[[Actuarial Notes Wiki|Wiki]] / [[Exam 5 (CAS)]] / **Loss Development Triangle** ## Definition ==Loss Development Triangle== A loss development triangle is a two-dimensional array that displays cumulative loss data by accident year (or policy year) and development age, used to identify patterns in how losses emerge and develop over time. ## Structure ### Rows (Accident Years) Each row represents losses for a specific accident year: - AY 2020 - AY 2021 - AY 2022 - AY 2023 - AY 2024 ### Columns (Development Ages) Each column represents age of losses in months (or years): - 12 months - 24 months - 36 months - 48 months - 60 months ### Values (Cumulative Losses) Each cell contains cumulative losses as of that age: - Includes all paid and case reserves - Cumulative from accident date - Values generally increase left to right ## Example Triangle ``` Cumulative Incurred Losses ($000s) Development Age (months) AY 12 24 36 48 60 72 2019 500 750 875 925 950 960 2020 525 800 925 975 1,000 2021 550 825 975 1,025 2022 575 875 1,025 2023 600 900 2024 625 Latest Diagonal: Current evaluation of all years ``` ## Purpose and Uses ### Reserve Estimation - Project ultimate losses - Calculate IBNR - Estimate unpaid claims - Support financial statements ### Trend Analysis - Identify development patterns - Detect changes in reporting/settlement - Monitor case reserve adequacy - Assess operational changes ### Diagnostic Tool - Identify anomalies - Detect systematic changes - Validate assumptions - Support method selection ## Creating a Triangle ### Step 1: Organize Data ``` Extract data by: - Accident year - Evaluation date (valuation date) - Loss value (paid, incurred, case, etc.) ``` ### Step 2: Calculate Age ``` Development Age = Evaluation Date - Accident Year Midpoint Example: Evaluation: 12/31/2024 AY 2023 midpoint: 7/1/2023 Age: 18 months ``` ### Step 3: Populate Triangle - Place each value in appropriate cell - Verify cumulative nature - Check for reasonableness ### Step 4: Complete Latest Diagonal ``` Latest diagonal shows most recent evaluation: - Oldest years: Most developed - Recent years: Less developed - Current year: Least developed ``` ## Development Patterns ### Age-to-Age Factors ``` Link Ratio = Value at age (n+1) / Value at age n Example: 24-month value: $800 12-month value: $500 12-24 factor: $800/$500 = 1.600 ``` ### Cumulative Development Factors ``` CDF = Ultimate Value / Value at age n Example: Ultimate: $1,000 24-month value: $800 24-ultimate CDF: $1,000/$800 = 1.250 ``` ## Types of Triangles ### Incurred Loss Triangle - Paid losses + Case reserves - Most common for reserving - Reflects both payments and reserves ### Paid Loss Triangle - Only actual cash payments - Less volatile than incurred - Slower development ### Case Reserve Triangle - Outstanding case reserves only - Shows reserve adequacy trends - Useful diagnostic ### Claim Count Triangle - Number of claims - Used for frequency analysis - Combined with loss triangles ### ALAE Triangle - Allocated loss adjustment expenses - Often developed separately - Different patterns than losses ## Special Considerations ### Tail Development ``` Losses continue to develop beyond triangle: - Long-tail lines need tail factor - Short-tail lines may be fully developed - Actuarial judgment required ``` ### Large Losses - Can distort patterns - May need separate treatment - Consider capping or excluding ### Changing Environment - Rate changes affect reported values - Mix changes affect patterns - Calendar year effects - Operational changes ## Diagnostic Analysis ### Review Patterns ``` Check for: 1. Smoothness - Are factors relatively stable? 2. Trends - Are factors increasing/decreasing? 3. Outliers - Any unusual values? 4. Diagonal effects - Consistent evaluation dates ``` ### Common Issues - Negative development - Erratic patterns - Unexplained jumps - Missing data ## Related Concepts - [[Development Factor]] - [[Age-to-Age Factor]] - [[Chain Ladder Method]] - [[IBNR Reserves]] - [[Tail Factor]] - [[Cumulative Development Factor]] ## References - Friedland, Chapters 2-4 - Werner & Modlin, Chapter 6