Calculating Payments Across Multiple Coverage Layers

Why Calculate Each Layer Separately?

A layered coverage arrangement divides loss amounts into successive bands. Each band has an attachment point, where that layer starts to pay, and a limit, the maximum amount it can pay. To allocate a claim correctly, calculate each layer’s payment from the original loss amount and that layer’s own terms. The amount paid by one layer is not the attachment point of the next.

This distinction matters even when all layers sit directly next to one another. A loss may be below the first layer, fall partly within one layer, exhaust several layers, or exceed the top of the entire stack. The same calculation handles each case. The examples treat each claim as a single hypothetical loss measured in dollars; they do not apply coinsurance, an aggregate cap, or any other claim adjustment.

Payment From One Layer

Definition: For a loss amount $X$, a layer with attachment point $a$ and limit $L$ pays $Y(X)=\min\big((X-a)^+,L\big)$, where $z^+=\max(z,0)$. The layer begins to pay when $X>a$, and it reaches its maximum payment $L$ when $X\geq a+L$. Its covered band of loss amounts runs from $a$ to $a+L$.

The attachment point and the limit describe different things. The attachment point is measured on the scale of the original loss. The limit is the maximum payment by that particular layer. Thus a layer attaching at $200,000 with a $300,000 limit covers the band from $200,000 to $500,000. It pays $300,000 once the loss reaches $500,000; it does not require a loss of $300,000 to begin paying.

For a stack of $n$ layers, write the attachment and limit of layer $j$ as $a_j$ and $L_j$. Its payment is

Y_j(X)=\min\big((X-a_j)^+,L_j\big),\qquad j=1,\ldots,n.

Calculate every $Y_j(X)$ using the same original loss $X$. The payment by layer $j$ is not calculated by first subtracting the payments made by lower layers from $X$.

Adjacent Layers and the Total Allocation

Two successive layers are adjacent when the upper boundary of the lower layer equals the attachment point of the next layer. For an ordered stack, this means

a_{j+1}=a_j+L_j,\qquad j=1,\ldots,n-1.

This equality means the layers neither leave a gap in the covered bands nor cover the same band twice. If the first layer attaches at $a_1$, the stack begins above $a_1$; any loss below that point is retained. If the stack’s upper boundary is $a_n+L_n$, any loss above that boundary is not paid by these layers.

Theorem: For adjacent layers, the sum of layer payments is $\sum_{j=1}^n Y_j(X)=(X-a_1)^+-(X-(a_n+L_n))^+$. If $X$ is at least $a_1$, this is the portion of the loss above the stack’s first attachment point, capped at the stack’s total height.

A useful way to apply this result is to calculate each layer separately, add the payments, and then reconcile the total with the part of the loss that the stack can cover. The check does not replace the individual calculations: it confirms that the allocation is consistent.

Worked Example: A Loss That Exhausts the Stack

Hypothetical claim. A single loss is $735,000. Three layers have terms: layer 1 attaches at $0 with a $200,000 limit; layer 2 attaches at $200,000 with a $300,000 limit; layer 3 attaches at $500,000 with a $150,000 limit. Assume the stated loss is the amount to which these per-loss layer terms apply, with no other adjustment. Find each payment and the amount above the stack.

Method. Apply the layer payment formula to the original $735,000 loss for each layer. The attachments are adjacent because $0 + $200,000 = $200,000 and $200,000 + $300,000 = $500,000.

For layer 1, the amount above its attachment is $735,000 − $0 = $735,000. Its limit is $200,000, so its payment is $\min($735,000,$200,000)=$200,000$.

For layer 2, the amount above its attachment is $735,000 − $200,000 = $535,000. Its payment is $\min($535,000,$300,000)=$300,000$.

For layer 3, the amount above its attachment is $735,000 − $500,000 = $235,000. Its payment is $\min($235,000,$150,000)=$150,000$.

The total payment is $200,000 + $300,000 + $150,000 = $650,000. The top of the stack is $500,000 + $150,000 = $650,000, so the loss above the stack is $735,000 − $650,000 = $85,000. The check is $650,000 paid + $85,000 above the stack = $735,000, the original loss. The method avoids treating the third layer’s attachment as a payment already made by the lower layers.

Why the Total Formula Works

The layer formula can also be written as a difference of two positive-part amounts:

$$ Y_j(X)=(X-a_j)^+-(X-a_j-L_j)^+. $$

To verify this identity, consider three cases. If $X\leq a_j$, both positive-part terms are zero, so the layer pays zero. If $a_j<X<a_j+L_j$, the first term is $X-a_j$ and the second is zero, so the payment is the loss amount within the layer. If $X\geq a_j+L_j$, the difference is $(X-a_j)-(X-a_j-L_j)=L_j$, the layer’s full limit.

For adjacent layers, $a_j+L_j=a_{j+1}$. Therefore, the negative term for layer $j$ cancels the positive term for layer $j+1$ when payments are summed:

\begin{aligned} \sum_{j=1}^{n}Y_j(X) &=\sum_{j=1}^{n}\left[(X-a_j)^+-(X-a_j-L_j)^+\right]\\ &=(X-a_1)^+-(X-a_n-L_n)^+\\ &=(X-a_1)^+-(X-(a_n+L_n))^+. \end{aligned}

Only the bottom boundary and the top boundary remain. This proves the total-payment formula and explains why adjacent layer payments add to the covered portion of the loss without double-counting any band.

Worked Example: A Retained Amount Below the First Layer

Hypothetical claim. A single loss is $415,000. Layer 1 attaches at $50,000 with a $100,000 limit; layer 2 attaches at $150,000 with a $125,000 limit; and layer 3 attaches at $275,000 with a $200,000 limit. Assume no other adjustment applies. Calculate each layer payment, the total payment, and the retained amount below the stack.

