Combining Occurrence Payments With An Aggregate Limit

Why the Order of Calculation Matters

A policy can restrict insurer payments at more than one level. A per-occurrence deductible or limit determines what the insurer would pay for a particular occurrence. A policy-period aggregate limit then places an overall maximum on payments for all covered occurrences during the period. These terms do different jobs, so the order of calculation matters: first determine each occurrence payment under its own coverage terms, then apply the aggregate cap to their combined amount.

This tutorial concerns the total insurer payment when both kinds of terms apply. It assumes the policy has a single aggregate limit, no aggregate deductible, and no reinstatement of exhausted aggregate capacity. Claim grouping into occurrences is taken as given. The method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $L_j$ be the covered loss assigned to occurrence $j$, and let $g_j(L_j)$ be the insurer payment for that occurrence after applying its occurrence-level coverage terms. If the policy-period aggregate limit is $A$, the total insurer payment is $$ T=\min\left(A,\sum_{j=1}^{n}g_j(L_j)\right). $$ The aggregate limit caps the sum of occurrence payments; it does not replace the calculation of those payments.

Two Levels of Coverage

An occurrence payment function $g_j$ can reflect a deductible, a per-occurrence limit, coinsurance, or a combination of these terms. For example, with a per-occurrence deductible $D$ and a maximum insurer payment $U$ after the deductible, the payment for a nonnegative occurrence loss $L$ is

g(L)=\min\bigl(\max(L-D,0),U\bigr).

If a coinsurance percentage $c$ applies to loss above a deductible $D$, and the insurer's payment is also subject to a per-occurrence maximum $U$, the corresponding function is

g(L)=\min\bigl(c\max(L-D,0),U\bigr).

These functions are examples, not universal rules for every contract. The policy wording determines whether a limit is stated before or after coinsurance, how a deductible applies, and what loss is covered. Use the specified terms to calculate each $g_j(L_j)$; only after that calculation should the aggregate limit be applied.

Why the Aggregate Formula Holds

The occurrence terms produce a set of amounts the insurer would owe before the aggregate is considered. Their total is $G=\sum_{j=1}^{n}g_j(L_j)$. With no aggregate limit, the insurer pays $G$. With an aggregate limit $A$, it cannot pay more than $A$, but it also does not pay more than the occurrence terms allow. Therefore the total is the smaller of $G$ and $A$.

T= \begin{cases} G, & G\leq A,\\ A, & G>A, \end{cases} \qquad\text{or equivalently}\qquad T=\min(A,G).

This also explains how the calculation works when occurrences are processed in sequence. Before occurrence $j$, let $R_{j-1}$ be the remaining aggregate capacity. The amount paid for that occurrence is the smaller of its occurrence-level payment and the remaining capacity:

R_0=A,\qquad P_j=\min\bigl(g_j(L_j),R_{j-1}\bigr),\qquad R_j=R_{j-1}-P_j.

The sequential method gives $\sum_j P_j=\min(A,\sum_j g_j(L_j))$. If the occurrence payments use all the capacity, their sequential payments add to $A$; otherwise, all occurrence payments are paid and their sum is below $A$. The sequence can affect which occurrence payments fit under the cap, but not the total for a fixed set of occurrence payments.

Identify the occurrences.
Use the policy's grouping of covered losses. If an occurrence contains several claims, first combine those claims as required by the applicable coverage terms.

Calculate each occurrence payment.
Apply the stated deductible, coinsurance, and occurrence limit in the order specified by the policy. Record $g_j(L_j)$ separately for every occurrence.

Apply the aggregate limit once.
Add the occurrence payments and compare their sum with $A$. The insurer's total is the smaller amount.

Check the result.
The total cannot exceed the aggregate limit or the sum of the occurrence-level payments. If it is below the aggregate limit, the aggregate has not reduced the payment.

Worked Examples

Worked Example: Deductible and Occurrence Limit Before the Aggregate

A hypothetical policy covers three occurrences during one policy period. Each occurrence has a deductible of $2,000 and a maximum insurer payment of $8,000 after the deductible. The aggregate limit is $10,000. The covered occurrence losses are $5,000, $12,000, and $3,000. Find the total insurer payment.

