Question:

We are going to use the exponentially weighted moving average (EWMA) model. Assume lambda (i.e., the decay factor) is 90%, that yesterday’s (day n-1) volatility was 2.0% and that yesterday’s daily return was +3.0% (periodic return for day n-1).

(i) What is today’s volatility estimate (day n) under the exponentially weighted moving average (EWMA)?
(ii) If the decay factor is 90% (lambda = 0.9), what weight is effectively assigned to the most recent return (return-squared)?
(iii) If the decay factor is 90% (lambda = 0.9), what weight is effectively assigned to day n-5?
(iv) What is effectively the persistence parameter for the EWMA model?

Answer:

Spreadsheet workout here

(i)
Variance (EWMA,n) = (lambda)(previous variance) + (1-lambda)(previous squared return)
Variance (EWMA,n) = (90%)(2%^2) + (10%)(3%^2) = 0.00045
Volatility (EWMA,n) = SQRT[0.00045] = 2.12%

(ii)
The previous squared return is assigned a weight of (1-lambda) or 10%, where lambda is 90%
Note: the 90% weight assigned to the previous variance is the recursive feature

(iii)
The weights decline in constant proportion; lambda is the proportion.
The weight for each squared return is given by:
(1-lambda)(lambda^[t-1]) where t refers to the number of days prior.
The first, most recent weight is therefore given by:
(1-lambda)(lambda^[1-1]) = (1-lambda)
The weight assigned on t-2 is given by:
(1-lambda)(lambda^[2-1]) = (1-lambda)(lambda)

And for for (t-5) or (n-5), therefore, the weight is given by:
(10%)(90%^4) = 6.56%

(iv) EWMA does not incorporate mean reversion. It’s persistence is unity (1.0)

Hi David

Can you verify the third part of this question, my understanding is the weight for n-5 should be
(10%)(90%^4) = 6.5% instead of (10%)(90%^5) = 5.9%

Since when n-1 the power of lambda will be zero.

Pleas let me know if I am interpreting it incorrectly.

Thanks for your help on this.

Regards
Neha

Absolutely you are right, my answer lazily plugged in 5.
I corrected it above. Thank you!

David

Hi David,

I can`t workout the first question. Since the decay factor is 90%, answer should be something like that:

Variance n= 90% * Variance n-1 + (1-90%)* yesterday’s squared return.

Please let me kwow if I am wrong.

Ciao and congratulations on the superb web.

Hi padrino,

While the answer is correct, I indeed mucked up the explanation. You are absolutely correct in your calculation: lambda (decay factor) multiplied by previous variance. It is corrected above.

Just to check, I put the calc into a worksheet (below). And thank you very much for your kind feedback...David

Spreadsheet workout