Why Large Claims Need Special Attention
A small number of unusually large claims can materially change an observed loss trend. That change may reflect a real change in the underlying cost of claims, but it may also reflect random variation in the count or size of a few claims. A trend estimate based on only two periods can be especially sensitive to one event. The analyst should therefore compare summaries that retain all losses with summaries that cap or separate large claims, rather than treating any single summary as automatically correct.
The purpose of a cap or a separate large-claim analysis is not to make inconvenient losses disappear. It is to make the source of the observed change clearer. The chosen treatment must be applied consistently, and any resulting trend must be interpreted as describing the particular loss measure that was calculated.
For period \(t\), let \(E_t\) be the exposure measure, \(N_t\) the number of claims, and \(L_t\) the aggregate loss. The observed claim frequency is \(N_t/E_t\), the average severity is \(L_t/N_t\), and the pure premium is \(L_t/E_t\). These are descriptive summaries of the data. Unless a model is explicitly specified, no claim-size probability distribution is assumed. Interest and mortality assumptions are not needed for these calculations.
Three Views of the Same Experience
An unadjusted summary uses the full observed loss amounts. It is the direct measure of the experience, but a few large claims can dominate its change. A capped summary replaces each claim amount by the lesser of that amount and \(K\). It reduces the influence of extreme severities while retaining claim counts. A separated summary splits the experience into components, such as ordinary losses up to \(K\) and excess above \(K\), or ordinary claims and a separately defined large-claim group.
These approaches answer different questions. The unadjusted result describes the total observed loss cost. The capped result describes loss cost under a per-claim cap. The separated result shows how much of the total change is associated with each component. A cap is not an estimate of the actual total losses, unless an appropriate excess component is added back.
| Approach | What is summarized? | Interpretation and caution |
|---|---|---|
| Unadjusted | All claim amounts at full value | Represents observed total loss cost; sensitive to large claims. |
| Capped | Each claim amount limited to \(K\) | Reduces individual-claim influence; the result is not total loss cost. |
| Separated | Defined ordinary and large-claim components | Shows component contributions; trends may differ and should be recombined for a total. |
The claim count \(N_t\) cancels, leaving \(L_t/E_t\). Therefore, over two periods, the pure premium trend factor equals the frequency trend factor multiplied by the severity trend factor, provided the same claim definition and exposure basis are used in both periods. This identity also holds for capped losses when \(L_t\) is replaced by the aggregate capped loss and the claim count still includes all claims. It does not justify excluding large claims from the count in one calculation while retaining them in another.
Worked Example: Comparing Unadjusted and Capped Trends
Worked Example: Comparing Unadjusted and Capped Trends
A hypothetical insurer reviews two equal-length periods, each with 10,000 exposures. Period 1 has 200 claims: 196 claims totaling $800,000 and four large claims of $140,000, $100,000, $90,000, and $70,000. Period 2 has 220 claims: 215 claims totaling $852,000 and five large claims of $180,000, $140,000, $110,000, $90,000, and $80,000. Calculate the unadjusted and capped pure premium trends using a $100,000 per-claim cap. These are observed aggregate data; no claim-size distribution, interest rate, or mortality assumption is used.
Method and unadjusted result. Period 1 aggregate loss is \(800{,}000+140{,}000+100{,}000+90{,}000+70{,}000=\$1{,}200{,}000\). Its frequency is \(200/10{,}000=0.0200\), severity is \(\$1{,}200{,}000/200=\$6{,}000\), and pure premium is \(\$1{,}200{,}000/10{,}000=\$120\) per exposure.
Period 2 aggregate loss is \(\$852{,}000+\$180{,}000+\$140{,}000+\$110{,}000+\$90{,}000+\$80{,}000=\$1{,}452{,}000\). Its frequency is \(220/10{,}000=0.0220\), severity is \(\$1{,}452{,}000/220=\$6{,}600\), and pure premium is \(\$145.20\) per exposure. The unadjusted pure premium trend factor is \(145.20/120=1.2100\), or a 21.00% increase. Frequency increased by 10.00% and severity by 10.00%; their combined factor is \(1.10\times1.10=1.2100\).
