Compare Per-Claim And Aggregate Loss Limits

Why the Application Base Matters

A loss cap can limit each individual claim, or it can limit the converted loss total for the entire policy period. These are different plan designs. A per-claim cap responds to the size of each claim; an aggregate cap responds to the portfolio total. Even when both designs use the same claims and conversion factor, they need not produce the same limited losses or retrospective premium.

To compare the designs correctly, identify what amount the cap applies to and when it applies. In the calculations below, $X_j$ is the eligible loss for claim $j$, $k$ is the loss conversion factor (LCF), $c$ is the per-claim cap on unconverted losses, and $A$ is the aggregate cap on converted losses. Amounts are stated in dollars or thousands of dollars as indicated. The two caps are applied to different bases, so their numerical values are not directly interchangeable.

Definition: Under a per-claim loss cap $c$, each eligible claim loss $X_j$ is limited to $\min(X_j,c)$, and the limited claims are then converted. The resulting converted losses are $L_C=k\sum_j\min(X_j,c)$. Under an aggregate converted-loss cap $A$, eligible losses are first summed and converted, and the cap is then applied once: $L_A=\min(k\sum_jX_j,A)$. These definitions assume a positive LCF and that $c$ applies to each eligible unconverted claim amount while $A$ applies to the portfolio’s converted loss total.

Here, “converted losses” means eligible losses multiplied by the stated LCF. The examples treat the claim amounts as final for the rating calculation: there is no further development, discounting, or separate adjustment to loss adjustment expense. No mortality assumption or probability distribution is needed because the calculations use specified claim amounts rather than a random-loss expectation. If a plan instead caps losses before adding a separate LAE component, or applies a cap to a different loss base, that plan wording must be followed.

Calculate Each Design on Its Own Base

For a per-claim cap, apply the cap claim by claim, add the limited amounts, and then multiply by the LCF. A claim below the cap remains unchanged; a claim above the cap contributes only the cap amount. For an aggregate converted-loss cap, add the eligible losses without first limiting individual claims, multiply that total by the LCF, and compare the converted total with the aggregate cap.

L_C=k\sum_{j=1}^{n}\min(X_j,c) \qquad\text{and}\qquad L_A=\min\left(k\sum_{j=1}^{n}X_j,A\right)

The difference in order follows directly from what each contract provision limits. The per-claim rule replaces every claim amount $X_j$ by its own limited amount before adding the claims. The aggregate rule leaves the individual amounts intact through summation and conversion, then limits the single portfolio total. Thus the two calculations generally have different inputs at the point where the cap is applied.

Identify the cap base.
Determine whether the stated limit applies to each unconverted claim or to the converted portfolio total.

Apply the stated sequence.
For a per-claim cap, limit each claim before summing and converting. For an aggregate cap on converted losses, sum and convert before applying the aggregate limit.

Calculate premium consistently.
Use the limited converted-loss amount in the plan’s premium formula, keeping all other premium parameters unchanged when comparing alternatives.

Worked Example: Compare Both Caps on the Same Claims

A hypothetical policy has three eligible claims of $18,000, $42,000, and $75,000. Compare a per-claim cap of $40,000 with an aggregate cap of $125,000 on converted losses. The LCF is 1.10. For the premium comparison, the basic premium is $60,000 and the tax multiplier is 1.08. Assume there are no binding minimum or maximum premium provisions and that the stated amounts are final eligible losses. There is no discounting, mortality assumption, or probability distribution; the claim amounts are deterministic plan inputs.

Per-claim method. Apply the $40,000 cap to each claim:

Claim 1: $\min(18{,}000,40{,}000)=18{,}000$. Claim 2: $\min(42{,}000,40{,}000)=40{,}000$. Claim 3: $\min(75{,}000,40{,}000)=40{,}000$.

The limited unconverted total is $18{,}000+40{,}000+40{,}000=98{,}000$. Convert it: $98{,}000(1.10)=107{,}800$. Thus $L_C=\$107{,}800$.

