# Exponential Moving Average Filters

Exponential moving average filters (EMA) are another common type of filter used to filter price data for stocks, bonds and other assets. EMAs are infinite impulse response filters where the current filtered output is dependent upon all of the previous price values. With the EMA filter the impact of earlier data on the current filtered value falls off at an exponential rate. The exponential decay of the impact of earlier data may be fast or slow and is characterized by the smoothing factor, alpha.

PEMA(n) = *alpha*P(n) + (1-*alpha*)PEMA(n-1) where *alpha* is the smoothing factor and 0 < *alpha* < 1

The Price filtering action, or frequency response, of the exponential moving average filter with alpha of 0.125, 0.25 and 0.5 is shown in the figure below. The sample frequency is once per month (fs=1/month). The 3dB corner frequency for alpha = 0.125fs is 0.0215fs or 46.5 months. The 3dB corner frequency for alpha = 0.25fs is 0.047fs or 21.3 months. The 3dB corner frequency for alpha = 0.5fs is 0.115fs or 46.5 months.

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