Portfolio Correlation
When building a portfolio of investments, it is desirable to diversify them so that they are as uncorrelated as possible. In the absence of correlation, the largest scaled eigenvalue of the covariance matrix is expected to be distributed according to the Tracy–Widom distribution of . This allows stock combinations that are correlated to be identified.
Portfolio of 50 of the largest companies by market value in the S&P 500, together with their stock price history since 2000.
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"], 50, {"Discrete", 50}, {"Discrete", 1}, 1, {
ValueDimensions -> 1, DateFunction -> Automatic}}, True, 314.1];
Obtain the correlation.
val50 = td50["ValueList"];
{m, \[Nu]} = val50 // Dimensions;
cor = (\[Nu] - 1) Correlation[Transpose[val50]];
will be distributed according to the Tracy–Widom distribution of for uncorrelated data.
\[Mu] = (Sqrt[\[Nu] - 1/2] + Sqrt[m - 1/2])^2;
\[Sigma] = ((Sqrt[\[Nu]] + Sqrt[m]) (1/Sqrt[\[Nu] - 1/2] + 1/Sqrt[
m - 1/2])^(1/3));
eigs = Eigenvalues[cor];
x = (Max[eigs] - \[Mu])/\[Sigma];
A very small value indicates that the data is unlikely to be uncorrelated.
pvalue = SurvivalFunction[TracyWidomDistribution[1], x]
You can find out which portfolios have significant correlations by taking those eigenvalues with a small value.
SurvivalFunction[TracyWidomDistribution[1],
Take[(eigs - \[Mu])/\[Sigma], 5]]
correlated = Take[Eigenvectors[cor], 4];
You can see that the portfolio with the most correlation closely resembles a portfolio of the stock weighted by market value.