Why Measure Large-Loss Influence?
An overall rate indication can change materially when one unusually large claim is included. That sensitivity is worth measuring: it helps explain what is driving an indication and whether the result depends heavily on one observation. It does not, by itself, establish that the claim should be excluded or that the indicated change is unreasonable. The calculation is a controlled comparison: hold the indication’s other inputs constant, remove one claim’s projected contribution, and measure the difference.
The analysis is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. It uses the overall indication framework, including expense and profit provisions, to isolate one claim’s influence. The key discipline is to compare like with like: the claim amount removed must be on the same projected, trended, and developed basis as the loss total in the indication.
This model assumes a positive denominator, \(P(1-e_v-p)>0\), and that the stated provisions are expressed consistently with the premium base. If the indication includes other provisions that vary with premium, those provisions belong in the denominator as appropriate. If a fixed expense or another input is deliberately changed when the claim is removed, the simple leave-one-out comparison no longer isolates only that claim.
Deriving the Leave-One-Out Impact
The difference between the full factor and the leave-one-out factor measures the claim’s impact in rate-factor points. Starting from the same denominator in both calculations,
The equality follows because the common terms \(L+F\) and \(L-C_i+F\) differ by exactly \(C_i\). The indicated rate changes are each the corresponding factor minus one, so their difference is also \(C_i/[P(1-e_v-p)]\). Thus, when only the projected loss amount changes, the claim’s rate-change impact in percentage points is its projected amount divided by the premium base after variable expenses and profit provision.
This is a difference in percentage points, not necessarily a relative percentage change in the indication. For example, a difference between rate factors of 0.83 and 0.65 is 0.18, or 18 percentage points in rate change. It is not an 18% relative change in the factor.
Worked Example: Measuring One Claim’s Influence
A hypothetical portfolio has premium of $4,000 thousand, variable expenses of 20%, profit and contingency of 5%, fixed expenses of $100 thousand, and projected losses of $2,400 thousand, including one claim contributing $540 thousand. Calculate the full indication, the leave-one-out indication, the claim’s rate-change impact, and its share of projected losses. Use the same units throughout; report factors to four decimals and rate changes and shares to two decimal places.
Method and denominator. This is a leave-one-out comparison under the overall indication model. The premium remaining after the percentage provisions is $$ 4{,}000(1-0.20-0.05)=4{,}000(0.75)=3{,}000 $$ thousand.
Full indication. Add projected losses and fixed expenses: \(2{,}400+100=2{,}500\) thousand. Therefore, $$ R=\frac{2{,}500}{3{,}000}=0.8333. $$ The indicated rate change is \(0.833333\ldots-1=-0.166667\ldots\), or a 16.67% decrease.
Leave-one-out indication. Removing the claim leaves projected losses of \(2{,}400-540=1{,}860\) thousand. Add the unchanged fixed expenses: \(1{,}860+100=1{,}960\) thousand. Thus, $$ R_{-i}=\frac{1{,}960}{3{,}000}=0.6533. $$ The leave-one-out rate change is \(0.653333\ldots-1=-0.346667\ldots\), or a 34.67% decrease.
Impact and loss share. The factor difference is \(0.833333\ldots-0.653333\ldots=0.1800\), equivalent to an 18.00-percentage-point difference in indicated rate change. The claim’s share of projected losses is \(540/2{,}400=0.225\), or 22.50%.
Check and common mistake avoided. The direct impact check is \(540/3{,}000=0.1800\), matching the factor difference. The leave-one-out numerator is \(1{,}860+100=1{,}960\), not a value obtained by changing the denominator. The loss share and rate impact differ because they use different denominators: total losses for the share, and premium net of percentage provisions for the impact.
Three Quantities That Should Not Be Confused
A large-loss review is clearer when it reports distinct quantities rather than labeling all of them “impact.” For a claim with projected contribution \(C_i\), the loss share is \(C_i/L\). The rate-factor impact is \(C_i/[P(1-e_v-p)]\), under the fixed-input assumptions above. The indicated rate change itself is \(R-1\). These measure different things: the claim’s proportion of projected losses, its change to the rate factor, and the overall direction and size of the indicated change.
