Sample Covariance Matrices with More Assets Than Observations

Summary

The document raises a portfolio optimization problem in which tracking error is constrained using a sample covariance matrix. It describes a case with 1,000 assets but only 60 monthly return observations, producing a covariance estimate that is not positive semidefinite. That matters because the quadratic tracking-error constraint is generally expected to use a positive semidefinite matrix.

The author asks whether switching to daily returns would solve the issue by increasing the observation count beyond the number of assets, and notes that an attempted nearest-positive-semidefinite adjustment did not work in this case. The document does not provide an answer or evidence that daily sampling is appropriate. Increasing observation frequency changes the data and may introduce dependence or other estimation concerns; the sample count alone does not establish that the resulting matrix is reliable. It is therefore a useful problem statement about high-dimensional covariance estimation, rather than a complete method for repairing the optimizer input.

Key ideas

• A sample covariance matrix based on fewer observations than assets can be singular and may fail to be positive semidefinite.
• The example has many more assets than monthly observations.
• The optimization uses covariance in a quadratic tracking-error constraint.
• Using daily returns is posed as a question, not established as a solution.
• A positive-semidefinite adjustment may fail or may not address the underlying estimation limitations.

Tags

# "fix" a sample covariance matrix which is not positive semidefinite by using daily returns instead of monthly


# "fix" a sample covariance matrix which is not positive semidefinite by using daily returns instead of monthly












In the portfolio optimization problem at hand, one of the constraints is that the tracking error should not be greater than $\gamma$.

The constraint is therefore:

$(\textbf{x}-\textbf{w})^\mathrm{T}\Sigma(\textbf{x}-\textbf{w})\leq\gamma^2$

where $\Sigma$ is the (sample) covariance matrix, $\textbf{x}=(x_1,\dots, x_n)^\mathrm{T}$ is the vector of decision variables, and $\textbf{w}=(w_1,\dots, w_n)^\mathrm{T}$ are the weights of the benchmark portfolio.

Since in the problem at hand $n=1,000$ and $\Sigma$ was solely calculated on the basis of $T=60$ monthly return observations, the (sample) covariance matrix is unfortunately not positive semi-definite. This is certainly because of $n>T$. During my research I came across this thread. However, finding the nearest positive semi-definite matrix unfortunately did not work in my case. The result is still not positive semi-definite.

Now the question is whether it is advisable and reasonable to consider daily returns instead of monthly returns in order to (possibly) generate positive semi-definiteness as this would result in $T>n$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.