S_EllipsoidNormal.m

% this script represents covariance and expected value in terms 
% of the location-dispersion ellipsoid in a normal example
% see "Risk and Asset Allocation"-Springer (2005), by A. Meucci

clc; clear; close all
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% input parameters

Mu=[0 0]';
s=[1 1];
r=.99;

NumSimulations=20000;
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Sigma=[s(1)^2     r*s(1)*s(2)
    r*s(1)*s(2)    s(2)^2];

% generate sample
X = mvnrnd(Mu,Sigma,NumSimulations);

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% compute location and dispersion (analytical)
Expected_Value=Mu;
Covariance=Sigma;
Standard_Deviation=sqrt(diag(Covariance))
Correlation=Covariance(1,2)/(Standard_Deviation(1)*Standard_Deviation(2))

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% plots
figure 

% marginals
NumBins=round(10*log(NumSimulations));

subplot('Position',[.05 .3 .2 .6]) 
[n,D]=hist(X(:,2),NumBins);
barh(D,n,1);
[y_lim]=get(gca,'ylim');
set(gca,'xtick',[])
grid on

subplot('Position',[.3 .05 .6 .2]) 
[n,D]=hist(X(:,1),NumBins);
bar(D,n,1);
[x_lim]=get(gca,'xlim');
set(gca,'ytick',[])
grid on

% scatter plot
subplot('Position',[.3 .3 .6 .6]) 
h=plot(X(:,1),X(:,2),'.');
set(gca,'xlim',x_lim,'ylim',y_lim)
hold on 
Scale=2;
PlotEigVectors=1;
PlotSquare=1;
TwoDimEllipsoid(Expected_Value,Covariance,Scale,PlotEigVectors,PlotSquare)
grid on