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When $x_i$ is the return of the $i$th asset, the returns of portfolio $\vec{w}$ are $\sum_i w_i x_i$. The covariance of the returns of two portfolios, $\vec{w}$ and $\vec{v}$ are then $$ \sum_i \sum_j w_i v_j \operatorname{cov}\left(x_i, x_j\right). $$ Now note that $\Sigma_{i,j} = \operatorname{cov}\left(x_i,x_j\right)$. The rest is confirming that this ...


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The Hierarchical risk parity (HRP) portfolio, introduced by Lopez de Prado (2016), applies graph theory and machine learning to build a diversified portfolio. Like the traditional risk based allocation methods, HRP is also a function of the estimate of the covariance matrix, but it doesn't require its invertibility. Complete graph and tree graph figure, ...


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