Method. Calculate each payment from the original loss and its own attachment and limit. The layers are adjacent: $50,000 + $100,000 = $150,000, and $150,000 + $125,000 = $275,000.

Layer 1 has $415,000 − $50,000 = $365,000 of loss above its attachment. Its payment is $\min($365,000,$100,000)=$100,000$.

Layer 2 has $415,000 − $150,000 = $265,000 above its attachment. Its payment is $\min($265,000,$125,000)=$125,000$.

Layer 3 has $415,000 − $275,000 = $140,000 above its attachment. Its payment is $\min($140,000,$200,000)=$140,000$.

The total is $100,000 + $125,000 + $140,000 = $365,000. The loss below the first attachment is $50,000, and the loss above the stack is $0 because the top is $275,000 + $200,000 = $475,000, which exceeds the loss. The allocation check is $50,000 retained + $365,000 paid = $415,000.

Checking a Stack with Several Boundaries

A claim can cross multiple layer boundaries without reaching the top. In that case, lower layers may pay their full limits while the layer containing the loss’s upper endpoint pays only part of its limit. Layers above that point pay zero. Comparing the loss with each layer’s lower and upper boundaries makes these cases straightforward to identify.

List each band.
For layer $j$, record its attachment $a_j$, limit $L_j$, and upper boundary $a_j+L_j$.

Check adjacency.
Verify that each layer’s upper boundary equals the next layer’s attachment.

Calculate each payment.
Use $Y_j(X)=\min((X-a_j)^+,L_j)$, always using the original loss $X$.

Reconcile the allocation.
Add the payments and check them against the loss retained below the stack and any loss above its top.

Worked Example: A Loss Within the Top Layer

Hypothetical claim. A single loss is $1,280,000. Four adjacent layers are specified: layer 1 attaches at $0 with a $250,000 limit; layer 2 attaches at $250,000 with a $400,000 limit; layer 3 attaches at $650,000 with a $500,000 limit; and layer 4 attaches at $1,150,000 with a $250,000 limit. Assume these are the only adjustments. Determine all layer payments and the total.

Method. Find the loss amount above each attachment, then cap that amount at the corresponding layer limit. The boundaries match: $0 + $250,000 = $250,000; $250,000 + $400,000 = $650,000; and $650,000 + $500,000 = $1,150,000.

Layer 1 pays $\min($1,280,000-$0,$250,000)=\min($1,280,000,$250,000)=$250,000$.

Layer 2 pays $\min($1,280,000-$250,000,$400,000)=\min($1,030,000,$400,000)=$400,000$.

Layer 3 pays $\min($1,280,000-$650,000,$500,000)=\min($630,000,$500,000)=$500,000$.

Layer 4 pays $\min($1,280,000-$1,150,000,$250,000)=\min($130,000,$250,000)=$130,000$.

The total payment is $250,000 + $400,000 + $500,000 + $130,000 = $1,280,000. The top of the stack is $1,150,000 + $250,000 = $1,400,000, so none of the loss is above it. The bottom attachment is $0, so none is below the stack. The total payment therefore reconciles exactly to the original loss.

Common Mistake: Treating Payments as Attachment Points

An attachment point is a threshold on the original loss scale; it is not the sum paid by lower layers. In the second example, layer 3 attaches at $275,000. Its payment is based on $415,000 − $275,000 = $140,000. A wrong calculation would subtract the lower layers’ payments, $100,000 + $125,000 = $225,000, from the loss and assign the remainder, $415,000 − $225,000 = $190,000, to layer 3. That is incorrect because $225,000 is not layer 3’s attachment point. The correction is to use the stated $275,000 attachment and then apply layer 3’s $200,000 limit.

Another useful check is to compare the sum of payments with the portion of loss that falls inside the full stack. A sum greater than that portion suggests that some loss band has been counted twice; a sum that is too small may indicate a missed layer or a misapplied attachment.

Key takeaway: For every layer, use the original loss, subtract that layer’s attachment, floor the result at zero, and cap it at that layer’s limit. For adjacent layers, verify the boundaries and reconcile the sum of payments with the loss inside the stack.

Check Your Understanding

For each hypothetical single-loss arrangement, calculate the requested layer payments and check how they relate to the loss.

A $286,000 loss is subject to adjacent layers: layer 1 attaches at $0 with a $90,000 limit; layer 2 attaches at $90,000 with a $120,000 limit; layer 3 attaches at $210,000 with a $150,000 limit. Find each payment and the total. (Answer: $90,000, $120,000, and $76,000; total $286,000.)
A $640,000 loss has layers attaching at $40,000, $140,000, and $340,000, with respective limits of $100,000, $200,000, and $250,000. Find the payments, total payment, retained amount below the stack, and amount above it. (Answer: $100,000, $200,000, and $250,000; total $550,000, retained $40,000, and $50,000 above the stack.)
A $1,035,000 loss is allocated across adjacent layers from $0 to $300,000, from $300,000 to $700,000, and from $700,000 to $1,100,000. Find the three payments and the total. (Answer: $300,000, $400,000, and $335,000; total $1,035,000.)
A $510,000 loss is subject to layers attaching at $75,000, $200,000, and $380,000, with respective limits of $125,000, $180,000, and $250,000. Find the payment from each layer and reconcile the total with the loss below the stack. (Answer: $125,000, $180,000, and $130,000; total $435,000, with $75,000 below the stack.)
A $872,500 loss is allocated across four adjacent layers: $25,000 to $175,000, $175,000 to $425,000, $425,000 to $825,000, and $825,000 to $1,000,000. Find each payment, the total, and any amount below or above the stack. (Answer: $150,000, $250,000, $400,000, and $47,500; total $847,500, with $25,000 below and $0 above.)

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