Decomposing Claims at the Split Point

Why Decompose a Claim at a Split Point?

In an experience rating plan, a split point separates each claim into a primary portion and an excess portion. The primary portion is capped at the split point; any amount above that point is excess. This is a claim-by-claim calculation: the split does not assign a whole claim to one category or the other. It divides the amount of an individual claim between two components.

For example, with a split point of $10,000, a $7,000 claim contributes $7,000 to primary losses and nothing to excess losses. A $10,000 claim contributes $10,000 to primary losses and nothing to excess losses. A $16,000 claim contributes $10,000 to primary losses and $6,000 to excess losses. The boundary matters: reaching the split point does not create an excess amount; excess begins only above it.

Definition: Let $X$ be a nonnegative claim amount and let $S$ be a nonnegative split point. The primary loss for the claim is $P=\min(X,S)$. The excess loss for the claim is $E=\max(X-S,0)$. Both amounts are nonnegative, and they are measured in the same units as $X$ and $S$.

These formulas describe a decomposition of a claim amount, not a deductible or a policy limit. In particular, the excess formula subtracts the split point only when the claim exceeds it. The primary amount is the portion up to the split point, not a separate payment triggered only by small claims.

The Three Cases

For $X\geq 0$ and $S\geq 0$, the formulas can be written by cases. This makes the treatment of claims at the boundary explicit.

(P,E)= \begin{cases} (X,0), & 0\leq X\leq S,\\ (S,X-S), & X>S. \end{cases}

If $X<S$, the whole claim is primary and none is excess. If $X=S$, the claim is still wholly primary: $P=S$ and $E=0$. If $X>S$, the primary amount is exactly $S$, and only the amount above $S$ is excess. The same split point is applied separately to every claim in a list.

The decomposition has a useful check:

$$ P+E=X. $$

This identity holds in both cases. When $X\leq S$, $P+E=X+0=X$. When $X>S$, $P+E=S+(X-S)=X$. Thus, after calculating the components, their sum should recover each original claim and, by addition, the total of a portfolio’s claims.

A Direct Derivation and a Reliable Calculation Process

The formula for excess follows from the meaning of the split. The primary portion uses the first $\min(X,S)$ dollars of the claim. Whatever remains is $X-\min(X,S)$. If $X\leq S$, this remainder is $X-X=0$; if $X>S$, it is $X-S$. Equivalently, the remainder is $\max(X-S,0)$, which gives the stated excess formula. This reasoning also shows why using $X-S$ without a nonnegative floor is incorrect for claims below the split.

Fix the unit and split point.
Use the same currency and scale for every claim and for $S$. For example, do not compare claims stated in dollars with a split point stated in thousands of dollars without converting one of them.

Apply the formulas to each claim.
For each $X$, calculate $P=\min(X,S)$ and $E=\max(X-S,0)$. Do not calculate the split once on the portfolio total when the plan specifies a split for each claim.

Reconcile the results.
Check $P+E=X$ for each claim and check that total primary plus total excess equals total losses. Investigate any difference before reporting the components.

This procedure treats the reported claim amounts as fixed inputs. No probability distribution, interest rate, or mortality assumption is needed to split a given claim. If claim amounts are instead modeled as random variables, the same definitions apply to each possible value, but calculating expected component losses requires additional distributional information.

Worked Claim Decompositions

Worked Example: Claims Below, At, and Above the Split

A hypothetical property program uses a split point of $S=\$12{,}000$. Three reported claim amounts are $X_1=\$7{,}400$, $X_2=\$12{,}000$, and $X_3=\$19{,}500$. Find the primary and excess amounts for each claim and reconcile the totals. Treat the claim amounts as fixed; no probability distribution, interest rate, or mortality assumption is used. Report amounts to the cent; no rounding is needed.

Method. Apply the minimum and maximum formulas to each claim separately, then sum the components.

For claim 1, $X_1=\$7{,}400\leq \$12{,}000$. Therefore, $P_1=\min(\$7{,}400,\$12{,}000)=\$7{,}400$, and $E_1=\max(\$7{,}400-\$12{,}000,0)=\max(-\$4{,}600,0)=\$0$. Check: $\$7{,}400+\$0=\$7{,}400$.

