[[Actuarial Notes Wiki|Wiki]] / [[Exam 5 (CAS)]] / **Chain Ladder Method** ## Definition ==Chain Ladder Method== The chain ladder method (also called development method or link ratio method) is a reserving technique that projects ultimate losses by applying historical development patterns to current reported losses. ## Methodology ### Step 1: Create Development Triangle Organize cumulative incurred (or paid) losses by accident year and age. ### Step 2: Calculate Age-to-Age Factors ``` Age-to-Age Factor (Link Ratio) = Loss at age (n+1) / Loss at age n Example for 12-24 month factor: AY 2020: 800/500 = 1.600 AY 2021: 825/550 = 1.500 AY 2022: 875/575 = 1.522 Average: 1.541 ``` ### Step 3: Select Development Factors Methods for selection: - **Simple average** - All years equally weighted - **Weighted average** - Weight by volume - **Medians** - Reduce impact of outliers - **Latest periods** - More weight to recent - **Excluding outliers** - Remove abnormal years ### Step 4: Calculate Cumulative LDFs ``` Cumulative LDF = Product of age-to-age factors Example: 12-24: 1.541 24-36: 1.150 36-48: 1.055 48-Ultimate: 1.030 12-Ultimate CDF = 1.541 × 1.150 × 1.055 × 1.030 = 1.928 ``` ### Step 5: Project Ultimate Losses ``` Ultimate = Latest Reported × CDF Example for AY 2023 @ 12 months: Reported: $600,000 12-Ult CDF: 1.928 Ultimate: $600,000 × 1.928 = $1,156,800 ``` ### Step 6: Calculate IBNR ``` IBNR = Ultimate - Reported Example: Ultimate: $1,156,800 Reported: $600,000 IBNR: $556,800 ``` ## Complete Example ``` Incurred Loss Development Triangle: Age (months) AY 12 24 36 48 60 2020 500 800 920 968 990 2021 550 825 979 1,028 2022 575 875 1,025 2023 600 900 2024 625 Age-to-Age Factors: 12-24 24-36 36-48 48-60 2020 1.600 1.150 1.052 1.023 2021 1.500 1.187 1.050 2022 1.522 1.171 2023 1.500 Avg: 1.531 1.169 1.051 1.023 Selected LDFs: 12-24: 1.500 24-36: 1.170 36-48: 1.050 48-60: 1.023 60-Ult: 1.010 (tail) Cumulative LDFs: 12-Ult: 1.500 × 1.170 × 1.050 × 1.023 × 1.010 = 1.896 24-Ult: 1.170 × 1.050 × 1.023 × 1.010 = 1.264 36-Ult: 1.050 × 1.023 × 1.010 = 1.084 48-Ult: 1.023 × 1.010 = 1.033 60-Ult: 1.010 Ultimate Projections: AY 2020: 990 × 1.010 = 1,000 AY 2021: 1,028 × 1.033 = 1,062 AY 2022: 1,025 × 1.084 = 1,111 AY 2023: 900 × 1.264 = 1,138 AY 2024: 625 × 1.896 = 1,185 Total Ultimate: 5,496 IBNR Reserve: Total Ultimate: 5,496 Total Reported: 4,603 IBNR: 893 ``` ## Advantages 1. **Simple** - Easy to understand and implement 2. **Objective** - Based solely on historical data 3. **Industry standard** - Widely accepted 4. **Flexible** - Can use various averaging methods ## Disadvantages 1. **Assumes stability** - Past patterns continue 2. **No external data** - Ignores known changes 3. **Volatile** - Can be affected by outliers 4. **Requires volume** - Needs sufficient data ## Key Assumptions ### Pattern Stability - Historical development patterns will continue - No systematic changes in reporting or settlement - Consistent case reserving practices ### Homogeneity - All years develop similarly - Mix of business is consistent - No significant operational changes ### Adequate Data - Sufficient volume for stable patterns - Mature enough to observe development - Representative of future experience ## Variations and Refinements ### Weighted vs Unweighted ``` Volume-weighted average: Factor = Σ(Loss_n+1) / Σ(Loss_n) Advantages: - More credibility to larger years - Reduces impact of small years - Better when volume varies significantly ``` ### Outlier Treatment - Exclude unusual years - Cap extreme factors - Use medians instead of means ### Trend Adjustments ``` Adjust historical losses for trend before developing: Trended Loss = Historical Loss × Trend Factor ``` ### Separate Tail Factor ``` Tail Factor accounts for development beyond triangle: - Based on industry data - Actuarial judgment - Curve fitting methods ``` ## Diagnostics and Testing ### Reasonableness Checks 1. **Factor progression** - Should generally decline 2. **Year-over-year consistency** - No wild swings 3. **Ultimate loss ratios** - Within expected range 4. **IBNR patterns** - Reasonable by accident year ### Sensitivity Testing - Try different averaging methods - Test various tail factors - Exclude/include questionable years - Compare to other methods ## When to Use **Appropriate for:** - Stable development patterns - Sufficient historical data - Homogeneous books of business - Standard reserving analyses **Less appropriate for:** - Known operational changes - Unstable patterns - Limited historical data - Rapidly evolving lines ## Related Concepts - [[Loss Development Triangle]] - [[Development Factor]] - [[Age-to-Age Factor]] - [[Cumulative Development Factor]] - [[IBNR Reserves]] - [[Tail Factor]] ## References - Friedland, Chapter 4 - Werner & Modlin, Chapter 6