by Allan Vasenius

The best way to understand exponential moving averages (EMA) is to examine how they are calculated. Mathematically, EMAs are said to be recursive which simply means, that an EMA value depends in part on the previous value. Given EMA Mi, we wish to determine the next value Mi+1. The formula is:

Mi+1 = Mi + α (Xi+1 – Mi)

Xi+1 is the price or other value that serves as the basis for the EMA.

α is a constant less than 1 and is normally a fairly small value. Shortly, we will see how to determine an appropriate value for α.

So, let’s make sure that we see what is happening with the above formula. If the new price Xi+1 is the same as Mi, then no change takes place in our EMA. If price is greater, then a fraction of that price determined by α is added to the EMA. Correspondingly, if the price is less than the EMA, the new EMA value will decrease.

So, how do we determine the right value to use for α? The formula for α is simply:

α = 2 /(N + 1)

The symbol N represents the number period value. For instance, if we want the 9-period value for α, our formula indicates that it has a value of 0.2 = 2/(9 + 1).

We have one final detail to cover and that is how we calculate the first EMA value. The initial value is simply the initial price or X value.

EMAs are popular because the more recent values are weighted more heavily than the current value. With each update, all previous values are in effect multiplied by the quantity (1-α). So, the significance of a particular X value continues to become less and less important. Now you should see why an EMA is also known as a fading filter because the significance of all of the previous updates fades away at each update by a factor of (1-α). Note also that an EMA can only rise when price is above it and can only decline when price is below.

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