How Split Points Affect Component Weights

Why the Split Point Changes Component Weights

In a primary-excess experience plan, each claim is divided at a specified per-claim split point. The amount below the split point belongs to the primary component; the amount above it belongs to the excess component. Moving the split point does not change the claim’s total loss, but it changes how much expected loss is assigned to each component. When each component has its own credibility standard, that reallocation can change the two credibility weights differently.

The key calculation is performed claim by claim, before aggregation. For a nonnegative loss $X$ and split point $m$, the primary payment is the lesser of the loss and the split point; the excess payment is the amount, if any, by which the loss exceeds the split point. This tutorial focuses on how those expected amounts and their resulting weights change as $m$ changes. All examples are hypothetical and use undiscounted losses: no interest or mortality assumption is needed.

Definition: For a per-claim split point $m$, define primary loss as $X_P=\min(X,m)$, and excess loss as $X_E=(X-m)^+=\max(X-m,0)$. For a portfolio of $N$ claims, let $V_P=N E[X_P]$ and $V_E=N E[X_E]$ be the expected aggregate primary and excess losses. These quantities are the component expected-loss volumes used in the credibility calculations below.

Expected Losses and Separate Component Standards

For each possible claim amount, allocate the loss at the split point. A claim smaller than $m$ is entirely primary. A claim larger than $m$ contributes $m$ to primary and the remaining amount to excess. A claim equal to $m$ contributes $m$ to primary and zero to excess. If the severity distribution has possible values $x_k$ with probabilities $p_k$, then the component means are found by weighting each allocated amount by its probability.

E[X_P]=\sum_k p_k\min(x_k,m), \qquad E[X_E]=\sum_k p_k\max(x_k-m,0).

A common square-root plan-weight method uses a separate full-credibility expected-loss threshold for each component. Under that method, if $K_P$ and $K_E$ are the primary and excess thresholds, respectively, the weights are:

Z_P=\min\left(1,\sqrt{\frac{V_P}{K_P}}\right),\qquad Z_E=\min\left(1,\sqrt{\frac{V_E}{K_E}}\right).

This formula assumes the square-root credibility method is being used and that the component volumes and thresholds are expressed on the same basis and in the same currency. A component at or above its threshold receives full credibility, so its weight is capped at 1. The formula is not a statement that every rating plan must use these thresholds or this credibility method; a plan with different standards must use its specified weighting rule.

The component volumes add back to the expected total losses because every claim is partitioned into its primary and excess amounts. This is an important check: shifting the split point changes the allocation, not the sum.

X_P+X_E=X,\qquad V_P+V_E=N E[X].

Why the Reallocation Occurs

For any fixed claim amount $x$, raising the split point cannot reduce $\min(x,m)$, and it cannot increase $\max(x-m,0)$. The amount transferred to primary is exactly the amount removed from excess. Taking expected values preserves that relationship: expected primary loss is nondecreasing in $m$, expected excess loss is nonincreasing, and their sum remains $E[X]$.

For a continuous severity distribution, the rate of change gives an additional interpretation. Let $F_X(m)=P(X\le m)$, and suppose the relevant expectations are finite. A small increase in $m$ increases primary payment by approximately that increase for claims above $m$, and does not affect claims below $m$. Thus, where the derivative exists:

\frac{d}{dm}E[\min(X,m)]=P(X>m),\qquad \frac{d}{dm}E[(X-m)^+]=-P(X>m).

To justify the first equality, write $E[\min(X,m)]$ as the expected value of the capped loss. For a loss $x$, its derivative with respect to $m$ is 0 when $x<m$ and 1 when $x>m$. Averaging these derivatives gives $P(X>m)$, subject to the usual conditions for differentiating the expectation. For the excess payment, the corresponding derivative is $-1$ when $x>m$ and 0 when $x<m$, giving the negative of that probability. For a distribution with probability mass at the split point, derivative statements require care at that point; the claim-by-claim allocation formulas remain valid.

