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iid process with infinite variance - can I scale VaR with sqrt(t)?

Posted: February 21st, 2012, 8:08 am
by MHill
Spinning off from another thread:Say I have a time series, iid, infinite variance. I come up with a 1 day VaR number. Can I scale this to a 10 day VaR number by the square root of time?If so, why?, If not, why not? Edit: bad spelling

iid process with infinite variance - can I scale VaR with sqrt(t)?

Posted: February 21st, 2012, 8:40 am
by ACD
Someone should check I haven't screwed something up here, but if we're assuming iid and infinite variance we can pick a distribution, the Cauchy distribution, that satisfies these conditions and see what happens. This distribution has pdf:Let's choose x0 as 0 over 1 day. This is a stable distribution so behaves nicely under addition. The characteristic function of it is (under our assumptions):So after adding n of these variables we have cf:Which is a Cauchy with the gamma now n times the original. Comparing the 1 day and n day 0.95 VaR using the inverse cdf we have:1 day 95% VaR :n day 95% VaR :Which is n times more, so VaR in this case is scaling linearly with time.As to why, the sqrt(t) rule applies to standard deviation not to the quantile of a distribution (can be used on a normal rv but not usually other distributions) and also since variance (standard deviation) is infinite in this case scaling it doesn't have any meaning.

iid process with infinite variance - can I scale VaR with sqrt(t)?

Posted: February 21st, 2012, 8:25 pm
by tagoma
QuoteOriginally posted by: MHillSpinning off from another thread:Say I have a time series, iid, infinite variance. I come up with a 1 day VaR number. Can I scale this to a 10 day VaR number by the square root of time?If so, why?, If not, why not? Edit: bad spellingmy 2-centthe square root of time root applies when the returns are iid and normally distributednormal distribution has a sigma^2 variance, as you might know .this is not the distribution you describe. if you drew your VaR from historical simulation, VaR might correspond to a quantile of some unspecified empirical distribution , the quantiles of which don't obey the square root of time rule (C.Alexander).

iid process with infinite variance - can I scale VaR with sqrt(t)?

Posted: February 22nd, 2012, 5:28 pm
by MHill
Thanks guys - good start.I suppose I'm looking for a proof or a contradiction. I don't think the empirical distribution would count as an infinite variance iid process.I suppose getting suggestions of infinite variance iid processes would be useful, as we could then try & work out the validity of sqrt(t) rules on them.Any further suggestions?

iid process with infinite variance - can I scale VaR with sqrt(t)?

Posted: February 23rd, 2012, 9:42 am
by ACD
I think you have everything you need then. The Cauchy distribution is the building block for the Cauchy process, it is an iid and infinite variance process. By the argument below, you cannot scale VaR by sqrt(t). You have a proof from this that it's not valid to do this in general because you have a counter example.To be honest it's only really valid to do it with a normal distribution if you assume a zero mean (which most of us do over 1 day horizons), change that to a non-zero value and VaR will not scale with sqrt(t) even with a normal distribution (the rule hinges on VaR being determined solely by variance and nothing else).

iid process with infinite variance - can I scale VaR with sqrt(t)?

Posted: February 23rd, 2012, 10:23 am
by MHill
QuoteOriginally posted by: ACDI think you have everything you need then. The Cauchy distribution is the building block for the Cauchy process, it is an iid and infinite variance process. By the argument below, you cannot scale VaR by sqrt(t). You have a proof from this that it's not valid to do this in general because you have a counter example.D'oh!Was too busy multi-tasking to read your first post properly!Now I have egg on my face as well as baby-food on my suit!Cheers ACD - very helpful!

iid process with infinite variance - can I scale VaR with sqrt(t)?

Posted: February 23rd, 2012, 10:36 am
by ACD
No problem