Wolfram Language

Matrizes aleatórias

Correlação de portfólios

Na criação de um portfólio de investimentos, é desejável diversificá-los para que sejam o menos relacionados possível. Na falta de correlação, é esperado que o maior valor próprio escalado da matriz de covariância seja distribuído de acordo com a distribuição de Tracy-Widom de . Isso permite que combinações de ações que estejam correlacionados sejam identificadas.

Portfólio das 50 maiores empresas por valor de mercado no S&P 500, junto com o histórico de preços das ações desde 2000.

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Obtenha a correlação.

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]];

será distribuido de acordo com a distribuição de TracyWidom de para dados não correlacionados.

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];

Um valor muito pequeno indica que os dados provavelmente não estejam correlacionados.

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

Você pode encontrar quais portfólios tem correlações significativas pegando esses valores próprios com um valor pequeno.

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];
mostre o input completo da Wolfram Language
In[9]:=
Click for copyable input
ListLinePlot[-correlated.val50, ImageSize -> Medium, PlotTheme -> "Detailed", Frame -> None, PlotLegends -> Automatic]
Out[9]=

Você pode ver que o portfólio com a correlação mais próxima se assemelha a um portfólio das ações ponderadas pelo valor de mercado.

mostre o input completo da Wolfram Language
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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