Calculating Covariance for Stocks

The fields of mathematics and statistics offer a great many tools to help us evaluate stocks. One of these is covariance, which is a statistical measure of the directional relationship between two asset prices. One may apply the concept of covariance to anything, but here the variables are stock prices. Formulas that calculate covariance can predict how two stocks might perform relative to each other in the future. Applied to historical prices, covariance can help determine if stocks' prices tend to move with or against each other.

Using the covariance tool, investors might even be able to select stocks that complement each other in terms of price movement. This can help reduce the overall risk and increase the overall potential return of a portfolio. It is important to understand the role of covariance when selecting stocks.

Covariance applied to a portfolio can help determine what assets to include in the portfolio. It measures whether stocks move in the same direction (a positive covariance) or in opposite directions (a negative covariance). When constructing a portfolio, a portfolio manager will select stocks that work well together, which usually means these stocks would not move in the same direction. 

Calculating a stock's covariance starts with finding a list of previous prices or "historical prices" as they are called on most quote pages. Typically, you use the closing price for each day to find the return. To begin the calculations, find the closing price for both stocks and build a list. For example:

Next, we need to calculate the average return for each stock:

• For ABC, it would be (1.1 + 1.7 + 2.1 + 1.4 + 0.2) / 5 = 1.30.
• For XYZ, it would be (3 + 4.2 + 4.9 + 4.1 + 2.5) / 5 = 3.74.
• Then, we take the difference between ABC's return and ABC's average return and multiply it by the difference between XYZ's return and XYZ's average return.
• Finally, we divide the result by the sample size and subtract one. If it was the entire population, you could divide by the population size. 

This is represented by the following equation:

$\text{Covariance}=\frac{\sum{\left(Return_{ABC}-Average_{ABC}\right)*\left(Return_{XYZ}-Average_{XYZ}\right)}}{\left(\text{Sample Size}\right)-1}$Covariance=(Sample Size)−1∑(ReturnABC​−AverageABC​)∗(ReturnXYZ​−AverageXYZ​)​

Using our example of ABC and XYZ above, the covariance is calculated as:

= [(1.1 - 1.30) x (3 - 3.74)] + [(1.7 - 1.30) x (4.2 - 3.74)] + [(2.1 - 1.30) x (4.9 - 3.74)] + …

= [0.148] + [0.184] + [0.928] + [0.036] + [1.364]

= 2.66 / (5 - 1)

= 0.665

In this situation, we are using a sample, so we divide by the sample size (five) minus one.

The covariance between the two stock returns is 0.665. Because this number is positive, the stocks move in the same direction. In other words, when ABC had a high return, XYZ also had a high return. 

In Excel, you use one of the following functions to find the covariance:

= COVARIANCE.S() for a sample

or

= COVARIANCE.P() for a population

You will need to set up the two lists of returns in vertical columns as in Table 1. Then, when prompted, select each column. In Excel, each list is called an "array," and two arrays should be inside the brackets, separated by a comma. 

In the example, there is a positive covariance, so the two stocks tend to move together. When one stock has a high return, the other tends to have a high return as well. If the result were negative, then the two stocks would tend to have opposite returns—when one had a positive return, the other would have a negative return.

Finding that two stocks have a high or low covariance might not be a useful metric on its own. Covariance can tell how the stocks move together, but to determine the strength of the relationship, we need to look at their correlation. The correlation should, therefore, be used in conjunction with the covariance, and is represented by this equation:

$\begin{aligned} &\text{Correlation}=\rho=\frac{cov\left(X, Y\right)}{\sigma_X\sigma_Y}\\ &\textbf{Where:}\\ &cov\left(X, Y\right)=\text{covariance between X and Y}\\ &\sigma_X=\text{standard deviation of X}\\ &\sigma_Y=\text{standard deviation of Y}\\ \end{aligned}$​Correlation=ρ=σX​σY​cov(X,Y)​Where:cov(X,Y)=covariance between X and YσX​=standard deviation of XσY​=standard deviation of Y​

The equation above reveals that the correlation between two variables is the covariance between both variables divided by the product of the standard deviation of the variables. While both measures reveal whether two variables are positively or inversely related, the correlation provides additional information by determining the degree to which both variables move together. The correlation will always have a measurement value between -1 and 1, and it adds a strength value on how the stocks move together.

If the correlation is 1, they move perfectly together, and if the correlation is -1, the stocks move perfectly in opposite directions. If the correlation is 0, then the two stocks move in random directions from each other. In short, covariance tells you that two variables change the same way while correlation reveals how a change in one variable affects a change in the other. 

You also may use covariance to find the standard deviation of a multi-stock portfolio. The standard deviation is the accepted calculation for risk, which is extremely important when selecting stocks. Most investors would want to select stocks that move in opposite directions because the risk will be lower, although they'll provide the same amount of potential return.

Covariance is a common statistical calculation that can show how two stocks tend to move together. Because we can only use historical returns, there will never be complete certainty about the future. Also, covariance should not be used on its own. Instead, it should be used in conjunction with other calculations such as correlation or standard deviation.