Controlling for Claim Mix and Large Losses

Why Claim Mix Matters in Berquist-Sherman Analysis

Berquist-Sherman adjustments use observed claim experience to assess changes in measures such as closure rates and case-reserve adequacy. Those measures can change because claim handling or reserving changed. They can also change because the claims in the comparison changed: one evaluation may contain more severe injuries, broader coverage, or large claims than another. The distinction matters. An aggregate change caused by a different mix is not, by itself, evidence of a change in settlement speed or case-reserve adequacy.

A useful first step is to compare claims within reasonably homogeneous groups. Depending on the data and the question, groups might distinguish coverage type, injury type, or claim-size band. Large losses may need individual review or a separate category. The aim is not to create as many groups as possible; it is to avoid comparing unlike claims as if they were interchangeable.

Definition: A claim-mix effect is a change in an aggregate measure caused by a change in the relative numbers of claims in different segments, even when the measure within each segment is unchanged. A standardized measure combines segment results using a fixed set of weights, so comparisons hold the segment mix constant.

How Mix Changes Aggregate Measures

Suppose claims are divided into segments $s=1,\ldots,k$. Let $N_s$ be the number of claims in segment $s$, $C_s$ the number closed by a specified evaluation age, and $O_s$ the total case outstanding for open claims in that segment. The segment closure rate is $r_s=C_s/N_s$, and the average case outstanding per open claim is $a_s=O_s/(N_s-C_s)$, when there is at least one open claim.

The aggregate closure rate is a claim-count-weighted average of segment closure rates. The aggregate average case outstanding per open claim is weighted by each segment’s share of open claims. These weights can differ across evaluations, so aggregate measures can move even if the segment measures do not.

r_{\mathrm{agg}}=\frac{\sum_s C_s}{\sum_s N_s} =\sum_s \left(\frac{N_s}{\sum_j N_j}\right)r_s

a_{\mathrm{agg}}=\frac{\sum_s O_s}{\sum_s (N_s-C_s)} =\sum_s \left(\frac{N_s-C_s}{\sum_j (N_j-C_j)}\right)a_s

These equations follow by writing each segment total in terms of its segment rate or average. For closure, $C_s=N_sr_s$; substituting into total closures divided by total claims gives the first equation. For open-claim case reserves, $O_s=(N_s-C_s)a_s$; substituting gives the second. In both cases, the weights are shares of the relevant denominator. A change in those weights can alter the aggregate even when every $r_s$ or $a_s$ stays constant.

Key result: To compare closure or adequacy measures without a change in claim mix, select fixed segment weights $w_s$, with $\sum_s w_s=1$, and apply them consistently at each evaluation. For closure rates, weights should represent a fixed share of claims; for average case outstanding per open claim, they should represent a fixed share of open claims.

Worked Example: Separating Closure Change from a Mix Shift

A hypothetical insurer compares two evaluations at the same development age. Claims are grouped as routine or complex. At the first evaluation, there are 100 claims in each group: 60 routine claims and 20 complex claims are closed. At the second evaluation, there are 180 routine claims and 20 complex claims: 108 routine claims and 4 complex claims are closed. Calculate the aggregate closure rates, compare the segment rates, and standardize the second evaluation to the first evaluation’s 50/50 claim mix.

Method. Calculate each segment’s closed-to-reported rate, then calculate the aggregate rates using the actual claim counts. Finally, apply the first evaluation’s segment weights to the second evaluation’s segment rates.

At the first evaluation, the routine rate is $60/100=60\%$, and the complex rate is $20/100=20\%$. The aggregate rate is:

\frac{60+20}{100+100}=\frac{80}{200}=40\%.

At the second evaluation, the routine rate is $108/180=60\%$, and the complex rate is $4/20=20\%$. The aggregate rate is:

\frac{108+4}{180+20}=\frac{112}{200}=56\%.

The aggregate rate increased by 16 percentage points, but neither segment’s rate changed. The second evaluation has 90% routine claims, compared with 50% at the first evaluation. Because routine claims have the higher closure rate, the changed mix raises the aggregate.

Using the first evaluation’s 50/50 mix, the standardized second-evaluation rate is:

0.50(60\%)+0.50(20\%)=40\%.

The standardized rates match: 40% at both evaluations. This indicates no within-segment closure-rate change in the supplied data. Check: the second evaluation’s actual 56% rate is recovered with its actual weights: $0.90(60\%)+0.10(20\%)=56\%$. The common mistake is to call the 16-point aggregate increase faster settlement without checking the segment mix.

