When Minimum-Variance Portfolios Make Return Constraints Bind

Summary

The document addresses whether two Markowitz portfolio formulations are equivalent when one requires expected return to equal a target and the other requires it to be at least that target. Its argument considers a candidate minimum-variance portfolio whose expected return exceeds the target. Under the stated setup, a small reduction in every portfolio weight preserves return above the target while lowering variance, contradicting the candidate’s optimality.

This reasoning supports equality at the optimum when the return constraint is feasible and the covariance matrix is positive definite, along with the implicit assumptions behind the proposed weight adjustment. The excerpt is very brief and does not state the two formulations in full or provide a general proof covering arbitrary constraints, budget rules, short-sale limits, or other portfolio restrictions. The scaling argument should therefore be applied only where its assumptions make that perturbation valid.

Key ideas

• A minimum-variance solution with return strictly above its lower bound may permit a small weight reduction.
• With a positive definite covariance matrix, the proposed uniform weight reduction lowers portfolio variance.
• If the adjustment remains feasible and above the return target, strict excess return contradicts optimality.
• The argument depends on the portfolio constraints allowing the stated weight adjustment.

Tags

# Show that two formulations of Markowitz problem are equivalent


# Show that two formulations of Markowitz problem are equivalent












I would like to solve (as mathematically and formally as possible) that the following Markowitz problems are equivalent. The big point is: I want to show that it is equivalent to constrain the return of the portfolio to be greater than $m$ or equal to $m$.

Formulation I

Formulation II

## Answer by fni (score 2, accepted)

https://quant.stackexchange.com/a/33181

I guess it amounts to saying that you want to exclude the case when the optimal portfolio $w_*$ is such that $\mu'w_{*}>m$. Notice that, given that $\Sigma$ is positive definite, you can choose another portfolio $w_{**}=w_{*}-1\epsilon$, with $\epsilon>0$ and small enough, such that $\mu'w_{**}=\mu'w_* - \mu'1\epsilon>m$, but clearly $w_{**} {'}\Sigma w_{**}=w_{*} {'}\Sigma w_{*} - 1{'}\Sigma 1 \epsilon <w_{*} {'}\Sigma w_{*}$, because $1{'}\Sigma 1>0$. This contradicts the optimality of $w_*$

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