Policy Limits And Coinsurance As Piecewise Functions

From Policy Wording to a Payment Function

A policy with a deductible, coinsurance, and a maximum payment creates different payment rules in different loss ranges. Below the deductible, the insurer pays nothing. Above it, the insurer pays a fixed proportion of eligible loss until the payment reaches its cap. Further increases in loss do not increase the insurer’s payment. Writing these regions explicitly helps prevent errors about when coinsurance applies and what the limit caps.

Throughout this tutorial, let $X$ be the nonnegative loss and $Y^L$ the insurer’s payment per loss. Assume a deductible $d\geq 0$, an insurer coinsurance share $c$ with $0<c\leq 1$, and a maximum insurer payment $u>0$. The contract is defined so that the deductible is subtracted first, coinsurance is applied to the remaining eligible loss, and $u$ caps the insurer’s payment after coinsurance.

Definition: Under a deductible $d$, insurer coinsurance share $c$, and maximum insurer payment $u$, the payment per loss is $$ Y^L=\min\{c\max(X-d,0),u\}. $$ The deductible is applied before coinsurance. The cap $u$ is a maximum dollar payment by the insurer, not a cap on the loss eligible for coinsurance.

The expression $\max(X-d,0)$ is the loss remaining after the deductible, with negative amounts replaced by zero. Multiplying by $c$ gives the uncapped insurer payment. Taking the minimum of that amount and $u$ imposes the maximum payment. The formula describes the contract; to identify which values of $Y^L$ can actually occur, we must also know the support of $X$.

Deriving the Piecewise Function and Its Breakpoints

For losses no greater than $d$, the amount remaining after the deductible is zero. For losses above $d$, the uncapped payment is $c(X-d)$. The cap starts to apply when this amount reaches $u$. Solving for the corresponding loss gives the cap threshold

c(X-d)=u \quad\Longrightarrow\quad X=d+\frac{u}{c}.

Write $b=d+u/c$. Losses from $d$ up to $b$ produce payments that increase linearly; losses at or above $b$ produce the capped payment $u$. Therefore the payment function is

Y^L= \begin{cases} 0, & X\leq d,\\ c(X-d), & d<X<b,\\ u, & X\geq b, \end{cases} \qquad b=d+\frac{u}{c}.

At $X=d$, the middle expression would also give zero. At $X=b$, it would give $c(b-d)=u$. The chosen interval endpoints assign each boundary to one piece without changing the payment. They also make the regions easy to interpret: no payment, partial payment, and capped payment.

Apply the deductible.
Find the eligible amount $\max(X-d,0)$. Losses at or below $d$ produce no insurer payment.

Apply coinsurance.
Multiply eligible loss by the insurer’s share $c$. Before the cap, the payment is $c(X-d)$ when $X>d$.

Apply the payment cap.
Set the uncapped payment equal to $u$ and solve for the loss threshold $b=d+u/c$. At and above that threshold, payment is $u$.

Check the loss support.
Compare the loss values that can occur with the three regions. A policy region may exist algebraically but be unreachable under the specified loss model.

This derivation also explains why the threshold is not generally $d+u$. The amount added to the deductible must be large enough that the insurer’s fraction of it equals the cap. If $c<1$, that amount is $u/c$, which exceeds $u$.

Worked Example: Finding the Three Payment Regions

Hypothetical policy. A policy has a deductible of $250, the insurer pays 75% of eligible loss, and the maximum insurer payment is $1,500 per loss. Find the piecewise payment function and calculate the insurer’s payments for losses of $200, $250, $1,050, $2,250, and $3,100.

Method. Identify $d=250$, $c=0.75$, and $u=1{,}500$. First find the loss at which the uncapped payment reaches the cap:

b=d+\frac{u}{c} =250+\frac{1{,}500}{0.75} =250+2{,}000 =2{,}250.

Thus the payment function is zero for $X\leq250$, equals $0.75(X-250)$ for $250<X<2{,}250$, and equals $1,500 for $X\geq2{,}250$.

