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When a Short-Constrained Minimum Variance Portfolio Holds One Asset

Article Quant Q&A · Author: gerscorpion
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Summary

The document poses a two-asset portfolio question: under a no-short-selling constraint, what variance would make the lower-return asset alone the minimum-variance portfolio? It asks whether that asset would need to be risk-free to be preferable, and seeks a mathematical condition for the portfolio choice.

No answer, derivation, or empirical evidence is included, so the document does not establish a threshold variance or explain how expected returns affect the minimum-variance allocation. It is useful as a formulation of a constrained portfolio optimization problem, but readers would need to supply the covariance structure and solve the boundary condition themselves. The discussion concerns the minimum-variance portfolio, which minimizes risk subject to the constraints, rather than necessarily selecting the asset with the highest expected return.

Key ideas

  • The problem concerns a two-asset portfolio with a short-selling constraint.
  • It asks when the minimum-variance allocation lies entirely in the lower-return asset.
  • The document provides no derivation or threshold variance to answer the question.
  • The minimum-variance portfolio objective is distinct from maximizing expected return.

Tags

Portfolio constructionRisk managementStatistics
Full text
# Find variance of Asset with lesser return to make a pure portfolio of it the min-variance portfolio


# Find variance of Asset with lesser return to make a pure portfolio of it the min-variance portfolio












I need to solve the question mentioned above. For an asset with a worse payoff than another, I need to determine a variance for which the minimum-variance portfolio only consists of this asset. There are only two assets and a short-selling constraint.

My first guess would be that this asset would need to be risk-free for it to be preferable to the other one and lie on the efficient-frontier, but maybe there is a mathematical way to solve this I can't find in my script/books?

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