# Portfolio Variance: Definition, Formula, Calculation, and Example

## What Is Portfolio Variance?

Portfolio variance is a measurement of risk, 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.

### Key Takeaways

• Portfolio variance is a measure of a portfolio’s overall risk and is the portfolio’s standard deviation squared.
• Portfolio variance takes into account the weights and variances of each asset in a portfolio as well as their co-variances.
• A lower correlation between securities in a portfolio results in a lower portfolio variance.
• Portfolio variance (and standard deviation) define the risk-axis of the efficient frontier in modern portfolio theory (MPT).
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## Understanding Portfolio Variance

Portfolio variance looks at the co-variance or correlation co-efficients for the securities in a 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 co-variance 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, where the variance (or standard deviation) of the portfolio is the x-axis of the efficient frontier.

## Formula and Calculation of Portfolio Variance

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 co-variances. 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 formula for portfolio variance in a two-asset portfolio is as follows:

• Portfolio variance = w12σ12 + w22σ22 + 2w1w2Cov1,2

Where:

• w1 = the portfolio weight of the first asset
• w2 = the portfolio weight of the second asset
• σ1 = the standard deviation of the first asset
• σ2 = the standard deviation of the second asset
• Cov1,2 = the co-variance of the two assets, which can thus be expressed as p(1,2)σ1σ2, where p(1,2) is the correlation co-efficient between the two assets

The portfolio variance is equivalent to the portfolio standard deviation squared.

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.

## Portfolio Variance and Modern Portfolio Theory

Modern portfolio theory (MPT) is a framework for constructing an investment portfolio. MPT takes as its central premise the idea that rational investors want to maximize returns while also minimizing risk, sometimes measured using volatility. Investors seek what is called an efficient frontier, or the lowest level of risk and volatility at which a target return can be achieved.

Risk is lowered in MPT portfolios by investing in non-correlated assets. Assets that might be risky on their own can actually lower the overall risk of a portfolio by introducing an investment that will rise when other investments fall. This reduced correlation can reduce the variance of a theoretical portfolio.

In this sense, an individual investment’s return is less important than its overall contribution to the portfolio, in terms of risk, return, and diversification.

The level of risk in a portfolio is often measured using standard deviation, which is calculated as the square root of the variance. If data points are far away from the mean, then the variance is high, and the overall level of risk in the portfolio is high as well. Standard deviation is a key measure of risk used by portfolio managers, financial advisors, and institutional investors. Asset managers routinely include standard deviation in their performance reports.

## Example of Portfolio Variance

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 to the formula, the variance is calculated to be:

• Variance = (33.3%^2 × 20%^2) + (66.7%^2 × 10%^2) + (2 × 33.3% × 20% × 66.7% × 10% × 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 12.81%.

## What is portfolio variance?

Portfolio variance measures the risk in a given portfolio, based on the variance of the individual assets that make up the portfolio. The portfolio variance is equal to the portfolio’s standard deviation squared.

## How is variance used in constructing a portfolio?

Most portfolio managers seek to minimize risk and maximize value, along the lines of modern portfolio theory (MPT). The greater the variance in the portfolio indicates the greater the variance of the individual assets, and hence the greater the risk. Portfolio managers thus seek to reduce risk by incorporating assets with low correlations, meaning there is little relationship in the movement of the assets in the portfolio.

## Where does standard deviation fit in?

Most portfolio analysts focus on the standard deviation of the portfolio as a whole to get the best picture of the range of outcomes in the portfolio. Standard deviation is the square root of the variance and provides a more realistic look at the level of risk of the portfolio. The higher the standard deviation, the more volatile a portfolio is likely to be, and vice versa.

## The Bottom Line

Variance is a statistical measure of the volatility or risk of a portfolio and the individual securities in it. Variance itself is not the main number to pay attention to, but rather its standard deviation, which is the square root of a portfolio’s variance. The higher the standard deviation, the more risk the portfolio is carrying, while the opposite is true for a low standard deviation.

Standard deviation is in turn a factor of the variance and correlation of the securities in a portfolio. If the standard deviation is deemed too high or risky, the portfolio manager can adjust their holdings to incorporate lower correlation assets in the portfolio and potentially lower the standard deviation or risk of the portfolio.

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