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An index that represents all of the market is a CAPM assumption, but in reality $m$ is typically some stock index (like the S&P 500, which represent U.S. large cap stocks). It's not practical to build an index that would include all possible aset classes (stocks, bonds, FX, real estate...). Even a worldwide stock index isn't very practical. The ...

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No, because correlation is a unitless quantity. As you use volatilities to do the scaling, the $\sqrt{252}$ factor should already be taken into account in them. If you take a correlation of 1 between two assets, multiplying your correlation matrix by a factor $C \neq 1$ risks either to underestimate correlations (by hiding perfect (anti)correlations) or have ...

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A negative beta just means there is a negative covariance (and thus correlation) between your asset in question and your reference “market” portfolio. Perhaps the most intuitive example of this is your “market” being stocks, and adding bonds or gold to the portfolio. These have positive (expected) rates of return, driven by macro effects that are maybe ...

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Uncorrelated does not imply independent, hence a copula could capture the dependence for very small correlations if there is any. As an example, the student t copula with a degree of freedom of 0.4 and a correlation of 0 even has a tail-dependence of 0.4 and you can see the structure in the scatter plot of a sample. However, if there is not any structure in ...

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The question of significance is not about the correlation but about the precision of the estimation. If the value estimated with more data is still near the same value estimated with less data, that means you are more sure of that correlation being close to your estimate. We can test the hypothesis that the correlation is 0 just as easily as testing that the ...

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