Variance and Standard Deviation

Variance and standard deviation measure how spread out a set of values is around its average. In trading, they are the most common measures of risk and volatility: a stock whose daily returns swing widely has a high standard deviation; one that barely moves has a low one. Standard deviation underpins historical volatility, the Sharpe ratio, position sizing by volatility, Bollinger Bands and option pricing.

The formulas#

population variance σ² = Σ (x - μ)² / N
sample variance s² = Σ (x - x̄)² / (n - 1)
standard deviation = √variance

• μ or x̄: the mean
• N or n: number of values

The sample formula divides by n minus 1 (Bessel's correction) to avoid underestimating variance when using a sample. In trading, you almost always use the sample version.

Variance is in squared units; standard deviation is in the original units, such as percent returns, which makes it easier to interpret.

Worked example#

Annualising volatility#

If daily returns are roughly independent, variance scales with time, so standard deviation scales with the square root of time:

annual volatility ≈ daily standard deviation × √252
monthly volatility ≈ annual volatility / √12

A daily standard deviation of 1.35% implies annual volatility of about 1.35% × 15.87 ≈ 21.4%. See Historical and Realized Volatility.

Interpreting standard deviation#

For roughly normal data:

Real market returns have fatter tails than the normal distribution, so moves of 3 or more standard deviations happen more often than these numbers suggest. See Normal Distribution and Fat Tails.

Where standard deviation appears in trading#

Limits of standard deviation as risk#

• Treats upside and downside equally: a big gain counts as "risk". The Sortino ratio uses only downside deviation. See Sortino Ratio.
• Misses tail risk: strategies that sell options can show low standard deviation until a crash. See Theta Harvesting.
• Changes over time: volatility clusters. See GARCH.
• Depends on the window used to calculate it.

Variance of a portfolio#

Portfolio variance depends on each asset's variance and how they move together:

σ²_p = w1² σ1² + w2² σ2² + 2 w1 w2 ρ σ1 σ2

Combining assets with low or negative correlation reduces portfolio risk. See Covariance and Correlation and Diversification.

Downside deviation and semivariance#

Because traders care more about losses than gains, some measures only count returns below a target, often zero or the risk free rate. Downside deviation is the square root of the average squared shortfall below the target. It powers the Sortino ratio and gives a clearer picture of risk for strategies with skewed returns. See Sortino Ratio and Skewness and Kurtosis.

Variance in position sizing#

Many traders size positions so that each one contributes a similar amount of volatility to the portfolio. If one asset is twice as volatile as another, it gets roughly half the position size. This inverse volatility approach keeps a single volatile position from dominating results. See Volatility and ATR-Based Sizing and Risk Budgeting and Risk Parity.

Frequently asked questions#

What is standard deviation in trading?#

A measure of how much returns vary around their average, commonly used as a measure of volatility and risk.

What is the difference between variance and standard deviation?#

Variance is the average squared deviation from the mean; standard deviation is its square root, expressed in the same units as the data.

How do you annualise daily volatility?#

Multiply the daily standard deviation by the square root of the number of trading days in a year, usually √252.

Next, learn to measure how unusual a value is in Percentiles, Quantiles and Z-Scores.