Why Separate Loss Components?
An experience rating plan may treat smaller claim amounts differently from amounts above a specified threshold. Separating losses into primary and excess components makes that distinction explicit: primary loss is the portion up to the split point, while excess loss is the portion above it. The split is performed claim by claim, before the components are summed.
This matters when a plan assigns different credibility weights to the two components. A calculation that splits an aggregate total instead of individual claims can produce different primary and excess totals, and therefore a different modification. The examples here use a single fixed split point for all claims and hypothetical data. Interest, mortality, and a probability distribution are not assumed; the calculations are deterministic.
Partition Each Claim Before Aggregating
If a claim is at or below \(c\), all of it is primary and none is excess. If it is above \(c\), exactly \(c\) is primary and the remainder is excess. Thus, for every nonnegative claim amount,
This identity is a useful calculation check. Summing it across claims gives \(\sum_i L_i=A_P+A_X\). It also shows why the split point is applied to each claim rather than once to the sum: the function \(\min(L_i,c)\) is applied separately to every claim amount.
Worked Example: Splitting and Reconciling Four Claims
A hypothetical account has four included claims of $1,250, $4,800, $11,600, and $7,500. The plan specifies a fixed per-claim split point of $5,000. Calculate each primary and excess amount, then aggregate both components. All values are hypothetical; no interest, mortality, or loss distribution is assumed.
Method. Apply \(P_i=\min(L_i,5{,}000)\) and \(X_i=\max(L_i-5{,}000,0)\) to each claim.
| Claim amount | Primary calculation | Primary | Excess calculation | Excess |
|---|---|---|---|---|
| $1,250 | min($1,250, $5,000) | $1,250 | max($1,250 − $5,000, 0) | $0 |
| $4,800 | min($4,800, $5,000) | $4,800 | max($4,800 − $5,000, 0) | $0 |
| $11,600 | min($11,600, $5,000) | $5,000 | max($11,600 − $5,000, 0) | $6,600 |
| $7,500 | min($7,500, $5,000) | $5,000 | max($7,500 − $5,000, 0) | $2,500 |
Aggregate primary: $1,250 + $4,800 + $5,000 + $5,000 = $16,050.
Aggregate excess: $0 + $0 + $6,600 + $2,500 = $9,100.
Check: The original claims total $1,250 + $4,800 + $11,600 + $7,500 = $25,150. The components total $16,050 + $9,100 = $25,150, so they reconcile. The common mistake this avoids is capping the aggregate account loss at $5,000, which would incorrectly treat a per-claim threshold as an account-level threshold.
Why the Component Identity Holds
The partition formula follows by considering the two possible relationships between a claim and the split point. If \(L_i\le c\), then \(P_i=L_i\) and \(X_i=0\), so \(P_i+X_i=L_i\). If \(L_i>c\), then \(P_i=c\) and \(X_i=L_i-c\), so \(P_i+X_i=c+(L_i-c)=L_i\). These cases cover every nonnegative claim amount, including a claim exactly equal to \(c\).
Summing the claim-level identity gives the aggregate reconciliation:
This result is more than a convenient check. It confirms that the partition reallocates each included dollar between two components rather than discarding or counting a dollar twice. It relies on nonnegative claim amounts and on the two components being defined at the same split point and on the same loss basis.
Using Components in a Modification
A plan may compare actual primary and excess losses with their corresponding expected amounts and apply separate credibility weights. The exact experience modification formula is plan-specific. To make the calculation precise, the next example stipulates a particular formula rather than implying that every plan uses it.
The numerator is a credibility-weighted departure from expected loss. The denominator is the stipulated expected loss volume plus ballast. Matching actual and expected components is essential: a primary actual amount must be compared with primary expected loss, not with total expected loss or expected excess loss.
Worked Example: Applying Separate Component Weights
A hypothetical plan reports amounts in thousands of dollars and specifies the formula in the information box above. Actual primary and excess losses are $36 thousand and $24 thousand. Expected primary and excess losses are $40 thousand and $15 thousand. The primary and excess credibility weights are 0.80 and 0.40, and the ballast is $10 thousand. Calculate the modification. The values are hypothetical; no interest, mortality, or probability distribution is assumed.
Method. Use the already-separated component totals and keep all dollar quantities in thousands throughout.
Step 1: Calculate component deviations. Primary deviation \(=36-40=-4\) thousand. Excess deviation \(=24-15=9\) thousand.
Step 2: Apply the component weights. Weighted primary deviation \(=0.80(-4)=-3.2\) thousand. Weighted excess deviation \(=0.40(9)=3.6\) thousand.
Step 3: Add the weighted deviations. Numerator \(=-3.2+3.6=0.4\) thousand.
