[Maths Class Notes] on Covariance Formula Pdf for Exam

Statistics is a major phenomenon in mathematics and all of us have studied it in all forms. Covariance is a part of statistics and it is the measure of the relationship between two random variables or random problems. This helps us figure out the changes in different variables. As the word itself suggests, Co-variance, lets different variables correlate and vary.

Covariance is somewhat related to variance The difference is just that ‘variance’ talks about a single variable, whereas ‘covariance’ talks about two variables. One more thing to note about covariance is that it might produce positive as well as negative values of variables. These given values can also be known as the following:

1. Positive Covariance – This tells us that the two variables are supposed to go in the same direction.

2. Negative Covariance – This tells us that the two variables tend to go about in inverse directions.

Covariance Formula in Statistics

Definition: Suppose X and Y are random variables with means µXand µY. The covariance of X and Y is defined as  – 

cov(x,y) = [frac{sum _{i=1}^{n}left ( x_{i}-bar{x} right )left ( y_{i}-bar{y} right )}{n-1}]

where x_i= the values of the X- variable

y_i = the values of the X- variable

x = Mean or the average of the X variable

y = Mean or the average of the Y variable

N = Number of datapoints

Cov(x,y) = Covariance of variables x and y

In this Covariance formula in statistics, we can see that the covariance of the two variables x and y is equal to the sum of the products of the differences of each value and the mean of its variables and finally divided by one less than the total number of data points. The x and y with a bar represent the means of each variable.

As Statistics have a formula or a solution for everything, covariance has a formula that includes some important points that students should notice. Those points are mentioned below-

• Covariance is known to measure and observe the relationship between the two variables, their movement, and asset prices.

• Whenever two variables move together, they are considered to be positive covariance. Whereas, on the other hand, when they move inversely, they are considered to be negative covariance.

• By comparing or pairing assets with negative covariance, risks and volatility are reduced.

• It is also a significant tool that ascertains what securities to put in the portfolio in modern portfolio theory.

Covariance Evaluation

To understand the covariance one should know how it works, it evaluates the positioning of two variables moving together. When two stocks, let stock A and stock B move together, when stock A moves higher and also stock B moves higher equally and vice versa for the negative fall then these both stocks are said to be positive covariance. Security holdings are diversified and calculated through the covariance, in finance.

Covariance – Applications

The covariance has much-diversified application with respect to financing and the modern portfolio theory like – 

1. CAPM stands for a capital asset pricing model that is calculated to expect the return that an asset would offer.

2. The formula of the model key variables is created through the covariance between a market and its security.

3. The covariance help to evaluate a systematic risk compared to the market as a whole.

Properties of Covariance 

1. CovX, c = 0 for any constant c. 

2. CovaX, Y = a · CovX, Y CovX, aY = a · CovX, Y 

3. CovX, Y = CovY, X 

4. CovX, X = VarX 

5. Bilinearity (a.k.a. distributive property): CovX+Y, Z = CovX, Z + CovY, Z CovX, Y+Z = CovX, Y + CovX, Z

What is the Formula for Covariance? (Population and Sample Covariance Formula)

Notations in the Formula for Covariance

Key Takeaways (Covariance in Finance)

• Covariance is known to be a statistical tool that can be used to determine the relationship between the movement of any two asset prices.

• When two stocks tend to move together, then they are seen as having a positive covariance; when they move inversely, the covariance is basically negative.

• Covariance can be a significant tool in modern portfolio theory used to ascertain what securities to put in a portfolio.

• Risk and volatility can also be reduced in a portfolio by pairing assets that have a negative covariance.

Questions to be Solved

1: Compute the value of covariance i.e Cov(x,y) for the given data set.

Solution: First, Let’s find the mean of each variable. We know the formula for covariance

Thus, x̄ =(98+87+90+85+95+75)/6 = 88. 33

ȳ =(15+12+10+10+16+7)/6 = 11.67

Now, we subtract each value from its respective mean and then multiply these new values together.

The next step is to add all the products together, which is 125.66.

Now, divide the above value by (n-1) i.e by (6 – 1) i.e. 5.

Therefore, Cov(x,y) = 125.66/5 = 25.132