Large Claims And Credibility

Why Large Claims Matter in Small Segments

A small rating segment can have only a few claims in its experience period. In that setting, one unusually large claim may account for most of the measured losses. The resulting average severity and loss per exposure can change sharply, even though the number of claims—and therefore the observed frequency—has not changed. A sound analysis keeps these measures separate and states exactly how the large claim is treated.

The examples below use hypothetical claim records. They are empirical calculations: no probability distribution is assumed, and no interest or mortality assumption is used. The loss amounts are on a common measurement basis, and exposures refer to the same segment and period as the claims. In practice, experience also needs to be sufficiently complete and comparable before it is used to assess a segment.

Definition: For a segment with $E$ exposure units, $N$ claims, and measured claim losses $x_1,\ldots,x_N$, the observed frequency is $N/E$, average severity is $\sum_{i=1}^{N}x_i/N$ when $N>0$, and loss per exposure is $\sum_{i=1}^{N}x_i/E$. These quantities must use a clearly specified and consistent claim population and loss basis. When the same claims and losses are used in all three measures, loss per exposure equals frequency multiplied by average severity.

Frequency answers how often claims occurred per exposure unit. Severity answers how large the measured losses were per claim. Loss per exposure, also called the observed pure premium on this basis, combines both. A change in loss per exposure does not by itself show whether frequency, severity, or both changed.

Why One Claim Can Dominate the Result

Suppose a segment has a fixed exposure count and a small number of claims. Adding a large claim increases the loss total and the number of claims by one. The size of the claim can make the loss total increase far more than the count. Consequently, severity can rise substantially; frequency also rises by one claim divided by exposure, but it does not rise in proportion to the claim’s dollar amount.

\text{Frequency}=\frac{N}{E},\qquad \text{Average severity}=\frac{L}{N},\qquad \text{Loss per exposure}=\frac{L}{E}

Here $L$ is the total measured loss for the claims included under the selected treatment. The identity connecting the measures follows directly from cancellation of $N$, provided $N>0$ and all three quantities use the same $L$, $N$, and $E$:

\frac{N}{E}\times\frac{L}{N}=\frac{L}{E}.

If no claims occur, observed frequency and loss per exposure are both zero, while average severity is undefined because its denominator is zero. The identity should not be used to assign a severity in that case.

Worked Example: Measuring a Large Claim With a Per-Claim Cap

A hypothetical equipment-service segment has 80 exposure units and four claims of $2,400, $3,600, $4,000, and $30,000. For a sensitivity measure, losses are capped at $8,000 per claim, but each claim remains in the count. Calculate uncapped and capped frequency, severity, and loss per exposure. No probability distribution, interest rate, or mortality assumption is used.

Method. First calculate the uncapped measures from all four claims. Then replace each claim amount $x_i$ by $\min(x_i,8000)$ for the capped loss measures, without changing $N$ or $E$.

Uncapped total. The losses sum to $2,400+3,600+4,000+30,000=40,000$ dollars. There are $N=4$ claims and $E=80$ exposures.

Uncapped frequency. $4/80=0.05$ claims per exposure.

Uncapped average severity. $40,000/4=10,000$ dollars per claim.

Uncapped loss per exposure. $40,000/80=500$ dollars per exposure. The component check gives $0.05\times10,000=500$.

Capped total. The capped claim amounts are $2,400, $3,600, $4,000, and $8,000. Their sum is $2,400+3,600+4,000+8,000=18,000$ dollars.

Capped frequency. The claim count has not changed, so $4/80=0.05$.

Capped average severity. $18,000/4=4,500$ dollars per claim.

Capped loss per exposure. $18,000/80=225$ dollars per exposure. The check is $0.05\times4,500=225$.

Check and mistake avoided. The cap reduces the largest claim’s measured amount by $22,000, which reduces the total from $40,000 to $18,000. It does not erase the claim from the count. Recomputing frequency with only three claims would apply a different treatment from the one specified.

