Diagnosing Influential Class Loss Experience

Why Diagnose Influential Class Experience?

A class indication can move sharply because of a small number of large losses, especially when exposure is limited. The observed class loss cost shows the full movement in experience; a credibility-weighted indication reflects only the portion assigned to that class’s own experience. Diagnosing both movements helps distinguish a genuinely large raw fluctuation from a comparably large change in the indicated cost.

The calculations here hold the class’s earned exposure, credibility weight, and collective complement fixed. Loss amounts are assumed to be on a consistent coverage, valuation, and trend basis. The examples use hypothetical insurance classes; they require no interest or mortality assumptions. Where a credibility rule is specified, it is applied as stated: there is no single universal formula for every class credibility plan.

Definition: Let $n$ be a class’s exposure count, $S$ its aggregate loss amount, and $x=S/n$ its observed loss cost per exposure. If $m$ is the collective complement and $Z$ is the class’s credibility weight, the credibility-weighted class indication is $I=Zx+(1-Z)m$. An influence is the change in $x$ or $I$ caused by a specified loss, loss adjustment, or other specified change to class experience. Unless stated otherwise, the diagnostics below hold $n$, $Z$, and $m$ fixed.

From a Loss Change to an Indicated Change

Suppose a claim amount $c$ is added to aggregate losses while the class’s exposure count stays at $n$. The observed loss cost rises by $c/n$. With the credibility weight and complement fixed, the indicated cost rises by $Zc/n$. If the claim is removed, the changes have the opposite sign.

\Delta x=\frac{\Delta S}{n} \qquad\text{and}\qquad \Delta I=Z\Delta x=\frac{Z\Delta S}{n}

This result follows directly from the credibility formula. The original indication is $I=Zx+(1-Z)m$; after a loss-cost change $\Delta x$, it is $I'=Z(x+\Delta x)+(1-Z)m$. Subtracting gives $I'-I=Z\Delta x$, because the complement term is unchanged. The class therefore receives its full observed change only when $Z=1$; when $Z=0$, the indication does not move with that class’s experience.

A useful special case arises under the specified rule $Z=n/(n+K)$, where $K>0$ is a fixed credibility parameter. For a single claim amount $c$, the weighted change simplifies to $c/(n+K)$. This simplification is specific to that rule; it should not be applied if a different credibility formula is in use.

\Delta I=\frac{n}{n+K}\frac{c}{n}=\frac{c}{n+K}

A Practical Diagnostic Sequence

For an influential-experience check, identify the change being tested before doing arithmetic. A claim deletion, a newly reported claim, and a reserve adjustment can have different signs and different amounts. Then calculate the raw movement and the weighted movement separately. That separation makes it easier to catch unit errors and to communicate whether a class’s apparent sensitivity comes from its loss experience, its exposure volume, or its credibility weight.

Fix the comparison.
State whether the diagnostic adds, removes, or adjusts a loss. Specify the amount and whether earned exposure is held fixed.

Find the raw loss-cost movement.
Divide the aggregate loss change by the exposure count. Keep the result in dollars per exposure, not total dollars.

Apply credibility once.
Multiply the raw movement by $Z$ to obtain the movement in the credibility-weighted indication, provided $Z$ and the complement are fixed.

Check direction and scale.
An added positive loss cannot reduce the indicated cost under a fixed, nonnegative $Z$. The absolute indicated change cannot exceed the raw change when $0\leq Z\leq1$.

Worked Example: Measuring the Effect of a Newly Added Claim

A hypothetical class has 400 earned exposures and aggregate losses of $288,000, so its observed loss cost is $720 per exposure. The collective complement is $600. The plan specifies $Z=n/(n+K)$, with $K=600$. A newly included claim is $120,000. Holding exposure, $Z$, and the complement fixed, calculate the changes in observed and indicated loss cost, and the new indication.

