Aggregating Primary Losses Across Claims

Why Aggregate Losses by Claim?

Experience rating plans may distinguish losses below a specified per-claim split point from losses above it. The first component is called primary loss; the amount above the point is excess loss. Since the split is applied separately to each claim, the primary and excess totals must be built by splitting each claim first and summing the resulting components afterward.

This order matters. A large claim may contribute both a primary amount and an excess amount, while a small claim contributes only primary loss. Applying one split point to the total losses for a policyholder would not preserve that claim-by-claim allocation. Once the two actual totals are calculated, they can be used as the actual-loss inputs to the plan’s modification formula.

Definition: For claim $j$ with loss amount $X_j$ and per-claim split point $s$, its primary loss is $P_j=\min(X_j,s)$, and its excess loss is $E_j=\max(X_j-s,0)$. For $n$ claims, aggregate primary loss is $A_P=\sum_{j=1}^n P_j$, and aggregate excess loss is $A_E=\sum_{j=1}^n E_j$.

Here, $X_j$ is the loss amount included under the plan’s stated loss basis. A calculation must use a consistent basis for all claims—for example, the plan’s specified incurred or paid amounts—and the split point must be applied to each such amount. The examples below assume the listed amounts are already the amounts to be used. No interest rate or probability distribution is needed: these are deterministic allocations of a supplied claim set.

Splitting and Summing

For each claim, compare its loss with the split point. If the loss does not exceed the point, the entire amount is primary and excess is zero. If the loss exceeds the point, the primary amount equals the point and the excess is the difference.

P_j=\min(X_j,s), \qquad E_j=(X_j-s)^+=\max(X_j-s,0)

A useful check follows directly from these definitions. For every claim,

P_j+E_j=\min(X_j,s)+\max(X_j-s,0)=X_j.

Therefore, when all claims are included on the same basis, their aggregated components must reconcile to the original aggregate loss:

A_P+A_E=\sum_{j=1}^n X_j.

This identity is an audit, not a substitute for splitting each claim. Two incorrect allocations can sometimes have the correct combined total, so also verify that no individual primary amount exceeds $s$, and that excess is zero whenever $X_j\leq s$.

Worked Applications

Worked Example: Aggregate a Four-Claim Set and Calculate the Modification

A hypothetical account has four losses of $\$4{,}800$, $\$12{,}500$, $\$37{,}000$, and $\$9{,}900$. The plan uses a per-claim split point of $\$10{,}000$. Its modification formula is

M=1+\frac{Z_P(A_P-E_P)+Z_E(A_E-E_E)}{E_P+E_E+B}.

The plan supplies expected primary loss $E_P=\$36{,}000$, expected excess loss $E_E=\$24{,}000$, weights $Z_P=0.80$ and $Z_E=0.30$, and ballast $B=\$20{,}000$. Find the actual component totals and $M$. Use the stated formula and round $M$ to four decimal places.

Method. Split every claim at $\$10{,}000$, then sum each component.

For the $\$4{,}800$ claim, primary loss is $\min(4{,}800,10{,}000)=\$4{,}800$, and excess loss is $\max(4{,}800-10{,}000,0)=\$0$.

For the $\$12{,}500$ claim, primary loss is $\$10{,}000$, and excess loss is $\$12{,}500-\$10{,}000=\$2{,}500$.

For the $\$37{,}000$ claim, primary loss is $\$10{,}000$, and excess loss is $\$37{,}000-\$10{,}000=\$27{,}000$.

For the $\$9{,}900$ claim, primary loss is $\$9{,}900$, and excess loss is $\$0$. Thus,

A_P=4{,}800+10{,}000+10{,}000+9{,}900=\$34{,}700, \qquad A_E=0+2{,}500+27{,}000+0=\$29{,}500.

The original total is $\$4{,}800+\$12{,}500+\$37{,}000+\$9{,}900=\$64{,}200$, and $A_P+A_E=\$34{,}700+\$29{,}500=\$64{,}200$, so the allocation reconciles.

The primary deviation is $A_P-E_P=\$34{,}700-\$36{,}000=-\$1{,}300$. The excess deviation is $A_E-E_E=\$29{,}500-\$24{,}000=\$5{,}500$. The weighted numerator is

0.80(-1{,}300)+0.30(5{,}500)=-1{,}040+1{,}650=\$610.

