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EWMA Model

  • In an exponentially weighted moving average model, the weights assigned to the u2 decline exponentially as we move back through time
  • This leads to:
  • Apply the recursive relationship:

  • Hence we have

  • Variance estimate for next day (n) is given by (1-λ) weight to recent squared return and λ weight to the previous variance estimate
  • Risk-metrics (by JP Morgan) assumes a Lambda of 0.94
  • Since returns are squared, their direction is not considered. Only the magnitude is considered
  • In EWMA, we simply need to store 2 data points: latest return & latest volatility estimate
  • Consider the equation:
  • In this equation, variance for time ‘t’ was also an estimate. So we can substitute for it as follows:

  • What are the weights for old returns and variance?
  • λ is called ‘Persistence factor’ or even “Decay Factor”. Higher λ gives more weight to older data (impact of older data is allowed to persist). Lower λ gives higher weight to recent data (i.e. previous data impacts are not allowed to persist)
  • Higher λ means higher persistence or lower decay
  • Since, (1- λ) is weight given to latest square return, it is called ‘Reactive factor’
  • Example 1: On Tuesday, return on a stock was 4%.  Volatility (Std. deviation) estimate for Tuesday was 1%. Find volatility estimate for Wednesday using λ of 0.94
    • Variance estimate for Wednesday = (1-0.94)*(4%)^2 +(0.94)*(1%)^2 = 1.9%2
    • Std. Dev. = sqrt (1.9%2)=1.378%
    • Tuesday volatility (Std. Dev.) estimate was 1%. Actual return on Tuesday was 4%. Therefore, volatility estimate for Wednesday is estimated upwards than Tuesday i.e. 1.378% as compared to 1%.
  • Notice how the volatility estimate has been revised due to high return.



Example 1: Continuing the previous example, volatility estimate for Wednesday was 1.378%. Assume that actual return on Wednesday was 0%. What is the variance estimate for Thursday?


Variance estimate for Thursday = (1-0.94)*(0%)^2 + 0.94*(1.378%)^2 =  1.78%2 Std. Dev. = 1.34%
In very short-term like daily returns, estimated volatility is the expected return
Since latest return of 0% was lesser than estimated volatility (and estimated return) of 1.378%, volatility for next day is revised downward from 1.378% to 1.34%
Notice the downward revision in the estimate due to lower return

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