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CAS Exam 5 · Berquist-Sherman Adjustments · Tutorial 777 of 989

Development Factors from Adjusted Triangles

Calculate development factors on a consistent adjusted basis, compare them with raw factors, and explain what the differences do—and do not—show.

Advanced 12 min read

What You'll Learn

  • Define age-to-age factors for adjusted paid and incurred cumulative triangles
  • Calculate volume-weighted factors using only accident periods observed at both ages
  • Derive the relationship between an aggregate factor and individual link ratios
  • Compare raw and adjusted factors for the same development interval
  • Interpret factor changes in light of reserve adequacy and settlement-rate adjustments
  • Avoid common errors involving unmatched rows, reversed ratios, and simple averages

Why Calculate Factors from Adjusted Triangles?

A development factor summarizes the movement in cumulative claims from one development age to the next. In a Berquist-Sherman analysis, the goal is to estimate that movement after addressing changes in case reserve adequacy for incurred losses or settlement rates for paid losses. The factor calculated from an adjusted triangle therefore describes movement on the adjusted basis; it is not simply a factor copied from the raw triangle.

The calculation itself is straightforward, but the comparison requires care. Raw and adjusted factors must use the same development interval and the same accident periods. A difference between them can indicate that an adjustment changed the development pattern represented by the data. It does not, by itself, establish that the adjustment is correct or determine which factor should ultimately be selected.

Definition: Let \(C_{i,j}\) denote cumulative claims for accident period \(i\) at development age \(j\), measured in consistent units. The age-to-age factor from age \(j\) to age \(j+1\) is the ratio of the aggregate cumulative claims at age \(j+1\) to the aggregate cumulative claims at age \(j\), using only accident periods observed at both ages. For adjusted data, write the factor as \(f_j^{\text{adj}}\); for raw data, write it as \(f_j^{\text{raw}}\). Apply the definition separately to paid or incurred triangles.

Calculating an Adjusted Age-to-Age Factor

For a paid triangle, \(C_{i,j}\) can be written \(P^{\text{adj}}_{i,j}\), the adjusted cumulative paid amount for accident period \(i\) at age \(j\). For an incurred triangle, use \(I^{\text{adj}}_{i,j}\), the adjusted cumulative incurred amount. The development age might be 12, 24, or 36 months. The values at the two ages must be on the same monetary and adjustment basis.

$$ f_j^{\text{adj}} = \frac{\sum_{i \in \mathcal{I}_j} C^{\text{adj}}_{i,j+1}} {\sum_{i \in \mathcal{I}_j} C^{\text{adj}}_{i,j}} $$

Here \(\mathcal{I}_j\) is the set of accident periods with observations at both ages \(j\) and \(j+1\). The factor is dimensionless: both numerator and denominator are amounts of claims. If amounts are expressed in dollars, thousands of dollars, or another consistent unit, the ratio is unchanged. The denominator must be positive for the factor to be defined.

The formula is a volume-weighted average of the individual accident-period link ratios, not generally their simple arithmetic average. This weighting makes larger amounts at age \(j\) contribute more to the aggregate movement.

Why the Aggregate Factor Is Volume-Weighted

Suppose every included accident period has a positive cumulative amount at age \(j\), and define its individual link ratio as \(r_{i,j}=C^{\text{adj}}_{i,j+1}/C^{\text{adj}}_{i,j}\). Substituting \(C^{\text{adj}}_{i,j+1}=C^{\text{adj}}_{i,j}r_{i,j}\) into the aggregate factor gives:

$$ f_j^{\text{adj}} = \frac{\sum_i C^{\text{adj}}_{i,j}r_{i,j}} {\sum_i C^{\text{adj}}_{i,j}} = \sum_i \left( \frac{C^{\text{adj}}_{i,j}}{\sum_k C^{\text{adj}}_{k,j}} \right)r_{i,j} $$

The weights are each accident period's share of the total cumulative amount at age \(j\). They are nonnegative and sum to one, so the aggregate factor lies between the smallest and largest included individual link ratios. This identity also explains why an unweighted average can produce a different answer: it assigns equal weight to accident periods even when their cumulative amounts differ substantially.

Worked Example: Adjusted Incurred Factor Versus Raw Factor

A hypothetical liability insurer has three accident periods with incurred amounts at 12 and 24 months, shown in thousands of dollars. An adequacy adjustment has rebased the incurred values to a common case reserve adequacy basis. Calculate the raw and adjusted 12-to-24-month factors and compare them. The data are deterministic claim amounts; no interest or mortality model is involved.

