Covariance Formula

What is Covariance?

Covariance is a statistical measure used to find the relationship between two assets and is calculated as the standard deviation of the return of the two assets multiplied by its correlation. If it gives a positive number then the assets are said to have positive covariance i.e. when the returns of one asset goes up, the return of second assets also goes up and vice versa for negative covariance.

In the financial parlance, the term “covariance” is primarily used in portfolio theory, and it refers to the measurement of the relationship between the returns of two stocks or other assets and can be calculated based on returns of both the stocks at different intervals and the sample size or the number of intervals.

Covariance Formula

Mathematically, it is represented as,

where

• R_Ai =Return of stock A in the i^th interval
• R_Bi =Return of stock B in the i^th interval
• R_A=Mean of the return of stock A
• R_B=Mean of the return of stock B
• n = Sample size or the number of intervals

The calculation of covariance between stock A and stock B can also be derived by multiplying the standard deviation of returns of stock A, the standard deviation of returns of stock B, and the correlation between returns of stock A and stock B. Mathematically, it is represented as,

Cov (R_A, R_B) = ρ(A, B) * ơ_A * ơ_B

where ρ(A, B) = Correlation between returns of stock A and stock B

•  ơ_A = Standard deviation of returns of stock A
• ơ_B = Standard deviation of returns of stock B

Explanation

The calculation of covariance between stock A and stock B can be derived by using the first method in the following steps:

• Step 1: Firstly, determine the returns of stock A at different intervals, and they are denoted by R_Ai, which is the return in the i^th interval, i.e., R_A1, R_A2, R_A3,….., R_An are the returns for 1^st, 2^nd, 3^rd,….. and n^th interval.
• Step 2: Next, determine the returns of stock B at the same intervals and they are denoted by R_Bi 
• Step 3: Next, calculate the mean of the returns of stock A by adding all the returns of stock A and then dividing the result by the number of intervals. It is denoted by R_A.

• Step 4: Next, calculate the mean of the returns of stock B by adding all the returns of stock B and then dividing the result by the number of intervals. It is denoted by R_B

• Step 5: Finally, the calculation of covariance is derived based on returns of both the stocks, their mean returns, and the number of intervals, as shown above.

The calculation of covariance between stock A and stock B can also be derived by using the second method in the following steps:

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• Step 1: Firstly, determine the standard deviation of the returns of stock A based on the mean return, returns at each interval, and several intervals. It is denoted by ơ_A.
• Step 2: Next, determine the standard deviation of the returns of stock B, and it is denoted by ơ_B.
• Step 3: Next, determine the correlation between the returns of stock A and that of stock B by using statistical methods such as the Pearson R test. It is denoted by ρ(A, B).
• Step 4: Finally, the calculation of covariance between stock A and stock B can be derived by multiplying the standard deviation of returns of stock A, the standard deviation of returns of stock B, and the correlation between returns of stock A and stock B as shown below.

Cov (R_A, R_B) = ρ(A, B) * ơ_A * ơ

Example

Determine the covariance between stock A and stock B.

Given, R_A1 = 1.2%,R_A2 = 0.5%,R_A3 = 1.0%

R_B1= 1.7%,R_B2 = 0.6%,R_B3 = 1.3%

Therefore, the calculation will be as follows,

Now, Mean Return of stock A,R_A= (R_A1 + R_A2 + R_A3 ) / n

• R_A= (1.2% + 0.5% + 1.0%) / 3
• R_A= 0.9%

Mean Return of Stock B, R_B= (R_B1 +R_B2+ R_B3 ) / n

• R_B= (1.7% + 0.6% + 1.3%) / 3
• R_B= 1.2%

Therefore, the covariance between stock A and stock B can be calculated as,

= [(1.2 – 0.9) * (1.7 – 1.2) + (0.5 – 0.9) * (0.6 – 1.2) + (1.0 – 0.9) * (1.3 – 1.2)] / (3 -1)

Covariance between Stock A and Stock B will be –

• Cov (R_A, R_B) = 0.200

Therefore, the correlation between stock A and stock B is 0.200 which is positive and as such it means that both returns move in the same direction i.e. either both have positive returns or both have negative returns.

Relevance and Uses

From the perspective of a portfolio analyst, it is vital to grasp the concept of covariance because it is primarily used in portfolio theory to decide which assets are to be included in the portfolio. It is a statistical tool to measure the directional relationship between the price movement of two assets, such as stocks. It can also be used to ascertain the movement of a stock vis-à-vis the benchmark index, i.e., whether the stock price goes up or goes down with the increase in the benchmark index or vice versa. This metric helps a portfolio analyst to reduce the overall risk for a portfolio. A positive value indicates that the assets move in the same direction, while a negative value indicates that the assets move in opposite directions.

Recommended Articles

This article has been a guide to Covariance and its definition. Here we discuss how to calculate covariance using its formula along with a practical example and downloadable excel template. You can learn more about financing from the following articles –

• Population Variance Formula
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• Calculate Portfolio Variance
• Correlation vs. Covariance
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