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When Minimum-Variance Portfolio Weights Can Be Zero or Negative

Article Quant Q&A · Author: Mr.Price
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Summary

The document explains that a minimum-variance portfolio does not necessarily assign a nonzero weight to every risky asset. A weight can be zero, and unconstrained optimization can also produce negative weights when short selling is allowed.

Its example uses two perfectly positively correlated assets with different variances: the minimum-variance choice holds only the lower-variance asset. The answer adds that without perfect correlation or a no-short-selling constraint, zero weights are not expected. This is a brief conceptual response rather than a general derivation; it does not discuss broader covariance structures or other constraints that can also affect which assets receive zero weight.

Key ideas

  • Minimum-variance optimization can assign zero weight to an asset.
  • Unconstrained minimum-variance portfolios may include negative weights.
  • With perfect positive correlation, the lower-variance of two assets can alone form the minimum-variance portfolio.
  • Constraints such as prohibiting short sales affect the possible weights.

Tags

Portfolio constructionRisk managementStatistics
Full text
# Do the wallet weights with the minimum variance need to be nonzero?


# Do the wallet weights with the minimum variance need to be nonzero?












I wonder if we have n risky assets, does the portfolio with the minimum variance always have non-zero weights or can any weight be 0?

## Answer by Oscar (score 3, accepted)

https://quant.stackexchange.com/a/53831

There can be zero weights, and for that matter there can be (and often will be) negative weights as well unless you specifically have a constraint saying there can't be. Consider the case where you have 2 risky assets with different variance that are perfectly positively correlated with each other. The minimum variance portfolio will then consist only of the asset with lower variance. Unless you have perfectly correlated asset or the constraint that you don't allow short-selling you won't see zero weights.

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.