# Understanding Portfolio Variance: Key Concepts and Calculation Formula

## What Is Portfolio Variance?

Portfolio [variance](https://www.investopedia.com/terms/v/variance.asp) evaluates the risk reflected by fluctuations in the combined [actual returns](https://www.investopedia.com/terms/a/actualreturn.asp) of securities within a [portfolio](https://www.investopedia.com/terms/p/portfolio.asp). It relies on the [standard deviations](https://www.investopedia.com/terms/s/standarddeviation.asp) and correlations between each asset in the portfolio. A crucial aspect of modern portfolio management involves leveraging these metrics to optimize risk and return, emphasizing strategic asset selection based on their [correlation](https://www.investopedia.com/terms/c/correlation.asp) coefficients.

### Key Takeaways

*   Portfolio variance is a measure of risk that captures how much the returns of securities within a portfolio fluctuate, using standard deviations and correlations between securities.
*   The formula for calculating portfolio variance involves a weighted combination of individual variances and co-variances among securities, leading to a lower variance than a mere weighted average of individual variances.
*   Modern Portfolio Theory suggests that adding non-correlated assets can reduce portfolio variance, enhancing an investment's efficient frontier by achieving maximum returns with minimal risk.
*   Standard deviation, the square root of variance, is used by analysts as a key metric to assess a portfolio’s volatility, with higher values indicating greater risk levels.
*   Portfolio managers often seek to minimize risk by including low-correlation assets to balance variability and enhance diversification for optimal portfolio performance.

![Portfolio Variance: A measure of the volatility of an investment portfolio.](https://www.investopedia.com/thmb/vkEuF78VGro52eA2updod56AK1I=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/portfolio-variance.asp-Final-a5efd061d3ff4a28a72b63b2b0ffc0f9.png)

*— Investopedia / Theresa Chiechi*

## How Portfolio Variance Measures Risk in Your Investments

Portfolio variance looks at the [co-variance](https://www.investopedia.com/terms/c/covariance.asp) or [correlation co-efficients](https://www.investopedia.com/terms/c/correlationcoefficient.asp) 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](https://www.investopedia.com/terms/w/weightedaverage.asp) weight multiplied by the co-variance of all individual security pairs.

[Modern portfolio theory](https://www.investopedia.com/terms/m/modernportfoliotheory.asp) says that portfolio variance can be reduced by choosing [asset classes](https://www.investopedia.com/terms/a/assetclasses.asp) with a low or [negative correlation](https://www.investopedia.com/terms/n/negative-correlation.asp), such as stocks and bonds, where the variance (or standard deviation) of the portfolio is the x-axis of the [efficient frontier](https://www.investopedia.com/terms/e/efficientfrontier.asp).

## Calculating Portfolio Variance: Formula and Process

The key aspect of portfolio variance is that it combines individual [asset](https://www.investopedia.com/terms/a/asset.asp) variances, adjusted by their covariances, into a weighted sum. As a result, portfolio variance is lower than when just averaging the variances of individual stocks.

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

*   **Portfolio variance** = w1σ1 \+ w2σ2 \+ 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

> **Important**
>
> 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. [Using software like Excel](https://www.investopedia.com/ask/answers/041615/how-do-you-calculate-variance-excel.asp) can make the calculation of these numbers easier.

## How Modern Portfolio Theory Influences Portfolio Variance

Modern Portfolio Theory (MPT) is a guideline for building investment portfolios. MPT is based on the idea that rational [investors](https://www.investopedia.com/terms/i/investor.asp) aim to maximize [returns](https://www.investopedia.com/terms/r/return.asp) while minimizing [risk](https://www.investopedia.com/terms/f/financialrisk.asp), often shown by [volatility](https://www.investopedia.com/terms/v/volatility.asp). Investors aim for the efficient frontier, where they can achieve a target return with the lowest risk and volatility.

MPT reduces risk 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. Lower correlation decreases portfolio variance.

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](https://www.investopedia.com/terms/d/diversification.asp).

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](https://www.investopedia.com/ask/answers/021915/how-standard-deviation-used-determine-risk.asp) used by portfolio managers, financial advisors, and institutional investors. Asset managers routinely include standard deviation in their performance reports.

## A Practical Example: Calculating Portfolio Variance

For instance, consider a portfolio 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.