Why a Corrected Loss Requires a Recalculation
An experience modification depends on the losses recognized under the rating plan, not simply on the first amounts recorded for a policy period. A claim correction or recovery can move a claim across a per-claim split point. That changes not only the claim’s total but also how much is counted as primary loss and how much as excess loss. The modification must therefore be recomputed from the revised claim amounts and components.
This tutorial uses a hypothetical plan-specific formula. Actual rating plans can define loss valuation, recovery treatment, expected losses, credibility, and other inputs differently. Follow the terms of the plan in the problem. The calculations below assume expected-loss inputs, credibility, and ballast remain fixed when an individual claim amount is corrected. The method is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
The split point applies separately to each claim. After calculating each claim’s components, add the primary components to obtain aggregate actual primary losses \(A_P\), and add the excess components to obtain aggregate actual excess losses \(A_X\). In general, splitting the total of all claims is not equivalent to splitting each claim and then adding the components.
A Plan-Specific Modification Formula
For the examples, let \(E_P\) and \(E_X\) denote the plan’s expected primary and expected excess losses, \(Z\) its excess-loss credibility, and \(B\) its ballast. Assume \(0\leq Z\leq1\), \(E_P+E_X+B>0\), and all these inputs are fixed during a claim correction. The hypothetical plan calculates the modification as follows:
The formula uses actual primary losses in full, gives weight \(Z\) to actual excess losses, and gives weight \(1-Z\) to expected excess losses. The expected primary losses appear in the denominator but not as a separate term in this numerator. This is the specified formula for these examples; other plans may use a different formula, so its structure must not be assumed without being given.
Recompute Components Before Updating the Factor
A reliable update starts with the claim records, not with the old aggregate components. For each corrected claim, determine its revised recognized amount and calculate the primary and excess components anew. Then aggregate the revised components and substitute them in the formula. This avoids errors when a correction or recovery crosses the split point.
Establish the corrected claim amount and any recovery recognized under the plan. Confirm whether the recovery reduces the loss before the split is applied.
For each claim, calculate \(\min(L,S)\) as primary and \(\max(L-S,0)\) as excess.
Sum the revised primary and excess components separately. Recalculate the modification using the plan formula and any inputs that the correction actually changes.
Why the Change in the Factor Has This Form
Let \(\Delta A_P\) and \(\Delta A_X\) be the differences between the revised and original aggregate actual primary and excess losses. Under the stated fixed-input assumptions, the denominator and all expected-loss terms remain unchanged. Subtracting the original formula from the revised formula therefore gives:
This result follows directly: the numerator changes by \(\Delta A_P+Z\Delta A_X\), while the denominator stays at \(E_P+E_X+B\). For one claim changing from \(L\) to \(L'\), the component changes are
These expressions show why the full dollar change in a claim is not always a change in primary loss, or always a change in excess loss. A change entirely below the split point affects primary loss only; a change entirely above it affects excess loss only. A change that crosses the split point affects both components.
Worked Example: Correcting One Claim and Recognizing a Recovery
A hypothetical plan has a per-claim split point of \(10\) thousand dollars. The two initially recorded claims are \(8\) and \(14\) thousand dollars. The first is corrected to \(12\) thousand dollars, and a \(3\)-thousand-dollar recovery is recognized on the second, making its net amount \(11\) thousand dollars. The plan inputs are \(E_P=18\), \(E_X=6\), \(Z=0.60\), and \(B=6\), all in thousands of dollars where applicable. Find the modification before and after the changes.
Method. Split each claim separately, aggregate primary and excess losses, and substitute the totals into the specified formula. The fixed denominator is \(18+6+6=30\).
Before the changes. The \(8\)-thousand-dollar claim has primary \(8\) and excess \(0\). The \(14\)-thousand-dollar claim has primary \(10\) and excess \(4\). Thus \(A_P=8+10=18\) and \(A_X=0+4=4\). The numerator is \(18+0.60(4)+0.40(6)+6=18+2.4+2.4+6=28.8\), so \(M=28.8/30=0.9600\).
After the changes. The corrected \(12\)-thousand-dollar claim has primary \(10\) and excess \(2\). The second claim’s net amount is \(14-3=11\), so it has primary \(10\) and excess \(1\). Therefore \(A_P=10+10=20\) and \(A_X=2+1=3\). The revised numerator is \(20+0.60(3)+0.40(6)+6=20+1.8+2.4+6=30.2\). The revised modification is \(30.2/30=1.0067\), to four decimal places.
Check. The primary total rises by \(2\) and the excess total falls by \(1\), so the numerator changes by \(2+0.60(-1)=1.4\). The factor change is \(1.4/30=0.0467\); adding this to \(0.9600\) gives \(1.0067\), as calculated directly.
Worked Example: A Recovery Moves a Claim Below the Split Point
A hypothetical plan uses a \(10\)-thousand-dollar split point. Its claims are \(15\) and \(6\) thousand dollars. A \(7\)-thousand-dollar recovery is recognized on the \(15\)-thousand-dollar claim, reducing its net amount to \(8\) thousand dollars. Let \(E_P=16\), \(E_X=4\), \(Z=0.50\), and \(B=4\), in thousands of dollars where applicable. Calculate the factor before and after the recovery.
