Bornhuetter ferguson method

The Bornhuetter Ferguson method is a crucial technique in the field of actuarial science, specifically within insurance loss reserving.

What Is the Bornhuetter Ferguson Method?

The Bornhuetter Ferguson method is an actuarial technique used by insurance companies to estimate future claim liabilities, particularly for claims that have been incurred but not reported (IBNR). This method is a hybrid approach, combining an insurer's initial expectation of losses, often based on a pre-determined a priori loss ratio, with the actual claims experience that has emerged to date⁵⁶, ⁵⁷. The Bornhuetter Ferguson method aims to provide more stable and reliable estimates, especially for recently originated policy periods where actual claims data may be limited or immature⁵³, ⁵⁴, ⁵⁵. It serves as a cornerstone in risk management for insurers, helping them set adequate liabilities on their balance sheet.

History and Origin

The Bornhuetter Ferguson method was developed by U.S. actuaries Ronald L. Bornhuetter and Ronald E. Ferguson. It was first introduced to the actuarial community in their seminal 1972 paper titled "The Actuary and IBNR"⁵¹, ⁵². At the time of its introduction, the insurance industry faced challenges in accurately estimating outstanding claims, particularly for lines of business with significant reporting delays, known as long-tail lines of insurance⁴⁹, ⁵⁰. The method emerged as a response to the need for a more robust approach than relying solely on historical claims development, which could be highly volatile for immature periods. It quickly gained widespread adoption due to its balanced incorporation of both expert judgment (through the expected loss ratio) and emerging claims data, becoming a universally recognized methodology taught in actuarial exams⁴⁸. The original paper can be accessed for further historical context. 1972 Bornhuetter-Ferguson Paper

Key Takeaways

• The Bornhuetter Ferguson method estimates ultimate insurance losses, including Incurred But Not Reported (IBNR) claims⁴⁷.
• It blends an initial expected loss ratio with actual reported claims experience⁴⁵, ⁴⁶.
• This method is particularly useful for immature claim periods or volatile lines of business where limited data might otherwise lead to unstable estimates⁴³, ⁴⁴.
• It provides a more stable estimate compared to methods relying solely on historical claims development, especially when actual losses are not a strong indicator of IBNR⁴².
• The method assigns more weight to actual reported claims as the policy period matures and claims become more fully developed⁴¹.

Formula and Calculation

The Bornhuetter Ferguson method estimates the ultimate loss for a given policy year by combining reported losses with a projection of unreported losses. There are two algebraically equivalent ways to express the ultimate loss calculation:

Approach 1:

Where:

• Reported Losses (L): The cumulative claims that have been reported and paid or reserved as of the valuation date for a specific policy year⁴⁰.
• Expected Loss Ratio (ELR): An a priori estimate of the ratio of total ultimate losses to earned premium for the policy year³⁹. This is often based on historical data, industry benchmarks, or internal business plans³⁸.
• Earned Premium (Exposure): The portion of premium for which the insurance coverage has already been provided over a given period³⁷.
• Percent Unreported (1-w): The estimated percentage of total ultimate losses that are yet to be reported, often derived from historical loss development factors³⁶.

Approach 2 (alternative, but equivalent, formulation often used in conjunction with the chain ladder method):

In this formulation, the "Percent Reported" is the reciprocal of the Age-to-Ultimate Loss Development Factor (LDF). So, $\text{Percent Reported} = \frac{1}{\text{LDF}}$ and $\text{Percent Unreported} = 1 - \frac{1}{\text{LDF}}$.

This second approach highlights the blend of actual experience (Reported Losses * Percent Reported) and the a priori expectation (Expected Loss Ratio * Earned Premium * Percent Unreported).

Interpreting the Bornhuetter Ferguson Method

The interpretation of the Bornhuetter Ferguson method's output centers on understanding the balance it strikes between observed data and initial expectations. For new or "immature" accident years, where very few claims have been reported, the Bornhuetter Ferguson method places more emphasis on the a priori expected loss ratio. This is because the limited actual reported claims may not be credible enough to reliably project the ultimate outcome³⁴, ³⁵.

