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Key Takeaways
- Variance is the average of the squared deviations of returns from their mean, so it quantifies how widely outcomes spread around the average.
- Standard deviation is the square root of variance and expresses risk in the same units as the returns themselves.
- The variance of a sum equals the summed variances plus every cross-covariance term. Uncorrelated additive increments give the familiar square-root volatility scaling.
- Squaring the deviations makes variance sensitive to large moves and outliers, and it counts upside and downside dispersion equally.
- Divide by n to describe the observed set or by n − 1 for the usual sample estimate. The latter is unbiased for independent, identically distributed observations with finite variance.
What is variance?
Variance is a measure of dispersion that averages the squared deviations of observations, such as returns, from their mean. Squaring makes every deviation non-negative and gives large moves more weight. Its square root is the standard deviation, which restores the original units, so analysts calculate with variance and usually report standard deviation.
The unit matters. If returns are entered as percentage-point numbers, variance is in squared percentage points. If returns are decimals, variance is in squared decimal-return units. Changing from percentage points to decimals divides variance by 10,000.
Formula
Population variance = (1/N) × sum of (Ri − Rbar)²; sample variance = (1/(n − 1)) × sum of (Ri − Rbar)²; standard deviation = square root of variance- Ri is each period return and Rbar is the mean return over the observed window.
- The n divisor describes dispersion in the complete observed set. The n − 1 divisor gives an unbiased variance estimate under independent sampling from a common finite-variance distribution.
- Variance is stated in squared return units, so standard deviation is reported instead when risk needs the original units.
How to calculate variance: a five-return example
A stock posts five monthly returns of +3%, −1%, +2%, −2%, and +3%. A trader computes the sample variance and standard deviation to gauge how dispersed the returns were around their average.
- Mean return = (3 − 1 + 2 − 2 + 3) / 5 = 1%
- Deviations from the mean: +2, −2, +1, −3, +2 (in percentage points)
- Squared deviations sum to 22 squared percentage points.
- Sample variance = 22 / (5 − 1) = 5.5 squared percentage points, or 0.00055 in squared decimal-return units.
- Sample standard deviation = square root of 5.5 = 2.35%
The five returns average 1%. Their squared deviations sum to 22. Dividing by four gives sample variance 5.5 squared percentage points, and the square root gives sample standard deviation 2.35 percentage points. Dividing by all five observations instead gives descriptive variance 4.4.
For a descriptive calculation using all five observations as the set of interest, variance is 22 / 5 = 4.4 and standard deviation is 2.10 percentage points. Mean absolute deviation is 2 points, a separate measure from either standard deviation.
Population vs sample variance
The n − 1 correction estimates population variance without bias under independent sampling from a common finite-variance distribution. The correction applies to variance; taking its square root does not generally produce an unbiased estimator of population standard deviation.
A sample calculation can be above or below the population value. Unbiasedness describes its expectation over repeated sampling, not the direction of every estimate.
Variance over time and square-root scaling
The variance of a sum includes all individual variances and twice every pairwise covariance. When increments are uncorrelated and share one variance, the variance of k summed increments is k times the individual variance.
This gives square-root-of-time scaling for their standard deviation. Log returns are additive. Compounded simple returns are multiplicative and do not generally obey the exact same aggregation formula.
Portfolio variance and covariance
For two positions, portfolio variance is wA² × varianceA + wB² × varianceB + 2 × wA × wB × covarianceAB.
Use the actual weights and covariance. The portfolio calculation includes squared-weight terms as well as the cross term; it is not a simple sum of individual asset variances.
Variance as a risk measure
Variance needs no normal or symmetric distribution. Equal distances on either side of the mean contribute equally, while larger distances contribute more through squaring.
Limitations of variance
- Take the square root before you read the number. Variance is in squared units, such as squared percentage points, and standard deviation restores the original units.
- Pair it with skewness, quantiles or downside deviation to see which way the risk runs, because variance counts upside and downside dispersion equally.
- Check the window for outliers before you rely on the number. Squaring lets a few large moves or outliers dominate the result.
- Recalculate across more than one window. A sample estimate depends on the window and the number of observations, and any single estimate can sit above or below the true value.
- Include the covariances when you combine periods or positions. Leaving them out misstates the total when the pieces are correlated, because correlation enters every cross term.
Bottom line
Variance measures how widely returns spread around their average and is the building block for standard deviation, portfolio risk and volatility scaling. Calculate it in stated units with a stated divisor, include covariances when combining periods or positions, and report standard deviation when people need to read the result.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- NIST/SEMATECH e-Handbook of Statistical Methods: Variance and Standard Deviation
- Fidelity: Standard Deviation Indicator
- Federal Reserve FEDS: Expected Stock Returns and Variance Risk Premia
- CFA Institute Financial Analysts Journal: Return Dispersion and Active Management
How entries are sourced, calculated and updated: editorial methodology.