Worked Example: Comparing Two Split Points

A hypothetical book has 100 claims. Each claim has a discrete severity distribution: $P(X=\$4{,}000)=0.50$, $P(X=\$12{,}000)=0.30$, and $P(X=\$30{,}000)=0.20$. There is no discounting or mortality component. Compare split points of \$10,000 and \$20,000. The separate full-credibility thresholds are $K_P=\$1{,}000{,}000$ and $K_E=\$400{,}000$. Use the square-root plan-weight method.

Method. Calculate the primary and excess amount at each severity, probability-weight those amounts to obtain per-claim means, multiply by 100 claims, and then apply each component’s own threshold.

At $m=\$10{,}000$. At severities of \$4,000, \$12,000, and \$30,000, primary amounts are \$4,000, \$10,000, and \$10,000. Therefore:

E[X_P]=0.50(\$4{,}000)+0.30(\$10{,}000)+0.20(\$10{,}000) =\$2{,}000+\$3{,}000+\$2{,}000=\$7{,}000.

The corresponding excess amounts are \$0, \$2,000, and \$20,000. Thus:

E[X_E]=0.50(\$0)+0.30(\$2{,}000)+0.20(\$20{,}000) =\$0+\$600+\$4{,}000=\$4{,}600.

For 100 claims, $V_P=100(\$7{,}000)=\$700{,}000$ and $V_E=100(\$4{,}600)=\$460{,}000$. The weights are:

Z_P=\sqrt{\frac{\$700{,}000}{\$1{,}000{,}000}} =\sqrt{0.70}=0.8367,\qquad Z_E=\min\left(1,\sqrt{\frac{\$460{,}000}{\$400{,}000}}\right)=1.0000.

The excess volume exceeds its threshold, so the excess weight is capped at 1.

At $m=\$20{,}000$. Primary amounts at the three severities are \$4,000, \$12,000, and \$20,000:

E[X_P]=0.50(\$4{,}000)+0.30(\$12{,}000)+0.20(\$20{,}000) =\$2{,}000+\$3{,}600+\$4{,}000=\$9{,}600.

Excess amounts are \$0, \$0, and \$10,000, so $E[X_E]=0.50(\$0)+0.30(\$0)+0.20(\$10{,}000)=\$2{,}000$. The portfolio volumes are $V_P=\$960{,}000$ and $V_E=\$200{,}000$. Therefore:

Z_P=\sqrt{\frac{\$960{,}000}{\$1{,}000{,}000}} =\sqrt{0.96}=0.9798,\qquad Z_E=\sqrt{\frac{\$200{,}000}{\$400{,}000}} =\sqrt{0.50}=0.7071.

Check. At either split point, total expected losses equal $100[0.50(\$4{,}000)+0.30(\$12{,}000)+0.20(\$30{,}000)]=\$1{,}160{,}000$. The component volumes sum to \$700,000 + \$460,000 = \$1,160,000 at \$10,000, and \$960,000 + \$200,000 = \$1,160,000 at \$20,000. Raising the split point increases the primary volume and decreases the excess volume. Common mistake avoided: applying the primary threshold to excess losses, or vice versa, would produce weights inconsistent with the specified separate standards.

Interpreting Changes in the Weights

If exposure and the severity distribution remain fixed, a higher split point transfers expected loss from excess to primary. Under separate square-root standards, the primary weight will not decrease and the excess weight will not increase. However, the size of either weight change depends on its threshold and on whether that component is already at full credibility. Once a component is capped at 1, additional expected volume does not raise its assigned weight.

The ratio of component volumes is not necessarily the ratio of their weights. Before either weight reaches its cap, the weights depend on both the volumes and their respective thresholds. For example, a component with a larger expected volume can still have a lower weight if its full-credibility threshold is sufficiently larger. Always calculate each weight using its own component’s volume and standard.

