Translating Contract Layers Into Claim Payment Maps

Why Payment Maps Matter

A contract description is not yet a calculation rule. To calculate a claim payment reliably, first translate the wording into a function of the ground-up loss $X$. The function should make clear which amounts are removed by a deductible, which losses are subject to a limit, how coinsurance is applied, and where a reinsurer’s layer attaches and exhausts.

The same deductible and numerical limit can produce different payments depending on what the limit caps. A limit may cap the ground-up loss considered by the contract, or it may cap the covered amount after the deductible. Those interpretations have different boundary points. State the interpretation before calculating.

Definition: A payment map is a function assigning a claim payment to each possible ground-up loss $x$. For a deductible $d$, the amount above the deductible is $(x-d)_+=\max(x-d,0)$. A payment limit and coinsurance modify this amount according to the contract wording. For an excess layer with attachment $a$ and limit $h$, the layer payment on an eligible amount $z$ is $\min((z-a)_+,h)$.

Deductibles, Limits, and Coinsurance

For a deductible $d$ and coinsurance percentage $c$, where $0\le c\le 1$, the payment before any limit is $c(x-d)_+$. Coinsurance multiplies the amount specified by the contract; it does not change the deductible unless the wording explicitly says otherwise.

Two common limit conventions are shown below. In the first, $u$ is a cap on gross loss considered: the eligible loss is $\min(x,u)$, and the deductible is then subtracted. In the second, $L$ is a cap on the covered amount after the deductible. When coinsurance applies, it is applied to that capped covered amount.

Y^L(x)=c\bigl(\min(x,u)-d\bigr)_+ \qquad\text{(gross-loss limit \(u\))}

Y^L(x)=c\min\bigl((x-d)_+,L\bigr) \qquad\text{(covered-amount limit \(L\))}

For a gross-loss limit with $u>d$, payment is zero up to $d$, increases at rate $c$ between $d$ and $u$, and is capped at $c(u-d)$ above $u$. For a covered-amount limit, payment reaches its cap $cL$ at $x=d+L$. These breakpoints follow directly from the arguments of the minimum and positive-part functions.

Worked Example: Coinsurance With a Gross-Loss Limit

A hypothetical property policy has a deductible of $d=\$400$, a gross-loss limit of $u=\$2{,}400$, and coinsurance of $c=0.75$. The limit caps the gross loss considered before the deductible is subtracted. Write the payment map and calculate the payment for ground-up losses of $\$200$, $\$1{,}000$, $\$2{,}400$, and $\$3{,}100$.

Method. Use $Y^L(x)=0.75(\min(x,2400)-400)_+$. The formula changes at $x=400$, where the deductible is exceeded, and at $x=2400$, where the gross-loss limit binds.

For $0\le x\le400$, the amount after the deductible is zero. For $400<x<2400$, payment is $0.75(x-400)$. For $x\ge2400$, payment is capped at $0.75(2400-400)=\$1{,}500$. Thus the piecewise map is

Y^L(x)= \begin{cases} 0, & 0\le x\le 400,\\ 0.75(x-400), & 400<x<2{,}400,\\ 1{,}500, & x\ge2{,}400. \end{cases}

At $x=\$200$, payment is $0.75(\min(200,2400)-400)_+=0.75(0)=\$0$. At $x=\$1{,}000$, it is $0.75(1{,}000-400)=0.75(600)=\$450$. At $x=\$2{,}400$, it is $0.75(2{,}400-400)=0.75(2{,}000)=\$1{,}500$. At $x=\$3{,}100$, the limit still caps the considered loss at $\$2{,}400$, so payment remains $\$1{,}500$.

Check and trap avoided. The largest payment is $0.75(2{,}400-400)=\$1{,}500$. Applying the deductible after coinsurance would incorrectly give $0.75(2{,}400)-400=\$1{,}400$; the stated contract subtracts the deductible first and then applies coinsurance.

