How a Policy Limit Caps a Payment
A policy limit places a maximum on what the insurer pays for a covered loss. If the covered loss is smaller than the limit, the insurer can pay the full loss. If it is larger, the insurer pays no more than the limit and the insured retains the excess. Correctly identifying which side of the limit a loss falls on is the central step in these calculations.
This tutorial considers a single covered loss with no deductible and no coinsurance. The limit applies separately to each loss. It does not describe a policy with an annual aggregate cap, a deductible, or a payment percentage; those contract terms change the payment rule.
The limit is a cap on the insurer’s payment, not a cap on the size of the loss. Thus, if \(X>u\), the loss remains \(X\); only the insurer’s liability is capped at \(u\). The retained amount \(R\) is the part of the loss above the limit, not the insurer’s payment.
Payment Rule in Cases
The minimum formula gives the payment in one expression. A piecewise form makes the boundary cases explicit:
For a loss exactly equal to the limit, \(X=u\), both expressions give \(Y=u=X\). The payment does not jump at the limit. The retained amount has the corresponding form:
At \(X=u\), the insured retains zero: the full loss is paid, but there is no room for an insurer payment above the cap. Above \(u\), every additional dollar of loss is retained by the insured.
Why the Minimum Formula Holds
Under the stated contract, the insurer’s payment cannot exceed the covered loss: paying more than \(X\) would over-indemnify the loss. It also cannot exceed the policy limit \(u\). The payment is the full loss whenever that amount is within the cap; otherwise it is the cap itself. These two requirements give the smaller of \(X\) and \(u\).
The retained-loss formula follows by subtracting the insurer’s payment from the total loss. If \(X\leq u\), then \(R=X-X=0\). If \(X>u\), then \(R=X-u\). Combining those cases gives \(R=\max(X-u,0)\).
Confirm that \(u\) is a per-loss limit and that the problem specifies no deductible, coinsurance, or other payment adjustment.
If \(X\leq u\), the insurer pays \(X\). If \(X>u\), the insurer pays \(u\).
Calculate \(R=X-Y\). Verify that \(Y+R=X\), \(0\leq Y\leq u\), and \(R\geq0\).
Worked Applications
In each example, amounts are hypothetical, and the policy limit applies to each covered loss separately. The method is to cap each insurer payment at \(u\), then find retained loss by subtracting the payment from the original loss.
Worked Example: Claims Below, At, and Above the Limit
A hypothetical policy has a per-loss limit of $18,500 and no deductible. Three separate covered claims are $12,740, $18,500, and $27,260. Find the insurer payment and retained loss for each claim, then find the totals.
Method. Apply \(Y=\min(X,18{,}500)\) to each claim separately, then calculate \(R=X-Y\).
For the first claim, \(X=$12{,}740<$18{,}500\). Therefore \(Y=$12{,}740\), and \(R=$12{,}740-$12{,}740=$0\).
For the second claim, \(X=$18{,}500=u\). Therefore \(Y=$18{,}500\), and \(R=$18{,}500-$18{,}500=$0\). Equality with the limit does not create an excess.
For the third claim, \(X=$27{,}260>$18{,}500\). Therefore \(Y=$18{,}500\), and \(R=$27{,}260-$18{,}500=$8{,}760\).
The total insurer payment is \($12{,}740+$18{,}500+$18{,}500=$49{,}740\). The total retained loss is \($0+$0+$8{,}760=$8{,}760\). The total covered loss is \($12{,}740+$18{,}500+$27{,}260=$58{,}500\), and the check is \($49{,}740+$8{,}760=$58{,}500\).
Common mistake avoided: Treating the claim exactly equal to the limit as an excess claim. At \(X=u\), the insurer pays the full loss and the insured retains zero.
Worked Example: Applying a Per-Loss Limit to Several Claims
A hypothetical repair policy pays up to $25,000 per covered claim, with no deductible. During a stated period, three separate covered losses are $8,400, $31,600, and $47,900. Calculate total insurer payments, total retained losses, and verify the totals.
Method. Since the limit is per loss, apply \(Y_j=\min(X_j,25{,}000)\) to each claim \(j\), rather than capping the sum of all losses.
For the $8,400 loss, the payment is \(\min($8{,}400,$25{,}000)=$8{,}400\), so the retained amount is \($8{,}400-$8{,}400=$0\).
For the $31,600 loss, the payment is \(\min($31{,}600,$25{,}000)=$25{,}000\), so the retained amount is \($31{,}600-$25{,}000=$6{,}600\).
