Frequency Severity Techniques

Frequency-Severity reserving methods project ultimate losses by modeling the number of claims (frequency or count) and the cost per claim (severity) separately.

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Methodological Approaches

Three distinct approaches are utilized depending on data availability and exposure details.

Approach 1: Simple Count & Severity Development (Constant Exposures)

Used when exposures are constant or not explicitly modeled.

1. Develop counts: Apply standard Chain Ladder to the claim count triangle to project ultimate claim counts.
2. Develop severity: Apply Chain Ladder to average severity (losses divided by counts) to project ultimate severity.
3. Multiply: $$\text{Ultimate Losses} = \text{Ultimate Claim Counts} \times \text{Ultimate Severity}$$

Approach 2: Trended Frequency & Severity (Variable Exposures)

Used when exposures vary over time and trends are expected to continue.

1. Develop frequency:
   • Project ultimate claim counts.
   • Divide by the exposures of that year to get the historical ultimate frequency.
   • Trend the frequencies to the target period.
2. Develop severity:
   • Develop historical average severities to ultimate.
   • Apply severity trends to project severity to the target period.
3. Multiply: $$\text{Ultimate Losses} = \text{Target Exposures} \times \text{Projected Frequency} \times \text{Projected Severity}$$

Approach 3: Incremental Severity & Claims Disposal (No Triangles)

Used to project unpaid claims directly using claim disposal patterns and incremental payments.

1. Claim counts remaining: Project remaining open claim counts using disposal rates ($\text{DR}_j = \frac{\text{Closed Claims}_j}{\text{Ultimate Claims}}$). $$\text{Incremental Claims in period } (j, j+1) = \frac{\text{Ultimate Counts} - \text{Latest Closed Counts}}{1 - \text{Latest DR}} \times (\text{DR}_{j+1} - \text{DR}_j)$$
2. Trend incremental severity: Apply severity trends to historical incremental paid severities for future maturities.
3. Multiply & sum: Multiply projected incremental counts by trended incremental severities for each future maturity, then sum them to calculate total unpaid claims.

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Selecting Tail Severity

At older maturities, sparse data can lead to highly volatile average severity estimates. Actuaries use several criteria to select tail severities:

1. Data Consolidation: Combine data at the age where results become erratic to improve credibility.
2. Materiality Check: If the selection of a tail factor has a negligible impact on the overall reserve estimate, a highly refined tail analysis is unnecessary.
3. Stability Criterion: Ensure there are enough closed claims in the tail period to produce a stable severity estimate, but not so many that earlier maturities are left with insufficient data.

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Technical Details & Common Pitfalls

• Volume-Weighted Averages for Severity LDFs: When averaging severity age-to-age factors, weight each factor by the number of claim counts at the starting period, not by the dollar amount of losses: $$\text{Weighted Severity LDF} = \frac{\sum \text{Severity LDF}_i \times \text{Claim Counts}_i}{\sum \text{Claim Counts}_i}$$
• Modeling Period Length: Using shorter periods (e.g., quarters instead of years) improves the recognition of seasonal claim patterns and accounts for shifting average accident dates.
• Anomalies: Disregard periods of temporary operational disruption (e.g., claims backlog), as these patterns are not representative of future development.

[!WARNING] Reserving Pitfall Do not multiply the cumulative severity development factor by cumulative paid losses. The severity LDF must be multiplied by average severity (Losses / Counts) to obtain the projected average ultimate severity.