For claim 2, $X_2=\$12{,}000=S$. Therefore, $P_2=\min(\$12{,}000,\$12{,}000)=\$12{,}000$, and $E_2=\max(\$12{,}000-\$12{,}000,0)=\$0$. Check: $\$12{,}000+\$0=\$12{,}000$.

For claim 3, $X_3=\$19{,}500>\$12{,}000$. Therefore, $P_3=\min(\$19{,}500,\$12{,}000)=\$12{,}000$, and $E_3=\max(\$19{,}500-\$12{,}000,0)=\$7{,}500$. Check: $\$12{,}000+\$7{,}500=\$19{,}500$.

The primary total is $\$7{,}400+\$12{,}000+\$12{,}000=\$31{,}400$. The excess total is $\$0+\$0+\$7{,}500=\$7{,}500$. The original claim total is $\$7{,}400+\$12{,}000+\$19{,}500=\$38{,}900$, and $\$31{,}400+\$7{,}500=\$38{,}900$, so the totals reconcile.

Common mistake avoided: treating the claim exactly at the split as partly excess. At $X=S$, the excess is zero, not a positive amount.

Worked Example: Applying One Split Consistently Across a Claim List

A hypothetical liability account has four claims of $\$2{,}600$, $\$8{,}500$, $\$11{,}300$, and $\$21{,}750$. The split point is $S=\$8{,}500$. Calculate the primary and excess amount for each claim and find both totals. These are fixed claim amounts, with no probability distribution, interest rate, or mortality assumption. Keep exact dollar amounts and round only if needed to cents.

Method. Compare each claim with $\$8{,}500$, compute its two components, and then sum. The split is applied to every individual claim, not just to the largest one.

For the $\$2{,}600$ claim, $P_1=\min(\$2{,}600,\$8{,}500)=\$2{,}600$, and $E_1=\max(\$2{,}600-\$8{,}500,0)=\$0$.

For the $\$8{,}500$ claim, $P_2=\min(\$8{,}500,\$8{,}500)=\$8{,}500$, and $E_2=\max(\$8{,}500-\$8{,}500,0)=\$0$.

For the $\$11{,}300$ claim, $P_3=\min(\$11{,}300,\$8{,}500)=\$8{,}500$, and $E_3=\max(\$11{,}300-\$8{,}500,0)=\$2{,}800$.

For the $\$21{,}750$ claim, $P_4=\min(\$21{,}750,\$8{,}500)=\$8{,}500$, and $E_4=\max(\$21{,}750-\$8{,}500,0)=\$13{,}250$.

The primary total is $\$2{,}600+\$8{,}500+\$8{,}500+\$8{,}500=\$28{,}100$. The excess total is $\$0+\$0+\$2{,}800+\$13{,}250=\$16{,}050$. The original total is $\$2{,}600+\$8{,}500+\$11{,}300+\$21{,}750=\$44{,}150$. The component total is $\$28{,}100+\$16{,}050=\$44{,}150$, as required.

Common mistake avoided: applying the split to the account total as if there were one claim of $\$44{,}150$. That would produce a single primary amount of $\$8{,}500$, which does not represent the sum of the primary portions of four separately split claims.

Worked Example: Reconciling a Portfolio With a Zero Claim

A hypothetical equipment account reports five claim amounts: $\$0$, $\$9{,}800$, $\$15{,}000$, $\$16{,}200$, and $\$43{,}100$. Its split point is $S=\$15{,}000$. Calculate both components for all five claims and verify the portfolio total. The entries are fixed amounts; no probability distribution, interest rate, or mortality assumption is used. No rounding is required.

Method. Use the case formula for each claim, including the zero claim and the claim equal to the split point.

For $X_1=\$0$, $P_1=\min(\$0,\$15{,}000)=\$0$, and $E_1=\max(\$0-\$15{,}000,0)=\$0$.

