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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?

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    $\begingroup$ Because of the no short selling constraint the answer is unfortunately yes, there is no analytic solution for this case. It requires applying an optimization algorithm. $\endgroup$
    – nbbo2
    Oct 29, 2023 at 12:24

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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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  • $\begingroup$ Thank you very much! I was indeed already moving towards quadratic programming but I thought maybe a closed form solution would be easier. $\endgroup$ Oct 30, 2023 at 13:04
  • $\begingroup$ @soulsbornefan you can give me an upvote if you feel my answer helped :) $\endgroup$
    – KaiSqDist
    Oct 30, 2023 at 13:46
  • $\begingroup$ @Kai I think he was asking explicitly for a solution with shortsale constraints..I was thinking about something similar to Kuhn-Tucker, however, this leaves the question unsanswered as one has to specify the Lagrange multipliers .. $\endgroup$
    – T123
    Nov 30, 2023 at 12:45

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