Why Long-Only Markowitz Portfolios Require Quadratic Optimization

Summary

The document asks how to modify the closed-form Markowitz efficient-portfolio weights so that every asset weight is positive. The supplied formula uses expected returns and the inverse covariance matrix to describe the unconstrained efficient portfolio frontier. The response explains that this analytical expression does not directly solve the long-only version.

Non-negativity constraints change the optimization problem, so the unconstrained formula’s building blocks cannot simply be adjusted to enforce positive weights. The suggested method is convex optimization, typically expressed as a quadratic program with constraints on the portfolio weights. The discussion is brief: it gives no worked example, solver guidance, or treatment of whether weights must be strictly positive or may equal zero. Its practical lesson is to solve the constrained problem directly rather than expect the unconstrained closed form to remain valid.

Key ideas

• The stated closed-form Markowitz weights solve an unconstrained portfolio problem.
• Non-negativity constraints prevent direct use of that analytical solution.
• A long-only efficient portfolio can be found with convex optimization such as quadratic programming.
• The response does not provide implementation details or a numerical example.

Tags

# Closed-form analytical solution for Markowitz efficient portfolio without short-selling


# Closed-form analytical solution for Markowitz efficient portfolio without short-selling












In a portfolio without risk-free assets I know that the efficient portfolio si given by: $\omega=\frac{1}{BC-A^2}[\mu(C\Sigma^{-1}R-A\Sigma^{-1}\mathbb{1})+B\Sigma^{-1}\mathbb{1}-A\Sigma^{-1}R]$, where:

$\mu$ is the portfolio return,

$R$ is the vector of the assets' return,

$A=\mathbb{1}'\Sigma^{-1}R$,

$B=R'\Sigma^{-1}R$,

$C=\mathbb{1}'\Sigma^{-1}\mathbb{1}$.

Now I also want that my weights $\omega_i$ are positive (i.e. $\omega_i>0$), I do not want go short.

How does $\omega$ become?

## Answer by develarist (score 7, accepted)

https://quant.stackexchange.com/a/59065

If you're asking how to use or modify the closed-form analytical solution you showed, consisting of the building blocks $A$, $B$ and $C$, as derived by Merton in 1972, you can't. That is intended for solving the unconstrained portfolio only. There is no analytical solution for the constrained portfolio because of frictions with the non-negativity requirement, therefore, it can only be solved with convex optimization (quadratic programming).

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.