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Given:

vector of portfolio weights $W = [w_1 w_1 ]$

correlation matrix $C = \left( \begin{array}{ccc} a & b \\ d & e \end{array} \right) $

standard deviation of the asset returns $S = [s_1 s_2]$

How can I calculate the portfolio volatility?

If I had the covariance I'd just say $W* Cov * W^{T}$, correct?

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You are correct in your basic approach. Given the correlation matrix $\textbf{C}$ and standard deviation matrix $\textbf{S}$ where standard deviations occupy the diagonal and zeros the rest (i.e. $s_{i,j} = \sigma_i | i = j$ and $s_{i,j} = 0 | i \neq j$), the covariance matrix can be found as $\textbf{R} = \textbf{SCS}$. Then your portfolio standard deviation is $\sqrt{\textbf{w}^T\textbf{Rw}}$.

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