Building a Positive-Definite Covariance Matrix from Unequal Return Histories
Summary
The document clarifies that mean-variance portfolio optimization does not require every asset’s return history to have the same length. It requires a covariance matrix, which may be estimated from data, proxies, or judgment. The practical problem is how to construct a usable covariance estimate when some assets have shorter histories than others.
For a very new asset, the answer suggests using a proxy or leaving the asset out of the universe. When a shorter series still contains useful observations, it proposes estimating pairwise correlations from the available overlapping data. Such estimates may yield a matrix that is not positive definite; the suggested repair is to use singular value decomposition, discard nonpositive or very small eigenvalues, then rescale the retained components to form a positive-definite matrix. A least-squares adjustment of matrix elements is offered as a further refinement. The eigenvalue cutoff is a heuristic from the answer, not a general guarantee of statistical quality. The document gives no empirical validation, and the resulting portfolio remains sensitive to proxy choices and estimation error.
Key ideas
- Mean-variance optimization requires a covariance matrix, not equal-length return histories.
- Use proxies or exclude assets whose return histories are too short to support estimation.
- Pairwise correlations can use available overlapping observations when histories differ in length.
- Singular value decomposition can help convert an indefinite estimate into a positive-definite covariance matrix.
- The suggested eigenvalue cutoff is a heuristic and should not be treated as universally optimal.
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Full text
# Portfolio optimization with multivariate returns of different length # 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.) ## Answer by Dimitri Vulis (score 1) https://quant.stackexchange.com/a/54596 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.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.