Calculating Payments Under A Per-Loss Limit

What a Per-Loss Limit Does

A policy can cover a loss while still restricting how much the insurer will pay for it. A per-loss limit sets the maximum payment the insurer makes for each covered loss. When a covered loss is below the limit, the insurer pays the loss amount; when it exceeds the limit, the insurer pays only up to the limit.

This tutorial considers a per-loss limit on its own. Assume the loss is covered, is measured before applying the limit, and has no deductible, coinsurance, or other payment adjustment. These assumptions isolate the effect of the cap. If a policy contains additional terms, its payment rule must account for those terms as well.

Definition: Let $X$ be the nonnegative covered amount of one loss, and let $u\geq 0$ be the maximum insurer payment per loss. The insurer payment per loss, denoted $Y^L$, is the smaller of the loss amount and the limit: $$ Y^L=\min(X,u). $$ The amount of the covered loss not paid by the insurer is $X-Y^L$. Under these assumptions, this retained amount is $\max(X-u,0)$.

The phrase per loss matters: the limit applies separately to each loss. It is not automatically a single cap shared by all losses under a policy. This tutorial assumes no separate policy-wide aggregate limit. Also distinguish the point at which the limit is reached from the point at which it reduces the payment: at $X=u$, the insurer pays exactly $u$, but no part of the loss is left unpaid because of the limit. The limit reduces the payment when $X>u$.

Writing and Interpreting the Payment Rule

The minimum formula is compact, but its piecewise form makes the threshold explicit. For losses at or below $u$, the insurer pays the full covered loss. For losses above $u$, the payment stays fixed at $u$.

Y^L= \begin{cases} X, & 0\leq X\leq u,\\ u, & X>u. \end{cases}

The payment never exceeds $u$, and it never exceeds $X$. The payment increases dollar for dollar with the loss until the loss reaches $u$; after that, the payment is constant. The part not paid by the insurer is zero up to the limit, then increases dollar for dollar with any further increase in the loss.

X-Y^L=\max(X-u,0)

This retained amount is a way to check the calculation, not a second insurer payment. In particular, for $X>u$, the insurer pays $u$ and the remaining $X-u$ is not paid under this limited coverage.

Why the Minimum Formula Holds

The formula follows directly from the two restrictions on payment. The insurer cannot pay more than the covered loss, so $Y^L\leq X$. The insurer also cannot pay more than the per-loss limit, so $Y^L\leq u$. A payment that exhausts the covered amount when $X\leq u$ is $X$; a payment that exhausts the limit when $X>u$ is $u$. Thus the payment is the smaller of $X$ and $u$.

Identify the covered loss.
Use $X$ for the covered amount to which the limit applies, not an amount from a different claim or a policy-wide total.

Compare loss with limit.
If $X\leq u$, the payment is $X$. If $X>u$, the payment is $u$.

Check the amounts.
Confirm that the insurer payment is no greater than either $X$ or $u$. If needed, subtract it from $X$ to find the unpaid portion.

Worked Applications

Worked Example: A Single Loss Above the Limit

A hypothetical equipment policy covers a repair loss of $26,400 and has a per-loss maximum payment of $19,000. Assume there is no deductible or other adjustment. Find the insurer payment and the amount of this covered loss not paid by the insurer. State whether the limit is reached and whether it reduces the payment.

Method. Compare the covered loss $X=\$26{,}400$ with the limit $u=\$19{,}000$. Since $X>u$, use the capped branch of the payment rule:

Y^L=\min(\$26{,}400,\$19{,}000)=\$19{,}000.

The amount not paid by the insurer is the loss minus the insurer payment:

X-Y^L=\$26{,}400-\$19{,}000=\$7{,}400.

The limit is reached, and it reduces the payment by $7,400 compared with paying the full covered loss. The check is $\$19{,}000+\$7{,}400=\$26{,}400$, which recovers the covered loss exactly. The common mistake to avoid is to report $7,400 as the insurer payment; it is the unpaid portion.

Worked Example: Comparing Losses at, Below, and Above the Limit

A hypothetical policy pays covered losses subject to a $24,000 limit per loss. Three separate covered losses are $16,800, $24,000, and $31,500. Find the insurer payment and unpaid portion for each, and identify which loss is reduced by the limit.

For the $16,800 loss, $X<u$, so the payment equals the loss:

Y^L=\min(\$16{,}800,\$24{,}000)=\$16{,}800, \qquad X-Y^L=\$0.

For the $24,000 loss, $X=u$, so the payment also equals the loss:

Y^L=\min(\$24{,}000,\$24{,}000)=\$24{,}000, \qquad X-Y^L=\$0.

For the $31,500 loss, $X>u$, so the payment is capped at $24,000:

Y^L=\min(\$31{,}500,\$24{,}000)=\$24{,}000, \qquad X-Y^L=\$31{,}500-\$24{,}000=\$7{,}500.

Only the $31,500 loss has its payment reduced. The $24,000 loss reaches the limit exactly, but its full amount is still paid. Across all three separate losses, the insurer pays $\$16{,}800+\$24{,}000+\$24{,}000=\$64{,}800$, and the total unpaid portion is $7,500. The check is that these amounts sum to the total covered losses, $\$16{,}800+\$24{,}000+\$31{,}500=\$72{,}300$.

