Purpose of a Split Point
Experience rating uses an insured’s loss experience to help inform its rating. A plan may separate that experience into primary losses and excess losses, rather than treating every dollar of every claim as the same kind of loss. A split point sets the claim-level boundary between those two portions. This distinction is useful when the plan intends to treat smaller losses differently from the part of a large loss above a specified amount.
For a given claim, the primary portion includes the loss up to the split point. The excess portion is the amount, if any, above it. The split point is applied to each claim individually, not once to the total losses for a year. That distinction is central to experience-rating calculations aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
The split point does not erase or exclude loss. It allocates each claim between two components. Nor does it, by itself, specify how the components affect a premium or how much credibility either component receives. Those are separate plan-design decisions. Here, the task is to understand the claim-level allocation and its arithmetic.
Applying the Split Point Claim by Claim
For each claim, compare its amount with \(M\). If the claim is smaller than \(M\), record the full claim as primary and zero as excess. If it equals \(M\), record \(M\) as primary and zero as excess. If it exceeds \(M\), record \(M\) as primary and subtract \(M\) from the claim to find its excess portion.
The allocation has a useful check: for every claim, the primary and excess portions add back to that claim. Therefore, after applying the split point to all claims, the aggregate primary and excess losses must add to the original aggregate loss. If they do not, an allocation or addition error has occurred.
Worked Example: Splitting Four Claims at One Threshold
A hypothetical account has four adjusted claim amounts of $8,000, $25,000, $42,000, and $61,000. The plan specifies a per-claim split point of $25,000. Find each claim’s primary and excess portions, then aggregate the components. The amounts are stipulated; no probability distribution, interest assumption, or mortality assumption is used.
Method. Apply the split point separately to each claim using \(X^P=\min(X,M)\) and \(X^E=\max(X-M,0)\).
Claim 1: $8,000. Since $8,000 is below $25,000, primary loss is \(\min(8{,}000,25{,}000)=\$8{,}000\). Excess loss is \(\max(8{,}000-25{,}000,0)=\$0\).
Claim 2: $25,000. The claim is exactly at the split point. Primary loss is \(\min(25{,}000,25{,}000)=\$25{,}000\), and excess loss is \(\max(25{,}000-25{,}000,0)=\$0\).
Claim 3: $42,000. Primary loss is \(\min(42{,}000,25{,}000)=\$25{,}000\). Excess loss is \(\max(42{,}000-25{,}000,0)=\$17{,}000\).
Claim 4: $61,000. Primary loss is \(\min(61{,}000,25{,}000)=\$25{,}000\). Excess loss is \(\max(61{,}000-25{,}000,0)=\$36{,}000\).
The aggregate primary loss is \(8{,}000+25{,}000+25{,}000+25{,}000=\$83{,}000\). The aggregate excess loss is \(0+0+17{,}000+36{,}000=\$53{,}000\). The original aggregate is \(8{,}000+25{,}000+42{,}000+61{,}000=\$136{,}000\), and the component check gives \(83{,}000+53{,}000=\$136{,}000\).
Common mistake avoided: The excess portion of the $61,000 claim is not $61,000. That would count the first $25,000 again as excess. Only the amount above the split point, $36,000, is excess.
Why the Allocation Formula Works
The formula follows directly by considering the two possible positions of a nonnegative claim relative to \(M\). If \(X\le M\), then \(\min(X,M)=X\) and \(\max(X-M,0)=0\). The sum is \(X\). If \(X>M\), then \(\min(X,M)=M\) and \(\max(X-M,0)=X-M\). Their sum is \(M+(X-M)=X\). These cases also establish the boundary: when \(X=M\), the whole claim is primary and none is excess.
This claim-level identity explains why the aggregate check works. If an experience period contains claims \(X_1,\ldots,X_n\), apply the identity to each claim and add the resulting equalities. The total primary loss plus total excess loss must equal the total claim loss.
The formula is for nonnegative claim amounts on the plan’s specified measurement basis. If claim records are adjusted or restricted by other policy terms, first establish which amount the plan says to split. Do not silently switch between ground-up losses, insurer payments, or another measure. A different basis can produce different portions even when the split point is unchanged.
Worked Example: Aggregating Split Losses for an Account
A hypothetical account has five claim amounts, in thousands of dollars: 12, 39, 40, 57, and 88. The plan applies a $40,000 split point per claim to these stated amounts. Find aggregate primary and excess losses and verify the total. No probability distribution, interest assumption, or mortality assumption is used; all claim amounts are given.
Method. Work in thousands of dollars and split each claim at \(M=40\).
For the $12,000 claim, primary is \(\min(12,40)=12\) and excess is \(\max(12-40,0)=0\). For $39,000, primary is 39 and excess is 0. For $40,000, primary is 40 and excess is 0. For $57,000, primary is 40 and excess is \(57-40=17\). For $88,000, primary is 40 and excess is \(88-40=48\).
Add the primary portions: \(12+39+40+40+40=171\), or $171,000. Add the excess portions: \(0+0+0+17+48=65\), or $65,000.
The original claims total \(12+39+40+57+88=236\), or $236,000. The split totals reconcile because \(171+65=236\). Thus the aggregate primary and excess amounts are $171,000 and $65,000.
