Per-Loss Coverage Versus Per-Occurrence Coverage

Why the Unit of Coverage Matters

A contract may apply the same kind of coverage rule at different levels. A deductible might reduce the insurer’s payment for each individual claim, or it might reduce the payment once for the combined losses from one occurrence. A limit might cap payment on each claim or cap the total payment for the occurrence. The amounts of the claims alone do not determine the insurer’s payment: the contract’s unit of application matters too.

For the calculations here, assume that the policy identifies the claims being treated as one occurrence. The focus is not on deciding which claims belong together, but on what happens after that grouping is specified. All loss amounts are ground-up amounts, before deductibles or limits. The examples are hypothetical and deterministic: no interest rate, mortality assumption, or probability distribution is needed.

Definition: Under per-loss coverage, the stated rule is applied separately to each claim, and the resulting insurer payments are added. Under per-occurrence coverage, the claim amounts in the occurrence are added first, and the rule is applied once to that combined amount. A contract’s deductible and limit must each be read at their stated unit; they need not use the same unit.

Set Up the Two Calculations

Let $x_1,\ldots,x_n$ be the ground-up amounts of the $n$ claims specified as belonging to one occurrence. The occurrence total is

x_{\mathrm{occ}}=\sum_{j=1}^{n}x_j.

For a deductible $d$, the amount remaining after the deductible on a loss amount $x$ is $(x-d)^+=\max(x-d,0)$. For a limit $u$, this tutorial treats $u$ as a maximum insurer payment, applied after the deductible. Thus the payment under a deductible-and-limit rule on one amount $x$ is $\min((x-d)^+,u)$. This is a stated contract convention for these examples; a contract that defines its limit differently must be calculated according to its wording.

\text{Per-loss total payment} =\sum_{j=1}^{n}\min\bigl((x_j-d)^+,u\bigr), \qquad \text{Per-occurrence payment} =\min\left(\left(\sum_{j=1}^{n}x_j-d\right)^+,u\right).

These expressions make the key difference explicit. In the first expression, the coverage rule acts on every $x_j$, and then the payments are summed. In the second, the losses are summed first, and the coverage rule acts once. If the contract has no limit, omit the minimum with $u$; if it has no deductible, set $d=0$.

Why the Order Changes the Result

The formulas follow directly from the contract’s unit of application. Suppose the policy has a deductible but no limit. For a per-loss contract, claim $j$ produces payment $(x_j-d)^+$, so the total is the sum of those claim payments. For a per-occurrence contract, the covered amount before the deductible is the occurrence total $\sum_j x_j$, so the deductible is subtracted once from that total, subject to a payment floor of zero.

\sum_{j=1}^{n}(x_j-d)^+ \quad\text{versus}\quad \left(\sum_{j=1}^{n}x_j-d\right)^+.

This is not a matter of rearranging the same arithmetic: applying a nonlinear rule claim by claim generally differs from applying it to the sum. A deductible-only per-occurrence contract usually pays at least as much as a per-loss contract with the same deductible, because it does not impose a separate deductible on each claim. A limit-only per-loss contract can pay more than a per-occurrence contract with the same limit, because the per-loss design can provide a separate capped payment for each claim.

Read the unit.
Identify whether each deductible or limit applies per loss or per occurrence. Do not infer the unit from the number of claims.

Form the relevant amount.
For per-loss coverage, work with each claim amount separately. For per-occurrence coverage, add the claims in the specified occurrence first.

Apply the contract rule and add only where required.
Apply any deductible, then the stated payment limit. Sum individual payments for per-loss coverage; a per-occurrence rule produces one payment for the occurrence.

Worked Example: A Deductible Applied at Two Different Units

A hypothetical policy covers three claims from one specified occurrence, with ground-up amounts of $250, $700, and $1,050. Compare (a) a $400 deductible per loss with no limit and (b) a $400 deductible per occurrence with no limit. The loss amounts are fixed; no interest, mortality, or probability model applies. The method is to apply the deductible at the contract’s stated unit. The common mistake to avoid is subtracting the per-occurrence deductible separately from every claim.

(a) Per-loss deductible. Apply $400 to each claim separately:

Claim 1: $(250-400)^+=0$. Claim 2: $(700-400)^+=300$. Claim 3: $(1{,}050-400)^+=650$. Add the three payments: $0+300+650=\$950$.

(b) Per-occurrence deductible. First add the ground-up losses: $250+700+1{,}050=\$2{,}000$. Subtract the deductible once: $(2{,}000-400)^+=\$1{,}600$.

Check. Under (a), the insured retains $250, $400, and $400, for a total of $1,050; $1,050 plus the insurer’s $950 equals the $2,000 total loss. Under (b), the insured retains the $400 deductible and the insurer pays $1,600; these also total $2,000. The per-occurrence design pays $650 more here because one deductible is applied instead of three.

