# annualized volatility formula is an approximation?

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 $$\sqrt{PeriodLength}*StDev(Daily)$$ but the assumption behind this formula seems to be $$YearlyChange = \sum{DailyChange} $$ 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  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?

• Are you using discrete or log returns? – amdopt Sep 12 '19 at 14:09
• @amdopt I was thinking about discrete returns – SerhiiPoklonskyi Sep 12 '19 at 14:12
• That is the problem. The formula is exact only for logarithmic returns. – Alex C Sep 13 '19 at 15:24

As indicated by @AlexC and @amdopt, the formula is exact for log returns and approximate for discrete returns. Define the factor by which a price changes as $$k$$ so that price tomorrow $$P_{t+1}$$ is the price today times $$k$$ : $$P_{t}*k$$.Then the change in the price over a business year is $$\prod_{i \in [1, 252]}{k}$$ The log of the change is by properties of logarithms $$\sum_{i \in [1, 252]}{ln(k)}$$ and the formula for the variance then applies because the log returns are i.i.d. The daily change to use for this formula is necessarily the log change, not the discrete one
• Note the important caveat: log returns should be uncorrelated as only then $\text{Var}(X_1+\dots+X_n)=\text{Var}(X_1)+\dots+\text{Var}(X_n)$. (Serhii says log returns are i.i.d. This is a sufficient but not necessary condition.) – Richard Hardy Oct 20 '19 at 9:35
• @Richard thanks for correction, that's true. note, however, that, if the daily log returns are not identically distributed (independently distributed would be an overkill assumption as noted by @Richard), than formula $$ from the original question still does not apply because the daily volatilities are all different. moreover, they would be impossible to estimate. I would suggest that the assumption is: identically distributed uncorrelated daily log returns. or? – SerhiiPoklonskyi Oct 20 '19 at 9:55