Many elements of mathematics and statistics are applied to evaluate stocks. Covariance calculations show how two stocks might move together in the future. Looking at historical prices, covariance helps the investor to determine if the prices tend to move with each other or against each other. This can predict the potential price movement of a two-stock portfolio.

An investor might even be able to select complementary stocks, which can reduce the overall risk and increase the overall potential return of a portfolio. Understanding covariance is important when selecting stocks.

### What Is Covariance?

Covariance measures how two variables move together. It measures whether two variables move in the same direction (a positive covariance) or in opposite directions (a negative covariance). In this article, the variables are stock prices, but the covariance can be applied to anything.

In the stock market, a strong emphasis is placed on reducing the amount of assumed risk for the same amount of return. When constructing a portfolio, a portfolio manager will select stocks that work well together. This usually means that these stocks do not move in the same direction.

### Calculating Covariance

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, the closing price for each day is used to find the return from one day to the next. Do this for both stocks and build a list to begin the calculations.

For example:

Day |
ABC Returns (%) |
XYZ Returns (%) |

1 | 1.1 | 3 |

2 | 1.7 | 4.2 |

3 | 2.1 | 4.9 |

4 | 1.4 | 4.1 |

5 | 0.2 | 2.5 |

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.

Next, it's a matter of taking the differences between ABC's return and ABC's average return and multiplying it by the difference between XYZ's return and XYZ's average return. The last step is to divide the result by the sample size and subtract one. If it was the entire population, you could divide by the population size.

This can be represented by the following equation:

Using our example on 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.

You can see that 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.

### Using Microsoft Excel

In Excel, you can easily find the covariance by using one of the following functions:

= 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.

### Meaning

In the example, there is a positive covariance, so the two stocks tend to move together. When 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.

### Uses of Covariance

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 it is represented by this equation:

where:

- cov (X, Y) = covariance between X and Y
- σ
_{X}= standard deviation of X - σ
_{Y}= 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 X and Y. 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 effect a change in the other.

The covariance can also be used to find the standard deviation of a multi-stock portfolio. The standard deviation is the accepted calculation for risk, and this is extremely important when selecting stocks. Most investors would want to select stocks that move in opposite directions because the risk will be lower while providing the same amount of potential return.

### The Bottom Line

Covariance is a common statistical calculation that can show how two stocks tend to move together. We can only use historical returns, so there will never be complete certainty regarding 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.