Volatility Is Not Risk, and the Square Root of Time Is a Choice

Where the square root actually comes from

Volatility means the standard deviation of returns, but a standard deviation only exists relative to a period. A daily standard deviation and a monthly one are different numbers describing the same asset, so to compare anything you have to agree on a period, and the convention is one year. The bridge is: annualised volatility = period standard deviation × √(periods per year).

The derivation is two lines. The annual return is the sum of the period returns — in log terms, exactly. The variance of a sum of independent random variables is the sum of their variances, so if there are n periods in a year and each has variance σ², the annual variance is nσ². Take the square root and the annual standard deviation is σ√n. That is the whole thing. For daily data on an asset that trades every day, √365 = 19.105; for a market with 252 trading days, √252 = 15.875; weekly gives √52 = 7.211; monthly gives √12 = 3.464; quarterly gives exactly 2.

Notice what had to be assumed to get there: that consecutive returns are independent, so that the covariance terms in the variance of the sum all vanish. Nothing in the data guarantees that. The square root of time is a consequence of a model of how prices move — independent increments — and when the model is wrong the factor is wrong with it. That is the difference between a piece of arithmetic and a modelling choice, and almost every volatility figure you will read is quoted as if it were the first.

How much the answer moves when returns are not independent

Keep the assumption of a constant per-period variance but allow the returns to be serially correlated with coefficient ρ that decays geometrically. The variance of an n-period sum is then σ²·[n + 2·Σₖ₌₁^(n−1)(n − k)·ρᵏ] instead of nσ². Divide by nσ² and you get a variance ratio that tells you exactly how wrong the naive scaling is.

Take an asset whose daily standard deviation is 3.141 percent — exactly the figure that annualises to 60.0 percent under the independence assumption with 365 days. If the daily autocorrelation is 0.05, the honest annualised volatility is 63.1 percent and the published figure understates by 4.9 percent. At 0.1 it is 66.3 percent, an understatement of 9.5. At 0.2 it is 73.4 percent, an understatement of 18.3. Trending, momentum-driven markets sit on this side of the line, and the standard formula tells you they are calmer than they are.

The correction runs the other way for mean-reverting series. At a daily autocorrelation of −0.1 the honest annualised figure is 54.3 percent rather than 60.0, and at −0.2 it is 49.0: the standard formula overstates by 10.5 and 22.4 percent respectively. Neither direction is a small effect, and neither is visible in a volatility number quoted on its own. Whenever a volatility figure matters — sizing a position, pricing an option, comparing two assets — the autocorrelation of the underlying return series is worth measuring before the √t factor is applied.

Log returns, simple returns, and why the formula wants the first

A simple return is (P₁ − P₀) ÷ P₀. A log return is ln(P₁ ÷ P₀). The derivation above required the annual return to be the sum of the period returns, and that is true of log returns and false of simple ones. Two months of +50 percent and −50 percent sum to zero as simple returns; the actual outcome is 1.50 × 0.50 − 1 = −25 percent. As log returns they sum to ln(1.5) + ln(0.5) = −28.768 percent, and exp(−0.28768) − 1 = −25 percent exactly. Only one of the two conventions has the additivity the variance argument depends on.

For small moves the two coincide, which is why the choice is usually invisible. A 1 percent simple return is a 0.995 percent log return, a difference of 0.005 points. At 5 percent the gap is 0.121 points, at 10 percent it is 0.469, at 20 percent it is 1.768 and at 50 percent it is 9.453. Losses diverge too, and asymmetrically: a −20 percent simple return is a −22.314 percent log return, and a −50 percent simple return is −69.315 percent in logs. On a quiet index the convention hardly matters. On an asset that regularly moves 10 or 20 percent in a day, the volatility computed on simple returns and the volatility computed on log returns are different numbers, and only one of them belongs in the √t formula.

There is a second, quieter dependency: the sampling frequency you chose. An asset with a 60 percent annualised volatility has a daily standard deviation of 3.141 percent, a weekly one of 8.321 percent and a monthly one of 17.321 percent. Those are the same volatility expressed three ways, so long as the independence assumption holds — and when it does not hold, they annualise to three different numbers. Sampling hourly rather than daily on the same asset picks up microstructure noise that is not part of the risk anyone cares about, and √8760 = 93.595 magnifies it. Choosing a frequency is choosing what counts as a price movement.

Volatility drag: why a positive average return can still lose money

This is the part with money in it. The average of a set of returns and the rate at which money actually compounds through them are two different quantities, and the second is always the smaller. The cleanest demonstration is two periods: +50 percent then −50 percent. The arithmetic mean is zero. The compound outcome is 1.5 × 0.5 = 0.75, a loss of 25 percent, and the per-period geometric mean is √0.75 − 1 = −13.397 percent. Nothing went wrong; that is what averaging does when the quantities multiply rather than add.

The size of the gap is governed by the volatility. For symmetric moves of ±x with an arithmetic mean of zero, the geometric mean is √(1 − x²) − 1, and the population standard deviation is exactly x. At ±10 percent the geometric mean is −0.501 percent against a predicted −σ² ÷ 2 = −0.500. At ±20 percent it is −2.020 against −2.000; at ±30 percent, −4.606 against −4.500; at ±50 percent, −13.397 against −12.500. The rule of thumb — the arithmetic mean exceeds the compound rate by about σ² ÷ 2 — is very good at low volatility and starts to understate the damage at high volatility.