Step 4: Calculate the denominator. \(E_P+E_X+B=40+15+10=65\) thousand.
Step 5: Calculate and round the modification. \(M=1+0.4/65=1.006153846\ldots\), so the modification is 1.0062 to four decimal places.
Check and trap avoided. The excess component is above expected, while the primary component is below expected; their weighted deviations partially offset, leaving a positive $0.4 thousand numerator. Using \(A_P+A_X=60\) against \(E_P+E_X=55\) as one unweighted deviation would discard the different component weights and would not implement the stated formula.
Calculation Sequence and Data Checks
A disciplined sequence reduces errors when claim records and experience-rating inputs come from different sources. First confirm that the included claims and loss amounts are on the plan’s required basis. For example, whether amounts are paid, incurred, capped, or adjusted must come from the plan instructions; the split operation alone does not decide that basis. Then apply the split point claim by claim, sum each component, and reconcile the component totals to the included total loss.
Identify the included claims, the loss basis, the units, and the fixed split point. Use a consistent basis for actual and expected component amounts.
For each claim, calculate primary as the lesser of its amount and the split point, and excess as any amount above the split point.
Sum all primary amounts to obtain \(A_P\); independently sum all excess amounts to obtain \(A_X\).
Verify that \(A_P+A_X\) equals the included total claim amount. Insert the reconciled components into the specific modification formula.
Worked Example: Claims at, Below, and Above the Split Point
A hypothetical account has claims of $1,900, $5,000, $5,001, and $9,750, with a fixed per-claim split point of $5,000. Calculate aggregate primary and excess losses and verify the total. All amounts are hypothetical; no interest, mortality, or probability distribution is assumed.
Method. Apply the split separately, including to the claim exactly at the threshold.
For $1,900, primary is $1,900 and excess is $0. For $5,000, primary is $5,000 and excess is $0. For $5,001, primary is $5,000 and excess is $1. For $9,750, primary is $5,000 and excess is $4,750.
Aggregate primary: $1,900 + $5,000 + $5,000 + $5,000 = $16,900.
Aggregate excess: $0 + $0 + $1 + $4,750 = $4,751.
Check: Original claims total $1,900 + $5,000 + $5,001 + $9,750 = $21,651. Components total $16,900 + $4,751 = $21,651. A claim equal to the split point has no excess; the dollar above the point on the $5,001 claim is excess.
Common Mistakes
- Splitting the aggregate instead of the claims. For a per-claim split point, calculate the components for each claim first. The aggregate primary amount can exceed the split point because multiple claims each contribute a primary portion.
- Counting the threshold amount twice. A claim above \(c\) contributes \(c\) to primary and only \(L_i-c\) to excess. It does not contribute the full claim amount to both components.
- Putting a claim exactly at the split point into excess. When \(L_i=c\), primary is \(c\) and excess is zero under the definitions used here.
- Mixing loss bases or units. Do not compare actual primary losses measured on one basis with expected primary losses measured on another. Similarly, do not combine dollars with thousands of dollars in a single formula.
Wrong calculation and correction: With a $5,000 split point and claims of $4,000 and $8,000, treating the account total of $12,000 as one amount gives primary $5,000 and excess $7,000. That is an aggregate split, not a per-claim split. Correctly, the claims produce primary amounts of $4,000 and $5,000, for aggregate primary of $9,000; their excess amounts are $0 and $3,000, for aggregate excess of $3,000. The corrected components still total $12,000.
Check Your Understanding
For each problem, assume hypothetical deterministic claim amounts, no interest or mortality assumption, and no probability distribution. Apply the stated fixed per-claim split point.
- Claims are $900, $4,000, and $6,200, and the split point is $4,000. Find aggregate primary and excess losses and check the total. (Answer: Primary $8,900; excess $2,200; total $11,100.)
- Claims are $8,500, $19,000, $12,000, and $31,500, and the split point is $12,000. Find both aggregate components, including the claim exactly at the threshold. (Answer: Primary $44,500; excess $26,500; total $71,000.)
- In thousands of dollars, \(A_P=28\), \(E_P=30\), \(Z_P=0.75\), \(A_X=14\), \(E_X=10\), \(Z_X=0.50\), and \(B=20\). Under \(M=1+\{Z_P(A_P-E_P)+Z_X(A_X-E_X)\}/(E_P+E_X+B)\), calculate \(M\) to four decimal places. (Answer: \(M=1.0083\).)
- Claims are $9,900, $10,200, and $14,100, with a $10,000 split point. Calculate the aggregate components and reconcile them. (Answer: Primary $29,900; excess $4,300; total $34,200.)