Deriving Breakpoints In Combined Payment Functions

Why Breakpoints Matter

A deductible, a coinsurance percentage, and a payment limit each affect a claim differently. Below the deductible, the insurer pays nothing. Above it, the insurer pays a share of the loss in excess of the deductible. Once that payment reaches its limit, further increases in loss do not increase the insurer’s payment. The losses at which these changes occur are the breakpoints of the payment function.

Finding the breakpoints before substituting claim amounts is a reliable way to organize calculations. It also makes it possible to check whether a proposed formula behaves properly where its pieces meet. The methods here are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $X$ be a nonnegative loss, let $d\geq 0$ be a deductible, let $c$ be the insurer’s coinsurance share with $0<c\leq 1$, and let $u>0$ be a cap on the insurer’s payment. Under the convention used here, the insurer pays share $c$ of the loss above the deductible, subject to payment limit $u$. The payment per loss is $Y^L=\min\{c(X-d)_+,u\}$, where $(z)_+=\max(z,0)$.

The order of operations is important: first determine the loss above the deductible, then apply the insurer’s share $c$, and finally cap the resulting insurer payment at $u$. Thus, $u$ is a limit on the insurer’s payment, not a limit on the amount of loss considered before coinsurance. A policy using a different limit convention would have a different payment function and potentially different breakpoints.

Deriving the Two Breakpoints

The first breakpoint is the deductible $d$. If $0\leq x\leq d$, no loss remains above the deductible, so payment is zero. For $x>d$, payment before applying the limit is $c(x-d)$.

The second breakpoint occurs when the payment before the cap just reaches $u$. Solve $c(x-d)=u$ for $x$. Since $c>0$, division by $c$ is valid:

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

Write this second threshold as $b=d+u/c$. For losses below $b$, the uncapped payment is less than $u$; at $b$, it equals $u$; and for losses above $b$, it exceeds $u$, so the cap applies. Because $u>0$ and $c>0$, $b>d$, ensuring that the middle segment has positive width.

Theorem: For $x\geq 0$, the payment function is $Y^L(x)=0$ for $0\leq x\leq d$, $Y^L(x)=c(x-d)$ for $d<x<b$, and $Y^L(x)=u$ for $x\geq b$, where $b=d+u/c$. The insurer’s payment begins at $d$, increases with slope $c$ between $d$ and $b$, and remains at $u$ from $b$ onward.

The middle formula could also be included at either boundary: at $x=d$, it gives zero, and at $x=b$, it gives $u$. The half-open intervals in the theorem simply assign each boundary to one segment without changing the payment.

Why the Piecewise Formula Is Continuous

Continuity is a useful check, not an additional policy feature. At the deductible, the payment immediately below the threshold is zero. The middle expression approaches $c(d-d)=0$ as $x$ decreases toward $d$ from above. Thus, when $d>0$, both the value at $d$ and the left- and right-hand limits are zero.

When $d=0$, the loss domain is $x\geq 0$, so there are no allowed loss values to the left of the endpoint. In this case, continuity is checked from within the domain: $Y^L(0)=0$, and the right-hand limit is also zero. It would not be appropriate to state a left-hand limit at zero for a function defined only on nonnegative losses.

At the upper breakpoint $b$, the middle expression gives $c(b-d)=c(u/c)=u$. The capped segment also gives $u$. Since $b>d$, it is an interior point of the nonnegative loss domain, and both one-sided limits are defined and equal to the function value.

\begin{aligned} &d>0: &&\lim_{x\uparrow d}Y^L(x)=Y^L(d)=\lim_{x\downarrow d}Y^L(x)=0,\\ &d=0: &&Y^L(0)=\lim_{x\downarrow 0}Y^L(x)=0,\\ &b=d+\frac{u}{c}: &&\lim_{x\uparrow b}Y^L(x)=Y^L(b)=\lim_{x\downarrow b}Y^L(x)=u. \end{aligned}

These equalities follow directly from the adjacent pieces: below the deductible the function is zero, the uncapped formula meets zero at $d$, and the uncapped formula meets the cap at $b$. The slope changes at each breakpoint, but the payment itself does not jump.

