Wolfram Language

Matrices aleatorias

Correlación de portafolios

Cuando cree un portafolio de inversiones, es deseable diversificarlos para que sean lo menos relacionados posible. En la ausencia de correlación, se espera que el valor propio escalado de la covarianza más grande sea distribuido de acuerdo con la distribución de TracyWidom de . Esto permite combinaciones de acciones que estén relacionados para ser idénticos.

Un portafolio de 50 de las compañías más grandes de acuerdo con su valor de mercado en S&P 500, junto con su historia de precio de acciones desde el año 2000.

In[1]:=
Click for copyable input
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Obtenga la correlación.

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 acuerdo con la distribución de TracyWidom de para datos no 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];

Un valor muy pequeño indica que los datos probablemente no estén correlacionados.

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

Usted puede encontrar cuales portafolios tienen correlaciones significativas tomando dichos valores propios con un valor pequeño.

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];
muestre la entrada completa de Wolfram Language
In[9]:=
Click for copyable input
ListLinePlot[-correlated.val50, ImageSize -> Medium, PlotTheme -> "Detailed", Frame -> None, PlotLegends -> Automatic]
Out[9]=

Usted puede ver que el portafolio con la correlación más cerca semeja un portafolio de las acciones ponderadas por el valor del mercado.

muestre la entrada completa de 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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