Method. Calculate each occurrence payment as the loss above its deductible, limited to $8,000. Then add those payments and apply the $10,000 aggregate cap.

For the first occurrence, the loss above the deductible is $ \$5{,}000-\$2{,}000=\$3{,}000 $. This is below the per-occurrence maximum, so its payment is $3,000.

For the second occurrence, the loss above the deductible is $ \$12{,}000-\$2{,}000=\$10{,}000 $. The $8,000 per-occurrence maximum reduces its payment to $8,000.

For the third occurrence, the loss above the deductible is $ \$3{,}000-\$2{,}000=\$1{,}000 $, so its payment is $1,000.

G=\$3{,}000+\$8{,}000+\$1{,}000=\$12{,}000, \qquad T=\min(\$10{,}000,\$12{,}000)=\$10{,}000.

Check. The total is $2,000 below the sum of occurrence payments and equals, but does not exceed, the aggregate limit. A common mistake would be to stop at $12,000 without applying the policy-period cap.

Worked Example: Coinsurance, a Per-Occurrence Maximum, and an Aggregate

A hypothetical policy pays 80% of covered loss above a $1,000 deductible for each occurrence, subject to a maximum insurer payment of $6,000 per occurrence. Its aggregate limit is $9,000. The three occurrence losses are $4,000, $9,000, and $7,000. Calculate the total insurer payment.

Method. For each occurrence, subtract the deductible, multiply the positive remainder by 0.80, and then apply the $6,000 occurrence maximum. Apply the aggregate cap to the resulting sum.

For the $4,000 loss, the amount above the deductible is $\$4{,}000-\$1{,}000=\$3{,}000$. The coinsured payment is $0.80(\$3{,}000)=\$2{,}400$, below the $6,000 maximum.

For the $9,000 loss, the amount above the deductible is $\$9{,}000-\$1{,}000=\$8{,}000$. The coinsured payment is $0.80(\$8{,}000)=\$6{,}400$, which is reduced to the $6,000 maximum.

For the $7,000 loss, the amount above the deductible is $\$7{,}000-\$1{,}000=\$6{,}000$. The coinsured payment is $0.80(\$6{,}000)=\$4{,}800$, below the maximum.

G=\$2{,}400+\$6{,}000+\$4{,}800=\$13{,}200, \qquad T=\min(\$9{,}000,\$13{,}200)=\$9{,}000.

Check. The cap reduces the total by $\$13{,}200-\$9{,}000=\$4{,}200$. A common mistake is to apply 80% to the entire loss, rather than to the amount above the deductible, or to apply the aggregate before calculating the occurrence-level maximums.

Worked Example: Several Claims Within Each Occurrence

A hypothetical policy has a $3,000 deductible and a $7,000 maximum insurer payment per occurrence, followed by a $14,000 policy-period aggregate limit. The claims are grouped into three occurrences: occurrence 1 contains claims of $2,000, $5,000, and $5,000; occurrence 2 contains claims of $4,000 and $2,000; occurrence 3 contains one claim of $10,000. Find the total insurer payment.

Method. First sum the covered claims within each occurrence. Apply the deductible and per-occurrence maximum to that occurrence total, then apply the aggregate limit to the sum of the three payments.

Occurrence 1 has total loss $\$2{,}000+\$5{,}000+\$5{,}000=\$12{,}000$. After its $3,000 deductible, $9,000 remains, so its payment is $\min(\$9{,}000,\$7{,}000)=\$7{,}000$.

Occurrence 2 has total loss $\$4{,}000+\$2{,}000=\$6{,}000$. After the deductible, its payment is $\min(\$6{,}000-\$3{,}000,\$7{,}000)=\$3{,}000$.

Occurrence 3 has total loss of $10,000. After the deductible, $7,000 remains, which is exactly the per-occurrence maximum. Its payment is $7,000.