Aggregate method. First sum the unmodified claims: $18{,}000+42{,}000+75{,}000=135{,}000$. Convert the total: $135{,}000(1.10)=148{,}500$. Apply the aggregate converted-loss cap: $\min(148{,}500,125{,}000)=125{,}000$. Thus $L_A=\$125{,}000$.

Premium comparison. Use the stated formula $P=(B+L)T$, where $B$ is basic premium, $L$ is limited converted losses, and $T$ is the tax multiplier. Under the per-claim design, $P_C=(60{,}000+107{,}800)(1.08)=167{,}800(1.08)=\$181{,}224.00$. Under the aggregate design, $P_A=(60{,}000+125{,}000)(1.08)=185{,}000(1.08)=\$199{,}800.00$.

Check. The aggregate limited-loss amount exceeds the per-claim amount by $125{,}000-107{,}800=\$17{,}200$. The premium difference is $17{,}200(1.08)=\$18{,}576.00$, which agrees with $199{,}800-181{,}224=\$18{,}576.00$. The method avoids treating the $125,000 aggregate converted-loss cap as if it were a $125,000 cap on each claim.

Why the Two Calculations Differ

A per-claim limitation can remove amounts from multiple claims, while an aggregate limitation removes only the amount by which the converted portfolio total exceeds its one overall cap. The per-claim result depends on how the total is distributed among claims; the aggregate result depends on the total eligible loss, the LCF, and the aggregate cap. In particular, two portfolios with the same total loss can have different per-claim limited totals if their claim sizes differ.

This distinction is also why the two formulas cannot be substituted for one another. In general, one cannot replace a collection of claim-level limits with a single portfolio limit, or vice versa, without changing the contract terms. A cap’s amount has meaning only together with its application base and place in the calculation sequence.

Worked Example: Carry Each Limited Loss Through to Premium

A hypothetical plan has eligible claims of $9,500, $31,200, $46,700, and $80,400. Compare a per-claim cap of $35,000 with an aggregate cap of $130,000 on converted losses. The LCF is 1.05, the basic premium is $72,000, and the tax multiplier is 1.04. The plan’s minimum premium is $190,000 and its maximum premium is $215,000. Assume the claim amounts are final, with no discounting, mortality assumption, or probability distribution.

Per-claim method. The limited claims are $9{,}500$, $31{,}200$, $\min(46{,}700,35{,}000)=35{,}000$, and $\min(80{,}400,35{,}000)=35{,}000$. Their sum is $9{,}500+31{,}200+35{,}000+35{,}000=110{,}700$. The converted limited losses are $110{,}700(1.05)=116{,}235$.

Before applying the premium bounds, the indicated premium is $(72{,}000+116{,}235)(1.04)=188{,}235(1.04)=\$195{,}764.40$. This is above the $190,000 minimum and below the $215,000 maximum, so neither bound changes it. The per-claim premium is $\$195{,}764.40$.

Aggregate method. The uncapped claim total is $9{,}500+31{,}200+46{,}700+80{,}400=167{,}800$. Convert it: $167{,}800(1.05)=176{,}190$. Apply the aggregate cap: $\min(176{,}190,130{,}000)=130{,}000$.

The indicated premium is $(72{,}000+130{,}000)(1.04)=202{,}000(1.04)=\$210{,}080.00$. This also falls between the minimum and maximum, so the aggregate premium is $\$210{,}080.00$.

Check. The aggregate method produces $130{,}000-116{,}235=\$13{,}765$ more limited converted losses. Multiplying by 1.04 gives a premium difference of $\$14{,}315.60$, equal to $210{,}080.00-195{,}764.40$. The premium bounds were checked after applying the tax multiplier, as specified by the premium formula and plan assumptions.

Claim Allocation Can Change the Per-Claim Result

Hold the portfolio total fixed and change only the claim allocation. The aggregate converted-loss calculation is unchanged because the sum of eligible losses is unchanged. A per-claim cap may produce a different result because the amount above the cap is determined separately for each claim.

