Per-Claim Limits Versus Aggregate Limits

Why the Two Kinds of Limit Must Be Kept Separate

A per-claim limit caps the insurer’s payment on each individual claim. An aggregate limit instead caps the insurer’s total payment over a stated period, such as one policy year. A policy may have both: first the per-claim limit applies to each claim, then the aggregate cap applies to the resulting total. Because the aggregate cap can bind only when covered payments accumulate, its effect depends on the number and sizes of claims in the period.

An increased limits factor (ILF) compares expected limited loss per claim at two per-claim limits. It does not, by itself, measure the effect of a policy-period aggregate limit. The distinction is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: Let $X_j$ be the covered loss on claim $j$, and let $L$ be the per-claim limit. The insurer’s payment on that claim before any aggregate cap is $Y_j(L)=\min(X_j,L)$. For a policy period with claims $j=1,\ldots,N$ and aggregate limit $A$, the total insurer payment is $$ T(L,A)=\min\left(\sum_{j=1}^{N}Y_j(L),A\right). $$ A per-claim ILF for limit $L$ relative to base limit $B$ is $\operatorname{ILF}(L;B)=E[\min(X,L)]/E[\min(X,B)]$, assuming the same loss basis and a positive denominator. This ratio is for per-claim limited expected loss; it does not include an aggregate cap unless the calculation explicitly models one.

The order of operations matters. Apply the per-claim limit separately to each covered loss, add those limited payments, and then apply the aggregate limit. This definition assumes the aggregate cap applies to the sum of the covered payments being modeled, with no other policy provisions changing those payments. Actual policy wording determines which claims and payments count toward an aggregate.

Calculating the Per-Claim ILF

For a single claim, limited expected loss at limit $L$ is $E[\min(X,L)]$. If the same claim severity distribution is used at both limits, the ILF is the ratio of the two limited expected losses. For a fixed number of claims and no aggregate cap, expected total payment is the number of claims times expected payment per claim. This multiplication does not apply unchanged when an aggregate cap can bind.

Worked Example: A Per-Claim ILF and Two Claims Without an Aggregate

A hypothetical policy has exactly two claims. Claim severities are independent and identically distributed: $X=30$ thousand dollars with probability $0.4000$, and $X=70$ thousand dollars with probability $0.6000$. Find the ILF for an 80-thousand-dollar per-claim limit relative to a 40-thousand-dollar base limit. Then find expected total payments at both limits if there is no aggregate cap.

Method and assumptions. Calculate the expected capped payment for one claim at each limit. The claim count is fixed at two, and no aggregate cap or other payment adjustment applies.

At the base limit of 40 thousand dollars, the payments are 30 and 40 thousand dollars, respectively. Thus, $$ E[\min(X,40)]=0.4000(30)+0.6000(40)=12+24=36 $$ thousand dollars per claim.

At the 80-thousand-dollar limit, both possible losses are below the limit. Therefore, $$ E[\min(X,80)]=0.4000(30)+0.6000(70)=12+42=54 $$ thousand dollars per claim, and $$ \operatorname{ILF}(80;40)=\frac{54}{36}=1.500. $$

For two claims with no aggregate cap, multiply each per-claim expected payment by two. Expected total payments are $2(36)=72$ thousand dollars at the base limit and $2(54)=108$ thousand dollars at the higher limit. Check: the ratio of these expected totals is $108/72=1.500$, matching the per-claim ILF because the claim count is fixed and there is no aggregate cap. The common mistake avoided here is reporting these amounts as 72,000 and 108,000 while the calculations are stated in thousands; in the stated units, the answers are 72 and 108 thousand dollars.

How an Aggregate Cap Changes the Total

With both limits present, the aggregate is applied to the sum of per-claim-limited payments. It is not a new per-claim limit, and it does not change the payment on a claim when the period’s total remains below the cap. If the sum exceeds the aggregate, the policy’s total payment is reduced to the aggregate amount.

Worked Example: Two Claims Exhaust a Policy-Period Aggregate

Use the hypothetical severity distribution from the previous example, with exactly two independent claims and an 80-thousand-dollar per-claim limit. The policy also has a 90-thousand-dollar aggregate limit. Calculate expected total payment. All amounts are in thousands of dollars.

Method and assumptions. First cap each claim at 80, then cap the sum at 90. Each claim pays either 30 or 70, with probabilities $0.4000$ and $0.6000$. Independence gives the following three possible totals:

Both claims are 30: probability $0.4000^2=0.1600$; the pre-aggregate total is $30+30=60$, so the policy pays $\min(60,90)=60$.

One claim is 30 and one is 70: probability $2(0.4000)(0.6000)=0.4800$; the pre-aggregate total is $30+70=100$, so the policy pays $\min(100,90)=90$.

Both claims are 70: probability $0.6000^2=0.3600$; the pre-aggregate total is $70+70=140$, so the policy pays $\min(140,90)=90$.

The expected total payment is $$ 0.1600(60)+0.4800(90)+0.3600(90) =9.6+43.2+32.4 =85.2 $$ thousand dollars. Check: the probabilities sum to $0.1600+0.4800+0.3600=1.0000$. Without the aggregate cap, expected payment would be 108 thousand dollars, so the cap reduces expected payment by $108-85.2=22.8$ thousand dollars. The common mistake avoided is multiplying the per-claim expected payment by two and treating 108 as the aggregate-limited expected payment.

Why the Aggregate Cannot Be Applied to Expected Loss

In general, aggregate-limited expected payment is $E[\min(S,A)]$, where $S$ is the total after per-claim limits. It is not generally $\min(E[S],A)$. The aggregate applies to the realized total in each policy period; taking an expectation first loses information about how often the total falls above or below the cap.

