Empirical ILFs From Uncapped Claim Data

Why Empirical Limited Averages Matter

An increased limits factor (ILF) compares the expected loss per claim at one policy limit with the expected loss per claim at another. When a credible severity model is unavailable or an empirical check is needed, observed ground-up claim amounts can be capped at each limit and averaged. The order is essential: cap each claim separately, then average the capped amounts.

This calculation uses claim severities, not annual policy losses. A claim-weighted average gives each claim one contribution. A policy-year average instead gives each policy year one contribution, which may include several claims or none. The two quantities answer different questions and generally are not equal.

Definition: Given $n$ uncapped ground-up claim amounts $x_1,\ldots,x_n$ and a per-claim limit $u$, the empirical limited average loss is $$ \widehat{m}(u)=\frac{1}{n}\sum_{k=1}^{n}\min(x_k,u). $$ For two limits $u_2>u_1$, the empirical ILF at $u_2$ relative to $u_1$ is $$ \widehat{\operatorname{ILF}}(u_2;u_1)=\frac{\widehat{m}(u_2)}{\widehat{m}(u_1)}, $$ provided $\widehat{m}(u_1)>0$.

The observations must be ground-up amounts that are not already capped at a policy limit; otherwise the true severity above that cap is not present in the data. The formula treats each observed claim as one claim, uses the same monetary basis and claim definition at every limit, and assumes the observations are representative of the claim population being priced. It makes no parametric distribution assumption: the observed claims define an empirical severity distribution. There is no interest or mortality assumption in this calculation.

From Claim Amounts to Limited Averages

For every selected limit, replace each claim amount $x_k$ with $\min(x_k,u)$. A claim below the limit remains unchanged; a claim above the limit contributes exactly the limit. Then add those limited amounts and divide by the number of claims. To calculate an ILF, divide the limited average at the higher limit by the limited average at the lower limit.

Worked Example: Two Limits From Ground-Up Claims

Hypothetical scenario. An analyst has four uncapped ground-up claims of $20$, $20$, $80$, and $140$ thousand dollars. Calculate the limited average at limits of $50$ and $100$ thousand dollars, then calculate the ILF at $100$ relative to $50$. Treat the four claims as the complete empirical sample; no parametric severity distribution, interest rate, or mortality assumption is used.

At a limit of $50$. The limited amounts, in thousands of dollars, are $20,20,50,50$. Their sum is $20+20+50+50=140$. There are four claims, so the limited average is $140/4=35$ thousand dollars.

At a limit of $100$. The limited amounts are $20,20,80,100$. Their sum is $20+20+80+100=220$, so the limited average is $220/4=55$ thousand dollars.

ILF. The higher-limit average divided by the lower-limit average is $55/35=1.571428\ldots$, or 1.5714 to four decimal places. The result is greater than one, as expected when the limit increases. As a check, both averages lie between zero and their respective limits.

Method and trap avoided. This is a claim-by-claim cap followed by an arithmetic mean. Do not average the uncapped amounts first and then cap that average: in general, $\frac{1}{n}\sum_k\min(x_k,u)$ is not equal to $\min(\frac{1}{n}\sum_k x_k,u)$.

Why the Empirical Formula Works

The formula is the expected limited severity under the empirical distribution that places probability $1/n$ on each observed claim. This interpretation explains both the averaging step and its limits: it estimates a claim-level expectation for the population represented by the sample, rather than an annual aggregate loss.

Define the empirical distribution.
Let $F_n$ assign probability $1/n$ to each observed amount $x_k$. Repeated values receive their corresponding repeated probability mass.

Apply the payment cap.
At limit $u$, each observation contributes $\min(x_k,u)$. The expected value under $F_n$ is therefore $\int \min(x,u)\,dF_n(x)$.

Evaluate the empirical expectation.
Because each observation has mass $1/n$, the integral is $\sum_{k=1}^n (1/n)\min(x_k,u)$, which equals $\widehat{m}(u)$.

Form the factor.
For the same claim population and basis, the relative limited average is $\widehat{m}(u_2)/\widehat{m}(u_1)$. The claim count cancels when the two averages are written as sums divided by the same $n$.

This derivation establishes what the calculation measures; it does not guarantee that the sample is representative. A small or selectively reported sample can produce an unstable estimate, especially when relatively few claims approach the higher limit. The empirical calculation also does not fill in unsupported tail behavior beyond the observed data.

Claim-Weighted Averages and Policy-Year Averages

Suppose claims are split into groups, such as territories or classes. If group $g$ has $n_g$ claims and limited average $\widehat{m}_g(u)$, the combined claim-weighted limited average is $\sum_g n_g\widehat{m}_g(u)/\sum_g n_g$. A simple average of group averages gives every group equal weight, regardless of how many claims it contains; that is not the claim-level average unless the group sizes are equal.

For policy years, let $N_j$ be the number of claims in year $j$, and let $x_{jk}$ be claim $k$'s ground-up amount in that year. The average limited loss per policy-year across $J$ equally weighted years is $\frac{1}{J}\sum_{j=1}^J\sum_{k=1}^{N_j}\min(x_{jk},u)$. It includes zero-claim years as zero. The claim-level limited average instead divides the total limited claim amount by $\sum_j N_j$. When there are claims, the policy-year average equals the claim-level average multiplied by the average number of claims per year, $(\sum_jN_j)/J$. Thus, the two measures are not interchangeable.