Deriving the Breakpoints

A useful way to derive a map is to identify where each operation changes the expression. Consider a gross-loss limit $u>d$. If $x\le d$, then $(\min(x,u)-d)_+=0$. If $d<x<u$, then $\min(x,u)=x$, so the payment is $c(x-d)$. If $x\ge u$, then $\min(x,u)=u$, so payment is the constant $c(u-d)$. This yields the three pieces above.

For a covered-amount limit $L>0$, the amount $(x-d)_+$ first becomes positive at $x=d$. It reaches $L$ when $x-d=L$, or $x=d+L$. Therefore the payment increases as $c(x-d)$ only between those two points, and is capped at $cL$ thereafter. The breakpoint is $d+L$, not $L$: the limit applies after the deductible.

Worked Example: Limit on the Covered Amount

A hypothetical equipment policy has deductible $d=\$300$, covered-amount limit $L=\$1{,}800$, and coinsurance $c=0.60$. Its limit applies after the deductible. Write the map and calculate payments for losses of $\$250$, $\$1{,}300$, $\$2{,}100$, and $\$5{,}000$.

Method. The covered amount is $\min((x-300)_+,1800)$, and the insurer pays 60% of it. The two boundaries are $x=300$ and $x=300+1800=\$2{,}100$.

Y^L(x)= \begin{cases} 0, & 0\le x\le 300,\\ 0.60(x-300), & 300<x<2{,}100,\\ 1{,}080, & x\ge2{,}100. \end{cases}

At $x=\$250$, the loss is below the deductible, so payment is $\$0$. At $x=\$1{,}300$, the amount above the deductible is $1{,}300-300=\$1{,}000$; payment is $0.60(1{,}000)=\$600$. At $x=\$2{,}100$, the covered amount has reached its limit: $0.60(1{,}800)=\$1{,}080$. At $x=\$5{,}000$, the same cap applies, so payment remains $\$1{,}080$.

Check and trap avoided. The capped payment is $cL=0.60(1{,}800)=\$1{,}080$. Treating $L$ as a gross-loss limit would incorrectly make the cap bind at $x=\$1{,}800$, rather than at $x=d+L=\$2{,}100$.

Adding an Excess Layer

An excess-of-loss layer is defined relative to an eligible amount. If the eligible amount is $z$, an excess layer attaches at $a$ and has limit $h$, then its recovery is $R(z)=\min((z-a)_+,h)$. It pays nothing through $a$, increases dollar-for-dollar above $a$, and is capped at $h$ once $z\ge a+h$.

When the layer reinsures a primary insurer’s payment $Z(x)$, apply the treaty to $Z(x)$, not automatically to the ground-up loss $x$. The primary insurer’s net retained payment is $Z(x)-R(Z(x))$. The treaty attachment and exhaustion points in ground-up-loss units must therefore be found by solving for the $x$ values at which $Z(x)=a$ and $Z(x)=a+h$.

Worked Example: Primary Coverage With a Reinsured Layer

A hypothetical primary policy has a $\$1{,}000$ deductible and a gross-loss limit of $\$10{,}000$, with no coinsurance. A reinsurer covers the layer of the primary payment from $\$4{,}000$ excess of $\$4{,}000$, meaning attachment $a=\$4{,}000$ and layer limit $h=\$4{,}000$. Calculate primary gross payment, reinsurance recovery, and primary net retention for $X=\$4{,}000$, $\$5{,}000$, $\$8{,}000$, and $\$10{,}000$.

Method. The primary payment is $Z(x)=(\min(x,10{,}000)-1{,}000)_+$. Its reinsurance recovery is $R(x)=\min((Z(x)-4{,}000)_+,4{,}000)$, and net retention is $Z(x)-R(x)$.

The primary payment reaches the treaty attachment when $Z(x)=4{,}000$, so $x-1{,}000=4{,}000$, or $x=\$5{,}000$. The layer exhausts when $Z(x)=4{,}000+4{,}000=\$8{,}000$, so $x=\$9{,}000$. The primary gross-loss limit then binds at $x=\$10{,}000$.

