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2

You indeed seem to have an error in your calculation somewhere. Let the covariance matrix be $$\Sigma=\begin{pmatrix}0.0324&0.021016&0\\ 0.021016 & 0.0256 & 0 \\ 0 & 0 & 0.0256\end{pmatrix}$$ and the vector of (excess) returns are $$\mu-r_f=\begin{pmatrix}0.05 \\ 0.02 \\ 0.03\end{pmatrix}$$ Ultimately, the weights are then computed ...

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The key phrase is "adjusted by their covariances". The formula for the variance of a portfolio of two assets is $\sigma _{p}^{2}=w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+2w_{A}w_{B}\sigma _{A}\sigma _{B}\rho _{AB}$ , which, since $\rho _{AB} <= 1$, is always less than $w_{A}^{2}\sigma _{A}^{2}+w_{B}^{2}\sigma _{B}^{2}+2w_{A}w_{B}\sigma ... 0 You can handle this problem with scenario optimization: assume a matrix$R$of returns, in which the rows are the scenarios and the columns are assets. For given portfolio weights$w$, you can compute the portfolio returns as$Rw$. You can now evaluate an objective function such as the MAD, so your objective becomes$\min\ \mathrm{mean}(|Rw|)\$. Now feed ...

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The third approach is the correct one. In general, one cannot aggregate partial moments of single assets into partial moments of the portfolio, as discussed for instance in this paper: @ARTICLE{Grootveld1999, author = {Henk Grootveld and Winfried Hallerbach}, title = {Variance vs downside risk: Is there really that much ...

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