For the $47,900 loss, the payment is \(\min($47{,}900,$25{,}000)=$25{,}000\), so the retained amount is \($47{,}900-$25{,}000=$22{,}900\).
Total insurer payments are \($8{,}400+$25{,}000+$25{,}000=$58{,}400\). Total retained losses are \($0+$6{,}600+$22{,}900=$29{,}500\). Total losses are \($8{,}400+$31{,}600+$47{,}900=$87{,}900\), and \($58{,}400+$29{,}500=$87{,}900\), as required.
Common mistake avoided: Applying the $25,000 cap once to the combined $87,900 total would give $25,000, but that would describe an aggregate cap, not the per-loss limit stated here.
Worked Example: Inferring a Limit From Payments
A hypothetical policy has a single per-loss limit \(u\) and no deductible. A covered loss of $41,200 results in an insurer payment of $32,000. A separate covered loss of $26,000 is paid in full. Determine \(u\), calculate the retained amount on each claim, and find the fraction of the combined loss paid by the insurer.
Method. Because the first loss exceeds the payment, the payment rule \(Y=\min(X,u)\) implies that the limit is $32,000. The second loss provides a consistency check: if the limit is $32,000, a $26,000 loss should be paid in full.
For the first claim, \(X=$41{,}200\) and \(Y=$32{,}000\). Thus \(R=$41{,}200-$32{,}000=$9{,}200\). The payment equals the limit, as expected for a loss above it.
For the second claim, \(X=$26{,}000<$32{,}000\). The payment is \(Y=$26{,}000\), and the retained amount is \(R=$26{,}000-$26{,}000=$0\).
Combined losses are \($41{,}200+$26{,}000=$67{,}200\), and combined insurer payments are \($32{,}000+$26{,}000=$58{,}000\). The fraction paid is \($58{,}000/$67{,}200=0.863095\ldots\), or \(86.31\%\) to two decimal places. The combined retained amount is \($9{,}200\), and \($58{,}000+$9{,}200=$67{,}200\).
Common mistake avoided: A payment equal to the limit does not reveal that the loss equals the limit. The first payment tells us the loss is at least $32,000; the stated $41,200 loss confirms it is above the limit.
Reading Payments Carefully
The payment rule also clarifies what can be inferred from a payment when the loss amount is not separately known. If the observed insurer payment is below \(u\), then the loss must equal that payment under this contract. If the payment equals \(u\), the loss may equal \(u\) or may be larger. A capped payment therefore does not reveal the full amount of a loss above the limit.
Common Exam Trap
A frequent error is to use \(\max(X-u,0)\) as the insurer’s payment. That expression is the insured’s retained excess. For example, with \(X=$27{,}260\) and \(u=$18{,}500\), the incorrect calculation gives \(\max($27{,}260-$18{,}500,0)=$8{,}760\) as the insurer payment. The correct insurer payment is \(\min($27{,}260,$18{,}500)=$18{,}500\), and $8,760 is retained by the insured. The check is \($18{,}500+$8{,}760=$27{,}260\).
Another wording trap is the distinction between “per loss” and “in total.” A per-loss limit is applied claim by claim. A limit applying to an entire period or a group of losses is a different contract condition and must not be substituted without evidence in the problem statement.
Check Your Understanding
For each problem, apply the stated per-loss limit and distinguish the insurer’s payment from the insured’s retained amount.
- A hypothetical policy has a $14,000 limit and no deductible. A covered loss is $9,350. Find the payment and retained amount. (Answer: payment $9,350; retained amount $0)
- A covered loss is exactly $22,500 under a no-deductible policy with a $22,500 per-loss limit. Find the payment and retained amount. (Answer: payment $22,500; retained amount $0)
- A hypothetical equipment policy has a $36,000 per-loss limit and no deductible. For a covered loss of $52,750, calculate the insurer payment and insured retention, then check their sum. (Answer: payment $36,000; retention $16,750; sum $52,750)
- A policy with a $19,000 per-loss limit covers two separate losses of $15,500 and $28,400, with no deductible. Find the total insurer payment and total retained loss. (Answer: total payment $34,500; total retained loss $9,400)
- A no-deductible policy pays $24,000 on a covered loss known to exceed the limit. What is the limit, and what can be concluded about the loss amount from the payment alone? (Answer: limit $24,000; the loss is at least $24,000, but the payment alone does not determine how much it exceeds the limit)