Closed-Form Minimum-Variance Weights and Long-Only Constraints

Summary

The question asks whether mean-variance portfolio weights have a closed-form solution when weights must sum to one and each asset weight must lie between zero and one. The response gives the familiar inverse-covariance expression for the global minimum-variance portfolio under a full-investment constraint. That expression uses the inverse covariance matrix and a vector of ones, then normalizes the resulting weights so they sum to one.

The response distinguishes this minimum-variance formula from the broader mean-variance objective, which also includes expected returns, and notes that the cited result does not establish a closed form for the long-only constrained problem. In general, adding bound constraints changes the optimization problem, so quadratic programming is a practical approach. The exchange is brief and does not derive the result, specify conditions such as an invertible covariance matrix, or fully resolve the thesis question. Its formula should not be mistaken for a solution that automatically satisfies nonnegative weight bounds.

Key ideas

• The inverse-covariance formula gives fully invested global minimum-variance weights when short positions are allowed.
• Mean-variance optimization also depends on expected returns, unlike pure minimum-variance optimization.
• Long-only bounds can prevent the unconstrained formula from satisfying the required portfolio constraints.
• Quadratic programming is a practical way to handle portfolio weight bounds.

Tags

# Closed form solution for Mean-Variance optimization without short-selling


# Closed form solution for Mean-Variance optimization without short-selling












So I am writing my bachelor thesis about the naive portfolio vs mean-variance portfolio and I am currently a bit stuck at the part about describing the mean-variance portfolio. I know that if there are only constraints in the form of equalities, you can use the Lagrangian method in order to find a closed form solution for the problem: $$ min\frac{\lambda}{2}w\Sigma w^T-w^T\mu $$ But in my problem I have the following constraints: $e^Tw=1$ and $0\leq w_i\leq1$, with e being the vectors of ones. Can I only find a optimal weight allocation by using quadratic programming?

## Answer by KaiSqDist (score 1)

https://quant.stackexchange.com/a/77186

There is a closed-form solution for this in DeMiguel, Plyakha, Uppal, Vilkov (2013) in equation (8). You can take a look at it yourself in the paper as well. However, this is only for the minimum-variance portfolio, I am not sure if there is one for the mean-variance. For this minimum-variance case,

\begin{equation} w_{min} = \frac{\Sigma^{-1} e}{e^T \Sigma^{-1} e} \end{equation}

with $e=[1, 1, \cdots, 1]$. The only difference being that you can short securities, so you no longer have the purely positive weights constraints. However, I am more of a fan of the quadratic programming approach. Hopefully this gives you some ideas?

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