The fixed expense \(F\) affects both full and leave-one-out factors, but cancels in their difference when it is held constant. It does not belong in the loss-share denominator, either. A loss share is not a claim’s fraction of total indicated costs when fixed expenses are present.
Worked Example: A Different Premium and Expense Mix
A hypothetical program has premium of $6,000 thousand, variable expenses of 15%, profit and contingency of 5%, fixed expenses of $300 thousand, and projected losses of $3,600 thousand, including a $720 thousand claim. Calculate both factors, the difference in rate changes, and the loss share. Report factors to four decimals and percentages to two decimal places.
Method and denominator. Hold all inputs other than the claim’s loss contribution fixed. The net premium denominator is $$ 6{,}000(1-0.15-0.05)=6{,}000(0.80)=4{,}800 $$ thousand.
Full indication. The numerator is \(3{,}600+300=3{,}900\) thousand, giving $$ R=\frac{3{,}900}{4{,}800}=0.8125. $$ The indicated rate change is \(0.8125-1=-0.1875\), a 18.75% decrease.
Leave-one-out indication. Losses excluding the claim are \(3{,}600-720=2{,}880\) thousand. Including unchanged fixed expenses gives a numerator of \(2{,}880+300=3{,}180\) thousand, so $$ R_{-i}=\frac{3{,}180}{4{,}800}=0.6625. $$ The leave-one-out rate change is \(0.6625-1=-0.3375\), a 33.75% decrease.
Impact and loss share. The factor difference is \(0.8125-0.6625=0.1500\), or 15.00 percentage points. The loss share is \(720/3{,}600=0.2000\), or 20.00%.
Check and common mistake avoided. The impact check is \(720/4{,}800=0.1500\). Do not treat the 20.00% loss share as the rate impact: the loss share uses $3,600 thousand as its denominator, while the rate impact uses $4,800 thousand. The full factor is \(3{,}900/4{,}800=0.8125\), not a factor formed from losses alone.
Worked Example: A Claim That Changes the Direction of the Indication
A hypothetical portfolio has premium of $2,500 thousand, variable expenses of 25%, profit and contingency of 5%, fixed expenses of $75 thousand, and projected losses of $1,800 thousand, including a claim contributing $450 thousand. Calculate the full and leave-one-out rate changes, the claim’s impact, and its loss share. Report factors to four decimals and percentages to two decimal places.
Method and denominator. Use a common denominator and remove only the claim’s projected contribution: $$ 2{,}500(1-0.25-0.05)=2{,}500(0.70)=1{,}750 $$ thousand.
Full indication. The full numerator is \(1{,}800+75=1{,}875\) thousand. Hence $$ R=\frac{1{,}875}{1{,}750}=1.0714, $$ and the full indicated rate change is \(1.071428\ldots-1=0.071428\ldots\), or a 7.14% increase.
Leave-one-out indication. Losses excluding the claim are \(1{,}800-450=1{,}350\) thousand. Adding $75 thousand of fixed expenses gives a numerator of \(1{,}425\) thousand, so $$ R_{-i}=\frac{1{,}425}{1{,}750}=0.8143. $$ The leave-one-out rate change is \(0.814285\ldots-1=-0.185714\ldots\), or an 18.57% decrease.
Impact and loss share. The factor difference is \(1.071428\ldots-0.814285\ldots=0.257142\ldots\), or 25.71 percentage points. The loss share is \(450/1{,}800=0.25\), or 25.00%.
Check and common mistake avoided. The direct impact check is \(450/1{,}750=0.257142\ldots\). The sign of the full indication does not change the definition of the impact: it is the full factor minus the leave-one-out factor. Do not describe this 25.71-point difference as a 25.71% relative increase; it is the change in the indicated rate change when the claim is included.
How to Interpret the Result
A large leave-one-out difference is evidence of sensitivity, not an automatic reason to discard a claim. A claim may be large because of random variation, an unusual event, a concentration of exposure, or a feature that is genuinely part of the risk being priced. The calculation describes what happens under one controlled change to the inputs. It does not decide which losses belong in the projection, how much weight to give an indication, or what rate should ultimately be selected.
Use the result to guide review. Confirm that the claim is included once, that its contribution is measured on the same basis as \(L\), and that the comparison leaves the same premium and provisions in place. If the indication uses projected ultimate losses, subtract the claim’s projected ultimate contribution—not an inconsistent paid amount or an unadjusted historical amount. Any development, trend, or other adjustments already embedded in the selected loss basis must be reflected consistently.