As an accident year matures and more claims are reported, the weight shifts, and the actual reported losses become increasingly influential in the ultimate loss estimate. The method's strength lies in its ability to provide stable estimates for immature periods by incorporating a foundational expectation, while still being responsive to actual ultimate claims experience as it emerges³², ³³. Actuaries use the resulting ultimate loss estimates to set appropriate financial reporting reserves.

Hypothetical Example

Consider an insurance company estimating ultimate losses for a new policy year in its property and casualty insurance division.

Given Data:

• Reported Losses (L) to date: $100,000
• Earned Premium (Exposure) for the policy year: $2,000,000
• Expected Loss Ratio (ELR): 60%
• Loss Development Factor (LDF) for this maturity period: 1.25 (meaning 80% of losses are currently reported, since $1/1.25 = 0.80$)

Step-by-Step Calculation:

1. Calculate Percent Reported (w):

   $$w = \frac{1}{\text{LDF}} = \frac{1}{1.25} = 0.80$$

2. Calculate Percent Unreported (1-w):

   $$1 - w = 1 - 0.80 = 0.20$$

3. Calculate the Bornhuetter Ferguson Ultimate Loss:
   Using the first approach for simplicity:

   $$\text{BF Ultimate Loss} = \text{Reported Losses} + (\text{Expected Loss Ratio} \times \text{Earned Premium} \times \text{Percent Unreported})$$ $$\text{BF Ultimate Loss} = \$100,000 + (0.60 \times \$2,000,000 \times 0.20)$$ $$\text{BF Ultimate Loss} = \$100,000 + (\$1,200,000 \times 0.20)$$ $$\text{BF Ultimate Loss} = \$100,000 + \$240,000$$ $$\text{BF Ultimate Loss} = \$340,000$$

4. Calculate Incurred But Not Reported (IBNR) Losses:

   $$\text{IBNR} = \text{BF Ultimate Loss} - \text{Reported Losses}$$ $$\text{IBNR} = \$340,000 - \$100,000 = \$240,000$$

In this scenario, the Bornhuetter Ferguson method estimates that the total ultimate losses for this policy year will be $340,000, with $240,000 of these still needing to be reported. This calculation demonstrates how the method combines the limited actual experience ($100,000 reported) with an expected future development ($240,000 IBNR based on the initial expected loss ratio and development pattern).

Practical Applications

The Bornhuetter Ferguson method is widely used in the insurance industry for several practical applications:

• Claims Reserving: It is a primary tool for actuaries to estimate the amount of money an insurer needs to set aside for future claim payments, ensuring solvency and compliance with regulatory obligations³⁰, ³¹. It is particularly effective for estimating reserves in workers compensation and other liability lines²⁹.
• New Lines of Business: When an insurer enters a new line of business where historical internal claims data is scarce or non-existent, the Bornhuetter Ferguson method allows actuaries to leverage industry benchmarks or expert judgment (through the expected loss ratio) to make initial, yet stable, loss reserve estimates²⁸.
• Immature Data Periods: For the most recent accident years, where claims have not had sufficient time to fully develop and report, the Bornhuetter Ferguson method provides more stable estimates than methods that rely purely on emerging claims patterns from incomplete claims development triangle data²⁶, ²⁷.
• Pricing Decisions: Accurate loss reserving directly impacts pricing. By providing reliable ultimate loss estimates, the Bornhuetter Ferguson method assists underwriters in setting appropriate premiums that reflect the true cost of risk for various insurance products.
• Financial Auditing and Reporting: Actuaries often present results from the Bornhuetter Ferguson method in actuarial reports that support the financial statements of insurance companies, providing auditors and regulators with a reasoned basis for loss reserve adequacy²⁵.

