Boundary Conventions for Deductibles and Limits

Why Boundary Conventions Matter

A deductible and a limit divide ground-up losses into regions with different insurance payments. The exact boundary wording matters whenever the loss distribution assigns positive probability to a threshold. For example, a loss exactly equal to the deductible produces no payment under an ordinary deductible, while a loss exactly at the amount needed to exhaust the limit produces the full capped payment.

For a continuous loss distribution, the probability of an exact point is zero, so changing an interval endpoint from strict to inclusive usually does not change its probability. For a discrete or mixed distribution, an endpoint may carry positive probability. Assigning that mass to the wrong payment category can produce incorrect probabilities, even if the interval calculations otherwise look reasonable.

Definition: Let $X$ be a nonnegative ground-up loss, let $d\ge 0$ be an ordinary deductible, and let $u>0$ be a limit on the insurer’s payment per loss. The payment is $$ Y^L=\min\bigl((X-d)_+,u\bigr), $$ where $(z)_+=\max(z,0)$. Thus $Y^L=0$ when $X\le d$, $Y^L=X-d$ when $d<X<d+u$, and $Y^L=u$ when $X\ge d+u$.

In this tutorial, $u$ limits the payment after applying the deductible. The ground-up loss that just exhausts this limit is therefore $d+u$, not $u$. A different contract that places a limit on the covered ground-up loss would have a different payment transformation; the thresholds must always be read from the specified contract.

Translate Payment Categories Into Loss Events

The payment formula gives three disjoint events. A loss equal to $d$ belongs to the zero-payment category: after subtracting the deductible, the amount payable is still zero. A loss equal to $d+u$ belongs to the capped category: the payment has reached $u$. Only losses strictly between those thresholds produce a partial payment.

Payment category	Ground-up loss event	Payment
Zero payment	$X\le d$	$Y^L=0$
Partial payment	$d<X<d+u$	$0<Y^L<u$
Capped payment	$X\ge d+u$	$Y^L=u$

The inequalities in the table follow directly from the payment formula, not from a convention that can be chosen separately for each calculation. In particular, writing $X<d$ for zero payment would leave the loss $X=d$ unclassified, while writing $X>d+u$ for a capped payment would leave the loss $X=d+u$ unclassified.

Theorem: With $F_X(x)=P(X\le x)$, define the left limit $F_X(c-)=P(X<c)$. For $c=d+u$, the payment-category probabilities are $$ P(Y^L=0)=F_X(d),\qquad P(0<Y^L<u)=F_X(c-)-F_X(d),\qquad P(Y^L=u)=1-F_X(c-). $$ The left limit is needed at the cap threshold because the cap event includes $X=c$.

Why the CDF Left Limit Is Needed

The CDF is inclusive: $F_X(c)$ counts both losses below $c$ and any probability mass exactly at $c$. But the partial-payment event stops strictly before $c$, so its probability is $P(X<c)-P(X\le d)$, or $F_X(c-)-F_X(d)$. The cap-payment event includes the mass at $c$, so it is $1-P(X<c)=1-F_X(c-)$.

These formulas also provide a useful check. Since $c=d+u>d$, the three categories partition all possible losses:

F_X(d)+\bigl[F_X(c-)-F_X(d)\bigr]+\bigl[1-F_X(c-)\bigr]=1.

If a distribution is continuous at both thresholds, then there is no point mass at either one. In that case the corresponding left limits equal the CDF values, and using strict or inclusive endpoints does not change the probabilities. That simplification is not valid merely because a distribution has a density over part of its range: a mixed distribution can have a continuous component and still have atoms at the thresholds.

Identify the contract thresholds.
For an ordinary deductible $d$ and payment limit $u$, use $d$ and $c=d+u$.

Assign equality correctly.
Put $X=d$ in the zero-payment event and $X=c$ in the capped-payment event.

Calculate each event from the distribution.
Use $F_X(d)$, $F_X(c-)-F_X(d)$, and $1-F_X(c-)$, preserving any atoms at the boundaries.

Check the partition.
The three probabilities must be nonnegative and sum to 1.

Worked Examples

Worked Example: A Discrete Loss With Atoms at Both Boundaries

A hypothetical insurer covers a discrete loss $X$. The loss is $\$800$, $\$1{,}000$, $\$1{,}800$, $\$2{,}500$, or $\$3{,}000$, with respective probabilities $0.10$, $0.20$, $0.25$, $0.15$, and $0.30$. The policy has an ordinary deductible $d=\$1{,}000$ and a payment limit $u=\$1{,}500$. Find the probabilities of zero, partial, and capped payments.

