EWMA

[limhuisi]

Hi David,



I am referring to your spreadsheet EWMA and I have done another EWMA calculation spreadsheet by using John Hull's example (my department bible...) however, by using the same input and parameter, I find that the result varies by around 0.05%. I am not sure what went wrong, but can you please have a look, just to make sure I have did it correctly?



Additional query:



1. What is the difference between annualised vol and 250 days vol?

2. When do we use daily vol * sqrt (250)?

3. By starting with ln(row 252/ row 1), then apply the EWMA formula, what do we call the outcome? Annualised vol or 250 moving window vol? What is the difference between this two?



Thank you very much in advance.



Regards,

Hui Si

 

[David Harper CFA FRM]

Hi Hui,

Did you mean to attach an XLS?

If you used the recursive EWMA, there can be no difference: lambda(prev var) + (1-lambda)(prev return^2).
As this is the solution to an infinite series, tiny differences typically arise when comparing to "manual" series; i.e.,

1st weight = (1-lambda)
2nd weight = 1st weight*lambda
3rd weight = 2nd weight*lambda

In an XLS, a finite such series will sum to weights < 100%. In my XLS you may note *below* i have another version of weights that do sum to 100%. With my params, the difference is not noticeable. But, strickly speaking, as Linda Allen (FRM assigned) shows, it is proper to take the extra tiny bit of weight (not captured above) and redistributed it into the finite series. So:

1st weight = (1-lambda)/(1-lambda^Number in finite series)
and so on...

The latter weight series, being ever so slightly, proportionately increased by the "gap" will sum to exactly 100%. IMO, this typically explains perceived tiny differences, but i'll take a look if you upload the xls

1. It's funny how sometimes the basics lack consistent notation! To me, annualized vol implies we have scaled the units into annual; e.g., daily vol * SQRT[250] = annualized volatility (which, for some, "volatility" already implies annualized standard deviation). And 250 days vol is where the measurement PERIOD is one year; e.g., 250-day Vol = LN(Price at end of 2007/Price at end of 2006). I like to refer to the "periodicity" to distinguish between the measurement interval (i.e., are we measuring price changes at one day or one year) and the horizon...

2. This is relevant to FRM candidate (see AIMS for Linda Allen) - we do this to annualize a daily volatility per the square root rule BUT only under unrealistic condition of i.i.d. (independent with constant variance). Empirically, as the process is typically not i.i.d., we are typically abusing the square root rule

3. I think i need the XLS for this?

David