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

Question

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 1

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?