###### Key Properties of the Normal Distribut ...

A random variable is said to have the normal distribution (Gaussian curve) if... **Read More**

Covariance is a measure of the degree of co-movement between two random variables. For instance, we could be interested in the degree of co-movement between interest rate and inflation rate. The general formula used to calculate the covariance between two random variables, X and Y, is:

$$ \ text {cov} [X,Y] = E [(X – E[X ])(Y – E[Y])] $$

While the formula for covariance given above is correct, we use a slightly modified formula to calculate the covariance of returns from a joint probability model. It is based on the probability-weighted average of the cross-products of the random variables’ deviations from their expected values for each possible outcome. Therefore, if we have two assets, I and J, with returns R_{i }and R_{j} respectively, then:

$$ { \sigma }_{ { R }_{ i },{ R }_{ j } }=\sum _{ i=1 }^{ n }{ P\left( { R }_{ i } \right) \left[ { R }_{ i }-E\left( { R }_{ i } \right) \right] \left[ { R }_{ j }-E\left( { R }_{ j } \right) \right] } $$

The covariance between two random variables can be positive, negative, or zero. A positive number indicates co-movement (i.e. the variables tend to move in the **same** **direction**); a value of zero indicates **no** **relationship**, and a negative value shows that the variables move in **opposite directions**.

Correlation is the ratio of the covariance between two random variables and the product of their two standard deviations i.e.

$$ { \text{Correlation} }\left( { R }_{ i },{ R }_{ j } \right) =\frac { \text{Covariance}\left( { R }_{ i },{ R }_{ j } \right) }{ \text{Standard deviation}\left( { R }_{ i } \right) \ast \text{Standard deviation}\left( { R }_{ j } \right) } $$

It measures the strength of the **linear relationship** between two variables. While the covariance can take on any value between negative infinity and positive infinity, the correlation is **always a value between -1 and +1**.

You should note the following:

First, -1 indicates a **perfect inverse relationship** (i.e., a unit change in one means that the other will have a unit change in the opposite direction). Secondly, +1 indicates **a perfect linear relationship** (i.e., the two variables move in the same direction with equal unit changes). Finally, if there is** ****no** **linear relationship**** **at all, then the correlation will be zero.

We anticipate a 15% chance that next year’s stock returns for ABC Corp will be 6%, a 60% probability that they will be 8%, and a 25% probability of 10% return. We already know that the expected value of returns is 8.2%, and the standard deviation is 1.249%.

We also anticipate that the same probabilities and states are associated with a 4% return for XYZ Corp, a 5% return, and a 5.5% return. Then the expected value of returns is 4.975, and the standard deviation is 0.46%.

Suppose we wish to calculate the covariance and the correlation between ABC and XYZ returns. Then:

$$ \begin{align*} \text{Covariance}, \text{cov}(\text R_{\text{ABC}},\text R_{\text{XYZ}}) & = 0.15(0.06 – 0.082)(0.04 – 0.04975) \\ & + 0.6(0.08 – 0.082)(0.05 – 0.04975) \\ & + 0.25(0.10 – 0.082)(0.055 – 0.04975) \\ & = 0.0000561 \\ \end{align*} $$

$$ { \text{Correlation} }\left( { R }_{ i },{ R }_{ j } \right) =\frac { \text{Covariance}\left( { R }_{ i },{ R }_{ j } \right) }{ \text{Standard deviation}\left( { R }_{ i } \right) \ast \text{Standard deviation}\left( { R }_{ j } \right) } $$

Therefore:

$$ \begin{align*} \text{Correlation} & =\cfrac {0.0000561}{(0.01249 * 0.0046)} \\ & = 0.976 \\ \end{align*} $$

Interpretation: the correlation between the returns of the two companies is very strong (almost +1) and the returns move linearly in the same direction.