Why the Definition of Severity Matters
A severity estimate is not fully defined by a dollar amount and a count. The dollar amount might represent payments made so far, incurred losses at a valuation date, or estimated ultimate losses. The count might include every reported claim, only claims with a payment, or claims estimated to have occurred ultimately. Changing either choice can change the average, even when the underlying claim experience has not changed.
For an actuarial analysis, therefore, name both parts of the calculation: the loss measure in the numerator and the population counted in the denominator. This tutorial focuses on that definition and its consequences. It does not assume that different measures will produce the same severity, or that one is always preferable.
The ratio is an arithmetic average: it allocates the aggregate loss amount evenly across the selected count for purposes of summarizing experience. It does not say that every claim has that amount, nor does it describe the spread of individual claim sizes. Unless the analysis states otherwise, loss totals should have a consistent definition—for example, whether allocated loss adjustment expenses are included—and should be stated on a consistent valuation and currency basis.
Paid, Incurred, and Ultimate Loss Measures
These three measures refer to different views of the amount associated with claims. Their meanings should be fixed before dividing by a claim count.
| Loss measure | What the total represents | Interpretive caution |
|---|---|---|
| Paid loss | Payments made by the valuation date for the claims in scope | Claims with no payment yet can still be included in the claim population, if that is the stated denominator. |
| Incurred loss | Paid loss to date plus outstanding case reserves at the valuation date, under the stated accounting definition | It is a current estimate, not necessarily the final amount paid for the claims. |
| Ultimate loss | The estimated total loss after all future payments and claim development for the claims in scope | It is an estimate of the final amount, not simply the cash paid so far. |
A reported-claim count generally counts claims reported by a specified valuation date. A paid-claim count must also state what “paid” means: for example, claims with at least one payment by that date. An ultimate claim count is an estimate of the number of claims attributable to the experience period after allowing for claims not yet reported, when that is relevant to the analysis. Each count is a different denominator and answers a different question.
Why the Ratio Is an Average
Suppose the selected claim population contains \(n\) claims, and let \(x_j^{(M)}\) be the amount assigned to claim \(j\) under loss measure \(M\). If the aggregate for that measure is the sum of those claim amounts, then \(L_M=\sum_{j=1}^{n}x_j^{(M)}\). The arithmetic mean of the per-claim amounts is, by definition, their sum divided by their number:
This justifies aggregate loss divided by claim count as average severity when the aggregate and count refer to the same stated population and measure. If the numerator includes losses for one population and the denominator counts a different population, the ratio may still be a calculable statistic, but it is not the simple mean of the per-claim amounts for either population without additional interpretation.
Calculating Severity Under Different Definitions
Worked Example: Three Loss Measures for the Same Reported Claims
A hypothetical insurer reviews 16 reported equipment claims at a single valuation date. The aggregate paid loss is $96,000, incurred loss is $144,000, and estimated ultimate loss is $192,000. Each total refers to the same 16 claims and uses the same stated loss definition. Calculate all three average severities per reported claim.
Method and assumptions. Divide each loss measure by the 16 reported claims. The totals are already expressed at the same valuation date; no interest or mortality assumption is relevant, and no claim-size probability distribution is assumed.
Paid severity per reported claim:
Incurred severity per reported claim:
Ultimate severity per reported claim:
Check. Multiplying each result by 16 recovers its original total: \(16(\$6{,}000)=\$96{,}000\), \(16(\$9{,}000)=\$144{,}000\), and \(16(\$12{,}000)=\$192{,}000\). The increasing averages reflect the stated differences among paid, incurred, and ultimate totals, not different claim counts.
Common mistake avoided. Calling all three numbers simply “claim severity” hides their different loss bases. Each calculation is valid as severity per reported claim, but the label should include the measure—paid, incurred, or ultimate.
An important nuance is that the denominator does not have to be a count of claims that already have positive payments. For example, paid dollars divided by all reported claims measures paid dollars per reported claim, including claims that have not yet been paid. Dividing by only claims with a payment instead measures the average paid amount among that selected subset. Neither definition is inherently wrong; they describe different quantities.
Worked Example: Reported-Claim Paid Severity Versus Paid-Claim Severity
A hypothetical liability portfolio has 24 reported claims. By the valuation date, 18 have received at least one payment, and cumulative paid loss across all 24 claims is $90,000. Incurred loss for the same 24 claims is $150,000. Calculate paid loss per reported claim, paid loss per claim with a payment, and incurred loss per reported claim.
Method and assumptions. Treat the $90,000 as total paid across the full set of 24 claims, including any claims with no payment. Use 18 only for the explicitly conditional paid-claim denominator. Amounts are on the same valuation basis; no interest, mortality, or claim-size distribution assumption is used.
Paid per reported claim:
Paid per claim with a payment:
This second average is conditional on belonging to the 18 claims with a payment by the valuation date. It is not the average paid amount across all 24 reported claims.
Incurred per reported claim:
Check. The first average times 24 gives $90,000, and the second average times 18 also gives $90,000. The incurred average times 24 gives $150,000. The two paid averages differ because their denominators differ: \(18<24\), while the numerator is the same.
