Using Limited Loss Data in Severity Estimates

Why the Observation Limit Matters

Claim records do not always preserve each claim’s full ground-up amount. A data extract may record each loss only up to a specified per-claim observation limit, $u$. In that case, a claim larger than $u$ appears as $u$, not as its actual amount. The resulting average is a valid severity measure, but it measures limited losses rather than full ground-up losses.

That distinction matters when severity is combined with frequency. A frequency count of all claims can be paired with average limited severity to estimate limited aggregate losses. It cannot, without an additional adjustment, be paired with that severity to represent uncapped aggregate losses. The count and the severity must describe the same claim population and payment basis.

Definition: Let $X$ be a nonnegative ground-up loss and $u$ a per-claim observation limit. The recorded limited loss is $Y=\min(X,u)$. For $n$ claims, the limited average severity is $\sum_{j=1}^{n}\min(X_j,u)/n$. It is the average amount recorded per claim under the cap, not necessarily the average ground-up loss.

This tutorial treats the limit as a top-coding rule: every claim remains in the count, but its recorded amount cannot exceed $u$. It is not a deductible, which removes an initial portion of a loss, and it is not truncation, where claims past a threshold may be omitted from the data entirely. Those data-handling rules require different calculations.

Calculating Limited Severity from Capped Records

When a record equal to $u$ represents a claim that reached or exceeded the cap, sum the recorded limited amounts and divide by the number of claims in the extract. The denominator includes claims recorded at the limit. If the extract reports only claims below a threshold, rather than retaining all claims with capped amounts, this calculation does not describe severity for all claims.

Worked Example: Average Severity with One Capped Claim

A hypothetical insurer studies six claims. Amounts are recorded in thousands of dollars and capped at $u=\$20{,}000$. The recorded values are $\$3{,}200$, $\$5{,}600$, $\$8{,}400$, $\$11{,}800$, $\$17{,}500$, and $\$20{,}000$. The last record is flagged as having reached the cap. Calculate the limited total and limited average severity.

Method. Treat each recorded amount as $Y_j=\min(X_j,u)$, retain all six claims in the denominator, and calculate the arithmetic mean of the limited amounts.

The limited total is

3{,}200+5{,}600+8{,}400+11{,}800+17{,}500+20{,}000 =\$66{,}500.

There are $n=6$ claims, including the one recorded at the limit. Therefore,

\text{limited average severity} =\frac{\$66{,}500}{6} =\$11{,}083.33.

The answer is $\$11{,}083.33$ per claim, rounded to the nearest cent. This is not an estimate of the uncapped mean: the recorded $\$20{,}000$ does not reveal how far the underlying claim exceeded the cap. Common mistake avoided: dropping the capped claim would use five claims as the denominator and would no longer calculate the average limited loss for all six claims.

A useful check is that limited severity cannot exceed the limit: $0\leq Y_j\leq u$, so $0\leq \bar{Y}\leq u$. If a calculated average exceeds $u$, either the amounts were not actually limited at $u$, the units were mixed, or the calculation is wrong.

Keeping Frequency and Severity on the Same Basis

Suppose a portfolio has $N$ exposures and $C$ claims, with every claim represented in both the count and the limited-loss total. Define claim frequency as $C/N$ and limited severity as the limited total divided by $C$. Their product equals limited loss per exposure, because the claim count cancels:

\frac{C}{N}\times\frac{\sum_{j=1}^{C}\min(X_j,u)}{C} = \frac{\sum_{j=1}^{C}\min(X_j,u)}{N}.

This identity is a consistency check, not a way to recover losses above the cap. Its left side estimates limited aggregate losses per exposure. For a ground-up estimate, the severity component must represent ground-up losses, or a defensible adjustment for losses above $u$ must be added.

Worked Example: Limited Pure Premium and the Basis of the Result

A hypothetical book has 4,000 earned exposures and 120 claims. Its claim records are top-coded at $\$25{,}000$, and the sum of the 120 recorded amounts is $\$1{,}440{,}000$. Assume the claim count includes all claims in the limited-loss total. Calculate frequency, limited severity, and limited loss per exposure.

