# 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?

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).
• I presented all the formulas in order to let you know exactly what I was referring to. I was asking if there exists an analytical formula for $\omega$ in the case of positive weights Nov 1 '20 at 11:15