# Portfolio Variance

## What is 'Portfolio Variance'

Portfolio variance is a measurement of how the aggregate actual returns of a set of securities making up a portfolio fluctuate over time. This portfolio variance statistic is calculated using the standard deviations of each security in the portfolio as well as the correlations of each security pair in the portfolio.

## BREAKING DOWN 'Portfolio Variance'

Portfolio variance looks at the covariance or correlation coefficients for the securities in the portfolio. Generally, a lower correlation between securities in a portfolio results in a lower portfolio variance.

Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding twice the weighted average weight multiplied by the covariance of all individual security pairs.

Modern portfolio theory says that portfolio variance can be reduced by choosing asset classes with a low or negative correlation, such as stocks and bonds.

## Two-Asset Portfolio Variance Example

The most important quality of portfolio variance is that its value is a weighted combination of the individual variances of each of the assets adjusted by their covariances. This means that the overall portfolio variance is lower than a simple weighted average of the individual variances of the stocks in the portfolio.

The equation for the portfolio variance of a two-asset portfolio, the simplest portfolio variance calculation, takes into account five variables:

w(1) = the portfolio weight of the first asset

w(2) = the portfolio weight of the second asset

o(1) = the standard deviation of the first asset

o(2) = the standard deviation of the second asset

Cov(1,2) = the covariance of the two assets, which can be sampled to: q(1,2)o(1)o(2), where q(1,2) is the correlation between the two assets

The formula for variance in a two-asset portfolio is:

Variance = (w(1)^2 x o(1)^2) + (w(2)^2 x o(2)^2) + (2 x (w(1)o(1)w(2)o(2)q(1,2))

For example, assume there is a portfolio that consists of two stocks. Stock A is worth \$50,000 and has a standard deviation of 20%. Stock B is worth \$100,000 and has a standard deviation of 10%. The correlation between the two stocks is 0.85. Given this, the portfolio weight of Stock A is 33.3% and 66.7% for Stock B. Plugging in this information into the formula, the variance is calculated to be:

Variance = (33.3%^2 x 20%^2) + (66.7%^2 x 10%^2) + (2 x 33.3% x 20% x 66.7% x 10% x 0.85) = 1.64%

Variance is not a particularly easy statistic to interpret on its own, so most analysts calculate the standard deviation, which is simply the square root of variance. In this example, the square root of 1.64% is 12.82%.

As the number of assets in the portfolio grows, the terms in the formula for variance increase exponentially. For example, a three-asset portfolio has six terms in the variance calculation, while a five-asset portfolio has 15.