Portfolio Variance

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What is 'Portfolio Variance'

Portfolio variance is the measurement of how the actual returns of a group of securities making up a portfolio fluctuate. Portfolio variance looks at the standard deviation of each security in the portfolio as well as how those individual securities correlate with the others in the portfolio. In other words, portfolio variance looks at the covariance or correlation coefficient for the securities in the portfolio.

BREAKING DOWN 'Portfolio Variance'

Generally, the lower the correlation between securities in a portfolio, the lower the portfolio variance. Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding two times 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. This type of diversification is used to reduce overall risk.

Portfolio Variance Example

The most important quality of portfolio variance is it is a combination of the individual variances of each of the assets and their co-variances. This means that the overall portfolio variance will be lower than simply a weighted average of the constituents individual variances.

The equation for the portfolio variance of a two-asset portfolio takes into account five variables. They are:

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 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 $30,000 and has a standard deviation of 25%. Stock B is worth $70,000 and has a standard deviation of 15%. The correlation between the two stocks is 0.65. Given this, the portfolio weight of Stock A is 30% and 70% for Stock B. Plugging in this information into the formula, the variance is calculated to be:

Variance = (30%^2 x 25%^2) + (70%^2 x 15%^2) + (2 x 30% x 25% x 70% x 15% x 0.65) = 2.69%

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 2.69% is 16.4%.

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