Bornhuetter-Ferguson Method for Claims Reserving

The Core Idea: Known vs. Unknown Losses

The central insight of the Bornhuetter-Ferguson method is to split the estimate of ultimate loss into two distinct components:

1. The "Known" Portion: This is the actual, observed data—the cumulative claims (paid or incurred) that have been reported to date. This part is taken as given.
2. The "Unknown" Portion: This is the estimate for claims that are yet to emerge (the IBNR, or Incurred But Not Reported). Instead of projecting this from the "known" losses (as the chain ladder does), the BF method estimates it based on an independent, a priori view of the total expected loss.

The final estimate is simply the sum of these two parts. The following article provides a clear definition and presents the main formula.

As the article explains, the algebraic formulation is:

The "Expected Ultimate Loss" is the a priori estimate, typically calculated by multiplying the earned premium (or another measure of exposure) for that year by a selected Expected Loss Ratio (ELR). The "% Unreported" is derived from the same loss development factors (LDFs) used in the chain ladder method, where % Unreported = $1-1/f_{\text{ult}}$, with $f_{\text{ult}}$ being the age-to-ultimate development factor.

The Mechanics of the BF Method

The BF method exists in two primary forms: a "paid" version (using a triangle of paid claims) and an "incurred" version (using a triangle of incurred claims = paid claims + case reserves). The logic is the same, but the interpretation of the "unknown" portion differs slightly.

• Paid BF: The unknown portion, EUL * % Unpaid, represents the total outstanding liability (Case Reserves + IBNR).
• Incurred BF: The unknown portion, EUL * % Unreported, represents only the IBNR.

The following video provides an excellent visual walkthrough of both versions and compares them directly to the development methods we've already studied.

The key takeaway is that the BF estimate of the outstanding reserve does not depend on the amount of claims paid or incurred to date, only on the a priori estimate and the development pattern. This is what gives the method its stability.

Worked Example: Calculating BF Reserves

Let's apply the method to a practical example. Suppose we have the following data for an insurer. Our task is to calculate the total outstanding claims reserve as at the end of 2022 using the paid Bornhuetter-Ferguson method.

Given Data:

• Cumulative Paid Claims Triangle (£'000):

• Earned Premium (£'000):

• Selected Expected Loss Ratio (ELR): 75%
• Tail Factor (from Dev Year 3 to Ultimate): 1.05

Step 1: Calculate Loss Development Factors (LDFs)

First, we calculate the age-to-age LDFs from the historical triangle, just as we did for the chain ladder method.

• LDF(0-1): $\frac{4,500+4,900+5,300}{2,500+2,700+2,900}=\frac{14,700}{8,100}=1.8148$
• LDF(1-2): $\frac{5,500+6,000}{4,500+4,900}=\frac{11,500}{9,400}=1.2234$
• LDF(2-3): $\frac{6,000}{5,500}=1.0909$

Step 2: Calculate Age-to-Ultimate Factors and % Unpaid

Next, we chain the LDFs together from the end to get the age-to-ultimate factors ($f_{\text{ult}}$) and then find the percentage of claims yet to be paid.

• $f_{\text{ult}}(3)=1.05$ (the given tail factor)
• $f_{\text{ult}}(2)=f_{\text{ult}}(3)\times \text{LDF}(2-3)=1.05\times 1.0909=1.1454$
• $f_{\text{ult}}(1)=f_{\text{ult}}(2)\times \text{LDF}(1-2)=1.1454\times 1.2234=1.4014$
• $f_{\text{ult}}(0)=f_{\text{ult}}(1)\times \text{LDF}(0-1)=1.4014\times 1.8148=2.5432$

Now, we calculate the percentage expected to be unpaid at each development stage using $\%\text{Unpaid}=1-1/f_{\text{ult}}$.

• Dev Year 0: $1-1/2.5432=60.68\%$
• Dev Year 1: $1-1/1.4014=28.64\%$
• Dev Year 2: $1-1/1.1454=12.70\%$
• Dev Year 3: $1-1/1.05=4.76\%$

Step 3: Calculate the Reserve for Each Accident Year

We can now bring all the pieces together in a table to calculate the outstanding reserve for each incomplete accident year. The reserve is calculated as: (Earned Premium * ELR) * % Unpaid.

The total outstanding claims reserve to be held at the end of 2022 is £7,352,900.

A Completed Example

The final output of a BF calculation is often presented in a summary format. The image below shows a completed BF analysis. You can see how the components we just calculated—the claims data, the premium, the ELR (ultimate loss ratio), and the development pattern—all come together to produce the final reserve figure.

In this example, the development factors (f_2,3, f_1,2, f_0,1) would have been calculated separately, just as we did in our worked example. These are used to determine the % unreported for each accident year, which is then multiplied by the initial ultimate estimate (Premium * ultimate loss ratio) to get the reserve for that year. The o/s claims reserve of 4379.30 is the sum of these individual reserves.

Conclusion

This lesson concludes our study of deterministic claims reserving models. The Bornhuetter-Ferguson method is a crucial technique in the actuary's toolkit, providing a stable and sensible alternative when pure data-driven methods like the chain ladder are likely to produce unreliable results.

Key Takeaways:

• Hybrid Approach: The BF method combines observed data (Reported Loss) with an a priori view (Expected Ultimate Loss).
• Formula: Ultimate Loss = Reported Loss + Expected Ultimate Loss × (1 - 1/f_ult).
• Stability: The reserve estimate (Expected Ultimate Loss × (1 - 1/f_ult)) is independent of the actual claims reported to date, making the method robust against random fluctuations in early data.
• Application: It is particularly useful for immature accident years, new lines of business, or high-severity/low-frequency claims.

With this lesson, you have covered the main run-off triangle techniques required for the CM2 exam. In practice, an actuary will typically apply several methods—Basic Chain Ladder, Inflation-Adjusted Chain Ladder, Bornhuetter-Ferguson, and others—and use their professional judgment to select a final reserve figure from within the range of results.

As this is the final lesson in our structured course, your next step is to consolidate this knowledge by working through a wide variety of exam-style questions. This will help you build the speed and confidence needed to integrate concepts from financial engineering and loss reserving to solve complex problems under exam conditions.