Common Errors in Expected Losses and Plan Weights

Why Errors in Expected Losses and Plan Weights Matter

A plan weight is only meaningful when its expected-loss volume and credibility threshold use compatible units and refer to the same loss component and rate level. Likewise, a primary-and-excess allocation is only useful if both pieces reconcile to the total expected loss on the intended basis. A small-looking mismatch can materially change a weight or create an allocation that cannot be explained.

This tutorial focuses on diagnosing those errors. It uses the square-root partial-credibility convention $Z=\min(1,\sqrt{V/T})$, where $V$ is expected loss volume and $T$ is the full-credibility threshold expressed in the same units. A plan may specify different thresholds for primary and excess losses; do not substitute one for the other.

Definition: A consistent expected-loss calculation expresses the expected loss amount, its primary or excess component, and the applicable credibility threshold on a common basis: the same exposure volume, time period, currency units, rate level, and component definition. For a split point $M$, the primary payment per loss is $\min(X,M)$ and the excess payment per loss is $(X-M)_+=\max(X-M,0)$.

A Short Reconciliation Derivation

For any nonnegative loss $X$ and split point $M$, each individual loss can be decomposed exactly. If $X\leq M$, its primary portion is $X$ and its excess portion is zero. If $X>M$, its primary portion is $M$ and its excess portion is $X-M$. In either case, the portions add back to $X$:

X=\min(X,M)+(X-M)_+

Taking expectations preserves this equality, provided the expectations exist. Thus the expected primary and excess losses must sum to the expected total loss when they use the same loss distribution and basis:

E[X]=E[\min(X,M)]+E[(X-M)_+]

This identity is a practical check as well as a mathematical result. If allocated components do not add to the stated total, investigate the units, rate level, split point, and D-ratio basis before using the components in a plan-weight calculation.

Key result: If $D$ is the primary D-ratio for a specified loss distribution and split point, then $D=E[\min(X,M)]/E[X]$, for $E[X]>0$. The expected primary and excess losses are $D E[X]$ and $(1-D)E[X]$, respectively. These allocations are valid only for the distribution, split point, and rate level represented by $D$.

Worked Examples: Finding and Correcting the Error

Worked Example: Converting Per-Exposure Losses to Plan Volume

A hypothetical portfolio has 50 exposure units. Its expected loss is $3,600 per exposure unit, and the full-credibility threshold is $320,000 of expected loss volume. Under the square-root rule, calculate the plan weight. A draft calculation uses $3,600 directly as the volume.

Method. Convert the expected loss per exposure to total expected loss volume first. The 50 exposure units are the multiplier:

$$ V=50(3{,}600)=180{,}000 $$

The threshold and volume are both in dollars of expected loss. Therefore:

Z=\min\left(1,\sqrt{\frac{180{,}000}{320{,}000}}\right) =\min(1,\sqrt{0.5625}) =0.75

The correct weight is 0.75. The draft calculation compares a per-exposure amount with a portfolio-level threshold:

Z_{\text{wrong}}=\sqrt{\frac{3{,}600}{320{,}000}} =\sqrt{0.01125} \approx 0.1061

That calculation is dimensionally inconsistent and omits the 50 exposure units. The reconciliation check is $180{,}000/320{,}000=0.5625$, whose square root is exactly $0.75$, below the cap of 1. The common mistake avoided is treating an expected loss per exposure as though it were total expected loss volume.

Worked Example: Keeping a D-Ratio on the Target Rate Level

A hypothetical book has expected losses of $450,000 at a base rate level. For the target period, the applicable loss-level factor is 1.10. The D-ratio at the target level and selected split point is 0.60. Find target-level primary and excess expected losses.

Method. First bring the total expected losses to the target level:

E[X]_{\text{target}}=450{,}000(1.10)=495{,}000

The supplied D-ratio is for this target level, so apply it to the target-level total:

E[\text{primary}]=0.60(495{,}000)=297{,}000

The excess share is $1-0.60=0.40$, giving:

E[\text{excess}]=0.40(495{,}000)=198{,}000

Check the allocation: $297{,}000+198{,}000=495{,}000$, exactly the target-level total. A tempting but incorrect calculation applies the D-ratio to the base-level amount: primary $=0.60(450{,}000)=270{,}000$, excess $=0.40(450{,}000)=180{,}000$. Those amounts add to $450,000, not $495,000, so they remain at the base level. The common mistake avoided is combining a target-level D-ratio with base-level expected losses.

Worked Example: Applying the Split Point to Each Loss

In a hypothetical severity model, a claim has equal probability of taking each of the four values $3,000, $8,000, $12,000, or $17,000. The split point is $10,000 per claim. There are 100 claim exposures. Calculate the expected primary and excess loss volumes and the primary D-ratio.

