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Evaluation

Note Section 3.1 • Reading time: ~5 mins

Actuaries must continually evaluate the adequacy of reserve estimates by comparing actual claim emergence against expected emergence. This process is known as retroactive testing.

Retroactive Testing (Actual vs. Expected)

Retroactive testing acts as a diagnostic tool to assess whether the reserving methods and assumptions are functioning as expected, or if operational/environmental changes are distorting results.

Actual Development Exceeds Expected Development

If the actual claims reported or paid during a period exceed the expected emergence, the actuary has three interpretive choices depending on the underlying cause:

  1. Reduce IBNR (Reporting Speedup): If the higher emergence is due to an acceleration in claims reporting (claims are closed or processed faster, but ultimate counts are unchanged):
    • Reserving Response: Reduce future expected emergence (consistent with the Expected Claims or Berquist-Sherman adjustments).
  2. Leave IBNR Unchanged (Atypical / Black Swan Event): If the variance is caused by a one-time random shock or large claim that does not affect future development expectations:
    • Reserving Response: Leave future IBNR unchanged. The variance is absorbed, and future emergence is expected to revert to the baseline (consistent with the Bornhuetter-Ferguson assumption).
  3. Increase IBNR (Deterioration of Loss Ratio): If the variance represents a systematic increase in claim frequency or severity (loss environment is worse than expected):
    • Reserving Response: Increase future IBNR to reflect the higher expected ultimate losses (consistent with the Chain Ladder / Development method).

Expected Claim Emergence Calculations

Expected emergence projects the volume of payments or reported claims expected between two evaluation dates (tt and t+1t+1).

General Emergence Formula (Preserves Current Reserves)

To calculate the expected emergence between tt and t+1t+1 while preserving the current unpaid claim or IBNR estimate:

Expected Emergencet,t+1=(Projected Ultimate−Cumulative Observedt)×%Observedt+1−%Observedt1−%Observedt\text{Expected Emergence}_{t, t+1} = \left( \text{Projected Ultimate} - \text{Cumulative Observed}_t \right) \times \frac{\% \text{Observed}_{t+1} - \% \text{Observed}_t}{1 - \% \text{Observed}_t}

Where:

  • %Observedt=1CDFt\% \text{Observed}_t = \frac{1}{\text{CDF}_t}.

Chain Ladder Emergence Formula

If the ultimate estimates are derived strictly using the Chain Ladder development method, the general formula simplifies to two equivalent expressions:

  • Observed Basis: Expected Emergencet,t+1=Cumulative Observedt×(ft−1)\text{Expected Emergence}_{t, t+1} = \text{Cumulative Observed}_t \times \left( f_t - 1 \right) Where ftf_t is the age-to-age factor (LDF) from tt to t+1t+1.
  • Ultimate Basis: Expected Emergencet,t+1=Projected Ultimate×(%Observedt+1−%Observedt)\text{Expected Emergence}_{t, t+1} = \text{Projected Ultimate} \times \left( \% \text{Observed}_{t+1} - \% \text{Observed}_t \right)

Interpolation of Development Patterns

  • Linear Interpolation Within Quarters: Linear interpolation of cumulative development factors is reasonably accurate within short periods (e.g., quarterly). This reflects the fact that development occurs more rapidly in early periods and slows down over time.
  • Non-Linearity Across Prolonged Periods: Assuming a linear development pattern over an entire year is generally inappropriate. Loss development is typically front-loaded (convex curve), meaning most development occurs early in the period rather than uniformly throughout the year.