Actuarial Notes

Berquist-Sherman Method

Study guides: Exam 5, Exam GI-101

Berquist-Sherman Method restates a historical triangle so that every year reflects current claims practice, removing the distortion caused by a change in case reserve adequacy or in claim settlement rates. Development factors are then selected from the restated triangle.

\text{Adjusted Case}_{\text{AY},n} = \text{Case}_{\text{latest},n} \times \left(1 + t\right)^{-(\text{years before latest})}

\text{Adjusted Reported}_{\text{AY},n} = \text{Paid}_{\text{AY},n} + \text{Adjusted Case}_{\text{AY},n}

The case adequacy adjustment (for a reported triangle):

  1. Compute average case outstanding per open claim at each accident year and maturity.
  2. Select a severity trend t — the rate at which average case outstanding should grow if adequacy were unchanged.
  3. Trend the latest diagonal's average case outstanding backwards at t to restate each historical cell at today's adequacy level.
  4. Multiply the restated averages by the open claim counts to get adjusted case reserves, add paid, and rebuild the reported triangle.
  5. Select factors from the restated triangle.

The settlement rate adjustment (for a paid triangle):

  1. Compute disposal rates — closed claims ÷ ultimate (or reported) claims — at each accident year and maturity.
  2. Select the current disposal rate pattern as the target.
  3. For each historical cell, interpolate the paid losses that would have been paid had that year closed claims at the current rate — reading along the row's paid-versus-disposal-rate relationship.
  4. Rebuild the paid triangle at the common disposal pattern and select factors from it.

Further points:

  • The trigger for either adjustment is a diagonal effect: age-to-age factors on the latest diagonal all above (or all below) their column history. That is a calendar-year signal, and averaging it into the selections applies a one-time shift as though it were an ongoing pattern.
  • Which adjustment applies is decided by the diagnostic, not by preference: a jump in average case outstanding across all years points to adequacy; a jump in closure rates points to settlement speed. Both can be happening at once.
  • The adjustments are large and judgment-laden — the severity trend and the target disposal pattern are both selections — so the restated result should be compared against methods that are insensitive to the distortion in question (paid methods for a case-adequacy change; reported methods for a settlement-rate change).
  • Berquist-Sherman does not fix the diagonal itself. Strengthened reserves are on the books and belong in the estimate; what the restatement fixes is the factors, which would otherwise develop the strengthening a second time.

Case Adequacy Adjustment Average case outstanding per open claim at 12 months:

AY Avg case O/S Open claims
2022 \$4{,}000 250
2023 \$4{,}200 250
2024 \$5{,}500 250

Paid at 12 months is \$300{,}000 for each year. The selected severity trend is 5\%; average case outstanding grew 31\% in the latest year, indicating a strengthening.

Restate the 12-month reported column.

Answer Trend the latest average (\$5{,}500) backwards at 5\% to put every year on today's adequacy level:

\begin{align*}
2023: \; \$5{,}500 / 1.05 &= \$5{,}238 \\
2022: \; \$5{,}500 / 1.05^{2} &= \$4{,}989
\end{align*}

Adjusted case reserves and reported losses:

AY Adj. avg Adj. case Adj. reported
2022 \$4{,}989 \$1{,}247{,}250 \$1{,}547{,}250
2023 \$5{,}238 \$1{,}309{,}500 \$1{,}609{,}500
2024 \$5{,}500 \$1{,}375{,}000 \$1{,}675{,}000

Against the original reported figures (\$1{,}300{,}000, \$1{,}350{,}000, \$1{,}675{,}000), the older years have been raised by around 19\% while the latest is unchanged.

The consequence for the factors is the point: in the original triangle, the 12-month column was artificially low in the old years relative to their subsequent (post-strengthening) development, inflating the 12–24 factors. Restating the base removes that inflation, and the factors selected from the adjusted triangle no longer develop the strengthening a second time.

Which Adjustment Does the Data Call For? Diagnostics for a liability book:

AY at 24 mo Avg case O/S Closed ÷ reported counts Paid ÷ reported losses
2021 \$12{,}000 58\% 0.42
2022 \$12{,}600 59\% 0.43
2023 \$13{,}100 67\% 0.52
2024 \$13{,}700 74\% 0.61

Which Berquist-Sherman adjustment applies?

Answer The settlement rate adjustment, on the paid triangle.

Average case outstanding is growing \approx 4.5\% a year — steady, and consistent with ordinary severity trend. There is no sign of a case adequacy shift.

The disposal rate, though, has moved from 58\% to 74\% in two years, and the paid-to-reported ratio has moved with it. Claims are closing much faster.

The consequences:

  • Paid development factors are biased. Historical factors were built on a book that had paid only 42\% of reported losses by 24 months; AY 2024 has paid 61\%. Applying the old factors to the new diagonal over-develops it substantially.
  • Reported development is largely unaffected, since reported losses include case reserves whether or not the claim has closed.

So: restate the paid triangle to a common disposal rate before selecting paid factors, and use the reported chain ladder — which is insensitive to settlement speed — as the check on the result. Applying the case adequacy adjustment here would correct a distortion that is not present.