Method. Translate each payment category into a ground-up loss interval, then add the probabilities of the loss values in that interval. The boundary values must be assigned using the inclusive events in the payment definition.

The loss that exhausts the limit is $c=d+u=\$1{,}000+\$1{,}500=\$2{,}500$.

Zero payment: $X\le \$1{,}000$. The possible values are $\$800$ and $\$1{,}000$, so

$$P(Y^L=0)=0.10+0.20=0.30.$$

Partial payment: $\$1{,}000<X<\$2{,}500$. Only the value $\$1{,}800$ is in this interval, so

P(0<Y^L<\$1{,}500)=0.25.

Capped payment: $X\ge \$2{,}500$. Both $\$2{,}500$ and $\$3{,}000$ qualify, so

P(Y^L=\$1{,}500)=0.15+0.30=0.45.

The probabilities sum to $0.30+0.25+0.45=1.00$. The loss exactly equal to the deductible is included in zero payment, and the loss exactly equal to $\$2{,}500$ is included in capped payment.

Common mistake avoided: Excluding $X=\$2{,}500$ by using $X>\$2{,}500$ for the cap event would give $0.30$, incorrectly omitting the probability $0.15$ at the boundary.

Worked Example: A Mixed Loss With Mass at the Deductible and Cap Threshold

A hypothetical portfolio has a loss distribution with probability $0.10$ at $\$1{,}000$, probability $0.20$ at $\$2{,}500$, and probability $0.70$ distributed uniformly on $[\$0,\$3{,}500]$. A contract has an ordinary deductible $d=\$1{,}000$ and payment limit $u=\$1{,}500$. Find the probabilities of zero payment, capped payment, and partial payment.

Method. Separate the two point masses from the uniform component. The cap threshold is $c=d+u=\$2{,}500$. The uniform component assigns zero probability to any single point, but the specified atoms at both thresholds must be counted.

For zero payment, the event is $X\le \$1{,}000$. It includes the atom at $\$1{,}000$ and the uniform component from $\$0$ through $\$1{,}000$. The uniform interval has length $\$1{,}000$ out of a total length of $\$3{,}500$, so

P(Y^L=0)=0.10+0.70\left(\frac{1{,}000}{3{,}500}\right)=0.10+0.20=0.30.

For capped payment, the event is $X\ge \$2{,}500$. It includes the atom at $\$2{,}500$ and the uniform component from $\$2{,}500$ to $\$3{,}500$:

P(Y^L=\$1{,}500)=0.20+0.70\left(\frac{1{,}000}{3{,}500}\right)=0.20+0.20=0.40.

Partial payment occurs only when $\$1{,}000<X<\$2{,}500$. Neither point mass lies strictly inside this interval. The uniform contribution is

P(0<Y^L<\$1{,}500)=0.70\left(\frac{2{,}500-1{,}000}{3{,}500}\right)=0.70\left(\frac{1{,}500}{3{,}500}\right)=0.30.

The three results are $0.3000$, $0.4000$, and $0.3000$, respectively, and sum to $1.0000$. The atom at the deductible belongs to zero payment; the atom at the cap threshold belongs to capped payment.

Common mistake avoided: Counting only the uniform component below the deductible gives $0.20$, not the full zero-payment probability. The atom $0.10$ at $\$1{,}000$ must also be included. Similarly, the atom at $\$2{,}500$ is part of the cap probability, not the partial-payment probability.

Worked Example: A Continuous Loss and the Endpoint Check

A hypothetical loss $X$ is uniformly distributed on $[\$0,\$4{,}000]$. A policy has deductible $d=\$800$ and payment limit $u=\$1{,}200$. Find the three payment-category probabilities, and explain whether endpoint inclusivity affects the numerical results.

Method. The loss is continuous and uniform, so the probability of an interval is its length divided by $\$4{,}000$, and any exact point has probability zero. The cap threshold is $c=\$800+\$1{,}200=\$2{,}000$.

The zero-payment interval is $[0,\$800]$, with length $\$800$:

P(Y^L=0)=\frac{800}{4{,}000}=0.20.