Accident periodRaw at 12 monthsRaw at 24 monthsAdjusted at 12 monthsAdjusted at 24 months
A120156150180
B210273250300
C330396400480

Raw factor. The raw 12-month total is \(120+210+330=660\). The raw 24-month total is \(156+273+396=825\). Therefore:

$$ f_{12}^{\text{raw}}=\frac{825}{660}=1.25 $$

Adjusted factor. The adjusted 12-month total is \(150+250+400=800\). The adjusted 24-month total is \(180+300+480=960\). Thus:

$$ f_{12}^{\text{adj}}=\frac{960}{800}=1.20 $$

The adjusted factor is 0.05 lower than the raw factor. On these data, adjusted incurred claims grow by 20% from 12 to 24 months, compared with 25% for the raw data. This difference is consistent with the adjustment changing the development pattern represented in the triangle. It is not proof, on its own, that the adjusted factor is more appropriate for every purpose.

Method and mistake avoided. This is the aggregate, volume-weighted link ratio. The same three accident periods are included in both calculations. Do not divide the adjusted 24-month total by the raw 12-month total: the numerator and denominator of a link ratio must come from the same data basis.

Calculating Multiple Factors for Adjusted Paid Data

A paid settlement-rate adjustment standardizes the paid triangle to a common settlement-rate basis. Once that adjusted triangle is available, calculate each age-to-age factor from adjacent adjusted columns, using only rows observed at both ages. Do not use the adjustment step itself as a substitute for checking which accident periods enter each factor.

Worked Example: Comparing Two Adjusted Paid Development Intervals

A hypothetical insurer has raw and settlement-rate-standardized paid losses for four accident periods. All values are in thousands of dollars. Calculate the raw and adjusted factors for 12-to-24 and 24-to-36 months. The adjustment is assumed to have already been completed; the task is to measure development on each basis, not to estimate a settlement-rate adjustment.

Accident periodRaw 12Raw 24Raw 36Adjusted 12Adjusted 24Adjusted 36
A80120162100130156
B120180243150195234
C160240324200260312
D240360486300390468

Raw 12-to-24 factor. The raw totals are \(80+120+160+240=600\) at 12 months and \(120+180+240+360=900\) at 24 months. Hence \(900/600=1.50\).

Adjusted 12-to-24 factor. The adjusted totals are \(100+150+200+300=750\) at 12 months and \(130+195+260+390=975\) at 24 months. Hence \(975/750=1.30\).

Raw 24-to-36 factor. The raw 24-month total is \(900\), and the raw 36-month total is \(162+243+324+486=1{,}215\). Thus \(1{,}215/900=1.35\).

Adjusted 24-to-36 factor. The adjusted 24-month total is \(975\), and the adjusted 36-month total is \(156+234+312+468=1{,}170\). Thus \(1{,}170/975=1.20\).

The adjusted factors are 1.30 and 1.20, compared with raw factors of 1.50 and 1.35. In this hypothetical triangle, settlement-rate standardization reduces measured development in both intervals. The comparison is meaningful because each factor uses the corresponding adjacent ages and the same accident periods on its raw or adjusted basis.

Method and mistake avoided. This is a pair of adjacent-age aggregate ratios. A frequent error is to use the 12-month denominator for the 24-to-36 factor. Its denominator must be the 24-month total on the basis being analyzed.

Comparing Adjusted Paid and Incurred Factors

Paid and incurred factors can be compared as diagnostics when both triangles have been adjusted as intended and the age ranges, accident periods, and measurement units are aligned. They need not be equal. Paid amounts reflect payments made by the valuation date; incurred amounts combine paid amounts with case reserves. Thus the two triangles can respond differently to settlement patterns and reserve changes even after the relevant standardizations.

Worked Example: Paid and Incurred Factors on a Common Basis

For three hypothetical accident periods, adjusted paid and adjusted incurred losses are available at 12 and 24 months. Values are in thousands of dollars, and the same accident periods are included in both calculations. Find both factors and describe their difference. No probability distribution or discounting assumption is needed because the task is a deterministic ratio calculation.

