Berquist-Sherman Techniques

The Berquist-Sherman reserving techniques adjust historical development triangles to correct for distortions caused by operational shifts: changes in claim settlement speeds (disposal rates) or changes in case reserving adequacy.

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Paid Losses Adjustment (Settlement Speed Shift)

When an insurer accelerates or decelerates claim closing speeds, cumulative paid development patterns shift, rendering standard Chain Ladder projections inaccurate.

Methodology

1. Calculate Historical Disposal Rates ($DR_{i,j}$): $$DR_{i,j} = \frac{\text{Cumulative Closed Claim Counts}_{i,j}}{\text{Projected Ultimate Claim Counts}_i}$$
2. Select a Target Disposal Rate ($DR_j$) for each maturity $j$ (typically the latest year’s rate to reflect current operations).
3. Adjust Paid Losses: Interpolate unadjusted paid losses to match the selected target disposal rates.

1. Linear Interpolation Assumption

Assumes a linear relationship between cumulative paid claims and cumulative closed counts. If the selected target disposal rate $S$ at maturity $j$ lies between unadjusted rates $A$ (at maturity $j$) and $B$ (at maturity $j+1$), the adjusted paid loss is:

$$\text{Adjusted Paid Losses}_S = C_A + \frac{S - A}{B - A} \times (C_B - C_A)$$

Where:

• $C_A$: Paid losses at maturity $j$ (corresponding to $DR = A$).
• $C_B$: Paid losses at maturity $j+1$ (corresponding to $DR = B$).

2. Exponential Interpolation Assumption

Assumes an exponential relationship between paid claims ($Y$) and closed counts ($X$): $Y = a \cdot e^{b X}$.

1. Calculate adjusted paid claim counts: $\text{Adjusted Counts} = \text{Projected Ultimate Counts} \times \text{Selected } DR_j$.
2. Fit exponential curves for each adjacent maturity interval (e.g., $12$ to $24$ months): $$b_{24} = \frac{\ln(Y_{24}) - \ln(Y_{12})}{X_{24} - X_{12}}$$ $$a_{24} = \frac{Y_{24}}{e^{b_{24} X_{24}}}$$
3. Apply the parameters based on the relationship between Adjusted Counts and Actual Counts:
   • If Adjusted Counts > Actual Counts (Green Zone): Adjusted losses are projected to be higher. Interpolate forward using parameters from the subsequent interval (e.g., $24$ to $36$ months): $$\text{Adjusted Paid Losses} = a_{36} \cdot e^{b_{36} \times \text{Adjusted Counts}}$$
   • If Adjusted Counts < Actual Counts (Red Zone): Adjusted losses are projected to be lower. Interpolate backward using parameters from the current interval (e.g., $12$ to $24$ months): $$\text{Adjusted Paid Losses} = a_{24} \cdot e^{b_{24} \times \text{Adjusted Counts}}$$

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Case Reserve Adjustment (Reserve Adequacy Shift)

If claims adjusters increase or decrease the average adequacy of case reserves, reported loss development triangles will be distorted.

Methodology

1. Calculate Historical Case Severity ($S_{i,j}$): $$S_{i,j} = \frac{\text{Reported Losses}_{i,j} - \text{Paid Losses}_{i,j}}{\text{Open Claim Counts}_{i,j}} = \frac{\text{Case Outstanding}_{i,j}}{\text{Open Claim Counts}_{i,j}}$$
2. Select a Base Case Severity: Typically the latest diagonal’s severity for each maturity, representing current reserve adequacy.
3. Detrend/Trend Case Severities: Detrend the selected base severity back to the historical accident years using the selected paid severity trend ($g$): $$\text{Adjusted Case Severity}_{i,j} = \text{Selected Severity}_j \times (1 + g)^{-(n - i)}$$ Where $n$ is the latest calendar year and $n-i$ is the number of years to detrend.
4. Reconstruct Reported Losses: $$\text{Adjusted Reported Losses}_{i,j} = \text{Paid Losses}_{i,j} + \left( \text{Adjusted Case Severity}_{i,j} \times \text{Open Claim Counts}_{i,j} \right)$$
5. Apply standard Chain Ladder to the adjusted reported loss triangle.

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Technical Considerations

• Detrending Consistency: Detrend all historical periods back to their respective dates, even if the underlying loss trend has remained constant.
• Medical Malpractice Exception: Paid loss data cannot be used to estimate severity trends for case reserve adjustments in Medical Malpractice. Slow claim payment patterns reduce the volume of paid data at early maturities, making paid severity trends highly volatile and unreliable.