Bornhuetter Ferguson

The Bornhuetter-Ferguson (B-F) method is a reserving technique that combines the Chain Ladder development pattern with an a priori expected loss estimate.

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Key Assumptions

1. Observational Credibility: Claims observed to date are accurate and do not require adjustment.
2. Independence of Future Development: The volume of claims reported (or paid) to date does not alter the expectation of future development. Future unpaid losses develop strictly according to the a priori expected losses and the selected development pattern.

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Mathematical Formulation

The B-F method can be applied to either paid or reported losses.

1. General Formula

The ultimate losses for accident year $i$ are:

$$\text{Ultimate Losses}_i = C_{i, t} + \text{Unpaid (or Unreported) Losses}_i$$

Where:

• $C_{i,t}$: Cumulative observed (paid or reported) losses at maturity $t$.
• $\text{Unpaid (or Unreported) Losses}_i$: The projected remaining development.

2. Projecting Remaining Development

Let $\text{CDF}_t$ be the Cumulative Loss Development Factor at age $t$. The percent of ultimate losses expected to be observed is:

$$\% \text{Observed}_t = \frac{1}{\text{CDF}_t}$$

The percent expected to be unpaid/unreported is:

$$\% \text{Unobserved}_t = 1 - \frac{1}{\text{CDF}_t}$$

Using the a priori expected losses ($\text{Ultimate}_{\text{prior}}$):

$$\text{Unpaid Losses}_i = \text{Ultimate}_{\text{prior}} \times \left(1 - \frac{1}{\text{CDF}_t}\right)$$

If using the Expected Loss Ratio ($\text{ELR}$) and Earned Premium ($\text{EP}$):

$$\text{Unpaid Losses}_i = (\text{EP} \times \text{ELR}) \times \left(1 - \frac{1}{\text{CDF}_t}\right)$$

3. Ultimate Losses Summary

$$\text{Ultimate Losses}_i = C_{i, t} + (\text{EP} \times \text{ELR}) \times \left(1 - \frac{1}{\text{CDF}_t}\right)$$

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Bornhuetter-Ferguson: Paid vs. Incurred

The method can be implemented using two separate experience bases:

• Paid B-F Method: Projects unpaid claims using paid cumulative losses ($P_{i,t}$) and the paid CDF: $$\text{Ultimate (Paid BF)} = P_{i, t} + \text{Prior Expected Losses} \times \left(1 - \frac{1}{\text{CDF}_{t, \text{paid}}}\right)$$
• Incurred B-F Method: Projects IBNR using reported cumulative losses ($R_{i,t}$) and the reported CDF: $$\text{Ultimate (Incurred BF)} = R_{i, t} + \text{Prior Expected Losses} \times \left(1 - \frac{1}{\text{CDF}_{t, \text{reported}}}\right)$$

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Comparison of Reserving Methods

The B-F method functions as a compromise between the Chain Ladder and Expected Claims methods:

Feature	Chain Ladder	Expected Claims	B-F Method
Sensitivity to Observed Data	$100\%$ (Highly sensitive)	$0\%$ (Completely ignores)	Partial (Balances actual and expected)
Stability of Estimates	Low (High volatility at early ages)	High (Independent of actual losses)	High (Highly stable at early ages)
A Priori Dependency	None	$100\%$ dependent	Partially dependent on selection of ELR
Reserving Target	Projects total ultimate	Projects total ultimate	Projects future unpaid directly