ewma calculation
TRANSCRIPT
Nicholas Bucheleres October 31, 2010 Exponentially Weighted Moving Average Calculation Step 1: Determine Time/Price Series Step 2: Calculate Periodic Returns
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xi = ln Ri
Ri−1
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⎝ ⎜
⎞
⎠ ⎟
Natural Log of (today’s returns over yesterday’s returns). Step 3: Average Squared Returns
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σ2 =1m
xn−12
i=1
m
∑
The average of the summation of squared returns over period ‘m’ returns the simple (non-weighted) moving average. Now we have the average of the squared returns (simple moving average). In order to exponentially weight this average, we are going to assign a decreasingly weighted “tail” to our price series. We are going to use the lambda coefficient to apply a proportionally weighted significance to (n-‐1) trailing time periods. The most recent periodic return receives a weight of (1-‐
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λ), which means that today’s squared return receives a weight of (1-‐x%) of the series. Industry convention dictates that
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λ=94%, so today receives a 6% weighting, yesterday receives a weighting of (6%)*(
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λ=94%), so 5.6%, and so on. Each period receives 94% of the weighting that one more recent does. This process can be reduced into one streamlined formula:
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n2
σ = λ n−12
σ + (1− λ) n−12x
The weighted squared returns of period ‘n=today’ equals (yesterday’s variance times the lambda coefficient) plus (1 minus lambda) times yesterday’s squared return. Note:
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λ + (1-‐
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λ) = 1