Limitations and Criticisms

While highly valued for its stability, the Bornhuetter Ferguson method is not without its limitations and criticisms:

• Reliance on A Priori Loss Ratio: A significant drawback is its dependence on the accuracy of the initial expected loss ratio. If this initial estimate is flawed or not representative of the actual underlying loss experience, the ultimate loss estimate derived from the Bornhuetter Ferguson method will also be inaccurate²², ²³, ²⁴. The selection of this "Initial Expected Loss Ratio" often involves considerable actuarial judgment and can vary widely in practice²¹.
• Assumption of Consistent Reporting Patterns: The method assumes a certain consistency in how claims are reported over time (i.e., the percentage of claims reported by a given development period remains stable across different origin periods)²⁰. Changes in claims handling practices, policy benefits, or external factors can distort these reporting patterns, potentially leading to inaccurate estimates¹⁸, ¹⁹.
• Less Responsive to Extreme Deviations: For very immature years, where the actual reported losses are significantly different from the initial expectation, the Bornhuetter Ferguson method may be slow to fully reflect these emerging trends due to the strong weight given to the a priori estimate. This is considered a "flaw" by some critics in certain contexts, particularly when compared to methods that are purely data-driven¹⁷.
• Limited for Short-Tail Lines: For short-tail lines of insurance, where claims develop and close quickly, the Bornhuetter Ferguson method may offer little advantage over simpler development methods, as the actual reported claims quickly become the dominant factor, causing the method to approximate those simpler approaches¹⁶.

An academic paper evaluating various loss reserving methods provides further insights into the complexities and potential drawbacks of widely used actuarial techniques, including the Bornhuetter Ferguson method. Anatomy of Actuarial Methods of Loss Reserving

Bornhuetter Ferguson Method vs. Chain-Ladder Method

The Bornhuetter Ferguson method and the Chain-Ladder Method are two of the most popular actuarial techniques for estimating insurance loss reserves, particularly Incurred But Not Reported (IBNR) claims. While both use historical claims data presented in a claims development triangle, their fundamental approaches differ significantly.

The Chain-Ladder Method is primarily data-driven. It relies solely on observed historical patterns of claims development to project future losses. It assumes that future development patterns will mirror past patterns and uses "loss development factors" (or link ratios) derived directly from the historical data to project cumulative losses to their ultimate value¹⁴, ¹⁵. This method is highly effective for mature lines of business with stable historical data but can produce volatile and unreliable results for immature or volatile periods because small fluctuations in early data points are heavily extrapolated¹¹, ¹², ¹³.

In contrast, the Bornhuetter Ferguson method is a blend of data-driven and expectation-based approaches. It does not rely solely on the historical development pattern of reported claims. Instead, it combines the actual reported claims with an initial expectation of ultimate losses, usually derived from an a priori loss ratio and earned premium¹⁰. For immature periods, the Bornhuetter Ferguson method places greater weight on this initial expectation, providing more stable estimates. As claims mature, more weight is given to the actual reported losses⁹. The key difference lies in the source of information for the unreported portion: Chain-Ladder estimates it purely from historical development factors, while Bornhuetter Ferguson bases it on an expected percentage of the total ultimate loss. This makes the Bornhuetter Ferguson method more robust for situations where actual reported losses are not a good indicator of IBNR⁶, ⁷, ⁸.

FAQs

Q: Why is the Bornhuetter Ferguson method important for insurance companies?
A: It's important because it helps insurance companies accurately estimate their future liabilities for claims that have happened but haven't been fully reported yet. This ensures they hold enough capital to pay claims, which is vital for their financial stability and regulatory compliance.

Q: When is the Bornhuetter Ferguson method most effective?
A: It is particularly effective for estimating reserves for new policy years or for types of insurance where claims take a long time to be fully reported and settled (known as long-tail liabilities). In these situations, there isn't much actual claims data yet, so the method's ability to incorporate an initial expectation helps provide a more stable estimate⁴, ⁵.

Q: Can the Bornhuetter Ferguson method be used in conjunction with other methods?
A: Yes, actuaries often use the Bornhuetter Ferguson method alongside other techniques, such as the Chain-Ladder Method, to get a comprehensive view of loss reserves. It's common to use the Bornhuetter Ferguson method for more recent, immature periods and the Chain-Ladder Method for more mature periods³.

Q: What is an "a priori loss ratio" in the context of this method?
A: An a priori loss ratio is an initial, pre-determined estimate of what the total losses for a given period are expected to be, expressed as a percentage of the earned premiums. This estimate is typically based on historical experience, industry data, or an insurer's business plan and serves as a starting point for the Bornhuetter Ferguson calculation¹, ².