suddenly having troubles with the annualized volatility formula... is it really an approximation?
one usually writes the standard deviation of the yearly percentage change in the stock price as $$\sqrt{PeriodLength}*StDev(Daily)[1]$$ but the assumption behind this formula seems to be $$YearlyChange = \sum{DailyChange} [2]$$ and hence $$Var(YearlyChange) = Var(\sum{DailyChange}) = \sum{Var(DailyChange)}$$ and so on, under the assumption that daily change is i.i.d, one arrives at the formula.
But the formula [2] is clearly NOT true? because the yearly percentage change is not the sum of daily percentage changes but rather the accumulated percentage change? am I being blind and missing something?