[[Actuarial Notes Wiki|Wiki]] / [[Exam 5 (CAS)]] / **Exponential Trend** ## Definition ==Exponential Trend== (also called compound or geometric trend) is a trending method where values grow by a constant percentage each period, with growth compounding over time. ## Formula ``` Future Value = Historical Value × (1 + r)^n Where: r = annual trend rate (as decimal) n = number of years ``` ## Characteristics - Growth compounds each period - Percentage increase is constant - Dollar increase accelerates over time - Most common method in insurance ## Example ``` Historical loss: $1,000,000 Annual trend rate: 5% Trend period: 3 years Future value = $1,000,000 × (1.05)^3 = $1,000,000 × 1.157625 = $1,157,625 Year-by-year: Year 1: $1,000,000 × 1.05 = $1,050,000 Year 2: $1,050,000 × 1.05 = $1,102,500 Year 3: $1,102,500 × 1.05 = $1,157,625 ``` ## When to Use **Preferred for:** - Inflation-driven trends - Medical cost inflation - Wage inflation - Most insurance applications - Percentage changes constant ## Comparison to Linear ``` Exponential: Growth compounds Linear: Growth is additive For $1,000 base, 5% trend, 10 years: Exponential: $1,000 × 1.05^10 = $1,629 Linear: $1,000 × (1 + 0.05×10) = $1,500 Exponential grows faster long-term ``` ## Related Concepts - [[Linear Trend#Definition]] - [[Trend#Definition]] - [[Loss Trend#Definition]] ## References - Werner & Modlin, Chapter 8 - ASOP 13