Why Factor Selection Still Requires Judgment
A Berquist-Sherman adjustment is intended to place historical losses on a more consistent basis when case reserve adequacy or settlement rates have changed. After rebuilding adjusted triangles and calculating their link ratios, however, a calculation alone does not determine which factor to use. Different averaging methods can produce different answers, and a factor that summarizes past movement well may still be a poor estimate of future movement if the underlying claim mix or operation has changed.
This tutorial focuses on selecting factors from adjusted triangles. It does not repeat the mechanics of making the adjustments. The central task is to assess the evidence at each development age: how much loss volume supports each observation, whether the observations form a stable pattern, and whether any remaining differences have a plausible explanation relevant to future periods. A defensible selection makes those judgments explicit.
The simple average gives each selected origin period equal weight, regardless of its adjusted loss volume at the starting age. The volume-weighted average gives greater weight to origin periods with more starting loss. Neither method is automatically preferable in every situation: the analyst must consider the data and the intended use of the factor.
What Volume Weighting Measures
For a particular development age, write \(r_i=A_{i,j+1}/A_{i,j}\) for the observed ratio in origin period \(i\). The volume-weighted factor is:
The second equality follows because \(A_{i,j+1}=A_{i,j}r_i\) for each selected origin period. Thus, the weighted factor is a weighted average of the individual ratios, where the weight for origin period \(i\) is its adjusted starting loss \(A_{i,j}\). This explains both its appeal and its limitation: it summarizes the aggregate movement of the selected adjusted losses, but large-volume periods can dominate the result.
By contrast, the simple average is \(\widehat{f}_j^{\,\mathrm{simple}}=\sum_i r_i/n\), where \(n\) is the number of selected observations. It describes the average observed ratio across origin periods, not the ratio of their aggregate adjusted losses. If starting volumes differ substantially, the two summaries can differ materially.
Worked Example: Comparing Volume-Weighted and Simple Averages
A hypothetical commercial equipment portfolio has four adjusted origin-period amounts at 12 months and 24 months. Select a summary 12-to-24-month factor using both averaging methods and explain which method more directly represents aggregate movement.
| Origin period | Adjusted at 12 months | Adjusted at 24 months | Link ratio |
|---|---|---|---|
| A | 800 | 1,040 | 1.300 |
| B | 1,200 | 1,440 | 1.200 |
| C | 1,500 | 1,800 | 1.200 |
| D | 500 | 700 | 1.400 |
Step 1: Verify the individual ratios. The ratios are \(1{,}040/800=1.300\), \(1{,}440/1{,}200=1.200\), \(1{,}800/1{,}500=1.200\), and \(700/500=1.400\).
Step 2: Calculate the volume-weighted factor. The total adjusted amount at 12 months is \(800+1{,}200+1{,}500+500=4{,}000\). The total at 24 months is \(1{,}040+1{,}440+1{,}800+700=4{,}980\). Therefore, the volume-weighted factor is \(4{,}980/4{,}000=1.245\).
Step 3: Calculate the simple average. The sum of ratios is \(1.300+1.200+1.200+1.400=5.100\). Dividing by four gives \(5.100/4=1.275\).
Selection and check. The volume-weighted factor, 1.245, is exactly the ratio of the two aggregate amounts, \(4{,}980/4{,}000\). It is a natural summary if the goal is to describe the aggregate movement of these selected losses. The simple average, 1.275, gives the two smaller-volume observations greater influence relative to their starting amounts. Neither result establishes which factor best predicts future development; that also depends on stability and comparability. The common mistake avoided here is averaging the ratios while describing the result as the aggregate link ratio.
Assess Each Development Age on Its Own Evidence
A development pattern is a sequence of age-to-age factors, not one general factor applied at every age. The observations supporting 12-to-24 months may differ from those supporting 24-to-36 months in number, volume, maturity, and consistency. A selected factor should therefore be justified separately at each age.
A useful review begins with the individual ratios and their starting volumes. Look for a coherent pattern across origin periods; compare the volume-weighted and simple averages; and check whether a recent observation is unusually influential. An isolated high or low ratio is a prompt to investigate, not a reason by itself to include or exclude that observation. In particular, distinguish a genuine change relevant to future periods from random fluctuation or an artifact that the adjustment was intended to address.
