Applying Per-Claim Split Points

Why a Per-Claim Split Point Matters

Experience rating plans may separate eligible losses into a primary component and an excess component. The primary part is limited at a specified split point for each claim; the excess part is the amount of that claim above the point. The calculation is simple, but the unit of application matters: a per-claim split point is applied separately to every eligible claim, not once to a collection of claims.

This tutorial focuses on the mechanics of that split. It is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus and assumes the loss amount for each claim is already on the plan-required basis. The examples use fixed hypothetical amounts; no probability distribution, interest rate, or mortality assumption is needed.

Definition: Let $X_j$ be the nonnegative eligible loss amount for claim $j$, and let $C$ be the nonnegative per-claim split point. The limited primary loss for claim $j$ is $X_j^{\mathrm{prim}}=\min(X_j,C)$, and its excess loss is $X_j^{\mathrm{ex}}=\max(X_j-C,0)$. The same $C$ is applied separately to each claim in the calculation.

The loss basis is part of the problem setup. For example, the stated $X_j$ might be a plan-defined incurred loss amount after specified adjustments. Do not add or remove deductibles, expenses, recoveries, or other adjustments unless the problem tells you to do so. In the calculations below, each supplied amount is already the eligible loss amount to be split.

Reading the Formula at the Boundary

The minimum and maximum functions express three cases. If a claim is below the split point, all of it is primary. If it equals the point, it is still entirely primary. If it exceeds the point, primary loss is limited to $C$, and only the amount above $C$ is excess.

\left(X_j^{\mathrm{prim}},X_j^{\mathrm{ex}}\right) = \begin{cases} (X_j,0), & 0\leq X_j\leq C,\\ (C,X_j-C), & X_j>C. \end{cases}

The equality case belongs in the first line: when $X_j=C$, the excess is zero. The formulas also permit $C=0$. Then every nonnegative claim has zero primary loss and excess equal to the full claim amount. A negative split point is not allowed under this setup; using one would break the stated component bounds.

Claim-level checks: For $X_j\geq 0$ and $C\geq 0$, each calculation must satisfy $X_j^{\mathrm{prim}}+X_j^{\mathrm{ex}}=X_j$, $0\leq X_j^{\mathrm{prim}}\leq C$, and $X_j^{\mathrm{ex}}\geq 0$. These checks apply to each claim separately.

Why the Components Reconcile

The two formulas divide the original loss without either losing or counting any amount twice. This can be justified directly by considering whether the claim is at or below the split point.

Claim at or below the point.
If $0\leq X_j\leq C$, then $\min(X_j,C)=X_j$ and $\max(X_j-C,0)=0$, since $X_j-C\leq 0$. Therefore, $X_j^{\mathrm{prim}}+X_j^{\mathrm{ex}}=X_j+0=X_j$.

Claim above the point.
If $X_j>C$, then $\min(X_j,C)=C$ and $\max(X_j-C,0)=X_j-C$. Therefore, $X_j^{\mathrm{prim}}+X_j^{\mathrm{ex}}=C+(X_j-C)=X_j$.

Check the bounds.
In the first case, $X_j^{\mathrm{prim}}=X_j$, which is between zero and $C$, and excess is zero. In the second case, primary is $C$ and excess is positive. Thus the reconciliation and bounds hold in both cases.

This argument depends on both setup conditions: eligible claim losses are nonnegative, and the split point is nonnegative. If a data file contains negative adjustments or a plan specifies a different treatment for negative values, these formulas do not by themselves determine that treatment.

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

A hypothetical rating plan uses a per-claim split point of $C=\$5{,}000$. Three eligible claim losses are $\$3{,}750$, $\$5{,}000$, and $\$8{,}450$. Calculate the primary and excess components for each claim. The amounts are deterministic; no probability distribution or interest assumption is used.

Method. Apply the minimum and maximum formulas separately to each claim. This directly handles the exact-boundary case as well as claims on either side of it.

Claim 1: $X_1=\$3{,}750$. Since $\$3{,}750<\$5{,}000$, $X_1^{\mathrm{prim}}=\min(\$3{,}750,\$5{,}000)=\$3{,}750$, and $X_1^{\mathrm{ex}}=\max(\$3{,}750-\$5{,}000,0)=\max(-\$1{,}250,0)=\$0$. The check is $\$3{,}750+\$0=\$3{,}750$.

Claim 2: $X_2=\$5{,}000$. Since the loss equals the split point, $X_2^{\mathrm{prim}}=\min(\$5{,}000,\$5{,}000)=\$5{,}000$, and $X_2^{\mathrm{ex}}=\max(\$5{,}000-\$5{,}000,0)=\$0$. The check is $\$5{,}000+\$0=\$5{,}000$.