Worked Example: A Weight Rises to Full Credibility

A hypothetical portfolio has 80 claims. Claim severity is \$4,000 with probability 0.50, \$12,000 with probability 0.30, and \$30,000 with probability 0.20. There is no discounting or mortality component. Compare split points of \$5,000 and \$15,000. The separate thresholds are $K_P=\$500{,}000$ and $K_E=\$800{,}000$.

At $m=\$5{,}000$. The primary amounts are \$4,000, \$5,000, and \$5,000, giving:

E[X_P]=0.50(\$4{,}000)+0.30(\$5{,}000)+0.20(\$5{,}000) =\$2{,}000+\$1{,}500+\$1{,}000=\$4{,}500.

The excess amounts are \$0, \$7,000, and \$25,000, so $E[X_E]=0.50(\$0)+0.30(\$7{,}000)+0.20(\$25{,}000)=\$2{,}100+\$5{,}000=\$7{,}100$. Multiplying each mean by 80 gives $V_P=\$360{,}000$ and $V_E=\$568{,}000$. The weights are:

Z_P=\sqrt{\frac{\$360{,}000}{\$500{,}000}}=\sqrt{0.72}=0.8485,\qquad Z_E=\sqrt{\frac{\$568{,}000}{\$800{,}000}}=\sqrt{0.71}=0.8426.

At $m=\$15{,}000$. The primary amounts are \$4,000, \$12,000, and \$15,000, so $E[X_P]=\$2{,}000+\$3{,}600+\$3{,}000=\$8{,}600$. The excess amounts are \$0, \$0, and \$15,000, so $E[X_E]=\$3{,}000$. The portfolio volumes are $V_P=80(\$8{,}600)=\$688{,}000$ and $V_E=80(\$3{,}000)=\$240{,}000$. Thus:

Z_P=\min\left(1,\sqrt{\frac{\$688{,}000}{\$500{,}000}}\right)=1.0000,\qquad Z_E=\sqrt{\frac{\$240{,}000}{\$800{,}000}}=\sqrt{0.30}=0.5477.

Check. Expected total loss per claim is \$11,600, so expected total losses for 80 claims are \$928,000. The components sum to \$360,000 + \$568,000 = \$928,000 at the lower split and \$688,000 + \$240,000 = \$928,000 at the higher split. The primary weight rises to its cap; the excess weight falls. Common mistake avoided: the uncapped primary calculation at the higher split is greater than 1, but the assigned credibility weight is 1, not a number above 1.

A Split Point at a Possible Claim Amount

A split point can coincide with a severity value that has positive probability. The allocation rule still applies directly: a claim equal to the split point contributes that amount to primary and nothing to excess. Do not assign a positive excess amount merely because the claim is “at” the split; excess begins only above the split.

Worked Example: Handling Probability Mass at the Split

A hypothetical portfolio has 120 claims. Each claim is \$4,000 with probability 0.40, \$10,000 with probability 0.35, or \$25,000 with probability 0.25. Use a \$10,000 split point. There is no discounting or mortality component. The primary and excess thresholds are $K_P=\$1{,}200{,}000$ and $K_E=\$600{,}000$.

At the three severities, primary amounts are \$4,000, \$10,000, and \$10,000. Therefore $E[X_P]=0.40(\$4{,}000)+0.35(\$10{,}000)+0.25(\$10{,}000)=\$1{,}600+\$3{,}500+\$2{,}500=\$7{,}600$. The \$10,000 claim is entirely primary because it equals the split point.

The excess amounts are \$0, \$0, and \$15,000. Hence $E[X_E]=0.40(\$0)+0.35(\$0)+0.25(\$15{,}000)=\$3{,}750$. For 120 claims, $V_P=\$912{,}000$ and $V_E=\$450{,}000$. Applying the separate standards:

Z_P=\sqrt{\frac{\$912{,}000}{\$1{,}200{,}000}} =\sqrt{0.76}=0.8718,\qquad Z_E=\sqrt{\frac{\$450{,}000}{\$600{,}000}} =\sqrt{0.75}=0.8660.