Worked Example: A Limit Applied to Each Claim or the Occurrence

A hypothetical policy covers claims of $600, $1,000, and $500 from one specified occurrence. There is no deductible. Compare a limit of $800 per loss with a limit of $800 per occurrence. The losses are fixed and deterministic, so no interest, mortality, or probability distribution is involved. The method is to cap each claim payment in the first design and cap the combined payment in the second. The common mistake to avoid is counting an $800 occurrence limit once for each claim.

Per-loss limit. The payments are $\min(600,800)=\$600$, $\min(1{,}000,800)=\$800$, and $\min(500,800)=\$500$. Their sum is $600+800+500=\$1{,}900$.

Per-occurrence limit. The occurrence total is $600+1{,}000+500=\$2{,}100$. The one occurrence payment is $\min(2{,}100,800)=\$800$.

Check. The per-loss payments total $1,900, which is less than the $2,100 total loss. The per-occurrence payment is $800, not $3\times \$800$, because the contract provides one cap for the specified occurrence. The remaining $1,300 is retained by the insured.

Worked Example: A Deductible and Limit at Both Units

A hypothetical policy covers three claims of $250, $700, and $1,100 from one specified occurrence. Compare (a) a deductible of $300 and a payment limit of $900, both per loss, with (b) the same deductible and limit, both per occurrence. The amounts are fixed; no interest, mortality assumption, or probability distribution applies. In both designs the deductible is applied before the insurer-payment limit. The method is to calculate the payment for each claim in (a), but the single combined payment in (b). The common mistake to avoid is using an occurrence-level cap more than once.

(a) Both terms per loss. For the $250 claim, $(250-300)^+=0$, and the capped payment is $\min(0,900)=\$0$. For the $700 claim, $(700-300)^+=400$, so the payment is $\min(400,900)=\$400$. For the $1,100 claim, $(1{,}100-300)^+=800$, so the payment is $\min(800,900)=\$800$. The total is $0+400+800=\$1{,}200$.

(b) Both terms per occurrence. Add the losses: $250+700+1{,}100=\$2{,}050$. Subtract the deductible once: $2{,}050-300=\$1{,}750$. Apply the single $900 payment limit: $\min(1{,}750,900)=\$900$.

Check. The per-loss payments sum to $1,200 and the per-occurrence payment is $900. Both are below the $2,050 ground-up total. Here, the occurrence limit restricts the combined payment to $900, whereas the per-loss design allows separate claim payments. This example also shows why there is no universal rule that one design always pays more when both deductibles and limits are involved.

Common Mistakes and What the Comparison Tells You

A frequent error is to calculate each claim’s net payment and then call the sum a per-occurrence payment. That calculation has already applied the rule per loss. For instance, in the first worked example, $0+300+650=\$950$ is the per-loss result. The per-occurrence calculation is instead $(250+700+1{,}050-400)^+=\$1{,}600$. The two answers differ because the deductible is used once in the second contract, not because of rounding or a different ground-up loss total.

Another error is to treat a per-occurrence limit as a separate limit for every claim. In the second worked example, multiplying $800 by three would contradict the contract: the one occurrence-level cap is applied to the $2,100 combined amount, producing an $800 payment. Always determine the unit before doing the arithmetic.

With both deductibles and limits, neither design necessarily pays more in every situation. A high combined occurrence can reach its single limit even when the separate per-loss payments sum to more, as in the third example. Conversely, several claims that individually fall below a deductible can have a combined amount above a per-occurrence deductible, so the occurrence design can pay when the per-loss design pays nothing. This distinction is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus: the calculation depends on identifying the covered unit and applying each term at that level.

Key takeaway: Per-loss coverage means “apply the rule to each claim, then add payments.” Per-occurrence coverage means “add the specified claims, then apply the rule once.” Deductibles and limits can have different units, so identify the unit of every term before calculating.

Check Your Understanding

For each hypothetical contract, calculate the requested total insurer payments. Unless stated otherwise, all losses are fixed ground-up amounts from one specified occurrence, with no interest, mortality, or probability assumption.

Claims are $180, $520, and $900. There is a $250 deductible and no limit. Find the per-loss total and the per-occurrence payment. (Answer: Per loss $920; per occurrence $1,350.)
Claims are $350, $700, and $1,200. There is no deductible and the limit is $600. Find the per-loss total and the per-occurrence payment. (Answer: Per loss $1,550; per occurrence $600.)
Claims are $200, $500, and $1,000. A $300 deductible and a $700 payment limit both apply per loss; then compare with both applying per occurrence. (Answer: Per loss $900; per occurrence $700.)
Claims are $400, $900, and $1,500. There is a $500 deductible and no limit. Compare a per-loss deductible with a per-occurrence deductible. (Answer: Per loss $1,400; per occurrence $2,300.)
Claims are $100, $400, and $700. There is a $250 deductible and a $900 payment limit. Both terms apply per loss in contract A and per occurrence in contract B. Calculate both totals and state which pays more. (Answer: Contract A pays $600; contract B pays $900, so contract B pays $300 more.)

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