Standardizing Case-Reserve Adequacy Measures

For incurred-loss Berquist-Sherman analysis, a commonly examined diagnostic is average case outstanding per open claim at a given development age. It can be affected by the types of claims still open. For example, complex injury claims may remain open longer and have larger case reserves than routine claims. A shift toward complex open claims can raise the aggregate average even if case-reserve levels within both groups are stable.

To make a controlled comparison, calculate the average within each segment and then apply a fixed set of open-claim weights. Use the same segment definitions and weights at each evaluation, and make sure the measures being compared refer to a consistent development age and claim basis.

Worked Example: Standardizing Average Case Outstanding

At a fixed development age, a hypothetical insurer records average case outstanding per open claim for routine and complex claims. At the earlier evaluation, the averages are $2,000 and $10,000, respectively; 75% of open claims are routine and 25% are complex. At the later evaluation, the averages are $2,400 and $9,000; only 25% of open claims are routine. Calculate both actual aggregate averages and compare them with averages standardized to the earlier 75/25 mix.

Method. Weight each evaluation’s segment averages by its actual open-claim mix to obtain the actual aggregate. Then use the earlier evaluation’s weights for both standardized values.

The earlier actual aggregate is:

0.75(\$2{,}000)+0.25(\$10{,}000) =\$1{,}500+\$2{,}500=\$4{,}000.

At the later evaluation, the actual mix is 25% routine and 75% complex. Its actual aggregate is:

0.25(\$2{,}400)+0.75(\$9{,}000) =\$600+\$6{,}750=\$7{,}350.

The actual aggregate rose by $\$7{,}350-\$4{,}000=\$3{,}350$, or $3{,}350/4{,}000=83.75\%$. But this comparison includes a substantial shift toward complex open claims. Standardizing both evaluations to the earlier 75/25 mix gives:

\begin{aligned} \text{Earlier standardized average} &=0.75(\$2{,}000)+0.25(\$10{,}000)=\$4{,}000,\\ \text{Later standardized average} &=0.75(\$2{,}400)+0.25(\$9{,}000)\\ &=\$1{,}800+\$2{,}250=\$4{,}050. \end{aligned}

On the fixed mix, the average increased by $\$50$, or $50/4{,}000=1.25\%$. Thus, most of the aggregate increase is associated with the changed mix, not the segment-level changes alone. Check: applying the actual later weights to the later segment values gives $7,350, as calculated above. The trap is to interpret the 83.75% aggregate increase as an across-the-board change in case-reserve adequacy.

Large Losses Need Deliberate Treatment

A small number of large claims can dominate aggregate paid, incurred, or case-outstanding amounts. Their concentration can also change between evaluations. An average “per claim” may therefore respond more to one large claim’s development than to a broad shift in typical claim handling or reserving.

A practical analysis can separate ordinary claims from large claims, use appropriate size bands, or examine material claims individually. The threshold and treatment should be defined consistently for the comparison. If large claims are removed from a routine-claim measure, their amounts must still be tracked and reconciled to the overall totals; excluding them from one diagnostic does not make them disappear from the loss estimate.

Worked Example: Separating Large-Loss Concentration from Routine Claims

At the same development age, a hypothetical portfolio has 100 routine open claims and one large open claim at each of two evaluations. At the first evaluation, average routine case outstanding is $8,000 and the large claim’s case outstanding is $200,000. At the second evaluation, average routine case outstanding is $8,800 and the large claim’s case outstanding is $500,000. Calculate the aggregate average per open claim at each evaluation, the aggregate percentage change, and the routine-claim percentage change.

Method. First total the case outstanding across all claims and divide by 101 open claims. Then compare routine averages separately. This shows how an aggregate per-claim average combines ordinary and concentrated large-claim movement.

At the first evaluation, routine case outstanding totals $100(\$8{,}000)=\$800{,}000$. Including the large claim gives total case outstanding of $\$800{,}000+\$200{,}000=\$1{,}000{,}000$. The aggregate average is:

\frac{\$1{,}000{,}000}{101}=\$9{,}900.99

At the second evaluation, routine case outstanding totals $100(\$8{,}800)=\$880{,}000$. Total case outstanding is $\$880{,}000+\$500{,}000=\$1{,}380{,}000$, so the aggregate average is:

\frac{\$1{,}380{,}000}{101}=\$13{,}663.37.

