# Portfolio optimization with multivariate returns of different length

The mean variance model of Markowitz that uses multivariate covariance matrix requires the length of each of the N assets return time series under consideration to be of equal length. Are there any techniques to do asset allocation without this requirement, with or without the covariance matrix? (This is not a question on how to truncate time series to equal length.)

## 1 Answer

Markowitz does not require time series of equal length. Markowitz does not veen require that the covariance matrix be based on time series. Markowitz just requires a covariance matrix. The covariance matrix could, for all Markowitz cares, be based entirely on judgment, on proxies, etc.

So your question really is, how do we construct a covariance matrix (non-negative definite, or better yet positive definite) from time series when some time series are short?

Heuristiclaly, if a time series is too short (you're looking for 3 years of stock returns, but one of your stocks only had an IPO a week ago), you use proxies for it. (Or just don't include it in your universe).

If a time series is shorter than most, but long enough that you don't want to throw out the data that you have, then you calculate pairwise correlations using the data that you have.

You use singular value decomposition to get eigenvalues. Drop the eigenvalues that are negative, zero, or too small (numerically, a positive eigenvalue that's less than 2% of the largest positive one is just noise). Renormalize the eigenvalues and eigenfunctions to obtain a positvive definite matrix. You can stop here and use the latter, or you can use move matrix elements in a least squares fit to get yet another positive definite matrix that'll be a little closer to your original matrix.