[[Actuarial Notes Wiki|Wiki]] / [[Exam 5 (CAS)]] / **Bornhuetter-Ferguson Method** ## Definition ==Bornhuetter Ferguson Method== The Bornhuetter-Ferguson (BF) method is a reserving technique that combines actual reported losses with an a priori expected loss estimate, weighted by the percentage of losses unreported. ## Formula ``` Ultimate Losses = Reported Losses + (Expected Losses × % Unreported) Where: % Unreported = 1 - (1 / CDF) IBNR = Expected Losses × % Unreported ``` ## Methodology ### Step 1: Determine Expected Losses ``` Expected Losses = Expected Loss Ratio × Earned Premium Or: Expected Losses = Pure Premium × Exposures Sources for expected loss ratio: - Pricing assumptions - Industry benchmarks - Recent historical experience - Budgeted loss ratio ``` ### Step 2: Calculate % Unreported ``` % Unreported = 1 - (1/CDF) Example: 12-month CDF to ultimate: 1.896 % Unreported at 12 months: 1 - (1/1.896) = 0.473 or 47.3% Interpretation: 47.3% of ultimate losses not yet reported at 12 months ``` ### Step 3: Calculate IBNR ``` IBNR = Expected Losses × % Unreported Example: Expected Losses: $1,000,000 % Unreported: 47.3% IBNR: $1,000,000 × 0.473 = $473,000 ``` ### Step 4: Calculate Ultimate ``` Ultimate = Reported + IBNR Example: Reported: $600,000 IBNR: $473,000 Ultimate: $1,073,000 ``` ## Complete Example ``` AY 2023 Analysis: Given Information: - Earned Premium: $2,000,000 - Reported Losses @ 12 months: $600,000 - Expected Loss Ratio: 65% - 12-Ult CDF (from development): 1.896 Step 1: Expected Losses = $2,000,000 × 0.65 = $1,300,000 Step 2: % Unreported = 1 - (1/1.896) = 1 - 0.528 = 0.472 or 47.2% Step 3: IBNR = $1,300,000 × 0.472 = $613,600 Step 4: Ultimate = $600,000 + $613,600 = $1,213,600 Comparison to Chain Ladder: CL Ultimate: $600,000 × 1.896 = $1,137,600 BF Ultimate: $1,213,600 Difference: $76,000 (6.7% higher) ``` ## Key Features ### Blending Actuals and Expected The BF method gives: - **100% weight** to reported losses - **0% weight** to expected losses for reported portion - **100% weight** to expected losses for unreported portion ### Credibility Perspective ``` BF is equivalent to: - Chain Ladder with 100% credibility on reported - Expected losses with credibility on unreported Reported portion: Fully credible (actual data) Unreported portion: Use a priori estimate ``` ## Advantages 1. **Stability** - Less volatile than chain ladder for immature years 2. **Uses external info** - Incorporates expected losses 3. **Exposure responsive** - Reflects premium/exposure changes 4. **Credibility-based** - Logical weighting of information 5. **Works with limited data** - Effective for new years ## Disadvantages 1. **Requires expected losses** - Need credible a priori estimate 2. **Ignores actual emergence** - Doesn't respond to unusual reporting 3. **Selection risk** - Results depend on expected loss ratio choice 4. **Less responsive** - Slower to react to changes ## When to Use **Preferred for:** - Immature accident years - Limited development history - Known operational changes - Unstable development patterns - New products or lines **Less suitable for:** - Mature accident years (use Chain Ladder) - When actual emergence is credible - Stable, well-developed books - When expected losses uncertain ## Comparison to Chain Ladder | Aspect | Chain Ladder | Bornhuetter-Ferguson | |--------|--------------|---------------------| | Data used | Historical only | Historical + Expected | | Volatility | Higher for immature years | Lower | | Responsiveness | More responsive | Less responsive | | Best for | Mature years | Immature years | | Stability | Less stable | More stable | ## Selecting Expected Losses ### Pricing Indication ``` Use loss ratio from rate filing: Expected LR = Target Loss Ratio from pricing ``` ### Historical Average ``` Average loss ratio from recent stable years: Expected LR = Avg(Historical Loss Ratios) ``` ### Blend of Methods ``` Expected LR = w₁(Pricing) + w₂(Historical) + w₃(Industry) ``` ### Considerations - Credibility of each source - Recent trends - Known changes - Regulatory requirements ## Practical Application ### By Accident Year Maturity **Recent Years (12-24 months)** - BF typically preferred - Limited credibility of actual emergence - Expected losses more reliable **Middle Years (24-48 months)** - Blend of BF and CL - Increasing credibility of actuals - May use Benktander **Mature Years (48+ months)** - Chain Ladder typically preferred - Actual emergence highly credible - Less reliance on expected ## Related Concepts - [[Chain Ladder Method]] - [[Expected Loss Method]] - [[Benktander Method]] - [[Cape Cod Method]] - [[IBNR Reserves]] - [[Development Factor]] ## References - Friedland, Chapter 5 - Bornhuetter & Ferguson, "The Actuary and IBNR"