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Trying to answer: in the blog post that you mention the author looks at three equity funds and one REIT fund. One could say that these markets are different to FX markets (for various reasons but let's start with the question whether there is a risk premium in FX markets). what he does is the usual PCA analysis on the data. You find various questions in ...


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If $\Sigma$ is the covariance matrix of all assets and $w$ is the column vector of weightings of the asset in a certain portfolio. Then $$ w^T \Sigma w = VAR $$ is the variance of the portfolio. The contribution to volatility of asset $i$ is given by $$ w_i (\Sigma w)_i/\sqrt{VAR}, $$ where $(\Sigma w)_i$ is the $i_{th}$ entry in the vector $\Sigma w$. Note ...


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Another way to look at it is that: $$\begin{align} \beta &= \frac{cov(R_p,R_M)}{var(R_M)}\\ &= \rho(R_p,R_M)\frac{\sigma(R_p)}{\sigma(R_M)} \end{align}$$ In other words, the beta is the product of the correlation between your portfolio and the market and the ratio of their volatility. You can then see why the inverse beta is not what you expected. ...



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