A per-claim cap is one way to examine how much a large claim drives a result. It is not a claim that the capped amount is the ultimate cost, nor does the calculation establish that the claim should be ignored. The threshold and treatment should be chosen for the purpose of the analysis and applied consistently. A large claim may be an unusual random outcome, but it may also indicate a genuine feature of the segment’s risk.

Excluding a Loss Is Not the Same as Excluding a Claim

The phrase “exclude the large claim” can refer to different calculations. An analyst might remove the claim’s dollars from the loss measure while retaining the reported claim in the count. Alternatively, the analyst might remove the claim from both the loss amount and the claim count. Those choices produce different severity and frequency measures. The exposure denominator normally remains the segment’s exposure for the period; removing one claim does not remove the policy or exposure that generated it.

The central discipline is to state the rule before calculating. If the loss amount is excluded but the claim remains counted, severity is a defined measure of retained losses per counted claim; it is not the average severity of only the retained-loss claims. The product identity still holds if loss per exposure is calculated from those same retained losses and severity uses the retained claim count.

Worked Example: Excluding a Claim’s Loss but Retaining Its Count

A hypothetical small-business segment has 40 exposure units and three claims of $1,000, $2,000, and $17,000. The analysis excludes the $17,000 amount from measured losses but explicitly retains that claim in the count. Calculate the measured total, frequency, measured average severity, and measured loss per exposure. No probability distribution, interest rate, or mortality assumption is used.

Method. Apply the stated loss-only exclusion: the measured losses are $1,000 and $2,000, while the count remains $N=3$. The exposure denominator is $E=40$.

Measured total. $1,000+2,000=3,000$ dollars.

Frequency. Since all three claims remain counted, $3/40=0.075$ claims per exposure.

Measured average severity. Divide the $3,000 measured loss total by the specified count of three: $3,000/3=1,000$ dollars per counted claim.

Measured loss per exposure. $3,000/40=75$ dollars per exposure. The consistency check is $0.075\times1,000=75$.

Check and mistake avoided. The excluded claim contributes no dollars to the measured total, but it still contributes one to the specified count. Using $20,000 as the loss total would mistakenly put the excluded $17,000 back into the measure. Using a count of two would change the stated treatment. If the claim were instead excluded from both losses and count, frequency would be $2/40=0.05$ and severity would be $3,000/2=1,500$, while loss per exposure would remain $3,000/40=75$.

The last comparison illustrates why labels matter. “Average severity” can mean different things under different large-claim treatments. Always identify the numerator, the claim count in the denominator, and whether the count includes claims whose losses were excluded or capped.

Comparing Frequency and Severity Effects

A large claim directly affects a loss-based severity measure through its dollar amount. It affects observed frequency only through the fact that it is one claim. A claim of $38,000 and a claim of $3,800 each add one claim to the count, even though their loss amounts differ by a factor of ten. This distinction is important when comparing segments: a high loss per exposure may reflect more claims, larger claims, or both.

Worked Example: Separating Frequency From Severity

A hypothetical specialty repair segment has 100 exposures and two claims of $2,000 and $38,000. For a limited-loss view, cap each claim at $10,000 and retain both claims in the count. A comparison segment has 100 exposures and five claims, each with measured losses of $4,000. Calculate frequency, average severity, and loss per exposure for the first segment before and after the cap, then compare the capped results with the second segment. No probability distribution, interest rate, or mortality assumption is used.

Uncapped first segment. Total losses are $2,000+38,000=40,000$ dollars. Frequency is $2/100=0.02$, average severity is $40,000/2=20,000$ dollars, and loss per exposure is $40,000/100=400$ dollars. The check is $0.02\times20,000=400$.

Capped first segment. The capped losses are $2,000 and $10,000, so the measured total is $2,000+10,000=12,000$ dollars. Frequency remains $2/100=0.02$. Capped average severity is $12,000/2=6,000$ dollars, and capped loss per exposure is $12,000/100=120$ dollars. The check is $0.02\times6,000=120$.

Comparison segment. Its total losses are $5\times4,000=20,000$ dollars. Frequency is $5/100=0.05$, average severity is $20,000/5=4,000$ dollars, and loss per exposure is $20,000/100=200$ dollars. The check is $0.05\times4,000=200$.

