Scaling volatility over time: data frequency and the root-of-time law

Why does data frequency change the numbers?

The longer the gap between observations, the more a price can move in between. So returns measured over longer periods are larger, and so are their variance and standard deviation. Daily returns are small; annual returns are large.

Volatility is the everyday word for the standard deviation of returns. Data frequency is how often you observe the series: daily, weekly, monthly, yearly. Match it to the question: a trading desk managing overnight risk needs daily figures; a pension fund planning its contributions needs annual ones. Very frequent data carry many small moves that may be noise rather than a lasting change in value.

Illustrative figures for an equity index, not market data:

Frequency	Trading days per period	Average return	Standard deviation
Daily	1	0.03%	1.00%
Weekly	5	0.15%	2.15%
Monthly	22	0.66%	4.40%
Annual	252	7.6%	14.2%

Same asset, four very different “volatilities”. None is wrong; each answers a different horizon. The table returns in a later section.

What is the root-of-time law?

If each period’s return adds a similar, independent amount of uncertainty, the variance over T periods is T times the one-period variance, so the standard deviation is √T times the one-period figure. This is the root-of-time law .

$$\sigma^2_T \approx T \times \sigma^2_1$$

$$\sigma_T \approx \sqrt{T} \times \sigma_1$$

Here σ₁ is the standard deviation over one short period (say a day) and T is the number of short periods in the long one. This variance scaling is what makes the standard deviation grow only with √T. Use trading days , not calendar days: about 22 in a month and 252 in a year (some sources use 250). Five trading days make a week; 12 months a year; 52 weeks a year.

Exam tip Four times the horizon, twice the volatility: √4 = 2. A year of 252 days has about 15.87 times the daily volatility, not 252 times.

Returns are annualized by compounding; volatility is scaled with the square root of time. Two different tools for two different numbers (see annualized returns).

Return over T short periodsT = number of trading days or months in the horizon

• Variance Memorize × T
• Standard deviation Memorize × √T
• Mean Know the idea roughly × T
• Trading days Memorize about 22 a month, 252 a year

How do you go from a long horizon back to a short one?

Divide by the same square root.

$$\sigma_1 \approx \dfrac{\sigma_T}{\sqrt{T}}$$

Going the other way, monthly to annual: 4.0% × √12 = 13.86%.

Do real returns follow the law?

Only roughly. In practice the variance of longer-horizon returns often grows more slowly than the law predicts, so a scaled-up daily figure tends to overstate the volatility you actually observe over a year, and a scaled-down annual figure understates daily volatility.

Longer-horizon variance often grows more slowly than T × short-horizon varianceso scaled-up figures tend to overstate

• Market frictions trading costs, prices that do not update instantly
• Changing market behavior economic cycles, sentiment
• Jumps sudden large price changes
• Fat tails leptokurtosis

Back to the table: daily 1.00% scaled up gives 1.00% × √22 = 4.69% a month and 1.00% × √252 = 15.87% a year, against 4.40% and 14.2% observed. Going the other way, 14.2% ÷ √252 = 0.89% a day, about 10% below the 1.00% observed.

Treat a scaled figure as an estimate. If you need annual risk and you have annual data, measure it directly as well.

Why compute statistics on returns, not prices?

Price levels are usually non-stationary : their mean and variance drift over time, so statistics computed on them describe no stable distribution. Returns, monthly or more frequent, are much closer to stationary : their mean and variance stay broadly the same over time, which is what the formulas assume.

An index that rose from 1,000 to 4,000 over twenty years has a meaningless “average level”; its monthly returns have a mean and standard deviation you can compare across decades. Returns, being percentages, also compare across assets, periods and currencies, where price levels cannot.

What happens to the coefficient of variation?

Know the idea

Over T periods the mean grows roughly in line with T but the standard deviation only with √T, so the coefficient of variation (CV) of daily data is far larger than an annual one. Annualize both before comparing (lesson 4).

From the table: the daily CV is 1.00 ÷ 0.03 = 33.3. Annualized, the mean is 0.03% × 252 = 7.56%, the standard deviation is 1.00% × √252 = 15.87%, and the CV is 15.87 ÷ 7.56 = 2.10 (33.3 ÷ √252 gives the same).

Go deeper Why the square root?

If one day’s return tells you nothing about the next, the variance of the sum of two days’ returns is the sum of their variances. Over T similar days the variance is T times one day’s, and the standard deviation √T times. Continuously compounded returns add across periods exactly (Learning Module 1), which is why this reasoning is usually applied to them. This is background, not needed for the exam.