A Worked ILF Indication From Claim Data

From Claim Records to an ILF Indication

An increased limits factor (ILF) expresses how the loss cost at a higher policy limit compares with the loss cost at a selected base limit. Claim records make the calculation concrete: for each claim, determine the amount payable under each limit, total those amounts, and compare the totals. The calculation is simple only after the limits, claim basis, and premium component have been made consistent.

This tutorial develops an empirical indication from hypothetical claim data. It assumes that the claims are ground-up amounts on a common coverage and evaluation basis, and that the limits apply per claim. A real indication also depends on whether the data represent the risks being priced and whether observed claims support the limits being considered.

Definition: For a ground-up claim amount $X$ and a per-claim limit $L$, the limited loss is $Y(L)=\min(X,L)$. Given the same $n$ claims and base and higher limits $L_b<L_h$, the claim-data ILF indication is

\widehat{\mathrm{ILF}}(L_b,L_h) = \frac{\sum_{j=1}^{n}\min(X_j,L_h)} {\sum_{j=1}^{n}\min(X_j,L_b)}.

The denominator must be positive. This ratio compares aggregate limited losses on the specified data; it is not an estimate of every possible cost or premium component by itself.

Why the Ratio of Totals Is the Indication

Under a common claim population, the average limited loss per claim at limit $L$ is the total limited loss divided by the number of claims. Taking the ratio of the higher-limit and base-limit averages therefore cancels the common claim count. This gives a direct empirical estimate of the relative limited loss cost.

\frac{\frac{1}{n}\sum_{j=1}^{n}\min(X_j,L_h)} {\frac{1}{n}\sum_{j=1}^{n}\min(X_j,L_b)} = \frac{\sum_{j=1}^{n}\min(X_j,L_h)} {\sum_{j=1}^{n}\min(X_j,L_b)}.

The justification depends on using the same claims in both sums. If the claim counts or exposure bases differ, the count does not simply cancel; the analyst must first establish a comparable basis. Likewise, if limits include different deductibles, coverage terms, or other adjustments, capping the same ground-up amounts is not enough to make the comparison valid.

For the empirical indication to be useful, the records should be appropriate for the risks and coverage being priced. Claim amounts must be measured consistently, and the base and higher limits must be applied to the same per-claim loss basis. The resulting factor is an indication from the supplied data, not a guarantee that the future loss ratio will equal the observed ratio.

Worked Calculation and Premium Application

The method is to calculate each claim's limited amount at both limits, total each column, divide the higher-limit total by the base-limit total, and apply that factor only to the premium component it is intended to scale. The first example carries out all of these steps.

Worked Example: Five Claims and a Higher-Limit Premium

A hypothetical liability portfolio has five ground-up claims, in thousands of dollars: $12, 28, 45, 75,$ and $130$. Estimate the ILF from a base limit of $\$50{,}000$ to a higher limit of $\$100{,}000$. The base-limit premium for the exposure is $\$2{,}400.00$; assume the full amount is the limit-sensitive premium component to which the ILF is to be applied.

Method: claim-by-claim capping. At the base limit, the limited amounts in thousands are $12, 28, 45, 50,$ and $50$. Their total is $12+28+45+50+50=185$.

At the higher limit, the corresponding amounts are $12, 28, 45, 75,$ and $100$, totaling $12+28+45+75+100=260$.

The indicated factor is $\widehat{\mathrm{ILF}}=260/185=1.405405\ldots$, or $1.4054$ to four decimal places. The higher-limit indication is $1.4054$ times the base-limit amount.

Applying the unrounded ratio to the stated premium gives $\$2{,}400.00(260/185)=\$3{,}372.9729\ldots$, or $\$3{,}372.97$ to the nearest cent.

Check. The higher-limit total, $\$260{,}000$, exceeds the base-limit total, $\$185{,}000$, so the factor is above 1, as expected. The indicated premium also exceeds $\$2{,}400.00$. The common mistake avoided is treating the ILF as a percentage increase: a factor of $1.4054$ means about 140.54% of the base amount, not an increase of 140.54% on top of it.

Aggregation, Weighting, and the Premium Basis

When the records are grouped, calculate the ratio from the aggregate limited losses if the groups together represent the claim population on a comparable basis. Do not generally take an unweighted average of group-level ILFs: that gives a small group's factor the same influence as a large group's factor, regardless of their base-limit loss amounts.

Worked Example: Combining Unequal Claim Groups

A hypothetical insurer reviews two groups of claims, all in thousands of dollars, using a base limit of $\$40{,}000$ and a higher limit of $\$100{,}000$. Group A has ground-up claims $10, 20, 40,$ and $80$. Group B has one claim of $150$. The base-limit premium for the combined portfolio is $\$6{,}000.00$, and the indicated factor applies to the full premium amount.

Method: pool the limited losses before dividing. For Group A, base-limited losses are $10,20,40,40$, totaling $110$. Higher-limited losses are $10,20,40,80$, totaling $150$.

For Group B, the one claim is limited to $40$ at the base limit and $100$ at the higher limit. Thus the portfolio totals are $110+40=150$ at base and $150+100=250$ at the higher limit.

The pooled ILF is $250/150=1.666666\ldots$, or $1.6667$ to four decimal places. Applying the unrounded ratio gives $\$6{,}000.00(250/150)=\$10{,}000.00$.