At $X=\$4{,}000$, $Z=4{,}000-1{,}000=\$3{,}000$, $R=\min((3{,}000-4{,}000)_+,4{,}000)=\$0$, and net retention is $\$3{,}000$. At $X=\$5{,}000$, $Z=\$4{,}000$, $R=\$0$, and net retention is $\$4{,}000$. At $X=\$8{,}000$, $Z=\$7{,}000$, $R=\min(3{,}000,4{,}000)=\$3{,}000$, and net retention is $\$4{,}000$. At $X=\$10{,}000$, $Z=\$9{,}000$, $R=\min(5{,}000,4{,}000)=\$4{,}000$, and net retention is $\$5{,}000$.

Check and trap avoided. In every case, primary gross payment equals reinsurer recovery plus net retention. The treaty attaches to $Z$, so using $X=\$8{,}000$ directly as the treaty’s eligible amount would give an incorrect recovery.

Boundary Checks and Common Errors

At a boundary, either adjoining expression should give the same payment. For example, at $x=d+L$, the increasing piece $c(x-d)$ equals the capped piece $cL$. Such checks are quick ways to detect a misplaced deductible or a limit applied to the wrong quantity.

Confusing limit conventions. A gross-loss limit $u$ binds at $x=u$; a covered-amount limit $L$ binds at $x=d+L$.
Applying coinsurance in the wrong order. Under the maps defined here, first determine the covered amount after deductible and limit, then multiply by $c$.
Using the wrong input to a treaty. Identify whether the treaty applies to ground-up loss, primary covered payment, or another explicitly defined amount.
Forgetting exhaustion. A layer with attachment $a$ and limit $h$ pays at most $h$, even when the eligible amount is much larger than $a+h$.

Payment maps also support later calculations by making the contractual transformation explicit. Here the task is to establish the map and its boundaries; calculating a probability distribution or expected payment after transformation requires additional steps.

Key takeaway: Write the eligible amount first, apply each contract term in its stated order, and solve for the ground-up losses at which each deductible, limit, attachment, or exhaustion point takes effect. Then verify the map at its boundaries and check that gross payment equals ceded payment plus net retention.

Check Your Understanding

For each hypothetical contract, write down the relevant payment calculation before evaluating it.

A policy has deductible $d=\$250$, gross-loss limit $u=\$1{,}750$, and coinsurance $c=0.80$. What does it pay on a ground-up loss of $\$1{,}000$? (Answer: $\$600$)
A policy has deductible $d=\$400$, covered-amount limit $L=\$1{,}200$, and coinsurance $c=0.75$. What is its maximum payment, and at what ground-up loss does that maximum first apply? (Answer: $\$900$, first applies at $X=\$1{,}600$)
A primary policy pays $(\min(X,6{,}500)-500)_+$. A treaty covers the first $\$1{,}500$ of primary payment excess of $\$2{,}500$. For $X=\$4{,}500$, find primary gross payment, recovery, and net retention. (Answer: $\$4{,}000$, $\$1{,}500$, and $\$2{,}500$, respectively)
A primary policy has deductible $\$1{,}000$ and gross-loss limit $\$9{,}000$. A treaty has attachment $\$3{,}000$ and limit $\$2{,}000$, applied to primary payment. For $X=\$6{,}000$, find recovery and net retention. (Answer: recovery $\$2{,}000$; net retention $\$3{,}000$)
A primary policy pays $Z(X)=(\min(X,5{,}600)-600)_+$. A treaty covers a layer of limit $\$1{,}000$ excess of $\$2{,}500$ of $Z$. State the ground-up loss at which the treaty attaches and the loss at which it exhausts. (Answer: attaches at $X=\$3{,}100$; exhausts at $X=\$4{,}100$)

What You'll Learn