Worked Example: Selecting Factors When the Latest Period Differs
A hypothetical specialty liability portfolio has adjusted 12-to-24-month observations with ratios \(1.18, 1.20, 1.19, 1.21,\) and \(1.22\), and corresponding 12-month starting amounts of \(1{,}000, 1{,}400, 1{,}200, 1{,}600,\) and \(800\). At 24-to-36 months, four older origin periods have ratios \(1.10, 1.12, 1.09,\) and \(1.11\), with starting amounts \(1{,}800, 2{,}000, 1{,}600,\) and \(1{,}400\). The latest 24-to-36-month observation has ratio \(1.24\) on a starting amount of \(1{,}000\). The latest period is known to have a different claim mix, and there is no evidence that this mix will continue. Select working factors and show the effect of including the latest observation.
Step 1: Select the 12-to-24-month factor. The weighted next-age amounts are \(1{,}000(1.18)=1{,}180\), \(1{,}400(1.20)=1{,}680\), \(1{,}200(1.19)=1{,}428\), \(1{,}600(1.21)=1{,}936\), and \(800(1.22)=976\). Their sum is \(7{,}200\), and the starting amounts sum to \(6{,}000\). The volume-weighted factor is \(7{,}200/6{,}000=1.200\). The ratios are also tightly grouped, so 1.200 is a reasonable working selection.
Step 2: Summarize the older 24-to-36-month observations. Their weighted next-age amounts are \(1{,}800(1.10)=1{,}980\), \(2{,}000(1.12)=2{,}240\), \(1{,}600(1.09)=1{,}744\), and \(1{,}400(1.11)=1{,}554\). These sum to \(7{,}518\); their starting amounts sum to \(6{,}800\). The volume-weighted factor for those periods is \(7{,}518/6{,}800=1.105588\ldots\), or 1.106 to three decimals.
Step 3: Measure the impact of including the latest period. Its weighted next-age amount is \(1{,}000(1.24)=1{,}240\). Including it gives a numerator of \(7{,}518+1{,}240=8{,}758\) and a denominator of \(6{,}800+1{,}000=7{,}800\). The resulting factor is \(8{,}758/7{,}800=1.122820\ldots\), or 1.123 to three decimals.
Selection and check. Given the stated evidence that the latest period has a different mix that is not expected to continue, select 1.106 as the 24-to-36-month working factor, subject to monitoring. The all-period factor of 1.123 demonstrates how the latest observation raises the result. If that mix were expected to persist, the case for relying on the older-period pattern would need to be reconsidered. The mistake avoided is excluding a recent observation merely because it is inconvenient: the selection here rests on the stated comparability evidence, not on the direction of the result.
When an Unusual Ratio Has High or Low Volume
Volume weighting does not make an observation reliable; it determines its influence in an aggregate summary. A large-volume observation can dominate even when its ratio is unusual, while a small-volume observation can have little influence on the weighted average but a large effect on the simple average. The analyst should investigate unusual observations using available context, including whether the adjustment was applied consistently and whether claim mix or operational conditions differ.
The selection should also be appropriate to the purpose of the pattern. If a change is temporary and not expected to recur, a factor based heavily on that change may not be representative of future development. If the change is expected to persist, older observations may be less relevant. Where the evidence does not resolve the issue, a range of reasonable selections can be more informative than false precision. Record the selected factor, the observations used, and the rationale.
Worked Example: A High Ratio on a Small Starting Volume
For a hypothetical adjusted 36-to-48-month development age, three comparable origin periods have ratios \(1.06, 1.08,\) and \(1.20\), on starting amounts of \(5{,}000, 4{,}500,\) and \(500\), respectively. The third period's high ratio reflects a one-time operational disruption that is not expected to recur. Calculate both averages and select a working factor consistent with that information.
Step 1: Calculate the aggregate amounts. The weighted next-age amounts are \(5{,}000(1.06)=5{,}300\), \(4{,}500(1.08)=4{,}860\), and \(500(1.20)=600\). Their sum is \(10{,}760\), while starting amounts sum to \(5{,}000+4{,}500+500=10{,}000\).
Step 2: Calculate the volume-weighted factor. The result is \(10{,}760/10{,}000=1.076\).