Method and assumptions. This is a fixed-exposure claim-addition diagnostic. The amount is added to aggregate losses; it does not add exposures. First calculate the credibility weight:

Z=\frac{400}{400+600}=\frac{400}{1{,}000}=0.4

The claim increases observed loss cost by its amount divided by exposure:

\Delta x=\frac{\$120{,}000}{400}=\$300

Before the claim, the indication is:

I=0.4(\$720)+0.6(\$600)=\$288+\$360=\$648

After adding the claim, aggregate losses are $\$288{,}000+\$120{,}000=\$408{,}000$, and the observed loss cost is $\$408{,}000/400=\$1{,}020$. The new indication is:

I'=0.4(\$1{,}020)+0.6(\$600)=\$408+\$360=\$768

Thus the observed loss cost rises by $300 and the indicated cost rises by $\$768-\$648=\$120$. The check $Z\Delta x=0.4(\$300)=\$120$ agrees. The common mistake is to report the full $300 as the indicated change; that ignores the 0.4 credibility weight.

Leave-One-Claim-Out Influence

A leave-one-claim-out calculation asks how the observed cost or indication would differ if a specified claim were excluded from the loss total. For this diagnostic, the claim is removed from aggregate losses, but the earned exposure remains in the denominator: the policyholder or exposure still existed even though the claim is being omitted from the experience calculation. With $Z$ and $m$ held fixed, removing a claim of amount $c$ reduces observed loss cost by $c/n$ and the indication by $Zc/n$.

This is an influence check, not a claim-frequency model or a prediction of future losses. It answers a deliberately narrow question: how much does this claim contribute to the current class indication under the specified fixed-weight calculation? If exposure or credibility is recalculated, the fixed-weight formula no longer captures the full difference.

Worked Example: Excluding a Large Claim Without Deleting Exposure

A hypothetical class has 250 exposures, total losses of $250,000, complement $m=\$800$, and credibility rule $Z=n/(n+250)$. One $50,000 claim is being examined. Find the current indication and the leave-one-claim-out indication, and quantify the claim’s influence.

Method and assumptions. Keep all 250 earned exposures, the complement, and the credibility weight fixed. The observed loss cost is $\$250{,}000/250=\$1{,}000$, and:

Z=\frac{250}{250+250}=0.5

The current indication is:

I=0.5(\$1{,}000)+0.5(\$800)=\$500+\$400=\$900

Excluding the claim leaves aggregate losses of $\$250{,}000-\$50{,}000=\$200{,}000$. With exposure held at 250, the revised observed loss cost is $\$200{,}000/250=\$800$. The revised indication is:

I'=0.5(\$800)+0.5(\$800)=\$400+\$400=\$800

The claim’s leave-one-out influence is a decrease of $\$900-\$800=\$100$ in the indicated cost. The direct check gives $Zc/n=0.5(\$50{,}000/250)=0.5(\$200)=\$100$.

Common mistake. Deleting one exposure as well as the claim would use $200{,}000/249$, about $803.21, as the revised loss cost. That answers a different question: it removes both a claim and an exposure. Here only the loss is excluded, so the correct denominator remains 250.

Comparing Sensitivity Across Classes

A dollar amount does not have the same influence in every class. Under the rule $Z=n/(n+K)$, and with the same $K$, a single claim’s indicated impact is $c/(n+K)$. Thus a claim of the same amount generally moves the indication more in a lower-volume class. Comparisons must still use the same units and the same definition of the loss change. A larger claim in a larger class can outweigh the effect of a smaller exposure count.

Worked Example: Comparing Two Classes Under the Same Credibility Rule

Two hypothetical classes each have an observed loss cost of $700 and share complement $m=\$600$. The credibility rule is $Z=n/(n+400)$. Class A has 100 exposures; Class B has 400. A $40,000 claim is added to each class in turn, with exposure and credibility held fixed. Compare the observed and indicated movements and the resulting indications.

Method and assumptions. Calculate each class’s weight and raw movement separately. For Class A:

Z_A=\frac{100}{100+400}=0.2, \qquad \Delta x_A=\frac{\$40{,}000}{100}=\$400

Its initial indication is $0.2(\$700)+0.8(\$600)=\$140+\$480=\$620$. After the claim, its observed cost is $\$700+\$400=\$1{,}100$, and its indication is $0.2(\$1{,}100)+0.8(\$600)=\$220+\$480=\$700$. Its indicated movement is $\$700-\$620=\$80$, also $0.2(\$400)=\$80$.