When a selection appears heavily influenced by one claim, consider whether the exposure and premium base are appropriate for the losses being compared, whether the claim is properly classified in the data, and whether the underlying assumptions are suitable for the intended prospective period. Also consider whether a claim’s unusual size is relevant to the portfolio’s future risk. These are investigation questions; leave-one-out analysis alone supplies no answer to them.
A Practical Review Sequence
State the premium base, percentage provisions, fixed expenses, and projected loss total. Check that units and periods are consistent.
Use the amount included in projected losses on the selected basis, not a different raw amount from another valuation basis.
Subtract that contribution from losses, leave the other inputs unchanged, and calculate the leave-one-out factor.
Verify that the factor difference equals the claim contribution divided by the net premium denominator. Report the loss share separately and investigate the dependence before drawing a selection conclusion.
Common Exam Traps
- Changing more than one input. If premium, provisions, or fixed expenses also change in the leave-one-out calculation, the difference no longer isolates the loss’s influence.
- Subtracting from the wrong quantity. Remove the claim contribution from projected losses, not from premium, total expenses, or the full indication numerator without accounting for fixed expenses.
- Using the wrong denominator. Under this model, the impact denominator is \(P(1-e_v-p)\). The loss-share denominator is \(L\), so the two percentages generally differ.
- Confusing percentage points with relative percent. Subtract the two indicated rate changes to obtain percentage points. A relative change requires a separately specified comparison base.
- Treating sensitivity as a recommendation. A high influence measure calls for review; it does not prove that a claim is anomalous or should be excluded.
Check Your Understanding
For each problem, use the overall indication model in this tutorial. Hold all inputs other than the specified claim contribution fixed. Work in thousands of dollars where stated; round factors to four decimals and percentages to two decimal places.
- A hypothetical portfolio has premium of $5,000 thousand, variable expenses of 18%, profit and contingency of 7%, fixed expenses of $150 thousand, and projected losses of $3,000 thousand, including a $500 thousand claim. Find the full factor, leave-one-out factor, factor impact in percentage points, and loss share. (Answer: Full factor 0.8400; leave-one-out factor 0.7067; impact 13.33 percentage points; loss share 16.67%.)
- A hypothetical book has premium of $8,000 thousand, variable expenses of 20%, profit and contingency of 5%, fixed expenses of $200 thousand, and projected losses of $5,000 thousand, including a $1,000 thousand claim. Find both rate changes and the claim’s impact. (Answer: Full rate change: 13.33% decrease; leave-one-out rate change: 30.00% decrease; impact: 16.67 percentage points.)
- A hypothetical portfolio has premium of $3,000 thousand, variable expenses of 22%, profit and contingency of 8%, fixed expenses of $120 thousand, and projected losses of $2,400 thousand, including a $360 thousand claim. Find the full and leave-one-out factors, impact, and loss share. (Answer: Full factor 1.2000; leave-one-out factor 1.0286; impact 17.14 percentage points; loss share 15.00%.)
- A hypothetical program has premium of $10,000 thousand, variable expenses of 20%, profit and contingency of 5%, fixed expenses of $250 thousand, and projected losses of $7,000 thousand. Two claims contribute $1,200 thousand and $400 thousand, respectively. Calculate each claim’s leave-one-out factor impact using the same denominator and identify which claim has greater influence. (Answer: Denominator $7,500 thousand; first claim impact 16.00 percentage points; second claim impact 5.33 percentage points; the $1,200 thousand claim has greater influence.)
- A hypothetical portfolio has premium of $6,000 thousand, variable expenses of 20%, profit and contingency of 5%, fixed expenses of $150 thousand, and projected losses of $3,300 thousand, including a $900 thousand claim. A reviewer flags claims whose impact exceeds 15 percentage points. Calculate this claim’s impact and the claim amount corresponding exactly to the 15-point threshold, then decide whether it exceeds the threshold. (Answer: Impact \(900/4{,}500=20.00\) percentage points; threshold amount \(0.15(4{,}500)=$675\) thousand; yes, the claim exceeds the threshold.)