# Portfolio Optimization : Shrinkage of Covariance Matrix when data is available

It seems that shrinking the covariance matrix is especially useful if the number of individual stocks is greater than the number of data points. However is there any special gain if you're not constrained by the data ?

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Just as a short comment: if the number of stocks is greater than the number of data points (e.g. days) then the sample covariance matrix is singular. Optimizers will reject such matrices as they allow for pathological results (if constraints do not prevent this). Furthermore if I have e.g. 250 days of data and 400 stocks (this something like $400*399/2$ correlations to estimate) how can I do this in a consistent way on 250 days - we should not believe too much in such an estimate. –  Richard Jan 30 at 8:51
So it really seems like shrinkage is useful in this setup but not when the number of days is greater than the number of stocks. I don't really understand how, given the incredible amount of ressources online, one could be in the situation where one has more stocks than days. –  user1627466 Jan 30 at 11:24
No, there are 2 misunderstandings: shrinkage (or some other reduction technique) is necessary if you have more stocks than days AND (!) it is useful in general. For your second sentence: it must not depend on your sources. If you are a professional portflio manager then you have data for thousands of stocks. Obviously you might want to do some analysis on say 500 stocks using 1 year of look-back period. Then you need shrinkage. Count the stocks in big indices. You universe is not limited by online ressources but it is detemerined by the market that you want to cover and the look-back period. –  Richard Jan 30 at 12:44
You don't want to use 10 years of data just because your universe contains a lot of stocks. It is debatable whether you want to use pre-2008 data for an optimization with a holding period of say 6 months from now. –  Richard Jan 30 at 12:47
Thank you, the paper didn't make it clear it was useful in general. I thought the greater the look back period the better but you're raising an interesting point. –  user1627466 Jan 30 at 13:39

When using the estimated covariance in the context of mean-variance optimization, then, yes, shrinking the covariance matrix is useful even when you have sufficient data.

A good reference is Golts and Jones, A Sharper Angle on Optimization, who discuss convariance shrinkage among other techniques and give two examples of the usefulness of shrunk covariance estimates in forming (unconstrained) optimal portfolios. The first is desensitizing the optimizer to small variations in alphas of highly correlated assets. The second is controlling leverage.

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Thanks for the reference. –  user1627466 Jan 30 at 12:15