## Tuesday, 27 January 2015

### How to build a variance-covariance matrix in Python

Recently I wrote a script to calculate the VaR of a portfolio of stocks given historical prices and returns and, in order to do that, I had to study the basics of a variance-covariance matrix.

First of all, what is it?
The formal definition should sound something like this: a variance-covariance matrix is a matrix whose element in the i,j position is the covariance between the ith and jth elements of a vector of random variables. Informally, we may say that a variance-covariance matrix is the matrix of the covariances and since the covariance of a random variable with itself is its variance, the main diagonal of the matrix is filled with the variances of the random variables (hence the fancy name).

What is it useful for?
When calculating VaR, say for a single stock, you need to gather the standard deviation of the returns of your stock. However, when calculating the VaR of a portfolio, things get pretty messy pretty quick, since you cannot simply add or subtract variances. In a more intuitive way, we may say that the variance-covariance matrix generalizes the notion of variance to multiple dimensions.

So how can we build it in Python?
Here is a simple template of how I built mine. I used only two stocks, but in the script I talked about earlier I used 500 stocks, you can easily imagine what a mess it can be if you miss some numbers.

Before showing the code, let’s take a quick look at relationships between variance, standard deviation and covariance:

Standard deviation is the square root of the variance

Covariance can be obtained given correlation (check how to build a correlation matrix) and standard deviations.

Now we can look at the script:

And here is the output:

Hope this was interesting.