Question on Editgrid - Parametric Volatilities

[papillonring]

Hi David

I was using the same example in your editgrid to calculate the Moving Average on my own spreadsheet. I learn that you have derived Moving Average Variance (also called Historical Variance) by take the average of the squared returns.

Hence the formula is
(sum of u^2) / 10

In the course slide, the formula provided by you is
(sum of u^2) / m

So my question is, what is the value of m. I would have thought it is 11.

Hence, to calculate Sample Variance, its
(sum of u^2) / (m-1)
which also = (sum of u^2) / 10

 

[David Harper CFA FRM]

Hi Papillonring

Good question (2008 FRM dropped Hull CH 19 for volatility, which is better than Allen on this)

m = number of observations. In the spreadsheet, 10 datapoints, so not 11 but rather two choices:

(sum of u^2)/(m=10) or
(sum of u^2)/(m-1=9) or

But please note in the deck I am careful to back into Jorion's moving average (MA) by showing it comes from two practical simplifications. Start with the true unbiased variance (and this matters b/c it relates to the assigned Gujarati):

sample variance = (sum of [u - average u]^2)/(m-1)

When computing historical variance/volatility, this complete calc can never be wrong. Then, in the XLS, see i added that we then make TWO PRACTICAL SIMPLIFYING assumptions:
"Simplifies because (i) we assume average return is zero and (ii) we divide sum of squared returns by n instead of (n-1)"
We are okay to do this for daily observations.

So, those two simplifying get us to:
(sum of u^2)/(m)


So, given 10 datapoints, dividing by 11 is wrong, but 9 or 10 would be okay.
To divide by 11 would shrink the sample variance which is the opposite of greater uncertainty.

For exam purposes, we want to distinguish between POPULATION VARIANCE...
population variance = (sum of [u - average u]^2)/(m)

...and SAMPLE VARIANCE
sample variance = (sum of [u - average u]^2)/(m-1)

conceptually, even if we don't really expect to ever take a population variance of a financial series (we are always sampling)

David