Accident periodAdjusted paid at 12Adjusted paid at 24Adjusted incurred at 12Adjusted incurred at 24
A90117180216
B150180250300
C210273370444

Paid factor. The adjusted paid totals are \(90+150+210=450\) at 12 months and \(117+180+273=570\) at 24 months. Therefore:

$$ f_{12}^{\text{paid, adj}}=\frac{570}{450}=1.2667 $$

Incurred factor. The adjusted incurred totals are \(180+250+370=800\) at 12 months and \(216+300+444=960\) at 24 months. Therefore:

$$ f_{12}^{\text{incurred, adj}}=\frac{960}{800}=1.20 $$

The paid factor exceeds the incurred factor by approximately \(1.2667-1.20=0.0667\). In this data set, adjusted paid losses show greater proportional movement than adjusted incurred losses over the interval. That observation can prompt review of the underlying patterns, but it does not identify a cause without considering the adjustments, claim experience, and data quality.

Method and mistake avoided. Calculate paid and incurred factors separately from their own totals. Do not add paid and incurred amounts together: incurred amounts already include paid claims, so adding the two measures would double count part of the claim estimate.

Interpretation, Conditions, and Common Traps

An adjusted factor summarizes the development pattern in adjusted data over one interval. In an incurred triangle, a change from the raw factor may be consistent with rebasing case reserves to constant adequacy. In a paid triangle, a change may be consistent with standardizing settlement rates. The factor comparison is a diagnostic description of the data after adjustment, not a test that validates the adjustment or a stand-alone causal analysis.

Before interpreting a factor, verify the following:

  • Matching periods: include only accident periods with values at both ages. If the observed rows differ across two calculations, the factor comparison may reflect a change in the row set as well as an adjustment.
  • Adjacent ages: a 12-to-24 factor uses columns 12 and 24; a 24-to-36 factor uses columns 24 and 36.
  • Consistent basis: use either raw values for a raw factor or adjusted values for an adjusted factor. Do not mix bases in a single ratio.
  • Usable denominator: the aggregate amount at the earlier age must be positive. Investigate zero or negative amounts before interpreting the ratio.
  • Meaning of the measure: cumulative incurred amounts can decrease if estimates are revised downward, and paid amounts can be affected by recoveries or data corrections. A factor is not required to exceed 1 in every data set.

For example, an incorrect calculation might use a simple average of individual ratios when the stated method is the aggregate factor. If two accident periods have values of \(100\) and \(300\) at the earlier age, with next-age values of \(150\) and \(360\), their individual factors are 1.50 and 1.20. The simple average is 1.35. The aggregate factor is instead \((150+360)/(100+300)=510/400=1.275\), because the second period carries three times the earlier-age volume of the first. Use the method requested and state it clearly.

These calculations produce candidate development factors from adjusted data. They do not determine which factors should be selected for an estimate or how an ultimate loss estimate changes; those are separate decisions requiring additional analysis.

Key takeaway: Calculate each adjusted age-to-age factor as the later-age aggregate divided by the earlier-age aggregate, using matching accident periods and a consistent adjusted basis. Compare it with the corresponding raw factor to describe how the adjustment changes measured development, without treating that comparison as proof of validity or as a final selection decision.

Check Your Understanding

For each question, use only accident periods observed at both ages and report factors to four decimal places where needed.

  1. A hypothetical adjusted incurred triangle has 12-month values of 140, 210, and 350, and corresponding 24-month values of 168, 252, and 455, all in thousands of dollars. Calculate the adjusted 12-to-24 factor. (Answer: \(875/700=1.2500\))
  2. Two adjusted paid accident periods move from 100 to 150 and from 300 to 360. Calculate the aggregate factor and the simple average of the individual factors. Which is the volume-weighted factor? (Answer: Aggregate factor \(510/400=1.2750\); simple average \(1.3500\); the volume-weighted factor is \(1.2750\).)
  3. Adjusted paid totals at 12 and 24 months are 240 and 296. Raw totals at the same ages are 200 and 280. Calculate both 12-to-24 factors and state which is larger. (Answer: Adjusted \(296/240=1.2333\); raw \(280/200=1.4000\); the raw factor is larger.)
  4. For the same four accident periods, adjusted incurred totals at 12 and 24 months are 800 and 920, while adjusted paid totals are 500 and 625. Calculate the factors and their difference. (Answer: Incurred \(920/800=1.1500\); paid \(625/500=1.2500\); paid exceeds incurred by \(0.1000\).)
  5. For a 24-to-36 factor, adjusted values at 24 months are 200, 300, and 500, while only the first two accident periods have 36-month observations, of 240 and 390. Calculate the factor using the eligible periods, and explain why the third period is excluded. (Answer: \((240+390)/(200+300)=1.2600\); the third period is excluded because it has no 36-month observation.)

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