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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

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?

Solution

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