Realized Variance: Formula, Annualization, and Measurement Choices

For educational purposes only; not investment advice.

Direct answer

Realized variance measures price variation that actually occurred over a stated observation window. A common estimator adds squared log returns and then annualizes the result. Realized volatility is the square root of realized variance.

There is no context-free realized-variance number. The answer depends on the asset, price source, start and end observations, return definition, sampling interval, treatment of the mean, annualization factor, missing data, and corporate actions. A contract’s formal specification controls its settlement calculation; a research estimate may use a different convention.

Formula and conventions

For prices P₀, P₁, …, Pₙ, define each log return as:

rᵢ = ln(Pᵢ / Pᵢ₋₁)

One zero-mean daily-return convention is:

RV = (A / n) × Σᵢ₌₁ⁿ rᵢ²

where A is the assumed number of observations per year, often 252 for daily U.S. trading observations. Annualized realized volatility is √RV. Under this convention, a variance of 0.04 corresponds to volatility of 0.20, or 20%.

Other estimators subtract the sample mean, divide by n − 1, use simple returns, or sample intraday prices. These are not interchangeable. Higher-frequency sampling can capture more intraday movement, but bid-ask bounce, discreteness, asynchronous quotes, and bad ticks create market-microstructure noise. Realized-kernel and related estimators are designed to address parts of that problem.

Cboe’s S&P 500 Variance Futures specification provides a concrete contract convention: daily S&P 500 Index log returns, zero assumed daily mean, 252-day annualization, and specified closing values plus a final special opening quotation. That convention should not be silently substituted for another product or research series.

Five-return calculation

Assume five consecutive daily log returns are +1.00%, −0.50%, +0.75%, −1.25%, and +0.25%. Written as decimals, their squared sum is:

0.0100² + (−0.0050)² + 0.0075² + (−0.0125)² + 0.0025² = 0.00034375

Using the zero-mean convention and 252 trading days:

RV = (252 / 5) × 0.00034375 = 0.017325

realized volatility = √0.017325 = 0.1316 = 13.16%

Variance is 0.017325, not 13.16%. If quoted in squared volatility points, the same result is approximately 173.25 because (13.16)² ≈ 173.19; small differences arise from rounding. The five-day sample is intentionally short and unstable, so it illustrates arithmetic rather than a dependable risk estimate.

One large observation matters disproportionately because returns are squared. Replacing the −1.25% day with −5.00% increases annualized variance to 0.13545 and realized volatility to about 36.80%, even though the other four returns do not change.

Measurement and use checklist

Fix the exact start, end, time zone, calendar, and observation timestamp before calculating.
Use adjusted or unadjusted prices consistently and handle splits, distributions, stale values, and missing observations explicitly.
State whether returns are logarithmic or simple and whether the mean is assumed zero or estimated.
Match the divisor and annualization factor to the selected convention.
Do not compare a 5-minute estimator directly with a close-to-close estimator without reconciling overnight returns and noise.
Preserve full precision through the calculation and round only the reported output.
Separate realized variance to date from expected future variance; before expiration, a variance-linked instrument can contain both.
Read the governing contract for observation disruptions, corrections, final settlement, multiplier, and quotation units.
Treat historical realized variance as a measurement, not a forecast or a guaranteed future range.
Stress jumps and data errors; both can dominate a sum of squared returns.

Common misconceptions

“Realized variance and realized volatility are the same number.” Volatility is the square root of variance.
“A variance of 400 means 400% volatility.” In volatility-point-squared notation, 400 corresponds to 20 volatility points.
“More observations always improve the estimate.” Excessive sampling can amplify microstructure noise.
“Daily return variance includes intraday and overnight risk equally.” Close-to-close returns combine them but do not identify their separate contributions.
“Subtracting the sample mean never matters.” It may be small for short-horizon financial returns, but it is a defined methodological choice.
“Realized variance is implied volatility after expiration.” One is computed from observed returns; the other is inferred from option prices under a model.
“Annualization creates more information.” It rescales an estimate and can make a short, noisy sample look deceptively comparable.
“Every variance contract uses the textbook formula.” Observation schedules, price sources, disruption rules, units, and settlement terms can differ.