Worked Example: Combining Groups With Unequal Claim Counts

Hypothetical scenario. A portfolio has one claim of $20$ thousand dollars in Group A and three claims of $70$, $90$, and $130$ thousand dollars in Group B. At a limit of $50$ thousand dollars, calculate the overall claim-weighted limited average and compare it with the unweighted average of the two group means. The observed amounts are the full empirical sample; no parametric distribution, interest rate, or mortality assumption is used.

Group A. Its one claim is below the limit, so the limited amount is $20$, and its group mean is $20/1=20$ thousand dollars.

Group B. Its three limited amounts are $50,50,50$, with total $150$. The group mean is $150/3=50$ thousand dollars.

Claim-weighted result. Across all four claims, the limited amounts are $20,50,50,50$. Their sum is $20+50+50+50=170$ thousand dollars, so the overall claim-weighted mean is $170/4=42.5$ thousand dollars.

Unweighted group result. Giving the two groups equal weight gives $(20+50)/2=35$ thousand dollars. This differs from $42.5$ because Group A has one claim while Group B has three. The unweighted result answers a different question: the mean of the two group means, not the mean limited severity across claims.

Method and trap avoided. Weight each group mean by its claim count when the target is the average claim. The arithmetic check is $170$ for the four limited amounts, not $220$.

Worked Example: Claim Average Versus Policy-Year Average

Hypothetical scenario. Consider three equally weighted policy years. Year 1 has claims of $30$ and $80$ thousand dollars; Year 2 has no claims; Year 3 has claims of $100$ and $20$ thousand dollars. At a per-claim limit of $40$ thousand dollars, calculate the claim-level limited average and the average limited loss per policy-year. The ground-up claims are uncapped; no parametric severity distribution, interest rate, or mortality assumption is used.

Limit each claim. Year 1 contributes $30$ and $40$, for a limited annual total of $30+40=70$. Year 2 contributes $0$. Year 3 contributes $40$ and $20$, for a limited annual total of $40+20=60$. Across all years, the limited total is $70+0+60=130$ thousand dollars, and the claim count is four.

Claim-level limited average. Divide by the four claims: $130/4=32.5$ thousand dollars per claim.

Policy-year limited average. Divide the same total by the three policy years, including the zero-claim year: $130/3=43.333\ldots$ thousand dollars per policy-year, or $43.33$ to the nearest cent.

Check and interpretation. The average claim count is $4/3$ per policy-year, and $32.5(4/3)=43.333\ldots$, matching the policy-year calculation. The two averages differ because their denominators differ. Dividing by four when asked for a policy-year average, or omitting the zero-claim year, would answer the wrong question.

Practical Checks and Common Errors

Before accepting an empirical ILF, verify the data basis and the arithmetic. The comparison is meaningful only when both limited averages use the same claim population, claim definition, and dollar basis. If observations are already capped, they cannot reveal how much a claim above that cap would have been; an uncapped ground-up severity file is required for this direct calculation.

Cap each claim, not the mean. Apply $\min(x_k,u)$ to every observation before summing.
Use the right denominator. Divide by claim count for average limited severity per claim; divide by policy-year count for average limited loss per policy-year.
Weight groups by claim count. An unweighted average of subgroup means generally does not equal the combined claim mean.
Keep units consistent. If amounts are in thousands of dollars, the limited averages are also in thousands; the ratio is dimensionless.
Check the ratio direction. With a positive lower-limit average and $u_2>u_1$, each limited payment at $u_2$ is at least the corresponding payment at $u_1$. The resulting ILF should therefore be at least one.

The empirical ILF is useful as a direct summary of observed claim severities, but it can be sensitive to the number and size of claims near the limits. Report the claim count and the limits alongside the result so that the basis is clear. This empirical calculation is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, calculate from the stated ground-up claims and use the specified denominator. Amounts are in thousands of dollars unless stated otherwise.

A hypothetical sample has claims of $12,38,75,130$. Find the limited averages at $60$ and $100$, then the ILF at $100$ relative to $60$. (Answer: limited averages $42.5$ and $56.25$; ILF $1.3235$.)
At a limit of $50$, Group A has claims of $25,75$, and Group B has claims of $10,20,30,90$. Find each group mean and the combined claim-weighted mean. (Answer: Group A $37.5$, Group B $27.5$, combined $30.8333$.)
Four equally weighted policy years have claims as follows: Year 1: $30,80$; Year 2: none; Year 3: $100,20$; Year 4: $60$. At a limit of $40$, find the claim-level limited average and average limited loss per policy-year. (Answer: $34$ per claim and $42.5$ per policy-year.)
A hypothetical uncapped sample contains claims of $40,80,120$. Find the limited averages at $50$ and $100$, and the ILF at $100$ relative to $50$. (Answer: limited averages $46.6667$ and $73.3333$; ILF $1.5714$.)
At a limit of $35$, a sample contains claims of $15,45,60,90,120$. Find the total limited loss and the empirical limited average per claim. (Answer: total $155$; limited average $31$.)

What You'll Learn