Estimating volatilities and correlation--unweighting variance

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Hi David, please help me with these two questions;

Q1) when calculating the periodic retuns which is an input for the calculating the variance rate per day,

Linda Allen used Ui=Ln(Si/Si-1) while John Hull also used Ui=(Si-Si-1/Si)

John Hulls formular is for short periods, BUT a short period may relative to whoever is pursuing this calculation. is short period, a period within a year or two years or...........?

i would appreciate it if you clarify this for me and state when to use Linda Allens formular.


Q2) when it comes to unweighte scheme we have 2 formulars to assist to pursue this calculation;

a) Var(n)=1/m-1 zigma (Ui-mean returns)exp2-----unbiased estimator(denominator)

John Hull made two adjusment to the above formula; m-1=m and mean returns=0

b) Var(n)=1/m zigma (Ui)exp2---maximum liklihood estimator

in the exams when we are asked to calculate the variance rate per day and the question is silent on whether to use the maximum likelihood estimator or unbiased estimator, what do we do, since all the inputs in the formular can be given in the question?

thanks

 

[David Harper CFA FRM]

Hi baffour,

Q1)
We've corresponded recently with GARP (using other forum threads) on this very point and GARP has agreed that they should explicitly indicate whether continuous log returns (aka, geometric) or simple (aka, arithmetic). I don't think this choice is a function of short periods; rather, it is just that for short periods (daily), the difference is not material. The choice tends to concern our preference with respect to adding returns: 1. do we want the returns to add over time (then log returns per time additivity) or 2. do we want to add returns for positions "cross-sectionally" across the portfolio (then simple arithmetic).

Short periods (i.e., where Hull makes his simplifications) tends to refer to ONE DAY or LESS; and, if daily, the choice doesn't make a big difference.

But GARP is well aware that they need to distinguish)

Q2)
If they do not indicate, you are generally fine to follow Hull's simplification and use your simpler MLE; your (b). Especially if daily, it is common to use the MLE and treat the variance as "the average of the squared returns" rather than go to the trouble of including the mean and dividing by (m-1). So I would ASSUME the simpler MLE estimate unless told otherwise

I hope that helps, Thanks, David