Covariance

What is Covariance?

Covariance is a statistical term used in security and portfolio evaluation that measures the degree to which two assets move together. A positive covariance means the assets move in the same direction, and the larger the value, the greater one asset moves in relation to the other.

An example of covariance is as follows:

• Stock A has a covariance to Stock B of +1.4
• For every $1 increase of Stock B, Stock A will increase by $1.4

Holding assets with negative or low covariance is preferable for a well-diversified portfolio because it reduces the risk that all assets will move in the same direction at the same time.

Covariance is a statistical measure that shows how two variables move together.

Examples of positive and negative covariance 

The following table illustrates the relationships between various economic variables and their impacts:

Formula for Covariance

The covariance formula can be used to determine the direction of the relationship between two variables or stocks:

For a sample data:

cov(X, Y) = Σ((x – x̄) * (y – ȳ)) / (n-1)

For population data, the formula is:

cov(X, Y) = Σ((x – x̄) * (y – ȳ)) / n

Where:

• cov(X, Y) represents the covariance between variables X and Y
• Σ denotes the summation symbol, which indicates that you need to sum up all the values calculated within the brackets
• x and y represent the values of variables X and Y, respectively
• x̄ and ȳ represent the means of variables X and Y, respectively
• n represents the number of observation

Types of Covariance

There are three types of Covariance:

1. Positive Covariance

When two variables tend to move in the same direction, they are said to have positive covariance. In other words, one variable goes up; the other one usually follows suit. Vice versa. Therefore, in this instance, the value of covariance will be positive.

Example

Step 1: List the Data Points

The data points for X and Y are:

X=[1,2,3,4,5] 

Y=[2,4,6,8,10]

Step 2: Calculate the Means of X and Y

x̄=(1+2+3+4+5)/5=3

ȳ=(2+4+6+8+10)/5=6

Step 3: Calculate the Deviations from the Mean

Σ(x – x̄)=[1−3,2−3,3−3,4−3,5−3]=[−2,−1,0,1,2]

Σ(y – ȳ)=[2−6,4−6,6−6,8−6,10−6]=[−4,−2,0,2,4]

Step 4: Multiply the Deviations and Sum Them

Multiply each deviation of X by the corresponding deviation of Y: 

(−2)×(−4)=8

(−1)×(−2)=2

0×0=0

1×2=2

2×4=8

Sum these products: 20

Step 5: Calculate the Covariance

cov(X, Y) = Σ((x – x̄) * (y – ȳ)) / n

=20/5=4

2. Negative Covariance

Negative covariance is a term we use when two variables tend to move in opposite directions. This implies that if one variable increases, the other will decrease. When this happens, the covariance value will be negative.

Example

X: [1, 2, 3, 4, 5]

Y: [10, 8, 6, 4, 2]

Cov(X,Y)=4−8−2+0−2−8​=4−20​=−4

3. Zero Covariance

When the two variables don't have a straight-line relationship, their covariance is zero. This means the variables might be related in some way. But it's hard to predict how they'll move together. In this case, the covariance value will be very close to zero.

Example

X: [1, 2, 3, 4, 5]

Y: [7, 1, 9, 3, 5]

Cov(X,Y)=4−4+4+0−2+0​=4−2​=−0.5≈0

Covariance Vs. Variance Vs. Correlation

Knowing dissimilarities among covariance variance and correlation is essential in finance and statistics.

Covariance and variance are both used to measure the extent to which data points fluctuate from each other. However, they differ in their uses—for instance, variance measures spread along a single axis.

Covariance checks out whether there is any relationship between two variables. When you have different investments in your portfolio, it is important to know how they perform relative to one another.

On the other hand, the correlation coefficient measures how strong or weak a linear relationship exists between two variables, hence, giving more standardized indicators as compared to covariance.

The table below provides a detailed comparison of covariance, variance, and correlation, outlining their definitions, ranges, indications, interpretations, normalization, usage, and examples:

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