When Square-Root-of-Time Volatility Scaling Is Exact
Summary
This note clarifies the assumptions behind scaling daily volatility by the square root of the number of periods. The key distinction is between discrete percentage returns and log returns: discrete returns compound multiplicatively, so their annual change is not generally the sum of daily changes. Log returns add across time, making the variance of the aggregate equal to the sum of daily variances when those returns are uncorrelated.
Under constant daily standard deviation, this yields the familiar square-root-of-time relationship for log-return volatility. Identical distribution is sufficient but not required; lack of correlation and appropriate treatment of changing daily variance are the relevant conditions stated in the amendment. The rule is therefore exact under those log-return assumptions and only approximate when applied to discrete returns. The discussion gives a conceptual derivation, not empirical testing or guidance for serially dependent returns.
Key ideas
- Discrete returns compound, while log returns add across periods.
- Square-root-of-time scaling applies exactly to aggregate log-return volatility when returns are uncorrelated and variance is constant or properly weighted.
- Independent and identically distributed daily returns are sufficient but not necessary for the scaling condition.
- Applying the rule to discrete percentage returns is an approximation.
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Full text
# annualized volatility formula is an approximation?
# 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 $$\sqrt{PeriodLength}*StDev(Daily)[1]$$ 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?
## Answer by SerhiiPoklonskyi (score 4, accepted)
https://quant.stackexchange.com/a/48683
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.