Check. The expected loss per claim is $0.40(\$4{,}000)+0.35(\$10{,}000)+0.25(\$25{,}000)=\$1{,}600+\$3{,}500+\$6{,}250=\$11{,}350$. For 120 claims, this is \$1,362,000, equal to the component sum $\$912{,}000+\$450{,}000$. Common mistake avoided: counting the \$10,000 claims as having excess losses would violate the definition $X_E=\max(X-m,0)$.

Common Mistakes and Practical Checks

When comparing split points, keep the severity distribution, claim count, currency basis, and component standards fixed unless the problem explicitly changes them. The following checks help isolate allocation errors from credibility errors:

Allocate per claim before aggregating. Apply $\min(X,m)$ and $\max(X-m,0)$ to each severity, then take expected values.
Check conservation. Confirm that the two component means sum to $E[X]$, and that their aggregate volumes sum to total expected losses.
Use the matching threshold. Compute $Z_P$ from $V_P$ and $K_P$, and $Z_E$ from $V_E$ and $K_E$.
Apply the cap. A component whose volume exceeds its threshold has weight 1 under the stated method.
Keep the split convention clear. A claim exactly at $m$ has zero excess loss.

A tempting but incorrect calculation is to use the full expected loss volume for both component weights. For instance, in the \$10,000 split of the first example, total expected losses are \$1,160,000. Substituting that total for both component volumes would give $\sqrt{\$1{,}160{,}000/\$1{,}000{,}000}$, capped at 1, for primary and $\sqrt{\$1{,}160{,}000/\$400{,}000}$, also capped at 1, for excess. Those are not the component weights: the correct volumes are \$700,000 and \$460,000, giving $Z_P=0.8367$ and $Z_E=1.0000$. The total volume is a useful reconciliation check, not a substitute for either component volume.

Key takeaway: Changing the split point redistributes expected losses between primary and excess while preserving their total. With separate credibility standards, calculate each component’s expected volume first, then apply its own square-root weight and cap.

Check Your Understanding

For each problem, allocate losses at the stated split point before calculating the component weights. Round weights to four decimal places.

A hypothetical portfolio has 50 claims with severity \$6,000 (probability 0.50), \$18,000 (probability 0.30), or \$40,000 (probability 0.20). At a \$12,000 split, calculate both expected component volumes and weights if $K_P=\$900{,}000$ and $K_E=\$400{,}000$. (Answer: $V_P=\$450{,}000,\ V_E=\$370{,}000;\ Z_P=0.7071,\ Z_E=0.9618$.)
A hypothetical book has 100 claims with severity \$2,000 (probability 0.40), \$8,000 (probability 0.40), or \$20,000 (probability 0.20). At a \$6,000 split, calculate component volumes and weights using $K_P=\$400{,}000$ and $K_E=\$500{,}000$. (Answer: $V_P=\$440{,}000,\ V_E=\$360{,}000;\ Z_P=1.0000,\ Z_E=0.8485$.)
A hypothetical portfolio has 80 claims with severity \$1,000 (probability 0.25), \$9,000 (probability 0.50), or \$21,000 (probability 0.25). At a \$10,000 split, find component volumes and weights for $K_P=\$800{,}000$ and $K_E=\$400{,}000$. (Answer: $V_P=\$580{,}000,\ V_E=\$220{,}000;\ Z_P=0.8515,\ Z_E=0.7416$.)
A hypothetical portfolio has 40 claims with severity \$5,000 (probability 0.50), \$15,000 (probability 0.30), or \$35,000 (probability 0.20). Compare weights at split points of \$10,000 and \$20,000, using $K_P=\$350{,}000$ and $K_E=\$300{,}000$. (Answer: at \$10,000, $V_P=\$300{,}000,\ V_E=\$260{,}000;\ Z_P=0.9258,\ Z_E=0.9309$. At \$20,000, $V_P=\$440{,}000,\ V_E=\$120{,}000;\ Z_P=1.0000,\ Z_E=0.6325$.)

What You'll Learn