modelling risk distributions - garch

[aditya]

dear david,

thanx for making all the complicated problems so easy to understand which really saves our time.........

plz tell which of the statement is correct between the two:

watever i have read till now nothing has been said about mean return in theory of garch

The GARCH can produce fat tails in the return distribution.
The GARCH imposes a positive conditional mean return.
thanx
adi

 

[David Harper CFA FRM]

Hi adi,

sure thing. You recollection is good here, the second is false; the mean is not restricted (an asset can be consistently dropping in value--negative mean return--and still fit GARCH. It will give positive variance; variance/volatility must always be positive but not mean).

The first is arguably a bit imprecise (I'd be surprised to see this phrasing on the exam). It wants to be true, as GARCH() is considered a "fat tail" process. The conditional in GARCH(1,1) is why it helps to internalize conditional versus unconditional in Gujarati, because in GARCH(1,1):

* the conditional return is normally distributed, and (however)
* the unconditional distribution of returns is fat-tailed; i.e., kurtosis > 3

Both are true because, as Linda Allen says, the volatility (under GARCH, e.g.) is time-varying. It may be a lame metaphor, but imagine a lawnmower with wobbly wheels. The wheels wobble with a normal distribution. Now push the lawnmower forward but swerve, left then right etc. When you are done and you look back at the path you've cut (looking over your whole path is an "unconditional" look), it will appear to have fat tails due to the left/right movement. Put another way, if you measure your deviations, you will find non-normal extremes (created by a normal wheels) due to the swerving. Similarly, the normal return is fluctuating thru time, and this creates unconditional fat tails. Hope this helps...

David