Now apply it to something that looks like an investment. Take an asset whose returns have an arithmetic mean of 8.00 percent a year — a perfectly respectable number, and unambiguously positive — and vary only the volatility. Under lognormal returns the compound growth rate is exact rather than approximate, and the table above reports both. At 10 percent volatility the asset compounds at 7.46 percent. At 30 percent it compounds at 3.25 percent, having lost more than half its average return to variability alone. At 39.23 percent volatility it compounds at exactly zero. At 60 percent it compounds at −9.79 percent a year: a positive average return, and $10,000 becomes $3,569 over ten years.

A 400,000-draw simulation confirms the closed form rather than replacing it: at 30 percent volatility the simulated mean simple return came out at 7.97 percent with a compound rate of 3.23; at 60 percent, 8.24 percent and −9.65. The break-even volatility for an 8 percent average return is 39.23 percent — above that, the average return is positive and the money shrinks. This is the mechanism behind the standard warning that daily-rebalanced leveraged products do not track their index over long holding periods: multiplying the daily return by two multiplies the drag by four.

Why volatility is not the same thing as risk

Volatility is a measure of how much prices move. Risk is the possibility of an outcome you cannot accept. These overlap but they are not the same, and three gaps between them matter. First, volatility is symmetric: it counts a 10 percent rise and a 10 percent fall identically, which is the point made at length in the companion article on the Sharpe ratio. Second, volatility is measured on the returns that occurred, so it is blind to a loss that has not yet happened — an asset can be perfectly calm right up to the moment it is not, and no amount of past data will contain the event. Third, volatility says nothing about permanence: a diversified index falling 30 percent and a single position falling 30 percent register the same, though only one of them has a mechanism for coming back.

What volatility is genuinely good for is position sizing and comparison. An asset with 60 percent annualised volatility will, under a normal-distribution assumption that is itself imperfect, spend most of its time within roughly a 60 percent band around its trend, and that is enough to decide how much of it you can hold without the portfolio's whole outcome depending on it. It is also the correct input to option pricing, and the correct denominator when comparing two returns. It is not a statement about how much you can lose.

What to state whenever you quote a volatility figure

Four things, none of them optional. The sampling frequency, because hourly, daily, weekly and monthly data give different answers for the same asset. The window, because a volatility computed over 30 days and one computed over two years describe different periods and are not interchangeable. The return convention, log or simple, because they diverge as soon as the moves are large. And the annualisation factor actually used, since 365 and 252 differ by 20 percent and both are defensible depending on whether the asset trades every day.

One last distinction worth keeping straight: everything above is realised volatility, computed backwards from prices that already happened. Implied volatility is a different object entirely — it is the volatility figure that makes an option pricing model reproduce a traded option's price, so it is a forward-looking number extracted from what people are paying rather than a measurement of what occurred. The two are quoted in the same units and are routinely confused, and they answer different questions.

Frequently asked questions

Should I annualise with 365 days or 252?

Use the number of periods in which the asset can actually move. A market that closes at weekends and holidays produces roughly 252 observations a year, so √252 = 15.875 is right; an asset that trades continuously produces 365, so √365 = 19.105 is right. The two differ by 20.4 percent, which is far too large to leave unstated: the same daily standard deviation of 3.141 percent annualises to 60.0 percent on 365 days and 49.9 percent on 252. The one thing you must not do is compare a figure computed one way with a figure computed the other, which is exactly what happens when a continuously traded asset is put beside an exchange-traded one without checking the convention.

Does a higher volatility always mean a worse investment?

No, but it always means a larger gap between the average return and the compound one, and that gap grows with the square of the volatility. Doubling volatility from 20 to 40 percent does not double the drag from 2.00 to 4.00 points — it quadruples it to 8.00. That is why the question is not whether an asset is volatile but whether its average return is large enough to pay for its own volatility. At 8 percent average return the break-even is 39.23 percent volatility. An asset averaging 25 percent could carry 60 percent volatility and still compound positively; an asset averaging 8 percent could not. The volatility figure alone answers nothing.

Why do two sites report different volatility figures for the same asset?

Because at least four choices go into the number and almost nobody publishes all four. The sampling frequency changes the answer whenever returns are not independent. The lookback window changes it always — 30 days and 730 days describe different periods. The return convention changes it as soon as the moves are large, since a −20 percent simple return is a −22.314 percent log return. And the annualisation factor changes it by 20.4 percent between 252 and 365. Add the choice between sample and population standard deviation, which matters on short windows, and two honest calculations of the same asset over the same rough period can differ by a third. Before comparing two volatility figures, check that all four conventions match; if a page does not state them, its number cannot be reproduced.

Is volatility drag a real cost or an accounting illusion?

It is entirely real, in the sense that it is the difference between the money you end up with and the money the average return suggests you should have. It is not a fee, and nobody collects it: it is a property of multiplication. The two-period example settles the question — +50 percent then −50 percent averages to zero and leaves you with 75 percent of what you started with, and you cannot spend the average. What is illusory is the arithmetic mean itself, which describes a typical single period rather than the outcome of a sequence of them. If the question is how much money you will have, the compound growth rate is the only relevant number, and it is always at or below the arithmetic mean, with equality only if every period's return is identical.

How many observations do I need for a volatility estimate to be usable?

More than most people use, and the requirement grows if the returns are far from normally distributed. The standard error of an estimated standard deviation falls roughly as one over the square root of the sample size, so going from 30 observations to 120 halves the uncertainty and going from 120 to 480 halves it again. That is the reason a 30-day volatility figure moves so much from week to week: most of the movement is estimation noise rather than a change in the underlying asset. There is a genuine trade-off, since a longer window is more precise about a period that is further in the past and may no longer describe the present. The practical compromise is to look at two windows — a short one and a long one — and to treat a large divergence between them as information rather than as a number to average.