Tracing a Loss Through Coverage Layers
A layered coverage program divides the amount of a loss into ranges. A primary policy may pay first, after a deductible, up to its limit. An excess layer may then pay amounts above the primary policy’s exhaustion point, up to its own limit. To calculate a claim correctly, trace the loss through the layers in order: determine how much reaches each layer, then apply that layer’s payment rule.
The calculations in this tutorial use hypothetical coverage terms. Amounts are stated in thousands of dollars where indicated. Treat each loss as a covered loss to which the stated terms apply; ignore expenses, coinsurance, and aggregate limits unless a problem expressly includes them.
The primary deductible and the primary limit have different roles. The deductible is the initial part of a loss that the primary policy does not pay. The limit is the maximum amount the primary policy pays after the deductible. Thus, a primary policy with a deductible of 10 and a limit of 90 pays at most 90, and is exhausted at a loss of 100, not at a loss of 90.
Why the Layer Formula Works
A layer attaching at \(a\) responds only to the part of the loss above \(a\). That amount is zero when \(x\le a\), and \(x-a\) when \(x>a\); in one expression it is \(\max(x-a,0)\). The layer cannot pay more than its width \(w\), so the payment is the smaller of that amount and \(w\).
This piecewise form makes the attachment and exhaustion points explicit. Below attachment, payment is zero. Between attachment and exhaustion, each additional dollar of loss increases payment by one dollar. At and above exhaustion, payment remains at the layer width. The same reasoning gives the primary payment: first remove the deductible, then cap the result at the primary limit.
Worked Examples
Worked Example: Find the Primary and Excess Payments
A hypothetical primary policy has a deductible of $10,000 and a limit of $90,000. An excess layer attaches at $100,000 and has a limit of $150,000. Calculate the two payments for a covered loss of $175,000, and identify both exhaustion points. Amounts in this example are in thousands of dollars.
Method. Calculate the primary payment first, then apply the excess layer formula at its stated attachment. The primary exhaustion point is the deductible plus the primary limit: \(10+90=100\). The excess layer exhaustion point is its attachment plus its width: \(100+150=250\).
The primary payment is \(\min(\max(175-10,0),90)=\min(165,90)=90\). The primary is therefore exhausted at this loss.
The excess payment is \(\min(\max(175-100,0),150)=\min(75,150)=75\). The loss is between the excess layer’s attachment of 100 and exhaustion point of 250, so the excess layer pays the amount above its attachment.
The total payment is \(90+75=165\). The unpaid amount is \(175-165=10\), exactly the primary deductible. Check: the excess payment of 75 is no greater than its width of 150, and total payments of 165 do not exceed the covered loss of 175.
Worked Example: Trace Several Losses Across Three Layers
A hypothetical program has a primary policy with a deductible of 5 and limit of 45. A middle layer attaches at 50 and has width 75. An upper layer attaches at 125 and has width 75. For each loss below, calculate all three payments and locate the relevant attachment and exhaustion points. Amounts are in thousands of dollars.
The primary exhausts at \(5+45=50\). The middle layer attaches at 50 and exhausts at \(50+75=125\). The upper layer attaches at 125 and exhausts at \(125+75=200\). Apply each layer to the original loss amount using its own attachment, rather than subtracting the previous payment from the loss.
| Loss | Primary calculation | Middle calculation | Upper calculation | Total paid |
|---|---|---|---|---|
| 3 | \(\min(\max(3-5,0),45)=0\) | \(\min(\max(3-50,0),75)=0\) | \(\min(\max(3-125,0),75)=0\) | 0 |
| 28 | \(\min(28-5,45)=23\) | 0 | 0 | 23 |
| 80 | \(\min(80-5,45)=45\) | \(\min(80-50,75)=30\) | 0 | 75 |
| 125 | 45 | \(\min(125-50,75)=75\) | 0 | 120 |
| 170 | 45 | 75 | \(\min(170-125,75)=45\) | 165 |
| 220 | 45 | 75 | 75 | 195 |
At a loss of 125, the middle layer has reached its exhaustion point and pays its full width of 75. At 220, the upper layer is also exhausted because the loss exceeds its exhaustion point of 200. The total payment is then \(45+75+75=195\), less than the loss of 220. The remaining 25 consists of the initial deductible of 5 and the 20 of loss above the program’s final exhaustion point.
Check: at a loss of 170, the retained amount is \(170-165=5\), the primary deductible. At 220, the retained amount is \(220-195=25\). In every row, the total paid is no greater than the loss.
Worked Example: Reconcile Payments Across Several Claims
Use the three-layer program from the previous example to trace four separate covered losses of 42, 98, 156, and 214. Find each claim’s layer payments, the total paid by each layer, and the overall amount retained. Amounts are in thousands of dollars. There is no aggregate limit.