The partial-payment interval is $(\$800,\$2{,}000)$, with length $\$1{,}200$:

P(0<Y^L<\$1{,}200)=\frac{2{,}000-800}{4{,}000}=\frac{1{,}200}{4{,}000}=0.30.

The capped-payment interval is $[\$2{,}000,\$4{,}000]$, with length $\$2{,}000$:

P(Y^L=\$1{,}200)=\frac{4{,}000-2{,}000}{4{,}000}=0.50.

The probabilities sum to $0.20+0.30+0.50=1.00$. Although the exact thresholds are assigned to particular categories by the contract, including or excluding either endpoint would not change these probabilities because a continuous uniform distribution assigns probability zero to a single point.

Common mistake avoided: Treating this continuous-distribution simplification as a general rule. It works here because there are no atoms; it cannot be used for a discrete or mixed loss without checking the point masses.

Common Boundary Errors

Boundary mistakes often arise from switching between a statement about a loss and a statement about a payment. For example, “the loss exceeds the deductible” means $X>d$, but the zero-payment event under an ordinary deductible is $X\le d$. The equality case matters because $X=d$ produces no payment.

Using the wrong cap threshold. When $u$ limits the insurer’s payment after the deductible, the ground-up cap threshold is $d+u$, not $u$.
Omitting the deductible atom. The zero-payment event includes $X=d$, so its probability is $F_X(d)$, not $F_X(d-)$.
Omitting the cap atom. The capped-payment event includes $X=d+u$, so it uses $1-F_X((d+u)-)$, not $1-F_X(d+u)$.
Putting endpoint mass in the partial category. Partial payments require $d<X<d+u$; neither boundary belongs in that interval.
Forgetting the probability check. Zero, partial, and capped payment events are exhaustive and disjoint, so their probabilities must add to 1.

A reliable calculation begins with the contract’s payment function, translates each payment event into an event for $X$, and only then applies the distribution. This is especially important for mixed distributions, where a continuous density can make endpoint probabilities seem negligible even though separate atoms at those endpoints are not.

Key takeaway: For ordinary deductible $d$ and insurer-payment limit $u$, the correct loss regions are $X\le d$ for zero payment, $d<X<d+u$ for partial payment, and $X\ge d+u$ for capped payment. Inclusive endpoints determine where any boundary atoms belong.

Check Your Understanding

For each problem, identify the boundary events before calculating. Each loss distribution and policy below is hypothetical.

A loss takes values $\$500,\$900,\$1{,}000,\$1{,}800,\$2{,}500$ with probabilities $0.15,0.20,0.10,0.25,0.30$, respectively. The ordinary deductible is $\$1{,}000$, and the payment limit is $\$1{,}500$. Find the zero, partial, and capped payment probabilities. (Answer: $0.45,\ 0.25,\ 0.30$, respectively.)
A loss is uniform on $[\$0,\$4{,}000]$. The deductible is $\$800$, and the payment limit is $\$1{,}200$. Find the probability of a positive payment and the probability of a capped payment. (Answer: $0.80$ and $0.50$, respectively.)
A mixed loss has probability $0.15$ at $\$600$, probability $0.10$ at $\$1{,}000$, probability $0.25$ at $\$2{,}200$, and remaining probability $0.50$ uniformly distributed on $[\$0,\$3{,}000]$. With deductible $d=\$1{,}000$ and payment limit $u=\$1{,}200$, find the zero, partial, and capped payment probabilities. (Answer: $0.4167,\ 0.2000,\ 0.3833$, respectively.)
For any loss distribution with $c=d+u$, express the three payment-category probabilities using $F_X$ and its left limit. (Answer: $P(Y^L=0)=F_X(d)$, $P(0<Y^L<u)=F_X(c-)-F_X(d)$, and $P(Y^L=u)=1-F_X(c-)$.)
A loss distribution has $P(X<\$2{,}000)=0.72$, $P(X=\$2{,}000)=0.08$, and $P(X>\$2{,}000)=0.20$. If $\$2{,}000=d+u$, find the capped-payment probability. (Answer: $0.28$.)

Payment category	Ground-up loss event	Payment
Zero payment	\(X\le d\)	\(Y^L=0\)
Partial payment	\(d<X<d+u\)	\(0<Y^L<u\)
Capped payment	\(X\ge d+u\)	\(Y^L=u\)

What You'll Learn