Covariance – Definition

Cite this article as:"Covariance – Definition," in The Business Professor, updated December 20, 2019, last accessed October 20, 2020,


Covariance Definition

Covariance is a metric used in statistics and probability theory to measure the directional relationship between the returns of two risky assets (two variables). What the metric does is to evaluate to what extent and how much the variables move together. However, it does not measure the dependency between those variables.

A Little More on What is Covariance

Generally, the covariance sign illustrates the propensity in the linear relationship between two variables. It is tricky to interpret covariance’s magnitude because there is normalization. So, the interpretation of this will depend on the variables’ magnitude. Note that when the version of covariance and the correlation coefficient are normalized, the magnitude of the strength of the linear relation becomes evident.

Covariance is measured using units. You can compute the units by multiplying two-variable units. The variance can take either a positive or a negative value. Covariance assesses how and to what extent the mean value of two variables move. A variance is positive when the returns on the asset are moving in the same direction. On the other hand, the movement of the returns in the opposite direction (inverse) means that the variance is negative.

For instance, in a stock market, if the returns of ABC stock happens to move higher each time XYZ stock moves higher, and you witness the same behavior when the returns of each stock decrease, it means that the stock has a positive covariance.

Individuals in finance use the concept of portfolio theory. Through the method of diversification, those individuals are able to assess covariance between security holdings in a portfolio. By selecting security holdings that do not show a high positive covariance with each other, it is possible to eliminate some of the risks that prove impossible to diversify.

How Covariance Works

To calculate covariance, you will need to analyze standard deviations from the expected return. You can also arrive at covariance by multiplying the correlation between the two variables and then multiply it by the standard deviation of each variable.

When you have a set of data (x and y values), you can calculate covariance by using five variables from that particular data. They are as follows:

  • xi = a  given x value in a set of data
  • xm= the x values’ average or mean
  • yi= the value of y in the data set that corresponds with xi
  • ym= the mean, or average, of the y values
  • n= the number of data points

When you use the above information, the covariance formula will be as follows:

Cov(x,y) = SUM [(xi – xm) * (yi – ym)] / (n-1)

Note that the fact that covariance does measure the relationship of returns between two assets, but does not show the strength between the two assets. So, the coefficient of correlation becomes the best indicator for measuring strength.

Covariance Applications

Covariance is an important application in both modern portfolio theory and finance. For instance, there is a capital asset pricing model that they use to calculate the anticipated return on the asset. The covariance between a market and security is used in the formula in one of the important variables of a model known as a beta.

Beta is used to measure a security’s volatility (systematic risk) in comparison to the overall market. It is a practical metrics drawn from the covariance to measure the risk exposure of an investor specific to a single security.

To statistically reduce a portfolio’s overall risk, investors use covariance. Covariance protects portfolios against volatility during portfolio diversification. Note that when a diversified portfolio contains financial assets combination with varying covariance, it ensures more diversification. However, this is not the same as those assets with the same covariance that provides little diversification.

Covariance versus Correlation

A major similarity between covariance and correlation is that they both evaluate the relationship between two variables, with the closest similarity being the relationship between standard deviation and variance.

Unlike correlation that measures the strength of two variables, covariance measures the relationship between the movements of two variables. It shows the direction in which the variables are moving, either in the same direction or in the opposite direction.

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