Why the Deductible’s Level Matters
When one event gives rise to several claims, the word “deductible” is not enough to determine the insurer’s payment. The policy must also specify the unit to which the deductible applies. A deductible applied to each claim is subtracted repeatedly; a deductible applied to each occurrence is subtracted once from the combined covered losses of that occurrence. Confusing these units can materially change the result.
The calculations in this tutorial isolate that distinction. Assume covered losses are nonnegative, the deductible is a fixed dollar amount, and there are no coinsurance terms, limits, exclusions, or aggregate caps affecting payment. These assumptions let us compare the deductible designs directly. In practice, the policy wording determines how claims are grouped into occurrences and which payment provisions also apply. This distinction is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Write the Payment Rule Before Adding Numbers
Let \(X_j\) be the covered loss amount for claim \(j\), and let \(d\) be the deductible. Under a per-claim deductible, the payment for claim \(j\) is the part of that claim above the deductible, but never less than zero. For \(n\) claims, total payment is the sum of those claim payments.
Under a per-occurrence deductible, first add the covered losses from the claims assigned to the occurrence. Then subtract the deductible once, again with a floor of zero. For a single occurrence containing \(n\) claims, the occurrence payment is:
The order of operations is the essential difference: per claim means apply the deductible before summing claim payments; per occurrence means sum the losses before applying the deductible. If there are multiple occurrences, calculate each occurrence’s payment separately and then add those payments.
Read the policy term carefully: does the deductible apply to each claim, or once to each occurrence?
For a per-occurrence deductible, identify which covered claims belong to each occurrence and total their losses.
Use the maximum of the relevant amount minus \(d\) and zero. Do not apply the deductible at both levels.
Check that the payment is nonnegative and does not exceed the covered loss total under the assumptions used here.
Worked Example: Two Designs for One Occurrence
A hypothetical water-main failure is treated as one occurrence and produces three covered claims of $900, $1,600, and $2,500. The deductible is $1,000. There are no other payment terms. Calculate the total payment under each deductible design.
Method. For the per-claim design, apply the deductible to each claim separately. For the per-occurrence design, combine all three losses and apply the deductible once. The common mistake to avoid is subtracting the deductible once from the total when the policy says “per claim.”
Per claim. The three payments are:
Adding them gives \(0+600+1{,}500=\$2{,}100\).
Per occurrence. The combined loss is \(900+1{,}600+2{,}500=\$5{,}000\). Apply the deductible once: \(\max(5{,}000-1{,}000,0)=\$4{,}000\).
Check. Both payments are nonnegative and below the total covered loss of $5,000. The per-occurrence payment exceeds the per-claim total by $1,900 because the deductible is subtracted once rather than three times.
Why the Per-Occurrence Payment Is Often Larger
For nonnegative claim amounts and the same deductible, the per-occurrence payment for a group of claims is at least as large as the sum of per-claim payments. This comparison assumes the same claims are covered and grouped together, and that no other terms change payment.
To see why, first consider the claims that individually exceed \(d\). If none do, the per-claim total is zero and cannot exceed the nonnegative per-occurrence payment. If \(k\) claims exceed \(d\), the per-claim total is the sum of those \(k\) losses less \(kd\). The total occurrence loss is at least the sum of those losses, while the occurrence deductible is subtracted only once. Since \(k\geq 1\), \(kd\geq d\), so subtracting \(kd\) from the qualifying claims cannot produce more than subtracting \(d\) from the total occurrence loss. The occurrence payment is therefore no smaller.
This comparison does not mean that every per-occurrence result must be larger in every real policy calculation. Different coverage terms, exclusions, or groupings can affect the comparison. It is the result for the simple deductible-only assumptions stated above.
Worked Example: The Same Claims, Reversed Conclusion From a Unit Error
A hypothetical equipment failure is one occurrence with four covered claims of $1,200, $1,200, $1,200, and $5,000. The deductible is $1,500. Calculate the payment under a per-occurrence deductible and under a per-claim deductible.
Method. Calculate both designs independently from the payment rules. The common mistake is to subtract $1,500 once from the combined loss and call that a per-claim result, or to subtract it from each claim and call that per-occurrence.
Per occurrence. Add the four claims:
Subtract the $1,500 deductible once: \(\max(8{,}600-1{,}500,0)=\$7{,}100\).
