Variance (Statistics)

Variance, in statistics, is a measure of dispersion: it describes how far the values of a data set or a random variable typically lie from their average. It is defined as the mean of the squared differences between each value and the mean. A variance of zero means every value is identical, and a larger variance means the values are more spread out. The term was introduced by the statistician Ronald Fisher in 1918.

In finance, the variance of returns is the standard mathematical measure of risk. Its square root, the standard deviation, is what market practitioners call volatility. This statistical meaning is unrelated to the use of the word in management accounting, where a variance is the difference between a budgeted and an actual figure.

For a random variable $X$ with mean $\mu$, the variance is:

When it is estimated from a sample of $n$ observations, the formula is:

• $x_i$ is the $i$-th observation.
• $\bar{x}$ is the mean of the sample.
• $n - 1$ is used instead of $n$ to correct the tendency of a sample to understate the spread of the population it came from.

Take five monthly returns of 2, -1, 3, 0 and 1 percent. The mean is 1. The deviations are 1, -2, 2, -1 and 0, and their squares are 1, 4, 4, 1 and 0, which sum to 10. The sample variance is $10 / 4 = 2.5$, and the population formula, dividing by 5, would give 2.0. The sample standard deviation is the square root of 2.5, about 1.58 percent. Because the deviations are squared, variance is expressed in squared units, here percent squared, which is why the standard deviation is usually quoted instead.

Three properties account for most of the practical use of variance.

• Scaling. Multiplying a variable by a constant $a$ multiplies its variance by $a^2$. Doubling a position quadruples the variance of its profit and loss and doubles its standard deviation.
• Sums. The variance of a sum of two variables is the sum of their variances plus twice their covariance. If the variables are independent, the variances simply add.
• Time. If returns in successive periods are independent, the variance over several periods is the sum of the single-period variances. Annual variance is therefore about 252 times daily variance, and annual volatility is daily volatility multiplied by the square root of 252.

The sum rule is the mathematics of diversification. For a portfolio of two assets:

• $w_1$ and $w_2$ are the portfolio weights of the two assets.
• $\sigma_1$ and $\sigma_2$ are their standard deviations.
• $\rho$ is the correlation between their returns.

Suppose 60 percent is held in an asset with volatility of 20 percent and 40 percent in an asset with volatility of 10 percent, and the correlation is 0.25. The three terms are $0.36 \times 400 = 144$, $0.16 \times 100 = 16$ and $2 \times 0.6 \times 0.4 \times 0.25 \times 20 \times 10 = 24$. The portfolio variance is 184 and its volatility is about 13.6 percent. The weighted average of the two volatilities is 16 percent, so combining imperfectly correlated assets has reduced risk. Harry Markowitz made this calculation the foundation of modern portfolio theory in 1952, and portfolio optimization in its classic form chooses weights to minimize variance for a given expected return.

Variance appears throughout quantitative finance. The realized volatility of an asset is built from the sum of its squared returns. Models such as GARCH describe how the variance of returns changes from day to day. Variance can be traded directly through a variance swap, a contract that pays the difference between the variance realized over a period and a level agreed at the start. In regression analysis, the share of variance in one variable that is explained by others is the usual measure of fit.

Variance treats gains and losses alike, although investors are concerned mainly with losses. Measures based on downside deviations, such as the semi-variance used in the Sortino ratio, address this. Because deviations are squared, a few extreme observations dominate the result, and financial returns have fat tails that make sample variance unstable. Variance also summarizes only the width of a distribution. It says nothing about asymmetry or the weight of the tails, which are described by skewness and kurtosis.

The variance of market returns is not constant. It rises in crises and falls in calm periods, so an estimate depends heavily on the window of data used. Risk figures based on a quiet period will understate the variability that follows when conditions change.