Interpretation and check. After the stated cap, the first segment has lower observed frequency, $0.02$ versus $0.05$, but higher average severity, $6,000 versus $4,000. Its capped loss per exposure is $120, compared with $200 for the second segment. The comparison separates the two components; it does not, by itself, establish that either segment’s future experience will match its observed values.

Interpreting Large Claims in Credibility Work

Large claims create volatility in small segments because each claim is a substantial share of the observed total. That volatility is relevant to how much weight to place on segment experience, but the size of a claim alone does not determine the appropriate credibility treatment. A large loss can be random fluctuation around a stable risk level, or it can reveal structural differences in exposure, coverage, operations, or claim handling. The data and the intended measure must be examined before deciding how to use it.

For a transparent analysis, report the ordinary experience and any sensitivity view separately. For example, show uncapped losses alongside losses capped at a stated amount, identify whether the claim count is unchanged, and explain the effect on frequency, severity, and loss per exposure. This makes clear what the adjustment does and avoids presenting a sensitivity calculation as if it were the unmodified observed experience.

Define the measure.
Specify the exposure base, claim population, loss basis, and period being analyzed.

Choose the large-claim treatment.
State whether the claim is retained in full, capped, or excluded, and say explicitly whether it remains in the count.

Recalculate each component.
Compute frequency, severity, and loss per exposure using the chosen definitions; do not infer one component from a change in another.

Check consistency and interpret cautiously.
When numerator and denominators align, verify that frequency times severity equals loss per exposure. Treat the result as experience evidence, not automatic proof of a structural change.

Common Exam Trap

The most common error is to change the loss numerator but leave the claim count implicit. A second error is to treat a large dollar amount as if it were many claims, thereby confusing severity with frequency. Before calculating, write down what counts as a claim and what dollars enter the loss total.

Key takeaway: A large claim can dominate measured severity and loss per exposure while adding only one claim to frequency. Capping or excluding its dollars does not automatically remove it from the claim count; the count treatment must be stated and used consistently.

Check Your Understanding

For each problem, use the specified treatment and distinguish the effect on claim count from the effect on measured losses.

A hypothetical segment has 60 exposures and claims of $1,500, $2,500, and $20,000. Cap losses at $5,000 per claim and retain every claim in the count. Find capped total losses, frequency, capped average severity, and capped loss per exposure. (Answer: Total = $9,000; frequency = 0.05; capped average severity = $3,000; capped loss per exposure = $150.)
A hypothetical segment has 30 exposures and four claims of $800, $1,200, $2,000, and $10,000. Exclude the $10,000 loss amount but retain its claim in the count. Find measured losses, frequency, measured average severity, and measured loss per exposure. (Answer: Measured losses = $4,000; frequency = 4/30 = 0.1333; measured average severity = $1,000; measured loss per exposure = $133.33.)
A hypothetical segment has 80 exposures and two claims of $3,000 and $27,000. Exclude the $27,000 claim from both count and losses. Find the remaining frequency, average severity, and loss per exposure. (Answer: Frequency = 1/80 = 0.0125; average severity = $3,000; loss per exposure = $37.50.)
Segment A has 100 exposures and four claims with total measured losses of $32,000. Segment B has 100 exposures and eight claims with total measured losses of $40,000. Find frequency, average severity, and loss per exposure for each, then identify which has higher frequency and which has higher severity. (Answer: A: frequency = 0.04, severity = $8,000, loss per exposure = $320; B: frequency = 0.08, severity = $5,000, loss per exposure = $400. B has higher frequency; A has higher severity.)
A hypothetical segment has 50 exposures and three claims of $2,000, $6,000, and $22,000. Cap each loss at $10,000 while retaining all claims. Compare capped and uncapped severity and loss per exposure, and state whether frequency changes. (Answer: Uncapped total = $30,000, severity = $10,000, loss per exposure = $600; capped total = $18,000, severity = $6,000, loss per exposure = $360; frequency remains 3/50 = 0.06.)

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