Step 1: Calculate frequency. Divide the 120 claims by 4,000 exposures:

\text{frequency}=\frac{120}{4{,}000}=0.03 \quad\text{claims per exposure}.

Step 2: Calculate limited severity. Divide the limited total by the 120 claims:

\text{limited severity} =\frac{\$1{,}440{,}000}{120} =\$12{,}000 \text{ per claim}.

Step 3: Combine the components.

0.03\times \$12{,}000 =\$360 \text{ per exposure}.

Check directly: $\$1{,}440{,}000/4{,}000=\$360$ per exposure. The agreement confirms the arithmetic and the basis: $\$360$ is limited loss per exposure. Common mistake avoided: labeling this amount as ground-up loss per exposure. The calculation contains no information about the portions of claims above $\$25{,}000$.

Key distinction: An uncapped claim count multiplied by limited severity estimates limited aggregate losses per exposure, provided the count covers exactly the claims represented in the limited-loss total. It does not automatically estimate uncapped aggregate losses. If the count also omits claims, or uses a different claim definition, even the limited-loss interpretation may fail.

What Capped Data Can and Cannot Tell You

If the observation rule is $Y=\min(X,u)$, values below $u$ reveal their ground-up losses, while records at $u$ generally reveal only that the loss reached the cap. Thus, capped records identify the limited average but do not, by themselves, identify the uncapped average. Many different sets of losses above $u$ could produce exactly the same capped records.

The distinction can be written as a decomposition. For a nonnegative loss,

X=\min(X,u)+(X-u)_+, \qquad (X-u)_+=\max(X-u,0).

Taking expectations gives $E[X]=E[\min(X,u)]+E[(X-u)_+]$. For a continuous severity distribution, or where records separately flag claims exceeding the cap, the excess component can be expressed as

E[(X-u)_+] =P(X>u)\,E[X-u\mid X>u].

The first factor is the probability of exceeding the cap; the second is the mean amount above the cap conditional on exceeding it. Capped data can help estimate the first factor if the exceedance flag and all claims are retained. They do not show the conditional excess amount. Estimating that part requires supplementary uncapped data or an explicit tail model. If losses can have a point mass exactly at $u$, a record equal to $u$ alone may not distinguish a loss exactly at the limit from one above it; an exceedance indicator or an appropriate model is then needed.

Estimating Ground-Up Severity Requires an Explicit Tail Assumption

One possible assumption is that, conditional on exceeding $u$, the excess $X-u$ has an exponential distribution with mean $\theta$. Under that model, $E[X-u\mid X>u]=\theta$. This is a model for the excess above the limit, not a conclusion that can be read directly from the capped amounts. The estimate is appropriate only if the assumed tail is suitable for the portfolio and period being studied.

Worked Example: Adjusting Limited Severity with a Stated Tail Model

In a hypothetical portfolio, 10 claims are all retained and capped at $u=\$50{,}000$. Their recorded limited amounts sum to $\$280{,}000$, and five claims are flagged as exceeding the cap. Assume, solely for this estimate, that conditional excesses above $\$50{,}000$ follow an exponential distribution with mean $\theta=\$12{,}000$. Estimate the mean ground-up severity using the empirical capped data and this tail assumption.

Step 1: Calculate observed limited severity.

\bar{Y}=\frac{\$280{,}000}{10}=\$28{,}000.

Step 2: Estimate the exceedance proportion. Five of the 10 claims exceeded the limit, so the empirical proportion is $5/10=0.5$.

Step 3: Add estimated excess per claim. Under the stated exponential model, the mean excess among exceeders is $\$12{,}000$. Averaged across all claims, the estimated excess is $0.5\times \$12{,}000=\$6{,}000$ per claim.

Step 4: Estimate ground-up severity.

\widehat{E[X]} =\bar{Y}+\left(\frac{5}{10}\right)\theta =\$28{,}000+\$6{,}000 =\$34{,}000.