For a $200 loss, $200\leq250$, so the payment is $0. For a $250 loss, the loss is exactly the deductible, so the payment is also $0. For a $1,050 loss, the eligible amount is $1{,}050-250=800$, and the payment is $0.75(800)=\$600$. For a $2,250 loss, the uncapped payment is $0.75(2{,}250-250)=0.75(2{,}000)=\$1{,}500$, exactly the cap. For a $3,100 loss, the uncapped amount is $0.75(3{,}100-250)=0.75(2{,}850)=\$2{,}137.50$, so the insurer pays the capped amount of $1,500.

Check. At the threshold, $c(b-d)=0.75(2{,}000)=\$1{,}500=u$. The function is consistent at both boundaries: payment is zero at the deductible and reaches the cap at $2,250. The common mistake avoided here is using $d+u=\$1{,}750$ as the cap threshold; at that loss the insurer’s uncapped payment would be only $0.75(1{,}500)=\$1{,}125$.

Loss Regions and Payment Support

A payment function divides the possible losses into regions, but those regions are not themselves the support of the payment. The support of a random variable is the set of values it can take (or, for a continuous variable, the range that has positive probability in every neighborhood). To find the payment support, map the actual support of $X$ through the payment function.

If $X$ can take every nonnegative value, the function maps losses at or below $d$ to zero, losses between $d$ and $b$ to values between zero and $u$, and losses at or above $b$ to $u$. If instead $X$ has a finite or restricted support, some or all of those payment values may be unattainable. For instance, if every possible loss is below $d$, the only possible payment is zero, even though the policy’s linear and capped regions are defined.

For a continuous loss with positive density across the deductible and cap threshold, the payment can take values throughout $[0,u]$. The payment may also have probability concentrated at zero or at $u$: losses at or below $d$ all map to zero, and losses at or above $b$ all map to $u$. Those concentrations can occur even if $X$ itself is continuous. For a discrete loss, the payment support is found by evaluating the function at each possible loss; it need not fill the interval between zero and $u$.

Worked Example: Mapping a Discrete Loss Support

Hypothetical policy. A device-repair policy has a $400 deductible, a 60% insurer share of eligible loss, and a maximum insurer payment of $660. The loss $X$ has support $\{0,250,400,700,1{,}500,1{,}800,2{,}400\}$. Find the cap threshold, identify the losses in each payment region, and determine the support of $Y^L$.

The threshold is

b=400+\frac{660}{0.60}=400+1{,}100=1{,}500.

The losses 0, 250, and 400 are at or below the deductible, so each produces payment zero. The loss 700 is in the linear region and produces $0.60(700-400)=0.60(300)=\$180$. The loss 1,500 is exactly at the cap threshold, so its payment is $0.60(1{,}500-400)=0.60(1{,}100)=\$660$. Losses 1,800 and 2,400 exceed the threshold and each produce the capped payment of $660.

The losses in the zero-payment region are $\{0,250,400\}$, the loss in the linear region is $\{700\}$, and the losses in the capped region are $\{1{,}500,1{,}800,2{,}400\}$. Therefore the payment support is $\{0,180,660\}$, not every value from zero to $660. This follows from mapping the specified discrete support, rather than treating the payment function’s linear segment as proof that every intermediate payment is possible.

Check. The payment at the threshold is exactly the cap, $0.60(1{,}100)=\$660$. Each listed loss has been assigned to the correct region, and every resulting payment belongs to the stated support.

Read Carefully: What Does the Limit Cap?

Policy wording must specify whether the limit caps the insurer’s payment or the eligible loss before coinsurance. These are different contracts. In the model used above, $u$ is the maximum insurer payment, so the threshold is $d+u/c$. If a contract instead caps the eligible loss at $u$ before applying coinsurance, the insurer’s maximum payment is $cu$, and the threshold is $d+u$.

