Selecting Split Points for Experience Plans

Why the Split Point Matters

An experience plan may separate losses into a primary component and an excess component at a selected split point. That point affects how much expected loss is assigned to each component and how individual claims influence each component’s experience. A low split point places more expected loss in excess; a high split point places more in primary.

The split point is a plan-design choice, not automatically a policy deductible, coverage limit, or reinsurance attachment point. For the calculations here, $X$ is the severity of one claim on the loss basis used by the experience plan. If the plan uses covered losses after a deductible or limit, use that covered-loss amount as the input severity rather than treating the plan split as another policy term.

Definition: For a nonnegative claim severity $X$ and split point $c$, the primary loss per claim is $\min(X,c)$, and the excess loss per claim is $(X-c)_+=\max(X-c,0)$. Their expectations are the expected primary and excess losses per claim. The D-ratio at $c$ is the expected primary loss divided by expected total loss, provided $E[X]>0$: $$ D(c)=\frac{E[\min(X,c)]}{E[X]}. $$

The allocation is exhaustive: for every claim severity, $\min(X,c)+(X-c)_+=X$. Therefore the expected primary and excess losses add to $E[X]$. When the plan uses a fixed expected number of claims $N$, multiply each per-claim expectation by $N$ to obtain the corresponding aggregate expected loss.

Calculating and Comparing Expected Components

A useful way to calculate the primary component is to use the survival function $S_X(x)=P(X>x)$. For a nonnegative severity with finite mean, the layer-cake identity gives:

E[\min(X,c)]=\int_0^c S_X(x)\,dx,\qquad E[(X-c)_+]=\int_c^\infty S_X(x)\,dx.

The first integral accumulates the probability that a claim reaches each amount below the split; the second accumulates the probability that it reaches amounts above the split. Their sum is $E[X]=\int_0^\infty S_X(x)\,dx$.

Derivation: For each fixed claim amount $X$, the quantity $\min(X,c)$ is the length of the interval from zero to $c$ over which $X>x$. In indicator notation, $\min(X,c)=\int_0^c \mathbf{1}_{\{X>x\}}\,dx$. Taking expectations and interchanging the nonnegative integral and expectation gives $E[\min(X,c)]=\int_0^c P(X>x)\,dx$. The same argument applied to $(X-c)_+$ gives $\int_c^\infty S_X(x)\,dx$.

If an expected-loss rate or D-ratio is already available from a severity model, it can support comparison of candidate split points. But the arithmetic does not, by itself, determine the best plan design. The selected point should also be considered in light of the experience that will be assigned to each component and the plan’s intended responsiveness to large claims.

Worked Example: Comparing Three Split Points Under an Exponential Model

A hypothetical plan models ground-up claim severity $X$ as exponential with mean $\theta=\$20{,}000$, so $S_X(x)=e^{-x/20{,}000}$ for $x\geq0$. There is no deductible or policy limit in this example. Calculate the expected primary and excess losses per claim and the D-ratio at split points of $\$10{,}000$, $\$20{,}000$, and $\$40{,}000$. No interest rate or mortality assumption applies; the calculation is undiscounted claim severity.

Method. For an exponential severity with mean $\theta$, integrate the survival function: $$ E[\min(X,c)]=\int_0^c e^{-x/\theta}\,dx =\theta(1-e^{-c/\theta}), \qquad E[(X-c)_+]=\theta e^{-c/\theta}. $$ The second expression also follows by subtracting the primary expectation from $E[X]=\theta$.

At $c=\$10{,}000$. Here $c/\theta=0.5$, and $e^{-0.5}=0.6065307$. Thus expected primary loss is $\$20{,}000(1-0.6065307)=\$7{,}869.39$, and expected excess loss is $\$20{,}000(0.6065307)=\$12{,}130.61$. The D-ratio is $\$7{,}869.39/\$20{,}000=0.3935$.

At $c=\$20{,}000$. Here $c/\theta=1$, and $e^{-1}=0.3678794$. Expected primary loss is $\$20{,}000(1-0.3678794)=\$12{,}642.41$, and expected excess loss is $\$20{,}000(0.3678794)=\$7{,}357.59$. The D-ratio is $0.6321$.

At $c=\$40{,}000$. Here $c/\theta=2$, and $e^{-2}=0.1353353$. Expected primary loss is $\$20{,}000(1-0.1353353)=\$17{,}293.29$, and expected excess loss is $\$20{,}000(0.1353353)=\$2{,}706.71$. The D-ratio is $0.8647$.

