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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?

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  • $\begingroup$ Are you using discrete or log returns? $\endgroup$
    – amdopt
    Sep 12, 2019 at 14:09
  • $\begingroup$ @amdopt I was thinking about discrete returns $\endgroup$
    – SerhiiPoklonskyi
    Sep 12, 2019 at 14:12
  • $\begingroup$ That is the problem. The formula is exact only for logarithmic returns. $\endgroup$
    – Alex C
    Sep 13, 2019 at 15:24

1 Answer 1

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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

AMENDMENT 2019.20.10

As added by @Richard, it is necessary and sufficient for the annualization formula that the log returns are linearly independent (uncorrelated) and the standard deviation of the daily returns is either assumed to be constant or must be apropriately weighted. Independent and identical distribution is a sufficient, but not necessary, assumption.

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    $\begingroup$ 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.) $\endgroup$
    – Richard Hardy
    Oct 20, 2019 at 9:35
  • $\begingroup$ @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 $[1]$ 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? $\endgroup$
    – SerhiiPoklonskyi
    Oct 20, 2019 at 9:55
  • $\begingroup$ I might have posted my comment too quickly without thinking much. Uncorrelatedness and constant variance should be enough for the annualization to work. I doubt identical distribution is needed. Constant variance might not be strictly necessary if an appropriately weighted daily standard deviation is chosen in the [1] formula. $\endgroup$
    – Richard Hardy
    Oct 20, 2019 at 10:06
  • $\begingroup$ @Richard thanks, I've amended. Hope, my formulation is correct $\endgroup$
    – SerhiiPoklonskyi
    Oct 20, 2019 at 11:24
  • $\begingroup$ Your answer was actually fine given that the OP specified i.i.d.'ness. So instead of trying to correct you (you were not wrong), I just wanted to add a comment for those interested in the non-i.i.d. case. Your edit says the variance should be averaged appropriately, while I said standard deviation (which is multiplied by the square root of time). $\endgroup$
    – Richard Hardy
    Oct 20, 2019 at 11:36

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