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Covariance Matrix Question

June 14th, 2023, 9:09 pm

Apologies if this question doesn't belong here.

When performing mean-variance optimization on a portfolio of stocks, assuming the optimization and/or investment horizon is 30 days, should I compute the covariance matrix on rolling 30 day historical returns or on 30 day historical returns from non-overlapping periods? And what is the justification for whatever the answer is?

Using the former method (i.e. rolling), I'd have a lot more returns data points for the covariance matrix, whereas using the latter method (i.e. non-overlapping e.g. just month ends), I'd have far fewer data points for the covariance matrix.

Thanks immensely.

Kind Regards.
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Re: Covariance Matrix Question

June 15th, 2023, 5:12 pm

Probably rolling is ok, as the main issue with mean-variance optimization is the return estimates.

A more careful answer is that it depends on whether or not you want to actually test what you are doing. 
Implicitly, you are forming an estimator for a future expected covariance [$]\hat{C}_{ij}[$]. 
Your hypothesis might be that

(*) [$] \hat{C}_{ij} = \beta \, C^H_{ij} + \epsilon[$]

where "H" denotes a historical value, [$]\epsilon[$] is an error, and your hypothesis is that [$]\beta = 1[$].
You can run regressions using (*) to both check your hypothesis and assign confidence intervals to your estimates.

But, both the regression R-squares and error estimates on [$]\beta[$] will be distorted if you use rolling data periods, as the regression framework requires independent samples. In other words, many traditional stats in a regression results print-out, including your confidence intervals, will be untrustworthy.
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Re: Covariance Matrix Question

October 25th, 2023, 4:40 pm

If you are using covariance matrix in mean-variance optimization and the number of stocks is larger than time window used to estimate cov matrix, your covariance matrix will have 0 eigenvalues and numerically optimization will become meaningless. In this case the industry practice is to use multi-factor risk models to approximate covariance matrix. 
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Re: Covariance Matrix Question

October 25th, 2023, 6:43 pm

Thinking about it, and considering what Alan wrote above, I'd go for the rolling approach for short-term prediction horizon and with the non-overlapping one for a long term (30D+). It's a bias-variance tradeoff.