Bornhuetter-Ferguson (B-F) method

From the previous two sections, we have found the IBRN reserve with basic chain ladder and average cost per cliam. However, those two methods do not integrate the average and total information well.

The B-F method stands out because it cleverly combines two fundamental approaches:

1. The Chain Ladder Method: This method relies on historical claims development patterns to project future claims. It’s like using past trends to predict the future. However, it can be unreliable in the early stages of claims development or when historical data is scarce.
2. The Expected Claims Ratio Method (or a priori method): This method uses an initial estimate of the expected loss ratio (the ratio of claims to premiums) to predict ultimate claims. It incorporates expert judgment and external information, offering a benchmark for what claims are expected to be.

The B-F method brings these two approaches together, creating a more robust and stable estimate of IBNR. Think of it as getting a second opinion – it’s always better to have multiple perspectives when dealing with uncertainty.

Key Concepts: Insurance Ratios

Before we dive into the B-F calculations, let’s refresh our understanding of some essential insurance ratios:

• Loss Ratio: This is a key indicator of an insurer’s profitability. It measures the incurred claims (claims paid plus reserves for reported but not yet settled claims) against the earned premiums over a defined period.

  $$\text{Loss Ratio} = \frac{\text{Incurred Claims}}{\text{Earned Premiums}}$$

  • Interpretation: A loss ratio below 100% suggests that the insurance premiums are sufficient to cover the incurred claims, while a ratio above 100% indicates that claims are exceeding the earned premium.

• Combined Ratio: This ratio provides a broader view of profitability by including underwriting expenses (costs associated with acquiring and servicing policies).

  $$\text{Combined Ratio} = \frac{\text{Incurred Claims} + \text{Expenses}}{\text{Earned Premiums}}$$

  • Interpretation: Similar to the loss ratio, a combined ratio below 100% indicates an underwriting profit. A combined ratio above 100% suggests that the insurer is paying out more in claims and expenses than it’s collecting in premiums.

• Trading Ratio (or Operating Ratio): This is the most comprehensive measure of an insurer’s overall profitability, as it takes into account investment income earned on premiums and reserves. However, for IBNR calculation purposes, the loss ratio is most crucial.

  $$\text{Trading Ratio} = \frac{\text{Incurred Claims} + \text{Expenses} - \text{Investment Income}}{\text{Earned Premiums}}$$

  • Interpretation: This ratio reflects the overall financial performance of an insurance operation.

The Bornhuetter-Ferguson Method: A Step-by-Step Guide

In the following sections, we will breakdown the steps of B-F method to calculate the reserve. Before we start, we need to give some denotations:

• $r_i$ is the loss ratio for the $i_{\text{th}}$ original year or the accident year.
• $P_i$ is the earned premium for the $i_{\text{th}}$ original year or the accident year.
• $U_i^I$ is the initial estimate of the ultimate claims for the originating $i_{\text{th}}$ year.
• $d_{j|k}$ is the development factor from the $j_{\text{th}}$ year to the $k_{\text{th}}$ year.
• $d_j$ is the development factor from the $j_{\text{th}}$ year to the ultimate year.

1. Development Factors:

• Calculate Age-to-Age Development Factors, $d_{j|k}$: Using historical claims data, we track how claims develop over time and calculate the development factors.
• Average Development Factors: We then calculate the average development factor for each development period.
• Cumulative Development Factors (CDFs, $d_j$): To project claims to their ultimate settlement value, we multiply the average development factors sequentially. This creates a chain of factors that show the expected total development from a given period to the ultimate value.

2. Expected Ultimate Claims: The Initial Prediction

• Initial Expected Loss Ratio (IELR): This is a crucial input, representing the actuary’s initial estimate of the loss ratio for a particular line of business or accident year. It’s based on experience, industry data, pricing assumptions, and expert judgment.