The denominator is $\$36{,}000+\$24{,}000+\$20{,}000=\$80{,}000$. Therefore,

M=1+\frac{610}{80{,}000}=1.007625 \approx 1.0076.

Check. Primary is below expected while excess is above expected; their weighted effects partially offset, leaving a small positive adjustment. The common mistake avoided is applying the split point to the aggregate $\$64{,}200$, rather than to each claim.

Worked Example: Audit the Component Totals Before Substitution

A hypothetical account has claim losses of $\$6{,}000$, $\$14{,}000$, $\$26{,}000$, $\$8{,}000$, and $\$42{,}000$, with split point $s=\$12{,}000$. The supplied formula is the same as in the preceding example. The plan specifies $E_P=\$54{,}000$, $E_E=\$38{,}000$, $Z_P=0.70$, $Z_E=0.40$, and $B=\$25{,}000$. Calculate the component totals and modification, rounding the modification to four decimal places.

Method. Cap each claim’s primary component at $\$12{,}000$, and record only the amount above $\$12{,}000$ as excess. The claim-by-claim allocations, in thousands of dollars, are:

\begin{array}{c|r|r} X_j & P_j & E_j\\ 6 & 6 & 0\\ 14 & 12 & 2\\ 26 & 12 & 14\\ 8 & 8 & 0\\ 42 & 12 & 30 \end{array}

Thus, $A_P=\$6{,}000+\$12{,}000+\$12{,}000+\$8{,}000+\$12{,}000=\$50{,}000$, and $A_E=\$0+\$2{,}000+\$14{,}000+\$0+\$30{,}000=\$46{,}000$. The original claim total is $\$96{,}000$, matching $A_P+A_E=\$96{,}000$.

The primary deviation is $\$50{,}000-\$54{,}000=-\$4{,}000$. The excess deviation is $\$46{,}000-\$38{,}000=\$8{,}000$. The weighted numerator is $0.70(-\$4{,}000)+0.40(\$8{,}000)=-\$2{,}800+\$3{,}200=\$400$. The denominator is $\$54{,}000+\$38{,}000+\$25{,}000=\$117{,}000$. Hence,

M=1+\frac{400}{117{,}000}=1.0034188\ldots \approx 1.0034.

Check. The component totals sum to the $\$96{,}000$ gross total, and the formula’s numerator is positive because the weighted excess increase exceeds the weighted primary decrease. The common mistake avoided is treating the primary and excess weights as interchangeable; each deviation is multiplied by its own supplied weight.

Worked Example: Recompute a Primary Total Carefully

A hypothetical account has losses of $\$3{,}200$, $\$9{,}800$, $\$15{,}200$, $\$25{,}000$, $\$12{,}000$, and $\$41{,}000$. The per-claim split point is $\$15{,}000$. The plan supplies $E_P=\$68{,}000$, $E_E=\$40{,}000$, $Z_P=0.85$, $Z_E=0.25$, and $B=\$30{,}000$. Using the stated modification formula, calculate $A_P$, $A_E$, and $M$, rounded to four decimal places.

Method. Apply $P_j=\min(X_j,\$15{,}000)$ and $E_j=\max(X_j-\$15{,}000,0)$ to each claim. In thousands of dollars, the primary values are $3.2, 9.8, 15, 15, 12,$ and $15$. Their sum is

A_P=3{,}200+9{,}800+15{,}000+15{,}000+12{,}000+15{,}000=\$70{,}000.

The excess values are $0,0,200,10{,}000,0,$ and $26{,}000$, so

A_E=0+0+200+10{,}000+0+26{,}000=\$36{,}200.

The original total is $\$106{,}200$; the component sum is also $\$70{,}000+\$36{,}200=\$106{,}200$. The primary deviation is $\$70{,}000-\$68{,}000=\$2{,}000$, and the excess deviation is $\$36{,}200-\$40{,}000=-\$3{,}800$. The weighted numerator is $0.85(\$2{,}000)+0.25(-\$3{,}800)=\$1{,}700-\$950=\$750$. The denominator is $\$68{,}000+\$40{,}000+\$30{,}000=\$138{,}000$. Thus,

M=1+\frac{750}{138{,}000}=1.0054348\ldots \approx 1.0054.

Check. The primary total is $\$70{,}000$, not $\$71{,}100$: adding its six entries gives $3{,}200+9{,}800=13{,}000$, $13{,}000+15{,}000=28{,}000$, $+15{,}000=43{,}000$, $+12{,}000=55{,}000$, and $+15{,}000=70{,}000$. The common mistake avoided is reporting a primary sum before checking the arithmetic; an incorrect component total changes both the deviation and the final modification.