Claim 3: $X_3=\$8{,}450$. Since $\$8{,}450>\$5{,}000$, $X_3^{\mathrm{prim}}=\min(\$8{,}450,\$5{,}000)=\$5{,}000$, and $X_3^{\mathrm{ex}}=\max(\$8{,}450-\$5{,}000,0)=\$3{,}450$. The check is $\$5{,}000+\$3{,}450=\$8{,}450$.

Result. The component pairs $(\text{primary},\text{excess})$ are $(\$3{,}750,\$0)$, $(\$5{,}000,\$0)$, and $(\$5{,}000,\$3{,}450)$. The common mistake avoided here is treating a claim exactly at $C$ as having a positive excess amount; it has none.

Calculate, Then Reconcile Each Claim

A reliable exam procedure is to record the claim amount and split point first, calculate the primary amount, calculate excess as the nonnegative amount above the point, and then reconcile the two components to the original loss. The table is useful because it makes the claim-level unit explicit.

Worked Example: Applying a Larger Split Point Across Several Claims

A hypothetical commercial plan specifies $C=\$12{,}000$ per claim. Three eligible claim amounts are $\$9{,}650$, $\$12{,}000$, and $\$19{,}275$. Find both components for each claim and verify the results. No probability distribution or interest assumption is involved.

Step 1: Identify the claim’s position relative to $C$. The first claim is below $\$12{,}000$, the second is exactly $\$12{,}000$, and the third is above it.

Step 2: Calculate components. For the first claim, $X_1^{\mathrm{prim}}=\min(\$9{,}650,\$12{,}000)=\$9{,}650$, while $X_1^{\mathrm{ex}}=\max(\$9{,}650-\$12{,}000,0)=\max(-\$2{,}350,0)=\$0$. For the second, $X_2^{\mathrm{prim}}=\min(\$12{,}000,\$12{,}000)=\$12{,}000$, while $X_2^{\mathrm{ex}}=\max(\$12{,}000-\$12{,}000,0)=\$0$. For the third, $X_3^{\mathrm{prim}}=\min(\$19{,}275,\$12{,}000)=\$12{,}000$, while $X_3^{\mathrm{ex}}=\max(\$19{,}275-\$12{,}000,0)=\$7{,}275$.

Step 3: Reconcile. The checks are $\$9{,}650+\$0=\$9{,}650$, $\$12{,}000+\$0=\$12{,}000$, and $\$12{,}000+\$7{,}275=\$19{,}275$. Each primary amount is between zero and $\$12{,}000$, and each excess amount is nonnegative.

Result and trap avoided. The three component pairs are $(\$9{,}650,\$0)$, $(\$12{,}000,\$0)$, and $(\$12{,}000,\$7{,}275)$. Do not calculate excess as the full claim amount whenever a claim reaches the split point: excess is only the amount above the point.

Recovering an Amount From Its Components

A plan worksheet may provide a primary amount and an excess amount rather than the original eligible claim. The reconciliation identity makes it possible to recover that original amount by adding the two components. The split point still provides a consistency check: primary cannot exceed it, and a positive excess implies that primary is at the split point under this method.

Worked Example: Checking Reported Components and Recovering the Claim

A hypothetical audit uses a per-claim split point of $C=\$8{,}000$. A worksheet reports primary and excess components of $(\$8{,}000,\$2{,}350)$ for one claim and $(\$6{,}700,\$0)$ for another. Recover each eligible claim amount and check whether the reported components follow the per-claim split rule. Amounts are deterministic, with no probability distribution or interest assumption.

First claim. Add the reported components to recover the eligible loss: $X_1=\$8{,}000+\$2{,}350=\$10{,}350$. Because $X_1>\$8{,}000$, the formula gives primary $\min(\$10{,}350,\$8{,}000)=\$8{,}000$ and excess $\max(\$10{,}350-\$8{,}000,0)=\$2{,}350$. The worksheet is consistent.

Second claim. The recovered loss is $X_2=\$6{,}700+\$0=\$6{,}700$. Because $X_2<\$8{,}000$, the formula gives primary $\min(\$6{,}700,\$8{,}000)=\$6{,}700$ and excess $\max(\$6{,}700-\$8{,}000,0)=\$0$. This worksheet entry is also consistent.

Result and check. The recovered eligible claims are $\$10{,}350$ and $\$6{,}700$, and both reported pairs reproduce under the formulas. A useful audit trap is to accept a component pair just because its two entries add to a plausible loss: the primary cap and the excess rule must also be satisfied.