Per claim. The first three claims each fall below the deductible, so each produces no payment. The fourth produces \(\max(5{,}000-1{,}500,0)=\$3{,}500\). Thus, total payment is \(0+0+0+3{,}500=\$3{,}500\).
Check. Total covered loss is $8,600, and both calculated payments are no greater than that amount. The difference between the designs is \(7{,}100-3{,}500=\$3{,}600\).
When There Is More Than One Occurrence
A per-occurrence deductible is applied once to each occurrence, not once to the whole policy period and not once to every claim. Thus, when multiple occurrences are present, group claims by occurrence, calculate a payment for each group, and add the group payments. This step prevents a second common error: treating several separate occurrences as though they shared one deductible.
Worked Example: Applying One Deductible to Each Occurrence
A hypothetical storm causes three claims of $200, $600, and $1,000 in Occurrence A. A separate equipment incident causes claims of $700 and $200 in Occurrence B. The deductible is $400. Calculate total payment under a per-occurrence design and, for comparison, under a per-claim design. Assume all listed losses are covered.
Method. For the per-occurrence calculation, total losses within each event before applying the deductible. For the per-claim calculation, apply the deductible to all five claims separately. Do not combine separate occurrences before subtracting the per-occurrence deductible.
Per occurrence, Occurrence A. Its total loss is \(200+600+1{,}000=\$1{,}800\). Its payment is \(\max(1{,}800-400,0)=\$1{,}400\).
Per occurrence, Occurrence B. Its total loss is \(700+200=\$900\). Its payment is \(\max(900-400,0)=\$500\). Total per-occurrence payment is \(1{,}400+500=\$1{,}900\).
Per claim. The payments are \(\max(200-400,0)=0\), \(\max(600-400,0)=\$200\), \(\max(1{,}000-400,0)=\$600\), \(\max(700-400,0)=\$300\), and \(\max(200-400,0)=0\). Their sum is \(0+200+600+300+0=\$1{,}100\).
Check. The total loss is \(1{,}800+900=\$2{,}700\). The per-occurrence payment of $1,900 and per-claim payment of $1,100 are both nonnegative and below total loss. The per-occurrence calculation subtracts $400 twice, once for each occurrence.
Common Mistakes and a Reliable Check
- Applying the deductible at the wrong level. “Per claim” means calculate each claim’s payment separately. “Per occurrence” means combine losses within the occurrence first.
- Subtracting a per-occurrence deductible from every claim. This incorrectly converts the term into a per-claim deductible and can understate payment.
- Subtracting a per-claim deductible only once. This incorrectly treats all claims as one loss, and can overstate payment.
- Grouping separate events together. A per-occurrence deductible applies once to each defined occurrence, so claim grouping is part of the calculation.
- Allowing a negative payment. A loss below the deductible produces zero, not a negative insurer payment. The \(\max(\cdot,0)\) in each formula enforces this.
A useful final check is to compare the payment with the total covered loss. With only a deductible and nonnegative losses, payment cannot be negative or exceed that loss total. Also check the number of deductible applications: one for each claim in the per-claim design, or one for each occurrence in the per-occurrence design.
Check Your Understanding
For each problem, use the stated grouping and apply the deductible only at the specified level. All losses are covered, with no other payment terms.
- One occurrence produces claims of $500, $1,000, and $1,800. The deductible is $800. Find the per-occurrence payment and the per-claim total. (Answer: Per occurrence, $2,500; per claim, $1,200.)
- One occurrence produces claims of $400 and $900. The deductible is $500. Find the payment under each design. (Answer: Per occurrence, $800; per claim, $400.)
- Occurrence A has claims of $200, $600, and $1,000; Occurrence B has claims of $300 and $500. The deductible is $400. Find the total under each design. (Answer: Per occurrence, $1,800; per claim, $900.)
- One occurrence produces claims of $1,200, $1,200, $1,200, and $5,000. The deductible is $1,500. Find the per-occurrence payment and the per-claim total. (Answer: Per occurrence, $7,100; per claim, $3,500.)
- Occurrence A has claims of $250, $750, and $1,500; Occurrence B has claims of $900 and $1,100. The deductible is $1,000. Find the total under each design. (Answer: Per occurrence, $2,500; per claim, $600.)