Comparing Per-Loss And Per-Occurrence Designs

Why the Level of Application Matters

A policy may respond separately to each individual loss, or it may combine losses arising from one event and apply its terms to that combined amount. The distinction matters whenever an occurrence produces multiple claims. Even if the deductible and limit amounts look similar, applying them once per loss rather than once per occurrence can produce a different total payment.

For example, a hypothetical storm might damage several buildings insured under one policy. Each building has an individual loss, while the storm may be treated as one occurrence. The calculations below focus on the payment design, not on deciding which claims legally belong to an occurrence. Assume each scenario specifies the grouping.

Definition: A loss is an individual amount of covered damage or expense being evaluated. An occurrence is the event or grouping of losses to which occurrence-level policy terms apply. A single occurrence can contain one loss or several losses.

To compare designs precisely, this tutorial uses a common limit convention: the limit is the maximum amount the insurer pays, after the deductible. There is no coinsurance, aggregate policy-period limit, or other coverage term. All amounts are in dollars.

Payment formulas: For a single loss $x$ subject to a per-loss deductible $d$ and per-loss payment limit $u$, the insurer pays $\min\{(x-d)^+,u\}$, where $z^+=\max(z,0)$. For an occurrence with total loss $T$, subject to occurrence deductible $D$ and occurrence payment limit $U$, the insurer pays $\min\{(T-D)^+,U\}$. If the occurrence contains losses $x_1,\ldots,x_n$, then $T=\sum_{j=1}^n x_j$.

The level is part of the term. A per-loss deductible applies to each individual loss; a per-occurrence deductible applies once to the total loss from that occurrence. Likewise, the per-loss limit caps each individual payment, while the per-occurrence limit caps the payment for the occurrence as a whole. The amounts $d,u,D,U$ need not be equal.

How the Formulas Follow from the Contract Terms

For a per-loss deductible, the insured bears the first $d$ dollars of a loss $x$. If $x\le d$, nothing remains for the insurer to pay. If $x>d$, the amount above the deductible is $x-d$. The insurer’s payment cannot exceed $u$, so the payment is the smaller of $u$ and the positive part of $x-d$. Applying this reasoning separately to every loss gives the total per-loss payment.

Y_{\mathrm{loss}}=\sum_{j=1}^{n}\min\{(x_j-d)^+,u\}

For an occurrence-level design, first combine all losses in the occurrence. The insured bears the first $D$ dollars of that combined amount, and the insurer’s payment is then capped at $U$. Thus:

Y_{\mathrm{occ}}=\min\left\{\left(\sum_{j=1}^{n}x_j-D\right)^+,U\right\}

These formulas are not interchangeable. The per-loss formula applies a deductible and cap repeatedly, while the occurrence formula applies each term once to the combined loss. A useful consequence follows directly: if the occurrence membership and total loss stay fixed, the occurrence-level payment stays fixed, even if the total is divided among individual claims differently. The per-loss payment need not stay fixed because each claim is tested against its own deductible and limit.

Matched Calculations

Worked Example: Three Property Claims from One Event

A hypothetical wind event causes three covered building losses of $1,200, $3,100, and $6,200. Compare (a) a per-loss design with deductible $d=\$1,000$ and payment limit $u=\$3,000$ per loss, and (b) a per-occurrence design with deductible $D=\$3,000$ and payment limit $U=\$7,000$ for the event. Assume all three claims arise from the same occurrence.

Method. Calculate each per-loss payment separately. For the occurrence design, add the losses first, then apply its deductible and limit.

Per-loss design. For the $1,200 loss, the amount above the deductible is $\$1,200-\$1,000=\$200$, which is below the $3,000 limit. Payment: $200.

For the $3,100 loss, the amount above the deductible is $\$3,100-\$1,000=\$2,100$, also below the limit. Payment: $2,100.

For the $6,200 loss, the amount above the deductible is $\$6,200-\$1,000=\$5,200$. The $3,000 per-loss limit applies, so payment: $3,000.

The total per-loss payment is $\$200+\$2,100+\$3,000=\$5,300$.

Per-occurrence design. The total loss is $\$1,200+\$3,100+\$6,200=\$10,500$. After the $3,000 occurrence deductible, the amount is $\$10,500-\$3,000=\$7,500$. The $7,000 occurrence limit reduces payment to $7,000.

Check and trap. The occurrence payment is $1,700 greater than the per-loss payment. A common wrong calculation is to deduct $3,000 from each of the three losses, as though the occurrence deductible were per loss. That would incorrectly treat the deductible as $9,000 in total. The specified occurrence deductible is applied once, so the correct payment is $7,000.

Worked Example: Same Total, Different Claim Sizes

Two hypothetical claim patterns each total $10,000 and each arise from a single occurrence. Pattern A has losses of $2,000 and $8,000. Pattern B has losses of $5,000 and $5,000. Compare a per-loss design with $d=\$1,000$ and $u=\$6,000$ against an occurrence design with $D=\$2,000$ and $U=\$9,000$.

Method. Calculate both per-loss payments for each pattern. Then use the common total loss for the occurrence calculation.

Pattern A under the per-loss design. The $2,000 loss pays $\min(\$2,000-\$1,000,\$6,000)=\$1,000$. The $8,000 loss pays $\min(\$8,000-\$1,000,\$6,000)=\$6,000$. Total: $\$1,000+\$6,000=\$7,000$.