• Calculate Expected Ultimate Claims: For each accident year, multiply the earned premium by the IELR. This gives us the initial prediction of the total claims that will eventually be paid for that year.

  $$\text{Expected Ultimate Claims} = \text{Earned Premium} \times \text{Initial Expected Loss Ratio}$$

3. Adjusting for Development: Refining the Estimate

• Ultimate Claims in the Latest Development Year: We need to adjust the Expected Ultimate Claims to reflect the development observed up to the latest available period. We do this by dividing the Expected Ultimate Claims by the CDF that corresponds to the latest period for each accident year.

  $$\text{Ultimate Claims in Latest Development Year} = \frac{\text{Expected Ultimate Claims}}{\text{CDF to Latest Period}}$$

4. Emerging Liability: What’s Yet to Come

• Calculate Emerging Liability: This represents the portion of claims that we expect to emerge in the future but are not yet reflected in the reported claims. It’s the difference between the Expected Ultimate Claims and the Ultimate Claims in the Latest Development Year.

  $$\text{Emerging Liability} = \text{Expected Ultimate Claims} - \text{Ultimate Claims in Latest Development Year}$$

5. Putting it Together: The B-F Ultimate Claims

• Calculate Ultimate Liability: This is the core of the B-F method. We add the reported claims (the claims we already know about) to the Emerging Liability (the claims we expect to emerge). This gives us the B-F estimate of ultimate claims.

  $$\text{Ultimate Liability} = \text{Reported Claims} + \text{Emerging Liability}$$

6. The Final Piece: The IBNR Reserve

• Calculate Total Ultimate Liability: Sum the Ultimate Liability for all accident years.

• Calculate the Reserve: Finally, we subtract the total reported claims (across all accident years) from the Total Ultimate Liability. This difference is the estimated IBNR reserve – the amount the insurer needs to set aside to cover claims that have been incurred but not yet reported.

  $$\text{Reserve} = \text{Total Ultimate Liability} - \text{Total Reported Claims}$$

Example: Bringing the B-F Method to Life

Let’s walk through the previous example using the B-F method. We have the following data:

	Development year
Original year	0	1	2	3	Premium	Loss ratio	$U_i$
2020	100	180	240	280	324	86%
2021	120	220	300		365	86%
2022	140	260			380	86%
2023	160				580	86%

Step 1 & 2: Calculate Development Factors, Average Development Factors, and CDFs

In the first, we should calculate all development factors for each period. If we assume the calculation using the average development factor method (In the first section, we have already given the calculation steps for the development factor), then we can get the following table:

	Development year
Original year	0 → 1	1 → 2	2 → 3
2020	1.80	1.33	1.17
2021	1.83	1.36
2022	1.86
Average	1.83	1.345	1.17

Following the Arithmetic average method, we can calculate the average development factor for each period.

$$\begin{array}{rl} d_{3} = CDF_{3|3} &= 1 \\ \\ d_{2} = CDF_{2|3} &= 1.17 \\ \\ d_{1} = CDF_{1|3} &= CDF_{1|2} \times CDF_{2|3} = 1.345 \times 1.17 = 1.57 \\ \\ d_{0} = CDF_{0|3} &= CDF_{0|1} \times CDF_{1|3} = 1.83 \times 1.57 = 2.87 \end{array}$$

Step 3: Estimate Expected Ultimate Claims

Following the definition of the loss ratio, we can calculate the expected ultimate claims (Ultimate losses) for each year. $U_I = P_I \times r_I$, where $I$ is the original year.

$$\begin{array}{rl} \text{Expected Ultimate Claims}_{2020} = U_{2020} &= 324 \times 0.86 = 278.64 \\ \text{Expected Ultimate Claims}_{2021} = U_{2021} &= 365 \times 0.86 = 313.90 \\ \text{Expected Ultimate Claims}_{2022} = U_{2022} &= 380 \times 0.86 = 326.80 \\ \text{Expected Ultimate Claims}_{2023} = U_{2023} &= 580 \times 0.86 = 498.80 \end{array}$$

These numbers are the estimated ultimate claims when all claims happened in the original year are settled.

Step 4: Calculate Ultimate Claims in the Latest Development Year

Then, we should convert the ultimate claims to the latest available development year, which can correspond to the reported claims.

