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.
Last edited by ACD
on February 20th, 2012, 11:00 pm, edited 1 time in total.