What is Covariance?

In financial parlance, the term “covariance” is primarily in portfolio theory. It refers to measuring the relationship between the returns of two stocks or other assets. It can be calculated based on returns of the stocks at different intervals and the sample sizeSample SizeThe sample size formula depicts the relevant population range on which an experiment or survey is conducted. It is measured using the population size, the critical value of normal distribution at the required confidence level, sample proportion and margin of error.read more or the number of intervals.

Covariance Formula

Mathematically, it is represented as:

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where

  • RAi =Return of stock A in the ith intervalRBi =Return of stock B in the ith intervalRA=Mean of the return of stock ARB=Mean of the return of stock Bn = Sample size or the number of intervals

The covariance calculation between stock A and stock B can also be obtained by multiplying the standard deviation of returns of stock A, the standard deviation of returns of stock B, and the correlation between stock A’s and stock B’s returns. Mathematically, it represents as,

Cov (RA, RB) = ρ(A, B) * ơA * ơB

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

  • ơA = Standard deviation of returns of stock AơB = Standard deviation of returns of stock B

Explanation

Example

  • Firstly, determine the returns of stock A at different intervals, and they are denoted by RAi, which is the return in the ith interval, i.e., RA1, RA2, RA3,….., RAn are the returns for 1st, 2nd, 3rd,….. and nth interval. Next, determine the returns of stock B at the same intervals and they are denoted by RBi  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 RA. 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 RB 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: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 ơA.Step 2: Next, determine the standard deviation of the returns of stock B, and it is denoted by ơB.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 covariance calculation 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 (RA, RB) = ρ(A, B) * ơA * ơ

The calculation of covariance between stock A and stock B can also be derived by using the second method in the following steps: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 ơA.Step 2: Next, determine the standard deviation of the returns of stock B, and it is denoted by ơB.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 covariance calculation 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 (RA, RB) = ρ(A, B) * ơA * ơ

Determine the covariance between stock A and stock B.

Given, RA1 = 1.2%,RA2 = 0.5%,RA3 = 1.0%

RB1= 1.7%,RB2 = 0.6%,RB3 = 1.3%

Therefore, the calculation will be as follows,

Now, mean return of stock A,RA= (RA1 + RA2 + RA3 ) / n

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

Mean Return of Stock B, RB= (RB1 +RB2+ RB3 ) / n

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

Therefore, one can calculate the covariance between stock A and stock B 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 (RA, RB) = 0.200

Therefore, the correlation between stock A and stock B is 0.200, which is positive. As such, both returns move in the same direction, i.e., either with positive or 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. One can also use it 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 reduce a portfolio’s overall risk. A positive value indicates that the assets move in the same direction, while a negative value indicates that the assets move in opposite directions.

This article has been a guide to Covariance and its definition. Here, we discuss calculating covariance using its formula, a practical example, and a downloadable Excel template. You can learn more about financing from the following articles: –

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