We can apply the following formula to calculate the expected ultimate claims for each year:

$$U_i = \frac{U_i}{d_{j}}$$

to the ultimate year, $U_i$ is the expected ultimate claims for the $i_{\text{th}}$ year.

$$\begin{array}{rl} U_3^{2020} &= \frac{U^{2020}}{d_{3|3}} = \frac{278.64}{1} = 278.64 \\ \\ U_2^{2021} &= \frac{U^{2021}}{d_{2|3}} = \frac{313.9}{1.17} = 268.3 \\ \\ U_1^{2022} &= \frac{U^{2022}}{d_{1|3}} = \frac{326.8}{1.57} = 208.15 \\ \\ U_0^{2023} &= \frac{U^{2023}}{d_{0|3}} = \frac{498.8}{2.87} = 173.8 \end{array}$$

From the above calculation, we can get the expected ultimate claims for each year in the latest available development year.

Step 5: Calculate Emerging Liability

From the previous section calculation, we can get the following emerging liability:

$$\begin{array}{rl} \text{Emerging Liability}_{2020} &= 278.64 - 278.64 = 0.00 \\ \text{Emerging Liability}_{2021} &= 313.90 - 268.3 = 45.60 \\ \text{Emerging Liability}_{2022} &= 326.80 - 208.15 = 118.65 \\ \text{Emerging Liability}_{2023} &= 498.80 - 173.8 = 325 \end{array}$$

Step 6: Calculate Ultimate Liability

The ultimate liability is the sum of the reported claims and the emerging liability.

$$\begin{array}{rl} \text{Ultimate Liability}_{2020} &= 280 + 0.00 = 280.00 \\ \text{Ultimate Liability}_{2021} &= 300 + 45.60 = 345.60 \\ \text{Ultimate Liability}_{2022} &= 260 + 118.65 = 378.65 \\ \text{Ultimate Liability}_{2023} &= 160 + 325 = 485 \end{array}$$

Step 7: Calculate Total Ultimate Liability

The total ultimate liability is the sum of the ultimate liability for each year.

$$\text{Total Ultimate Liability} = 280 + 345.60 + 378.65 + 485 = 1489.25$$

Step 8: Calculate the Reserve (IBNR)

The reserve is the difference between the total ultimate liability and the total reported claims.

$$\text{Total Reported Claims} = 280 + 300 + 260 + 160 = 1000$$

$$\text{Reserve} = 1489.25 - 1000 = 489.25$$

Conclusion: Based on our calculations, the estimated IBNR reserve using the B-F method is 489.25. This is the amount the insurer should set aside to cover claims that have occurred but have not yet been reported.

Summary

In this section, we have introduced the B-F method to calculate the reserve. The B-F method is a more comprehensive method to calculate the reserve, which integrates the average and total information well. The calculation steps are similar to the chain ladder method, but the B-F method can provide more information about the emerging liability and ultimate liability.

Strengths and Weaknesses of the B-F Method

Like any actuarial method, the B-F method has its pros and cons:

Strengths:

• Blends Historical Data and Expert Judgment: By combining the Chain Ladder and Expected Claims Ratio methods, the B-F approach leverages both past experience and informed expectations about the future. This makes it more reliable than relying on just one of these methods alone.
• Stability: The B-F method tends to produce more stable estimates, especially in the early years of development or when dealing with volatile lines of business. It’s less sensitive to random fluctuations in the early data than the Chain Ladder method.
• Reduced Reliance on Tail Factors: In the Chain Ladder method, actuaries often need to estimate a “tail factor” to project claims beyond the observed development periods. The B-F method, by incorporating expected ultimate claims, reduces the dependence on these potentially subjective tail factors.

Weaknesses:

• Sensitivity to the Initial Expected Loss Ratio (IELR): The accuracy of the B-F method hinges on the chosen IELR. If the IELR is inaccurate, the resulting reserve estimates will be biased.
• Requires Careful Judgment: While the B-F method reduces reliance on some subjective elements compared to the Chain Ladder method, it still requires careful judgment in selecting the IELR and development factors.
• May Not Fully Capture Changes in Development Patterns: If there are significant shifts in the underlying claims development patterns (e.g., due to changes in legislation or claims handling practices), and the IELR is not updated to reflect these, the B-F method may not accurately capture these changes.