Check. At each split, the two components sum to $\$20{,}000$. Raising the split increases expected primary loss and its D-ratio, while reducing expected excess loss. The common mistake avoided here is treating the split as if it changed total expected loss; it reallocates the expected loss between components.

The comparison shows how a D-ratio summarizes one aspect of a split-point choice: the share of expected loss assigned to primary. It does not summarize the number of claims, the variance of either component, or how much historical experience is available to estimate each component. Those considerations matter when assessing whether the resulting plan can use its components as intended.

Worked Example: Comparing Split Points for a Discrete Severity Model

A hypothetical equipment-repair plan uses the following ground-up severity model: $X=\$4{,}000$ with probability 0.50, $X=\$16{,}000$ with probability 0.30, and $X=\$40{,}000$ with probability 0.20. There is no deductible, limit, interest, or mortality adjustment. Compare splits of $\$8{,}000$ and $\$20{,}000$, including expected component losses and D-ratios.

Step 1: Find total expected severity. The expected loss per claim is $0.50(\$4{,}000)+0.30(\$16{,}000)+0.20(\$40{,}000)=\$2{,}000+\$4{,}800+\$8{,}000=\$14{,}800$.

Step 2: Split at $\$8{,}000$. The primary amounts for the three possible severities are $\$4{,}000$, $\$8{,}000$, and $\$8{,}000$. Expected primary loss is $0.50(\$4{,}000)+0.30(\$8{,}000)+0.20(\$8{,}000)=\$2{,}000+\$2{,}400+\$1{,}600=\$6{,}000$. The excess amounts are $\$0$, $\$8{,}000$, and $\$32{,}000$, so expected excess loss is $0.50(\$0)+0.30(\$8{,}000)+0.20(\$32{,}000)=\$0+\$2{,}400+\$6{,}400=\$8{,}800$. The D-ratio is $\$6{,}000/\$14{,}800=0.4054$.

Step 3: Split at $\$20{,}000$. The primary amounts are $\$4{,}000$, $\$16{,}000$, and $\$20{,}000$. Expected primary loss is $0.50(\$4{,}000)+0.30(\$16{,}000)+0.20(\$20{,}000)=\$2{,}000+\$4{,}800+\$4{,}000=\$10{,}800$. The excess amounts are $\$0$, $\$0$, and $\$20{,}000$, giving expected excess loss $0.50(\$0)+0.30(\$0)+0.20(\$20{,}000)=\$4{,}000$. The D-ratio is $\$10{,}800/\$14{,}800=0.7297$.

Check and interpretation. Both allocations total $\$14{,}800$. At the higher split, the $\$16{,}000$ claim moves entirely into primary, while the $\$40{,}000$ claim contributes $\$20{,}000$ to each component. A common error is to count the entire $\$40{,}000$ claim in excess at the $\$20{,}000$ split; only the amount above the split, $\$20{,}000$, is excess.

Connect the Split to the Plan’s Intended Sensitivity

A split point determines how the same claim enters the two loss components. For a claim of size $x$, primary loss is capped at $c$; excess loss is zero when $x\leq c$ and equals $x-c$ when $x>c$. Consequently, increasing $c$ raises the primary contribution of claims between the old and new split points, while reducing their excess contribution. A claim far above both split points contributes the full lower split to primary and the remaining amount to excess.

The plan’s purpose helps frame the choice. If the primary component is intended to reflect more frequent, less severe losses, a lower split may help isolate that portion, but it also sends more expected loss to the excess component. If the plan is intended to let individual large claims influence primary experience more, a higher split allows more of each such claim into primary. These are design implications, not universal rules: the actual suitability depends on the severity pattern, available data, and how the plan uses its components.

Worked Example: Choosing Between Two Splits Given Sparse Large-Claim Experience

A hypothetical fleet plan has 100 expected claims. For planning, claim severity is modeled as $X=\$5{,}000$ with probability 0.60, $X=\$15{,}000$ with probability 0.25, and $X=\$45{,}000$ with probability 0.15. A review of the experience record indicates that only five of the 100 observed claims exceeded $\$25{,}000$, so experience above that amount is sparse. Compare split points of $\$10{,}000$ and $\$25{,}000$, and explain which better fits a design objective of limiting the effect of individual large claims on primary experience. No interest or mortality adjustment applies.

