Empirical Exposure Curves from Ground-Up Claims

From Ground-Up Claims to an Empirical Exposure Curve

An exposure curve describes how the expected loss per claim changes as the applicable limit changes. When complete ground-up claim severities are available, the curve can be estimated directly from the observed claims: cap every claim at each limit, average those capped amounts, and compare each average with the average at a chosen base limit. This provides a transparent alternative to assuming a parametric severity distribution.

The calculation is simple, but the data basis matters. The observations must represent ground-up severities that are not truncated by a policy limit or otherwise missing their upper portions. The method below treats each claim as one equally weighted observation. It does not adjust for policy-level weighting, development, trend, or censoring; those are separate data considerations.

Definition: Given $n$ complete ground-up claim severities $x_1,\ldots,x_n$, the empirical limited expected loss at limit $L$ is the average of the claim amounts capped at $L$: $$ \widehat{E}_n[\min(X,L)] = \frac{1}{n}\sum_{j=1}^{n}\min(x_j,L). $$ For a positive base limit $b$, the empirical exposure curve is $$ \widehat{EC}_n(L;b) = \frac{\widehat{E}_n[\min(X,L)]} {\widehat{E}_n[\min(X,b)]}, $$ provided the denominator is positive. Thus the curve equals 1 at the base limit.

Here $X$ denotes ground-up loss, and $L$ and $b$ must be expressed in the same monetary units as the observations. The calculation uses the empirical distribution that assigns probability $1/n$ to each observed claim. It estimates limited expected loss conditional on the observed claim sample; it is not a count-weighted estimate of total annual losses. If the sample contains zero ultimate losses, they are included as zero observations when that is the intended claim basis.

Calculation Procedure

For each requested limit, apply the cap to each severity separately before calculating the average. A claim below the limit contributes its full ground-up amount; a claim above the limit contributes exactly the limit. The base-limit average is calculated in the same way, and every limited average is divided by that same base average.

Check the data basis.
Confirm that each observation is a complete ground-up severity and identify the common currency unit. A paid amount already capped by a policy limit is not automatically a complete ground-up severity.

Cap each claim.
For each limit $L$, calculate $\min(x_j,L)$ for every observation $j$. Do not cap the sample average; capping after averaging does not generally give the same result.

Average at each limit.
Add the capped amounts and divide by the number of observations $n$. Use the same observations and denominator at every limit.

Normalize to the base limit.
Divide each limited average by the base-limit average. Retain sufficient precision during the calculations and round only the reported result.

Why the Empirical Formula Works

The empirical distribution treats every observed severity as one point of probability $1/n$. Under this distribution, the expectation of the loss capped at $L$ is the probability-weighted average of the $n$ capped observations. Since each has probability $1/n$, that expectation is their arithmetic mean:

\widehat{E}_n[\min(X,L)] = \sum_{j=1}^{n}\frac{1}{n}\min(x_j,L) = \frac{1}{n}\sum_{j=1}^{n}\min(x_j,L).

The exposure curve is defined as limited expected loss at $L$ relative to limited expected loss at $b$. Substituting the empirical averages into that definition gives the ratio in the formula above. The common factor $1/n$ cancels, so an equivalent calculation is the ratio of the sums of capped amounts:

\widehat{EC}_n(L;b) = \frac{\sum_{j=1}^{n}\min(x_j,L)} {\sum_{j=1}^{n}\min(x_j,b)}.

This equivalence is useful as a check, but the averages still help make the meaning clear: the curve compares average capped loss per observed claim, not total loss for a portfolio with a different number of claims. The empirical distribution is the model assumption here; no interest rate, mortality assumption, or parametric severity distribution is used.

Worked Examples

Worked Example: Building a Curve at Three Limits

A hypothetical insurer has five complete ground-up claim severities, recorded in thousands of dollars: $20, 50, 80, 150,$ and $300$. Estimate the empirical exposure curve at limits $50, 100,$ and $200$ thousand dollars, using $50$ thousand dollars as the base. Treat the observations as equally weighted; no interest or mortality assumption applies, and no parametric severity distribution is assumed.

Method. Cap each severity at each limit, average the five capped amounts, and divide by the average at the base limit.

At the base limit of $50$, the capped amounts are $20, 50, 50, 50,$ and $50$. Their sum is $220$, so the base average is $220/5=44$ thousand dollars.

At limit $100$, the capped amounts are $20, 50, 80, 100,$ and $100$. Their sum is $350$, giving an average of $350/5=70$ thousand dollars. The exposure-curve value is $70/44=1.5909$.

At limit $200$, the capped amounts are $20, 50, 80, 150,$ and $200$. Their sum is $500$, giving an average of $500/5=100$ thousand dollars. The curve value is $100/44=2.2727$.

The resulting values are $1.0000$ at $50$, $1.5909$ at $100$, and $2.2727$ at $200$. The base value is exactly 1 because its average is divided by itself. Values are rounded to four decimal places. The calculation avoids the common mistake of dividing each capped claim by the base limit rather than dividing the average capped loss by the base-limit average.

Worked Example: Changing the Base Limit

A hypothetical equipment portfolio has four complete ground-up severities, in thousands of dollars: $10, 40, 90,$ and $160$. Calculate the empirical exposure curve at limits $75$ and $150$ thousand dollars using a base limit of $25$ thousand dollars. Each observation has equal weight; there is no interest or mortality assumption and no fitted distribution.

