Covariance Formula | Step by Step Calculation of Covariance

Formula to Calculate Covariance

Covariance is a statistical measure used to find the relationship between two assets and its formula calculates this by looking at the standard deviation of the return of the two assets multiplied by the correlation, if this calculation 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. The covariance between stock A and stock B can be calculated on the basis of returns of both the stocks at different intervals and the sample size or the number of intervals.

Mathematically, it is represented as,

where

R_{A_i}=Return of stock A in the i^th interval
R_{B_i}=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_{A_i}which is the return in the i^th interval i.e. R_A₁, R_A₂, R_A₃,….., R_{A_n}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_{B_i}
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 on the basis of 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 on the basis of the mean return, returns at each interval and number of 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 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

You can download this Covariance Formula Excel Template here – Covariance Formula Excel Template

Let us take the example of stock A and stock B with the following daily returns for three days.

Below is given data for the calculation of covariance.

Using the above information we will do the calculation of covariance as follows,

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, calculation of the covariance in excel 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,

Covariance = [(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 of Covariance Formula

From the perspective of a portfolio analyst, it is important 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 covariance indicates that the assets move in the same direction, while a negative covariance indicates that the assets move in opposite directions.

Formula to Calculate Covariance

Explanation

Example

Relevance and Uses of Covariance Formula

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