For $X_2=\$9{,}800$, $P_2=\min(\$9{,}800,\$15{,}000)=\$9{,}800$, and $E_2=\max(\$9{,}800-\$15{,}000,0)=\$0$.

For $X_3=\$15{,}000$, $P_3=\$15{,}000$ and $E_3=\max(\$15{,}000-\$15{,}000,0)=\$0$.

For $X_4=\$16{,}200$, $P_4=\$15{,}000$ and $E_4=\$16{,}200-\$15{,}000=\$1{,}200$. For $X_5=\$43{,}100$, $P_5=\$15{,}000$ and $E_5=\$43{,}100-\$15{,}000=\$28{,}100$.

Summing the primary amounts gives $\$0+\$9{,}800+\$15{,}000+\$15{,}000+\$15{,}000=\$54{,}800$. Summing the excess amounts gives $\$0+\$0+\$0+\$1{,}200+\$28{,}100=\$29{,}300$. The original losses total $\$0+\$9{,}800+\$15{,}000+\$16{,}200+\$43{,}100=\$84{,}100$. The component check is $\$54{,}800+\$29{,}300=\$84{,}100$.

Common mistake avoided: dropping a zero claim from the working table without checking the remaining entries’ alignment. A zero contributes nothing to either component, but retaining it can help ensure that every reported claim is accounted for exactly once.

Common Errors and Practical Checks

A common incorrect calculation is to define excess as $X-S$ for every claim. For example, with $X=\$6{,}000$ and $S=\$10{,}000$, that calculation gives an excess amount of $-\$4{,}000$. A negative excess amount is not a valid component of this split. The correction is $E=\max(X-S,0)=\max(-\$4{,}000,0)=\$0$; primary is $\min(\$6{,}000,\$10{,}000)=\$6{,}000$, and the components add to the claim.

Another error is to classify an entire claim as either primary or excess. Under this definition, a claim above $S$ has both components: $S$ is primary and $X-S$ is excess. A third error is to apply the split only after summing claims. The functions $\min(X,S)$ and $\max(X-S,0)$ are applied at the claim level, so the total components must be calculated by summing the results for individual claims.

Before finalizing a calculation, verify that the split point and all claims use the same units, that no claim has been assigned a negative excess amount, and that each claim’s primary and excess amounts sum to its original value. Then reconcile the portfolio totals. These checks catch many arithmetic and setup errors without requiring any assumptions about the distribution of losses.

Key takeaway: For each claim, primary is the lesser of the claim and the split point; excess is the amount above the split point, floored at zero. Apply the split claim by claim and verify that primary plus excess recovers the original loss.

Check Your Understanding

For each problem, calculate the requested components using the stated split point. Answers are shown for checking.

A hypothetical account has one claim of $\$4{,}250$ and a split point of $\$6{,}000$. Find primary and excess. (Answer: primary $\$4{,}250$; excess $\$0$)
A claim is exactly $\$9{,}750$, equal to the split point. What are its two components? (Answer: primary $\$9{,}750$; excess $\$0$)
With a split point of $\$7{,}200$, decompose claims of $\$5{,}000$, $\$8{,}900$, and $\$12{,}400$. Find the component totals and reconcile to total losses. (Answer: claim components $(\$5{,}000,\$0)$, $(\$7{,}200,\$1{,}700)$, and $(\$7{,}200,\$5{,}200)$; primary total $\$19{,}400$, excess total $\$6{,}900$, total losses $\$26{,}300$)
A report lists two claims as $\$14.6$ thousand and $\$21.2$ thousand, with a split point of $\$16{,}000$. Convert consistently, then find each claim’s components and the excess total. (Answer: components $(\$14{,}600,\$0)$ and $(\$16{,}000,\$5{,}200)$; excess total $\$5{,}200$)
A hypothetical portfolio has claims of $\$0$, $\$10{,}000$, $\$13{,}500$, and $\$31{,}000$, and a split point of $\$10{,}000$. Find both component totals and confirm the reconciliation. (Answer: primary total $\$30{,}000$, excess total $\$24{,}500$, total losses $\$54{,}500$)

What You'll Learn