Per-Claim Is Not an Aggregate Cap

A per-claim split point does not cap the primary amount for a whole policy or group of claims. For each claim below the point, the entire eligible loss is primary, even if the primary amounts from several claims together exceed $C$. This distinction is about where the formula is applied; it does not change the calculation for any individual claim.

Worked Example: Avoiding a Policy-Level Cap Error

A hypothetical account has two eligible claims of $\$8{,}600$ and $\$7{,}900$, with a per-claim split point $C=\$10{,}000$. Calculate each claim’s components. Then assess a proposed worksheet that treats $\$10{,}000$ as a single cap on the combined amount. These are fixed losses; no probability distribution or interest assumption is used.

Claim 1. Since $\$8{,}600<\$10{,}000$, $X_1^{\mathrm{prim}}=\min(\$8{,}600,\$10{,}000)=\$8{,}600$, and $X_1^{\mathrm{ex}}=\max(\$8{,}600-\$10{,}000,0)=\$0$.

Claim 2. Since $\$7{,}900<\$10{,}000$, $X_2^{\mathrm{prim}}=\min(\$7{,}900,\$10{,}000)=\$7{,}900$, and $X_2^{\mathrm{ex}}=\max(\$7{,}900-\$10{,}000,0)=\$0$.

Check the proposed method. Applying one cap to the combined $\$8{,}600+\$7{,}900=\$16{,}500$ would label $\$10{,}000$ as primary and $\$16{,}500-\$10{,}000=\$6{,}500$ as excess. That is not the specified per-claim calculation: it assigns excess even though neither individual claim exceeds $\$10{,}000$.

Result and trap avoided. The correct individual pairs are $(\$8{,}600,\$0)$ and $(\$7{,}900,\$0)$. The error is applying the split point once to the combined losses instead of applying it separately to each claim.

Practical Checks and Common Errors

In a rating calculation, a clean component split depends on using the correct claim-level input and applying the stated point consistently. Before accepting a result, check that the amount being split is the plan-defined eligible loss, that the split point is nonnegative, and that the calculation has been performed claim by claim. Then verify the identity and bounds for every claim.

Subtracting the point without a zero floor. For a loss below $C$, $X_j-C$ is negative; excess must be $\max(X_j-C,0)$, not $X_j-C$.
Using the entire loss as excess. For a loss above $C$, excess is $X_j-C$, not $X_j$; the first $C$ dollars remain primary.
Misclassifying equality. At $X_j=C$, primary is $C$ and excess is zero.
Applying one cap to several claims. A per-claim split is calculated independently for each loss; it is not a single cap on their combined amount.
Ignoring the input basis. Do not silently change the stated loss for a deductible, recovery, or other adjustment that is not part of the supplied instructions.

Key takeaway: For each nonnegative eligible claim $X_j$ and nonnegative per-claim split point $C$, calculate $\min(X_j,C)$ as primary and $\max(X_j-C,0)$ as excess. Confirm that the components add back to that claim’s amount.

Check Your Understanding

For each problem, treat the stated claim amounts as nonnegative eligible losses and apply the stated split point separately to each claim. Amounts are deterministic; no probability distribution or interest assumption is required.

A plan has $C=\$4{,}200$. For one claim of $\$3{,}950$, calculate primary and excess. (Answer: primary $\$3{,}950$; excess $\$0$.)
For $C=\$9{,}500$, calculate the components of a claim exactly equal to $\$9{,}500$, and state the reconciliation check. (Answer: primary $\$9{,}500$; excess $\$0$; $\$9{,}500+\$0=\$9{,}500$.)
With $C=\$6{,}750$, split claims of $\$5{,}125$ and $\$8{,}900$ separately. (Answer: $\$5{,}125$ claim: primary $\$5{,}125$, excess $\$0$; $\$8{,}900$ claim: primary $\$6{,}750$, excess $\$2{,}150$.)
A worksheet reports primary $\$7{,}200$ and excess $\$1{,}460$ under $C=\$7{,}200$. Recover the claim amount and check the split. (Answer: claim $\$8{,}660$; primary $\$7{,}200$; excess $\$1{,}460$, consistent.)
Two claims are $\$6{,}400$ and $\$5{,}850$, with a per-claim point of $\$7{,}000$. A reviewer applies one $\$7{,}000$ cap to their combined amount. Give the correct per-claim pairs and explain why the reviewer’s method is wrong. (Answer: pairs $(\$6{,}400,\$0)$ and $(\$5{,}850,\$0)$; the reviewer’s combined cap is not a per-claim split and incorrectly assigns excess although neither claim exceeds $\$7{,}000$.)

What You'll Learn