Question: Correlation

Currencies \(A\) and \(B\) have a correlation of $0.8$, and currencies \(B\) and \(C\) a correlation of $0.9 .$ What is the least the correlation between \(A\) and \(C\) can be? (Hint: write down the \(3 \times 3\) correlation matrix \(\left(\Sigma_{i j}\right)_{i, j=1 \ldots .3}\) between the currencies. If we denote by \(x\) the correlation between \(A\) and \(C\), then the only unknown term in \(\Sigma\) is \(\Sigma_{13}=\Sigma_{31}=x\). Since \(\Sigma\) is a correlation matrix, it is symmetric and positive definite. So if you determine the interval \(\left(x_{0}, x_{1}\right)\) so that \(\Sigma\) has eigenvalues \(>0\) provided \(x \in\left(x_{0}, x_{1}\right)\). The answer to the question is \(\left.x_{0} .\right)\)

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