At the base limit $25$, the capped amounts are $10, 25, 25,$ and $25$. Their sum is $85$, so the base average is $85/4=21.25$ thousand dollars.

At limit $75$, the capped amounts are $10, 40, 75,$ and $75$. The sum is $200$, so the average is $200/4=50$ thousand dollars. Dividing by the base average gives $50/21.25=2.3529$.

At limit $150$, the capped amounts are $10, 40, 90,$ and $150$. The sum is $290$, so the average is $290/4=72.5$ thousand dollars. The curve value is $72.5/21.25=3.4118$.

Thus the estimated values are $2.3529$ at $75$ and $3.4118$ at $150$, rounded to four decimal places. The base itself would have value $1.0000$. This method avoids assuming that the base limit equals the smallest observed claim or the mean severity; it is the capped average at the stated base that belongs in the denominator.

Worked Example: Check the Claim-Level Caps and the Ratio

A hypothetical liability sample consists of complete ground-up severities of $40, 100, 100, 260,$ and $500$ thousand dollars. Estimate values at limits $250$ and $500$ thousand dollars, using $100$ thousand dollars as the base. Use equal claim weights and the empirical distribution only; no interest or mortality assumption applies.

At the base limit $100$, the capped observations are $40, 100, 100, 100,$ and $100$. The sum is $440$, and the average is $440/5=88$ thousand dollars.

At limit $250$, the capped observations are $40, 100, 100, 250,$ and $250$. The sum is $740$, and the average is $740/5=148$ thousand dollars. Therefore, the curve value is $148/88=1.6818$.

At limit $500$, the capped observations are $40, 100, 100, 260,$ and $500$. The sum is $1{,}000$, and the average is $1{,}000/5=200$ thousand dollars. The curve value is $200/88=2.2727$.

The estimates are $1.6818$ at $250$ and $2.2727$ at $500$, rounded to four decimal places. As a check, the uncapped sample mean is also $1{,}000/5=200$ thousand dollars, matching the average at the highest limit because no observation exceeds $500$. The method avoids counting the $500$ claim as $500$ at the $250$ limit: it contributes $250$ there.

Interpretation, Checks, and Common Traps

An empirical curve describes the capped-loss behavior of the observed claim sample. It can be used to compare limited expected losses on a consistent claim basis. It does not, by itself, resolve whether the sample represents the prospective book, whether claims are developed to ultimate, or whether exposure should be weighted at the policy level. Those questions affect whether the curve is appropriate for a particular pricing application.

Several checks follow directly from the construction. At the base limit, the value must equal 1. For nonnegative severities, increasing the limit cannot decrease any individual capped amount, so the empirical average and curve cannot decrease as the limit increases. If every observed severity is at or below a selected limit, the average at that limit is the ordinary sample mean. The curve should also be dimensionless: changing all monetary amounts from dollars to thousands of dollars changes the averages but not their ratio.

Cap before averaging. The observation-by-observation operation is $\min(x_j,L)$. Capping the average severity afterward is generally a different calculation.
Use the same denominator. Every curve value uses the average at the selected base limit, not the average at the limit currently being evaluated.
Keep the sample consistent. Use the same complete observations at all limits. A changing claim count can make a ratio reflect a sample change rather than a limit change.
Do not mistake censored payments for ground-up losses. If a recorded amount is already limited, its unobserved ground-up excess cannot be recovered by treating the recorded amount as complete. This method assumes complete ground-up observations.
Do not confuse per-claim averages with aggregate losses. The empirical formula averages severities. It does not multiply by claim frequency or account for differences in policy exposure.

Common exam trap: Suppose the sample is $30, 70, 180$, with a base limit of $50$ and a comparison limit of $100$. The wrong calculation first averages the raw losses to $280/3$, then caps that average at $100$, producing $93.33$. The correct base capped average is $(30+50+50)/3=43.33$, and the correct comparison average is $(30+70+100)/3=66.67$. The exposure-curve value is $66.67/43.33=1.5385$, not $93.33/43.33$. Each loss must be capped before averaging.

The empirical approach is especially useful when a sufficiently relevant sample of complete ground-up claims is available and the analyst wants a direct, auditable calculation. Its simplicity should not be mistaken for a guarantee of credibility or representativeness: a small sample can be highly sensitive to a few large claims. For exam calculations, show the capped observations, their averages, and the normalization explicitly so that both the data basis and arithmetic can be checked.

Check Your Understanding

For each problem, assume the stated observations are complete ground-up severities with equal weights and are recorded in thousands of dollars. No interest, mortality, or parametric severity assumption is used.

Severities are $12, 30, 70,$ and $100$. Calculate the empirical exposure-curve value at limit $60$, using $30$ as the base limit. (Answer: $1.5882$)
Severities are $25, 60, 100,$ and $220$. Find the average capped loss at the $50$ base limit and the curve value at limit $100$. (Answer: base average $43.75$; curve value $1.6286$)
Severities are $10, 50, 120,$ and $300$. Using $50$ as the base, calculate the curve values at limits $100$ and $200$. (Answer: $1.6250$ and $2.3750$)
Severities are $80, 80, 160, 400,$ and $500$. Calculate the base average at limit $100$ and the curve value at limit $300$. (Answer: base average $92$; curve value $2.0000$)
Ultimate ground-up severities, including one zero amount, are $0, 20, 60,$ and $120$. Using a base limit of $40$, calculate the average capped loss at limit $100$ and the empirical curve value there. (Answer: average $45$; curve value $1.8000$)

What You'll Learn