For Class B:

Z_B=\frac{400}{400+400}=0.5, \qquad \Delta x_B=\frac{\$40{,}000}{400}=\$100

Its initial indication is $0.5(\$700)+0.5(\$600)=\$350+\$300=\$650$. After the claim, its observed cost is $\$700+\$100=\$800$, and its indication is $0.5(\$800)+0.5(\$600)=\$400+\$300=\$700$. Its indicated movement is $\$700-\$650=\$50$, matching $0.5(\$100)=\$50$.

The same dollar claim produces a $400 raw movement and $80 indicated movement in Class A, versus a $100 raw movement and $50 indicated movement in Class B. Class A is more sensitive in indicated dollars, even though its credibility weight is lower, because the claim is spread over fewer exposures. A tempting but incorrect comparison is to look only at $Z_A$ and $Z_B$: credibility weight alone does not determine the claim’s impact; the per-exposure loss-cost change matters too.

Interpreting and Reporting the Diagnostic

A concise diagnostic should state the quantity being compared. “The claim changes experience by $400 per exposure” describes the raw loss-cost movement; “the claim changes the indication by $80 per exposure” describes the credibility-weighted movement. Reporting both avoids ambiguity. For comparing claims or classes, use absolute movements when the question is sensitivity, and retain the sign when the question concerns the direction of the indicated change.

The fixed-weight calculation is not appropriate without qualification when the comparison changes exposure, $Z$, or the collective complement. If a rating plan recalculates credibility after a change in exposure, the difference between indications includes both the altered loss cost and the altered weight. Likewise, a change to the complement changes the part of the indication not assigned to the class’s own experience. The simple influence formula isolates one effect by holding those other inputs fixed; it should not be presented as a complete recalculation when they also change.

A useful reasonableness check is that, for $0\leq Z\leq1$, the absolute indicated movement cannot exceed the absolute raw movement when both use the same per-exposure units. A violation usually points to a unit mismatch, a sign error, an incorrect weight, or a comparison in which other inputs changed.

Key takeaway: Under a fixed-weight class credibility indication, a loss change $\Delta S$ over $n$ exposures moves observed loss cost by $\Delta S/n$ and moves the indicated cost by $Z\Delta S/n$. Diagnose the raw and weighted movements separately, and be explicit about which inputs are held fixed.

Check Your Understanding

For each hypothetical class, assume the stated exposure, credibility rule, and complement are fixed during the specified change.

A class has 180 exposures and uses $Z=n/(n+270)$. A newly included claim is $27,000. What are the raw and credibility-weighted increases in loss cost? (Answer: raw increase $150; indicated increase $60.)
A class has 300 exposures, observed loss cost $950, complement $750, and $Z=n/(n+300)$. Exclude a $36,000 claim while retaining all earned exposures. What are the original and revised indications? (Answer: original $850; revised $790, a decrease of $60.)
Two classes use $Z=n/(n+360)$. Class A has 120 exposures and Class B has 480. A $54,000 claim is added to either class. Which class has the larger indicated increase, and by how much? (Answer: Class A; its increase is $112.50 versus $64.29 for Class B, a difference of $48.21.)
A class has 200 exposures and $Z=0.4$. A reserve revision reduces aggregate losses by $10,000, while a newly recognized loss adds $25,000. With both changes applied and all else fixed, what is the net indicated change? (Answer: increase of $30 per exposure, since the net loss change is $15,000 and $0.4(\$15{,}000/200)=\$30$.)
Under $Z=n/(n+300)$, compare a $30,000 claim in a class with 100 exposures to a $100,000 claim in a class with 500 exposures. Which claim has the larger indicated impact, and what are the two impacts? (Answer: the $100,000 claim has the larger impact; the impacts are $75 and $125, respectively.)

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