Capped result. In Period 1, the four large claims become $100,000, $100,000, $90,000, and $70,000, totaling $360,000. Capped aggregate loss is \(\$800{,}000+\$360{,}000=\$1{,}160{,}000\), so capped pure premium is \(\$116\) per exposure. In Period 2, the five large claims become $100,000, $100,000, $100,000, $90,000, and $80,000, totaling $470,000. Capped aggregate loss is \(\$852{,}000+\$470{,}000=\$1{,}322{,}000\), so capped pure premium is \(\$132.20\) per exposure.
The capped trend factor is \(132.20/116=1.139655\), or a 13.97% increase, rounded to two decimal places. The claim counts remain 200 and 220, so capped frequencies are still 0.0200 and 0.0220. Capped severities are \(\$1{,}160{,}000/200=\$5{,}800\) and \(\$1{,}322{,}000/220=\$6{,}009.09\), rounded to cents. The frequency factor of 1.10 times the capped severity factor of \(6{,}009.09/5{,}800\) reconciles to approximately 1.1397.
Check and common mistake avoided. The capped trend is lower than the unadjusted trend because the cap limits the impact of the largest claims; both calculations still include every claim in frequency. A common mistake is to divide capped loss by 10,000 and then call that a frequency trend. The corrected approach calculates capped pure premium in each period, then compares the two results. Capping loss amounts does not cap or remove claim counts.
Separating Ordinary and Large-Claim Components
Separation can be useful when the analyst wants to see whether a total change came from ordinary claim experience, the frequency of large claims, or their severity. One approach classifies claims above a threshold as large and tracks their count and total amount separately. Another splits each claim at a threshold: the amount up to \(K\) is the capped component, and the amount above \(K\) is the excess component. These definitions are not interchangeable. State which one is being used, and use it consistently.
When the components partition the same total losses, their pure premiums add. For example, if \(L_t=L_{O,t}+L_{G,t}\), where \(L_O\) and \(L_G\) are the ordinary and large-claim loss components, then:
The aggregate trend is not generally the simple average of the component trend rates. It depends on the starting-period contribution of each component. If the large-claim component is highly volatile, its observed trend may be unsuitable for mechanical projection. A selected trend can instead use judgment, additional periods, or a separate treatment for large claims, but the resulting assumption should be explicit.
Worked Example: Recombining Separated Trends
A hypothetical portfolio has 20,000 exposures in each of two periods. Claims below the stated large-claim threshold are classified as ordinary. In Period 1, there are 400 ordinary claims totaling $1,600,000 and four large claims totaling $600,000. In Period 2, there are 440 ordinary claims totaling $1,848,000 and six large claims totaling $900,000. Compare the component and aggregate pure premiums. The values are observed data; no claim-size distribution, interest rate, or mortality assumption is used.
Ordinary component. Period 1 ordinary pure premium is \(\$1{,}600{,}000/20{,}000=\$80\). Period 2 is \(\$1{,}848{,}000/20{,}000=\$92.40\). The factor is \(92.40/80=1.155\), a 15.50% increase. This also reconciles to frequency and severity: ordinary frequency rises from \(400/20{,}000=0.0200\) to \(440/20{,}000=0.0220\), a 10% increase; severity rises from \(\$1{,}600{,}000/400=\$4{,}000\) to \(\$1{,}848{,}000/440=\$4{,}200\), a 5% increase. Thus \(1.10\times1.05=1.155\).
Large-claim component. Period 1 large-claim pure premium is \(\$600{,}000/20{,}000=\$30\). Period 2 is \(\$900{,}000/20{,}000=\$45\), a factor of \(45/30=1.50\), or a 50% increase. The large-claim count rises from four to six, while average large-claim severity is \(\$600{,}000/4=\$150{,}000\) in both periods. The observed component increase is therefore attributable to the count in this example, not to average severity.
Reconciliation. Period 1 total pure premium is \(\$80+\$30=\$110\); Period 2 is \(\$92.40+\$45=\$137.40\). The aggregate factor is \(137.40/110=1.24909\), or a 24.91% increase, rounded to two decimal places. It is not the arithmetic average of 15.50% and 50.00%, because the two components have different starting-period weights.
Check and common mistake avoided. The ordinary and large components add back to each period’s total, so the split is internally consistent. A common mistake is to average the component trend percentages without weighting them. The correction is to project or compare component pure premiums in dollars per exposure, then add those amounts before calculating the aggregate trend.
Keep the Threshold on a Consistent Cost Basis
A nominal dollar threshold can represent different levels of claim severity in periods with different price levels. If a cap is meant to represent the same claim layer in every period, express losses and the threshold on a common cost basis before comparing capped results. The conversion should be based on the cost-level adjustment selected for the analysis, not on an unexplained change in the threshold.
