How to Calculate Covariance

By AJ Smith · March 29, 2022 · 6 minute read

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How to Calculate Covariance

Covariance is a measure of how two variables — such as the price of two securities — move together (positive covariance) or move in opposite directions (negative covariance). It’s calculated using a specific formula that captures the degree to which the mean of the two variables move in tandem.

Calculating covariance is important in finance because it’s one way to help insure that a portfolio is well-diversified. The concept, which comes from statistics, is one of the key tenets of modern portfolio theory: that risk can be reduced by investing in a variety of assets, particularly those with a negative covariance.

Effective diversification involves not just making qualitative judgments about different assets but understanding their statistical relationship to each other. For optimal diversification in their own portfolios, investors need to understand covariance.

What Is Covariance?

Covariance is very much as it sounds: variations in the average price of two securities, considered in relation to each other. In other words, if Stock X moves higher whenever Stock Y moves higher, and the same pattern occurs when each stock’s return decreases, then the two securities have a positive covariance. Negative covariance is when the mean values move inversely to one another.

Thus covariance is a measure of how two securities tend to move together or not, not a measure of the direction in which they move.

By considering the covariance of securities in a portfolio, investors may be able to enhance their portfolio’s diversification. It’s a way of handling investment risks.

Another way investors can gauge risk in their holdings is learning how to find portfolio beta, or the sensitivity to price swings in the broader market.

Can Covariance Be Negative?

When the variables move together, the covariance is positive (i.e., they move above or below their respective averages together). But when they move in opposite directions, the covariance is negative.

When you pair assets that have a negative covariance, volatility and therefore risk may be reduced in the portfolio.

Difference Between Covariance and Correlation

Covariance may sound similar to the concept of correlation. The two are related but they’re not identical. The correlation coefficient measures the strength of a correlative relationship between two stocks, for example.

Covariance Formula

Covariance has a formula:

Cov (X,Y) = [SUM(x-xa) (y-ya)] / (n-1)

In the formula:

•   x is the value of the x variable. y is the value of the y variable.

•   xa is the average of the x variable, while ya is the average of the y variable.

•   n is the number of data points.

Example of Calculating Covariance

For this example we will use the hypothetical performance of two stocks, AAA Corp and BBB Corp. Below are AAA Corp’s returns over five years.

2017: -15%

2018: -7%

2019: 25%

2020: 40%

2021: 50%

Average five-year returns = 18.6%

Here are BBB Corps returns.

2015: 5%

2016: 9%

2017: 11%

2018: 8%

2019: 12%

Average returns = 9

For each stock, we subtract the average return from the return of a given year and then multiply the two results together, and add them up for every year.

[(-15-18.6) x (5-9)] + [(-7-18.6) x (9-9)] + [(25-18.6) x (11-9)] + [(40-18.6) x (8-9)] + [(50-18.6) x (12-9)] / (5-1)

Interpreting Covariance Results

After doing the math (or running it through Excel), the covariance of these two stocks is 55. That it’s positive means the stocks tend to move in the same direction at the same time. When you look at the two sets of returns, there are years where AAA has a negative return, and then a high positive return, whereas BBB has steady positive returns in the five-year sample.

So you can see that covariance isn’t a measure of returns, but how returns vary relative to their mean. In the two years BBB had above average returns, AAA had very high and above average returns. For there to be negative covariance, the below and above average years would have to be not paired at all.

While these examples are not necessarily reflective of a real-world comparison between two assets in a portfolio, it shows how covariance can work and how to think about it when managing your portfolio.

Covariance Comparisons

In order to make use of covariance, it’s important for investors to distinguish between some common statistical measures.

Covariance vs Correlation

While correlation is another metric used to measure the relationship between two variables, covariance measures the direction of that relationship whereas correlation measures its strength. It is often expressed via a correlation coefficient, which ranges from -1 to +1.

The correlation between two variables is considered strong if the correlation coefficient is close to +1 (positive) or -1 (negative)

A correlation is considered to be strong if the correlation coefficient has a value that is close to +1 (positive correlation) or -1 (negative correlation). When the coefficient is close to zero, there is only a weak relationship between the two variables.

Covariance vs Variance

Both variance and covariance measure how data points are distributed around a calculated mean. However, variance is often used in data sets with only one variable, to gauge how closely data points group around the average, while covariance examines the directional relationship between two variables.

In finance, covariance is used to gauge whether that relationship is positive or negative covariance. Negative covariance is considered more desirable when trying to diversify the holdings in a portfolio.

How Investors Can Use Covariance

For an investor, the covariance formula is used when you want to determine the relationship between two financial assets.

For example, investors and portfolio managers can look at the returns of different securities in order to gauge their covariance. Let’s say you own stock in AAA Corp from the example above. When considering whether to buy BBB Corp, or another other stock, you can use covariance to determine the relationship between the new stock’s returns over time and the ones of a given stock in our portfolio.

The Takeaway

Covariance is a way of measuring whether the mean performance of two securities moves together, or inversely to each other. While covariance is similar to correlation, which measures the strength of the relationship between two variables (or in finance, two securities), covariance simply captures the direction: Are the two variables moving in tandem or not?

This statistical measure can be useful for creating a well-diversified portfolio, since securities with a negative covariance can help manage the impact of volatility on portfolio returns.

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