Applying Coinsurance Within A Coverage Layer

Why Coinsurance Changes a Layer Payment

A coverage layer responds only to loss in a specified range. Coinsurance changes the amount the layer pays for that eligible loss: the insurer pays a stated percentage, while the balance of the loss within the layer is not paid by that layer. The calculation therefore has two separate steps: first find the part of the loss in the layer, then apply the coinsurance percentage to that amount.

This distinction matters when a loss is larger than the layer’s upper boundary. The excess is not automatically part of the coinsured amount. Likewise, a loss below the attachment point does not become covered merely because the policy has a coinsurance percentage. The method below assumes a single hypothetical loss and no other policy terms, such as a separate deductible, aggregate cap, or sublimit.

Define the Layer Before Applying Coinsurance

Definition: Let $X$ be a nonnegative covered loss. Suppose a layer attaches at $A$ and covers a loss band of width $L$, so its upper boundary is $A+L$. If the layer pays a fraction $c$, where $0\leq c\leq 1$, of eligible loss within that band, its payment is \[ Y^L=c\min\{\max(X-A,0),L\}. \] Here $A$ and $L$ describe the loss band, not the amount paid. Under this convention, the maximum payment is $cL$.

The expression inside the formula first removes the amount below attachment: $\max(X-A,0)$ is zero if $X\leq A$, and is the loss above attachment otherwise. The minimum then prevents the layer from responding to more than its width $L$. Finally, $c$ is applied to the eligible portion. The quantities $X$, $A$, and $L$ must be measured in the same units, such as dollars.

Equivalently, the payment is zero below attachment, increases at rate $c$ while the loss moves through the band, and stays at $cL$ once the loss reaches the upper boundary. This piecewise form can be useful for checking a calculation.

Y^L= \begin{cases} 0, & X\leq A,\\ c(X-A), & A<X<A+L,\\ cL, & X\geq A+L. \end{cases}

A contract may instead describe its limit as a maximum payment, rather than as the width of a loss band. Those are different specifications. The formula above applies when $L$ is the width of the covered loss band. If the contract gives a maximum payment directly, first interpret the contract’s wording and determine the applicable loss range; do not automatically treat that payment limit as $L$.

Why the Formula Works

The layer cannot pay for loss below $A$, so only $X-A$, when positive, can be considered. But the layer stops responding after $L$ dollars of loss have entered the band. Thus the eligible loss is the smaller of the amount above attachment and $L$, after replacing any negative amount by zero. Since the insurer pays fraction $c$ of this eligible amount, multiplying by $c$ gives the layer payment.

Find loss above attachment.
Calculate $X-A$. If this is negative, the layer has no eligible loss.

Apply the layer width.
Take the smaller of the nonnegative amount above attachment and $L$. This is the loss within the layer.

Apply coinsurance.
Multiply the loss within the layer by $c$. The result is this layer’s payment.

When $A<X<A+L$, the loss is only partly through the band, so the layer pays $c(X-A)$. When $X\geq A+L$, the whole band is occupied and the payment cannot exceed $cL$. These cases also provide a direct check: the payment must be between zero and $cL$.

Worked Calculations

Worked Example: A Loss Partway Through the Layer

A hypothetical commercial property policy has a layer attaching at $A=\$40{,}000$, with a loss-band width of $L=\$60{,}000$. The layer pays 75% of loss in its band. A covered loss is $X=\$70{,}000$. Find the layer payment and the unpaid portion of the loss within the band.

Method. Find the eligible loss first, then apply the layer’s 75% share. This avoids applying coinsurance to the entire claim.

The upper boundary is \[ A+L=\$40{,}000+\$60{,}000=\$100{,}000. \] Since $\$40{,}000<\$70{,}000<\$100{,}000$, the loss is partway through the layer. The amount above attachment is \[ X-A=\$70{,}000-\$40{,}000=\$30{,}000. \] This is less than the layer width of $\$60{,}000$, so the loss within the band is $\$30{,}000$. The payment is \[ Y^L=0.75(\$30{,}000)=\$22{,}500. \] The unpaid portion within the band is \[ \$30{,}000-\$22{,}500=\$7{,}500. \] As a check, the maximum layer payment is $0.75(\$60{,}000)=\$45{,}000$, and the calculated payment of $\$22{,}500$ is between zero and that maximum.

Common mistake avoided: Applying 75% to the whole $\$70{,}000$ loss would give $\$52{,}500$. That includes the $\$40{,}000$ below attachment, which is not in this layer.

Worked Example: A Loss Beyond the Upper Boundary

A hypothetical equipment policy provides a layer with attachment $A=\$100{,}000$, loss-band width $L=\$80{,}000$, and coinsurance rate $c=65\%$. A covered loss is $\$185{,}000$. Calculate the layer payment and identify how much of the loss is above the layer.

Method. Compare the loss with the upper boundary before applying coinsurance. The layer cannot respond to loss beyond its band.

