The Bornhuetter-Ferguson (BF) method is a widely used technique in actuarial science for estimating the ultimate losses of an insurance or reinsurance company. It's particularly valuable when dealing with claims that have a long reporting tail, meaning it can take a significant amount of time for all claims to be reported and settled.

The fundamental formula for the BF method is:

Reserve = (Ultimate Loss Estimate) * (1 - Cumulative Reported Percentage) + Cumulative Reported Losses

Let's break down the key components:

This is the initial, forward-looking estimate of the total amount of losses that will eventually be paid for a given accident year or underwriting period. It's often derived from prior experience, industry benchmarks, or other actuarial models. The accuracy of the ULE significantly impacts the BF reserve estimate.

This represents the proportion of total claims that are expected to have been reported by a specific point in time. It's typically derived from historical claim development patterns, often expressed as a percentage of ultimate claims reported by each subsequent valuation date. For example, if 80% of claims are typically reported by the end of the first year after the accident, the CRP for that point would be 0.80.

This is the actual amount of losses that have been reported and recorded up to the valuation date. It includes both paid losses and outstanding loss reserves for reported claims.

The BF method essentially estimates the unreported portion of ultimate losses and adds it to the already reported losses. The 'unreported portion' is calculated as the ULE multiplied by the percentage of claims not yet reported (1 - CRP). This approach provides a more stable estimate than methods that solely rely on extrapolating from observed development, especially when the observed data is sparse or volatile.

When applying the BF method, actuaries must carefully consider:

• Selection of the Ultimate Loss Estimate (ULE): This is the most critical input. It should be based on sound actuarial principles and reflect current underwriting and economic conditions.
• Determination of the Cumulative Reported Percentage (CRP): Historical data must be analyzed to establish reliable patterns of claim reporting. Changes in claims handling or reporting practices need to be accounted for.
• Consistency: The method should be applied consistently over time to allow for meaningful comparisons and trend analysis.

The BF method is often used in conjunction with other reserving techniques, such as the Chain-Ladder method. Actuaries typically use multiple methods to triangulate on a reasonable reserve range, providing a more robust estimate. The BF method can be particularly useful for providing a 'sanity check' on estimates derived from methods that are purely retrospective.

Consider an insurer with the following data for Accident Year 2022:

• Ultimate Loss Estimate (ULE) = $1,000,000
• Cumulative Reported Losses (CRL) as of December 31, 2023 = $600,000
• Historical data suggests that by 18 months after the accident year end (which is December 31, 2023 for AY 2022), 75% of claims are typically reported (CRP = 0.75).

Using the BF formula: Reserve = ($1,000,000 * (1 - 0.75)) +$600,000 Reserve = ($1,000,000 * 0.25) +$600,000 Reserve = $250,000 +$600,000 Reserve = $850,000

This means the estimated reserve needed for Accident Year 2022 as of December 31, 2023, is $850,000. This includes the$600,000 already reported and an additional $250,000 estimated for claims yet to be reported.