# Tag Info

If you look at longer time returns (monthly or weekly as compared to daily) then these can be seen as the sum of daily (log-)returns: $$r^w = \sum r_1^d + \cdots r_5^d.$$ It is in general not true that the $r_i$ are iid because they are not independent. If they were then $r_i^2$ would be iid too and we know that volatility clusters. Even without ...
Sounds a little bit similar to variance ratio - the seminal paper of which was Lo, MacKinlay 1988, however that deals with variance, not differences of extrema. That you find "risk" narrows over time is odd, given that $Var[x + y] = Var[x] + Var[y]$ for independent variables $x \& y$, and one could look at days as being independent, and similarly ...