Wolfram Language

Random Matrices

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 TracyWidom 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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Obtain the correlation.

In[2]:=
Click for copyable input
val50 = td50["ValueList"]; {m, \[Nu]} = val50 // Dimensions;
In[3]:=
Click for copyable input
cor = (\[Nu] - 1) Correlation[Transpose[val50]];

will be distributed according to the TracyWidom distribution of for uncorrelated data.

In[4]:=
Click for copyable input
\[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));
In[5]:=
Click for copyable input
eigs = Eigenvalues[cor]; x = (Max[eigs] - \[Mu])/\[Sigma];

A very small value indicates that the data is unlikely to be uncorrelated.

In[6]:=
Click for copyable input
pvalue = SurvivalFunction[TracyWidomDistribution[1], x]
Out[6]=

You can find out which portfolios have significant correlations by taking those eigenvalues with a small value.

In[7]:=
Click for copyable input
SurvivalFunction[TracyWidomDistribution[1], Take[(eigs - \[Mu])/\[Sigma], 5]]
Out[7]=
In[8]:=
Click for copyable input
correlated = Take[Eigenvectors[cor], 4];
show complete Wolfram Language input
In[9]:=
Click for copyable input
ListLinePlot[-correlated.val50, ImageSize -> Medium, PlotTheme -> "Detailed", Frame -> None, PlotLegends -> Automatic]
Out[9]=

You can see that the portfolio with the most correlation closely resembles a portfolio of the stock weighted by market value.

show complete Wolfram Language input
In[10]:=
Click for copyable input
weights = {0.418273, 0.321269, 0.265727, 0.229489, 0.222083, 0.203792, 0.189571, 0.179218, 0.17362, 0.169012, 0.163313, 0.162299, 0.148874, 0.147221, 0.125398, 0.125165, 0.12214, 0.121672, 0.113446, 0.111451, 0.111076, 0.109088, 0.104356, 0.1013, 0.101011, 0.0999586, 0.0951326, 0.091055, 0.089831, 0.0889266, 0.0860497, 0.0851375, 0.0847399, 0.084467, 0.0828297, 0.0820423, 0.0750021, 0.0708699, 0.0678371, 0.0677046, 0.0672134, 0.0639545, 0.0624965, 0.0623718, 0.0621223, 0.0603369, 0.0601888, 0.0566882, 0.0532499, 0.0530472}; ListLinePlot[{weights, -First[correlated]}.val50, ImageSize -> Medium, PlotTheme -> "Detailed", Frame -> None, PlotLegends -> {"Market Weighted Fund", "Maximally Correlated Portfolio"}]
Out[10]=

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