Common mistake avoided: The split point is per claim. Applying $40,000 once to the $236,000 account total would not produce the plan’s primary and excess claim components.
Changing the Split Point Changes the Allocation
For a fixed set of claims, increasing the split point cannot reduce the primary portion of any claim: more of a large claim remains in the primary component. It cannot increase the excess portion either: less of that claim lies above the higher boundary. The total loss, however, is unchanged. A split point reallocates losses between components; it does not alter the underlying claim amounts.
Worked Example: Comparing Two Split Points
A hypothetical account has adjusted claims of $18,000, $44,000, $76,000, and $130,000. Compare aggregate primary and excess losses at split points of $30,000 and $50,000 per claim. Claim amounts are stipulated; no probability distribution, interest assumption, or mortality assumption is used.
Method. Apply the minimum and positive-part formulas at each threshold, without changing the claim amounts.
At \(M=\$30{,}000\): The primary portions are $18,000, $30,000, $30,000, and $30,000, totaling \(18+30+30+30=\$108{,}000\). The excess portions are $0, $14,000, $46,000, and $100,000, totaling \(0+14+46+100=\$160{,}000\). Together they equal $268,000.
At \(M=\$50{,}000\): The primary portions are $18,000, $44,000, $50,000, and $50,000, totaling \(18+44+50+50=\$162{,}000\). The excess portions are $0, $0, $26,000, and $80,000, totaling \(0+0+26+80=\$106{,}000\). Together they again equal $268,000.
Raising the split point from $30,000 to $50,000 increases aggregate primary losses by \(162-108=\$54{,}000\) and reduces aggregate excess losses by \(160-106=\$54{,}000\). The total remains unchanged. This is the expected result: only the allocation changes.
Common mistake avoided: A higher split point does not change the claim total or make the excess vanish for every large claim. For example, the $130,000 claim still has $80,000 of excess at a $50,000 split point.
What a Split Point Is—and Is Not
A split point is a threshold for each claim. It is not an annual aggregate deductible: an annual threshold would be applied to accumulated losses across the period, which is a different calculation. A split point is also not automatically a policy deductible or a coverage limit. A deductible determines the portion of a covered loss retained by the insured under the policy terms; a policy limit caps payment under those terms. A split point instead labels portions of the loss amount selected by the experience-rating plan.
The plan’s stated loss basis matters. If it says to split ground-up claim amounts, use those amounts. If it specifies a different measure, such as amounts paid under a stated coverage basis, use that measure instead. Do not subtract a deductible, apply a limit, or change a claim value unless the defined basis and applicable policy terms call for it.
A practical calculation sequence is:
Identify the claim amount the plan specifies and the unit in which it is reported.
Confirm that it is a per-claim threshold and use the same currency and scale as the losses.
Assign the lesser of the claim and the split point to primary; assign only the amount above the split point to excess.
Sum each component separately, then verify that primary plus excess equals the original aggregate claim amount.
Common Errors to Avoid
- Splitting the annual total instead of individual claims. Apply the boundary once to every claim, then aggregate the portions.
- Counting the whole large claim as excess. Excess is only the amount above \(M\); the primary portion already contains the first \(M\) dollars, or the entire smaller claim.
- Using inconsistent units. If claims are shown in thousands, express the split point in thousands too. A threshold of 40 in a table stated in thousands means $40,000, not $40.
- Changing the loss basis without instruction. Do not assume a split applies before or after a deductible or limit unless the plan specifies the relevant claim measure.
- Failing to reconcile totals. For the same claim basis, aggregate primary plus aggregate excess must equal aggregate losses.
Check Your Understanding
For each problem, use the stated claim basis and split point. Check your component totals against the original aggregate.
- A hypothetical account has claims of $19,000, $35,000, and $62,000, with a $35,000 per-claim split point. Find the primary and excess portion of each claim and the aggregate of each component. (Answer: Primary $89,000; excess $27,000; total $116,000.)
- Four stipulated claims, in thousands of dollars, are 47, 63, 110, and 40. The per-claim split point is $60,000. Calculate the aggregate primary and excess amounts and verify the total. (Answer: Primary $207,000; excess $53,000; total $260,000.)
- A hypothetical account has claims of $20,000, $52,000, and $75,000 in Year A, and $30,000, $45,000, and $90,000 in Year B. Apply the same $45,000 per-claim split point to all six claims. Find combined primary and excess losses and check their sum. (Answer: Primary $230,000; excess $82,000; total $312,000.)
- For claims of $18,000, $44,000, $76,000, and $130,000, an analyst raises the per-claim split point from $30,000 to $50,000. By how much do aggregate primary losses change, by how much do aggregate excess losses change, and does the total loss change? (Answer: Primary increases by $54,000; excess decreases by $54,000; total remains $268,000.)
- A plan specifies a $30,000 split point applied to stated ground-up losses. Two hypothetical ground-up claims are $22,000 and $42,000. A reader incorrectly subtracts a $5,000 policy deductible before splitting. Give the correct primary and excess totals on the specified basis, and identify the reader’s basis error. (Answer: Correct primary $52,000 and excess $12,000, totaling $64,000; subtracting the deductible changes the specified ground-up basis.)