Worked Example: Same Total, Different Claim Sizes

Compare two hypothetical portfolios, each with total eligible losses of $100,000. Portfolio A has claims of $10,000, $10,000, $10,000, and $70,000. Portfolio B has four claims of $25,000 each. The per-claim cap is $30,000, the LCF is 1.20, and the aggregate cap on converted losses is $110,000. For premium, the basic premium is $50,000 and the tax multiplier is 1.05; minimum and maximum premium provisions do not bind. There is no discounting, mortality assumption, or probability distribution: these are fixed claim amounts.

Portfolio A under the per-claim cap. The limited claims are $10{,}000$, $10{,}000$, $10{,}000$, and $30{,}000$. Their sum is $60{,}000$, so converted limited losses are $60{,}000(1.20)=\$72{,}000$.

Portfolio B under the per-claim cap. Each $25,000 claim is below the $30,000 cap, so the limited total is $4(25{,}000)=100{,}000$. Converted limited losses are $100{,}000(1.20)=\$120{,}000$.

Both portfolios under the aggregate cap. Each has an uncapped total of $100,000. Conversion gives $100{,}000(1.20)=\$120{,}000$, and the aggregate cap reduces this to $\min(120{,}000,110{,}000)=\$110{,}000$.

Premiums and check. Portfolio A’s per-claim premium is $(50{,}000+72{,}000)(1.05)=\$128{,}100$; Portfolio B’s is $(50{,}000+120{,}000)(1.05)=\$178{,}500$. Under the aggregate design, each portfolio’s premium is $(50{,}000+110{,}000)(1.05)=\$168{,}000$. The aggregate results match because both portfolios have the same total loss; the per-claim results differ because the claim sizes differ. The arithmetic also confirms that each limited-loss amount was multiplied by 1.20 before being used in the premium calculation.

Common Mistakes and Practical Checks

A frequent error is to apply the aggregate cap separately to each converted claim. That changes an aggregate portfolio limit into a claim-level limit. Another is to apply a cap to the correct amount but at the wrong point in the sequence—for example, limiting the unconverted portfolio total when the contract specifies a cap on converted losses.

Common exam trap: Suppose total eligible losses are $100,000, the LCF is 1.20, and the aggregate cap is $110,000 on converted losses. A wrong calculation applies the cap before conversion: $\min(100{,}000,110{,}000)(1.20)=\$120{,}000$. The correction is to convert first and then cap: $\min(100{,}000(1.20),110{,}000)=\min(120{,}000,110{,}000)=\$110{,}000$. The stated cap base determines the order.

Before finalizing a comparison, check that every cap is attached to the correct unit: one claim or the entire portfolio. Confirm whether the amount is pre-conversion or post-conversion, preserve the same basic premium and tax multiplier across alternatives, and apply any minimum or maximum premium only where the plan specifies. These checks make the comparison aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus and help prevent a mathematically neat calculation from representing the wrong plan design.

Check Your Understanding

For each problem, identify the cap base and follow its stated sequence.

A hypothetical portfolio has claims of $12,000, $28,000, and $51,000. The per-claim cap is $30,000, the LCF is 1.10, and the aggregate cap on converted losses is $95,000. Find the limited converted losses under each design. (Answer: Per-claim cap: $77,000; aggregate cap: $95,000.)
Use the losses and parameters in Question 1. If basic premium is $40,000 and the tax multiplier is 1.06, with no binding premium bounds, calculate premium under each cap design. (Answer: Per-claim: $124,020.00; aggregate: $143,100.00.)
Two hypothetical portfolios each total $90,000. Portfolio A has claims of $10,000, $20,000, $25,000, and $35,000; Portfolio B has four claims of $22,500. With a $25,000 per-claim cap, LCF 1.20, and aggregate converted-loss cap $100,000, find both limited converted-loss amounts for each portfolio. (Answer: Per-claim: Portfolio A $96,000 and Portfolio B $108,000; aggregate: $100,000 for each portfolio.)
A hypothetical portfolio has $150,000 of eligible losses, an LCF of 0.90, and an aggregate cap of $125,000 on converted losses. Basic premium is $55,000 and the tax multiplier is 1.04, with no binding minimum or maximum. Calculate the capped converted losses and premium. (Answer: Capped converted losses $125,000; premium $187,200.00.)

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