As a check, the limited total is $\$280{,}000$, and the estimated total excess is $5\times \$12{,}000=\$60{,}000$. The estimated ground-up total is therefore $\$340{,}000$, or $\$34{,}000$ per claim. The result exceeds limited severity, as it must under this positive-excess model. It is a model-based estimate, not an observed average. Common mistake avoided: adding $\$12{,}000$ to every claim. The assumed mean excess applies only to claims above the cap, so its portfolio-wide contribution is weighted by the exceedance proportion.

Data Rules and Exam Traps

Before calculating a severity, identify what the observation limit did to the claim records. A top-coded extract retains every claim but caps its amount. A truncated extract may omit claims above a threshold; its claim count and average are then conditional on inclusion unless missing claims are restored or otherwise accounted for. Confusing these rules changes both the numerator and denominator.

Also check whether the comparison basis is consistent. The limit, amounts, and claim count must refer to the same coverage, claim definition, and dollar basis. For example, a capped historical amount cannot be treated as an uncapped amount simply by combining it with an uncapped count. If a limit is stated in a different dollar basis from the losses, the limit must first be interpreted consistently with the amounts being analyzed.

Wrong: divide limited losses by only claims below the cap. This excludes capped claims from the denominator. For all-claim limited severity, divide by all claims represented in the limited total.
Wrong: use limited severity as ground-up severity. The capped data conceal excess amounts. Add an excess estimate only when the tail information or model is stated and appropriate.
Wrong: treat a capped value as the exact underlying loss. A record at $u$ may stand for any loss at or above the cap. The cap does not reveal the amount beyond it.
Wrong: use a truncated claim count with a capped-loss total as if the bases matched. Confirm that the count includes the same claims represented in the loss total.

Key takeaway: The average of $\min(X,u)$ is a limited severity estimate. Pair it with a claim count that covers the same claims to obtain limited aggregate losses. Estimating ground-up severity requires information about, or an explicit assumption for, losses above the observation limit.

Check Your Understanding

For each problem, identify the loss basis as well as the requested numerical result.

A hypothetical extract contains five claims with limited amounts of $\$2{,}000$, $\$4{,}500$, $\$9{,}000$, $\$15{,}000$, and $\$15{,}000$, under a $\$15{,}000$ cap. The last two claims are flagged as capped. Calculate limited total, limited average severity, and the proportion capped. (Answer: limited total $\$45{,}500$; limited severity $\$9{,}100$; proportion capped $0.4000$.)
A hypothetical portfolio has 2,000 exposures and 84 claims. All claims are retained, capped at $\$20{,}000$, and the limited total is $\$756{,}000$. Calculate frequency, limited severity, and limited loss per exposure. (Answer: frequency $0.042$ claims per exposure; limited severity $\$9{,}000$; limited loss per exposure $\$378$.)
Twenty claims are retained in a hypothetical extract capped at $\$30{,}000$. The limited total is $\$320{,}000$, and five claims are flagged as exceedances. Assume the mean excess conditional on exceeding the cap is $\$10{,}000$. Estimate ground-up average severity using this tail assumption. (Answer: limited severity $\$16{,}000$; estimated ground-up severity $\$18{,}500$.)
A capped record equals $\$40{,}000$, and the extract does not flag exceedances. Explain why this single record cannot establish that the ground-up loss was exactly $\$40{,}000$, and name the additional information needed to estimate an uncapped mean. (Answer: it may represent any loss at or above $\$40{,}000$; an exceedance indicator plus information about the excess amount, or an explicit tail model, is needed.)
A hypothetical file reports 30 claims below a $\$10{,}000$ threshold, omits claims at or above it, and gives the sum of the 30 retained losses as $\$120{,}000$. Calculate the average among retained claims and explain whether it is the all-claim average. (Answer: $\$4{,}000$ per retained claim; it is not the all-claim average because claims at or above the threshold are truncated from the file.)

What You'll Learn