Common mistake avoided. Dividing $90,000 by 18 and reporting the result as paid severity “per reported claim” would be incorrect. The result of $5,000 is paid severity per claim with a payment; per reported claim it is $3,750.
When the Claim Count Is Also Estimated
A loss total may be projected to ultimate while the observed count is still only the number reported by the valuation date. Dividing those two quantities is possible, but the result should be named accurately: it is estimated ultimate loss per reported claim, not ultimate loss per ultimate claim. If both ultimate loss and ultimate claim count are available for the same experience scope, their ratio gives ultimate severity per ultimate claim.
The distinction matters because a projected claim count can differ from the reported count. Using a larger ultimate count with the same total reduces the calculated average; using a smaller denominator increases it. The arithmetic is straightforward, but the interpretation depends on the population represented by each number.
Worked Example: Reported and Ultimate Counts Give Different Severities
For a hypothetical storm coverage period, an insurer has 40 claims reported by the valuation date and incurred losses of $360,000 on those reported claims. A separate estimate for the same period is $420,000 of ultimate loss and 60 ultimate claims. Calculate incurred loss per reported claim, ultimate loss per reported claim, and ultimate loss per ultimate claim.
Method and assumptions. Keep each numerator and denominator explicit. The incurred total is paired with the 40 reported claims. For the other two calculations, use the $420,000 ultimate estimate, first with the 40 reported claims and then with the 60 ultimate claims. Amounts are at one valuation basis; no interest, mortality, or claim-size distribution assumption is used.
Incurred loss per reported claim:
Ultimate loss per reported claim:
Ultimate loss per ultimate claim:
Check. The first two results multiplied by 40 recover $360,000 and $420,000, respectively. The last result multiplied by 60 recovers $420,000. Thus the two ultimate-loss ratios differ because one is per reported claim and the other is per ultimate claim.
Common mistake avoided. Reporting $10,500 as ultimate severity without stating “per reported claim” could lead a reader to assume the denominator was the 60 ultimate claims. With that denominator, the ultimate severity is $7,000.
Exam Traps and Practical Interpretation
When a problem asks for “average severity,” first look for qualifiers in both the loss total and the count. Words such as “paid to date,” “incurred,” “ultimate,” “reported,” and “with a payment” determine what is being measured. A robust calculation names the ratio before substituting numbers.
- Do not silently change the denominator. A count of claims with payments is not interchangeable with the count of all reported claims.
- Do not label a mixed-basis result as ultimate severity per ultimate claim. Both the loss measure and count basis must support that label.
- Keep the loss definition consistent. If one total includes a specified expense component and another does not, the resulting averages are not directly comparable without acknowledging the difference.
- Check the units. If a total is stated in thousands of dollars, keep that scale through the division and convert to dollars only if requested.
- Do not infer individual claim amounts from the average. Severity is an aggregate average, not a statement that each claim equals the calculated amount.
In applied work, the selected severity measure should serve the question being asked. A paid amount per reported claim summarizes cash paid to date across the reported population. Incurred severity summarizes current paid-plus-reserved amounts under the stated definition. Ultimate severity estimates final loss per claim on the selected count basis. These are related views, but they are not interchangeable. Precise labels make comparisons interpretable and keep calculations aligned with the intended analysis.
Check Your Understanding
For each problem, state the loss measure and count basis in your answer as well as the calculated average.
- A hypothetical insurer has 12 reported claims, with $36,000 paid and $54,000 incurred on those same claims. Calculate paid and incurred severity per reported claim. (Answer: Paid, $3,000 per reported claim; incurred, $4,500 per reported claim.)
- A hypothetical glass portfolio has 25 reported claims. Fifteen have received a payment, and total paid loss is $80,000. Calculate paid severity per reported claim and per claim with a payment. (Answer: $3,200 per reported claim; $5,333.33 per claim with a payment.)
- For a hypothetical coverage period, incurred loss is $315,000 on 45 reported claims. Estimated ultimate loss is $420,000 and estimated ultimate count is 60 claims. Calculate incurred severity per reported claim and ultimate severity per ultimate claim. (Answer: $7,000 per reported claim; $7,000 per ultimate claim.)
- A hypothetical insurer estimates ultimate loss of $550,000 and ultimate claim count of 50. It also has $360,000 of incurred loss on 40 reported claims. Calculate both stated-basis severities and the ultimate loss per reported claim. (Answer: $11,000 ultimate per ultimate claim; $9,000 incurred per reported claim; $13,750 ultimate per reported claim.)
- A hypothetical book has 30 reported claims, of which 24 have a payment. Total paid loss is $120,000 and incurred loss on the 30 reported claims is $210,000. How much larger is paid severity per claim with a payment than paid severity per reported claim, and what is incurred severity per reported claim? (Answer: Paid severity is $5,000 per claim with a payment versus $4,000 per reported claim, a difference of $1,000; incurred severity is $7,000 per reported claim.)