Check against an incorrect average. Group A's own ratio is $150/110=1.3636\ldots$, while Group B's is $100/40=2.5$. Their unweighted average is $(1.3636\ldots+2.5)/2=1.9318\ldots$, which is not the pooled indication. The pooled result correctly reflects base-limited loss amounts of $110$ and $40$; it can also be written as $(110/150)(150/110)+(40/150)(100/40)=250/150$. The common mistake avoided is averaging subgroup factors without appropriate weights.

Pooling alone does not correct for differences in exposure, claim reporting, or risk mix between groups. If groups are not representative in proportion to the portfolio being priced, the data need to be made comparable before pooling or weighting. A numerical ratio cannot repair a data-basis mismatch.

There is also an important distinction between an ILF for loss cost and an ILF applied to premium. If a base premium includes a fixed expense or other component that does not vary with the limit, multiplying the entire premium by the loss ILF may overstate the higher-limit premium. Apply the factor to the intended limit-sensitive portion, then add any components that remain unchanged. A problem may explicitly prescribe applying the ILF to the full base premium, as in the examples above.

A Second Claim-Data Indication

Capping should be done separately at each stated limit; a claim below the base limit contributes the same amount to both totals, while a claim above the base limit may contribute more at the higher limit. Showing the capped values explicitly is a useful audit trail.

Worked Example: Different Limit Gap and a Low Claim Count

A hypothetical specialty portfolio has four ground-up claims, in thousands of dollars: $8, 18, 32,$ and $90$. Calculate the empirical ILF from a base limit of $\$25{,}000$ to a higher limit of $\$75{,}000$. The base-limit premium is $\$1{,}800.00$, and for this indication the entire premium is assumed limit-sensitive.

Method: cap each observation at each limit. At $\$25{,}000$, the amounts are $8,18,25,25$, with total $8+18+25+25=76$ thousand dollars.

At $\$75{,}000$, they are $8,18,32,75$, with total $8+18+32+75=133$ thousand dollars. Therefore $\widehat{\mathrm{ILF}}=133/76=1.75$.

The premium indication is $\$1{,}800.00(1.75)=\$3{,}150.00$. The limited-loss totals increase from $\$76{,}000$ to $\$133{,}000$, and the premium indication is exactly 1.75 times the base premium, so both checks are consistent.

Common mistake avoided. It would be incorrect to leave the $\$90{,}000$ claim uncapped at the higher limit: its contribution is $\$75{,}000$, not $\$90{,}000$. The policy limit caps the amount included in that limited-loss total.

Interpreting and Checking the Indication

An ILF is a relative factor for a specified pair of limits and a specified basis. It is not the additional limit divided by the base limit, and it is not calculated by dividing the higher limit by the base limit. The claims determine how much of the higher limit is reflected in the observed limited losses.

For a valid calculation with $L_h>L_b$, every claim has $\min(X_j,L_h)\geq\min(X_j,L_b)$. If the denominator is positive, the empirical ILF therefore cannot be less than 1. Equality is possible when no observed claim exceeds the base limit. A factor above 1 indicates that at least one claim contributes more at the higher limit; it does not by itself establish that the sample adequately reflects the future tail.

Check the limits: confirm that the higher limit really exceeds the base limit and that both are per-claim limits on the same coverage.
Check the capping: each claim contributes no more than the applicable limit to its limited-loss total.
Check the denominator: the base limited-loss total must be positive for the ratio to be defined.
Check the premium: identify whether the indicated factor applies to the whole base premium or only to its limit-sensitive component.
Check the result: under these assumptions, the higher-limit total and factor should be at least as large as their base-limit counterparts.

A claim-data ILF indication is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. Its reliability still rests on a sound match between the data and the risks, coverage terms, limits, and premium basis under consideration.

Check Your Understanding

For each problem, use the indicated per-claim limits and claim basis. Round ILFs to four decimal places and premium amounts to the nearest cent where needed.

A hypothetical account has ground-up claims of $\$5{,}000,\$15{,}000,\$40{,}000,$ and $\$90{,}000$. Find the empirical ILF from $\$30{,}000$ to $\$60{,}000$. (Answer: $1.5000$)
Claims are $\$12{,}000,\$22{,}000,\$55{,}000,$ and $\$95{,}000$. Calculate the ILF from a $\$40{,}000$ base limit to an $\$80{,}000$ higher limit. (Answer: $169/114=1.4825$)
At limits of $\$50{,}000$ and $\$100{,}000$, three claims are $\$20{,}000,\$50{,}000,$ and $\$70{,}000$. If the base-limit premium is $\$3{,}600.00$ and the full premium is limit-sensitive, calculate the higher-limit premium. (Answer: $\$4{,}200.00$)
Group A has base- and higher-limit limited-loss totals of $\$100{,}000$ and $\$120{,}000$; Group B has totals of $\$80{,}000$ and $\$160{,}000$. Find the pooled ILF. (Answer: $280/180=1.5556$)
Four claims are $\$40{,}000,\$60{,}000,\$90{,}000,$ and $\$130{,}000$. Using limits of $\$50{,}000$ and $\$100{,}000$, find the ILF and apply it to a fully limit-sensitive base premium of $\$2{,}750.00$. (Answer: ILF $=290/190=1.5263$; higher-limit premium $=\$4{,}197.37$)

What You'll Learn