[[Actuarial Notes Wiki|Wiki]] / [[Exam 5 (CAS)]] / **Linear Trend** ## Definition ==Linear Trend== (also called additive trend) is a trending method where values grow by a constant dollar amount each period, with growth not compounding. ## Formula ``` Future Value = Historical Value × (1 + r × n) Where: r = annual trend rate (as decimal) n = number of years ``` ## Characteristics - Growth is additive, not compound - Dollar increase is constant each period - Percentage increase declines over time - Less common than exponential ## Example ``` Historical loss: $1,000,000 Annual trend rate: 5% Trend period: 3 years Future value = $1,000,000 × (1 + 0.05 × 3) = $1,000,000 × 1.15 = $1,150,000 Year-by-year (constant $50,000 increase): Year 1: $1,000,000 + $50,000 = $1,050,000 Year 2: $1,050,000 + $50,000 = $1,100,000 Year 3: $1,100,000 + $50,000 = $1,150,000 ``` ## When to Use **May be appropriate for:** - Short trend periods - Flat inflation environment - Legislative changes (fixed amount) - Minimum premium increases **Less appropriate for:** - Long-term trending - Inflationary environments - Most insurance applications ## Comparison to Exponential ``` For $1,000 base, 5% trend, 3 years: Exponential: $1,000 × 1.05^3 = $1,158 Linear: $1,000 × (1 + 0.05×3) = $1,150 Difference: $8 (0.7%) Difference grows with longer periods ``` ## Related Concepts - [[Exponential Trend#Definition]] - [[Trend#Definition]] ## References - Werner & Modlin, Chapter 8