Worked Example: Applying a Cap on a Common Cost Basis
A hypothetical portfolio has 1,000 exposures and one claim in each of two periods. The earlier-period claim is $90,000, and an independently selected cost-level index converts earlier-period dollars to the comparison basis by a factor of 1.10. The later-period claim is $110,000 on that comparison basis. Compare capped pure premium using a $100,000 threshold on the common basis. No probability distribution, interest rate, or mortality assumption is used.
Convert and cap. The earlier claim converts to \(\$90{,}000\times1.10=\$99{,}000\). Capping at $100,000 leaves it at $99,000. The later claim is already on the comparison basis, and its capped amount is $100,000. With 1,000 exposures, capped pure premiums are \(\$99{,}000/1{,}000=\$99\) and \(\$100{,}000/1{,}000=\$100\). The capped trend factor is \(100/99=1.010101\), or a 1.01% increase rounded to two decimal places.
Check and common mistake avoided. On the common basis, unadjusted pure premiums are $99 and $110, a factor of \(110/99=1.111111\). Applying a $100,000 cap to the original nominal amounts instead would produce $90,000 and $100,000, suggesting an 11.11% capped increase. That comparison uses a different cost basis for the earlier claim and does not apply the stated common-basis cap. The correction is to convert first, then cap.
Interpretation and Exam Traps
Large-claim treatment should be selected to answer a defined question, not just to produce a smoother number. A capped trend describes capped losses; it is not automatically the trend in total losses. A separated trend describes the components as defined; it needs a recombination step to describe the total. An unadjusted trend preserves the actual observed total but may be unstable when a few claims dominate.
- Do not silently delete large claims. Removing their losses but leaving their counts, or removing both losses and counts, changes the measure being calculated. Explain the treatment and apply it consistently.
- Do not confuse a cap with a large-claim exclusion. A cap limits each claim’s counted amount; it does not remove the claim from frequency.
- Do not compare unlike thresholds. A fixed nominal cap may not represent the same cost layer in different periods. Use a consistent basis or explain why a nominal threshold is intended.
- Do not average component trends without weights. Recombine component pure premiums before calculating an aggregate trend.
- Do not assume that capping removes all volatility. A capped result can still vary because claim frequency and the number of claims near the cap vary.
A disciplined analysis reports the unadjusted total, a capped or separated view where useful, the threshold and cost basis, and the effect of the treatment on the conclusion. Sensitivity to a reasonable alternative threshold can help show whether the selected trend is driven by a few large claims. These methods are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each hypothetical problem, identify the measure being calculated and keep claim counts, exposure bases, and thresholds consistent.
- A portfolio has 5,000 exposures and 100 claims totaling $500,000 in one period. In the next period it has 5,000 exposures and 110 claims totaling $605,000. Calculate the unadjusted frequency, severity, and pure premium trend factors. (Answer: frequency 1.10; severity 1.10; pure premium 1.21, a 21% increase.)
- Three claims are $40,000, $120,000, and $200,000. Calculate aggregate capped loss at a $100,000 per-claim cap, and the aggregate excess above the cap. (Answer: capped loss $240,000; excess $120,000; total $360,000.)
- Two periods each have 10,000 exposures. Ordinary loss is $700,000 then $770,000; large-claim loss is $300,000 then $450,000. Calculate each component pure-premium trend factor and the aggregate factor. (Answer: ordinary 1.10; large-claim 1.50; aggregate \(122/100=1.22\), a 22% increase.)
- An earlier-period claim is $80,000 and is converted to a common cost basis by a factor of 1.25. A later-period claim is $105,000 on that basis. Each period has 1,000 exposures. With a $100,000 cap on the common basis, calculate both capped pure premiums and their trend factor. (Answer: $100 and $100 per exposure; factor 1.0000, or no change.)
- Period 1 has 200 claims, $1,000,000 of uncapped loss, and 10,000 exposures. Period 2 has 220 claims, $1,210,000 of uncapped loss, and the same exposures. A cap reduces the aggregate loss to $950,000 and $1,050,000, respectively. Calculate the capped pure-premium trend factor and explain whether claim frequency changes under this cap. (Answer: \(105/95=1.1053\), or a 10.53% increase; capped claim frequency still increases from 0.0200 to 0.0220 because the cap changes amounts, not counts.)