Worked Example: Expected Payment Under a Discrete Loss Model

A hypothetical insured faces one covered loss during a specified period. Its amount $X$, in thousands of dollars, has this distribution: $X=8$ with probability 0.20, $X=20$ with probability 0.30, and $X=35$ with probability 0.50. The policy has a per-loss limit of $24,000. Find the probability that the limit reduces payment, the expected insurer payment, and the expected unpaid portion. Assume no other payment adjustment.

Method. Apply $Y^L=\min(X,24)$ separately to each possible loss amount. Since amounts are stated in thousands, the payment values below are also in thousands of dollars.

If $X=8$, the payment is $\min(8,24)=8$, and the unpaid portion is $8-8=0$. If $X=20$, the payment is $\min(20,24)=20$, and the unpaid portion is $20-20=0$. If $X=35$, the payment is $\min(35,24)=24$, and the unpaid portion is $35-24=11$.

The limit reduces payment only for $X=35$, so the probability of a reduction is 0.50. The expected insurer payment is:

$$ E[Y^L] =0.20(8)+0.30(20)+0.50(24) =1.6+6.0+12.0 =19.6. $$

Thus, the expected insurer payment is $19,600. The expected unpaid portion is:

$$ E[X-Y^L] =0.20(0)+0.30(0)+0.50(11) =5.5, $$

or $5,500. As a check, the expected covered loss is $0.20(8)+0.30(20)+0.50(35)=25.1$, or $25,100. The expected payment and expected unpaid portion sum to this amount: $\$19{,}600+\$5{,}500=\$25{,}100$. The common mistake is to apply the cap to the expected loss alone: $\min(E[X],24)=\min(25.1,24)=24$ is not the expected payment, which is $19,600. The cap must be applied to each possible loss before averaging.

Worked Example: Applying the Limit Separately to Two Losses

A hypothetical business has two distinct covered losses, $10,500 and $17,200. Each loss is subject to a $15,000 per-loss limit, with no other adjustment and no shared aggregate cap. Find the insurer payment for each loss and the total payment.

For the first loss:

Y^L_1=\min(\$10{,}500,\$15{,}000)=\$10{,}500.

For the second loss:

Y^L_2=\min(\$17{,}200,\$15{,}000)=\$15{,}000.

Because the limit applies per loss, calculate each payment separately and then add them:

Y^L_1+Y^L_2=\$10{,}500+\$15{,}000=\$25{,}500.

The first loss is paid in full, while $2,200 of the second loss is not paid under this limit. A common mistake is to treat $15,000 as a single maximum for the two losses combined. That would confuse a per-loss limit with a shared cap; under the stated terms, the correct total payment is $25,500.

Common Exam Traps

A per-loss limit is easy to misread when the question asks for both the payment and the amount left unpaid. Keep the two amounts separate and label them. Under the assumptions in this tutorial, the payment is $\min(X,u)$, whereas the unpaid portion is $\max(X-u,0)$. They are not interchangeable.

Subtracting the limit from every loss. For $X\leq u$, the insurer pays $X$, not $X-u$. The limit is a maximum payment, not an amount subtracted from the loss.
Calling the limit binding too early. At $X=u$, the limit is reached but does not reduce payment. A reduction occurs only when $X>u$.
Applying one cap to separate losses. A per-loss limit is applied to each covered loss. Do not replace it with one cap on their sum unless the policy states a shared limit.
Capping an average instead of each loss. For a random loss, find the payment at each possible loss amount and then calculate its expected value.
Ignoring the terms of the coverage. If a deductible, coinsurance percentage, or another limit is also specified, the standalone formula here is not the full payment rule. Interpret the stated contract terms before calculating.

Key takeaway: For one nonnegative covered loss $X$ subject only to a per-loss maximum payment $u$, the insurer pays $\min(X,u)$. The limit is reached at $X=u$ and reduces payment only for $X>u$. For separate losses, apply the rule to each loss individually.

Check Your Understanding

For each problem, use the stated standalone per-loss limit; assume no deductible or other payment adjustment unless the problem says otherwise.

A covered loss is $26,500 and the per-loss limit is $18,000. Find the insurer payment and the amount not paid by the insurer. (Answer: insurer payment $18,000; unpaid portion $8,500.)
A covered loss is exactly $32,000, equal to the per-loss limit. Find the insurer payment and state whether the limit reduces the payment. (Answer: insurer payment $32,000; the limit is reached but does not reduce payment.)
A loss $X$, in thousands of dollars, is 9 with probability 0.25, 22 with probability 0.45, and 40 with probability 0.30. The per-loss limit is $28,000. Find the probability the limit reduces payment, the expected payment, and the expected unpaid portion. (Answer: probability 0.3000; expected payment $20,550; expected unpaid portion $3,600.)
Two distinct covered losses are $11,000 and $19,000, each subject to a $14,000 per-loss limit. Find each payment and their total. (Answer: payments $11,000 and $14,000; total payment $25,000.)
A single covered loss is $X$, and the per-loss limit is $40,000. Give the payment rule for $0\leq X\leq \$40{,}000$ and for $X>\$40{,}000$, then calculate the payment for losses of $15,000, $40,000, and $57,000. (Answer: payment is $X$ when $0\leq X\leq \$40{,}000$ and $40,000 when $X>\$40{,}000$; the three payments are $15,000, $40,000, and $40,000.)

What You'll Learn