Controlling Large-Claim Effects on Severity

Why Measure Large-Claim Effects?

A small number of unusually large claims can materially raise the average severity in a period. That increase may reflect the underlying claim process, but it can also make an estimate based on a limited set of claims especially sensitive to individual observations. An analytical cap provides a way to measure that sensitivity: it limits the amount attributed to any one claim for a specified calculation, without removing the claim from the count.

The capped result is not automatically an estimate of total ground-up claim cost. It estimates the cost under the selected cap. If the intended measure is uncapped cost, the amounts above the cap must be considered separately; simply replacing uncapped severity with capped severity understates total cost. The cap is an analytical tool here, not a description of a hypothetical policy’s coverage or a reinsurance arrangement.

Definition: For claim amounts $X_1,\ldots,X_N$ and an analytical cap $C>0$, the capped amount for claim $j$ is $\min(X_j,C)$. The uncapped average severity is the sum of all claim amounts divided by $N$. The capped average severity is the sum of capped claim amounts divided by the same $N$. Claims above $C$ remain in the denominator and in the claim count; only their amounts above $C$ are excluded from the capped total.

In this tutorial, claim amounts are treated as observed deterministic values for the calculations shown. No probability distribution, mortality assumption, or discounting is used. All example amounts are hypothetical. Where amounts are stated in thousands, calculations stay in thousands until the final interpretation.

Deriving the Capped Pure Premium

Let $e$ be the exposure base, $N$ the number of claims arising from that exposure, and $X_j$ the amount of claim $j$ on the valuation basis being used. Define frequency as claims per exposure, $N/e$, and severity as dollars per claim. Pure premium is claim cost per exposure. The same identity applies to capped amounts, provided the frequency and severity use the same claim population.

\text{Uncapped severity} =\frac{\sum_{j=1}^{N}X_j}{N}, \qquad \text{Capped severity} =\frac{\sum_{j=1}^{N}\min(X_j,C)}{N}

\text{Pure premium} =\frac{\text{claim cost}}{e} =\frac{N}{e}\times\frac{\text{claim cost}}{N}

The justification is direct. Multiplying the claim count per exposure by the average amount per claim cancels the claim count: $(N/e)(S/N)=S/e$, where $S$ is the corresponding total claim amount. Using $S=\sum X_j$ gives uncapped pure premium. Using $S=\sum \min(X_j,C)$ gives capped pure premium. Thus, a capped severity must be paired with the frequency for the same claims, not with a count that silently omits claims above the cap.

PP_{\text{uncapped}} =\frac{N}{e}\left(\frac{\sum_{j=1}^{N}X_j}{N}\right), \qquad PP_{\text{capped}} =\frac{N}{e}\left(\frac{\sum_{j=1}^{N}\min(X_j,C)}{N}\right)

For each claim, $X_j=\min(X_j,C)+\max(X_j-C,0)$. Consequently, the difference between uncapped and capped totals is exactly the sum of amounts above the cap. This identity explains what the capped calculation removes: not whole large claims, but only the portion of each claim exceeding $C$.

Calculating the Effect of a Cap

Worked Example: Measuring the Contribution Above a Cap

A hypothetical portfolio has five claims, with amounts of $20,000, $30,000, $40,000, $60,000, and $150,000. Use an analytical cap of $50,000. The exposure is 1,000 units. Calculate uncapped and capped severity and pure premium, and identify the effect of the claims above the cap.

Method and assumptions. Use the empirical arithmetic average, with all five claims retained in both severity denominators and in the frequency. The amounts and cap are on the same cost basis. No trend or discounting is applied.

1. Uncapped total and severity. The total is $20{,}000+30{,}000+40{,}000+60{,}000+150{,}000=\$300{,}000$. Dividing by five gives uncapped severity of $\$300{,}000/5=\$60{,}000$ per claim.

2. Capped total and severity. The capped amounts are $\$20{,}000,\ \$30{,}000,\ \$40{,}000,\ \$50{,}000,$ and $\$50{,}000$. Their total is $\$190{,}000$, so capped severity is $\$190{,}000/5=\$38{,}000$ per claim.

3. Frequency and pure premiums. Frequency is $5/1{,}000=0.005$ claims per exposure. Uncapped pure premium is $0.005\times\$60{,}000=\$300$ per exposure. Capped pure premium is $0.005\times\$38{,}000=\$190$ per exposure.