Worked Applications

Worked Example: Finding Both Thresholds and Evaluating Claims

A hypothetical equipment policy has a deductible of $500, pays 80% of the loss above the deductible, and has a $6,000 limit on the insurer’s payment. Find both breakpoints and the payment for losses of $450, $2,000, $7,000, and $9,000. Check the values at the breakpoints.

Method. Use $d=500$, $c=0.80$, and $u=6{,}000$. The attachment breakpoint is $d=500$. The limit breakpoint is $b=d+u/c=500+6{,}000/0.80=500+7{,}500=8{,}000$.

Therefore, payment is zero through a loss of $500, is $0.80(x-500)$ for losses between $500 and $8,000, and is capped at $6,000 from $8,000 onward.

For a $450 loss, $450<500$, so $Y^L=0$.

For a $2,000 loss, the loss is in the middle segment: $Y^L=0.80(2{,}000-500)=0.80(1{,}500)=\$1{,}200$.

For a $7,000 loss, the loss is also in the middle segment: $Y^L=0.80(7{,}000-500)=0.80(6{,}500)=\$5{,}200$.

For a $9,000 loss, the uncapped amount would be $0.80(9{,}000-500)=0.80(8{,}500)=\$6{,}800$. That exceeds $6,000, so the payment is $Y^L=\$6{,}000$.

At $x=500$, the middle formula gives $0.80(500-500)=0$, matching the lower segment. At $x=8{,}000$, it gives $0.80(8{,}000-500)=0.80(7{,}500)=\$6{,}000$, matching the cap. The calculations avoid the common mistake of treating $6,000 as a limit on loss before applying the 80% share.

Worked Example: Finding the Loss That Reaches the Limit

A hypothetical policy has a $1,200 deductible, pays 75% of loss above the deductible, and limits the insurer’s payment to $9,000. Find the loss at which the limit first applies. Then calculate the insurer’s payment for losses of $800, $6,400, $9,200, and $13,200.

Method. Here $d=1{,}200$, $c=0.75$, and $u=9{,}000$. The deductible breakpoint is $1,200. The limit breakpoint is $b=1{,}200+9{,}000/0.75=1{,}200+12{,}000=\$13{,}200$.

A loss of $800 is below the deductible, so its payment is zero.

At a loss of $6,400, the amount above the deductible is $6{,}400-1{,}200=\$5{,}200$. The insurer pays $0.75(5{,}200)=\$3{,}900$.

At a loss of $9,200, the amount above the deductible is $9{,}200-1{,}200=\$8{,}000$. The insurer pays $0.75(8{,}000)=\$6{,}000$, which is below the $9,000 limit.

At a loss of $13,200, the amount above the deductible is $12,000. The uncapped payment is $0.75(12{,}000)=\$9{,}000$, exactly the limit. A loss of $13,200 is therefore the first loss at which the cap is reached; losses above it remain capped at $9,000.

The breakpoint check is $0.75(13{,}200-1{,}200)=0.75(12{,}000)=\$9{,}000$. A common mistake is to divide the limit by the deductible or to add the limit directly to the deductible without accounting for coinsurance. The required excess loss is $u/c$, not $u$.

Worked Example: Zero Deductible and a Target Payment

A hypothetical contract has no deductible, pays 60% of each loss, and caps the insurer’s payment at $2,400. Find the loss at which the cap is reached. Then calculate the payment for losses of $0, $2,500, $4,000, and $5,100. Explain how continuity is checked at the zero-loss endpoint.

Method. Set $d=0$, $c=0.60$, and $u=2{,}400$. The cap breakpoint is $b=0+2{,}400/0.60=4{,}000$. Thus, payment is $0.60x$ for $0\leq x<4{,}000$, and $2,400 for $x\geq4{,}000$.

At $x=0$, payment is $0.60(0)=\$0$. At $x=2{,}500$, payment is $0.60(2{,}500)=\$1{,}500$. At $x=4{,}000$, the uncapped calculation is $0.60(4{,}000)=\$2{,}400$, exactly the cap. At $x=5{,}100$, the uncapped amount is $0.60(5{,}100)=\$3{,}060$, so the actual payment is capped at $2,400.