The aggregate average increased by $\$13{,}663.37-\$9{,}900.99=\$3{,}762.38$. Using the unrounded totals, the percentage increase is $(\$1{,}380{,}000/\$1{,}000{,}000)-1=38.00\%$. The routine average increased by $(\$8{,}800/\$8{,}000)-1=10.00\%$. The large claim’s case outstanding increased by $(\$500{,}000/\$200{,}000)-1=150.0\%$.

The aggregate 38.00% increase is not a reasonable summary of the routine-claim movement: the large claim accounts for much of the increase. A sound diagnostic reports the routine measure separately and assesses the large claim on its own merits, while retaining both in the overall reconciliation. Check: the later total of $1,380,000 equals $880,000 for routine claims plus $500,000 for the large claim. The trap is either to let one large claim represent all claims or to omit it without accounting for its amount elsewhere.

A Practical Segmentation Workflow

The purpose of segmentation is to make comparisons more meaningful, not to conceal unfavorable movement. Apply the same definitions at each evaluation and preserve a bridge back to the full portfolio. A disciplined sequence is:

Specify the measure and comparison basis.
For example, define the closure-rate denominator, the open-claim population, and the development age. Confirm that the evaluations use comparable claim status and valuation conventions.

Identify plausible mix drivers.
Consider coverage, injury type, and claim size where they can materially affect closure patterns or case-reserve levels. Keep categories clear enough that claims are classified consistently.

Compare within segments and standardize.
Calculate segment measures first. If an aggregate comparison is needed, apply the same stated segment weights at each evaluation and identify whether weights represent claims or open claims.

Review large claims separately and reconcile.
Check whether a small number of claims dominate amounts or distort averages. Explain their treatment and verify that segment totals, including separately reviewed claims, tie back to portfolio totals.

Segmentation requires judgment. Splitting a sparse group into many small categories can make the resulting comparisons unstable or difficult to interpret. Conversely, a broad segment can hide meaningful differences in coverage, injury, or size. Use distinctions that are relevant to the measure and supported by the available claim information. When a segment’s counts or amounts are small, disclose that limitation rather than treating its result as a precise indication.

Common Interpretation Traps

Comparing aggregate rates without comparing weights. A higher closure rate may reflect a larger share of claims from a segment that typically closes faster.
Using the wrong denominator for standardization. Closure rates are weighted by claim counts. Average case outstanding per open claim is weighted by open-claim counts, not all reported claims.
Changing segment definitions between evaluations. If claims move between categories because the classification rule changed, the comparison mixes a data-definition change with an experience change.
Treating a large claim as representative. A material movement in one claim can dominate an aggregate average; report its effect separately and reconcile it to the total.
Assuming standardization proves causation. A fixed-mix comparison isolates the arithmetic effect of changing weights. It does not, on its own, explain why segment results changed or establish that claim handling caused the change.

Key takeaway: Before interpreting a Berquist-Sherman closure or adequacy measure, ask whether the evaluations contain comparable claims. Compare like with like, use fixed and appropriate weights when standardizing, examine concentrated large losses explicitly, and reconcile all components to the aggregate.

Check Your Understanding

For each problem, identify the mix effect before drawing a conclusion from the aggregate.

A hypothetical portfolio has 50 standard claims and 50 complex claims at the first evaluation. Their closure rates are 60% and 20%, respectively. At the second evaluation there are 150 standard claims and 50 complex claims, with the same segment closure rates. Calculate both aggregate closure rates and the second rate standardized to the first mix. (Answer: first aggregate 40%; second aggregate 50%; standardized second rate 40%.)
At a fixed age, routine and complex open claims have average case outstanding of $1,500 and $7,000 at the first evaluation, with an 80/20 open-claim mix. At the second evaluation, their averages are $1,800 and $7,600, with a 40/60 mix. Calculate the second actual aggregate and both evaluations standardized to the first mix. (Answer: second actual $5,280; first standardized $2,600; second standardized $2,960, a 13.85% increase.)
A portfolio has 80 routine open claims and two large open claims at each evaluation. Initially, routine average case outstanding is $5,000 and the two large claims are $100,000 each. Later, the routine average is $5,500 and the large claims are $150,000 and $50,000. Calculate the total case outstanding at each evaluation and the percentage changes in total and routine average. (Answer: totals $600,000 and $640,000; total increase 6.67%; routine average increase 10.00%.)
Two claim segments have closure rates of 70% and 30%. The first evaluation’s mix is 40% in the first segment; the second evaluation’s mix is 70% in that segment. Calculate the aggregate closure rates and state the standardized second rate using the first evaluation’s mix. (Answer: first aggregate 46%; second aggregate 58%; standardized second rate 46%.)

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