Solving Minimum-Variance Portfolio Weights with Bounds

Summary

The document frames portfolio construction as minimizing variance, expressed with a covariance matrix and portfolio weights that sum to one. It asks how to add lower and upper limits on the weights when implementing a constrained minimum-variance problem, referring to a paper’s treatment of a shrunk covariance matrix and minimum global variance portfolios. The response recommends Matlab’s fmincon as a solver for the referenced constraints.

This is a practical pointer to numerical constrained optimization, not a worked implementation. It does not specify the complete objective and constraint functions, solver settings, treatment of infeasible bounds, or how the covariance estimate should be obtained or shrunk. No portfolio data, comparisons, or performance evidence are provided, so the note teaches the problem setup and names a tool, while leaving implementation choices to the reader.

Key ideas

• A minimum-variance portfolio minimizes the quadratic form defined by weights and the covariance matrix.
• The basic full-investment constraint requires portfolio weights to sum to one.
• Lower and upper bounds can be added to restrict individual asset weights.
• The response identifies fmincon as a Matlab solver for the constrained problem.
• No implementation details or empirical portfolio results are supplied.

Tags

# constrained portfolio optimization in matlab


# constrained portfolio optimization in matlab












I am working through this paper, http://www.nber.org/papers/w8922.pdf

I want to implement the portfolio weight constraints see page 6-7.

Here is the brief overview of my problem:

Let `w` be the set of weights representing a portfolio. Then, mean-variance problem is to find the portfolio weights that minimizes portfolio variance, `argmin w'Sw` subject to `w'I = 1` which represents weights sum up to 1 and S is the estimated covariance matrix.

In this framework, portfolio weights are constrained by lower and upper bounds such as:

Then, the authors show that the following proposition is for the symmetric and positive semi-definite covariance matrix for the minimum global variance portfolios:

Here new covariance matrix is the shrunk version of S. I am trying to implement this in Matlab. My question is therefore, is there a method to implement a constrained optimization such as this or any suggestions as to how I could go about doing this?

Thank you for suggestions.

## Answer by Phun (score 2)

https://quant.stackexchange.com/a/16487

Try fmincon for solving (1)-(4).

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