Because losses are restricted to $x\geq0$, check the endpoint using the value and right-hand limit: $Y^L(0)=0$, and $\lim_{x\downarrow0}0.60x=0$. There is no left-hand limit within the stated domain. At the other breakpoint, the middle formula gives $0.60(4{,}000)=\$2{,}400$, equal to the capped value.

This example avoids the common mistake of asserting continuity by comparing values at negative and nonnegative losses when the function has only been defined for nonnegative losses.

A Reliable Breakpoint Procedure

For a new combined payment function, first make the contract convention explicit. In particular, identify whether the limit caps the insurer’s payment or some other quantity. Under the convention in this tutorial, the calculation can be organized as follows.

State the domain and terms.
Use $x\geq0$, deductible $d\geq0$, insurer share $0<c\leq1$, and payment limit $u>0$.

Find where payment begins.
Losses at or below $d$ produce no payment, so the lower breakpoint is $d$.

Set the uncapped payment equal to the cap.
Solve $c(x-d)=u$, giving the upper breakpoint $b=d+u/c$.

Write the segments and check the joins.
Use zero, then $c(x-d)$, then $u$; substitute each breakpoint into its adjacent expressions to confirm matching values.

The same calculation also clarifies the shape of the payment function. Before the deductible, each additional dollar of loss adds no insurer payment. Between the deductible and the cap, each additional dollar adds $c$ dollars of insurer payment. After the cap, additional loss adds no payment. This slope pattern is a quick reasonableness check on a graph or a table of calculated values.

Common Mistakes and Checks

Using $d+u$ as the cap breakpoint. The insurer’s payment grows at rate $c$, so it takes an excess loss of $u/c$ to generate payment $u$. The threshold is $d+u/c$.
Capping the wrong amount. For the convention here, calculate $c(x-d)$ first and then cap that payment at $u$. Do not cap $x-d$ at $u$ unless the contract defines the limit that way.
Applying coinsurance below the deductible. The insurer pays zero for $x\leq d$; coinsurance applies only to loss above $d$.
Assuming the payment jumps at a breakpoint. The formula changes, but the values match: the middle segment is zero at $d$ and equals $u$ at $b$.
Taking a left-hand limit at zero. When $d=0$, the loss function is defined only for $x\geq0$. Check the value at zero and the right-hand limit instead.

Key takeaway: Under a deductible $d$, insurer share $c$, and cap $u$ on insurer payment, the two thresholds are $d$ and $d+u/c$. The piecewise payments meet continuously at both thresholds; at the domain endpoint $d=0$, continuity is checked from the right.

Check Your Understanding

For each hypothetical policy, identify the relevant breakpoints and use the stated limit convention. Answers are included so you can check both the threshold and payment calculations.

A policy has $d=\$300$, $c=0.50$, and an insurer-payment limit $u=\$1{,}500$. Find the cap breakpoint and payment for a $2,300 loss. (Answer: Breakpoints $300 and $3,300; payment $1,000.)
A policy has $d=\$900$, $c=0.80$, and $u=\$4{,}800$. Find the loss at which the cap first applies and the payment for a $6,000 loss. (Answer: Cap breakpoint $6,900; payment $4,080.)
A policy has $d=0$, $c=0.40$, and $u=\$1{,}200$. Give the cap breakpoint and the payment at losses of $2,000 and $4,000. State the appropriate endpoint continuity check. (Answer: Cap breakpoint $3,000; payments $800 and $1,200; $Y^L(0)=\lim_{x\downarrow0}Y^L(x)=0$, with no left-hand limit on the stated domain.)
A policy has $d=\$1{,}500$, $c=0.60$, and $u=\$3{,}600$. A claim produces an insurer payment of $2,700. Find the loss amount, and verify that the claim is below the cap breakpoint. (Answer: Loss $6,000; cap breakpoint $7,500, so the payment is below the cap.)
A policy has $d=\$700$, $c=0.25$, and $u=\$2{,}000$. Find the cap breakpoint, then calculate payments at losses of $6,000 and $9,000. (Answer: Cap breakpoint $8,700; payments $1,325 and $2,000.)

What You'll Learn