Pattern B under the per-loss design. Each $5,000 loss pays $\min(\$5,000-\$1,000,\$6,000)=\$4,000$. Total: $\$4,000+\$4,000=\$8,000$.

Occurrence design. Each pattern has total loss $10,000. After the $2,000 occurrence deductible, the amount is $8,000, below the $9,000 limit. The occurrence payment is therefore $8,000 for either pattern.

Check and trap. The occurrence payment is unchanged because the total loss and occurrence terms are unchanged. The per-loss payment differs by $1,000. A tempting but incorrect approach is to conclude that equal total losses must always produce equal insurance payments. That conclusion fails here because the per-loss deductible and limit are tested against each claim separately.

Worked Example: One Event or Two Occurrences?

A hypothetical equipment failure results in two covered losses, $1,500 and $2,500. Compare the payments if they are treated as one occurrence and if they are treated as two separate occurrences. The per-loss terms are $d=\$1,000$ and $u=\$2,500$ per loss. The occurrence terms are $D=\$1,000$ and $U=\$2,500$ per occurrence.

Method. First calculate the per-loss total, which is based on the individual losses. Then apply the occurrence terms under each stated grouping.

Per-loss design. The $1,500 loss pays $\min(\$1,500-\$1,000,\$2,500)=\$500$. The $2,500 loss pays $\min(\$2,500-\$1,000,\$2,500)=\$1,500$. Total: $\$500+\$1,500=\$2,000$. This total is the same whether the losses are grouped into one occurrence or two, because the per-loss terms have not changed.

One occurrence under the occurrence design. The combined loss is $\$1,500+\$2,500=\$4,000$. After one $1,000 deductible, the amount is $3,000. The $2,500 occurrence limit applies, giving a payment of $2,500.

Two occurrences under the occurrence design. The $1,500 occurrence pays $\min(\$1,500-\$1,000,\$2,500)=\$500$. The $2,500 occurrence pays $\min(\$2,500-\$1,000,\$2,500)=\$1,500$. Total: $2,000.

Check and trap. Treating the two losses as one occurrence produces $2,500, while treating them as two occurrences produces $2,000. The mistake to avoid is applying the $1,000 occurrence deductible only once when the scenario specifies two separate occurrences. With two occurrences, the deductible applies once to each occurrence.

What to Track in a Comparison

A reliable comparison starts by identifying the unit to which each term applies, rather than by immediately substituting numbers. In particular, keep the following distinctions clear:

Claim count versus occurrence count: several losses can belong to one occurrence, and one occurrence can therefore have several claim amounts.
Repeated deductible versus single deductible: a per-loss deductible is applied to every loss; an occurrence deductible is applied once to the total for each occurrence.
Repeated cap versus single cap: a per-loss limit caps each individual payment; an occurrence limit caps the total payment for that occurrence.
Grouping information: occurrence-level calculations require knowing which losses are grouped together. A list of loss amounts alone may not provide enough information.

Identify the payment unit.
Mark which terms apply per individual loss and which apply per occurrence.

Keep the occurrence groups intact.
For each occurrence, add its losses before applying an occurrence deductible or limit.

Apply the correct formula.
For per-loss terms, calculate each claim payment and sum them. For occurrence terms, calculate one payment per occurrence and then sum across occurrences.

Check the scale of the result.
A per-loss limit can apply several times within one occurrence; an occurrence limit caps that occurrence’s total payment once.

These calculations are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The central exam discipline is to preserve the stated level of every term: do not turn a per-loss deductible into a per-occurrence deductible, or vice versa, simply because the losses arise from the same event.

Key takeaway: Per-loss payments are calculated claim by claim and then added. Per-occurrence payments are calculated from the combined loss within each occurrence. The two designs can differ because deductibles and limits operate at different levels.

Check Your Understanding

For each problem, identify the correct unit of application before calculating.

A single occurrence produces losses of $900 and $2,300. Under per-loss terms $d=\$500$, $u=\$1,800$, calculate the total payment. Under occurrence terms $D=\$1,000$, $U=\$4,000$, calculate the payment for the same occurrence. (Answer: per-loss $2,200; per-occurrence $2,200)
One occurrence contains losses of $1,200 and $4,800. Find the per-loss payment with $d=\$800$, $u=\$2,500$ per loss. Then find the occurrence payment with $D=\$1,600$, $U=\$5,000$. (Answer: per-loss $2,900; per-occurrence $4,400)
Two different claim patterns, each from one occurrence, are $(\$1,000,\$6,000)$ and $(\$3,500,\$3,500)$. Find the per-loss payment for each pattern with $d=\$1,000$, $u=\$4,000$, and the occurrence payment for either pattern with $D=\$2,000$, $U=\$6,000$. (Answer: per-loss payments $4,000 and $5,000; per-occurrence payment $5,000 for either pattern)
Losses of $900 and $2,100 form one occurrence, while a separate $4,000 loss forms a second occurrence. Calculate the total under occurrence terms $D=\$1,500$, $U=\$5,000$ per occurrence. (Answer: $4,000)
Three losses of $2,000, $3,000, and $5,000 belong to one occurrence. Compare per-loss terms $d=\$1,000$, $u=\$3,500$ with occurrence terms $D=\$2,500$, $U=\$6,000$. (Answer: per-loss $6,500; per-occurrence $6,000)

What You'll Learn