Derivation: For any realized total $S$, the identity $\min(S,A)=S-(S-A)_+$ holds, where $(z)_+=\max(z,0)$. Taking expectations gives $$ E[\min(S,A)]=E[S]-E[(S-A)_+]. $$ The second term is the expected amount by which the realized total exceeds the aggregate. It depends on the distribution of $S$, not merely on $E[S]$. Also, because $\min(S,A)\le S$ and $\min(S,A)\le A$, $E[\min(S,A)]\le\min(E[S],A)$. Equality need not hold.

The claim count and the dependence between claim severities can both affect the distribution of $S$. Consequently, two portfolios with the same expected per-claim limited loss can have different aggregate-limited expected payments if they have different claim counts or different patterns of joint losses. A per-claim ILF alone does not supply this information.

Worked Example: The Per-Claim ILF Is Not the Aggregate-Limited Factor

A hypothetical policy has exactly three independent claims. Each claim is 25 thousand dollars with probability $0.5000$ and 60 thousand dollars with probability $0.5000$. Compare a 25-thousand-dollar per-claim limit with a 50-thousand-dollar per-claim limit. Both options have a 90-thousand-dollar aggregate limit. Find the per-claim ILF and the ratio of aggregate-limited expected totals.

Method and assumptions. Calculate per-claim expected payments first. Then enumerate the number $K$ of claims with severity 60 thousand dollars. Since claims are independent, $K$ has probabilities $P(K=0)=0.1250$, $P(K=1)=0.3750$, $P(K=2)=0.3750$, and $P(K=3)=0.1250$.

At the 25-thousand-dollar limit, every claim pays 25 thousand dollars, so the expected total is $3(25)=75$ thousand dollars. At the 50-thousand-dollar limit, a claim pays 25 or 50 thousand dollars with equal probability. Expected payment per claim is $$ 0.5000(25)+0.5000(50)=12.5+25=37.5 $$ thousand dollars, giving the per-claim ILF $37.5/25=1.500$.

At the higher limit, before applying the aggregate, the total for $K$ high-severity claims is $25(3-K)+50K=75+25K$. For $K=0$, the total is 75 and the policy pays 75. For $K=1,2,$ or $3$, the total is at least 100, so the policy pays the 90-thousand-dollar aggregate. Therefore, $$ E[T(50,90)]=0.1250(75)+0.3750(90)+0.3750(90)+0.1250(90) =9.375+33.75+33.75+11.25 =88.125 $$ thousand dollars. At the 25-thousand-dollar limit, the 75-thousand-dollar total is below the aggregate, so the aggregate-limited total remains 75. The ratio of aggregate-limited totals is $88.125/75=1.175$, not the per-claim ILF of 1.500. Check: without the aggregate cap, the higher-limit expected total would be $3(37.5)=112.5$ thousand dollars; the 90-thousand-dollar cap appropriately reduces it. The common mistake avoided is applying the per-claim ILF to the base total without checking whether an aggregate cap binds.

Practical Checks and Common Traps

When comparing limits, state clearly whether the indicated quantity is expected payment per claim, expected total payment before an aggregate cap, or expected total payment after it. These are different quantities. The per-claim ILF formula is appropriate for comparing per-claim limited losses on a consistent basis. An aggregate-limited indication additionally requires a model or data for the distribution of the period’s total covered payments.

Do not treat an aggregate as a per-claim cap. Apply it to the policy-period total after applying each per-claim limit.
Do not multiply an ILF-adjusted base total through an aggregate cap without modeling outcomes. The cap is applied to each realized period total before taking the expectation.
Do not assume the aggregate binds just because expected total exceeds it. Whether it binds depends on the distribution of actual totals; expected total alone is insufficient.
Keep units consistent. If losses are stated in thousands of dollars, calculations and answers should remain in thousands unless converted explicitly.

A sound calculation starts by writing the payment formula and identifying the policy period, per-claim limit, aggregate limit, claim-count assumptions, and loss distribution. If an aggregate can bind, enumerate scenarios for a small discrete example or use an appropriate probability model for a larger portfolio.

Check Your Understanding

For each problem, distinguish the per-claim calculation from the aggregate calculation. Amounts are in thousands of dollars unless stated otherwise.

A claim is 20 with probability $0.5000$ and 60 with probability $0.5000$. Find the ILF for limit 60 relative to limit 30. For exactly two claims and no aggregate, find the expected totals at both limits. (Answer: ILF $=1.600$; expected totals are 50 and 80.)
Two claims have per-claim-limited payments of 35 and 45. The aggregate limit is 70. Find the policy payment and the reduction from the total before the aggregate. (Answer: payment $=70$; reduction $=10$.)
Three independent claims are each 10 with probability $0.5000$ and 30 with probability $0.5000$. The per-claim limit is 25 and the aggregate limit is 50. Find expected total payment. (Answer: $45.625$.)
A period’s pre-aggregate total $S$ is 60 with probability $0.5000$ and 140 with probability $0.5000$. If $A=100$, compare $E[\min(S,A)]$ with $\min(E[S],A)$. (Answer: $E[\min(S,A)]=80$, while $\min(E[S],A)=100$.)
A fixed number of claims has expected total payment before an aggregate of 120. Is the expected payment with aggregate limit 100 necessarily 100? State the strongest general bound available from this information alone. (Answer: No; the expected aggregate-limited payment is at most 100, but its exact value cannot be determined from the pre-aggregate expectation alone.)

What You'll Learn