G=\$7{,}000+\$3{,}000+\$7{,}000=\$17{,}000, \qquad T=\min(\$14{,}000,\$17{,}000)=\$14{,}000.

Check. The total is capped at $14,000. A common mistake is to subtract the deductible once from every individual claim in an occurrence. Here the policy says the deductible is per occurrence, so it is applied once to each occurrence total.

Worked Example: Tracking Unused Aggregate Capacity

A hypothetical policy has occurrence-level payments of $4,500, $3,000, and $2,000, already calculated under its occurrence terms. The policy-period aggregate limit is $8,000. Process the occurrences in the stated order and find the total insurer payment.

Method. At each occurrence, pay no more than the remaining aggregate capacity.

Initially, the remaining capacity is $8,000. The first occurrence payment is $\min(\$4{,}500,\$8{,}000)=\$4{,}500$, leaving $\$8{,}000-\$4{,}500=\$3{,}500$.

The second occurrence payment is $\min(\$3{,}000,\$3{,}500)=\$3{,}000$, leaving $\$3{,}500-\$3{,}000=\$500$.

The third occurrence has a potential payment of $2,000, but only $500 remains. Its payment is $\min(\$2{,}000,\$500)=\$500$, leaving no aggregate capacity.

T=\$4{,}500+\$3{,}000+\$500=\$8{,}000.

Check. The pre-aggregate sum is $\$4{,}500+\$3{,}000+\$2{,}000=\$9{,}500$, so the $8,000 aggregate is exhausted. The result agrees with $\min(\$8{,}000,\$9{,}500)=\$8{,}000$. This sequential check identifies which occurrence is reduced, while the total formula is sufficient when only the total is requested.

Common Traps and a Final Check

The most important distinction is between a limit that applies to each occurrence and one that applies once across the policy period. A per-occurrence maximum is applied separately when calculating each $g_j(L_j)$. The aggregate maximum is applied to the sum of those results. Confusing the two levels can either understate or overstate the insurer's payment.

Applying the aggregate separately to every occurrence. The expression $\sum_j \min(A,g_j(L_j))$ is generally wrong. It can exceed $A$, because it effectively gives the policy a new aggregate limit for each occurrence.
Applying occurrence terms to the aggregate loss. In general, calculate each occurrence under its own terms before adding. Applying one deductible or limit to the total loss can change the result.
Subtracting an occurrence deductible more than once. If the deductible is per occurrence, apply it once to the loss assigned to that occurrence, not once per claim within it.
Assuming the aggregate always reduces payment. If $\sum_j g_j(L_j)\leq A$, the aggregate does not bind and the insurer pays the full sum of occurrence payments.

Key takeaway: Calculate occurrence payments using the applicable occurrence-level terms, add those payments, and cap the sum at the policy-period aggregate limit. Thus, the total is $\min(A,\sum_j g_j(L_j))$.

Check Your Understanding

For each hypothetical policy, calculate the insurer's total payment after applying the occurrence terms and the aggregate limit.

A policy has a $2,000 deductible and a $5,000 maximum payment per occurrence. The occurrence losses are $1,000, $6,000, and $9,000, and the aggregate limit is $8,000. (Answer: $8,000)
A policy pays 75% of loss above a $4,000 per-occurrence deductible, subject to a $7,000 maximum per occurrence. Occurrence losses are $8,000, $16,000, and $10,000, and the aggregate limit is $12,000. (Answer: $12,000)
A $1,000 deductible and $6,000 maximum payment apply per occurrence. The three occurrences contain claims of (a) $2,000 and $3,000, (b) $10,000, and (c) $500 and $2,000. The aggregate limit is $10,000. (Answer: $10,000)
Occurrence-level payments have already been calculated as $5,000, $4,000, and $6,000. The aggregate limit is $9,000. Find the total payment without allocating the cap among individual claims. (Answer: $9,000)
A policy has an occurrence-level deductible of $5,000 and a maximum insurer payment of $10,000 per occurrence. Its occurrence losses are $4,000, $11,000, and $8,000, and its aggregate limit is $12,000. (Answer: $9,000)

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