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Standard deviation of returns

The square root of variance and a measure of how returns are dispersed around their mean, interpretable only with a stated frequency, sample and convention.

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Who this is for — Readers comparing the variability of instruments, strategies or portfolios who need to know which sampling and annualisation choices sit behind the number.

Standard deviation measures the dispersion of a series around its mean and is expressed in the same unit as the variable. Applied to periodic returns, it is often called historical volatility. It is the square root of variance; it does not directly measure maximum loss, liquidity, default risk or tail dependence.

A value is not interpretable without the series, frequency, window, currency, return definition and statistical convention. Daily and monthly dispersion cannot be compared directly. Net returns are not the same as gross returns. Infrequent or stale valuations can create artificially smooth volatility.

Standard deviation: from sample to measure Frequency, window, calendar and convention change the number Standard deviation: from sample to measure Frequency, window, calendar and convention change the number s = √[Σ(rₜ − r̄)² / (n − 1)] different windowdifferent frequencysample mean Return definition Simple or log, gross or net,currency and total return. Sample Missing dates, overlap andnon-stationarity affect theestimate. Annualisation Square-root-of-time scalingrequires stated scalingassumptions. Cyclepedia · educational diagram: state conventions, period and data
The same economic history can produce different estimates when frequency, window, valuation and data treatment change.

Population, sample and returns

For N observations rₜ and arithmetic mean r̄, descriptive population standard deviation is:

σ = √[(1/N) × ∑ₜ(rₜ − r̄)²]

When the observations are treated as a sample from a wider population, a common estimator uses N−1:

s = √[(1/(N−1)) × ∑ₜ(rₜ − r̄)²]

The formulas implement different conventions. With few observations the difference may be material, and software defaults vary. A reproducible report states which one it uses.

For a portfolio, standard deviation can be calculated from observed portfolio returns or estimated ex ante from weights and covariance:

σₚ = √(wᵀΣw)

One describes a realised history; the other is conditional on model inputs. They should not both be labelled simply “risk” without their nature and date.

Simple, logarithmic and total returns

The source series comes first. Simple returns compound geometrically and represent percentage change. Log returns add through time for positive values but cannot generally be averaged across assets to obtain portfolio return. Price returns exclude income, while total returns include dividends and coupons under a convention. Local- and base-currency returns carry different risks. Gross- and net-return distributions differ when costs vary.

Calculating the statistic on trending price levels rather than returns can produce a quantity without the intended financial meaning.

Annualisation and the square-root-of-time rule

A common approximation is:

σ_annual ≈ σ_period × √m

For monthly observations m may be 12; daily choices depend on the market and calendar. The rule requires assumptions that make variance scale linearly, often including negligible serial correlation and a sufficiently stable distribution. Autocorrelation, changing volatility, missing data, smoothing and continuously traded markets can make it misleading.

Andrew Lo shows how standard annualisation of the Sharpe ratio fails with serially correlated returns; the same problem enters through its standard- deviation denominator. Writing “252” or “365” without a calendar does not make two results comparable.

What it captures—and what it does not

Standard deviation gives the same squared weight to observations above and below the mean. That symmetry is useful when any deviation represents unpredictability, but may not match an investor who distinguishes unexpected gains from losses. Sortino ratio uses downside deviation relative to a target and introduces a different set of conventions.

Standard deviation does not identify tail shape, show return order, distinguish a gradual loss from a gap, remain stable across regimes, or forecast future dispersion with certainty. It can be understated by stale prices or smoothed models. It should be read alongside distributions, drawdowns, scenarios, liquidity and data quality.

Reproducible example

Five hypothetical percentage returns are 1, −1, 2, 0, −2. Their mean is zero and squared deviations sum to 10.

  • population convention: √(10/5) ≈ 1.414%;
  • sample convention: √(10/4) ≈ 1.581%.

Both can be correct under the stated convention. The sample is far too small for an operational conclusion and implies no threshold.

Comparison checklist

Before comparing values, align frequency and interval; calendar and missing data; currency and FX policy; simple or log returns; price or total return; gross or net basis; population or sample denominator; rolling or expanding window; annualisation assumptions; and ex-ante or ex-post status.

Common error — Calling volatility “low” without stating frequency, window and valuation method. The number may be low because risk is limited—or because the data conceal it.

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