Before the recovery. The first claim has primary \(10\) and excess \(5\); the second has primary \(6\) and excess \(0\). Thus \(A_P=16\), \(A_X=5\), and the denominator is \(16+4+4=24\). The numerator is \(16+0.50(5)+0.50(4)+4=16+2.5+2+4=24.5\). The original factor is \(24.5/24=1.0208\), to four decimal places.
After the recovery. The recovered claim’s net amount is \(15-7=8\). Its revised components are primary \(8\) and excess \(0\); the other claim remains at primary \(6\) and excess \(0\). Consequently, \(A_P=14\) and \(A_X=0\). The new numerator is \(14+0.50(0)+0.50(4)+4=20\), so the revised factor is \(20/24=0.8333\), to four decimal places.
Check. The recovery changes primary losses by \(-2\) and excess losses by \(-5\). The numerator falls by \(-2+0.50(-5)=-4.5\), from \(24.5\) to \(20\). Dividing the change by \(24\) gives a factor change of \(-0.1875\), consistent with \(0.8333\) after rounding.
Worked Example: Revising a Claim That Crosses the Split Point
A hypothetical plan has a split point of \(8\) thousand dollars and three claims of \(6\), \(11\), and \(8\) thousand dollars. The \(11\)-thousand-dollar claim is corrected to \(9\) thousand dollars. Expected losses are \(E_P=20\) and \(E_X=8\), credibility is \(Z=0.75\), and ballast is \(B=4\), all in thousands of dollars where applicable. Find the original and revised factors.
Original components. The claims have primary components \(6\), \(8\), and \(8\), and excess components \(0\), \(3\), and \(0\). Therefore \(A_P=6+8+8=22\) and \(A_X=3\). The denominator is \(20+8+4=32\). The numerator is \(22+0.75(3)+0.25(8)+4=22+2.25+2+4=30.25\), giving \(M=30.25/32=0.9453125\).
Revised components. The corrected \(9\)-thousand-dollar claim has primary \(8\) and excess \(1\). The other claims are unchanged, so \(A_P=22\) and \(A_X=1\). The revised numerator is \(22+0.75(1)+0.25(8)+4=22+0.75+2+4=28.75\). The revised factor is \(28.75/32=0.8984375\). To four decimal places, the original and revised factors are \(0.9453\) and \(0.8984\).
Check. The correction reduces excess losses by \(2\) and leaves primary losses unchanged. The numerator must fall by \(0.75(2)=1.5\); indeed, \(30.25-28.75=1.5\). The factor falls by \(1.5/32=0.046875\), matching the directly calculated difference.
Common Mistake: Assigning the Entire Change to One Component
A common error is to change aggregate primary or excess losses by the full dollar correction, even when a claim crosses the split point. In the first worked example, the \(8\)-thousand-dollar claim increases to \(12\). Its total rises by \(4\), but its primary component rises only from \(8\) to \(10\), a change of \(2\), while excess rises from \(0\) to \(2\), also a change of \(2\). The second claim’s recognized recovery reduces its excess component from \(4\) to \(1\), a change of \(-3\).
The correct revised totals are therefore \(A_P=20\) and \(A_X=3\). An incorrect shortcut would add the full \(4\) to primary and subtract the full \(3\) from excess, producing \(A_P=22\) and \(A_X=1\). That would give the incorrect numerator \(22+0.60(1)+0.40(6)+6=31\), and the incorrect factor \(31/30=1.0333\). Recomputing the claim components gives the correct factor \(1.0067\). The mistake is not merely a rounding difference: it assigns dollars to the wrong components, which the formula weights differently.
Check Your Understanding
For each problem, recalculate claim components before applying the stated plan formula. Amounts are in thousands of dollars where applicable.
- A plan has split point \(10\), claims of \(7\) and \(16\), and a correction reducing the second claim to \(13\). Let \(E_P=15\), \(E_X=5\), \(Z=0.50\), and \(B=5\). Find the factor before and after the correction. (Answer: \(1.1000\) before; \(1.0400\) after.)
- With split point \(10\), claims are \(12\) and \(9\), and a \(4\)-thousand-dollar recovery is recognized on the first claim, reducing it to \(8\). Let \(E_P=18\), \(E_X=6\), \(Z=0.75\), and \(B=6\). Find the factor before and after recovery. (Answer: \(0.9333\) before; \(0.8167\) after.)
- A split point of \(8\) applies to claims \(6\) and \(11\). The first is corrected to \(10\), and the second to \(9\). Let \(E_P=20\), \(E_X=5\), \(Z=0.40\), and \(B=5\). Find the original and revised factors. (Answer: \(0.7733\) before; \(0.8400\) after.)
- With split point \(15\), claims are \(22\) and \(12\); a recognized recovery of \(6\) reduces the first claim to \(16\). Let \(E_P=25\), \(E_X=10\), \(Z=0.70\), and \(B=5\). Find the factor before and after recovery. (Answer: \(0.9975\) before; \(0.8925\) after.)
- A plan has split point \(10\) and claims \(9\) and \(17\). The first is corrected to \(13\), while a recognized recovery reduces the second to \(14\). Let \(E_P=20\), \(E_X=8\), \(Z=0.50\), and \(B=4\). Find the factor before and after both changes. (Answer: \(0.9531\) before; \(0.9844\) after.)