Why the Allocation Identity Holds

The per-claim identity can be justified by considering the two possible cases. If $X_j\leq s$, then $P_j=X_j$ and $E_j=0$, giving $P_j+E_j=X_j$. If $X_j>s$, then $P_j=s$ and $E_j=X_j-s$, so $P_j+E_j=s+(X_j-s)=X_j$. Summing this equality across all $n$ claims proves the aggregate reconciliation:

\sum_{j=1}^n(P_j+E_j)=\sum_{j=1}^n X_j \quad\Longrightarrow\quad A_P+A_E=\sum_{j=1}^n X_j.

The formula’s numerator also has a direct interpretation: it adds the primary actual-to-expected deviation, multiplied by its plan weight, to the similarly weighted excess deviation. The denominator combines expected primary loss, expected excess loss, and ballast, all on the plan’s stated dollar basis. The result is a modification relative to 1; it is not obtained by dividing actual losses by expected losses unless the plan specifically defines a different formula.

Common Mistakes and a Reliable Workflow

A frequent error is to calculate $\max(\sum_jX_j-s,0)$ as aggregate excess. That subtracts the split point only once, whereas the plan applies it to every claim. Another is to set every primary amount equal to $s$, including claims below the point. Both errors violate the claim-level definitions.

Use this sequence to keep the allocation and modification calculation separate:

Confirm the inputs.
Use the claim amounts and split point on the exact loss basis specified by the plan.

Split each claim.
Calculate $\min(X_j,s)$ and $\max(X_j-s,0)$ separately for every claim.

Sum and reconcile.
Find $A_P$ and $A_E$; check each claim’s split and confirm $A_P+A_E=\sum_jX_j$.

Substitute into the plan formula.
Calculate the two actual-minus-expected deviations, apply their respective weights, then use the stated denominator and ballast.

A further trap is reversing the sign of a deviation. In the formula used here, the component term is $A_P-E_P$ or $A_E-E_E$, not expected minus actual. Before evaluating the formula, write each difference explicitly and preserve its sign. Do not round intermediate dollar totals; round the reported modification only at the final step.

Key takeaway: A per-claim split point creates two components for each claim. Aggregate those components only after splitting each claim, reconcile them to the gross total, and then apply the plan’s supplied modification formula using the correct component weights and signs. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, split claims individually before calculating any aggregate. Use the modification formula stated in the question where one is provided.

A hypothetical account has losses of $\$5{,}500$, $\$9{,}000$, $\$17{,}000$, and $\$31{,}000$, with split point $\$9{,}000$. Find $A_P$ and $A_E$. (Answer: $A_P=\$32{,}500$, $A_E=\$30{,}000$.)
Claims are $\$18{,}000$, $\$24{,}000$, $\$35{,}000$, and $\$9{,}000$, split at $\$20{,}000$. Given $E_P=\$65{,}000$, $E_E=\$24{,}000$, $Z_P=0.60$, $Z_E=0.50$, and $B=\$11{,}000$, calculate $M$ using $M=1+[Z_P(A_P-E_P)+Z_E(A_E-E_E)]/(E_P+E_E+B)$. (Answer: $A_P=\$67{,}000$, $A_E=\$19{,}000$, $M=0.9870$.)
Claims are $\$7{,}000$, $\$14{,}000$, $\$19{,}000$, and $\$33{,}000$, split at $\$10{,}000$. Given $E_P=\$40{,}000$, $E_E=\$30{,}000$, $Z_P=0.75$, $Z_E=0.20$, and $B=\$10{,}000$, calculate the modification from the formula in question 2. Round to four decimal places. (Answer: $A_P=\$37{,}000$, $A_E=\$36{,}000$, $M=0.9869$.)
Claims are $\$12{,}000$, $\$27{,}000$, $\$40{,}000$, and $\$65{,}000$, split at $\$25{,}000$. Given $E_P=\$90{,}000$, $E_E=\$50{,}000$, $Z_P=0.80$, $Z_E=0.30$, and $B=\$20{,}000$, calculate the modification from the formula in question 2. Round to four decimal places. (Answer: $A_P=\$87{,}000$, $A_E=\$57{,}000$, $M=0.9981$.)

What You'll Learn