Covariance Tutorial
Formula:
$$Cov (X,Y) =\frac{\sum (X_i - \overline X)(Y_i - \overline Y)}{n-1}$$
Where :
- xi = Data variable of x
- yi = Data variable of y
- x = Mean of x
- y = Mean of y
- n = Number of data variables
Example :
Suppose that the closing prices of two stocks daily basis is given below:
| day | abc return | xyz return |
|---|---|---|
| 1 | 2.8 | 1.5 |
| 2 | 2.5 | 3.3 |
| 3 | 1.1 | 3.6 |
| 4 | 1.4 | 3.1 |
| 5 | 0.4 | 2.2 |
Calculate the mean(average):
$$\text{Mean(abc)} =$$
$$\frac{2.8 + 2.5 + 1.1 + 1.4 + 0.4}{5}$$
$$\text{Mean(abc)} = 1.64 $$
$$\text{Mean(xyz)}\;=$$
$$\frac{1.5 + 3.3 + 3.6 + 3.1 + 2.2}{5} $$
$$\text{Mean(xyz)} = 2.74 $$
$$Cov (X,Y) =\frac{\sum (X_i - \overline X)(Y_i - \overline Y)}{n-1}$$
Put the values into the formula
Cov(x,y) =(((2.8 – 1.64) * (1.5 – 2.74)) + ((2.5 – 1.64)*
(3.3 – 2.74)) + ((1.1 – 1.64) * (3.6 – 2.74)) +
(1.4 – 1.64) * (3.1 –2.74) + ((0.4 – 1.64) *
(2.2 – 2.74))) / (5 – 1)
$$Cov(x,y)\;=$$
$$\frac{-1.44 + 0.48 - 0.46 - 0.08 + 0.67}{4}$$
$$Cov(x,y) = \frac{- 0.83}{4}$$
$$Cov(x,y) = -0.20$$
