500,000. What is the target payment? (Answer: $0)
- Gross loss is $800,000 with probability 0.25 and $1,200,000 with probability 0.75. A 20% quota share applies first, followed by a $300,000 limit excess of $200,000 on the retained loss. The target is $200,000 excess of $500,000. Find expected target payment per occurrence. (Answer: $120,000)