Worked Example: A Limit on Eligible Loss Before Coinsurance

Hypothetical policy. A policy has a $500 deductible, pays 80% of eligible loss, and limits the eligible loss to $1,000 before coinsurance is applied. Find the payment function, its maximum, and the payment for losses of $1,300 and $1,700.

Method. Here the $1,000 limit applies before coinsurance. The amount subject to the insurer’s share is $\min(\max(X-500,0),1{,}000)$, so

Y^L=0.80\min\{\max(X-500,0),1{,}000\} = \begin{cases} 0, & X\leq500,\\ 0.80(X-500), & 500<X<1{,}500,\\ 800, & X\geq1{,}500. \end{cases}

The threshold is $500+1{,}000=\$1{,}500$, and the maximum payment is $0.80(1{,}000)=\$800$. For a $1,300 loss, eligible loss is $1{,}300-500=\$800$, below the limit, so payment is $0.80(800)=\$640$. For a $1,700 loss, eligible loss before the limit is $1,200, but only $1,000 is subject to coinsurance, so payment is $0.80(1{,}000)=\$800$.

Check. The payment reaches its $800 maximum at a $1,500 loss and remains $800 for the $1,700 loss. Treating the $1,000 limit as a maximum insurer payment would instead produce a maximum payment of $1,000 and a different threshold; that interpretation would contradict the stated wording.

Common Exam Traps

A reliable solution separates the contract’s operations and checks their order. For the standard payment cap defined here, the correct expression is $\min\{c\max(X-d,0),u\}$. The following mistakes change the contract rather than merely changing notation.

Subtracting the deductible after coinsurance. For $d=\$300$, $c=0.80$, and $X=\$1{,}200$, the correct payment before any cap is $0.80(1{,}200-300)=\$720$. The incorrect calculation $0.80(1{,}200)-300=\$660$ applies the deductible in the wrong order.
Using the wrong threshold. If $u$ caps insurer payment, solve $c(X-d)=u$, giving $d+u/c$, not $d+u$. The latter is appropriate only when $u$ caps eligible loss before coinsurance.
Assuming the payment support is continuous. A continuous range of payments is not guaranteed when $X$ is discrete or has gaps in its support. Map the actual possible losses through the function.
Misclassifying the boundary. At $X=d$, payment is zero; at $X=b$, payment is the cap. Either neighboring piece can include its boundary if the formula agrees there, but the boundary value must be evaluated consistently.

Key takeaway: State what the limit caps, write the payment function in the same order as the contract, and solve for the loss threshold from the uncapped payment. Then map the actual loss support through the piecewise function to find the payment support.

Check Your Understanding

For each hypothetical policy, identify the relevant payment regions and use the stated loss support when determining possible payment values.

A policy has deductible $d=\$180$, insurer share $c=0.50$, and maximum insurer payment $u=\$720$. Find the cap threshold and the payments for losses of $100, $980, $1,620, and $2,000. (Answer: threshold $1,620; payments $0, $400, $720, and $720.)
A contract has deductible $250, insurer share 80%, and a maximum insurer payment of $1,000. Loss support is $\{0,250,500,1{,}250,1{,}500,2{,}200\}$. Find the threshold and payment support. (Answer: threshold $1,500; payment support $\{0,200,800,1{,}000\}$.)
A policy has a $100 deductible, pays 75% of eligible loss, and caps eligible loss at $600 before coinsurance. Find the loss threshold, maximum insurer payment, and payments for losses of $650 and $900. (Answer: threshold $700; maximum payment $450; payments $412.50 and $450.)
A contract has deductible $400, insurer share 60%, and a maximum insurer payment of $900. Calculate the threshold and the payment for a $1,600 loss. (Answer: threshold $1,900; payment $720.)
A loss has support $[0,1{,}000]$, and the policy has deductible $200, insurer share 50%, and maximum insurer payment $600. Determine whether the cap can bind and find the payment support. (Answer: the cap cannot bind because the threshold is $1,400; payment support is $[0,400]$.)

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