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The following lecture talks about the Markowitz Portfolio Optimization problem in convex optimization. Indeed, many variants of this problem exists, but the classical one looks like this


    \begin{equation*}\begin{aligned}& \underset{w}{\text{minimize}} & w^T \Sigma w \\ & \text{subject to} &   w^T \mu = r_{\text{min}}\end{aligned}\end{equation*}


where w is an n-sized vector containing the amount of assets to invest in. The vector \mu is the mean of the relative price asset change and the matrix \Sigma is the matrix of variance-covariance of assets. The parameter r_{\text{min}} is minimum accepted returns.

The problem is that in some scenarios, the matrix \Sigma is ill-conditioned and so the solution to the above Markowitz problem, which takes the following form

    \begin{equation*}w^* = \frac{r_{\text{min}}}{\mu^T \Sigma^{-1} \mu} \Sigma^{-1} \mu \end{equation*}

will give misleading allocation values. This lecture deals with a conditioned Markowitz type problem where a penalty is added as follows

    \begin{equation*}\begin{aligned}& \underset{w}{\text{minimize}} & w^T \Sigma w + \lambda \Vert w \Vert^2 \\ & \text{subject to} & w^T \mu = r_{\text{min}}\end{aligned}\end{equation*}


where in the lecture, we show, gives the following solution

    \begin{equation*}w^* = \frac{r_{\text{min}}}{\mu^T (\Sigma + \lambda I )^{-1} \mu} (\Sigma + \lambda I )^{-1} \mu\end{equation*}

Learn more on why the additional penalty term \lambda \Vert w \Vert^2 is one way to fix the ill-conditioned disease of matrix \Sigma.

The source of this solution is given in Brodie, Joshua, et al. “Sparse and stable Markowitz portfolios.” Proceedings of the National Academy of Sciences 106.30 (2009): 12267-12272.

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