Method. Apply the split point to each possible claim amount. The primary amounts are $3,000, $8,000, $10,000, and $10,000. Their mean is:

E[\min(X,10{,}000)] =\frac{3{,}000+8{,}000+10{,}000+10{,}000}{4} =7{,}750

The excess amounts are $0, $0, $2,000, and $7,000. Their mean is:

E[(X-10{,}000)_+] =\frac{0+0+2{,}000+7{,}000}{4} =2{,}250

The mean total claim loss is $(3{,}000+8{,}000+12{,}000+17{,}000)/4=10{,}000$. The component check is $7{,}750+2{,}250=10{,}000$. Thus:

D=\frac{7{,}750}{10{,}000}=0.775

Across 100 claim exposures, expected primary volume is $100(7{,}750)=\$775{,}000$, and expected excess volume is $100(2{,}250)=\$225{,}000$. They sum to $1,000,000, which also equals $100(10{,}000)$. Applying the split point once to the combined expected loss volume would be wrong: the split point is per claim, so the cap must be applied to each claim amount before aggregation. The common mistake avoided is splitting an aggregate amount as though it were one claim.

Worked Example: Using Separate Component Thresholds

A hypothetical experience plan has expected primary losses of $225,000 and expected excess losses of $100,000. Its full-credibility thresholds are $100,000 for primary losses and $400,000 for excess losses. Calculate each component’s square-root weight.

Method. Use each component’s own expected volume and threshold. For primary losses:

Z_{\text{primary}} =\min\left(1,\sqrt{\frac{225{,}000}{100{,}000}}\right) =\min(1,\sqrt{2.25})=1

For excess losses:

Z_{\text{excess}} =\min\left(1,\sqrt{\frac{100{,}000}{400{,}000}}\right) =\sqrt{0.25}=0.50

The primary component is at full credibility under its stated threshold; the excess component has a weight of 0.50 under its own threshold. The incorrect approach of combining the volumes and thresholds produces $\sqrt{325{,}000/500{,}000}=\sqrt{0.65}\approx0.8062$, which is not either component’s weight and does not answer the component-specific question. The common mistake avoided is applying a blended threshold where separate thresholds are specified.

Why the Square-Root Rule Is Sensitive to Units

The square-root weight reflects how uncertainty scales with volume under the assumptions behind the plan’s credibility convention. In a simple setting with independent, identically distributed losses having finite variance, the variance of a sum of $n$ losses is $n$ times the variance of one loss. Its standard deviation therefore scales as $\sqrt{n}$. Relative to a full-credibility volume $n_0$, the corresponding fraction is $\sqrt{n/n_0}$. A dollar-volume version expresses the same convention through $V/T$, if the plan defines $V$ and $T$ consistently.

The formula is not permission to mix unlike quantities. A per-exposure expected loss cannot be compared directly with a portfolio dollar threshold; a primary volume cannot be compared with an excess threshold; and a base-level amount cannot be silently paired with a target-level component ratio. The weight is also capped at 1 under the convention used here. If a plan specifies another credibility method, use that method rather than assuming the square-root rule.

Identify the quantity.
Determine whether each expected loss is per claim, per exposure, or for an entire portfolio or period.

Align the basis.
Confirm that the expected loss and threshold use the same currency units, exposure period, rate level, and primary or excess definition.

Allocate before weighting.
Apply the split point to individual losses or use a D-ratio appropriate to the stated distribution, split point, and level.

Reconcile and calculate.
Check that primary plus excess equals total expected loss, then apply the applicable component threshold and cap the weight as specified.

Common Exam Traps

Dropping exposure volume. Multiplying a per-exposure amount by the number of exposures may be necessary before comparing with a portfolio-level threshold.
Applying a D-ratio at the wrong level. A D-ratio based on target-level severities should not be applied to a base-level expected loss total.
Using the split point on the aggregate. A per-claim split point acts claim by claim, not once on the portfolio’s total expected loss.
Swapping the primary and excess shares. If $D$ is defined as the primary share, primary is $D E[X]$ and excess is $(1-D)E[X]$.
Using one threshold for both components. When distinct primary and excess thresholds are given, calculate distinct weights.
Forgetting the cap. Under the convention in this tutorial, a calculated value above 1 is set to 1.

Key takeaway: Before accepting a plan weight or allocation, write down the unit, rate level, exposure period, component, and threshold for every input. Then verify that the primary and excess expected losses reconcile to the same total expected loss.

Check Your Understanding

For each problem, keep the stated units and component definitions intact, and check any allocation against its total.

A hypothetical group has 36 exposure units, expected losses of $2,500 per exposure unit, and a full-credibility threshold of $360,000. Using the square-root rule, find the weight. (Answer: 0.500)
A hypothetical book has target-level expected losses of $640,000 and a primary D-ratio of 0.65 at the target level. Find target-level primary and excess expected losses. (Answer: Primary $416,000; excess $224,000)
Base-level expected losses are $280,000. The target-level factor is 1.05, and the target-level primary D-ratio is 0.75. Find the target-level primary and excess expected losses. (Answer: Primary $220,500; excess $73,500)
A plan has expected primary losses of $160,000 and expected excess losses of $90,000. The respective thresholds are $400,000 and $100,000. Find both square-root weights, capped at 1 and rounded to three decimals. (Answer: Primary 0.632; excess 0.949)
A hypothetical severity distribution assigns equal probability to $2,000, $6,000, $10,000, and $14,000. With an $8,000 per-claim split point and 500 claim exposures, find the expected primary and excess loss volumes and check their sum. (Answer: Primary $3,000,000; excess $1,000,000; total $4,000,000)

What You'll Learn