# 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 '14 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 '14 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 '14 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 '14 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 '14 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 '14 at 12:15

There's more than one way to shrink a covariance matrix. You can think of shrinking a covariance matrix as part of general class of estimators that limit the norms of a matrix. You could alternately think of shrinkage as a form of Bayesian analysis. Given the broad set of techniques one could use, it can be more helpful to think in terms of techniques to reduce estimation risk.

For instance, suppose you simulate some data (really you want to simulate a mean and covariance with error and then simulate data using those parameters) and then apply techniques that reduce the impact of estimation error while constructing an efficient frontier. If you do this many times, you will find that the techniques that reduce the estimation error will be closer to what the frontier would look like if you knew the true mean and covariance than if you used the sample parameters. So to this extent, techniques do reduce estimation error would be a good thing.

In practice, the techniques lead to more diversified (less concentrated) portfolios and increase stability over time, which tends to reduce turnover. It's not necessarily clear that the techniques would lead to better returns, however.

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Well thanks a lot for all the info. But I'm still wondering if it's useful if I have enough data. I'm using the technique in "Honey, I shrunk the sample covariance matrix" –  user1627466 Jan 29 '14 at 21:45
You seem to be focused on shrinkage only to ensure the covariance matrix is positive definite. That's not the only reason to use shrinkage. Reduction of estimation error is another reason. My answer focused on the benefits of reducing estimation error. –  John Jan 29 '14 at 22:04
Oh but that's exactly what I'm trying to do. I'm contemplating the idea of shrinkage to reduce the estimation error of the sample covariance matrix, I'm just not sure if it's useful. –  user1627466 Jan 29 '14 at 22:23