S_AnalyzeLognormalCorrelation.m

% this script analyzes the correlation of two jointly lognormal variables
% in terms of the condition ratio of the eigenvalues
% 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];
rhos=[-.99 : .01 : .99];

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for n=1:length(rhos)
    rho=rhos(n);
    Sigma=[s(1)^2     rho*s(1)*s(2)
        rho*s(1)*s(2)    s(2)^2];

    [Expected_Value,Covariance,Standard_Deviation,Correlation]=LogNormalParam2Statistics(Mu,Sigma);
    Lambda=eig(Covariance);

    Cs(n)=Correlation(1,2);
    CRs(n)=min(Lambda)/max(Lambda);
end


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figure

subplot(2,1,1)
plot(rhos,Cs)
ylim([-1 1])
grid on
xlabel('rho')
ylabel('correlation')

subplot(2,1,2)
plot(rhos,CRs)
ylim([0 1])
grid on
xlabel('rho')
ylabel('condition ratio')