4. Check. The two claims above the cap contribute $(\$60{,}000-\$50{,}000)+(\$150{,}000-\$50{,}000)=\$110{,}000$ above the cap. The total difference is $\$300{,}000-\$190{,}000=\$110{,}000$, as required. The pure-premium difference is $0.005\times\$22{,}000=\$110$ per exposure.

Common mistake avoided: removing the $150,000 claim from the data rather than capping its amount would discard $150,000, not just the $100,000 above the cap. The capped calculation retains that claim and counts its first $50,000.

Why the Claim-Count Basis Must Match

Capping amounts and excluding claims are different operations. If claims above $C$ are deleted, the remaining average is calculated over fewer claims. Pairing that average with a frequency that still counts all claims does not reconstruct either the capped total or the uncapped total. Conversely, if the frequency is also reduced to count only the retained claims, the resulting product describes only that selected subset unless the excluded claims are added back through a separate, explicit estimate.

Worked Example: Capping Claims Rather Than Deleting Them

A hypothetical book has eight claims with amounts, in thousands of dollars, of 10, 20, 30, 40, 50, 60, 150, and 250. The exposure is 200 units, and the analytical cap is $100,000. Calculate the capped and uncapped estimates. Then show the error from deleting claims above the cap while retaining the all-claim frequency.

Method and assumptions. Use all eight claims for both severity denominators and for frequency. Amounts are already on a common valuation basis, with no trend applied.

1. Uncapped estimate. The total in thousands is $10+20+30+40+50+60+150+250=610$. Uncapped severity is $610/8=76.25$ thousand dollars per claim.

2. Capped estimate. The capped amounts are 10, 20, 30, 40, 50, 60, 100, and 100 thousand dollars. Their total is $410$, and capped severity is $410/8=51.25$ thousand dollars per claim.

3. Pure premiums on the consistent basis. Frequency is $8/200=0.04$ claims per exposure. Uncapped pure premium is $0.04\times\$76{,}250=\$3{,}050$ per exposure. Capped pure premium is $0.04\times\$51{,}250=\$2{,}050$ per exposure.

4. Diagnose the deletion error. Deleting the two claims above $100,000 leaves six claims totaling $210,000, so the remaining-claim average is $\$210{,}000/6=\$35{,}000$. Multiplying that by the all-claim frequency gives $0.04\times\$35{,}000=\$1{,}400$ per exposure. This is not the capped pure premium of $2,050: the calculation has discarded the first $100,000 of each of the two large claims as well as their amounts above the cap.

Check. The amounts above the cap total $(150-100)+(250-100)=200$ thousand dollars. The difference in totals, $610-410=200$, agrees. The pure-premium difference is $0.04\times\$25{,}000=\$1{,}000$ per exposure, also equal to $\$200{,}000/200$.

Applying Trends Before a Prospective-Level Cap

When a projection applies trends, the cap must be defined on the same cost level as the amounts being capped. If the cap is intended to limit projected, prospective-level claim amounts, first trend the claim amounts to that level and then apply the prospective-level cap. Applying a future-dollar cap to historical amounts, or trending amounts after capping them at a fixed future-dollar threshold, can produce amounts above the intended future cap.

Worked Example: Projecting Capped and Uncapped Pure Premium

A hypothetical experience period has four claims of $18,000, $34,000, $48,000, and $72,000, with 1,000 exposure units. For the next period, apply a 5% frequency trend and a 10% severity trend. The intended cap is $60,000 in next-period dollars. Calculate next-period uncapped and capped severities and pure premiums.

Method and assumptions. The four observed claims and the frequency share a consistent exposure and claim definition. Apply each trend once for one period. Apply the cap after trending because it is specified in next-period dollars. No additional mix changes or adjustments are assumed.

1. Trend the frequency. Historical frequency is $4/1{,}000=0.004$ claims per exposure. Next-period frequency is $0.004(1.05)=0.0042$.

2. Trend each claim amount. The next-period amounts are $\$18{,}000(1.10)=\$19{,}800$, $\$34{,}000(1.10)=\$37{,}400$, $\$48{,}000(1.10)=\$52{,}800$, and $\$72{,}000(1.10)=\$79{,}200$. Their total is $\$189{,}200$, giving uncapped severity of $\$189{,}200/4=\$47{,}300$.