Step 1: Calculate expected severity per claim. $E[X]=0.60(\$5{,}000)+0.25(\$15{,}000)+0.15(\$45{,}000)=\$3{,}000+\$3{,}750+\$6{,}750=\$13{,}500$.

Step 2: Allocate expected loss at $\$10{,}000$. Primary amounts are $\$5{,}000$, $\$10{,}000$, and $\$10{,}000$. Expected primary loss per claim is $\$3{,}000+\$2{,}500+\$1{,}500=\$7{,}000$, so for 100 expected claims it is $\$700{,}000$. Expected excess per claim is $0.60(\$0)+0.25(\$5{,}000)+0.15(\$35{,}000)=\$1{,}250+\$5{,}250=\$6{,}500$, or $\$650{,}000$ for 100 claims.

Step 3: Allocate expected loss at $\$25{,}000$. Primary amounts are $\$5{,}000$, $\$15{,}000$, and $\$25{,}000$. Expected primary per claim is $\$3{,}000+\$3{,}750+\$3{,}750=\$10{,}500$, or $\$1{,}050{,}000$ for 100 claims. Expected excess per claim is $0.60(\$0)+0.25(\$0)+0.15(\$20{,}000)=\$3{,}000$, or $\$300{,}000$ for 100 claims.

Selection and check. Both choices allocate $\$1{,}350{,}000$ in total expected loss. The $\$10{,}000$ split caps each claim’s primary contribution at a lower amount and assigns more expected loss to excess, so it better fits the stated objective of limiting the influence of individual large claims on primary experience. The record’s five claims above $\$25{,}000$ also cautions against treating the observed excess experience as abundant. This is a design judgment, not a credibility-weight calculation: the example does not establish that $\$10{,}000$ is optimal for other objectives or data.

Practical Selection Checks

A disciplined split-point comparison should make the trade-offs explicit rather than select a threshold from a single statistic. In particular, check whether the severity model is on the same loss basis as the experience plan; whether policy deductibles and limits have been applied consistently; whether the proposed split sends a reasonable share of expected loss to each component; and whether the experience supporting each component is sufficiently informative for the plan’s intended use.

Where several candidate points are plausible, calculate expected primary loss, expected excess loss, and the D-ratio for each. Then inspect how representative individual claims move between components. The D-ratio is an expected-loss allocation measure, not a direct measure of claim-count credibility or claim volatility. Sparse large-claim data may make a design more sensitive to model assumptions, but that concern should be assessed alongside the plan’s objectives rather than treated as a formula that dictates the split.

Key takeaway: Select a split point by combining severity-based expected-loss allocations with the plan’s intended treatment of individual claims and the experience available for each component. Verify that primary plus excess equals total expected loss, and do not confuse a plan split with a policy deductible or limit.

Check Your Understanding

For each problem, use the stated severity basis and compare the requested split allocations.

A hypothetical severity model has $X=\$6{,}000$ with probability 0.50 and $X=\$18{,}000$ with probability 0.50. At a split of $\$10{,}000$, calculate expected primary and excess losses per claim. (Answer: Primary $\$8{,}000$; excess $\$4{,}000$.)
For exponential severity with mean $\$12{,}000$, calculate the D-ratio at $c=\$12{,}000$. No deductible, limit, interest, or mortality adjustment applies. (Answer: $1-e^{-1}=0.6321$.)
A hypothetical severity model has $X=\$2{,}000,\$10{,}000,\$30{,}000$ with probabilities $0.40,0.40,0.20$, respectively. Calculate expected primary and excess loss at a $\$10{,}000$ split. (Answer: Primary $\$6{,}800$; excess $\$4{,}000$.)
A model has $E[X]=\$25{,}000$, and two candidate splits have D-ratios 0.44 and 0.72. State the expected primary and excess loss per claim at each split, and identify which split assigns more expected loss to excess. (Answer: At 0.44, primary $\$11{,}000$ and excess $\$14{,}000$; at 0.72, primary $\$18{,}000$ and excess $\$7{,}000$; the 0.44 split assigns more to excess.)
A plan is intended to cap primary contribution from each claim at $\$15{,}000$. A $\$50{,}000$ claim occurs. State its primary and excess contributions under this split, and explain why assigning all $\$50{,}000$ to excess is incorrect. (Answer: Primary $\$15{,}000$, excess $\$35{,}000$; only the amount above the split is excess.)

What You'll Learn