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Risk and Asset Allocation

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Risk and Asset Allocation

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16 Nov 2005 (Updated )

Software for quantitative portfolio and risk management

S_NormalCopulaSimul.m
% this script simulates the copula of a bi-variate normal distribution
% see "Risk and Asset Allocation"-Springer (2005), by A. Meucci

clc; clear; close all
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% input parameters
Mu=[.04  0.05]';     
r=.0;            
sigma=[.25 .3]';    

NumSimul=40000;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Sigma=diag(sigma)*[1 r;r 1]*diag(sigma);

X=mvnrnd(Mu,Sigma,NumSimul); % bi-variate normal simulation
X_1=X(:,1);
X_2=X(:,2);

U_1=normcdf(X(:,1),Mu(1),sigma(1)); % grade 1 simulation
U_2=normcdf(X(:,2),Mu(2),sigma(2)); % grade 2 simulation
Copula = [U_1 U_2];		            % copula

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% plots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

figure 
% marginals
NumBins=round(10*log(NumSimul));

subplot('Position',[.05 .3 .2 .6]) 
[n,D]=hist(Copula(:,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(Copula(:,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(Copula(:,1),Copula(:,2),'.');
set(gca,'xlim',x_lim,'ylim',y_lim)
grid on
xlabel('grade 1');
ylabel('grade 2');

% 3-d histogram (~rescaled pdf)
NumBins3d=round(sqrt(NumSimul)/5);
figure
hist3(Copula(:,[1 2]),[NumBins3d NumBins3d]);

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