Exponential Smoothing Methods for Actuarial Forecasting

Simple Exponential Smoothing

Simple exponential smoothing (SES) forecasts a level series (no trend, no seasonality) using a weighted average that gives exponentially decreasing weights to past observations: S_t = alpha*Y_t + (1-alpha)*S_{t-1}, where 0 < alpha < 1 is the smoothing parameter. The forecast for all future periods is S_T (the last smoothed value). Larger alpha gives more weight to recent data (responsive but noisy); smaller alpha gives smoother forecasts.

SES is equivalent to an ARIMA(0, 1, 1) model with theta = 1 - alpha. This connection between exponential smoothing and ARIMA models is testable on Exam SRM.

Holt's Linear Trend Method

For series with a trend, Holt's method uses two smoothing equations: one for the level (L_t = alpha*Y_t + (1-alpha)*(L_{t-1} + T_{t-1})) and one for the trend (T_t = beta*(L_t - L_{t-1}) + (1-beta)*T_{t-1}). The h-step ahead forecast is Y_hat_{T+h} = L_T + h*T_T. The parameters alpha and beta control how quickly the level and trend estimates adapt to new data.

Holt-Winters Seasonal Method

For series with both trend and seasonality, add a seasonal component. The additive method: Y_hat_{T+h} = L_T + h*T_T + S_{T+h-m}, where m is the seasonal period and S_t is the seasonal factor. The multiplicative method: Y_hat_{T+h} = (L_T + h*T_T) * S_{T+h-m}. Three smoothing parameters (alpha, beta, gamma) are needed. For Exam SRM, understand the difference between additive and multiplicative seasonality: additive when seasonal fluctuations are roughly constant in size, multiplicative when they are proportional to the level.