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I’ve been thinking a little bit about covariance matrices recently, particularly how to estimate them from sample data. A typical problem in finance is to estimate a covariance matrix for a universe of stocks from a finite sample of returns data. The problem is that the number of parameters quickly becomes large, and can completely dwarf the number of samples you have available.

For example, consider trying to estimate the correlation of the stocks in the S&P 500 from a year of daily returns data — overall that’s about $500 \times 250 = 125,000$ pieces of information, to estimate a total of $\tfrac{1}{2} \times 500 \times 501 = 125,250$ parameters. Oops. As if that wasn’t bad enough, you are building a matrix with $500$ rows out of only $250$ columns — this means that your matrix is going to be of rank $250$ at best, and possibly lower. In particular, it won’t be invertible. If your application requires an invertible covariance matrix — say, if you’re building a Markowitz portfolio — then you’re already out of luck.

Fortunately there are several techniques that can be employed to improve the estimation of the covariance matrix, and ensure that it has full rank and is invertible. The simplest of these is called shrinkage, which is a fancy name for something simple: we transform to the correlation matrix, scale that toward the identity matrix, and then transform back into correlation space:

$\Sigma_{\rm new} = \lambda \Sigma_{\rm sample} + (1-\lambda) I$

for some $0\leq\lambda\leq 1$. This ensures that the correlation matrix, and hence the covariance matrix, is invertible. Over the next few posts I’m going to look into some more advanced techniques, including techniques that allow practitioners to incorporate their prior beliefs about market structure into the model.