The upper boundary is \[ A+L=\$100{,}000+\$80{,}000=\$180{,}000. \] The loss exceeds this boundary by \[ \$185{,}000-\$180{,}000=\$5{,}000. \] The full $\$80{,}000$ loss band is reached, so eligible loss is \[ \min\{\max(\$185{,}000-\$100{,}000,0),\$80{,}000\} =\min\{\$85{,}000,\$80{,}000\} =\$80{,}000. \] The layer payment is \[ Y^L=0.65(\$80{,}000)=\$52{,}000. \] The maximum payment is also $0.65(\$80{,}000)=\$52{,}000$, confirming that the layer is capped. The $\$5{,}000$ above the upper boundary is outside this layer; this calculation does not assign it to any other coverage.

Common mistake avoided: Applying 65% to the full $\$85{,}000$ above attachment gives $\$55{,}250$. That incorrectly includes the $\$5{,}000$ beyond the layer’s width.

Worked Example: Separating the Layer Payment From Its Unpaid Share

A hypothetical liability contract has a layer attaching at $\$50{,}000$, with a loss-band width of $\$60{,}000$. It pays 80% of loss within the band. For a covered loss of $\$137{,}000$, calculate (a) the layer payment, (b) the unpaid share within the band, and (c) the loss below and above the band.

Method. Split the loss at the layer’s two boundaries, then apply coinsurance only to the amount between them.

The upper boundary is \[ \$50{,}000+\$60{,}000=\$110{,}000. \] The loss is $\$50{,}000$ below attachment, then $\$60{,}000$ in the layer, then \[ \$137{,}000-\$110{,}000=\$27{,}000 \] above the layer. These parts reconcile to the total loss: \[ \$50{,}000+\$60{,}000+\$27{,}000=\$137{,}000. \] The layer payment is \[ 0.80(\$60{,}000)=\$48{,}000. \] The unpaid share within the band is \[ (1-0.80)(\$60{,}000)=0.20(\$60{,}000)=\$12{,}000. \] Thus, the layer pays $\$48{,}000$; $\$12{,}000$ within the band is unpaid by this layer; $\$50{,}000$ lies below attachment; and $\$27{,}000$ lies above the band. The payment also equals the maximum $0.80(\$60{,}000)=\$48{,}000$.

Common mistake avoided: Treating the $\$27{,}000$ above the layer as part of the coinsured band would overstate eligible loss. Coinsurance applies to the loss in the band, not to every dollar above attachment.

Common Checks and Exam Traps

Before accepting a result, check the contract description and the arithmetic. In particular, “80% coinsurance” means the layer pays 80% only if that is how the contract defines its share. Do not reverse the insurer’s percentage and the unpaid percentage. If the layer pays $c$, the unpaid share within the band is $1-c$; it is not $c$.

Applying coinsurance too early: First determine the loss within the band. Do not multiply the total loss by $c$.
Ignoring attachment: Loss below $A$ does not enter this layer.
Ignoring the layer width: Eligible loss cannot exceed $L$, even if $X-A$ is larger.
Confusing a loss-band width with a payment cap: Under the stated convention, maximum payment is $cL$, not $L$.
Forgetting the unpaid share: The layer’s payment is not the entire loss within the band unless $c=1$.

Key takeaway: For a coinsured layer, identify the attachment and loss-band width, cap the loss above attachment at that width, and only then multiply by the insurer’s stated percentage. The resulting payment cannot exceed $cL$.

This calculation is useful whenever a claim is allocated to a layer that shares covered loss rather than paying the entire loss in its band. It also gives a reliable way to check a stated payment: it must be nonnegative, cannot exceed the layer’s maximum payment, and must be based only on loss between the attachment and upper boundary. This treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical contract, calculate the payment from the specified layer. Treat each stated limit as a loss-band width and assume no other policy terms apply.

A layer attaches at $\$25{,}000$, has width $\$45{,}000$, and pays 80% of loss in the band. The covered loss is $\$50{,}000$. What is the layer payment? (Answer: $\$20{,}000$)
A layer attaches at $\$70{,}000$, has width $\$30{,}000$, and pays 60%. The covered loss is $\$120{,}000$. What is the layer payment? (Answer: $\$18{,}000$)
A layer attaches at $\$12{,}000$, has width $\$28{,}000$, and pays 75%. For a covered loss of $\$30{,}000$, how much does the layer pay and how much of the loss within the band is unpaid by it? (Answer: Payment $\$13{,}500$; unpaid within the band $\$4{,}500$)
A layer covers losses from $\$90{,}000$ to $\$150{,}000$ and pays 90% of loss in that band. A covered loss is $\$210{,}000$. Find the layer payment and the amount above the layer. (Answer: Payment $\$54{,}000$; amount above the layer $\$60{,}000$)
A layer attaches at $\$40{,}000$, has width $\$50{,}000$, and pays 80%. A loss is known to be within the layer, and the layer payment is $\$24{,}000$. Find the loss within the band and the total loss. (Answer: Loss within the band $\$30{,}000$; total loss $\$70{,}000$)

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