3. Apply the prospective cap. The capped amounts are $\$19{,}800,\ \$37{,}400,\ \$52{,}800,$ and $\$60{,}000$. Their total is $\$170{,}000$, giving capped severity of $\$170{,}000/4=\$42{,}500$.

4. Calculate pure premiums. Next-period uncapped pure premium is $0.0042\times\$47{,}300=\$198.66$ per exposure. Next-period capped pure premium is $0.0042\times\$42{,}500=\$178.50$ per exposure.

5. Check the cap effect. Only the fourth trended claim exceeds $60,000, by $\$79{,}200-\$60{,}000=\$19{,}200$. The total difference is $\$189{,}200-\$170{,}000=\$19{,}200$, and the pure-premium difference is $0.0042\times\$4{,}800=\$20.16$ per exposure.

Common mistake avoided: capping the historical $72,000 amount at $60,000 and then trending it produces $\$60{,}000(1.10)=\$66{,}000$, which exceeds the stated next-period cap. Because the cap is in next-period dollars, trend first and cap the resulting $79,200 amount at $60,000.

Interpreting and Communicating the Result

A useful comparison reports the cap, the valuation basis, the number of claims, the uncapped and capped totals, and both averages. If a pure premium is also shown, state the frequency basis and exposure unit. This makes it possible to distinguish a change in severity from a change in claim count or exposure.

Capped severity can help quantify how much the selected high amounts influence an average. It does not by itself determine whether those amounts are unusual, whether the cap is appropriate, or what the best estimate of future total cost should be. The selected threshold and period should be documented, and comparisons across periods should use consistent monetary and claim-count bases. Where the objective is a full-cost projection, the portion above the cap cannot simply be forgotten: the capped estimate covers only the amounts up to the cap for each claim.

Key takeaway: Cap each claim amount at the stated threshold, retain every claim in the count, and pair capped severity with frequency from that same population. A capped pure premium is a capped-cost measure, not an uncapped-cost estimate. These methods are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, keep the claim count consistent with the severity calculation. Amounts in thousands are stated explicitly.

A hypothetical portfolio has claims of $12,000, $18,000, $25,000, and $90,000, with 200 exposure units. Use a $40,000 cap. Find capped and uncapped severity and pure premium. (Answer: Capped severity = $23,750; uncapped severity = $36,250; frequency = 0.02; capped pure premium = $475 per exposure; uncapped pure premium = $725 per exposure.)
Six hypothetical claims, in thousands of dollars, are 8, 15, 22, 35, 80, and 140. Exposure is 200 units, and the cap is $50,000. Find both average severities and pure premiums. (Answer: Capped severity = $30,000; uncapped severity = $50,000; frequency = 0.03; capped pure premium = $900 per exposure; uncapped pure premium = $1,500 per exposure.)
Three hypothetical claims are $24,000, $60,000, and $96,000. Use a $75,000 cap and exposure of 600 units. What is the capped pure premium, and how much lower is it than uncapped pure premium? (Answer: Capped severity = $53,000; frequency = 0.005; capped pure premium = $265 per exposure; uncapped pure premium = $300 per exposure, a difference of $35 per exposure.)
A hypothetical experience has frequency 0.02 claims per exposure and claims of $10,000, $30,000, and $70,000. Apply a 10% frequency trend and a 5% severity trend, then use a $60,000 cap in prospective dollars. Find prospective capped and uncapped pure premiums. (Answer: Prospective frequency = 0.022; trended severities are $10,500, $31,500, and $73,500; capped severity = $34,000; uncapped severity = $38,500; capped pure premium = $748 per exposure; uncapped pure premium = $847 per exposure.)
Five hypothetical claims are $20,000, $25,000, $40,000, $100,000, and $200,000, with 2,500 exposure units and a $75,000 cap. Calculate the capped severity and pure premium, then the uncapped pure premium. (Answer: Capped total = $235,000; capped severity = $47,000; frequency = 0.002; capped pure premium = $94 per exposure; uncapped total = $385,000; uncapped severity = $77,000; uncapped pure premium = $154 per exposure.)

What You'll Learn