Step 3: Calculate the simple average. The ratios sum to \(1.06+1.08+1.20=3.34\), so the simple average is \(3.34/3=1.113333\ldots\), or 1.113 to three decimals.
Selection and check. Select 1.076 as a working factor if the operational disruption is genuinely one-time and the other two observations are comparable to expected future conditions. The weighted factor is close to the two ordinary ratios because the disrupted observation has a small starting amount. That does not, by itself, justify ignoring the disruption: if its cause could recur, the selected pattern may understate future movement. The common mistake avoided is treating either the volume-weighted result or the simple average as a substitute for investigating a known operational change.
A Defensible Selection Process
A concise, repeatable process helps separate arithmetic from judgment. It also makes the reasoning reviewable when two analysts would otherwise select different factors from the same adjusted triangle.
Verify that the adjusted cumulative amounts being compared are on a consistent basis and that each ratio uses the intended consecutive development ages.
Inspect the individual link ratios and their starting amounts. Calculate the volume-weighted factor and, when useful, the simple average.
Consider whether the observations cluster around a plausible pattern, whether recent periods differ, and whether any single period materially changes the result.
Use available evidence about claim mix, operations, and adjustment limitations to decide whether an unusual observation is likely to represent future conditions.
Choose a factor separately for the development age, identify the supporting data, and record important uncertainty or an alternative reasonable selection.
Common Selection Traps
- Calling a simple average a volume-weighted factor. The volume-weighted factor is the ratio of summed next-age amounts to summed starting amounts. The simple average assigns equal weight to each observed ratio.
- Using one selection rule at every age. Different development ages can have different stability, volume, and data limitations. Assess each age on its own evidence.
- Assuming adjustment removes every source of change. Berquist-Sherman adjustments address specified distortions; they do not guarantee that claim mix or operating conditions are identical across all periods.
- Excluding an observation solely because it is unusual. An unusual ratio should trigger investigation. Exclusion or reduced reliance needs a reason tied to data quality or future relevance.
- Reporting excessive precision. A calculated ratio such as 1.105588 is not evidence that the selected factor is known to six decimal places. Round sensibly and explain the selection.
Check Your Understanding
For each problem, distinguish the calculation from the selection judgment. Round reported factors to three decimals unless the answer is exact.
- At a particular development age, two adjusted origin periods move from 2,000 to 2,300 and from 500 to 650. Calculate the individual ratios, the simple average, and the volume-weighted factor. (Answer: Ratios 1.150 and 1.300; simple average 1.225; volume-weighted factor \(2{,}950/2{,}500=1.180\).)
- Three comparable adjusted observations have ratios 1.04, 1.06, and 1.08 on starting amounts 1,000, 2,000, and 3,000. Calculate the volume-weighted factor and the simple average. (Answer: Volume-weighted factor \(6{,}400/6{,}000=1.067\); simple average \(1.060\).)
- At one age, two older observations have ratios 1.12 and 1.14 on starting amounts 2,500 and 1,500. A latest observation has ratio 1.30 on a starting amount of 1,000, and a documented one-time system issue affected only that latest observation. Calculate the factor including all periods and state a reasonable selection if the issue will not recur. (Answer: Including all periods, \((2{,}800+1{,}710+1{,}300)/5{,}000=1.162\); a reasonable selection based on the comparable older periods is \((2{,}800+1{,}710)/4{,}000=1.128\).)
- At 12-to-24 months, the adjusted observations are 900 to 1,080 and 2,100 to 2,520. At 24-to-36 months, they are 1,080 to 1,188 and 2,520 to 2,772. Calculate the volume-weighted factor at each age and identify whether the evidence supports a distinct change in factor by age. (Answer: 12-to-24 months: \(3{,}600/3{,}000=1.200\); 24-to-36 months: \(3{,}960/3{,}600=1.100\); the two observed ages support distinct factors of 1.200 and 1.100.)
- At one development age, adjusted starting amounts are 4,000 and 1,000, with ratios 1.05 and 1.25. The larger ratio is associated with a new claim mix expected to continue. Calculate the weighted and simple averages, then explain why the weighted result alone does not settle the selection. (Answer: Volume-weighted factor \((4{,}200+1{,}250)/5{,}000=1.090\); simple average \(1.150\); because the changed mix is expected to continue, assess whether the historical mix remains representative rather than relying on the weighted summary alone.)