For a loss of 42, the primary pays \(\min(42-5,45)=37\); neither excess layer attaches. The total paid is 37 and the retained amount is \(42-37=5\).
For a loss of 98, the primary pays its full 45. The middle layer pays \(\min(98-50,75)=48\), and the upper layer pays zero. The total paid is \(45+48=93\), leaving \(98-93=5\).
For a loss of 156, the primary pays 45 and the middle layer pays 75. The upper layer pays \(\min(156-125,75)=31\). The total paid is \(45+75+31=151\), leaving \(156-151=5\).
For a loss of 214, the primary and middle layers each pay their full amounts, 45 and 75. The upper layer also pays its full width of 75 because \(214\) exceeds its exhaustion point of \(200\). Total paid is \(45+75+75=195\), leaving \(214-195=19\).
The layer totals are: primary \(37+45+45+45=172\); middle \(0+48+75+75=198\); upper \(0+0+31+75=106\). Overall payment is \(172+198+106=476\). The total loss is \(42+98+156+214=510\), so the overall retained amount is \(510-476=34\). This also equals the sum of the individual retained amounts, \(5+5+5+19=34\). Check: each claim’s layer payments are within the corresponding layer widths, and the total payment of 476 is below total losses of 510.
Attachment, Exhaustion, and Reconciliation
The attachment point tells you where a layer begins to pay; it is not itself an additional payment. For example, if a layer attaches at 50, a loss of 50 produces no payment from that layer. A loss of 51 produces a payment of 1, provided the layer has not exhausted. At the exhaustion point, the layer has paid exactly its full width.
For consecutive layers, the next attachment commonly equals the preceding layer’s exhaustion point. This alignment means the layers respond to consecutive portions of the loss. When calculating payments, use the original loss as \(x\) in each layer’s formula. The formula already accounts for the amount below that layer’s attachment.
Add the primary deductible and primary limit. This is the loss amount at which the primary has paid its maximum.
For an excess layer attaching at \(a\) with width \(w\), its interval runs from \(a\) to \(a+w\).
Use the primary formula for the primary policy and \(C_{a,w}(x)\) for each excess layer.
Add all layer payments and compare the sum with the loss. The sum must not exceed the loss; any difference is unpaid or retained under the stated terms.
A useful check is to identify the portion of the loss not paid. With a deductible and consecutive layers, losses below the first attachment leave some or all of the deductible unpaid. Losses above the final exhaustion point can also leave an amount unpaid because the program has reached its maximum total payment. If the layers are consecutive and have no gaps, a loss between the primary exhaustion point and the final exhaustion point is allocated across the layers without overlap.
Common Mistake: Subtracting Payments Instead of Using Attachments
Suppose a loss is 170 under the three-layer program above. The middle layer attaches at 50, so it pays \(\min(170-50,75)=75\). A common incorrect calculation subtracts the primary payment of 45 from the loss and then calculates the middle payment as \(\min(170-45,75)=75\). This happens to give the same middle payment here because the layer is exhausted, but the method is wrong: the layer responds to the part of the loss above its attachment, not the part left after subtracting the previous payment.
The error changes the answer when the layer is not exhausted. At a loss of 80, the correct middle payment is \(\min(80-50,75)=30\). Subtracting the primary payment first would give \(\min(80-45,75)=35\), overstating the middle payment by 5. The correct total is \(45+30=75\), leaving the deductible of 5 unpaid; the incorrect total of \(45+35=80\) wrongly pays the entire loss.
Check Your Understanding
For each hypothetical program, calculate the requested payments and check the loss allocation.
- A primary policy has a deductible of 8 and a limit of 52. An excess layer attaches at 60 and has width 90. For a loss of 47, find the primary payment, excess payment, and retained amount. (Answer: primary 39, excess 0, retained 8; amounts in thousands.)
- A primary policy has a deductible of 12 and limit of 68. An excess layer attaches at 80 with width 120. For a loss of 155, find the two payments and total payment. (Answer: primary 68, excess 75, total 143; amounts in thousands.)
- A primary policy has a deductible of 10 and limit of 90. An excess layer attaches at 100 with width 125. For a loss of 265, find total payment and retained amount, and state whether the excess layer is exhausted. (Answer: total payment 215, retained 50, and the excess layer is exhausted; amounts in thousands.)
- A primary policy has a deductible of 4 and limit of 36. A middle layer attaches at 40 with width 60, and an upper layer attaches at 100 with width 80. For a loss of 137, find each layer payment and the retained amount. (Answer: primary 36, middle 60, upper 37, retained 4; amounts in thousands.)
- A primary policy has a deductible of 6 and limit of 44. A middle layer attaches at 50 with width 60, and an upper layer attaches at 110 with width 60. Three losses are 35, 95, and 160. Find the total paid by each layer and the overall retained amount. (Answer: primary 117, middle 105, upper 50, overall retained 18; amounts in thousands.)